Subgroups — C₂₃×#34527 #19182153

Order 5888.

2216 conjugacy classes of subgroups.

The subgroup lattice has 892 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#562	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2382	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2379	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2379	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2374	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2364	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2362	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2361	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2360	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2359	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2335	C₂	1	2	yes	maximal, complemented
12	C₂₃×#2333	C₂	1	2	yes	maximal, complemented
13	C₂₃×#2332	C₂	1	2	yes	maximal, complemented
14	C₂₃×#2329	C₂	1	2	yes	maximal, complemented
15	C₂₃×#2324	C₂	1	2	yes	maximal, complemented
16	C₂₃×#2322	C₂	1	2	yes	maximal, complemented
17	C₄×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
18	C₄×C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
19	C₄×C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂₃×#562	C₄	1	16	yes	complemented
21	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#123	C₂²	1	2	yes	—
25	C₂×C₂₃×#123	C₂²	1	2	yes	—
26	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₂×C₂₃×#121	C₂²	1	2	yes	—
29	C₂×C₂₃×#121	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
36	C₂×C₂₃×#117	C₂²	1	2	yes	—
37	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
38	C₂₃×#391	C₂²	1	1	yes	characteristic
39	C₂₃×#391	C₂²	1	1	yes	characteristic
40	C₂₃×#391	C₂²	1	2	yes	—
41	C₂₃×#391	C₂²	1	2	yes	—
42	C₂₃×#390	C₂²	1	1	yes	characteristic
43	C₂₃×#390	C₂²	1	1	yes	characteristic
44	C₂₃×#390	C₂²	1	2	yes	—
45	C₂₃×#390	C₂²	1	2	yes	—
46	C₂₃×#389	C₂²	1	1	yes	characteristic
47	C₂₃×#389	C₂²	1	1	yes	characteristic
48	C₂₃×#389	C₂²	1	2	yes	—
49	C₂₃×#389	C₂²	1	2	yes	—
50	C₂₃×#389	C₂²	1	2	yes	—
51	C₂₃×#389	C₂²	1	2	yes	—
52	C₂₃×#388	C₂²	1	2	yes	—
53	C₂₃×#388	C₂²	1	2	yes	—
54	C₂₃×#388	C₂²	1	2	yes	—
55	C₂₃×#388	C₂²	1	2	yes	—
56	C₂₃×#388	C₂²	1	2	yes	—
57	C₂₃×#388	C₂²	1	2	yes	—
58	C₂₃×#388	C₂²	1	2	yes	—
59	C₂₃×#388	C₂²	1	2	yes	—
60	C₂₃×#388	C₂²	1	2	yes	—
61	C₂₃×#388	C₂²	1	2	yes	—
62	C₂₃×#388	C₂²	1	2	yes	—
63	C₂₃×#388	C₂²	1	2	yes	—
64	C₂₃×#387	C₂²	1	2	yes	—
65	C₂₃×#387	C₂²	1	2	yes	—
66	C₂₃×#387	C₂²	1	2	yes	—
67	C₂₃×#387	C₂²	1	2	yes	—
68	C₂₃×#387	C₂²	1	2	yes	—
69	C₂₃×#387	C₂²	1	2	yes	—
70	C₂₃×#387	C₂²	1	2	yes	—
71	C₂₃×#387	C₂²	1	2	yes	—
72	C₂₃×#387	C₂²	1	2	yes	—
73	C₂₃×#387	C₂²	1	2	yes	—
74	C₂₃×#386	C₂²	1	2	yes	—
75	C₂₃×#386	C₂²	1	2	yes	—
76	C₂₃×#386	C₂²	1	2	yes	—
77	C₂₃×#386	C₂²	1	2	yes	—
78	C₂₃×#386	C₂²	1	2	yes	—
79	C₂₃×#386	C₂²	1	2	yes	—
80	C₂₃×#385	C₂²	1	2	yes	—
81	C₂₃×#385	C₂²	1	2	yes	—
82	C₂₃×#385	C₂²	1	2	yes	—
83	C₂₃×#385	C₂²	1	2	yes	—
84	C₂₃×#385	C₂²	1	2	yes	—
85	C₂₃×#385	C₂²	1	2	yes	—
86	C₂₃×#382	C₂²	1	1	yes	characteristic
87	C₂₃×#382	C₂²	1	1	yes	characteristic
88	C₂₃×#382	C₂²	1	1	yes	characteristic
89	C₂₃×#381	C₂²	1	2	yes	—
90	C₂₃×#380	C₂²	1	2	yes	—
91	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
92	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
93	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
94	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
95	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
96	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
97	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
98	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
99	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
100	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
101	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
102	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
103	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
104	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
105	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
106	C₂×C₂₃×#95	C₂²	1	2	yes	—
107	C₄³×C₂₃	C₂²	1	1	yes	characteristic
108	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
109	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
110	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
111	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
112	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
114	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
115	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
116	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
117	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
118	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
119	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
120	C₂₃×#125	C₂×C₄	1	8	yes	complemented
121	C₂₃×#124	C₂×C₄	1	8	yes	complemented
122	C₂₃×#124	C₂×C₄	1	8	yes	complemented
123	C₂₃×#123	C₂×C₄	1	16	yes	complemented
124	C₂₃×#123	C₂×C₄	1	16	yes	complemented
125	C₂₃×C₂²⋊Q₈	C₂×C₄	1	16	yes	complemented
126	C₂₃×C₂²⋊Q₈	C₂×C₄	1	16	yes	complemented
127	C₂₃×#121	C₂×C₄	1	16	yes	complemented
128	C₂₃×#121	C₂×C₄	1	16	yes	complemented
129	C₄×Q₈×C₂₃	—	4	2	no	—
130	C₄×Q₈×C₂₃	—	4	2	no	—
131	C₄×D₄×C₂₃	—	4	4	no	—
132	C₄×D₄×C₂₃	—	4	2	no	—
133	C₄×D₄×C₂₃	—	4	2	no	—
134	C₄×D₄×C₂₃	—	4	4	no	—
135	C₂₃×#117	—	4	4	no	—
136	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
137	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
138	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
139	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
140	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
141	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
142	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
143	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
144	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
145	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
146	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
147	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
149	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
150	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
151	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
152	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
153	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
154	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
155	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
156	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
157	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
158	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
159	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
160	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
161	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
162	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
163	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
164	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
165	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
166	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
167	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
168	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
169	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
170	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
171	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
172	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
173	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
174	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
175	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
176	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
177	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
178	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
179	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
180	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
181	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
182	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
183	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
184	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
185	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
186	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
187	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
188	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
189	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
190	C₂₃×#95	C₂³	1	2	yes	—
191	C₂₃×#95	C₂³	1	2	yes	—
192	C₂₃×#95	C₂³	1	2	yes	—
193	C₂₃×#95	C₂³	1	2	yes	—
194	C₂₃×#95	C₂³	1	2	yes	—
195	C₂₃×#95	C₂³	1	2	yes	—
196	C₂₃×#95	C₂³	1	2	yes	—
197	C₂₃×#95	C₂³	1	2	yes	—
198	C₂₃×#95	C₂³	1	2	yes	—
199	C₂₃×#95	C₂³	1	2	yes	—
200	C₂₃×#95	C₂³	1	2	yes	—
201	C₂₃×#95	C₂³	1	2	yes	—
202	C₂₃×#95	C₂³	1	2	yes	—
203	C₂₃×#95	C₂³	1	2	yes	—
204	C₂₃×#95	C₂³	1	2	yes	—
205	C₂₃×#95	C₂³	1	2	yes	—
206	C₂₃×#95	C₂³	1	2	yes	—
207	C₂₃×#95	C₂³	1	2	yes	—
208	C₂₃×#95	C₂³	1	2	yes	—
209	C₂₃×#95	C₂³	1	2	yes	—
210	C₂₃×#95	C₂³	1	2	yes	—
211	C₂₃×#95	C₂³	1	2	yes	—
212	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
213	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
214	C₂×Q₈×C₂₃	—	4	1	no	—
215	C₂×Q₈×C₂₃	—	4	1	no	—
216	C₂×Q₈×C₂₃	—	4	1	no	—
217	C₂×Q₈×C₂₃	—	4	1	no	—
218	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
219	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
220	C₂×D₄×C₂₃	—	4	2	no	—
221	C₂×D₄×C₂₃	—	4	1	no	—
222	C₂×D₄×C₂₃	—	4	1	no	—
223	C₂×D₄×C₂₃	—	4	2	no	—
224	C₂×D₄×C₂₃	—	4	2	no	—
225	C₂×D₄×C₂₃	—	4	1	no	—
226	C₂×D₄×C₂₃	—	4	1	no	—
227	C₂×D₄×C₂₃	—	4	2	no	—
228	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
229	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
230	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
231	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
232	C₂²×C₄×C₂₃	—	2	4	no	—
233	C₂²×C₄×C₂₃	—	2	4	no	—
234	C₂²×C₄×C₂₃	—	4	2	no	—
235	C₂²×C₄×C₂₃	—	4	2	no	—
236	C₂²×C₄×C₂₃	—	4	2	no	—
237	C₂²×C₄×C₂₃	—	4	2	no	—
238	C₂²×C₄×C₂₃	—	4	2	no	—
239	C₂²×C₄×C₂₃	—	4	2	no	—
240	C₂²×C₄×C₂₃	—	4	2	no	—
241	C₂²×C₄×C₂₃	—	4	2	no	—
242	C₂²×C₄×C₂₃	—	4	2	no	—
243	C₂²×C₄×C₂₃	—	4	2	no	—
244	C₂²×C₄×C₂₃	—	4	2	no	—
245	C₂²×C₄×C₂₃	—	4	2	no	—
246	C₂²×C₄×C₂₃	—	4	2	no	—
247	C₂²×C₄×C₂₃	—	4	2	no	—
248	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
249	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
250	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
251	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
252	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
253	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
254	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
255	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
256	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
257	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
258	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
259	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
260	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
261	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
262	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
263	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
264	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
265	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
266	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
267	C₄⋊C₄×C₂₃	—	2	4	no	—
268	C₄⋊C₄×C₂₃	—	2	4	no	—
269	C₄⋊C₄×C₂₃	—	2	4	no	—
270	C₄⋊C₄×C₂₃	—	4	2	no	—
271	C₄⋊C₄×C₂₃	—	4	2	no	—
272	C₄⋊C₄×C₂₃	—	4	1	no	—
273	C₄⋊C₄×C₂₃	—	4	1	no	—
274	C₄⋊C₄×C₂₃	—	4	2	no	—
275	C₄⋊C₄×C₂₃	—	4	2	no	—
276	C₄⋊C₄×C₂₃	—	4	2	no	—
277	C₄⋊C₄×C₂₃	—	4	1	no	—
278	C₄⋊C₄×C₂₃	—	4	1	no	—
279	C₄⋊C₄×C₂₃	—	4	1	no	—
280	C₄⋊C₄×C₂₃	—	4	1	no	—
281	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
282	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
283	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
284	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
285	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
286	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
287	C₂²⋊C₄×C₂₃	—	2	4	no	—
288	C₂²⋊C₄×C₂₃	—	2	4	no	—
289	C₂²⋊C₄×C₂₃	—	2	4	no	—
290	C₂²⋊C₄×C₂₃	—	2	4	no	—
291	C₂²⋊C₄×C₂₃	—	4	2	no	—
292	C₂²⋊C₄×C₂₃	—	4	2	no	—
293	C₂²⋊C₄×C₂₃	—	4	2	no	—
294	C₂²⋊C₄×C₂₃	—	4	2	no	—
295	C₂²⋊C₄×C₂₃	—	4	2	no	—
296	C₂²⋊C₄×C₂₃	—	4	2	no	—
297	C₂²⋊C₄×C₂₃	—	4	2	no	—
298	C₂²⋊C₄×C₂₃	—	4	2	no	—
299	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
300	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
301	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
302	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
303	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
304	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
305	C₄²×C₂₃	—	2	4	no	—
306	C₄²×C₂₃	—	2	2	no	—
307	C₄²×C₂₃	—	2	2	no	—
308	C₄²×C₂₃	—	2	2	no	—
309	C₄²×C₂₃	—	4	2	no	—
310	C₄²×C₂₃	—	4	1	no	—
311	C₄²×C₂₃	—	4	1	no	—
312	C₄²×C₂₃	—	4	1	no	—
313	C₄²×C₂₃	—	4	2	no	—
314	C₄²×C₂₃	—	4	2	no	—
315	C₄²×C₂₃	—	4	2	no	—
316	C₄²×C₂₃	—	4	2	no	—
317	C₄²×C₂₃	—	4	2	no	—
318	C₄²×C₂₃	—	4	2	no	—
319	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
320	#34527	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
321	C₂³×C₂₃	—	2	2	no	—
322	C₂³×C₂₃	—	2	2	no	—
323	C₂³×C₂₃	—	2	2	no	—
324	C₂³×C₂₃	—	2	2	no	—
325	C₂³×C₂₃	—	2	2	no	—
326	C₂³×C₂₃	—	2	2	no	—
327	C₂³×C₂₃	—	2	2	no	—
328	C₂³×C₂₃	—	2	2	no	—
329	C₂³×C₂₃	—	2	2	no	—
330	C₂³×C₂₃	—	2	2	no	—
331	C₂³×C₂₃	—	2	2	no	—
332	C₂³×C₂₃	—	2	2	no	—
333	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
334	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
335	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
336	Q₈×C₂₃	—	4	4	no	—
337	Q₈×C₂₃	—	4	4	no	—
338	D₄×C₂₃	—	4	8	no	—
339	D₄×C₂₃	—	4	8	no	—
340	D₄×C₂₃	—	4	4	no	—
341	D₄×C₂₃	—	4	4	no	—
342	C₂×C₄×C₂₃	—	2	1	no	—
343	C₂×C₄×C₂₃	—	2	1	no	—
344	C₂×C₄×C₂₃	—	2	2	no	—
345	C₂×C₄×C₂₃	—	2	1	no	—
346	C₂×C₄×C₂₃	—	2	1	no	—
347	C₂×C₄×C₂₃	—	2	2	no	—
348	C₂×C₄×C₂₃	—	2	2	no	—
349	C₂×C₄×C₂₃	—	2	2	no	—
350	C₂×C₄×C₂₃	—	2	1	no	—
351	C₂×C₄×C₂₃	—	2	1	no	—
352	C₂×C₄×C₂₃	—	2	1	no	—
353	C₂×C₄×C₂₃	—	2	1	no	—
354	C₂×C₄×C₂₃	—	2	1	no	—
355	C₂×C₄×C₂₃	—	2	1	no	—
356	C₂×C₄×C₂₃	—	2	2	no	—
357	C₂×C₄×C₂₃	—	2	2	no	—
358	C₂×C₄×C₂₃	—	2	2	no	—
359	C₂×C₄×C₂₃	—	2	1	no	—
360	C₂×C₄×C₂₃	—	2	1	no	—
361	C₂×C₄×C₂₃	—	2	2	no	—
362	C₂×C₄×C₂₃	—	2	2	no	—
363	C₂×C₄×C₂₃	—	2	2	no	—
364	C₂×C₄×C₂₃	—	2	2	no	—
365	C₂×C₄×C₂₃	—	2	2	no	—
366	C₂×C₄×C₂₃	—	2	2	no	—
367	C₂×C₄×C₂₃	—	2	2	no	—
368	C₂×C₄×C₂₃	—	2	2	no	—
369	C₂×C₄×C₂₃	—	2	2	no	—
370	C₂×C₄×C₂₃	—	2	2	no	—
371	C₂×C₄×C₂₃	—	2	2	no	—
372	C₂×C₄×C₂₃	—	2	2	no	—
373	C₂×C₄×C₂₃	—	2	2	no	—
374	C₂×C₄×C₂₃	—	2	1	no	—
375	C₂×C₄×C₂₃	—	2	1	no	—
376	C₂×C₄×C₂₃	—	2	2	no	—
377	C₂×C₄×C₂₃	—	2	2	no	—
378	C₂×C₄×C₂₃	—	2	1	no	—
379	C₂×C₄×C₂₃	—	2	2	no	—
380	C₂×C₄×C₂₃	—	2	2	no	—
381	C₂×C₄×C₂₃	—	2	2	no	—
382	C₂×C₄×C₂₃	—	2	2	no	—
383	C₂×C₄×C₂₃	—	4	8	no	—
384	C₂×C₄×C₂₃	—	4	8	no	—
385	C₂×C₄×C₂₃	—	8	4	no	—
386	C₂×C₄×C₂₃	—	8	4	no	—
387	C₂×C₄×C₂₃	—	8	4	no	—
388	C₂×C₄×C₂₃	—	8	4	no	—
389	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
390	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
391	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
392	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
393	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
394	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
395	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
396	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
397	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
398	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
399	C₂×#562	C₂×C₂₃	1	1	yes	characteristic
400	#2382	C₂×C₂₃	1	1	yes	characteristic, complemented
401	#2379	C₂×C₂₃	1	1	yes	characteristic, complemented
402	#2379	C₂×C₂₃	1	1	yes	characteristic, complemented
403	#2374	C₂×C₂₃	1	2	yes	complemented
404	#2364	C₂×C₂₃	1	2	yes	complemented
405	#2362	C₂×C₂₃	1	2	yes	complemented
406	#2361	C₂×C₂₃	1	2	yes	complemented
407	#2360	C₂×C₂₃	1	2	yes	complemented
408	#2359	C₂×C₂₃	1	2	yes	complemented
409	#2335	C₂×C₂₃	1	2	yes	complemented
410	#2333	C₂×C₂₃	1	2	yes	complemented
411	#2332	C₂×C₂₃	1	2	yes	complemented
412	#2329	C₂×C₂₃	1	2	yes	complemented
413	#2324	C₂×C₂₃	1	2	yes	complemented
414	#2322	C₂×C₂₃	1	2	yes	complemented
415	C₄×#125	C₂×C₂₃	1	1	yes	characteristic, complemented
416	C₄×#124	C₂×C₂₃	1	1	yes	characteristic, complemented
417	C₄×#124	C₂×C₂₃	1	1	yes	characteristic, complemented
418	C₂²×C₂₃	—	2	4	no	—
419	C₂²×C₂₃	—	2	4	no	—
420	C₂²×C₂₃	—	2	4	no	—
421	C₂²×C₂₃	—	2	4	no	—
422	C₂²×C₂₃	—	2	4	no	—
423	C₂²×C₂₃	—	2	4	no	—
424	C₂²×C₂₃	—	4	2	no	—
425	C₂²×C₂₃	—	4	2	no	—
426	C₂²×C₂₃	—	4	2	no	—
427	C₂²×C₂₃	—	4	2	no	—
428	C₂²×C₂₃	—	4	2	no	—
429	C₂²×C₂₃	—	4	2	no	—
430	C₂²×C₂₃	—	4	2	no	—
431	C₂²×C₂₃	—	4	2	no	—
432	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
433	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
434	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
435	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
436	C₂²×C₂₃	#585	1	1	yes	characteristic
437	C₂²×C₂₃	#585	1	1	yes	characteristic
438	C₂²×C₂₃	#585	1	1	yes	characteristic
439	C₄×C₂₃	—	2	4	no	—
440	C₄×C₂₃	—	2	4	no	—
441	C₄×C₂₃	—	2	4	no	—
442	C₄×C₂₃	—	2	4	no	—
443	C₄×C₂₃	—	2	2	no	—
444	C₄×C₂₃	—	2	2	no	—
445	C₄×C₂₃	—	2	2	no	—
446	C₄×C₂₃	—	2	2	no	—
447	C₄×C₂₃	—	2	2	no	—
448	C₄×C₂₃	—	2	2	no	—
449	C₄×C₂₃	—	4	2	no	—
450	C₄×C₂₃	—	4	2	no	—
451	C₄×C₂₃	—	4	2	no	—
452	C₄×C₂₃	—	4	2	no	—
453	C₄×C₂₃	—	4	2	no	—
454	C₄×C₂₃	—	4	2	no	—
455	C₄×C₂₃	—	4	1	no	—
456	#562	C₄×C₂₃	1	16	yes	complemented
457	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
458	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
459	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
460	C₂×#123	C₂²×C₂₃	1	2	yes	—
461	C₂×#123	C₂²×C₂₃	1	2	yes	—
462	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
463	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
464	C₂×#121	C₂²×C₂₃	1	2	yes	—
465	C₂×#121	C₂²×C₂₃	1	2	yes	—
466	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
467	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
468	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
469	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
470	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
471	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
472	C₂×#117	C₂²×C₂₃	1	2	yes	—
473	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
474	#391	C₂²×C₂₃	1	1	yes	characteristic
475	#391	C₂²×C₂₃	1	2	yes	—
476	#391	C₂²×C₂₃	1	1	yes	characteristic
477	#391	C₂²×C₂₃	1	2	yes	—
478	#390	C₂²×C₂₃	1	1	yes	characteristic
479	#390	C₂²×C₂₃	1	2	yes	—
480	#390	C₂²×C₂₃	1	2	yes	—
481	#390	C₂²×C₂₃	1	1	yes	characteristic
482	#389	C₂²×C₂₃	1	1	yes	characteristic
483	#389	C₂²×C₂₃	1	1	yes	characteristic
484	#389	C₂²×C₂₃	1	2	yes	—
485	#389	C₂²×C₂₃	1	2	yes	—
486	#389	C₂²×C₂₃	1	2	yes	—
487	#389	C₂²×C₂₃	1	2	yes	—
488	#388	C₂²×C₂₃	1	2	yes	—
489	#388	C₂²×C₂₃	1	2	yes	—
490	#388	C₂²×C₂₃	1	2	yes	—
491	#388	C₂²×C₂₃	1	2	yes	—
492	#388	C₂²×C₂₃	1	2	yes	—
493	#388	C₂²×C₂₃	1	2	yes	—
494	#388	C₂²×C₂₃	1	2	yes	—
495	#388	C₂²×C₂₃	1	2	yes	—
496	#388	C₂²×C₂₃	1	2	yes	—
497	#388	C₂²×C₂₃	1	2	yes	—
498	#388	C₂²×C₂₃	1	2	yes	—
499	#388	C₂²×C₂₃	1	2	yes	—
500	#387	C₂²×C₂₃	1	2	yes	—
501	#387	C₂²×C₂₃	1	2	yes	—
502	#387	C₂²×C₂₃	1	2	yes	—
503	#387	C₂²×C₂₃	1	2	yes	—
504	#387	C₂²×C₂₃	1	2	yes	—
505	#387	C₂²×C₂₃	1	2	yes	—
506	#387	C₂²×C₂₃	1	2	yes	—
507	#387	C₂²×C₂₃	1	2	yes	—
508	#387	C₂²×C₂₃	1	2	yes	—
509	#387	C₂²×C₂₃	1	2	yes	—
510	#386	C₂²×C₂₃	1	2	yes	—
511	#386	C₂²×C₂₃	1	2	yes	—
512	#386	C₂²×C₂₃	1	2	yes	—
513	#386	C₂²×C₂₃	1	2	yes	—
514	#386	C₂²×C₂₃	1	2	yes	—
515	#386	C₂²×C₂₃	1	2	yes	—
516	#385	C₂²×C₂₃	1	2	yes	—
517	#385	C₂²×C₂₃	1	2	yes	—
518	#385	C₂²×C₂₃	1	2	yes	—
519	#385	C₂²×C₂₃	1	2	yes	—
520	#385	C₂²×C₂₃	1	2	yes	—
521	#385	C₂²×C₂₃	1	2	yes	—
522	#382	C₂²×C₂₃	1	1	yes	characteristic
523	#382	C₂²×C₂₃	1	1	yes	characteristic
524	#382	C₂²×C₂₃	1	1	yes	characteristic
525	#381	C₂²×C₂₃	1	2	yes	—
526	#380	C₂²×C₂₃	1	2	yes	—
527	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
528	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
529	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
530	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
531	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
532	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
533	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
534	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
535	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
536	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
537	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
538	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
539	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
540	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
541	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
542	C₂×#95	C₂²×C₂₃	1	2	yes	—
543	C₄³	C₂²×C₂₃	1	1	yes	characteristic
544	C₂×C₂₃	—	4	4	no	—
545	C₂×C₂₃	—	4	4	no	—
546	C₂×C₂₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
547	C₂×C₂₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
548	C₂×C₂₃	C₄×2₋¹⁺⁴	1	1	yes	characteristic
549	C₂×C₂₃	#3558	1	1	yes	characteristic
550	C₂×C₂₃	#3559	1	1	yes	characteristic
551	C₂×C₂₃	#3561	1	1	yes	characteristic
552	C₂×C₂₃	#3567	1	1	yes	characteristic
553	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
554	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
555	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
556	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
557	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
558	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
559	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
560	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
561	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
562	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
563	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
564	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
565	#125	C₂×C₄×C₂₃	1	8	yes	complemented
566	#124	C₂×C₄×C₂₃	1	8	yes	complemented
567	#124	C₂×C₄×C₂₃	1	8	yes	complemented
568	#123	C₂×C₄×C₂₃	1	16	yes	complemented
569	#123	C₂×C₄×C₂₃	1	16	yes	complemented
570	C₂²⋊Q₈	C₂×C₄×C₂₃	1	16	yes	complemented
571	C₂²⋊Q₈	C₂×C₄×C₂₃	1	16	yes	complemented
572	#121	C₂×C₄×C₂₃	1	16	yes	complemented
573	#121	C₂×C₄×C₂₃	1	16	yes	complemented
574	C₄×Q₈	—	4	2	no	—
575	C₄×Q₈	—	4	2	no	—
576	C₄×D₄	—	4	4	no	—
577	C₄×D₄	—	4	2	no	—
578	C₄×D₄	—	4	2	no	—
579	C₄×D₄	—	4	4	no	—
580	#117	—	4	4	no	—
581	C₂×C₄⋊C₄	—	4	4	no	—
582	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
583	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
584	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
585	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
586	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
587	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
588	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
589	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
590	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
591	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
592	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
593	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
594	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
595	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
596	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
597	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
598	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
599	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
600	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
601	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
602	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
603	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
604	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
605	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
606	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
607	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
608	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
609	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
610	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
611	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
612	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
613	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
614	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
615	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
616	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
617	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
618	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
619	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
620	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
621	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
622	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
623	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
624	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
625	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
626	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
627	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
628	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
629	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
630	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
631	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
632	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
633	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
634	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
635	#95	C₂³×C₂₃	1	2	yes	—
636	#95	C₂³×C₂₃	1	2	yes	—
637	#95	C₂³×C₂₃	1	2	yes	—
638	#95	C₂³×C₂₃	1	2	yes	—
639	#95	C₂³×C₂₃	1	2	yes	—
640	#95	C₂³×C₂₃	1	2	yes	—
641	#95	C₂³×C₂₃	1	2	yes	—
642	#95	C₂³×C₂₃	1	2	yes	—
643	#95	C₂³×C₂₃	1	2	yes	—
644	#95	C₂³×C₂₃	1	2	yes	—
645	#95	C₂³×C₂₃	1	2	yes	—
646	#95	C₂³×C₂₃	1	2	yes	—
647	#95	C₂³×C₂₃	1	2	yes	—
648	#95	C₂³×C₂₃	1	2	yes	—
649	#95	C₂³×C₂₃	1	2	yes	—
650	#95	C₂³×C₂₃	1	2	yes	—
651	#95	C₂³×C₂₃	1	2	yes	—
652	#95	C₂³×C₂₃	1	2	yes	—
653	#95	C₂³×C₂₃	1	2	yes	—
654	#95	C₂³×C₂₃	1	2	yes	—
655	#95	C₂³×C₂₃	1	2	yes	—
656	#95	C₂³×C₂₃	1	2	yes	—
657	C₂₃	#34527	1	1	yes	characteristic, direct product factor, Sylow
658	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
659	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
660	C₂×Q₈	—	4	1	no	—
661	C₂×Q₈	—	4	1	no	—
662	C₂×Q₈	—	4	1	no	—
663	C₂×Q₈	—	4	1	no	—
664	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
665	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
666	C₂×D₄	—	4	2	no	—
667	C₂×D₄	—	4	1	no	—
668	C₂×D₄	—	4	1	no	—
669	C₂×D₄	—	4	2	no	—
670	C₂×D₄	—	4	2	no	—
671	C₂×D₄	—	4	1	no	—
672	C₂×D₄	—	4	1	no	—
673	C₂×D₄	—	4	2	no	—
674	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
675	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
676	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
677	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
678	C₂²×C₄	—	2	4	no	—
679	C₂²×C₄	—	2	4	no	—
680	C₂²×C₄	—	4	2	no	—
681	C₂²×C₄	—	4	2	no	—
682	C₂²×C₄	—	4	2	no	—
683	C₂²×C₄	—	4	2	no	—
684	C₂²×C₄	—	4	2	no	—
685	C₂²×C₄	—	4	2	no	—
686	C₂²×C₄	—	4	2	no	—
687	C₂²×C₄	—	4	2	no	—
688	C₂²×C₄	—	4	2	no	—
689	C₂²×C₄	—	4	2	no	—
690	C₂²×C₄	—	4	2	no	—
691	C₂²×C₄	—	4	2	no	—
692	C₂²×C₄	—	4	2	no	—
693	C₂²×C₄	—	4	2	no	—
694	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
695	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
696	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
697	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
698	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
699	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
700	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
701	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
702	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
703	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
704	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
705	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
706	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
707	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
708	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
709	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
710	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
711	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
712	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
713	C₄⋊C₄	—	2	4	no	—
714	C₄⋊C₄	—	2	4	no	—
715	C₄⋊C₄	—	2	4	no	—
716	C₄⋊C₄	—	4	2	no	—
717	C₄⋊C₄	—	4	2	no	—
718	C₄⋊C₄	—	4	2	no	—
719	C₄⋊C₄	—	4	2	no	—
720	C₄⋊C₄	—	4	1	no	—
721	C₄⋊C₄	—	4	1	no	—
722	C₄⋊C₄	—	4	1	no	—
723	C₄⋊C₄	—	4	1	no	—
724	C₄⋊C₄	—	4	1	no	—
725	C₄⋊C₄	—	4	1	no	—
726	C₄⋊C₄	—	4	2	no	—
727	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
728	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
729	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
730	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
731	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
732	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
733	C₂²⋊C₄	—	2	4	no	—
734	C₂²⋊C₄	—	2	4	no	—
735	C₂²⋊C₄	—	2	4	no	—
736	C₂²⋊C₄	—	2	4	no	—
737	C₂²⋊C₄	—	4	2	no	—
738	C₂²⋊C₄	—	4	2	no	—
739	C₂²⋊C₄	—	4	2	no	—
740	C₂²⋊C₄	—	4	2	no	—
741	C₂²⋊C₄	—	4	2	no	—
742	C₂²⋊C₄	—	4	2	no	—
743	C₂²⋊C₄	—	4	2	no	—
744	C₂²⋊C₄	—	4	2	no	—
745	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
746	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
747	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
748	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
749	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
750	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
751	C₄²	—	2	4	no	—
752	C₄²	—	2	2	no	—
753	C₄²	—	2	2	no	—
754	C₄²	—	2	2	no	—
755	C₄²	—	4	2	no	—
756	C₄²	—	4	2	no	—
757	C₄²	—	4	2	no	—
758	C₄²	—	4	2	no	—
759	C₄²	—	4	2	no	—
760	C₄²	—	4	1	no	—
761	C₄²	—	4	1	no	—
762	C₄²	—	4	1	no	—
763	C₄²	—	4	2	no	—
764	C₄²	—	4	2	no	—
765	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
766	C₂³	—	2	2	no	—
767	C₂³	—	2	2	no	—
768	C₂³	—	2	2	no	—
769	C₂³	—	2	2	no	—
770	C₂³	—	2	2	no	—
771	C₂³	—	2	2	no	—
772	C₂³	—	2	2	no	—
773	C₂³	—	2	2	no	—
774	C₂³	—	2	2	no	—
775	C₂³	—	2	2	no	—
776	C₂³	—	2	2	no	—
777	C₂³	—	2	2	no	—
778	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
779	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
780	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
781	Q₈	—	4	4	no	—
782	Q₈	—	4	4	no	—
783	D₄	—	4	8	no	—
784	D₄	—	4	8	no	—
785	D₄	—	4	4	no	—
786	D₄	—	4	4	no	—
787	C₂×C₄	—	2	1	no	—
788	C₂×C₄	—	2	1	no	—
789	C₂×C₄	—	2	2	no	—
790	C₂×C₄	—	2	1	no	—
791	C₂×C₄	—	2	1	no	—
792	C₂×C₄	—	2	2	no	—
793	C₂×C₄	—	2	2	no	—
794	C₂×C₄	—	2	2	no	—
795	C₂×C₄	—	2	1	no	—
796	C₂×C₄	—	2	1	no	—
797	C₂×C₄	—	2	1	no	—
798	C₂×C₄	—	2	1	no	—
799	C₂×C₄	—	2	1	no	—
800	C₂×C₄	—	2	1	no	—
801	C₂×C₄	—	2	2	no	—
802	C₂×C₄	—	2	2	no	—
803	C₂×C₄	—	2	2	no	—
804	C₂×C₄	—	2	1	no	—
805	C₂×C₄	—	2	1	no	—
806	C₂×C₄	—	2	2	no	—
807	C₂×C₄	—	2	2	no	—
808	C₂×C₄	—	2	2	no	—
809	C₂×C₄	—	2	2	no	—
810	C₂×C₄	—	2	2	no	—
811	C₂×C₄	—	2	2	no	—
812	C₂×C₄	—	2	2	no	—
813	C₂×C₄	—	2	2	no	—
814	C₂×C₄	—	2	2	no	—
815	C₂×C₄	—	2	2	no	—
816	C₂×C₄	—	2	2	no	—
817	C₂×C₄	—	2	2	no	—
818	C₂×C₄	—	2	2	no	—
819	C₂×C₄	—	2	1	no	—
820	C₂×C₄	—	2	1	no	—
821	C₂×C₄	—	2	2	no	—
822	C₂×C₄	—	2	2	no	—
823	C₂×C₄	—	2	1	no	—
824	C₂×C₄	—	2	2	no	—
825	C₂×C₄	—	2	2	no	—
826	C₂×C₄	—	2	2	no	—
827	C₂×C₄	—	2	2	no	—
828	C₂×C₄	—	4	8	no	—
829	C₂×C₄	—	4	8	no	—
830	C₂×C₄	—	8	4	no	—
831	C₂×C₄	—	8	4	no	—
832	C₂×C₄	—	8	4	no	—
833	C₂×C₄	—	8	4	no	—
834	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
835	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
836	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
837	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
838	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
839	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
840	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
841	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
842	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
843	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
844	C₂²	—	2	4	no	—
845	C₂²	—	2	4	no	—
846	C₂²	—	2	4	no	—
847	C₂²	—	2	4	no	—
848	C₂²	—	2	4	no	—
849	C₂²	—	2	4	no	—
850	C₂²	—	4	2	no	—
851	C₂²	—	4	2	no	—
852	C₂²	—	4	2	no	—
853	C₂²	—	4	2	no	—
854	C₂²	—	4	2	no	—
855	C₂²	—	4	2	no	—
856	C₂²	—	4	2	no	—
857	C₂²	—	4	2	no	—
858	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
859	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
860	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
861	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
862	C₂²	C₂₃×#585	1	1	yes	characteristic
863	C₂²	C₂₃×#585	1	1	yes	characteristic
864	C₂²	C₂₃×#585	1	1	yes	characteristic
865	C₄	—	2	4	no	—
866	C₄	—	2	4	no	—
867	C₄	—	2	4	no	—
868	C₄	—	2	4	no	—
869	C₄	—	2	2	no	—
870	C₄	—	2	2	no	—
871	C₄	—	2	2	no	—
872	C₄	—	2	2	no	—
873	C₄	—	2	2	no	—
874	C₄	—	2	2	no	—
875	C₄	—	4	2	no	—
876	C₄	—	4	2	no	—
877	C₄	—	4	2	no	—
878	C₄	—	4	2	no	—
879	C₄	—	4	2	no	—
880	C₄	—	4	2	no	—
881	C₄	—	4	1	no	—
882	C₂	—	4	4	no	—
883	C₂	—	4	4	no	—
884	C₂	C₄×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
885	C₂	C₄×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
886	C₂	C₄×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
887	C₂	C₂₃×#3558	1	1	yes	characteristic
888	C₂	C₂₃×#3559	1	1	yes	characteristic
889	C₂	C₂₃×#3561	1	1	yes	characteristic
890	C₂	C₂₃×#3567	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#562	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂₃×#2382	C₂	semidirect
3. C₂₃×#2379	C₂	semidirect
4. C₂₃×#2379	C₂	semidirect
5. C₂₃×#2374	C₂	semidirect
6. C₂₃×#2364	C₂	semidirect
7. C₂₃×#2362	C₂	semidirect
8. C₂₃×#2361	C₂	semidirect
9. C₂₃×#2360	C₂	semidirect
10. C₂₃×#2359	C₂	semidirect
11. C₂₃×#2335	C₂	semidirect
12. C₂₃×#2333	C₂	semidirect
13. C₂₃×#2332	C₂	semidirect
14. C₂₃×#2329	C₂	semidirect
15. C₂₃×#2324	C₂	semidirect
16. C₂₃×#2322	C₂	semidirect
17. C₄×C₂₃×#125	C₂	semidirect
18. C₄×C₂₃×#124	C₂	semidirect
19. C₄×C₂₃×#124	C₂	semidirect
20. C₂₃×#562	C₄	semidirect
120. C₂₃×#125	C₂×C₄	semidirect
121. C₂₃×#124	C₂×C₄	semidirect
122. C₂₃×#124	C₂×C₄	semidirect
123. C₂₃×#123	C₂×C₄	semidirect
124. C₂₃×#123	C₂×C₄	semidirect
125. C₂₃×C₂²⋊Q₈	C₂×C₄	semidirect
126. C₂₃×C₂²⋊Q₈	C₂×C₄	semidirect
127. C₂₃×#121	C₂×C₄	semidirect
128. C₂₃×#121	C₂×C₄	semidirect
320. #34527	C₂₃	direct (N × H)
400. #2382	C₂×C₂₃	semidirect
401. #2379	C₂×C₂₃	semidirect
402. #2379	C₂×C₂₃	semidirect
403. #2374	C₂×C₂₃	semidirect
404. #2364	C₂×C₂₃	semidirect
405. #2362	C₂×C₂₃	semidirect
406. #2361	C₂×C₂₃	semidirect
407. #2360	C₂×C₂₃	semidirect
408. #2359	C₂×C₂₃	semidirect
409. #2335	C₂×C₂₃	semidirect
410. #2333	C₂×C₂₃	semidirect
411. #2332	C₂×C₂₃	semidirect
412. #2329	C₂×C₂₃	semidirect
413. #2324	C₂×C₂₃	semidirect
414. #2322	C₂×C₂₃	semidirect
415. C₄×#125	C₂×C₂₃	semidirect
416. C₄×#124	C₂×C₂₃	semidirect
417. C₄×#124	C₂×C₂₃	semidirect
456. #562	C₄×C₂₃	semidirect
565. #125	C₂×C₄×C₂₃	semidirect
566. #124	C₂×C₄×C₂₃	semidirect
567. #124	C₂×C₄×C₂₃	semidirect
568. #123	C₂×C₄×C₂₃	semidirect
569. #123	C₂×C₄×C₂₃	semidirect
570. C₂²⋊Q₈	C₂×C₄×C₂₃	semidirect
571. C₂²⋊Q₈	C₂×C₄×C₂₃	semidirect
572. #121	C₂×C₄×C₂₃	semidirect
573. #121	C₂×C₄×C₂₃	semidirect
657. C₂₃	#34527	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#562	2
C₂₃×#2382	2
C₂₃×#2379	2
C₂₃×#2374	2
C₂₃×#2364	2
C₂₃×#2362	2
C₂₃×#2361	2
C₂₃×#2360	2
C₂₃×#2359	2
C₂₃×#2335	2
C₂₃×#2333	2
C₂₃×#2332	2
C₂₃×#2329	2
C₂₃×#2324	2
C₂₃×#2322	2
C₄×C₂₃×#125	2
C₄×C₂₃×#124	2
#34527	23

Subgroups — C23×#34527 #19182153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34527 #19182153