Subgroups — C₂₃×#60275 #19207901

Order 5888.

1818 conjugacy classes of subgroups.

The subgroup lattice has 772 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3480	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2995	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2994	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2993	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2992	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2990	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2989	C₂	1	2	yes	maximal, complemented
8	Q₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2935	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2935	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#2932	C₂	1	2	yes	maximal, complemented
13	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#2923	C₂	1	2	yes	maximal, complemented
15	C₂₃×#2922	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#2906	C₂	1	2	yes	maximal, complemented
17	C₂₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×C₂₃×#131	C₂²	1	2	yes	—
19	C₂×C₂₃×#131	C₂²	1	2	yes	—
20	C₂×C₂₃×#131	C₂²	1	2	yes	—
21	C₂₃×#559	C₄	1	1	yes	characteristic
22	C₂₃×#559	C₄	1	2	yes	—
23	C₂₃×#557	C₄	1	1	yes	characteristic
24	C₂₃×#556	C₄	1	1	yes	characteristic
25	C₂₃×#556	C₄	1	2	yes	—
26	C₂₃×#553	C₄	1	2	yes	—
27	C₂₃×#553	C₄	1	4	yes	—
28	C₂₃×#550	C₄	1	1	yes	characteristic
29	C₂₃×#548	C₄	1	2	yes	—
30	C₂₃×#532	C₂²	1	1	yes	characteristic
31	C₂₃×#532	C₂²	1	2	yes	—
32	C₂₃×#529	C₂²	1	4	yes	—
33	C₂₃×#529	C₂²	1	2	yes	—
34	C₄×#41×C₂₃	C₂²	1	2	yes	—
35	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
36	C₄×#41×C₂₃	C₂²	1	2	yes	—
37	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#446	C₂²	1	1	yes	characteristic
39	C₂₃×#446	C₂²	1	1	yes	characteristic
40	C₂₃×#446	C₂²	1	2	yes	—
41	C₂₃×#446	C₂²	1	1	yes	characteristic
42	C₂₃×#446	C₂²	1	1	yes	characteristic
43	C₂₃×#446	C₂²	1	2	yes	—
44	C₂₃×#446	C₂²	1	2	yes	—
45	C₂₃×#446	C₂²	1	4	yes	—
46	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
47	C₂₃×#436	C₂²	1	2	yes	—
48	C₂₃×#436	C₂²	1	2	yes	—
49	C₂₃×#436	C₂²	1	2	yes	—
50	C₂₃×#436	C₂²	1	4	yes	—
51	C₂₃×#435	C₂²	1	4	yes	—
52	C₂₃×#435	C₂²	1	4	yes	—
53	C₂₃×#435	C₂²	1	4	yes	—
54	C₂₃×#435	C₂²	1	4	yes	—
55	C₂₃×#435	C₂²	1	4	yes	—
56	C₂₃×#435	C₂²	1	4	yes	—
57	C₂₃×#435	C₂²	1	4	yes	—
58	C₈×D₄×C₂₃	C₂²	1	2	yes	—
59	C₈×D₄×C₂₃	C₂²	1	2	yes	—
60	C₈×D₄×C₂₃	C₂²	1	2	yes	—
61	C₈×D₄×C₂₃	C₂²	1	4	yes	—
62	C₂₃×#433	C₂²	1	2	yes	—
63	C₂₃×#433	C₂²	1	2	yes	—
64	C₂₃×#433	C₂²	1	4	yes	—
65	C₂₃×#433	C₂²	1	4	yes	—
66	C₂₃×#432	C₂²	1	4	yes	—
67	C₂₃×#432	C₂²	1	4	yes	—
68	C₂₃×#431	C₂²	1	2	yes	—
69	C₂₃×#431	C₂²	1	2	yes	—
70	C₂₃×#424	C₂²	1	1	yes	characteristic
71	C₂₃×#424	C₂²	1	1	yes	characteristic
72	C₂₃×#424	C₂²	1	2	yes	—
73	C₂₃×#424	C₂²	1	4	yes	—
74	C₂₃×#424	C₂²	1	4	yes	—
75	C₂₃×#424	C₂²	1	4	yes	—
76	C₂₃×#423	C₂²	1	1	yes	characteristic
77	C₂₃×#423	C₂²	1	2	yes	—
78	C₂₃×#423	C₂²	1	4	yes	—
79	C₂₃×#423	C₂²	1	2	yes	—
80	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
81	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
82	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
83	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
84	C₂₃×#408	C₂²	1	1	yes	characteristic
85	C₂₃×#408	C₂²	1	1	yes	characteristic
86	C₂₃×#408	C₂²	1	2	yes	—
87	C₂₃×#408	C₂²	1	2	yes	—
88	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
89	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
90	C₂×C₂₃×#97	C₂²	1	2	yes	—
91	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
92	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
93	C₂×#41×C₂₃	C₂³	1	2	yes	—
94	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
95	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
96	C₂₃×#131	—	4	4	no	—
97	C₂₃×#131	—	4	4	no	—
98	C₂₃×#131	—	4	4	no	—
99	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
100	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
103	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
104	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
105	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
106	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
107	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
108	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
109	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
110	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
111	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
112	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
113	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
114	C₂₃×#127	C₂×C₄	1	2	yes	—
115	C₂₃×#126	C₂×C₄	1	4	yes	—
116	C₂₃×#126	C₂×C₄	1	4	yes	—
117	C₂₃×#126	C₂×C₄	1	4	yes	—
118	C₂₃×#125	C₂×C₄	1	4	yes	—
119	C₂₃×#125	C₂×C₄	1	2	yes	—
120	C₂₃×#124	C₂×C₄	1	2	yes	—
121	C₂₃×#124	C₂×C₄	1	4	yes	—
122	C₂₃×#124	C₂×C₄	1	2	yes	—
123	C₂₃×#124	C₂×C₄	1	2	yes	—
124	C₂₃×#124	C₂×C₄	1	2	yes	—
125	C₂₃×#123	C₂×C₄	1	4	yes	—
126	C₂₃×#123	C₂×C₄	1	4	yes	—
127	C₂₃×#123	C₂×C₄	1	4	yes	—
128	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
129	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
130	C₂₃×#121	C₂×C₄	1	2	yes	—
131	C₂₃×#121	C₂×C₄	1	4	yes	—
132	C₂₃×#121	C₂×C₄	1	2	yes	—
133	C₂₃×#121	C₂×C₄	1	4	yes	—
134	C₂₃×#121	C₂×C₄	1	2	yes	—
135	C₂₃×#121	C₂×C₄	1	4	yes	—
136	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
137	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
138	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
139	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
140	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
141	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
142	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
143	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
144	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
145	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
146	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
147	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
148	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
149	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
150	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
151	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
152	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
153	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
154	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
155	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
156	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
157	C₄×D₄×C₂₃	C₂³	1	2	yes	—
158	C₄×D₄×C₂₃	C₂³	1	2	yes	—
159	C₄×D₄×C₂₃	C₂³	1	4	yes	—
160	C₄×D₄×C₂₃	C₂³	1	4	yes	—
161	C₂₃×#117	C₂×C₄	1	2	yes	—
162	C₂₃×#117	C₂×C₄	1	4	yes	—
163	C₂₃×#117	C₂³	1	1	yes	characteristic
164	C₂₃×#117	C₂³	1	4	yes	—
165	C₂₃×#117	C₂³	1	2	yes	—
166	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
167	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
168	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
169	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
170	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
171	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
172	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
173	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
174	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
175	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
176	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
177	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
178	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
179	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
180	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
181	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
182	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
183	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
184	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
185	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
186	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
187	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
188	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
189	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
190	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
191	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
192	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
193	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
194	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
195	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
196	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
197	C₂₃×#97	C₂³	1	2	yes	—
198	C₂₃×#97	C₂³	1	2	yes	—
199	C₂₃×#97	C₂³	1	2	yes	—
200	C₂₃×#97	C₂³	1	2	yes	—
201	C₂₃×#97	C₂³	1	4	yes	—
202	C₂₃×#97	C₂³	1	2	yes	—
203	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
204	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
205	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
206	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
207	C₄×C₈×C₂₃	C₂³	1	2	yes	—
208	C₄×C₈×C₂₃	C₂³	1	4	yes	—
209	#41×C₂₃	—	2	4	no	—
210	#41×C₂₃	—	2	2	no	—
211	#41×C₂₃	—	2	4	no	—
212	#41×C₂₃	—	2	2	no	—
213	#41×C₂₃	—	4	4	no	—
214	C₂×Q₈×C₂₃	—	4	2	no	—
215	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
216	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
217	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
218	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
219	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
220	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
221	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
222	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
223	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
224	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
225	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
226	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
227	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
228	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
229	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
230	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
231	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
232	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
233	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
234	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
235	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
236	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
237	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
238	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
239	M₁₆×C₂₃	—	2	4	no	—
240	M₁₆×C₂₃	—	2	2	no	—
241	M₁₆×C₂₃	—	2	2	no	—
242	M₁₆×C₂₃	—	4	4	no	—
243	M₁₆×C₂₃	—	4	4	no	—
244	M₁₆×C₂₃	—	4	4	no	—
245	C₂×C₈×C₂₃	—	2	2	no	—
246	C₂×C₈×C₂₃	—	2	4	no	—
247	C₂×C₈×C₂₃	—	2	2	no	—
248	C₂×C₈×C₂₃	—	4	4	no	—
249	C₂×C₈×C₂₃	—	4	4	no	—
250	C₂×C₈×C₂₃	—	4	4	no	—
251	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
252	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
253	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
254	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
255	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
256	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
257	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
258	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
259	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
260	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
261	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
262	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
263	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
264	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
265	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
266	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
267	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
268	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
269	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
270	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
271	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
272	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
273	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
274	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
275	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
276	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
277	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
278	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
279	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
280	C₂²⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
281	C₂²⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
282	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
283	C₂²⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
284	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
285	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
286	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
287	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
288	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
289	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
290	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
291	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
292	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
293	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
294	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
295	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
296	C₄²×C₂₃	C₂⁴	1	2	yes	—
297	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
298	C₄²×C₂₃	C₂⁴	1	2	yes	—
299	C₄²×C₂₃	C₂⁴	1	2	yes	—
300	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
301	#60275	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
302	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
303	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
304	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
305	Q₈×C₂₃	—	2	2	no	—
306	Q₈×C₂₃	—	2	4	no	—
307	Q₈×C₂₃	—	2	2	no	—
308	Q₈×C₂₃	—	2	2	no	—
309	Q₈×C₂₃	—	4	2	no	—
310	D₄×C₂₃	—	2	4	no	—
311	D₄×C₂₃	—	2	8	no	—
312	D₄×C₂₃	—	2	2	no	—
313	D₄×C₂₃	—	2	4	no	—
314	D₄×C₂₃	—	2	4	no	—
315	D₄×C₂₃	—	4	4	no	—
316	D₄×C₂₃	—	4	2	no	—
317	D₄×C₂₃	—	4	4	no	—
318	C₂×C₄×C₂₃	—	2	2	no	—
319	C₂×C₄×C₂₃	—	2	4	no	—
320	C₂×C₄×C₂₃	—	2	4	no	—
321	C₂×C₄×C₂₃	—	2	2	no	—
322	C₂×C₄×C₂₃	—	2	2	no	—
323	C₂×C₄×C₂₃	—	2	4	no	—
324	C₂×C₄×C₂₃	—	4	2	no	—
325	C₂×C₄×C₂₃	—	4	4	no	—
326	C₂×C₄×C₂₃	—	4	1	no	—
327	C₂×C₄×C₂₃	—	4	2	no	—
328	C₂×C₄×C₂₃	—	4	2	no	—
329	C₂×C₄×C₂₃	—	4	4	no	—
330	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
331	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
332	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
333	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
334	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
335	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
336	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
337	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
338	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
339	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
340	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
341	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
342	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
343	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
344	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
345	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
346	C₈×C₂₃	—	2	1	no	—
347	C₈×C₂₃	—	2	2	no	—
348	C₈×C₂₃	—	2	2	no	—
349	C₈×C₂₃	—	2	1	no	—
350	C₈×C₂₃	—	2	4	no	—
351	C₈×C₂₃	—	2	2	no	—
352	C₈×C₂₃	—	2	1	no	—
353	C₈×C₂₃	—	2	1	no	—
354	C₈×C₂₃	—	2	2	no	—
355	#3480	C₂×C₂₃	1	1	yes	characteristic
356	#2995	C₂×C₂₃	1	4	yes	complemented
357	#2994	C₂×C₂₃	1	2	yes	complemented
358	#2993	C₂×C₂₃	1	2	yes	complemented
359	#2992	C₂×C₂₃	1	4	yes	complemented
360	#2990	C₂×C₂₃	1	2	yes	complemented
361	#2989	C₂×C₂₃	1	2	yes	complemented
362	Q₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
363	C₂×#446	C₂×C₂₃	1	1	yes	characteristic, complemented
364	#2935	C₂×C₂₃	1	1	yes	characteristic, complemented
365	#2935	C₂×C₂₃	1	2	yes	complemented
366	#2932	C₂×C₂₃	1	2	yes	complemented
367	#2931	C₂×C₂₃	1	1	yes	characteristic, complemented
368	#2923	C₂×C₂₃	1	2	yes	complemented
369	#2922	C₂×C₂₃	1	1	yes	characteristic, complemented
370	#2906	C₂×C₂₃	1	1	yes	characteristic, complemented
371	#2906	C₂×C₂₃	1	2	yes	complemented
372	C₂²×C₂₃	—	2	1	no	—
373	C₂²×C₂₃	—	2	1	no	—
374	C₂²×C₂₃	—	2	2	no	—
375	C₂²×C₂₃	—	2	4	no	—
376	C₂²×C₂₃	—	2	2	no	—
377	C₂²×C₂₃	—	2	4	no	—
378	C₂²×C₂₃	—	2	2	no	—
379	C₂²×C₂₃	—	2	4	no	—
380	C₂²×C₂₃	#520	1	2	yes	—
381	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
382	C₄×C₂₃	—	2	4	no	—
383	C₄×C₂₃	—	2	2	no	—
384	C₄×C₂₃	—	2	2	no	—
385	C₄×C₂₃	—	2	1	no	—
386	C₄×C₂₃	—	2	2	no	—
387	C₄×C₂₃	—	2	2	no	—
388	C₄×C₂₃	—	2	2	no	—
389	C₄×C₂₃	—	2	1	no	—
390	C₄×C₂₃	—	2	2	no	—
391	C₄×C₂₃	—	2	1	no	—
392	C₄×C₂₃	—	2	2	no	—
393	C₄×C₂₃	#520	1	2	yes	—
394	C₄×C₂₃	#568	1	2	yes	—
395	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
396	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
397	C₂×#131	C₂²×C₂₃	1	2	yes	—
398	C₂×#131	C₂²×C₂₃	1	2	yes	—
399	C₂×#131	C₂²×C₂₃	1	2	yes	—
400	#559	C₄×C₂₃	1	1	yes	characteristic
401	#559	C₄×C₂₃	1	2	yes	—
402	#557	C₄×C₂₃	1	1	yes	characteristic
403	#556	C₄×C₂₃	1	1	yes	characteristic
404	#556	C₄×C₂₃	1	2	yes	—
405	#553	C₄×C₂₃	1	2	yes	—
406	#553	C₄×C₂₃	1	4	yes	—
407	#550	C₄×C₂₃	1	1	yes	characteristic
408	#548	C₄×C₂₃	1	2	yes	—
409	#532	C₂²×C₂₃	1	1	yes	characteristic
410	#532	C₂²×C₂₃	1	2	yes	—
411	#529	C₂²×C₂₃	1	4	yes	—
412	#529	C₂²×C₂₃	1	2	yes	—
413	C₄×#41	C₂²×C₂₃	1	2	yes	—
414	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
415	C₄×#41	C₂²×C₂₃	1	2	yes	—
416	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
417	#446	C₂²×C₂₃	1	1	yes	characteristic
418	#446	C₂²×C₂₃	1	1	yes	characteristic
419	#446	C₂²×C₂₃	1	2	yes	—
420	#446	C₂²×C₂₃	1	1	yes	characteristic
421	#446	C₂²×C₂₃	1	1	yes	characteristic
422	#446	C₂²×C₂₃	1	2	yes	—
423	#446	C₂²×C₂₃	1	2	yes	—
424	#446	C₂²×C₂₃	1	4	yes	—
425	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
426	#436	C₂²×C₂₃	1	2	yes	—
427	#436	C₂²×C₂₃	1	2	yes	—
428	#436	C₂²×C₂₃	1	2	yes	—
429	#436	C₂²×C₂₃	1	4	yes	—
430	#435	C₂²×C₂₃	1	4	yes	—
431	#435	C₂²×C₂₃	1	4	yes	—
432	#435	C₂²×C₂₃	1	4	yes	—
433	#435	C₂²×C₂₃	1	4	yes	—
434	#435	C₂²×C₂₃	1	4	yes	—
435	#435	C₂²×C₂₃	1	4	yes	—
436	#435	C₂²×C₂₃	1	4	yes	—
437	C₈×D₄	C₂²×C₂₃	1	2	yes	—
438	C₈×D₄	C₂²×C₂₃	1	2	yes	—
439	C₈×D₄	C₂²×C₂₃	1	2	yes	—
440	C₈×D₄	C₂²×C₂₃	1	4	yes	—
441	#433	C₂²×C₂₃	1	2	yes	—
442	#433	C₂²×C₂₃	1	2	yes	—
443	#433	C₂²×C₂₃	1	4	yes	—
444	#433	C₂²×C₂₃	1	4	yes	—
445	#432	C₂²×C₂₃	1	4	yes	—
446	#432	C₂²×C₂₃	1	4	yes	—
447	#431	C₂²×C₂₃	1	2	yes	—
448	#431	C₂²×C₂₃	1	2	yes	—
449	#424	C₂²×C₂₃	1	1	yes	characteristic
450	#424	C₂²×C₂₃	1	2	yes	—
451	#424	C₂²×C₂₃	1	4	yes	—
452	#424	C₂²×C₂₃	1	4	yes	—
453	#424	C₂²×C₂₃	1	4	yes	—
454	#424	C₂²×C₂₃	1	1	yes	characteristic
455	#423	C₂²×C₂₃	1	1	yes	characteristic
456	#423	C₂²×C₂₃	1	2	yes	—
457	#423	C₂²×C₂₃	1	4	yes	—
458	#423	C₂²×C₂₃	1	2	yes	—
459	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
460	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
461	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
462	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
463	#408	C₂²×C₂₃	1	1	yes	characteristic
464	#408	C₂²×C₂₃	1	1	yes	characteristic
465	#408	C₂²×C₂₃	1	2	yes	—
466	#408	C₂²×C₂₃	1	2	yes	—
467	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
468	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
469	C₂×#97	C₂²×C₂₃	1	2	yes	—
470	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
471	C₂×C₂₃	—	2	2	no	—
472	C₂×C₂₃	—	4	2	no	—
473	C₂×C₂₃	—	4	4	no	—
474	C₂×C₂₃	C₂×#520	1	1	yes	characteristic
475	C₂×C₂₃	C₂×#568	1	1	yes	characteristic
476	C₂×C₂₃	#3573	1	1	yes	characteristic
477	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
478	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
479	C₂×#41	C₂³×C₂₃	1	2	yes	—
480	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
481	#131	—	4	4	no	—
482	#131	—	4	4	no	—
483	#131	—	4	4	no	—
484	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
485	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
486	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
487	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
488	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
489	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
490	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
491	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
492	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
493	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
494	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
495	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
496	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
497	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
498	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
499	#127	C₂×C₄×C₂₃	1	2	yes	—
500	#126	C₂×C₄×C₂₃	1	4	yes	—
501	#126	C₂×C₄×C₂₃	1	4	yes	—
502	#126	C₂×C₄×C₂₃	1	4	yes	—
503	#125	C₂×C₄×C₂₃	1	4	yes	—
504	#125	C₂×C₄×C₂₃	1	2	yes	—
505	#124	C₂×C₄×C₂₃	1	2	yes	—
506	#124	C₂×C₄×C₂₃	1	4	yes	—
507	#124	C₂×C₄×C₂₃	1	2	yes	—
508	#124	C₂×C₄×C₂₃	1	2	yes	—
509	#124	C₂×C₄×C₂₃	1	2	yes	—
510	#123	C₂×C₄×C₂₃	1	4	yes	—
511	#123	C₂×C₄×C₂₃	1	4	yes	—
512	#123	C₂×C₄×C₂₃	1	4	yes	—
513	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
514	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
515	#121	C₂×C₄×C₂₃	1	2	yes	—
516	#121	C₂×C₄×C₂₃	1	4	yes	—
517	#121	C₂×C₄×C₂₃	1	2	yes	—
518	#121	C₂×C₄×C₂₃	1	4	yes	—
519	#121	C₂×C₄×C₂₃	1	2	yes	—
520	#121	C₂×C₄×C₂₃	1	4	yes	—
521	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
522	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
523	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
524	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
525	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
526	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
527	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
528	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
529	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
530	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
531	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
532	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
533	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
534	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
535	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
536	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
537	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
538	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
539	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
540	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
541	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
542	C₄×D₄	C₂³×C₂₃	1	2	yes	—
543	C₄×D₄	C₂³×C₂₃	1	4	yes	—
544	C₄×D₄	C₂³×C₂₃	1	2	yes	—
545	C₄×D₄	C₂³×C₂₃	1	4	yes	—
546	#117	C₂×C₄×C₂₃	1	2	yes	—
547	#117	C₂×C₄×C₂₃	1	4	yes	—
548	#117	C₂³×C₂₃	1	1	yes	characteristic
549	#117	C₂³×C₂₃	1	2	yes	—
550	#117	C₂³×C₂₃	1	4	yes	—
551	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
552	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
553	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
554	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
555	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
556	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
557	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
558	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
559	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
560	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
561	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
562	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
563	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
564	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
565	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
566	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
567	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
568	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
569	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
570	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
571	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
572	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
573	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
574	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
575	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
576	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
577	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
578	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
579	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
580	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
581	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
582	#97	C₂³×C₂₃	1	2	yes	—
583	#97	C₂³×C₂₃	1	2	yes	—
584	#97	C₂³×C₂₃	1	2	yes	—
585	#97	C₂³×C₂₃	1	2	yes	—
586	#97	C₂³×C₂₃	1	4	yes	—
587	#97	C₂³×C₂₃	1	2	yes	—
588	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
589	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
590	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
591	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
592	C₄×C₈	C₂³×C₂₃	1	2	yes	—
593	C₄×C₈	C₂³×C₂₃	1	4	yes	—
594	C₂₃	#60275	1	1	yes	characteristic, direct product factor, Sylow
595	#41	—	2	4	no	—
596	#41	—	2	2	no	—
597	#41	—	2	2	no	—
598	#41	—	2	4	no	—
599	#41	—	4	4	no	—
600	C₂×Q₈	—	4	2	no	—
601	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
602	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
603	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
604	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
605	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
606	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
607	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
608	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
609	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
610	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
611	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
612	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
613	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
614	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
615	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
616	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
617	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
618	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
619	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
620	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
621	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
622	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
623	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
624	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
625	M₁₆	—	2	4	no	—
626	M₁₆	—	2	2	no	—
627	M₁₆	—	2	2	no	—
628	M₁₆	—	4	4	no	—
629	M₁₆	—	4	4	no	—
630	M₁₆	—	4	4	no	—
631	C₂×C₈	—	2	2	no	—
632	C₂×C₈	—	2	4	no	—
633	C₂×C₈	—	2	2	no	—
634	C₂×C₈	—	4	4	no	—
635	C₂×C₈	—	4	4	no	—
636	C₂×C₈	—	4	4	no	—
637	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
638	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
639	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
640	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
641	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
642	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
643	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
644	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
645	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
646	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
647	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
648	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
649	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
650	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
651	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
652	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
653	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
654	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
655	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
656	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
657	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
658	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
659	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
660	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
661	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
662	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
663	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
664	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
665	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
666	C₂²⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
667	C₂²⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
668	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
669	C₂²⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
670	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
671	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
672	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
673	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
674	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
675	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
676	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
677	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
678	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
679	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
680	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
681	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
682	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
683	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
684	C₄²	C₂⁴×C₂₃	1	2	yes	—
685	C₄²	C₂⁴×C₂₃	1	2	yes	—
686	C₄²	C₂⁴×C₂₃	1	2	yes	—
687	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
688	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
689	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
690	Q₈	—	2	2	no	—
691	Q₈	—	2	2	no	—
692	Q₈	—	2	2	no	—
693	Q₈	—	2	4	no	—
694	Q₈	—	4	2	no	—
695	D₄	—	2	4	no	—
696	D₄	—	2	8	no	—
697	D₄	—	2	4	no	—
698	D₄	—	2	2	no	—
699	D₄	—	2	4	no	—
700	D₄	—	4	4	no	—
701	D₄	—	4	4	no	—
702	D₄	—	4	2	no	—
703	C₂×C₄	—	2	2	no	—
704	C₂×C₄	—	2	4	no	—
705	C₂×C₄	—	2	4	no	—
706	C₂×C₄	—	2	2	no	—
707	C₂×C₄	—	2	2	no	—
708	C₂×C₄	—	2	4	no	—
709	C₂×C₄	—	4	2	no	—
710	C₂×C₄	—	4	4	no	—
711	C₂×C₄	—	4	2	no	—
712	C₂×C₄	—	4	4	no	—
713	C₂×C₄	—	4	1	no	—
714	C₂×C₄	—	4	2	no	—
715	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
716	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
717	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
718	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
719	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
720	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
721	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
722	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
723	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
724	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
725	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
726	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
727	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
728	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
729	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
730	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
731	C₈	—	2	1	no	—
732	C₈	—	2	2	no	—
733	C₈	—	2	2	no	—
734	C₈	—	2	1	no	—
735	C₈	—	2	4	no	—
736	C₈	—	2	2	no	—
737	C₈	—	2	1	no	—
738	C₈	—	2	1	no	—
739	C₈	—	2	2	no	—
740	C₂²	—	2	1	no	—
741	C₂²	—	2	1	no	—
742	C₂²	—	2	2	no	—
743	C₂²	—	2	4	no	—
744	C₂²	—	2	2	no	—
745	C₂²	—	2	4	no	—
746	C₂²	—	2	2	no	—
747	C₂²	—	2	4	no	—
748	C₂²	C₂₃×#520	1	2	yes	—
749	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
750	C₄	—	2	4	no	—
751	C₄	—	2	2	no	—
752	C₄	—	2	2	no	—
753	C₄	—	2	1	no	—
754	C₄	—	2	2	no	—
755	C₄	—	2	2	no	—
756	C₄	—	2	2	no	—
757	C₄	—	2	1	no	—
758	C₄	—	2	2	no	—
759	C₄	—	2	1	no	—
760	C₄	—	2	2	no	—
761	C₄	C₂₃×#520	1	2	yes	—
762	C₄	C₂₃×#568	1	2	yes	—
763	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
764	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
765	C₂	—	2	2	no	—
766	C₂	—	4	2	no	—
767	C₂	—	4	4	no	—
768	C₂	C₂×C₂₃×#520	1	1	yes	characteristic
769	C₂	C₂×C₂₃×#568	1	1	yes	characteristic
770	C₂	C₂₃×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂⁴×C₄
prime radical	C₂₃×#3480	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2995	C₂	semidirect
3. C₂₃×#2994	C₂	semidirect
4. C₂₃×#2993	C₂	semidirect
5. C₂₃×#2992	C₂	semidirect
6. C₂₃×#2990	C₂	semidirect
7. C₂₃×#2989	C₂	semidirect
8. Q₈×M₁₆×C₂₃	C₂	semidirect
9. C₂×C₂₃×#446	C₂	semidirect
10. C₂₃×#2935	C₂	semidirect
11. C₂₃×#2935	C₂	semidirect
12. C₂₃×#2932	C₂	semidirect
13. C₂₃×#2931	C₂	semidirect
14. C₂₃×#2923	C₂	semidirect
15. C₂₃×#2922	C₂	semidirect
16. C₂₃×#2906	C₂	semidirect
17. C₂₃×#2906	C₂	semidirect
301. #60275	C₂₃	direct (N × H)
356. #2995	C₂×C₂₃	semidirect
357. #2994	C₂×C₂₃	semidirect
358. #2993	C₂×C₂₃	semidirect
359. #2992	C₂×C₂₃	semidirect
360. #2990	C₂×C₂₃	semidirect
361. #2989	C₂×C₂₃	semidirect
362. Q₈×M₁₆	C₂×C₂₃	semidirect
363. C₂×#446	C₂×C₂₃	semidirect
364. #2935	C₂×C₂₃	semidirect
365. #2935	C₂×C₂₃	semidirect
366. #2932	C₂×C₂₃	semidirect
367. #2931	C₂×C₂₃	semidirect
368. #2923	C₂×C₂₃	semidirect
369. #2922	C₂×C₂₃	semidirect
370. #2906	C₂×C₂₃	semidirect
371. #2906	C₂×C₂₃	semidirect
594. C₂₃	#60275	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3480	2
C₂₃×#2995	2
C₂₃×#2994	2
C₂₃×#2993	2
C₂₃×#2992	2
C₂₃×#2990	2
C₂₃×#2989	2
Q₈×M₁₆×C₂₃	2
C₂×C₂₃×#446	2
C₂₃×#2935	2
C₂₃×#2932	2
C₂₃×#2931	2
C₂₃×#2923	2
C₂₃×#2922	2
C₂₃×#2906	2
#60275	23

Subgroups — C23×#60275 #19207901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60275 #19207901