Subgroups — C₂₃×#34083 #19181709

Order 5888.

2346 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2337	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2336	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2334	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2329	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2325	C₂	1	2	yes	maximal
7	C₂₃×#2322	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2298	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2296	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2295	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2290	C₂	1	2	yes	maximal, complemented
12	C₂₃×#2290	C₂	1	2	yes	maximal, complemented
13	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄²×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
15	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#529	C₄	1	16	yes	complemented
17	C₂×C₂₃×#126	C₂²	1	2	yes	—
18	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#123	C₂²	1	2	yes	—
21	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#117	C₂²	1	2	yes	—
28	C₂×C₂₃×#117	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
30	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
33	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#391	C₂²	1	4	yes	—
35	C₂₃×#389	C₂²	1	4	yes	—
36	C₂₃×#389	C₂²	1	4	yes	—
37	C₂₃×#389	C₂²	1	4	yes	—
38	C₂₃×#388	C₂²	1	4	yes	—
39	C₂₃×#388	C₂²	1	4	yes	—
40	C₂₃×#388	C₂²	1	4	yes	—
41	C₂₃×#388	C₂²	1	4	yes	—
42	C₂₃×#387	C₂²	1	4	yes	—
43	C₂₃×#387	C₂²	1	4	yes	—
44	C₂₃×#387	C₂²	1	4	yes	—
45	C₂₃×#387	C₂²	1	4	yes	—
46	C₂₃×#387	C₂²	1	4	yes	—
47	C₂₃×#387	C₂²	1	4	yes	—
48	C₂₃×#386	C₂²	1	4	yes	—
49	C₂₃×#386	C₂²	1	4	yes	—
50	C₂₃×#385	C₂²	1	4	yes	—
51	C₂₃×#385	C₂²	1	4	yes	—
52	C₂₃×#385	C₂²	1	4	yes	—
53	C₂₃×#385	C₂²	1	4	yes	—
54	C₂₃×#383	C₂²	1	2	yes	—
55	C₂₃×#382	C₂²	1	2	yes	—
56	C₂₃×#381	C₂²	1	2	yes	—
57	C₂₃×#380	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
59	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
64	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
65	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
66	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
68	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
69	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
71	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
72	C₂₃×#376	C₂²	1	2	yes	—
73	C₂₃×#376	C₂²	1	2	yes	—
74	C₂×C₂₃×#95	C₂²	1	2	yes	—
75	C₄³×C₂₃	C₂²	1	2	yes	—
76	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
78	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
82	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
83	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
84	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
86	C₂₃×#126	C₂×C₄	1	16	yes	complemented
87	C₂₃×#125	C₂×C₄	1	8	yes	complemented
88	C₂₃×#124	C₂×C₄	1	8	yes	complemented
89	C₂₃×#123	C₂×C₄	1	16	yes	complemented
90	C₂₃×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
91	C₂₃×#121	C₂×C₄	1	8	yes	—
92	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
93	C₄×D₄×C₂₃	—	2	16	no	—
94	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	complemented
95	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
96	C₂₃×#117	—	4	4	no	—
97	C₂₃×#117	C₂×C₄	1	16	yes	complemented
98	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
121	C₂×C₄²×C₂₃	—	4	2	no	—
122	C₂×C₄²×C₂₃	—	4	4	no	—
123	C₂×C₄²×C₂₃	—	4	2	no	—
124	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
125	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
126	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
127	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
128	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
129	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
130	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
131	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
132	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
133	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
134	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
137	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
138	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
139	C₂₃×#95	C₂³	1	4	yes	—
140	C₂₃×#95	C₂³	1	4	yes	—
141	C₂₃×#95	C₂³	1	4	yes	—
142	C₂₃×#95	C₂³	1	4	yes	—
143	C₂₃×#95	C₂³	1	4	yes	—
144	C₂₃×#95	C₂³	1	4	yes	—
145	C₂₃×#95	C₂³	1	4	yes	—
146	C₂₃×#95	C₂³	1	4	yes	—
147	C₂₃×#95	C₂³	1	4	yes	—
148	C₂₃×#95	C₂³	1	4	yes	—
149	C₂₃×#95	C₂³	1	2	yes	—
150	C₂₃×#95	C₂³	1	2	yes	—
151	C₂₃×#95	C₂³	1	4	yes	—
152	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
153	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
154	C₂×Q₈×C₂₃	—	4	2	no	—
155	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
156	C₂×D₄×C₂₃	—	2	8	no	—
157	C₂×D₄×C₂₃	—	4	2	no	—
158	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
159	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
160	C₂²×C₄×C₂₃	—	2	4	no	—
161	C₂²×C₄×C₂₃	—	2	2	no	—
162	C₂²×C₄×C₂₃	—	2	4	no	—
163	C₂²×C₄×C₂₃	—	2	2	no	—
164	C₂²×C₄×C₂₃	—	2	4	no	—
165	C₂²×C₄×C₂₃	—	2	4	no	—
166	C₂²×C₄×C₂₃	—	2	8	no	—
167	C₂²×C₄×C₂₃	—	4	2	no	—
168	C₂²×C₄×C₂₃	—	4	4	no	—
169	C₂²×C₄×C₂₃	—	4	2	no	—
170	C₂²×C₄×C₂₃	—	4	2	no	—
171	C₂²×C₄×C₂₃	—	4	4	no	—
172	C₂²×C₄×C₂₃	—	4	2	no	—
173	C₂²×C₄×C₂₃	—	4	2	no	—
174	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
175	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
176	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
177	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
178	C₂²×C₄×C₂₃	#41	1	8	yes	—
179	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
186	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
191	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
192	C₄⋊C₄×C₂₃	—	2	8	no	—
193	C₄⋊C₄×C₂₃	—	2	8	no	—
194	C₄⋊C₄×C₂₃	—	2	8	no	—
195	C₄⋊C₄×C₂₃	—	4	4	no	—
196	C₄⋊C₄×C₂₃	—	4	2	no	—
197	C₄⋊C₄×C₂₃	—	4	2	no	—
198	C₄⋊C₄×C₂₃	—	4	4	no	—
199	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
200	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
201	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
202	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
203	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
204	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
205	C₂²⋊C₄×C₂₃	—	2	8	no	—
206	C₂²⋊C₄×C₂₃	—	2	8	no	—
207	C₂²⋊C₄×C₂₃	—	4	4	no	—
208	C₂²⋊C₄×C₂₃	—	4	4	no	—
209	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
210	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
211	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
212	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
213	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
214	C₄²×C₂₃	—	2	8	no	—
215	C₄²×C₂₃	—	2	4	no	—
216	C₄²×C₂₃	—	2	4	no	—
217	C₄²×C₂₃	—	2	4	no	—
218	C₄²×C₂₃	—	2	4	no	—
219	C₄²×C₂₃	—	4	2	no	—
220	C₄²×C₂₃	—	4	4	no	—
221	C₄²×C₂₃	—	4	2	no	—
222	C₄²×C₂₃	—	4	4	no	—
223	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
224	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
225	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
226	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
227	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
228	#34083	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂³×C₂₃	—	2	1	no	—
230	C₂³×C₂₃	—	2	1	no	—
231	C₂³×C₂₃	—	2	2	no	—
232	C₂³×C₂₃	—	2	4	no	—
233	C₂³×C₂₃	—	2	2	no	—
234	C₂³×C₂₃	—	2	2	no	—
235	C₂³×C₂₃	—	2	4	no	—
236	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
237	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
238	C₂³×C₂₃	C₂×#41	1	4	yes	—
239	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
240	Q₈×C₂₃	—	4	4	no	—
241	D₄×C₂₃	—	2	16	no	—
242	D₄×C₂₃	—	4	4	no	—
243	C₂×C₄×C₂₃	—	2	8	no	—
244	C₂×C₄×C₂₃	—	2	4	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	4	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	2	no	—
252	C₂×C₄×C₂₃	—	2	4	no	—
253	C₂×C₄×C₂₃	—	2	4	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	8	no	—
257	C₂×C₄×C₂₃	—	2	16	no	—
258	C₂×C₄×C₂₃	—	2	4	no	—
259	C₂×C₄×C₂₃	—	2	4	no	—
260	C₂×C₄×C₂₃	—	2	4	no	—
261	C₂×C₄×C₂₃	—	2	4	no	—
262	C₂×C₄×C₂₃	—	2	4	no	—
263	C₂×C₄×C₂₃	—	2	4	no	—
264	C₂×C₄×C₂₃	—	2	4	no	—
265	C₂×C₄×C₂₃	—	2	4	no	—
266	C₂×C₄×C₂₃	—	2	4	no	—
267	C₂×C₄×C₂₃	—	4	4	no	—
268	C₂×C₄×C₂₃	—	4	8	no	—
269	C₂×C₄×C₂₃	—	4	16	no	—
270	C₂×C₄×C₂₃	—	4	8	no	—
271	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
272	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
273	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
274	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
275	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
276	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
277	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
278	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
279	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
280	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
281	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
282	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
283	C₂×#529	C₂×C₂₃	1	1	yes	characteristic
284	#2337	C₂×C₂₃	1	2	yes	complemented
285	#2336	C₂×C₂₃	1	2	yes	complemented
286	#2334	C₂×C₂₃	1	4	yes	complemented
287	#2329	C₂×C₂₃	1	2	yes	complemented
288	#2325	C₂×C₂₃	1	2	yes	—
289	#2322	C₂×C₂₃	1	4	yes	complemented
290	#2298	C₂×C₂₃	1	2	yes	complemented
291	#2296	C₂×C₂₃	1	2	yes	complemented
292	#2295	C₂×C₂₃	1	2	yes	complemented
293	#2290	C₂×C₂₃	1	2	yes	complemented
294	#2290	C₂×C₂₃	1	2	yes	complemented
295	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
296	C₄²×D₄	C₂×C₂₃	1	2	yes	complemented
297	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
298	C₂²×C₂₃	—	2	2	no	—
299	C₂²×C₂₃	—	2	2	no	—
300	C₂²×C₂₃	—	2	4	no	—
301	C₂²×C₂₃	—	2	4	no	—
302	C₂²×C₂₃	—	2	4	no	—
303	C₂²×C₂₃	—	2	4	no	—
304	C₂²×C₂₃	—	2	4	no	—
305	C₂²×C₂₃	—	4	4	no	—
306	C₂²×C₂₃	—	4	4	no	—
307	C₂²×C₂₃	C₄×#41	1	4	yes	—
308	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
309	C₂²×C₂₃	C₂²×#41	1	2	yes	—
310	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
311	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
312	C₂²×C₂₃	#585	1	2	yes	—
313	C₄×C₂₃	—	2	2	no	—
314	C₄×C₂₃	—	2	2	no	—
315	C₄×C₂₃	—	2	4	no	—
316	C₄×C₂₃	—	2	2	no	—
317	C₄×C₂₃	—	2	4	no	—
318	C₄×C₂₃	—	2	2	no	—
319	C₄×C₂₃	—	2	2	no	—
320	C₄×C₂₃	—	2	4	no	—
321	C₄×C₂₃	—	2	8	no	—
322	C₄×C₂₃	—	4	4	no	—
323	C₄×C₂₃	—	4	4	no	—
324	C₄×C₂₃	—	4	4	no	—
325	C₄×C₂₃	#520	1	4	yes	—
326	#529	C₄×C₂₃	1	16	yes	complemented
327	C₂×#126	C₂²×C₂₃	1	2	yes	—
328	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×#123	C₂²×C₂₃	1	2	yes	—
331	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
333	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
334	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
335	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
336	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×#117	C₂²×C₂₃	1	2	yes	—
338	C₂×#117	C₂²×C₂₃	1	2	yes	—
339	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
340	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
341	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
342	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
343	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
344	#391	C₂²×C₂₃	1	4	yes	—
345	#389	C₂²×C₂₃	1	4	yes	—
346	#389	C₂²×C₂₃	1	4	yes	—
347	#389	C₂²×C₂₃	1	4	yes	—
348	#388	C₂²×C₂₃	1	4	yes	—
349	#388	C₂²×C₂₃	1	4	yes	—
350	#388	C₂²×C₂₃	1	4	yes	—
351	#388	C₂²×C₂₃	1	4	yes	—
352	#387	C₂²×C₂₃	1	4	yes	—
353	#387	C₂²×C₂₃	1	4	yes	—
354	#387	C₂²×C₂₃	1	4	yes	—
355	#387	C₂²×C₂₃	1	4	yes	—
356	#387	C₂²×C₂₃	1	4	yes	—
357	#387	C₂²×C₂₃	1	4	yes	—
358	#386	C₂²×C₂₃	1	4	yes	—
359	#386	C₂²×C₂₃	1	4	yes	—
360	#385	C₂²×C₂₃	1	4	yes	—
361	#385	C₂²×C₂₃	1	4	yes	—
362	#385	C₂²×C₂₃	1	4	yes	—
363	#385	C₂²×C₂₃	1	4	yes	—
364	#383	C₂²×C₂₃	1	2	yes	—
365	#382	C₂²×C₂₃	1	2	yes	—
366	#381	C₂²×C₂₃	1	2	yes	—
367	#380	C₂²×C₂₃	1	2	yes	—
368	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
369	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
370	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
371	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
372	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
373	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
374	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
375	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
376	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
377	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
378	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
379	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
380	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
381	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
382	#376	C₂²×C₂₃	1	2	yes	—
383	#376	C₂²×C₂₃	1	2	yes	—
384	C₂×#95	C₂²×C₂₃	1	2	yes	—
385	C₄³	C₂²×C₂₃	1	2	yes	—
386	C₂×C₂₃	—	2	4	no	—
387	C₂×C₂₃	—	4	4	no	—
388	C₂×C₂₃	C₂×C₄×#41	1	1	yes	characteristic
389	C₂×C₂₃	C₂×#520	1	1	yes	characteristic
390	C₂×C₂₃	#3428	1	1	yes	characteristic
391	C₂×C₂₃	#3459	1	2	yes	—
392	C₂×C₂₃	#3461	1	2	yes	—
393	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
394	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
395	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
396	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
398	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
399	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
400	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
401	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
402	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
403	#126	C₂×C₄×C₂₃	1	16	yes	complemented
404	#125	C₂×C₄×C₂₃	1	8	yes	complemented
405	#124	C₂×C₄×C₂₃	1	8	yes	complemented
406	#123	C₂×C₄×C₂₃	1	16	yes	complemented
407	C₂²⋊Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
408	#121	C₂×C₄×C₂₃	1	8	yes	—
409	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
410	C₄×D₄	—	2	16	no	—
411	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	complemented
412	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
413	#117	—	4	4	no	—
414	#117	C₂×C₄×C₂₃	1	16	yes	complemented
415	C₂×C₄⋊C₄	—	2	8	no	—
416	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
417	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
418	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
419	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
420	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
423	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
424	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
427	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
437	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
438	C₂×C₄²	—	4	2	no	—
439	C₂×C₄²	—	4	4	no	—
440	C₂×C₄²	—	4	2	no	—
441	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
442	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
443	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
444	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
445	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
446	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
447	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
448	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
449	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
450	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
451	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
452	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
453	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
454	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
455	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
456	#95	C₂³×C₂₃	1	4	yes	—
457	#95	C₂³×C₂₃	1	4	yes	—
458	#95	C₂³×C₂₃	1	4	yes	—
459	#95	C₂³×C₂₃	1	4	yes	—
460	#95	C₂³×C₂₃	1	4	yes	—
461	#95	C₂³×C₂₃	1	4	yes	—
462	#95	C₂³×C₂₃	1	4	yes	—
463	#95	C₂³×C₂₃	1	4	yes	—
464	#95	C₂³×C₂₃	1	4	yes	—
465	#95	C₂³×C₂₃	1	4	yes	—
466	#95	C₂³×C₂₃	1	2	yes	—
467	#95	C₂³×C₂₃	1	2	yes	—
468	#95	C₂³×C₂₃	1	4	yes	—
469	C₂₃	#34083	1	1	yes	characteristic, direct product factor, Sylow
470	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
471	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
472	C₂×Q₈	—	4	2	no	—
473	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
474	C₂×D₄	—	2	8	no	—
475	C₂×D₄	—	4	2	no	—
476	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
477	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	8	no	—
485	C₂²×C₄	—	4	2	no	—
486	C₂²×C₄	—	4	4	no	—
487	C₂²×C₄	—	4	2	no	—
488	C₂²×C₄	—	4	2	no	—
489	C₂²×C₄	—	4	4	no	—
490	C₂²×C₄	—	4	2	no	—
491	C₂²×C₄	—	4	2	no	—
492	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
493	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
494	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
495	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
496	C₂²×C₄	#41×C₂₃	1	8	yes	—
497	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
498	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
499	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
500	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
502	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
503	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
505	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
506	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
507	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
508	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
509	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
510	C₄⋊C₄	—	2	8	no	—
511	C₄⋊C₄	—	2	8	no	—
512	C₄⋊C₄	—	2	8	no	—
513	C₄⋊C₄	—	4	4	no	—
514	C₄⋊C₄	—	4	4	no	—
515	C₄⋊C₄	—	4	2	no	—
516	C₄⋊C₄	—	4	2	no	—
517	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
518	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
519	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
520	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
521	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
522	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
523	C₂²⋊C₄	—	2	8	no	—
524	C₂²⋊C₄	—	2	8	no	—
525	C₂²⋊C₄	—	4	4	no	—
526	C₂²⋊C₄	—	4	4	no	—
527	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
528	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
529	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
530	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
531	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
532	C₄²	—	2	8	no	—
533	C₄²	—	2	4	no	—
534	C₄²	—	2	4	no	—
535	C₄²	—	2	4	no	—
536	C₄²	—	2	4	no	—
537	C₄²	—	4	2	no	—
538	C₄²	—	4	4	no	—
539	C₄²	—	4	2	no	—
540	C₄²	—	4	4	no	—
541	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
542	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
543	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
544	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
545	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
546	C₂³	—	2	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	4	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	4	no	—
553	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
554	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
555	C₂³	C₂×#41×C₂₃	1	4	yes	—
556	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
557	Q₈	—	4	4	no	—
558	D₄	—	2	16	no	—
559	D₄	—	4	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	8	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	16	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	4	4	no	—
585	C₂×C₄	—	4	8	no	—
586	C₂×C₄	—	4	16	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
589	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
590	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
591	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
592	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
593	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
594	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
595	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
596	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
597	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
598	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
599	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
600	C₂²	—	2	2	no	—
601	C₂²	—	2	2	no	—
602	C₂²	—	2	4	no	—
603	C₂²	—	2	4	no	—
604	C₂²	—	2	4	no	—
605	C₂²	—	2	4	no	—
606	C₂²	—	2	4	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	4	4	no	—
609	C₂²	C₄×#41×C₂₃	1	4	yes	—
610	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
611	C₂²	C₂²×#41×C₂₃	1	2	yes	—
612	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
613	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
614	C₂²	C₂₃×#585	1	2	yes	—
615	C₄	—	2	2	no	—
616	C₄	—	2	2	no	—
617	C₄	—	2	4	no	—
618	C₄	—	2	2	no	—
619	C₄	—	2	4	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	4	no	—
623	C₄	—	2	8	no	—
624	C₄	—	4	4	no	—
625	C₄	—	4	4	no	—
626	C₄	—	4	4	no	—
627	C₄	C₂₃×#520	1	4	yes	—
628	C₂	—	2	4	no	—
629	C₂	—	4	4	no	—
630	C₂	C₂×C₄×#41×C₂₃	1	1	yes	characteristic
631	C₂	C₂×C₂₃×#520	1	1	yes	characteristic
632	C₂	C₂₃×#3428	1	1	yes	characteristic
633	C₂	C₂₃×#3459	1	2	yes	—
634	C₂	C₂₃×#3461	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2337	C₂	semidirect
3. C₂₃×#2336	C₂	semidirect
4. C₂₃×#2334	C₂	semidirect
5. C₂₃×#2329	C₂	semidirect
7. C₂₃×#2322	C₂	semidirect
8. C₂₃×#2298	C₂	semidirect
9. C₂₃×#2296	C₂	semidirect
10. C₂₃×#2295	C₂	semidirect
11. C₂₃×#2290	C₂	semidirect
12. C₂₃×#2290	C₂	semidirect
13. C₂₃×#2283	C₂	semidirect
14. C₄²×D₄×C₂₃	C₂	semidirect
15. C₂×C₄×C₄⋊C₄×C₂₃	C₂	semidirect
16. C₂₃×#529	C₄	semidirect
86. C₂₃×#126	C₂×C₄	semidirect
87. C₂₃×#125	C₂×C₄	semidirect
88. C₂₃×#124	C₂×C₄	semidirect
89. C₂₃×#123	C₂×C₄	semidirect
90. C₂₃×C₂²⋊Q₈	C₂×C₄	semidirect
92. C₄×Q₈×C₂₃	C₂×C₄	semidirect
94. C₄×D₄×C₂₃	C₂×C₄	semidirect
95. C₄×D₄×C₂₃	C₂×C₄	semidirect
97. C₂₃×#117	C₂×C₄	semidirect
124. C₂×C₄²×C₂₃	C₂×C₄	semidirect
228. #34083	C₂₃	direct (N × H)
284. #2337	C₂×C₂₃	semidirect
285. #2336	C₂×C₂₃	semidirect
286. #2334	C₂×C₂₃	semidirect
287. #2329	C₂×C₂₃	semidirect
289. #2322	C₂×C₂₃	semidirect
290. #2298	C₂×C₂₃	semidirect
291. #2296	C₂×C₂₃	semidirect
292. #2295	C₂×C₂₃	semidirect
293. #2290	C₂×C₂₃	semidirect
294. #2290	C₂×C₂₃	semidirect
295. #2283	C₂×C₂₃	semidirect
296. C₄²×D₄	C₂×C₂₃	semidirect
297. C₂×C₄×C₄⋊C₄	C₂×C₂₃	semidirect
326. #529	C₄×C₂₃	semidirect
403. #126	C₂×C₄×C₂₃	semidirect
404. #125	C₂×C₄×C₂₃	semidirect
405. #124	C₂×C₄×C₂₃	semidirect
406. #123	C₂×C₄×C₂₃	semidirect
407. C₂²⋊Q₈	C₂×C₄×C₂₃	semidirect
409. C₄×Q₈	C₂×C₄×C₂₃	semidirect
411. C₄×D₄	C₂×C₄×C₂₃	semidirect
412. C₄×D₄	C₂×C₄×C₂₃	semidirect
414. #117	C₂×C₄×C₂₃	semidirect
441. C₂×C₄²	C₂×C₄×C₂₃	semidirect
469. C₂₃	#34083	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×#529	2
C₂₃×#2337	2
C₂₃×#2336	2
C₂₃×#2334	2
C₂₃×#2329	2
C₂₃×#2325	2
C₂₃×#2322	2
C₂₃×#2298	2
C₂₃×#2296	2
C₂₃×#2295	2
C₂₃×#2290	2
C₂₃×#2283	2
C₄²×D₄×C₂₃	2
C₂×C₄×C₄⋊C₄×C₂₃	2
#34083	23

Subgroups — C23×#34083 #19181709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34083 #19181709