Subgroups — C₂₃×#8425 #19156051

Order 5888.

750 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3503	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1981	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1806	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1550	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1503	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1477	C₂	1	1	yes	maximal, characteristic
8	Q₈²×C₂₃	C₄	1	2	yes	complemented
9	C₂₃×#557	—	2	1	no	—
10	C₂₃×#554	—	2	2	no	—
11	C₂₃×#554	C₄	1	2	yes	complemented
12	D₄×Q₈×C₂₃	—	2	2	no	—
13	D₄×Q₈×C₂₃	—	2	1	no	—
14	C₂₃×#544	—	2	2	no	—
15	C₂₃×#544	—	2	1	no	—
16	C₂₃×#533	—	2	1	no	—
17	C₂₃×#533	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂₃×#520	—	2	1	no	—
21	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#431	C₂²	1	1	yes	characteristic
23	C₂₃×#423	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#104	—	2	1	no	—
26	C₂×C₂₃×#103	—	2	2	no	—
27	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#339	—	2	1	no	—
29	C₂₃×#333	—	2	2	no	—
30	C₂₃×#333	—	2	1	no	—
31	C₂₃×#332	—	2	1	no	—
32	C₂₃×#328	—	2	2	no	—
33	C₂₃×Q₈⋊C₈	—	2	1	no	—
34	C₂₃×D₄⋊C₈	—	2	1	no	—
35	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₂₃	—	2	2	no	—
37	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
38	C₂×M₁₆×C₂₃	—	2	1	no	—
39	C₂²×C₈×C₂₃	—	2	1	no	—
40	C₂₃×C₄⋊Q₈	—	2	2	no	—
41	C₂₃×C₄⋊Q₈	—	2	2	no	—
42	C₂₃×C₄⋊Q₈	—	2	4	no	—
43	C₂₃×C₄⋊Q₈	—	2	2	no	—
44	C₂₃×C₄⋊Q₈	—	2	2	no	—
45	C₂₃×C₄⋊Q₈	—	2	1	no	—
46	C₂₃×C₄⋊Q₈	—	2	1	no	—
47	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
48	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
49	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
50	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
51	C₂₃×#125	—	2	4	no	—
52	C₂₃×#125	—	2	2	no	—
53	C₂₃×#125	—	2	2	no	—
54	C₂₃×#125	—	2	1	no	—
55	C₂₃×C₂²⋊Q₈	—	2	2	no	—
56	C₂₃×C₂²⋊Q₈	—	2	4	no	—
57	C₂₃×C₂²⋊Q₈	—	2	4	no	—
58	C₂₃×C₂²⋊Q₈	—	2	4	no	—
59	C₂₃×C₂²⋊Q₈	—	2	2	no	—
60	C₂₃×C₂²⋊Q₈	—	2	1	no	—
61	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
62	C₄×Q₈×C₂₃	—	2	2	no	—
63	C₄×Q₈×C₂₃	—	2	2	no	—
64	C₄×Q₈×C₂₃	—	2	2	no	—
65	C₄×Q₈×C₂₃	—	2	1	no	—
66	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
67	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
69	C₄×D₄×C₂₃	—	2	4	no	—
70	C₄×D₄×C₂₃	—	2	2	no	—
71	C₄×D₄×C₂₃	—	2	1	no	—
72	C₄×D₄×C₂₃	—	2	1	no	—
73	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
74	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
75	C₂₃×#117	—	2	2	no	—
76	C₂₃×#117	—	2	1	no	—
77	C₂₃×#117	—	2	1	no	—
78	C₂₃×#117	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄²×C₂₃	—	2	1	no	—
85	C₂×C₄²×C₂₃	—	2	1	no	—
86	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
87	C₂₃×C₄⋊C₈	—	2	1	no	—
88	C₂₃×C₄⋊C₈	—	2	1	no	—
89	C₂₃×C₄⋊C₈	—	2	1	no	—
90	C₂₃×C₄⋊C₈	—	2	1	no	—
91	C₂₃×C₄⋊C₈	—	2	1	no	—
92	C₂₃×#104	—	4	2	no	—
93	C₂₃×#103	—	4	4	no	—
94	C₂₃×C₂²⋊C₈	—	2	1	no	—
95	C₄×C₈×C₂₃	—	2	1	no	—
96	C₂₃×#95	—	2	1	no	—
97	#41×C₂₃	—	2	1	no	—
98	#41×C₂₃	—	2	1	no	—
99	#41×C₂₃	SD₁₆	1	2	yes	—
100	#41×C₂₃	Q₁₆	1	2	yes	—
101	C₂×Q₈×C₂₃	—	2	4	no	—
102	C₂×Q₈×C₂₃	—	2	2	no	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×Q₈×C₂₃	—	4	2	no	—
105	C₂×Q₈×C₂₃	—	4	2	no	—
106	C₂×Q₈×C₂₃	—	8	2	no	—
107	C₂×Q₈×C₂₃	—	8	2	no	—
108	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×Q₈×C₂₃	C₂²⋊C₄	1	2	yes	—
110	C₂×Q₈×C₂₃	C₂²⋊C₄	1	4	yes	—
111	C₂×D₄×C₂₃	—	2	1	no	—
112	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	—	2	1	no	—
114	C₂²×C₄×C₂₃	—	2	4	no	—
115	C₂²×C₄×C₂₃	—	2	2	no	—
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	2	1	no	—
118	C₂²×C₄×C₂₃	—	4	1	no	—
119	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
122	M₁₆×C₂₃	—	4	1	no	—
123	M₁₆×C₂₃	—	4	1	no	—
124	C₂×C₈×C₂₃	—	4	2	no	—
125	C₂×C₈×C₂₃	—	4	1	no	—
126	C₂×C₈×C₂₃	—	4	1	no	—
127	C₂×C₈×C₂₃	—	4	1	no	—
128	C₄⋊C₄×C₂₃	—	2	4	no	—
129	C₄⋊C₄×C₂₃	—	2	4	no	—
130	C₄⋊C₄×C₂₃	—	2	4	no	—
131	C₄⋊C₄×C₂₃	—	2	4	no	—
132	C₄⋊C₄×C₂₃	—	2	2	no	—
133	C₄⋊C₄×C₂₃	—	2	2	no	—
134	C₄⋊C₄×C₂₃	—	2	1	no	—
135	C₄⋊C₄×C₂₃	—	2	2	no	—
136	C₄⋊C₄×C₂₃	—	2	2	no	—
137	C₄⋊C₄×C₂₃	—	2	1	no	—
138	C₄⋊C₄×C₂₃	—	2	2	no	—
139	C₄⋊C₄×C₂₃	—	2	1	no	—
140	C₄⋊C₄×C₂₃	—	2	1	no	—
141	C₄⋊C₄×C₂₃	—	2	1	no	—
142	C₄⋊C₄×C₂₃	—	2	2	no	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	—	4	1	no	—
145	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
146	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
147	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²⋊C₄×C₂₃	—	2	4	no	—
149	C₂²⋊C₄×C₂₃	—	2	2	no	—
150	C₂²⋊C₄×C₂₃	—	2	1	no	—
151	C₂²⋊C₄×C₂₃	—	2	1	no	—
152	C₂²⋊C₄×C₂₃	—	2	1	no	—
153	C₄²×C₂₃	—	2	4	no	—
154	C₄²×C₂₃	—	2	1	no	—
155	C₄²×C₂₃	—	2	2	no	—
156	C₄²×C₂₃	—	2	1	no	—
157	C₄²×C₂₃	—	2	1	no	—
158	C₄²×C₂₃	—	4	1	no	—
159	C₄²×C₂₃	—	4	1	no	—
160	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
161	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
164	#8425	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂³×C₂₃	—	2	1	no	—
166	C₂³×C₂₃	#120	1	1	yes	characteristic
167	Q₈×C₂₃	—	2	1	no	—
168	Q₈×C₂₃	—	4	4	no	—
169	Q₈×C₂₃	—	4	2	no	—
170	Q₈×C₂₃	—	4	2	no	—
171	Q₈×C₂₃	—	4	1	no	—
172	Q₈×C₂₃	—	4	2	no	—
173	Q₈×C₂₃	—	4	1	no	—
174	Q₈×C₂₃	—	8	4	no	—
175	Q₈×C₂₃	#103	1	2	yes	complemented
176	D₄×C₂₃	—	2	2	no	—
177	D₄×C₂₃	—	2	2	no	—
178	D₄×C₂₃	—	2	1	no	—
179	D₄×C₂₃	#103	1	2	yes	complemented
180	C₂×C₄×C₂₃	—	2	1	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	2	1	no	—
184	C₂×C₄×C₂₃	—	2	1	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	1	no	—
187	C₂×C₄×C₂₃	—	2	1	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	2	2	no	—
190	C₂×C₄×C₂₃	—	4	1	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	4	1	no	—
193	C₂×C₄×C₂₃	—	4	1	no	—
194	C₂×C₄×C₂₃	—	4	2	no	—
195	C₂×C₄×C₂₃	—	8	2	no	—
196	C₂×C₄×C₂₃	—	8	4	no	—
197	C₂×C₄×C₂₃	—	8	1	no	—
198	C₂×C₄×C₂₃	—	8	1	no	—
199	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
203	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
204	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
205	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
206	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
207	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
208	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
209	C₈×C₂₃	—	8	1	no	—
210	C₈×C₂₃	—	8	1	no	—
211	C₈×C₂₃	—	8	1	no	—
212	#3503	C₂×C₂₃	1	1	yes	characteristic
213	#1981	C₂×C₂₃	1	1	yes	characteristic, complemented
214	#1806	C₂×C₂₃	1	1	yes	characteristic
215	#1550	C₂×C₂₃	1	1	yes	characteristic, complemented
216	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
217	#1503	C₂×C₂₃	1	1	yes	characteristic, complemented
218	#1477	C₂×C₂₃	1	1	yes	characteristic
219	C₂²×C₂₃	—	2	1	no	—
220	C₂²×C₂₃	—	2	1	no	—
221	C₂²×C₂₃	—	2	2	no	—
222	C₂²×C₂₃	—	4	1	no	—
223	C₂²×C₂₃	—	4	1	no	—
224	C₂²×C₂₃	#379	1	1	yes	characteristic
225	C₂²×C₂₃	#448	1	1	yes	characteristic
226	C₂²×C₂₃	#451	1	1	yes	characteristic
227	C₄×C₂₃	—	2	1	no	—
228	C₄×C₂₃	—	2	1	no	—
229	C₄×C₂₃	—	2	1	no	—
230	C₄×C₂₃	—	2	2	no	—
231	C₄×C₂₃	—	4	1	no	—
232	C₄×C₂₃	—	4	4	no	—
233	C₄×C₂₃	—	4	2	no	—
234	C₄×C₂₃	—	4	1	no	—
235	C₄×C₂₃	—	4	2	no	—
236	C₄×C₂₃	—	8	2	no	—
237	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
238	C₄×C₂₃	#419	1	1	yes	characteristic
239	C₄×C₂₃	#448	1	1	yes	characteristic
240	C₄×C₂₃	#451	1	1	yes	characteristic
241	C₄×C₂₃	#453	1	1	yes	characteristic
242	C₄×C₂₃	#455	1	1	yes	characteristic
243	Q₈²	C₄×C₂₃	1	2	yes	complemented
244	#557	—	2	1	no	—
245	#554	—	2	2	no	—
246	#554	C₄×C₂₃	1	2	yes	complemented
247	D₄×Q₈	—	2	2	no	—
248	D₄×Q₈	—	2	1	no	—
249	#544	—	2	2	no	—
250	#544	—	2	1	no	—
251	#533	—	2	1	no	—
252	#533	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×C₄⋊Q₈	—	2	1	no	—
254	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
255	#520	—	2	1	no	—
256	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
257	#431	C₂²×C₂₃	1	1	yes	characteristic
258	#423	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
260	C₂×#104	—	2	1	no	—
261	C₂×#103	—	2	2	no	—
262	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
263	#339	—	2	1	no	—
264	#333	—	2	2	no	—
265	#333	—	2	1	no	—
266	#332	—	2	1	no	—
267	#328	—	2	2	no	—
268	Q₈⋊C₈	—	2	1	no	—
269	D₄⋊C₈	—	2	1	no	—
270	C₂×C₂₃	—	2	1	no	—
271	C₂×C₂₃	—	2	1	no	—
272	C₂×C₂₃	—	4	2	no	—
273	C₂×C₂₃	#1787	1	1	yes	characteristic
274	C₂×C₂₃	#1792	1	1	yes	characteristic
275	C₂×C₂₃	#1796	1	1	yes	characteristic
276	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
277	C₂²×Q₈	—	2	2	no	—
278	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
279	C₂×M₁₆	—	2	1	no	—
280	C₂²×C₈	—	2	1	no	—
281	C₄⋊Q₈	—	2	2	no	—
282	C₄⋊Q₈	—	2	2	no	—
283	C₄⋊Q₈	—	2	4	no	—
284	C₄⋊Q₈	—	2	2	no	—
285	C₄⋊Q₈	—	2	2	no	—
286	C₄⋊Q₈	—	2	1	no	—
287	C₄⋊Q₈	—	2	1	no	—
288	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
289	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
290	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
291	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
292	#125	—	2	2	no	—
293	#125	—	2	4	no	—
294	#125	—	2	2	no	—
295	#125	—	2	1	no	—
296	C₂²⋊Q₈	—	2	2	no	—
297	C₂²⋊Q₈	—	2	4	no	—
298	C₂²⋊Q₈	—	2	4	no	—
299	C₂²⋊Q₈	—	2	2	no	—
300	C₂²⋊Q₈	—	2	4	no	—
301	C₂²⋊Q₈	—	2	1	no	—
302	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
303	C₄×Q₈	—	2	2	no	—
304	C₄×Q₈	—	2	2	no	—
305	C₄×Q₈	—	2	2	no	—
306	C₄×Q₈	—	2	1	no	—
307	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
308	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
309	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
310	C₄×D₄	—	2	4	no	—
311	C₄×D₄	—	2	2	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	—	2	1	no	—
314	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
315	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
316	#117	—	2	2	no	—
317	#117	—	2	1	no	—
318	#117	—	2	1	no	—
319	#117	D₄×C₂₃	1	1	yes	characteristic
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
328	C₄⋊C₈	—	2	1	no	—
329	C₄⋊C₈	—	2	1	no	—
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	—	2	1	no	—
333	#104	—	4	2	no	—
334	#103	—	4	4	no	—
335	C₂²⋊C₈	—	2	1	no	—
336	C₄×C₈	—	2	1	no	—
337	#95	—	2	1	no	—
338	C₂₃	#8425	1	1	yes	characteristic, direct product factor, Sylow
339	#41	—	2	1	no	—
340	#41	—	2	1	no	—
341	#41	SD₁₆×C₂₃	1	2	yes	—
342	#41	Q₁₆×C₂₃	1	2	yes	—
343	C₂×Q₈	—	2	4	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	4	2	no	—
347	C₂×Q₈	—	4	2	no	—
348	C₂×Q₈	—	8	2	no	—
349	C₂×Q₈	—	8	2	no	—
350	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
351	C₂×Q₈	C₂²⋊C₄×C₂₃	1	2	yes	—
352	C₂×Q₈	C₂²⋊C₄×C₂₃	1	4	yes	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	4	1	no	—
361	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
364	M₁₆	—	4	1	no	—
365	M₁₆	—	4	1	no	—
366	C₂×C₈	—	4	2	no	—
367	C₂×C₈	—	4	1	no	—
368	C₂×C₈	—	4	1	no	—
369	C₂×C₈	—	4	1	no	—
370	C₄⋊C₄	—	2	4	no	—
371	C₄⋊C₄	—	2	4	no	—
372	C₄⋊C₄	—	2	4	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	4	1	no	—
387	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
388	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
389	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
390	C₂²⋊C₄	—	2	4	no	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	2	1	no	—
393	C₂²⋊C₄	—	2	1	no	—
394	C₂²⋊C₄	—	2	1	no	—
395	C₄²	—	2	4	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	1	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	4	1	no	—
401	C₄²	—	4	1	no	—
402	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
403	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
404	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
405	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
406	C₂³	—	2	1	no	—
407	C₂³	C₂₃×#120	1	1	yes	characteristic
408	Q₈	—	2	1	no	—
409	Q₈	—	4	4	no	—
410	Q₈	—	4	2	no	—
411	Q₈	—	4	2	no	—
412	Q₈	—	4	1	no	—
413	Q₈	—	4	2	no	—
414	Q₈	—	4	1	no	—
415	Q₈	—	8	4	no	—
416	Q₈	C₂₃×#103	1	2	yes	complemented
417	D₄	—	2	2	no	—
418	D₄	—	2	2	no	—
419	D₄	—	2	1	no	—
420	D₄	C₂₃×#103	1	2	yes	complemented
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	8	2	no	—
437	C₂×C₄	—	8	4	no	—
438	C₂×C₄	—	8	1	no	—
439	C₂×C₄	—	8	1	no	—
440	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
441	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
442	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
443	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
444	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
445	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
446	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
447	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
448	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
449	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
450	C₈	—	8	1	no	—
451	C₈	—	8	1	no	—
452	C₈	—	8	1	no	—
453	C₂²	—	2	1	no	—
454	C₂²	—	2	1	no	—
455	C₂²	—	2	2	no	—
456	C₂²	—	4	1	no	—
457	C₂²	—	4	1	no	—
458	C₂²	C₂₃×#379	1	1	yes	characteristic
459	C₂²	C₂₃×#448	1	1	yes	characteristic
460	C₂²	C₂₃×#451	1	1	yes	characteristic
461	C₄	—	2	2	no	—
462	C₄	—	2	1	no	—
463	C₄	—	2	1	no	—
464	C₄	—	2	1	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	4	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	1	no	—
469	C₄	—	4	2	no	—
470	C₄	—	8	2	no	—
471	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
472	C₄	C₂₃×#419	1	1	yes	characteristic
473	C₄	C₂₃×#448	1	1	yes	characteristic
474	C₄	C₂₃×#451	1	1	yes	characteristic
475	C₄	C₂₃×#453	1	1	yes	characteristic
476	C₄	C₂₃×#455	1	1	yes	characteristic
477	C₂	—	2	1	no	—
478	C₂	—	2	1	no	—
479	C₂	—	4	2	no	—
480	C₂	C₂₃×#1787	1	1	yes	characteristic
481	C₂	C₂₃×#1792	1	1	yes	characteristic
482	C₂	C₂₃×#1796	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#379
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1981	C₂	semidirect
4. C₂₃×#1550	C₂	semidirect
5. C₂₃×#1548	C₂	semidirect
6. C₂₃×#1503	C₂	semidirect
8. Q₈²×C₂₃	C₄	semidirect
11. C₂₃×#554	C₄	semidirect
164. #8425	C₂₃	direct (N × H)
175. Q₈×C₂₃	#103	semidirect
179. D₄×C₂₃	#103	semidirect
213. #1981	C₂×C₂₃	semidirect
215. #1550	C₂×C₂₃	semidirect
216. #1548	C₂×C₂₃	semidirect
217. #1503	C₂×C₂₃	semidirect
243. Q₈²	C₄×C₂₃	semidirect
246. #554	C₄×C₂₃	semidirect
338. C₂₃	#8425	direct (N × H)
416. Q₈	C₂₃×#103	semidirect
420. D₄	C₂₃×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3503	2
C₂₃×#1981	2
C₂₃×#1806	2
C₂₃×#1550	2
C₂₃×#1548	2
C₂₃×#1503	2
C₂₃×#1477	2
#8425	23

Subgroups — C23×#8425 #19156051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8425 #19156051