Subgroups — #15686118

Order 2944.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093139	C₂	1	1	yes	maximal, characteristic, complemented
2	#13093082	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092779	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092778	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092766	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092711	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092696	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092635	C₂	1	1	yes	maximal, characteristic
9	#13092633	C₂	1	1	yes	maximal, characteristic
10	#13092621	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092546	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092516	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092515	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092499	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₂₃	—	2	1	no	—
17	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
19	#10625118	C₂²	1	1	yes	characteristic
20	#10625117	—	2	1	no	—
21	#10625115	C₂²	1	1	yes	characteristic
22	C₂×#67274	C₂²	1	1	yes	characteristic
23	C₂×D₁₈₄	C₂²	1	1	yes	characteristic
24	C₂×#67263	C₂²	1	1	yes	characteristic
25	#10625069	—	2	1	no	—
26	#10625067	—	2	1	no	—
27	#10625067	—	2	1	no	—
28	#10625067	—	2	1	no	—
29	#10625064	C₂²	1	1	yes	characteristic
30	#10625064	C₂²	1	1	yes	characteristic
31	#10625064	C₂²	1	1	yes	characteristic
32	#10625059	C₂²	1	1	yes	characteristic
33	#10625059	C₂²	1	1	yes	characteristic
34	#10625058	C₂²	1	1	yes	characteristic
35	#10625047	—	2	1	no	—
36	#10625046	C₂²	1	1	yes	characteristic
37	C₄×D₉₂	C₂²	1	1	yes	characteristic
38	#10625043	—	2	1	no	—
39	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
40	C₂₃×#107	C₂²	1	1	yes	characteristic
41	C₂₃×#107	C₂²	1	1	yes	characteristic
42	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₂₃×#103	C₂²	1	1	yes	characteristic
44	C₂₃×#103	C₂²	1	1	yes	characteristic
45	#10625017	C₂²	1	1	yes	characteristic
46	#10625003	C₂²	1	1	yes	characteristic
47	#10624998	C₂²	1	1	yes	characteristic
48	#10624997	C₂²	1	1	yes	characteristic
49	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
52	#10624992	C₂²	1	1	yes	characteristic
53	#10624992	C₂²	1	1	yes	characteristic
54	#10624991	C₂²	1	1	yes	characteristic
55	#10624991	C₂²	1	1	yes	characteristic
56	#10624991	C₂²	1	1	yes	characteristic
57	#10624989	C₂²	1	1	yes	characteristic
58	#10624985	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	—	2	1	no	—
60	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
61	Q₈×D₂₃	—	4	4	no	—
62	C₂×D₉₂	C₂³	1	1	yes	characteristic
63	C₂×D₉₂	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₂₃	—	2	1	no	—
65	C₂×C₄×D₂₃	—	2	1	no	—
66	C₂×C₄×D₂₃	—	2	1	no	—
67	C₂×C₄×D₂₃	—	2	1	no	—
68	C₂×Dic₄₆	—	2	1	no	—
69	C₂×Dic₄₆	—	2	1	no	—
70	C₂×Dic₄₆	D₄	1	1	yes	characteristic
71	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₄²×C₂₃	—	2	1	no	—
80	C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	#67274	—	2	2	no	—
82	(C₂×D₂₃)⋊C₄	—	2	1	no	—
83	(C₂×D₂₃)⋊C₄	—	2	1	no	—
84	(C₂×D₂₃)⋊C₄	—	2	1	no	—
85	(C₂×D₂₃)⋊C₄	—	2	1	no	—
86	(C₂×D₂₃)⋊C₄	—	2	1	no	—
87	(C₂×D₂₃)⋊C₄	—	2	1	no	—
88	(C₂×D₂₃)⋊C₄	—	2	1	no	—
89	(C₂×D₂₃)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
91	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
92	Dic₂₃⋊C₄	—	2	1	no	—
93	Dic₂₃⋊C₄	—	2	1	no	—
94	Dic₂₃⋊C₄	—	2	1	no	—
95	Dic₂₃⋊C₄	—	2	1	no	—
96	Dic₂₃⋊C₄	—	2	1	no	—
97	Dic₂₃⋊C₄	D₄	1	1	yes	characteristic
98	Dic₂₃⋊C₄	D₄	1	1	yes	characteristic
99	C₄×Dic₂₃	—	2	1	no	—
100	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
101	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
104	D₁₈₄	—	2	2	no	—
105	#67263	—	2	2	no	—
106	C₂²×D₂₃	—	2	1	no	—
107	C₂²×D₂₃	—	2	1	no	—
108	Q₈×C₂₃	—	2	1	no	—
109	Q₈×C₂₃	—	2	1	no	—
110	Q₈×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₂₃	—	2	1	no	—
119	C₂×Dic₂₃	—	2	1	no	—
120	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
124	D₉₂	—	2	1	no	—
125	D₉₂	—	2	1	no	—
126	D₉₂	—	2	1	no	—
127	D₉₂	#41	1	2	yes	—
128	C₄×D₂₃	—	4	2	no	—
129	C₄×D₂₃	—	4	2	no	—
130	C₄×D₂₃	—	4	2	no	—
131	C₄×D₂₃	—	4	2	no	—
132	Dic₄₆	—	2	2	no	—
133	Dic₄₆	—	2	1	no	—
134	Dic₄₆	—	2	1	no	—
135	Dic₄₆	—	4	2	no	—
136	Dic₄₆	—	4	2	no	—
137	C₈×C₂₃	—	2	1	no	—
138	C₈×C₂₃	—	2	1	no	—
139	C₂₃⋊C₈	—	2	1	no	—
140	C₂₃⋊C₈	—	2	1	no	—
141	#3367	—	23	1	no	maximal, Sylow
142	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₂₃	—	2	2	no	—
144	C₂×D₂₃	—	4	1	no	—
145	C₂×D₂₃	—	4	1	no	—
146	C₂×D₂₃	—	4	1	no	—
147	C₂×D₂₃	—	4	1	no	—
148	C₂×D₂₃	—	4	1	no	—
149	C₄×C₂₃	—	2	1	no	—
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	4	1	no	—
152	C₄×C₂₃	—	4	1	no	—
153	C₄×C₂₃	—	4	1	no	—
154	C₄×C₂₃	—	4	1	no	—
155	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
156	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₂₃	—	2	1	no	—
158	Dic₂₃	—	2	1	no	—
159	Dic₂₃	—	2	1	no	—
160	Dic₂₃	—	4	1	no	—
161	Dic₂₃	—	4	1	no	—
162	Dic₂₃	AGL(1,ℤ/8)	1	2	yes	—
163	#556	—	23	1	no	—
164	D₄×Q₈	—	23	1	no	—
165	#489	—	23	1	no	—
166	#489	—	23	1	no	—
167	#487	—	23	1	no	—
168	#477	—	23	1	no	—
169	#477	D₂₃	1	1	yes	characteristic, complemented
170	#474	—	23	1	no	—
171	#463	—	23	1	no	—
172	#463	—	23	1	no	—
173	#461	—	23	1	no	—
174	#446	—	23	1	no	—
175	#440	—	23	1	no	—
176	#440	—	23	1	no	—
177	C₄×D₈	—	23	1	no	—
178	C₂×C₂₃	#548	1	1	yes	characteristic
179	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×C₂₃	#578	1	1	yes	characteristic
181	D₂₃	—	4	2	no	—
182	D₂₃	—	8	1	no	—
183	C₂²×Q₈	—	46	1	no	—
184	C₂×SD₁₆	—	23	1	no	—
185	C₂×SD₁₆	—	23	1	no	—
186	C₂×D₈	—	23	1	no	—
187	C₄⋊Q₈	—	23	1	no	—
188	C₄⋊Q₈	—	23	1	no	—
189	C₄⋊Q₈	—	23	1	no	—
190	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
191	#126	—	46	1	no	—
192	#126	—	46	1	no	—
193	#124	—	23	1	no	—
194	#124	—	23	1	no	—
195	C₂²⋊Q₈	—	46	1	no	—
196	C₂²⋊Q₈	—	46	1	no	—
197	C₂²⋊Q₈	—	46	1	no	—
198	C₂²⋊Q₈	—	46	1	no	—
199	C₄×Q₈	—	23	1	no	—
200	C₄×Q₈	—	23	1	no	—
201	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
202	C₄×D₄	—	23	1	no	—
203	C₄×D₄	—	23	1	no	—
204	C₄×D₄	—	23	1	no	—
205	C₄×D₄	—	23	1	no	—
206	#117	—	46	1	no	—
207	#107	—	23	1	no	—
208	#107	C₂×D₂₃	1	1	yes	characteristic
209	#107	C₂×D₂₃	1	1	yes	characteristic
210	C₄⋊C₈	—	23	1	no	—
211	C₄⋊C₈	—	23	1	no	—
212	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
213	#103	—	23	1	no	—
214	#103	—	23	1	no	—
215	#103	—	23	1	no	—
216	#103	—	23	1	no	—
217	#103	C₂×D₂₃	1	1	yes	characteristic
218	#103	C₂×D₂₃	1	1	yes	characteristic
219	#102	—	23	1	no	—
220	#102	—	23	1	no	—
221	#102	—	23	1	no	—
222	#102	—	23	1	no	—
223	#97	—	23	1	no	—
224	#97	—	23	1	no	—
225	C₄×C₈	—	23	1	no	—
226	C₂₃	#3367	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	23	1	no	—
229	C₂×Q₈	—	23	1	no	—
230	C₂×Q₈	—	46	1	no	—
231	C₂×Q₈	—	46	1	no	—
232	C₂×Q₈	—	92	4	no	—
233	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
234	C₂×D₄	—	23	1	no	—
235	C₂×D₄	—	23	1	no	—
236	C₂²×C₄	—	46	1	no	—
237	C₂²×C₄	—	46	1	no	—
238	C₂²×C₄	—	46	1	no	—
239	C₂²×C₄	—	46	1	no	—
240	SD₁₆	—	46	2	no	—
241	SD₁₆	—	46	2	no	—
242	D₈	—	46	2	no	—
243	C₂×C₈	—	23	1	no	—
244	C₂×C₈	—	23	1	no	—
245	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	23	1	no	—
250	C₄⋊C₄	—	23	1	no	—
251	C₄⋊C₄	—	23	1	no	—
252	C₄⋊C₄	—	23	1	no	—
253	C₄⋊C₄	—	46	1	no	—
254	C₄⋊C₄	—	46	1	no	—
255	C₄⋊C₄	—	46	1	no	—
256	C₄⋊C₄	—	46	1	no	—
257	C₄⋊C₄	—	46	1	no	—
258	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
261	C₂²⋊C₄	—	46	1	no	—
262	C₂²⋊C₄	—	46	1	no	—
263	C₂²⋊C₄	—	46	1	no	—
264	C₂²⋊C₄	—	46	1	no	—
265	C₂²⋊C₄	—	46	1	no	—
266	C₂²⋊C₄	—	46	1	no	—
267	C₂²⋊C₄	—	46	1	no	—
268	C₂²⋊C₄	—	46	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	23	1	no	—
271	C₄²	—	23	1	no	—
272	C₄²	—	46	1	no	—
273	C₄²	C₂²×D₂₃	1	1	yes	characteristic
274	C₂³	—	46	1	no	—
275	C₂³	—	46	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	46	2	no	—
280	Q₈	—	46	1	no	—
281	Q₈	—	46	1	no	—
282	Q₈	—	92	2	no	—
283	Q₈	—	92	2	no	—
284	D₄	—	23	2	no	—
285	D₄	—	46	1	no	—
286	D₄	—	46	1	no	—
287	D₄	—	46	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	23	1	no	—
293	C₂×C₄	—	23	1	no	—
294	C₂×C₄	—	23	1	no	—
295	C₂×C₄	—	23	1	no	—
296	C₂×C₄	—	46	1	no	—
297	C₂×C₄	—	46	1	no	—
298	C₂×C₄	—	92	2	no	—
299	C₂×C₄	—	92	2	no	—
300	C₂×C₄	—	92	2	no	—
301	C₂×C₄	—	92	2	no	—
302	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
303	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	46	1	no	—
308	C₈	—	46	1	no	—
309	C₂²	—	46	2	no	—
310	C₂²	—	92	1	no	—
311	C₂²	—	92	1	no	—
312	C₂²	—	92	1	no	—
313	C₂²	—	92	1	no	—
314	C₂²	—	92	1	no	—
315	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	23	2	no	—
323	C₄	—	46	1	no	—
324	C₄	—	46	1	no	—
325	C₄	—	46	1	no	—
326	C₄	—	92	1	no	—
327	C₄	—	92	1	no	—
328	C₄	#10625158	1	1	yes	characteristic
329	C₄	#41×D₂₃	1	1	yes	characteristic
330	C₂	—	92	2	no	—
331	C₂	—	184	1	no	—
332	C₂	#13093077	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₂₃	1	1	yes	characteristic
334	C₂	#13093230	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#477	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×D₁₈₄	C₂²
hypercenter	#477	D₂₃
hypocenter	C₂₃	#3367
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

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

N	H ≅ G/N	type
1. #13093139	C₂	semidirect
2. #13093082	C₂	semidirect
3. #13092779	C₂	semidirect
4. #13092778	C₂	semidirect
5. #13092766	C₂	semidirect
6. #13092711	C₂	semidirect
7. #13092696	C₂	semidirect
10. #13092621	C₂	semidirect
11. #13092546	C₂	semidirect
12. #13092516	C₂	semidirect
13. #13092515	C₂	semidirect
15. C₂₃×#477	C₂	semidirect
169. #477	D₂₃	semidirect
226. C₂₃	#3367	semidirect

Maximal subgroups

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

subgroup	index
#13093139	2
#13093082	2
#13092779	2
#13092778	2
#13092766	2
#13092711	2
#13092696	2
#13092635	2
#13092633	2
#13092621	2
#13092546	2
#13092516	2
#13092515	2
#13092499	2
C₂₃×#477	2
#3367	23