Subgroups — #37546158

Order 23168.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041970	C₂	1	1	yes	maximal, characteristic, complemented
2	#26041932	C₂	1	1	yes	maximal, characteristic
3	#26041685	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041613	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041608	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041606	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041594	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041580	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041528	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041511	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041377	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041349	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041320	C₂	1	1	yes	maximal, characteristic, complemented
14	#107×D₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#477×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15668450	—	2	1	no	—
17	C₂×#15668436	C₂²	1	1	yes	characteristic
18	C₂×#15668432	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₈₁	C₂²	1	1	yes	characteristic
20	#19153810	—	2	1	no	—
21	#19153808	C₂²	1	1	yes	characteristic
22	#19153806	—	2	1	no	—
23	#19153801	—	2	1	no	—
24	#19153801	—	2	1	no	—
25	#19153801	—	2	1	no	—
26	#19153792	—	2	1	no	—
27	#19153792	C₂²	1	1	yes	characteristic
28	#19153792	C₂²	1	1	yes	characteristic
29	#19153791	C₂²	1	1	yes	characteristic
30	#19153791	C₂²	1	1	yes	characteristic
31	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
32	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
33	#19153774	C₂²	1	1	yes	characteristic
34	#19153773	—	2	1	no	—
35	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
38	#19153750	C₂²	1	1	yes	characteristic
39	#19153749	C₂²	1	1	yes	characteristic
40	#19153749	C₂²	1	1	yes	characteristic
41	#19153749	C₂²	1	1	yes	characteristic
42	#19153748	C₂²	1	1	yes	characteristic
43	#19153747	C₂²	1	1	yes	characteristic
44	#19153747	C₂²	1	1	yes	characteristic
45	#19153740	C₂²	1	1	yes	characteristic
46	#19153738	C₂²	1	1	yes	characteristic
47	#19153738	C₂²	1	1	yes	characteristic
48	#19153736	C₂²	1	1	yes	characteristic
49	#19153728	C₂²	1	1	yes	characteristic
50	#19153728	C₂²	1	1	yes	characteristic
51	#19153727	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	#107×C₁₈₁	C₂²	1	1	yes	characteristic
55	#107×C₁₈₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
57	#103×C₁₈₁	C₂²	1	1	yes	characteristic
58	#103×C₁₈₁	C₂²	1	1	yes	characteristic
59	#15668450	—	4	4	no	—
60	C₂×D₇₂₄	—	2	1	no	—
61	C₂×D₇₂₄	—	2	1	no	—
62	C₂×D₇₂₄	D₄	1	1	yes	characteristic
63	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₈₁	—	2	1	no	—
65	C₂×C₄×D₁₈₁	—	2	1	no	—
66	C₂×C₄×D₁₈₁	—	2	1	no	—
67	C₂×C₄×D₁₈₁	—	2	1	no	—
68	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
70	#15668436	—	2	2	no	—
71	#15668432	—	2	2	no	—
72	C₂×#13089678	C₂³	1	1	yes	characteristic
73	C₂×#13089678	C₂³	1	1	yes	characteristic
74	C₈×D₁₈₁	—	2	2	no	—
75	C₄⋊Dic₁₈₁	D₄	1	1	yes	characteristic
76	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
79	#15668428	—	2	1	no	—
80	#15668428	—	2	1	no	—
81	#15668428	—	2	1	no	—
82	#15668428	—	2	1	no	—
83	#15668428	—	2	1	no	—
84	#15668428	—	2	1	no	—
85	#15668428	—	2	1	no	—
86	#15668426	—	2	1	no	—
87	#15668426	—	2	1	no	—
88	#15668426	—	2	1	no	—
89	#15668426	—	2	1	no	—
90	#15668426	—	2	1	no	—
91	#15668426	—	2	1	no	—
92	#15668426	D₄	1	1	yes	characteristic
93	#15668426	D₄	1	1	yes	characteristic
94	C₄×Dic₁₈₁	—	2	1	no	—
95	C₂×Q₈×C₁₈₁	—	2	1	no	—
96	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₈₁	—	2	1	no	—
100	C₄⋊C₄×C₁₈₁	—	2	1	no	—
101	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
104	C₄²×C₁₈₁	—	2	1	no	—
105	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
106	C₂²×D₁₈₁	—	2	1	no	—
107	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
109	D₇₂₄	—	2	2	no	—
110	D₇₂₄	—	2	1	no	—
111	D₇₂₄	—	2	1	no	—
112	D₇₂₄	—	4	2	no	—
113	D₇₂₄	—	4	2	no	—
114	C₂×Dic₁₈₁	—	2	1	no	—
115	C₂×Dic₁₈₁	—	2	1	no	—
116	C₂×Dic₁₈₁	—	2	1	no	—
117	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
119	C₄×D₁₈₁	—	2	1	no	—
120	C₄×D₁₈₁	—	2	1	no	—
121	C₄×D₁₈₁	—	2	1	no	—
122	C₄×D₁₈₁	—	2	1	no	—
123	C₄×D₁₈₁	—	4	2	no	—
124	C₄×D₁₈₁	—	4	2	no	—
125	C₄×D₁₈₁	—	4	2	no	—
126	C₄×D₁₈₁	—	4	2	no	—
127	#13089678	—	2	1	no	—
128	#13089678	#41	1	2	yes	complemented
129	Q₈×C₁₈₁	—	2	1	no	—
130	Q₈×C₁₈₁	—	2	1	no	—
131	Q₈×C₁₈₁	—	4	2	no	—
132	C₂×C₄×C₁₈₁	—	2	1	no	—
133	C₂×C₄×C₁₈₁	—	2	1	no	—
134	C₂×C₄×C₁₈₁	—	2	1	no	—
135	C₂×C₄×C₁₈₁	—	2	1	no	—
136	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₈₁	—	2	1	no	—
140	C₈×C₁₈₁	—	2	1	no	—
141	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₈₁	—	2	1	no	—
143	C₂×D₁₈₁	—	2	1	no	—
144	C₂×D₁₈₁	—	2	1	no	—
145	C₂×D₁₈₁	—	2	1	no	—
146	C₂×D₁₈₁	—	2	1	no	—
147	C₂×D₁₈₁	—	4	1	no	—
148	C₂×D₁₈₁	—	4	1	no	—
149	C₂×D₁₈₁	—	4	1	no	—
150	C₂×D₁₈₁	AGL(1,ℤ/8)	1	2	yes	—
151	C₄×C₁₈₁	—	2	1	no	—
152	C₄×C₁₈₁	—	2	1	no	—
153	C₄×C₁₈₁	—	4	1	no	—
154	C₄×C₁₈₁	—	4	1	no	—
155	C₄×C₁₈₁	—	4	1	no	—
156	C₄×C₁₈₁	—	4	1	no	—
157	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₁₈₁	—	2	1	no	—
160	Dic₁₈₁	—	2	1	no	—
161	Dic₁₈₁	—	4	1	no	—
162	Dic₁₈₁	—	4	1	no	—
163	Dic₁₈₁	—	4	1	no	—
164	C₂×C₁₈₁	#547	1	1	yes	characteristic
165	C₂×C₁₈₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₁₈₁	#578	1	1	yes	characteristic
167	D₁₈₁	—	2	2	no	—
168	D₁₈₁	—	4	1	no	—
169	D₁₈₁	—	8	1	no	—
170	C₁₈₁	#3343	1	1	yes	characteristic, complemented, Sylow
171	#3343	—	181	1	no	maximal, Sylow
172	#552	—	181	1	no	—
173	#547	—	181	1	no	—
174	#501	—	181	1	no	—
175	#484	—	181	1	no	—
176	#483	—	181	1	no	—
177	#482	—	181	1	no	—
178	#480	—	181	1	no	—
179	#478	—	181	1	no	—
180	#477	D₁₈₁	1	1	yes	characteristic, complemented
181	#468	—	181	1	no	—
182	#465	—	181	1	no	—
183	#440	—	181	1	no	—
184	#435	—	181	1	no	—
185	#428	—	181	1	no	—
186	C₂×#107	—	181	1	no	—
187	C₂×#41	—	362	1	no	—
188	C₂×SD₁₆	—	181	1	no	—
189	C₂×M₁₆	—	181	1	no	—
190	C₂²×C₈	—	181	1	no	—
191	C₄⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
192	#126	—	362	1	no	—
193	#125	—	362	1	no	—
194	#125	—	181	1	no	—
195	#123	—	362	1	no	—
196	#123	—	362	1	no	—
197	#123	—	362	1	no	—
198	C₂²⋊Q₈	—	362	1	no	—
199	C₂²⋊Q₈	—	181	1	no	—
200	C₂²⋊Q₈	—	181	1	no	—
201	#121	—	181	1	no	—
202	#121	—	181	1	no	—
203	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
204	C₄×D₄	—	181	1	no	—
205	C₄×D₄	—	181	1	no	—
206	#117	—	362	1	no	—
207	#117	—	181	1	no	—
208	C₂×C₄⋊C₄	—	181	1	no	—
209	C₂×C₄⋊C₄	—	181	1	no	—
210	C₂×C₄⋊C₄	—	181	1	no	—
211	#107	—	181	1	no	—
212	#107	—	181	1	no	—
213	#107	—	181	1	no	—
214	#107	—	181	1	no	—
215	#107	C₂×D₁₈₁	1	1	yes	characteristic
216	#107	C₂×D₁₈₁	1	1	yes	characteristic
217	#106	—	181	1	no	—
218	#106	—	181	1	no	—
219	#106	—	181	1	no	—
220	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
221	#103	—	181	1	no	—
222	#103	C₂×D₁₈₁	1	1	yes	characteristic
223	#103	C₂×D₁₈₁	1	1	yes	characteristic
224	#102	—	181	1	no	—
225	#102	—	181	1	no	—
226	#102	—	181	1	no	—
227	C₂²⋊C₈	—	181	1	no	—
228	C₂²⋊C₈	—	181	1	no	—
229	#97	—	181	1	no	—
230	#41	—	724	4	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
233	C₂×D₄	—	362	1	no	—
234	C₂×D₄	—	362	1	no	—
235	C₂×D₄	—	181	1	no	—
236	C₂×D₄	—	181	1	no	—
237	C₂²×C₄	—	362	1	no	—
238	C₂²×C₄	—	362	1	no	—
239	C₂²×C₄	—	362	1	no	—
240	C₂²×C₄	—	362	1	no	—
241	C₂²×C₄	—	181	1	no	—
242	C₂²×C₄	—	181	1	no	—
243	SD₁₆	—	362	2	no	—
244	M₁₆	—	362	2	no	—
245	C₂×C₈	—	362	2	no	—
246	C₂×C₈	—	181	1	no	—
247	C₂×C₈	—	181	1	no	—
248	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	362	1	no	—
253	C₄⋊C₄	—	362	1	no	—
254	C₄⋊C₄	—	362	1	no	—
255	C₄⋊C₄	—	362	1	no	—
256	C₄⋊C₄	—	362	1	no	—
257	C₄⋊C₄	—	362	1	no	—
258	C₄⋊C₄	—	362	1	no	—
259	C₄⋊C₄	—	181	1	no	—
260	C₄⋊C₄	—	181	1	no	—
261	C₄⋊C₄	—	181	1	no	—
262	C₄⋊C₄	—	181	1	no	—
263	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
266	C₂²⋊C₄	—	362	1	no	—
267	C₂²⋊C₄	—	362	1	no	—
268	C₂²⋊C₄	—	362	1	no	—
269	C₂²⋊C₄	—	362	1	no	—
270	C₂²⋊C₄	—	362	1	no	—
271	C₂²⋊C₄	—	362	1	no	—
272	C₂²⋊C₄	—	181	1	no	—
273	C₂²⋊C₄	—	181	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	362	1	no	—
276	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
277	C₂³	—	362	1	no	—
278	C₂³	—	181	1	no	—
279	C₂³	—	181	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	4	2	no	—
283	D₄	—	362	2	no	—
284	D₄	—	362	1	no	—
285	D₄	—	362	1	no	—
286	D₄	—	724	2	no	—
287	D₄	—	724	2	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	—	362	1	no	—
293	C₂×C₄	—	362	1	no	—
294	C₂×C₄	—	362	1	no	—
295	C₂×C₄	—	362	1	no	—
296	C₂×C₄	—	362	1	no	—
297	C₂×C₄	—	362	1	no	—
298	C₂×C₄	—	362	1	no	—
299	C₂×C₄	—	181	1	no	—
300	C₂×C₄	—	181	1	no	—
301	C₂×C₄	—	724	2	no	—
302	C₂×C₄	—	724	2	no	—
303	C₂×C₄	—	724	2	no	—
304	C₂×C₄	—	724	2	no	—
305	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
306	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	362	1	no	—
311	C₈	—	181	2	no	—
312	C₂²	—	362	1	no	—
313	C₂²	—	362	1	no	—
314	C₂²	—	362	1	no	—
315	C₂²	—	362	1	no	—
316	C₂²	—	362	1	no	—
317	C₂²	—	181	2	no	—
318	C₂²	—	724	1	no	—
319	C₂²	—	724	1	no	—
320	C₂²	—	724	1	no	—
321	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	362	1	no	—
329	C₄	—	362	1	no	—
330	C₄	—	724	1	no	—
331	C₄	—	724	1	no	—
332	C₄	—	724	1	no	—
333	C₄	#41×D₁₈₁	1	1	yes	characteristic
334	C₄	#19153864	1	1	yes	characteristic
335	C₂	—	362	2	no	—
336	C₂	—	1448	1	no	—
337	C₂	—	724	1	no	—
338	C₂	#26041931	1	1	yes	characteristic
339	C₂	AGL(1,ℤ/8)×D₁₈₁	1	1	yes	characteristic
340	C₂	#26042092	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#477×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	#26041932	C₂
hypercenter	#477	D₁₈₁
hypocenter	C₁₈₁	#3343
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

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

N	H ≅ G/N	type
1. #26041970	C₂	semidirect
3. #26041685	C₂	semidirect
4. #26041613	C₂	semidirect
5. #26041608	C₂	semidirect
6. #26041606	C₂	semidirect
7. #26041594	C₂	semidirect
8. #26041580	C₂	semidirect
9. #26041528	C₂	semidirect
10. #26041511	C₂	semidirect
11. #26041377	C₂	semidirect
12. #26041349	C₂	semidirect
13. #26041320	C₂	semidirect
14. #107×D₁₈₁	C₂	semidirect
15. #477×C₁₈₁	C₂	semidirect
128. #13089678	#41	semidirect
170. C₁₈₁	#3343	semidirect
180. #477	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041970	2
#26041932	2
#26041685	2
#26041613	2
#26041608	2
#26041606	2
#26041594	2
#26041580	2
#26041528	2
#26041511	2
#26041377	2
#26041349	2
#26041320	2
#107×D₁₈₁	2
#477×C₁₈₁	2
#3343	181