Subgroups — #37546253

Order 23168.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041999	C₂	1	1	yes	maximal, characteristic
2	#26041969	C₂	1	1	yes	maximal, characteristic
3	#26041688	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041621	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041621	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041621	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041615	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041587	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041575	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041537	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041520	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041387	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041321	C₂	1	1	yes	maximal, characteristic, complemented
14	#26041317	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₁₈₁	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₇₂₄	C₂²	1	1	yes	characteristic
17	#19153816	C₂²	1	1	yes	characteristic
18	#19153809	—	2	1	no	—
19	#19153809	—	2	1	no	—
20	#19153809	—	2	1	no	—
21	#19153808	C₂²	1	1	yes	characteristic
22	#19153806	—	2	1	no	—
23	#19153803	—	2	1	no	—
24	#19153797	—	2	1	no	—
25	#19153795	C₂²	1	1	yes	characteristic
26	#19153795	C₂²	1	1	yes	characteristic
27	#19153795	C₂²	1	1	yes	characteristic
28	#19153795	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
31	#19153781	—	2	1	no	—
32	#19153775	C₂²	1	1	yes	characteristic
33	#19153775	C₂²	1	1	yes	characteristic
34	#19153775	C₂²	1	1	yes	characteristic
35	#19153772	—	2	1	no	—
36	C₂×C₄⋊Dic₁₈₁	C₂²	1	1	yes	characteristic
37	#19153750	C₂²	1	1	yes	characteristic
38	#19153750	C₂²	1	1	yes	characteristic
39	#19153750	C₂²	1	1	yes	characteristic
40	#19153750	C₂²	1	1	yes	characteristic
41	#19153750	C₂²	1	1	yes	characteristic
42	#19153748	C₂²	1	1	yes	characteristic
43	#19153748	C₂²	1	1	yes	characteristic
44	#19153748	C₂²	1	1	yes	characteristic
45	#19153748	C₂²	1	1	yes	characteristic
46	#19153741	C₂²	1	1	yes	characteristic
47	#19153741	C₂²	1	1	yes	characteristic
48	#19153741	C₂²	1	1	yes	characteristic
49	#19153741	C₂²	1	1	yes	characteristic
50	#19153741	C₂²	1	1	yes	characteristic
51	#19153741	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₈₁	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
58	#97×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	1	no	—
60	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
61	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₈₁	—	2	1	no	—
63	Dic₇₂₄	—	2	2	no	—
64	C₄⋊Dic₁₈₁	—	2	1	no	—
65	C₄⋊Dic₁₈₁	—	2	1	no	—
66	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
72	#15668428	—	2	1	no	—
73	#15668428	—	2	1	no	—
74	#15668428	—	2	1	no	—
75	#15668428	—	2	1	no	—
76	#15668428	—	2	1	no	—
77	#15668428	—	2	1	no	—
78	#15668428	—	2	1	no	—
79	#15668428	—	2	1	no	—
80	C₂²⋊Dic₁₈₁	—	2	1	no	—
81	C₂²⋊Dic₁₈₁	—	2	1	no	—
82	C₂²⋊Dic₁₈₁	—	2	1	no	—
83	C₂²⋊Dic₁₈₁	—	2	1	no	—
84	C₂²⋊Dic₁₈₁	—	2	1	no	—
85	C₂²⋊Dic₁₈₁	—	2	1	no	—
86	C₂²⋊Dic₁₈₁	—	2	1	no	—
87	C₄×Dic₁₈₁	—	2	1	no	—
88	C₄×Dic₁₈₁	—	2	1	no	—
89	C₄×Dic₁₈₁	—	2	1	no	—
90	C₄×Dic₁₈₁	—	2	1	no	—
91	C₄×Dic₁₈₁	—	2	1	no	—
92	C₂×D₄×C₁₈₁	D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
95	M₁₆×C₁₈₁	—	2	2	no	—
96	C₂×C₈×C₁₈₁	—	2	2	no	—
97	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
103	C₂²⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
104	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
105	Dic₃₆₂	—	2	1	no	—
106	Dic₃₆₂	—	2	1	no	—
107	Dic₃₆₂	—	2	1	no	—
108	Dic₃₆₂	—	2	1	no	—
109	Dic₃₆₂	—	4	2	no	—
110	C₂×Dic₁₈₁	—	2	1	no	—
111	C₂×Dic₁₈₁	—	2	1	no	—
112	C₂×Dic₁₈₁	—	2	1	no	—
113	C₂×Dic₁₈₁	—	2	1	no	—
114	C₂×Dic₁₈₁	—	2	1	no	—
115	C₂×Dic₁₈₁	—	2	1	no	—
116	C₂×Dic₁₈₁	—	2	1	no	—
117	C₂×Dic₁₈₁	—	2	1	no	—
118	C₂×Dic₁₈₁	—	4	2	no	—
119	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
120	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
121	D₄×C₁₈₁	—	4	1	no	—
122	C₂×C₄×C₁₈₁	—	2	1	no	—
123	C₂×C₄×C₁₈₁	—	2	1	no	—
124	C₂×C₄×C₁₈₁	—	2	1	no	—
125	C₂×C₄×C₁₈₁	—	2	1	no	—
126	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
131	C₈×C₁₈₁	—	2	1	no	—
132	C₈×C₁₈₁	—	2	1	no	—
133	C₈×C₁₈₁	—	2	1	no	—
134	C₈×C₁₈₁	#41	1	2	yes	—
135	C₂²×C₁₈₁	—	2	1	no	—
136	C₂²×C₁₈₁	—	2	1	no	—
137	C₂²×C₁₈₁	—	2	1	no	—
138	C₂²×C₁₈₁	—	2	1	no	—
139	C₂²×C₁₈₁	—	2	1	no	—
140	C₂²×C₁₈₁	—	2	1	no	—
141	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₁₈₁	—	2	1	no	—
143	C₄×C₁₈₁	—	2	1	no	—
144	C₄×C₁₈₁	—	2	1	no	—
145	C₄×C₁₈₁	—	2	1	no	—
146	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
147	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₁₈₁	—	4	1	no	—
149	Dic₁₈₁	—	4	1	no	—
150	Dic₁₈₁	—	4	1	no	—
151	Dic₁₈₁	—	4	1	no	—
152	Dic₁₈₁	—	4	1	no	—
153	Dic₁₈₁	—	4	1	no	—
154	Dic₁₈₁	—	4	1	no	—
155	Dic₁₈₁	—	4	1	no	—
156	C₂×C₁₈₁	—	4	1	no	—
157	C₂×C₁₈₁	—	4	1	no	—
158	C₂×C₁₈₁	#547	1	1	yes	characteristic
159	C₂×C₁₈₁	#575	1	1	yes	characteristic
160	C₂×C₁₈₁	#578	1	1	yes	characteristic
161	C₁₈₁	#3349	1	1	yes	characteristic, complemented, Sylow
162	#3349	—	181	1	no	maximal, Sylow
163	#556	—	181	1	no	—
164	#552	—	181	1	no	—
165	#501	—	181	1	no	—
166	#485	—	181	1	no	—
167	#485	—	181	1	no	—
168	#485	—	181	1	no	—
169	#484	—	181	1	no	—
170	#479	—	181	1	no	—
171	#477	—	181	1	no	—
172	#470	—	181	1	no	—
173	#467	—	181	1	no	—
174	#441	—	181	1	no	—
175	#435	D₁₈₁	1	1	yes	characteristic
176	#428	—	181	1	no	—
177	#427	—	181	1	no	—
178	C₂×Q₁₆	—	181	1	no	—
179	C₂×M₁₆	C₂×D₁₈₁	1	1	yes	characteristic
180	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
181	C₄⋊Q₈	—	181	1	no	—
182	#126	—	362	1	no	—
183	#126	—	362	1	no	—
184	#126	—	362	1	no	—
185	#125	—	362	1	no	—
186	#125	—	181	1	no	—
187	#124	—	362	1	no	—
188	#123	—	362	1	no	—
189	C₂²⋊Q₈	—	181	1	no	—
190	C₂²⋊Q₈	—	181	1	no	—
191	C₂²⋊Q₈	—	181	1	no	—
192	C₂²⋊Q₈	—	181	1	no	—
193	C₄×Q₈	—	362	1	no	—
194	C₄×Q₈	—	181	1	no	—
195	C₄×Q₈	—	181	1	no	—
196	C₄×D₄	C₂×D₁₈₁	1	1	yes	characteristic
197	#117	—	362	1	no	—
198	#117	—	181	1	no	—
199	#117	—	181	1	no	—
200	#117	—	181	1	no	—
201	C₂×C₄⋊C₄	—	181	1	no	—
202	#107	—	181	1	no	—
203	#107	—	181	1	no	—
204	#107	—	181	1	no	—
205	#107	—	181	1	no	—
206	#107	—	181	1	no	—
207	#106	—	181	1	no	—
208	#106	—	181	1	no	—
209	#106	—	181	1	no	—
210	#106	—	181	1	no	—
211	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
212	#103	—	181	1	no	—
213	#103	—	181	1	no	—
214	#103	—	181	1	no	—
215	#103	—	181	1	no	—
216	#103	—	181	1	no	—
217	#103	—	181	1	no	—
218	C₂²⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
219	C₂²⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
220	#97	C₂×D₁₈₁	1	1	yes	characteristic
221	C₂×Q₈	—	362	1	no	—
222	C₂×Q₈	—	181	1	no	—
223	C₂×Q₈	—	181	1	no	—
224	C₂×D₄	D₇₂₄	1	1	yes	characteristic
225	C₂²×C₄	—	362	1	no	—
226	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
227	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
228	Q₁₆	—	362	2	no	—
229	M₁₆	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
232	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
235	C₄⋊C₄	—	362	1	no	—
236	C₄⋊C₄	—	362	1	no	—
237	C₄⋊C₄	—	362	1	no	—
238	C₄⋊C₄	—	362	1	no	—
239	C₄⋊C₄	—	362	1	no	—
240	C₄⋊C₄	—	362	1	no	—
241	C₄⋊C₄	—	362	1	no	—
242	C₄⋊C₄	—	362	1	no	—
243	C₄⋊C₄	—	362	1	no	—
244	C₄⋊C₄	—	362	1	no	—
245	C₄⋊C₄	—	181	1	no	—
246	C₄⋊C₄	—	181	1	no	—
247	C₄⋊C₄	—	181	1	no	—
248	C₄⋊C₄	—	181	1	no	—
249	C₄⋊C₄	—	181	1	no	—
250	C₄⋊C₄	—	181	1	no	—
251	C₄⋊C₄	D₇₂₄	1	1	yes	characteristic
252	C₂²⋊C₄	—	362	1	no	—
253	C₂²⋊C₄	—	362	1	no	—
254	C₂²⋊C₄	—	362	1	no	—
255	C₂²⋊C₄	—	362	1	no	—
256	C₂²⋊C₄	—	362	1	no	—
257	C₂²⋊C₄	—	362	1	no	—
258	C₂²⋊C₄	—	362	1	no	—
259	C₂²⋊C₄	D₇₂₄	1	1	yes	characteristic
260	C₂²⋊C₄	D₇₂₄	1	1	yes	characteristic
261	C₄²	—	362	1	no	—
262	C₄²	—	362	1	no	—
263	C₄²	—	362	1	no	—
264	C₄²	—	362	1	no	—
265	C₄²	—	362	1	no	—
266	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
267	C₂³	C₂×D₇₂₄	1	1	yes	characteristic
268	C₂³	C₂×D₇₂₄	1	1	yes	characteristic
269	Q₈	—	362	1	no	—
270	Q₈	—	362	1	no	—
271	Q₈	—	362	1	no	—
272	Q₈	—	362	1	no	—
273	Q₈	—	724	2	no	—
274	D₄	—	4	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	362	1	no	—
280	C₂×C₄	—	362	1	no	—
281	C₂×C₄	—	362	1	no	—
282	C₂×C₄	—	362	1	no	—
283	C₂×C₄	—	362	1	no	—
284	C₂×C₄	—	362	1	no	—
285	C₂×C₄	—	362	1	no	—
286	C₂×C₄	—	362	1	no	—
287	C₂×C₄	—	724	2	no	—
288	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
289	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
290	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
291	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	#15668448	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	C₂²×D₇₂₄	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	724	1	no	—
309	C₄	—	724	1	no	—
310	C₄	—	724	1	no	—
311	C₄	—	724	1	no	—
312	C₄	—	724	1	no	—
313	C₄	—	724	1	no	—
314	C₄	—	724	1	no	—
315	C₄	—	724	1	no	—
316	C₄	C₂×#15668448	1	1	yes	characteristic
317	C₄	#19153863	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	4	1	no	—
320	C₂	#26041933	1	1	yes	characteristic
321	C₂	#26042079	1	1	yes	characteristic
322	C₂	#26042096	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₂₄
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#435×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×D₄×C₁₈₁	D₄
hypercenter	#435	D₁₈₁
hypocenter	C₁₈₁	#3349
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
3. #26041688	C₂	semidirect
4. #26041621	C₂	semidirect
5. #26041621	C₂	semidirect
6. #26041621	C₂	semidirect
7. #26041615	C₂	semidirect
8. #26041587	C₂	semidirect
9. #26041575	C₂	semidirect
10. #26041537	C₂	semidirect
11. #26041520	C₂	semidirect
12. #26041387	C₂	semidirect
13. #26041321	C₂	semidirect
14. #26041317	C₂	semidirect
161. C₁₈₁	#3349	semidirect

Maximal subgroups

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

subgroup	index
#26041999	2
#26041969	2
#26041688	2
#26041621	2
#26041615	2
#26041587	2
#26041575	2
#26041537	2
#26041520	2
#26041387	2
#26041321	2
#26041317	2
#435×C₁₈₁	2
#3349	181