Subgroups — #37545174

Order 23168.

322 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041892	C₂	1	1	yes	maximal, characteristic
2	#26041887	C₂	1	1	yes	maximal, characteristic
3	#26041589	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041585	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041583	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041538	C₂	1	1	yes	maximal, characteristic
7	#26041536	C₂	1	1	yes	maximal, characteristic
8	#26041521	C₂	1	1	yes	maximal, characteristic
9	#26041519	C₂	1	1	yes	maximal, characteristic
10	#26041490	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041489	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041488	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041488	C₂	1	1	yes	maximal, characteristic, complemented
14	#26041304	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₇₂₄	C₂²	1	1	yes	characteristic
17	C₂×Dic₇₂₄	C₂²	1	1	yes	characteristic
18	C₂×#15668439	C₂²	1	1	yes	characteristic
19	C₂×#15668439	C₂²	1	1	yes	characteristic
20	C₂×#15668432	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₈₁	C₂²	1	1	yes	characteristic
22	#19153816	C₂²	1	1	yes	characteristic
23	#19153814	C₂²	1	1	yes	characteristic
24	#19153811	—	2	1	no	—
25	#19153811	—	2	1	no	—
26	#19153811	—	2	1	no	—
27	#19153810	—	2	1	no	—
28	#19153808	—	2	1	no	—
29	#19153807	—	2	1	no	—
30	#19153807	—	2	1	no	—
31	#19153807	C₂²	1	1	yes	characteristic
32	#19153806	—	2	1	no	—
33	#19153796	C₂²	1	1	yes	characteristic
34	#19153796	C₂²	1	1	yes	characteristic
35	#19153796	C₂²	1	1	yes	characteristic
36	#19153792	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
38	#19153781	C₂²	1	1	yes	characteristic
39	#19153781	C₂²	1	1	yes	characteristic
40	#19153773	C₂²	1	1	yes	characteristic
41	#19153749	C₂²	1	1	yes	characteristic
42	#19153747	C₂²	1	1	yes	characteristic
43	C₄⋊#13089678	C₂²	1	1	yes	characteristic
44	#19153745	C₂²	1	1	yes	characteristic
45	#19153745	C₂²	1	1	yes	characteristic
46	#19153741	C₂²	1	1	yes	characteristic
47	#19153741	C₂²	1	1	yes	characteristic
48	#19153739	C₂²	1	1	yes	characteristic
49	#19153739	C₂²	1	1	yes	characteristic
50	#19153739	C₂²	1	1	yes	characteristic
51	#19153739	C₂²	1	1	yes	characteristic
52	#125×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	#107×C₁₈₁	C₂²	1	1	yes	characteristic
55	#106×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	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
60	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
61	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
63	Dic₇₂₄	—	4	1	no	—
64	Dic₇₂₄	—	4	1	no	—
65	#15668439	—	4	1	no	—
66	#15668439	—	4	1	no	—
67	#15668432	—	4	1	no	—
68	C₂×#13089678	C₂³	1	1	yes	characteristic
69	C₂×#13089678	C₂³	1	1	yes	characteristic
70	C₈×D₁₈₁	—	4	1	no	—
71	C₄⋊Dic₁₈₁	—	2	1	no	—
72	C₄⋊Dic₁₈₁	—	2	1	no	—
73	C₄⋊Dic₁₈₁	—	2	1	no	—
74	C₄⋊Dic₁₈₁	—	2	1	no	—
75	C₄⋊Dic₁₈₁	—	2	1	no	—
76	C₄⋊Dic₁₈₁	—	2	1	no	—
77	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
78	#15668428	—	2	1	no	—
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	D₄	1	1	yes	characteristic
86	#15668428	D₄	1	1	yes	characteristic
87	#15668426	—	2	1	no	—
88	#15668426	—	2	1	no	—
89	#15668426	—	2	1	no	—
90	#15668426	—	2	1	no	—
91	#15668426	D₄	1	1	yes	characteristic
92	#15668426	D₄	1	1	yes	characteristic
93	C₄×Dic₁₈₁	—	2	1	no	—
94	C₄×Dic₁₈₁	—	2	1	no	—
95	C₄×Dic₁₈₁	—	2	1	no	—
96	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×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	C₄⋊C₄×C₁₈₁	—	2	1	no	—
101	C₄⋊C₄×C₁₈₁	—	2	1	no	—
102	C₄⋊C₄×C₁₈₁	—	2	1	no	—
103	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
106	C₄²×C₁₈₁	—	2	1	no	—
107	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
109	Dic₃₆₂	—	2	1	no	—
110	Dic₃₆₂	—	2	1	no	—
111	Dic₃₆₂	—	2	1	no	—
112	Dic₃₆₂	—	2	1	no	—
113	Dic₃₆₂	—	2	1	no	—
114	Dic₃₆₂	—	2	1	no	—
115	C₂×Dic₁₈₁	—	2	1	no	—
116	C₂×Dic₁₈₁	—	2	1	no	—
117	C₂×Dic₁₈₁	—	2	1	no	—
118	C₂×Dic₁₈₁	—	2	1	no	—
119	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
122	C₄×D₁₈₁	—	2	1	no	—
123	C₄×D₁₈₁	—	2	1	no	—
124	#13089678	—	2	1	no	—
125	#13089678	—	2	1	no	—
126	Q₈×C₁₈₁	—	2	1	no	—
127	Q₈×C₁₈₁	—	2	1	no	—
128	C₂×C₄×C₁₈₁	—	2	1	no	—
129	C₂×C₄×C₁₈₁	—	2	1	no	—
130	C₂×C₄×C₁₈₁	—	2	1	no	—
131	C₂×C₄×C₁₈₁	—	2	1	no	—
132	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
135	C₈×C₁₈₁	—	2	1	no	—
136	C₈×C₁₈₁	—	2	1	no	—
137	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₁₈₁	—	2	1	no	—
139	C₂×D₁₈₁	—	2	1	no	—
140	C₂×D₁₈₁	—	2	1	no	—
141	C₄×C₁₈₁	—	2	1	no	—
142	C₄×C₁₈₁	—	2	1	no	—
143	C₄×C₁₈₁	—	4	1	no	—
144	C₄×C₁₈₁	—	4	1	no	—
145	C₄×C₁₈₁	—	4	1	no	—
146	C₄×C₁₈₁	—	4	1	no	—
147	C₄×C₁₈₁	H(5,𝔽₂)	1	1	yes	characteristic
148	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₈₁	—	2	1	no	—
150	Dic₁₈₁	—	2	1	no	—
151	Dic₁₈₁	—	2	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	#537	1	1	yes	characteristic
157	C₂×C₁₈₁	#578	1	1	yes	characteristic
158	C₂×C₁₈₁	#578	1	1	yes	characteristic
159	D₁₈₁	—	4	1	no	—
160	C₁₈₁	#3261	1	1	yes	characteristic, complemented, Sylow
161	#3261	—	181	1	no	maximal, Sylow
162	#541	—	181	1	no	—
163	#541	—	181	1	no	—
164	#479	—	181	1	no	—
165	#479	—	181	1	no	—
166	#479	—	181	1	no	—
167	#479	D₁₈₁	1	1	yes	characteristic, complemented
168	#470	—	181	1	no	—
169	#470	—	181	1	no	—
170	#467	—	181	1	no	—
171	#467	—	181	1	no	—
172	#462	—	181	1	no	—
173	#462	—	181	1	no	—
174	#462	—	181	1	no	—
175	#462	—	181	1	no	—
176	#424	—	181	1	no	—
177	C₂×Q₁₆	—	181	1	no	—
178	C₂×Q₁₆	—	181	1	no	—
179	C₂×Q₁₆	—	181	1	no	—
180	C₂×Q₁₆	—	181	1	no	—
181	C₂×M₁₆	—	181	1	no	—
182	C₂²×C₈	—	181	1	no	—
183	C₄⋊Q₈	—	181	1	no	—
184	C₄⋊Q₈	—	181	1	no	—
185	#126	—	362	1	no	—
186	#126	—	362	1	no	—
187	#126	—	362	1	no	—
188	#126	—	362	1	no	—
189	#125	—	362	1	no	—
190	#125	—	362	1	no	—
191	#125	—	362	1	no	—
192	#125	—	362	1	no	—
193	#125	—	181	1	no	—
194	#125	C₂×D₁₈₁	1	1	yes	characteristic
195	C₂²⋊Q₈	—	181	1	no	—
196	C₂²⋊Q₈	—	181	1	no	—
197	C₂²⋊Q₈	—	181	1	no	—
198	C₂²⋊Q₈	—	181	1	no	—
199	C₄×Q₈	—	181	1	no	—
200	C₄×Q₈	—	181	1	no	—
201	C₄×Q₈	—	181	1	no	—
202	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
203	#117	—	181	1	no	—
204	#107	—	181	1	no	—
205	#107	C₂×D₁₈₁	1	1	yes	characteristic
206	#106	—	181	1	no	—
207	#106	C₂×D₁₈₁	1	1	yes	characteristic
208	C₄⋊C₈	—	181	1	no	—
209	C₄⋊C₈	—	181	1	no	—
210	C₄⋊C₈	—	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	#103	C₂×D₁₈₁	1	1	yes	characteristic
219	#103	C₂×D₁₈₁	1	1	yes	characteristic
220	C₂×Q₈	—	181	1	no	—
221	C₂×Q₈	—	181	1	no	—
222	C₂×Q₈	—	181	1	no	—
223	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
224	C₂²×C₄	—	181	1	no	—
225	Q₁₆	—	724	1	no	—
226	Q₁₆	—	724	1	no	—
227	Q₁₆	—	724	1	no	—
228	Q₁₆	—	724	1	no	—
229	M₁₆	—	724	1	no	—
230	C₂×C₈	—	181	1	no	—
231	C₂×C₈	—	181	1	no	—
232	C₂×C₈	—	724	1	no	—
233	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	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	—	362	1	no	—
246	C₄⋊C₄	—	362	1	no	—
247	C₄⋊C₄	—	362	1	no	—
248	C₄⋊C₄	—	362	1	no	—
249	C₄⋊C₄	—	362	1	no	—
250	C₄⋊C₄	—	362	1	no	—
251	C₄⋊C₄	—	181	1	no	—
252	C₄⋊C₄	—	181	1	no	—
253	C₄⋊C₄	—	181	1	no	—
254	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	362	1	no	—
258	C₂²⋊C₄	—	362	1	no	—
259	C₂²⋊C₄	—	362	1	no	—
260	C₂²⋊C₄	—	362	1	no	—
261	C₂²⋊C₄	—	181	1	no	—
262	C₂²⋊C₄	—	181	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	362	1	no	—
265	C₄²	—	362	1	no	—
266	C₄²	—	362	1	no	—
267	C₄²	—	181	1	no	—
268	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
269	C₂³	—	181	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	362	1	no	—
273	Q₈	—	362	1	no	—
274	Q₈	—	362	1	no	—
275	Q₈	—	362	1	no	—
276	Q₈	—	362	1	no	—
277	Q₈	—	362	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	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	—	362	1	no	—
288	C₂×C₄	—	181	1	no	—
289	C₂×C₄	—	181	1	no	—
290	C₂×C₄	—	181	1	no	—
291	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
292	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	362	1	no	—
297	C₈	—	362	1	no	—
298	C₂²	—	362	1	no	—
299	C₂²	—	362	1	no	—
300	C₂²	—	362	1	no	—
301	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	362	1	no	—
309	C₄	—	362	1	no	—
310	C₄	—	362	1	no	—
311	C₄	—	724	1	no	—
312	C₄	—	724	1	no	—
313	C₄	—	724	1	no	—
314	C₄	—	724	1	no	—
315	C₄	#19153862	1	1	yes	characteristic
316	C₄	#19153864	1	1	yes	characteristic
317	C₂	—	724	1	no	—
318	C₂	#26041859	1	1	yes	characteristic
319	C₂	#26042093	1	1	yes	characteristic
320	C₂	#26042095	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#479×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂²×D₁₈₁	C₂×D₄
hypercenter	#479	D₁₈₁
hypocenter	C₁₈₁	#3261
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #26041589	C₂	semidirect
4. #26041585	C₂	semidirect
5. #26041583	C₂	semidirect
10. #26041490	C₂	semidirect
11. #26041489	C₂	semidirect
12. #26041488	C₂	semidirect
13. #26041488	C₂	semidirect
15. #479×C₁₈₁	C₂	semidirect
160. C₁₈₁	#3261	semidirect
167. #479	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041892	2
#26041887	2
#26041589	2
#26041585	2
#26041583	2
#26041538	2
#26041536	2
#26041521	2
#26041519	2
#26041490	2
#26041489	2
#26041488	2
#26041304	2
#479×C₁₈₁	2
#3261	181