Subgroups — #37546674

Order 23168.

348 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	#26041990	C₂	1	1	yes	maximal, characteristic, complemented
2	#26041988	C₂	1	1	yes	maximal, characteristic, complemented
3	#26041687	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041685	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041680	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041582	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041581	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041580	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041577	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041558	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041408	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041389	C₂	1	1	yes	maximal, characteristic
13	#26041388	C₂	1	1	yes	maximal, characteristic
14	#26041359	C₂	1	1	yes	maximal, characteristic
15	#474×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15668434	C₂²	1	1	yes	characteristic
17	#19153815	C₂²	1	1	yes	characteristic
18	#19153808	C₂²	1	1	yes	characteristic
19	#19153807	C₂²	1	1	yes	characteristic
20	#19153807	C₂²	1	1	yes	characteristic
21	#19153807	C₂²	1	1	yes	characteristic
22	#19153796	—	2	1	no	—
23	#19153795	—	2	1	no	—
24	#19153793	—	2	1	no	—
25	#19153793	—	2	1	no	—
26	#19153792	—	2	1	no	—
27	#19153792	—	2	1	no	—
28	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
29	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
32	#19153778	C₂²	1	1	yes	characteristic
33	#19153778	C₂²	1	1	yes	characteristic
34	C₂×#15668428	—	2	1	no	—
35	C₄⋊C₄×D₁₈₁	—	2	1	no	—
36	#19153750	C₂²	1	1	yes	characteristic
37	#19153749	C₂²	1	1	yes	characteristic
38	#19153749	C₂²	1	1	yes	characteristic
39	#19153748	C₂²	1	1	yes	characteristic
40	#19153747	C₂²	1	1	yes	characteristic
41	#19153747	C₂²	1	1	yes	characteristic
42	#19153747	C₂²	1	1	yes	characteristic
43	#19153745	C₂²	1	1	yes	characteristic
44	#19153745	C₂²	1	1	yes	characteristic
45	#19153738	C₂²	1	1	yes	characteristic
46	#19153738	C₂²	1	1	yes	characteristic
47	#19153737	C₂²	1	1	yes	characteristic
48	#19153736	C₂²	1	1	yes	characteristic
49	#19153727	C₂²	1	1	yes	characteristic
50	#19153726	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×D₄×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	#102×C₁₈₁	C₂²	1	1	yes	characteristic
58	#102×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	1	no	—
60	C₂×Dic₃₆₂	D₄	1	1	yes	characteristic
61	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₈₁	—	2	1	no	—
63	C₂²×Dic₁₈₁	—	2	1	no	—
64	C₂×C₄×D₁₈₁	—	2	1	no	—
65	C₂×C₄×D₁₈₁	—	2	1	no	—
66	C₂×C₄×D₁₈₁	—	2	1	no	—
67	#15668434	Q₈	1	4	yes	complemented
68	C₂×#13089678	C₂³	1	1	yes	characteristic
69	C₂×#13089678	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₈₁	—	2	1	no	—
71	C₄⋊Dic₁₈₁	—	2	1	no	—
72	C₄⋊Dic₁₈₁	—	2	1	no	—
73	C₄⋊Dic₁₈₁	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
77	#15668428	—	2	1	no	—
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	D₄	1	1	yes	characteristic
84	#15668428	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₈₁	—	2	1	no	—
86	C₂²⋊Dic₁₈₁	—	2	1	no	—
87	#15668426	—	2	1	no	—
88	#15668426	—	2	1	no	—
89	#15668426	—	2	1	no	—
90	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
91	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₁₈₁	—	2	1	no	—
93	C₂×D₄×C₁₈₁	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₈₁	—	2	1	no	—
95	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₁₈₁	—	2	1	no	—
98	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
101	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
102	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
103	C₂²×D₁₈₁	—	2	1	no	—
104	Dic₃₆₂	—	4	1	no	—
105	Dic₃₆₂	—	8	1	no	—
106	D₇₂₄	—	2	1	no	—
107	D₇₂₄	C₂×Q₈	1	2	yes	—
108	C₂×Dic₁₈₁	—	2	1	no	—
109	C₂×Dic₁₈₁	—	2	1	no	—
110	C₂×Dic₁₈₁	—	2	1	no	—
111	C₂×Dic₁₈₁	—	4	2	no	—
112	C₂×Dic₁₈₁	—	8	1	no	—
113	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
117	C₄×D₁₈₁	—	4	2	no	—
118	C₄×D₁₈₁	—	8	1	no	—
119	C₄×D₁₈₁	—	8	1	no	—
120	#13089678	—	2	1	no	—
121	#13089678	C₂×Q₈	1	2	yes	—
122	C₂³×C₁₈₁	—	2	1	no	—
123	Q₈×C₁₈₁	—	8	1	no	—
124	D₄×C₁₈₁	—	2	1	no	—
125	D₄×C₁₈₁	C₂×Q₈	1	2	yes	—
126	C₂×C₄×C₁₈₁	—	2	1	no	—
127	C₂×C₄×C₁₈₁	—	2	1	no	—
128	C₂×C₄×C₁₈₁	—	2	1	no	—
129	C₂×C₄×C₁₈₁	—	8	1	no	—
130	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
133	C₈×C₁₈₁	—	2	1	no	—
134	C₈×C₁₈₁	—	2	1	no	—
135	C₂²×C₁₈₁	—	2	2	no	—
136	C₂²×C₁₈₁	—	4	1	no	—
137	C₂²×C₁₈₁	—	4	1	no	—
138	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₈₁	—	2	2	no	—
140	C₂×D₁₈₁	—	4	1	no	—
141	C₂×D₁₈₁	—	4	1	no	—
142	C₄×C₁₈₁	—	2	1	no	—
143	C₄×C₁₈₁	—	2	1	no	—
144	C₄×C₁₈₁	—	4	1	no	—
145	C₄×C₁₈₁	—	4	1	no	—
146	C₄×C₁₈₁	—	4	1	no	—
147	C₄×C₁₈₁	C₂²×Q₈	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	—	2	1	no	—
153	Dic₁₈₁	—	4	1	no	—
154	Dic₁₈₁	—	4	1	no	—
155	Dic₁₈₁	—	4	1	no	—
156	C₂×C₁₈₁	—	4	2	no	—
157	C₂×C₁₈₁	D₄×Q₈	1	1	yes	characteristic
158	C₂×C₁₈₁	#575	1	1	yes	characteristic
159	C₂×C₁₈₁	#576	1	1	yes	characteristic
160	D₁₈₁	—	4	2	no	—
161	C₁₈₁	#3389	1	1	yes	characteristic, complemented, Sylow
162	#3389	—	181	1	no	maximal, Sylow
163	#554	—	181	1	no	—
164	#554	—	181	1	no	—
165	#501	—	181	1	no	—
166	#501	—	181	1	no	—
167	#499	—	181	1	no	—
168	#478	—	181	1	no	—
169	#478	—	181	1	no	—
170	#478	—	181	1	no	—
171	#478	—	181	1	no	—
172	#474	—	181	1	no	—
173	#474	D₁₈₁	1	1	yes	characteristic, complemented
174	#446	—	181	1	no	—
175	#442	—	181	1	no	—
176	#442	—	181	1	no	—
177	C₄×D₈	—	181	1	no	—
178	C₂×D₈	—	181	1	no	—
179	C₄⋊Q₈	—	181	1	no	—
180	C₄⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
181	#125	—	181	1	no	—
182	#125	—	181	1	no	—
183	#125	—	181	1	no	—
184	#125	—	181	1	no	—
185	C₂²⋊Q₈	—	362	1	no	—
186	C₂²⋊Q₈	—	362	1	no	—
187	C₂²⋊Q₈	—	362	1	no	—
188	C₂²⋊Q₈	—	362	1	no	—
189	C₂²⋊Q₈	—	362	1	no	—
190	C₂²⋊Q₈	—	362	1	no	—
191	C₄×Q₈	—	181	1	no	—
192	C₄×D₄	—	181	1	no	—
193	C₄×D₄	—	181	1	no	—
194	C₄×D₄	—	181	1	no	—
195	C₄×D₄	—	181	1	no	—
196	C₄×D₄	—	181	1	no	—
197	C₄×D₄	C₂×D₁₈₁	1	1	yes	characteristic
198	C₂×C₄⋊C₄	—	362	1	no	—
199	C₂×C₄⋊C₄	—	362	1	no	—
200	#107	—	181	1	no	—
201	#107	—	181	1	no	—
202	#107	—	181	1	no	—
203	#107	C₂×D₁₈₁	1	1	yes	characteristic
204	#107	C₂×D₁₈₁	1	1	yes	characteristic
205	#106	—	181	1	no	—
206	#106	—	181	1	no	—
207	#106	—	181	1	no	—
208	#106	—	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	#102	—	181	1	no	—
213	#102	—	181	1	no	—
214	#102	—	181	1	no	—
215	#102	—	181	1	no	—
216	#102	C₂×D₁₈₁	1	1	yes	characteristic
217	#102	C₂×D₁₈₁	1	1	yes	characteristic
218	#97	—	181	1	no	—
219	#97	—	181	1	no	—
220	C₄×C₈	—	181	1	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	362	1	no	—
223	C₂×Q₈	—	181	1	no	—
224	C₂×D₄	—	181	1	no	—
225	C₂×D₄	C₂²×D₁₈₁	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	362	1	no	—
228	C₂²×C₄	—	362	1	no	—
229	C₂²×C₄	—	362	1	no	—
230	C₂²×C₄	—	362	1	no	—
231	C₂²×C₄	—	362	1	no	—
232	D₈	—	181	4	no	—
233	C₂×C₈	—	181	1	no	—
234	C₂×C₈	—	181	1	no	—
235	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
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	—	181	1	no	—
248	C₄⋊C₄	—	181	1	no	—
249	C₄⋊C₄	—	181	1	no	—
250	C₄⋊C₄	—	181	1	no	—
251	C₄⋊C₄	—	181	1	no	—
252	C₄⋊C₄	—	181	1	no	—
253	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
256	C₂²⋊C₄	—	2	1	no	—
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	—	362	1	no	—
262	C₄²	—	181	1	no	—
263	C₄²	—	181	1	no	—
264	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
265	C₂³	—	2	1	no	—
266	C₂³	—	362	1	no	—
267	Q₈	—	8	1	no	—
268	Q₈	—	1448	1	no	—
269	Q₈	—	724	1	no	—
270	D₄	—	2	1	no	—
271	D₄	—	362	1	no	—
272	D₄	—	181	2	no	—
273	D₄	Q₈×D₁₈₁	1	2	yes	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	362	1	no	—
279	C₂×C₄	—	362	1	no	—
280	C₂×C₄	—	362	1	no	—
281	C₂×C₄	—	1448	1	no	—
282	C₂×C₄	—	1448	1	no	—
283	C₂×C₄	—	1448	1	no	—
284	C₂×C₄	—	181	1	no	—
285	C₂×C₄	—	181	1	no	—
286	C₂×C₄	—	181	1	no	—
287	C₂×C₄	—	181	1	no	—
288	C₂×C₄	—	724	2	no	—
289	C₂×C₄	—	724	2	no	—
290	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
291	C₂×C₄	D₄×D₁₈₁	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	—	362	1	no	—
296	C₈	—	181	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	362	2	no	—
301	C₂²	—	724	1	no	—
302	C₂²	—	724	1	no	—
303	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	362	1	no	—
310	C₄	—	362	1	no	—
311	C₄	—	362	1	no	—
312	C₄	—	362	1	no	—
313	C₄	—	724	1	no	—
314	C₄	—	724	1	no	—
315	C₄	—	724	1	no	—
316	C₄	C₂×Q₈×D₁₈₁	1	1	yes	characteristic
317	C₄	#19153863	1	1	yes	characteristic
318	C₂	—	4	2	no	—
319	C₂	—	724	2	no	—
320	C₂	#26041942	1	1	yes	characteristic
321	C₂	#26042078	1	1	yes	characteristic
322	C₂	#26042080	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#474×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×#15668434	C₂²
hypercenter	#474	D₁₈₁
hypocenter	C₁₈₁	#3389
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

Complements

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

N	H ≅ G/N	type
1. #26041990	C₂	semidirect
2. #26041988	C₂	semidirect
3. #26041687	C₂	semidirect
4. #26041685	C₂	semidirect
5. #26041680	C₂	semidirect
6. #26041582	C₂	semidirect
7. #26041581	C₂	semidirect
8. #26041580	C₂	semidirect
9. #26041577	C₂	semidirect
10. #26041558	C₂	semidirect
11. #26041408	C₂	semidirect
15. #474×C₁₈₁	C₂	semidirect
67. #15668434	Q₈	semidirect
161. C₁₈₁	#3389	semidirect
173. #474	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041990	2
#26041988	2
#26041687	2
#26041685	2
#26041680	2
#26041582	2
#26041581	2
#26041580	2
#26041577	2
#26041558	2
#26041408	2
#26041389	2
#26041388	2
#26041359	2
#474×C₁₈₁	2
#3389	181