Subgroups — #37536031

Order 23168.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041712	C₂	1	2	yes	maximal, complemented
2	#26041116	C₂	1	2	yes	maximal, complemented
3	D₅₇₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#19153839	—	2	2	no	—
7	#19153759	—	2	2	no	—
8	D₂₈₉₆	C₂²	1	2	yes	—
9	C₁₆×D₁₈₁	C₂²	1	1	yes	characteristic
10	#19153722	C₂²	1	1	yes	characteristic
11	Q₃₂×C₁₈₁	C₂²	1	2	yes	—
12	C₃₂×C₁₈₁	C₂²	1	1	yes	characteristic
13	#15668450	—	4	2	no	—
14	#15668436	—	4	2	no	—
15	D₁₄₄₈	—	2	2	no	—
16	C₈×D₁₈₁	D₄	1	1	yes	characteristic
17	#15668424	D₄	1	1	yes	characteristic
18	Q₁₆×C₁₈₁	—	2	2	no	—
19	C₁₆×C₁₈₁	C₂³	1	1	yes	characteristic
20	D₇₂₄	—	4	2	no	—
21	D₇₂₄	—	8	2	no	—
22	C₄×D₁₈₁	—	8	2	no	—
23	C₄×D₁₈₁	D₈	1	1	yes	characteristic
24	#13089678	D₈	1	1	yes	characteristic
25	Q₈×C₁₈₁	—	4	2	no	—
26	C₈×C₁₈₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₈₁	—	8	2	no	—
28	C₂×D₁₈₁	D₁₆	1	1	yes	characteristic
29	C₄×C₁₈₁	—	8	2	no	—
30	C₄×C₁₈₁	C₂×D₈	1	1	yes	characteristic
31	Dic₁₈₁	D₁₆	1	1	yes	characteristic
32	C₂×C₁₈₁	C₂×D₁₆	1	1	yes	characteristic
33	D₁₈₁	—	2	1	no	—
34	D₁₈₁	—	16	2	no	—
35	C₁₈₁	#2262	1	1	yes	characteristic, complemented, Sylow
36	#2262	—	181	1	no	maximal, Sylow
37	#508	—	181	2	no	—
38	Q₆₄	D₁₈₁	1	1	yes	characteristic, complemented
39	SD₆₄	—	181	2	no	—
40	D₃₂	—	181	1	no	—
41	C₂×C₃₂	—	181	1	no	—
42	#135	—	362	2	no	—
43	Q₃₂	C₂×D₁₈₁	1	2	yes	—
44	SD₃₂	—	362	2	no	—
45	D₁₆	—	181	2	no	—
46	C₂×C₁₆	—	181	1	no	—
47	C₃₂	—	181	1	no	—
48	C₃₂	C₂×D₁₈₁	1	1	yes	characteristic
49	#41	—	724	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	724	2	no	—
52	D₈	—	362	2	no	—
53	C₂×C₈	—	181	1	no	—
54	C₁₆	—	181	1	no	—
55	C₁₆	C₂²×D₁₈₁	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	1448	2	no	—
58	D₄	—	724	2	no	—
59	C₂×C₄	—	1448	2	no	—
60	C₂×C₄	—	181	1	no	—
61	C₈	—	181	1	no	—
62	C₈	D₄×D₁₈₁	1	1	yes	characteristic
63	C₂²	—	1448	2	no	—
64	C₂²	—	181	1	no	—
65	C₄	—	8	2	no	—
66	C₄	—	181	1	no	—
67	C₄	D₈×D₁₈₁	1	1	yes	characteristic
68	C₂	—	2896	2	no	—
69	C₂	—	362	1	no	—
70	C₂	D₁₆×D₁₈₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₁₈₁
commutator, cosocle	C₁₆×C₁₈₁	C₂³
Frattini	C₁₆	C₂²×D₁₈₁
Fitting	Q₆₄×C₁₈₁	C₂
socle	C₂×C₁₈₁	C₂×D₁₆
hypercenter	Q₆₄	D₁₈₁
hypocenter	C₁₈₁	#2262
LCS	G ▷ C₁₆×C₁₈₁ ▷ C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #26041712	C₂	semidirect
2. #26041116	C₂	semidirect
3. D₅₇₉₂	C₂	semidirect
4. C₃₂×D₁₈₁	C₂	semidirect
5. Q₆₄×C₁₈₁	C₂	semidirect
35. C₁₈₁	#2262	semidirect
38. Q₆₄	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041712	2
#26041116	2
D₅₇₉₂	2
C₃₂×D₁₈₁	2
Q₆₄×C₁₈₁	2
#2262	181