Subgroups — #37535994

Order 23168.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041684	C₂	1	1	yes	maximal, characteristic
2	#26041104	C₂	1	4	yes	maximal
3	C₄×#15668424	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	#19153816	—	2	1	no	—
6	C₂×#15668424	C₂²	1	2	yes	—
7	#19153750	—	2	1	no	—
8	#19153750	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₈₁	—	2	1	no	—
10	#107×C₁₈₁	—	2	1	no	—
11	#107×C₁₈₁	C₂²	1	2	yes	—
12	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₆₂	—	4	1	no	—
14	C₄⋊Dic₁₈₁	—	2	2	no	—
15	C₄⋊Dic₁₈₁	—	4	1	no	—
16	#15668424	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₁₈₁	—	4	1	no	—
18	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₈₁	—	2	2	no	—
21	C₄⋊C₄×C₁₈₁	—	4	1	no	—
22	C₄²×C₁₈₁	D₄	1	1	yes	characteristic
23	Dic₃₆₂	—	8	2	no	—
24	C₂×Dic₁₈₁	—	4	2	no	—
25	Q₈×C₁₈₁	—	8	2	no	—
26	C₂×C₄×C₁₈₁	—	4	2	no	—
27	C₂×C₄×C₁₈₁	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₈₁	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
30	C₈×C₁₈₁	Q₁₆	1	2	yes	—
31	C₈×C₁₈₁	C₂×Q₈	1	2	yes	—
32	C₂²×C₁₈₁	C₂×D₈	1	1	yes	characteristic
33	C₄×C₁₈₁	—	8	2	no	—
34	C₄×C₁₈₁	D₁₆	1	2	yes	—
35	C₄×C₁₈₁	Q₃₂	1	2	yes	—
36	C₄×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₁₈₁	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₁₈₁	—	8	2	no	—
39	C₂×C₁₈₁	#500	1	1	yes	characteristic
40	C₂×C₁₈₁	C₂×D₁₆	1	1	yes	characteristic
41	C₂×C₁₈₁	C₂×Q₃₂	1	1	yes	characteristic
42	C₁₈₁	#2252	1	1	yes	characteristic, complemented, Sylow
43	#2252	—	181	1	no	maximal, Sylow
44	#500	—	181	1	no	—
45	#500	D₁₈₁	1	1	yes	characteristic
46	#366	—	181	4	no	—
47	C₄×C₁₆	—	181	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	362	1	no	—
50	C₂×C₁₆	—	181	2	no	—
51	#107	—	2	1	no	—
52	#107	—	362	1	no	—
53	#107	—	181	2	no	—
54	#107	C₂×D₁₈₁	1	2	yes	—
55	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	724	1	no	—
58	C₂×C₈	C₂²⋊D₁₈₁	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	362	2	no	—
63	C₄⋊C₄	—	724	1	no	—
64	C₄²	C₂²⋊D₁₈₁	1	1	yes	characteristic
65	C₁₆	—	181	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	1448	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	724	2	no	—
70	C₂×C₄	#15668434	1	1	yes	characteristic
71	C₂×C₄	#15668434	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
73	C₈	#15668439	1	2	yes	—
74	C₈	Q₈×D₁₈₁	1	2	yes	—
75	C₂²	C₂×#15668434	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	1448	2	no	—
78	C₄	#19153757	1	2	yes	—
79	C₄	#19153762	1	2	yes	—
80	C₄	#19153814	1	1	yes	characteristic
81	C₄	C₂×#15668439	1	1	yes	characteristic
82	C₂	#26041682	1	1	yes	characteristic
83	C₂	C₂×#19153757	1	1	yes	characteristic
84	C₂	C₂×#19153762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15668434
commutator, cosocle	C₂×C₈×C₁₈₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₈₁
Fitting	#500×C₁₈₁	C₂
socle, prime radical	C₂²×C₁₈₁	C₂×D₈
hypercenter	#500	D₁₈₁
hypocenter	C₁₈₁	#2252
LCS	G ▷ C₂×C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
16. #15668424	Q₈	semidirect
42. C₁₈₁	#2252	semidirect

Maximal subgroups

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

subgroup	index
#26041684	2
#26041104	2
C₄×#15668424	2
#500×C₁₈₁	2
#2252	181