Subgroups — #16840169

Order 3904.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519776	C₂	1	1	yes	maximal, characteristic
2	#14519768	C₂	1	1	yes	maximal, characteristic
3	#14519711	C₂	1	1	yes	maximal, characteristic
4	#14519710	C₂	1	1	yes	maximal, characteristic
5	#14519708	C₂	1	1	yes	maximal, characteristic
6	#14519687	C₂	1	1	yes	maximal, characteristic
7	#106×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₂₂	—	2	1	no	—
9	C₂×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₆₁	—	2	1	no	—
13	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
15	#11758196	—	2	1	no	—
16	#11758196	—	2	1	no	—
17	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
18	C₂×#92418	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₆₁	—	2	1	no	—
20	C₂×C₄×C₆₁	—	2	1	no	—
21	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×Dic₆₁	—	2	1	no	—
23	C₂×Dic₆₁	—	2	1	no	—
24	C₂×Dic₆₁	D₄	1	1	yes	characteristic
25	C₂×Dic₆₁	D₄	1	1	yes	characteristic
26	Dic₁₂₂	—	4	2	no	—
27	C₈×C₆₁	Q₈	1	2	yes	complemented
28	#92418	Q₈	1	2	yes	complemented
29	C₂²×C₆₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₆₁	—	4	1	no	—
31	C₄×C₆₁	—	4	1	no	—
32	C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
34	Dic₆₁	—	2	1	no	—
35	Dic₆₁	—	2	1	no	—
36	Dic₆₁	—	4	1	no	—
37	Dic₆₁	—	4	1	no	—
38	C₂×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₂×C₆₁	#137	1	1	yes	characteristic
41	#501	—	61	1	no	maximal, Sylow
42	C₆₁	#501	1	1	yes	characteristic, complemented, Sylow
43	C₄⋊Q₈	—	61	1	no	—
44	#125	—	61	1	no	—
45	#107	—	61	1	no	—
46	#107	—	61	1	no	—
47	#106	—	61	1	no	—
48	#106	D₆₁	1	1	yes	characteristic
49	#97	—	61	1	no	—
50	C₂×Q₈	—	122	1	no	—
51	C₂×C₈	—	61	1	no	—
52	C₂×C₈	C₂×D₆₁	1	1	yes	characteristic
53	C₄⋊C₄	—	61	1	no	—
54	C₄⋊C₄	—	61	1	no	—
55	C₄⋊C₄	—	122	1	no	—
56	C₄⋊C₄	—	122	1	no	—
57	C₄⋊C₄	—	122	1	no	—
58	C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
60	C₄²	—	61	1	no	—
61	Q₈	—	244	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	61	1	no	—
65	C₂×C₄	—	61	1	no	—
66	C₂×C₄	—	122	1	no	—
67	C₂×C₄	—	122	1	no	—
68	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
69	C₈	—	61	2	no	—
70	C₈	Dic₁₂₂	1	2	yes	complemented
71	C₂²	D₄×D₆₁	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	122	1	no	—
75	C₄	—	122	1	no	—
76	C₄	—	244	1	no	—
77	C₄	—	244	1	no	—
78	C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
79	C₄	Q₈×D₆₁	1	1	yes	characteristic
80	C₂	#14519776	1	1	yes	characteristic
81	C₂	#14519804	1	1	yes	characteristic
82	C₂	#14519806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₆₁
commutator, cosocle	C₂×C₄×C₆₁	C₂³
Frattini	C₂×C₄	C₂²×D₆₁
Fitting	#106×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	C₂×D₄
hypercenter	#106	D₆₁
hypocenter	C₆₁	#501
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
27. C₈×C₆₁	Q₈	semidirect
28. #92418	Q₈	semidirect
42. C₆₁	#501	semidirect
70. C₈	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#14519776	2
#14519768	2
#14519711	2
#14519710	2
#14519708	2
#14519687	2
#106×C₆₁	2
#501	61