Subgroups — #16840004

Order 3904.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#14519757	C₂	1	2	yes	maximal
4	#14519700	C₂	1	2	yes	maximal, complemented
5	#107×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₆₁	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₂₂	C₂²	1	2	yes	—
8	C₂×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
10	#11758198	—	2	2	no	—
11	C₄⋊Dic₆₁	—	2	2	no	—
12	C₂×#92418	C₂²	1	1	yes	characteristic
13	Dic₂₄₄	—	2	2	no	—
14	C₈×D₆₁	—	2	2	no	—
15	C₂²×D₆₁	D₄	1	1	yes	characteristic
16	C₂×C₄×C₆₁	—	2	2	no	—
17	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂×Dic₆₁	—	2	2	no	—
19	C₂×Dic₆₁	D₄	1	1	yes	characteristic
20	C₄×D₆₁	—	2	1	no	—
21	C₄×D₆₁	—	2	1	no	—
22	Dic₁₂₂	—	2	2	no	—
23	Dic₁₂₂	—	2	2	no	—
24	C₈×C₆₁	D₄	1	2	yes	—
25	#92418	—	2	1	no	—
26	C₂²×C₆₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₆₁	—	2	1	no	—
28	C₂×D₆₁	—	2	1	no	—
29	C₂×D₆₁	Q₁₆	1	2	yes	complemented
30	C₄×C₆₁	—	4	2	no	—
31	C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₄×C₆₁	#41	1	1	yes	characteristic
33	Dic₆₁	—	2	1	no	—
34	Dic₆₁	—	4	2	no	—
35	C₂×C₆₁	#121	1	1	yes	characteristic
36	C₂×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₆₁	#135	1	1	yes	characteristic
38	D₆₁	—	2	2	no	—
39	#467	—	61	1	no	maximal, Sylow
40	C₆₁	#467	1	1	yes	characteristic, complemented, Sylow
41	C₂×Q₁₆	—	61	1	no	—
42	C₂²×C₈	—	61	1	no	—
43	C₂²⋊Q₈	—	61	2	no	—
44	#107	D₆₁	1	1	yes	characteristic, complemented
45	#103	—	61	2	no	—
46	C₂×Q₈	—	61	2	no	—
47	C₂²×C₄	—	61	1	no	—
48	Q₁₆	—	122	2	no	—
49	C₂×C₈	—	61	1	no	—
50	C₂×C₈	—	122	2	no	—
51	C₂×C₈	C₂×D₆₁	1	1	yes	characteristic
52	C₄⋊C₄	—	122	2	no	—
53	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
54	C₂²⋊C₄	—	122	2	no	—
55	C₂³	—	61	1	no	—
56	Q₈	—	122	2	no	—
57	Q₈	—	122	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	61	1	no	—
60	C₂×C₄	—	122	1	no	—
61	C₂×C₄	—	122	1	no	—
62	C₂×C₄	—	122	2	no	—
63	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
64	C₈	—	122	1	no	—
65	C₈	D₂₄₄	1	2	yes	—
66	C₂²	—	61	2	no	—
67	C₂²	—	122	1	no	—
68	C₂²	—	122	1	no	—
69	C₂²	D₄×D₆₁	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	122	1	no	—
72	C₄	—	244	2	no	—
73	C₄	C₂×D₂₄₄	1	1	yes	characteristic
74	C₄	#11758226	1	1	yes	characteristic
75	C₂	—	122	2	no	—
76	C₂	#14519752	1	1	yes	characteristic
77	C₂	Q₁₆×D₆₁	1	1	yes	characteristic
78	C₂	#14519796	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₆₁
commutator, cosocle	C₂×C₄×C₆₁	C₂³
Frattini	C₂×C₄	C₂²×D₆₁
Fitting	#107×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₄
prime radical	C₂²×D₆₁	D₄
hypercenter	#107	D₆₁
hypocenter	C₆₁	#467
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₂₄₄	C₂	semidirect
4. #14519700	C₂	semidirect
5. #107×C₆₁	C₂	semidirect
29. C₂×D₆₁	Q₁₆	semidirect
40. C₆₁	#467	semidirect
44. #107	D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₄₄	2
C₂×C₈×D₆₁	2
#14519757	2
#14519700	2
#107×C₆₁	2
#467	61