Subgroups — #16840159

Order 3904.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519776	C₂	1	2	yes	maximal
2	#14519709	C₂	1	1	yes	maximal, characteristic
3	#14519708	C₂	1	2	yes	maximal
4	C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	#106×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₂₂	—	2	2	no	—
7	C₂×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
9	C₄⋊Dic₆₁	C₂²	1	2	yes	—
10	#11758196	—	2	2	no	—
11	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
12	C₂×#92418	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₆₁	—	2	2	no	—
14	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
15	C₂×Dic₆₁	—	2	2	no	—
16	C₂×Dic₆₁	D₄	1	2	yes	—
17	Dic₁₂₂	—	4	4	no	—
18	C₈×C₆₁	Q₈	1	2	yes	complemented
19	#92418	Q₈	1	2	yes	complemented
20	C₂²×C₆₁	C₂×D₄	1	1	yes	characteristic
21	C₄×C₆₁	—	4	2	no	—
22	C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
24	Dic₆₁	—	4	2	no	—
25	Dic₆₁	SD₁₆	1	4	yes	—
26	C₂×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₆₁	C₂×SD₁₆	1	2	yes	—
28	#498	—	61	1	no	maximal, Sylow
29	C₆₁	#498	1	1	yes	characteristic, complemented, Sylow
30	C₄⋊Q₈	—	61	2	no	—
31	#106	—	61	1	no	—
32	#106	—	61	2	no	—
33	#106	D₆₁	1	1	yes	characteristic
34	C₄×C₈	—	61	1	no	—
35	C₂×Q₈	—	122	2	no	—
36	C₂×C₈	—	61	1	no	—
37	C₂×C₈	C₂×D₆₁	1	1	yes	characteristic
38	C₄⋊C₄	—	61	2	no	—
39	C₄⋊C₄	—	122	2	no	—
40	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
41	C₄²	—	61	1	no	—
42	Q₈	—	244	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	61	2	no	—
45	C₂×C₄	—	122	2	no	—
46	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
47	C₈	—	61	2	no	—
48	C₈	Dic₁₂₂	1	2	yes	complemented
49	C₂²	D₄×D₆₁	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	61	4	no	—
52	C₄	—	244	2	no	—
53	C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
54	C₄	Q₈×D₆₁	1	1	yes	characteristic
55	C₂	#14519776	1	1	yes	characteristic
56	C₂	SD₁₆×D₆₁	1	2	yes	—

	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	#498
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
18. C₈×C₆₁	Q₈	semidirect
19. #92418	Q₈	semidirect
29. C₆₁	#498	semidirect
48. 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
#14519709	2
#14519708	2
C₈×Dic₆₁	2
#106×C₆₁	2
#498	61