Subgroups — #14519756

Order 1952.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊Dic₆₁	C₂	1	2	yes	maximal
4	C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#11758196	C₂	1	2	yes	maximal, complemented
6	C₂³×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₆₁	—	2	2	no	—
8	C₂×C₄×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₁	C₂²	1	2	yes	—
11	C₂×Dic₆₁	C₂²	1	2	yes	—
12	Dic₁₂₂	—	2	2	no	—
13	C₂²×C₆₁	—	2	1	no	—
14	C₂²×C₆₁	—	2	1	no	—
15	C₂²×C₆₁	Q₈	1	2	yes	complemented
16	C₂²×C₆₁	C₂³	1	1	yes	characteristic
17	C₄×C₆₁	—	2	1	no	—
18	C₄×C₆₁	D₄	1	2	yes	—
19	Dic₆₁	—	2	2	no	—
20	Dic₆₁	—	2	2	no	—
21	C₂×C₆₁	—	2	2	no	—
22	C₂×C₆₁	C₂×D₄	1	1	yes	characteristic
23	C₂×C₆₁	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₆₁	#41	1	1	yes	characteristic
25	C₆₁	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
26	C₂²⋊Q₈	—	61	1	no	maximal, Sylow
27	C₂×Q₈	—	61	1	no	—
28	C₂²×C₄	D₆₁	1	1	yes	characteristic
29	C₄⋊C₄	—	61	2	no	—
30	C₄⋊C₄	—	61	1	no	—
31	C₂²⋊C₄	—	61	2	no	—
32	C₂³	C₂×D₆₁	1	1	yes	characteristic
33	Q₈	—	122	2	no	—
34	C₂×C₄	—	2	2	no	—
35	C₂×C₄	—	61	2	no	—
36	C₂×C₄	—	61	2	no	—
37	C₂×C₄	C₂×D₆₁	1	1	yes	characteristic
38	C₂×C₄	C₂×D₆₁	1	1	yes	characteristic
39	C₂²	—	2	1	no	—
40	C₂²	—	2	1	no	—
41	C₂²	Dic₁₂₂	1	2	yes	complemented
42	C₂²	C₂²×D₆₁	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	—	122	2	no	—
45	C₄	—	122	2	no	—
46	C₄	C₂²⋊D₆₁	1	2	yes	—
47	C₂	—	2	2	no	—
48	C₂	C₂×Dic₁₂₂	1	1	yes	characteristic
49	C₂	#11758222	1	1	yes	characteristic
50	C₂	C₂×C₂²⋊D₆₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂³
Fitting	C₂²×C₄×C₆₁	C₂
prime radical	C₂³×C₆₁	C₂²
hypercenter	C₂²×C₄	D₆₁
hypocenter	C₆₁	C₂²⋊Q₈
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₂₂	C₂	semidirect
4. C₄⋊Dic₆₁	C₂	semidirect
5. #11758196	C₂	semidirect
15. C₂²×C₆₁	Q₈	semidirect
25. C₆₁	C₂²⋊Q₈	semidirect
41. C₂²	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₆₁	2
C₂×Dic₁₂₂	2
C₂²⋊Dic₆₁	2
C₄⋊Dic₆₁	2
#11758196	2
C₂²⋊Q₈	61