Subgroups — #16840168

Order 3904.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519769	C₂	1	1	yes	maximal, characteristic
2	#14519710	C₂	1	2	yes	maximal
3	#14519708	C₂	1	2	yes	maximal
4	#14519688	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₆₁	—	2	1	no	—
7	C₄⋊C₄×C₆₁	—	2	1	no	—
8	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
9	C₄²×C₆₁	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
12	#11758196	—	2	1	no	—
13	#11758196	—	2	1	no	—
14	C₂×#92418	C₂²	1	2	yes	—
15	Q₈×C₆₁	—	4	2	no	—
16	C₂×C₄×C₆₁	—	2	2	no	—
17	C₂×C₄×C₆₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₆₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂×Dic₆₁	—	2	1	no	—
21	C₂×Dic₆₁	—	2	1	no	—
22	#92418	Q₈	1	4	yes	complemented
23	C₂²×C₆₁	C₂×D₄	1	1	yes	characteristic
24	C₄×C₆₁	—	2	1	no	—
25	C₄×C₆₁	—	2	1	no	—
26	C₄×C₆₁	—	4	2	no	—
27	C₄×C₆₁	C₂×Q₈	1	2	yes	—
28	Dic₆₁	—	4	1	no	—
29	Dic₆₁	—	4	1	no	—
30	C₂×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₆₁	#137	1	1	yes	characteristic
33	#501	—	61	1	no	maximal, Sylow
34	C₆₁	#501	1	1	yes	characteristic, complemented, Sylow
35	C₄⋊Q₈	D₆₁	1	1	yes	characteristic
36	#125	—	61	1	no	—
37	#107	—	61	2	no	—
38	#106	—	61	2	no	—
39	#97	—	61	1	no	—
40	C₂×Q₈	—	2	1	no	—
41	C₂×C₈	—	61	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	61	1	no	—
44	C₄⋊C₄	—	61	1	no	—
45	C₄⋊C₄	—	122	1	no	—
46	C₄⋊C₄	—	122	1	no	—
47	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
48	C₄²	C₂×D₆₁	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	122	1	no	—
52	C₂×C₄	—	122	1	no	—
53	C₂×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
56	C₈	—	61	4	no	—
57	C₂²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	—	244	1	no	—
62	C₄	—	244	1	no	—
63	C₄	Q₈×D₆₁	1	2	yes	—
64	C₂	#14519775	1	1	yes	characteristic
65	C₂	#14519803	1	1	yes	characteristic
66	C₂	#14519808	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₆₁
commutator, cosocle	C₂×C₄×C₆₁	C₂³
Frattini	C₂×C₄	C₂²×D₆₁
Fitting	C₄⋊Q₈×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	C₂×D₄
hypercenter	C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#501
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
22. #92418	Q₈	semidirect
34. C₆₁	#501	semidirect

Maximal subgroups

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

subgroup	index
#14519769	2
#14519710	2
#14519708	2
#14519688	2
C₄⋊Q₈×C₆₁	2
#501	61