Subgroups — #16840830

Order 3904.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6923	C₂	1	8	yes	maximal, complemented
2	#14519856	C₂	1	6	yes	maximal
3	C₂×Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₆₁	C₄	1	1	yes	characteristic, complemented
5	Q₈×D₆₁	C₄	1	4	yes	complemented
6	Q₈×D₆₁	C₂²	1	4	yes	—
7	C₂×C₄×D₆₁	C₂²	1	3	yes	—
8	C₂×Dic₁₂₂	C₄	1	3	yes	complemented
9	C₂²⋊#6923	C₂²	1	4	yes	—
10	C₄⋊#6923	C₂²	1	12	yes	—
11	C₄×#6923	C₂²	1	12	yes	—
12	C₂²×D₆₁	C₂³	1	1	yes	characteristic
13	C₂×#6923	C₂³	1	8	yes	—
14	Q₈×C₆₁	C₂×C₄	1	4	yes	—
15	C₂×C₄×C₆₁	C₂×C₄	1	3	yes	—
16	C₂×Dic₆₁	C₂×C₄	1	3	yes	—
17	C₄×D₆₁	C₂×C₄	1	6	yes	—
18	C₄×D₆₁	C₂³	1	6	yes	—
19	Dic₁₂₂	C₂×C₄	1	12	yes	—
20	C₂²×C₆₁	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₆₁	—	2	2	no	—
22	C₂×D₆₁	C₂²×C₄	1	1	yes	characteristic
23	C₂×D₆₁	C₂⁴	1	1	yes	characteristic
24	#6923	—	2	8	no	—
25	C₄×C₆₁	C₂²×C₄	1	6	yes	—
26	Dic₆₁	C₂²×C₄	1	6	yes	—
27	C₂×C₆₁	—	2	1	no	—
28	C₂×C₆₁	C₂³×C₄	1	1	yes	characteristic
29	D₆₁	—	2	1	no	—
30	D₆₁	2₋¹⁺⁴	1	2	yes	—
31	#519	—	61	1	no	maximal, Sylow
32	C₆₁	#519	1	1	yes	characteristic, complemented, Sylow
33	C₂²×Q₈	—	61	1	no	—
34	C₄×Q₈	—	61	8	no	—
35	#117	—	61	6	no	—
36	C₂×Q₈	—	61	4	no	—
37	C₂×Q₈	—	61	4	no	—
38	C₂×Q₈	—	61	3	no	—
39	C₂×Q₈	#6923	1	1	yes	characteristic, complemented
40	C₂²×C₄	—	61	3	no	—
41	C₄⋊C₄	—	61	12	no	—
42	C₂²⋊C₄	—	61	4	no	—
43	C₄²	—	61	12	no	—
44	C₂³	—	61	1	no	—
45	Q₈	—	61	12	no	—
46	Q₈	C₂×#6923	1	4	yes	—
47	C₂×C₄	—	61	3	no	—
48	C₂×C₄	—	61	6	no	—
49	C₂×C₄	—	61	6	no	—
50	C₂×C₄	—	61	8	no	—
51	C₂×C₄	C₂×#6923	1	3	yes	—
52	C₂²	—	61	1	no	—
53	C₂²	—	61	1	no	—
54	C₂²	—	122	2	no	—
55	C₂²	C₂²×#6923	1	1	yes	characteristic
56	C₄	—	61	6	no	—
57	C₄	—	122	8	no	—
58	C₄	C₂²×#6923	1	6	yes	—
59	C₂	—	2	1	no	—
60	C₂	—	61	2	no	—
61	C₂	—	122	1	no	—
62	C₂	C₂³×#6923	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#6923
commutator, socle	C₂×C₆₁	C₂³×C₄
Fitting	C₂×Q₈×C₆₁	C₄
prime radical	C₂²×D₆₁	C₂³
cosocle	C₂×D₆₁	C₂⁴
hypercenter	C₂×Q₈	#6923
hypocenter	C₆₁	#519
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#6923	C₂	semidirect
4. C₂×Q₈×C₆₁	C₄	semidirect
5. Q₈×D₆₁	C₄	semidirect
8. C₂×Dic₁₂₂	C₄	semidirect
32. C₆₁	#519	semidirect
39. C₂×Q₈	#6923	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6923	2
#14519856	2
C₂×Q₈×D₆₁	2
#519	61