Subgroups — #16840488

Order 3904.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519776	C₂	1	3	yes	maximal
2	#14519769	C₂	1	3	yes	maximal
3	#14519768	C₂	1	3	yes	maximal
4	Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₂₂	C₂	1	3	yes	maximal
7	C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₂₂	C₂²	1	3	yes	—
10	C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
11	C₄²×C₆₁	C₂²	1	3	yes	—
12	C₄⋊Dic₆₁	C₂²	1	3	yes	—
13	C₄⋊Dic₆₁	C₂²	1	3	yes	—
14	C₄⋊Dic₆₁	C₂²	1	3	yes	—
15	#11758196	C₂²	1	1	yes	characteristic
16	#11758196	C₂²	1	3	yes	—
17	#11758196	C₂²	1	6	yes	—
18	C₄×Dic₆₁	C₂²	1	3	yes	—
19	C₄×Dic₆₁	C₂²	1	3	yes	—
20	Q₈×C₆₁	Q₈	1	4	yes	complemented
21	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₆₁	C₂³	1	3	yes	—
23	C₂×C₄×C₆₁	C₂³	1	3	yes	—
24	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₆₁	C₂³	1	3	yes	—
26	C₂×Dic₆₁	C₂³	1	3	yes	—
27	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
28	Dic₁₂₂	—	2	6	no	—
29	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
30	C₄×C₆₁	—	2	3	no	—
31	C₄×C₆₁	—	2	1	no	—
32	C₄×C₆₁	C₂×Q₈	1	6	yes	—
33	Dic₆₁	—	2	3	no	—
34	Dic₆₁	—	2	1	no	—
35	Dic₆₁	—	2	3	no	—
36	Dic₆₁	#41	1	2	yes	—
37	C₂×C₆₁	C₂²×Q₈	1	1	yes	characteristic
38	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
39	C₂×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
40	#557	—	61	1	no	maximal, Sylow
41	C₆₁	#557	1	1	yes	characteristic, complemented, Sylow
42	C₄⋊Q₈	—	61	3	no	—
43	#125	—	61	3	no	—
44	#125	—	61	3	no	—
45	C₄×Q₈	—	61	1	no	—
46	C₄×Q₈	—	61	1	no	—
47	C₄×Q₈	—	61	3	no	—
48	C₄×Q₈	D₆₁	1	1	yes	characteristic
49	C₂×Q₈	—	61	3	no	—
50	C₂×Q₈	C₂×D₆₁	1	1	yes	characteristic
51	C₄⋊C₄	—	61	1	no	—
52	C₄⋊C₄	—	61	3	no	—
53	C₄⋊C₄	—	61	6	no	—
54	C₄⋊C₄	—	61	3	no	—
55	C₄⋊C₄	—	61	3	no	—
56	C₄⋊C₄	—	61	3	no	—
57	C₄⋊C₄	C₂×D₆₁	1	3	yes	—
58	C₄²	—	61	3	no	—
59	C₄²	—	61	3	no	—
60	C₄²	C₂×D₆₁	1	3	yes	—
61	Q₈	—	122	6	no	—
62	Q₈	Dic₁₂₂	1	4	yes	complemented
63	C₂×C₄	—	61	1	no	—
64	C₂×C₄	—	61	1	no	—
65	C₂×C₄	—	61	3	no	—
66	C₂×C₄	—	61	3	no	—
67	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₆₁	1	3	yes	—
69	C₂×C₄	C₂²×D₆₁	1	3	yes	—
70	C₂²	C₂³×D₆₁	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	2	1	no	—
73	C₄	—	61	2	no	—
74	C₄	—	122	3	no	—
75	C₄	—	122	1	no	—
76	C₄	—	122	3	no	—
77	C₄	C₂×Dic₁₂₂	1	6	yes	—
78	C₂	C₂²×Dic₁₂₂	1	1	yes	characteristic
79	C₂	#41×D₆₁	1	1	yes	characteristic
80	C₂	#14519825	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, prime radical, cosocle	C₂²×C₆₁	C₂⁴
Fitting	C₄×Q₈×C₆₁	C₂
hypercenter	C₄×Q₈	D₆₁
hypocenter	C₆₁	#557
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
20. Q₈×C₆₁	Q₈	semidirect
41. C₆₁	#557	semidirect
62. Q₈	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
#14519769	2
#14519768	2
Q₈×Dic₆₁	2
C₄×Dic₁₂₂	2
C₄×Q₈×C₆₁	2
#557	61