Subgroups — #16840579

Order 3904.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519824	C₂	1	2	yes	maximal, complemented
2	#14519810	C₂	1	6	yes	maximal, complemented
3	#14519799	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₂₄₄	C₂	1	3	yes	maximal, complemented
5	#131×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₆₁	C₂²	1	1	yes	characteristic
7	Q₈×D₆₁	—	2	2	no	—
8	#11758224	—	2	6	no	—
9	#11758222	C₂²	1	6	yes	—
10	C₂×Dic₁₂₂	C₂²	1	6	yes	—
11	M₁₆×C₆₁	C₂²	1	3	yes	—
12	C₂×C₈×C₆₁	C₂²	1	3	yes	—
13	Dic₂₄₄	C₂²	1	6	yes	—
14	Dic₂₄₄	C₂²	1	3	yes	—
15	D₄₈₈	C₂²	1	1	yes	characteristic
16	#11758190	C₂²	1	6	yes	—
17	Q₈×C₆₁	D₄	1	1	yes	characteristic
18	D₄×C₆₁	D₄	1	3	yes	—
19	C₂×C₄×C₆₁	C₂³	1	3	yes	—
20	C₂²⋊D₆₁	—	2	6	no	—
21	C₂×Dic₆₁	—	2	6	no	—
22	D₂₄₄	C₂³	1	2	yes	—
23	C₄×D₆₁	—	2	6	no	—
24	Dic₁₂₂	—	2	6	no	—
25	Dic₁₂₂	C₂³	1	6	yes	—
26	C₈×C₆₁	C₂³	1	3	yes	—
27	C₈×C₆₁	C₂³	1	1	yes	characteristic
28	C₂²×C₆₁	C₂×D₄	1	3	yes	—
29	C₂×D₆₁	—	2	2	no	—
30	C₄×C₆₁	C₂×D₄	1	3	yes	—
31	C₄×C₆₁	C₂⁴	1	1	yes	characteristic
32	Dic₆₁	—	2	6	no	—
33	C₂×C₆₁	—	2	3	no	—
34	C₂×C₆₁	C₂²×D₄	1	1	yes	characteristic
35	D₆₁	—	4	2	no	—
36	#578	—	61	1	no	maximal, Sylow
37	C₆₁	#578	1	1	yes	characteristic, complemented, Sylow
38	2₋¹⁺⁴	—	61	2	no	—
39	#137	—	61	6	no	—
40	#135	—	61	3	no	—
41	C₂×Q₁₆	—	61	3	no	—
42	#131	D₆₁	1	1	yes	characteristic, complemented
43	#41	—	61	6	no	—
44	#41	—	122	6	no	—
45	#41	C₂×D₆₁	1	1	yes	characteristic
46	C₂×Q₈	—	61	6	no	—
47	C₂×Q₈	—	122	2	no	—
48	Q₁₆	—	61	3	no	—
49	Q₁₆	—	61	6	no	—
50	SD₁₆	—	61	6	no	—
51	D₈	—	61	1	no	—
52	M₁₆	C₂×D₆₁	1	3	yes	—
53	C₂×C₈	C₂×D₆₁	1	3	yes	—
54	Q₈	—	61	6	no	—
55	Q₈	—	122	6	no	—
56	Q₈	D₂₄₄	1	1	yes	characteristic
57	D₄	—	61	2	no	—
58	D₄	—	122	6	no	—
59	D₄	D₂₄₄	1	3	yes	—
60	C₂×C₄	—	122	6	no	—
61	C₂×C₄	—	122	6	no	—
62	C₂×C₄	C₂²×D₆₁	1	3	yes	—
63	C₈	C₂²×D₆₁	1	1	yes	characteristic
64	C₈	C₂²×D₆₁	1	3	yes	—
65	C₂²	—	122	2	no	—
66	C₂²	C₂×D₂₄₄	1	3	yes	—
67	C₄	—	122	6	no	—
68	C₄	C₂×D₂₄₄	1	3	yes	—
69	C₄	C₂³×D₆₁	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	244	2	no	—
72	C₂	C₂²×D₂₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₄₄
commutator, cosocle	C₄×C₆₁	C₂⁴
Frattini	C₄	C₂³×D₆₁
Fitting	#131×C₆₁	C₂
socle	C₂×C₆₁	C₂²×D₄
hypercenter	#131	D₆₁
hypocenter	C₆₁	#578
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #14519824	C₂	semidirect
2. #14519810	C₂	semidirect
3. #14519799	C₂	semidirect
4. C₂×Dic₂₄₄	C₂	semidirect
5. #131×C₆₁	C₂	semidirect
37. C₆₁	#578	semidirect
42. #131	D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#14519824	2
#14519810	2
#14519799	2
C₂×Dic₂₄₄	2
#131×C₆₁	2
#578	61