Subgroups — #16840122

Order 3904.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519777	C₂	1	1	yes	maximal, characteristic, complemented
2	#14519765	C₂	1	1	yes	maximal, characteristic
3	#14519702	C₂	1	2	yes	maximal, complemented
4	#14519699	C₂	1	2	yes	maximal
5	#97×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₄₄	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₂₂	—	2	1	no	—
8	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₆₁	C₂²	1	2	yes	—
10	C₄²×C₆₁	C₂²	1	1	yes	characteristic
11	#11758198	—	2	1	no	—
12	#11758198	—	2	1	no	—
13	C₄⋊Dic₆₁	—	2	1	no	—
14	C₄⋊Dic₆₁	C₂²	1	2	yes	—
15	C₂²×D₆₁	—	2	1	no	—
16	C₂×C₄×C₆₁	D₄	1	1	yes	characteristic
17	C₂×C₄×C₆₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
19	C₂×Dic₆₁	—	2	2	no	—
20	C₂×Dic₆₁	—	2	1	no	—
21	D₂₄₄	—	2	2	no	—
22	Dic₁₂₂	—	2	2	no	—
23	Dic₁₂₂	—	4	2	no	—
24	C₈×C₆₁	—	2	2	no	—
25	C₂²×C₆₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₆₁	—	4	1	no	—
27	C₂×D₆₁	—	4	1	no	—
28	C₂×D₆₁	—	4	1	no	—
29	C₄×C₆₁	—	2	1	no	—
30	C₄×C₆₁	—	2	1	no	—
31	C₄×C₆₁	#41	1	2	yes	—
32	Dic₆₁	—	4	1	no	—
33	Dic₆₁	—	4	2	no	—
34	C₂×C₆₁	#124	1	1	yes	characteristic
35	C₂×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₆₁	#137	1	1	yes	characteristic
37	D₆₁	—	8	1	no	—
38	#489	—	61	1	no	maximal, Sylow
39	C₆₁	#489	1	1	yes	characteristic, complemented, Sylow
40	C₄⋊Q₈	—	61	1	no	—
41	#124	—	61	1	no	—
42	#103	—	61	2	no	—
43	#102	—	61	2	no	—
44	#97	D₆₁	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	61	1	no	—
46	C₂×Q₈	—	122	1	no	—
47	C₂×D₄	—	61	1	no	—
48	C₂×C₈	C₂×D₆₁	1	2	yes	—
49	C₄⋊C₄	—	61	2	no	—
50	C₄⋊C₄	—	122	1	no	—
51	C₂²⋊C₄	—	122	1	no	—
52	C₂²⋊C₄	—	122	1	no	—
53	C₄²	C₂×D₆₁	1	1	yes	characteristic
54	C₂³	—	122	1	no	—
55	Q₈	—	122	2	no	—
56	Q₈	—	244	2	no	—
57	D₄	—	122	2	no	—
58	C₂×C₄	—	122	1	no	—
59	C₂×C₄	—	122	2	no	—
60	C₂×C₄	D₂₄₄	1	1	yes	characteristic
61	C₂×C₄	D₂₄₄	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₂²	—	244	1	no	—
65	C₂²	—	244	1	no	—
66	C₂²	—	244	1	no	—
67	C₂²	C₂×D₂₄₄	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	244	1	no	—
71	C₄	—	244	2	no	—
72	C₄	#11758222	1	2	yes	—
73	C₂	—	488	1	no	—
74	C₂	#14519765	1	1	yes	characteristic
75	C₂	#14519805	1	1	yes	characteristic
76	C₂	#14519810	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₄₄
commutator, cosocle	C₂×C₄×C₆₁	C₂³
Frattini	C₂×C₄	C₂²×D₆₁
Fitting	#97×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₄
prime radical	C₂×D₂₄₄	C₂²
hypercenter	#97	D₆₁
hypocenter	C₆₁	#489
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. #14519777	C₂	semidirect
3. #14519702	C₂	semidirect
5. #97×C₆₁	C₂	semidirect
39. C₆₁	#489	semidirect
44. #97	D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#14519777	2
#14519765	2
#14519702	2
#14519699	2
#97×C₆₁	2
#489	61