Subgroups — #16840607

Order 3904.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#92420	C₂	1	2	yes	maximal
2	C₂×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₆₁	C₄	1	1	yes	characteristic
4	C₂²×Dic₆₁	C₄	1	1	yes	characteristic
5	C₂²×Dic₆₁	C₂²	1	1	yes	characteristic
6	C₂×#92420	—	2	2	no	—
7	C₄⋊Dic₆₁	—	2	2	no	—
8	C₂³×C₆₁	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₆₁	—	2	1	no	—
10	C₂×C₄×C₆₁	—	2	1	no	—
11	C₂×C₄×C₆₁	C₈	1	2	yes	complemented
12	C₂×Dic₆₁	—	2	1	no	—
13	C₂×Dic₆₁	—	4	1	no	—
14	C₂×Dic₆₁	—	4	1	no	—
15	C₂×Dic₆₁	D₄	1	2	yes	—
16	#92420	—	4	2	no	—
17	C₂²×C₆₁	—	2	1	no	—
18	C₂²×C₆₁	—	2	1	no	—
19	C₂²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₆₁	C₂×C₈	1	1	yes	characteristic
21	C₂²×C₆₁	M₁₆	1	1	yes	characteristic
22	C₄×C₆₁	—	2	2	no	—
23	Dic₆₁	—	2	2	no	—
24	Dic₆₁	—	4	1	no	—
25	C₂×C₆₁	—	2	1	no	—
26	C₂×C₆₁	—	2	1	no	—
27	C₂×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
28	C₂×C₆₁	#99	1	1	yes	characteristic
29	C₂×C₆₁	#101	1	1	yes	characteristic
30	#324	—	61	1	no	maximal, Sylow
31	C₆₁	#324	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₄⋊C₄	—	61	1	no	—
33	C₂²⋊C₈	—	61	2	no	—
34	C₂²×C₄	—	61	1	no	—
35	C₂²×C₄	—	61	1	no	—
36	C₂²×C₄	#6923	1	1	yes	characteristic
37	C₂×C₈	—	122	2	no	—
38	C₄⋊C₄	—	122	2	no	—
39	C₂³	C₂×#6923	1	1	yes	characteristic
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	61	2	no	—
43	C₂×C₄	—	122	1	no	—
44	C₂×C₄	—	244	1	no	—
45	C₂×C₄	—	244	1	no	—
46	C₂×C₄	#92420	1	2	yes	complemented
47	C₈	—	244	2	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₂×#92420	1	1	yes	characteristic
51	C₂²	#11758217	1	1	yes	characteristic
52	C₂²	C₂²⋊#6923	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	122	2	no	—
55	C₄	—	244	1	no	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₂²⋊#92420	1	1	yes	characteristic
59	C₂	#14519836	1	1	yes	characteristic
60	C₂	#14519838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#6923
commutator	C₂²×C₆₁	C₂×C₈
Frattini	C₂³	C₂×#6923
Fitting	C₂²×C₄×C₆₁	C₄
socle	C₂²×C₆₁	C₂²⋊C₄
prime radical	C₂³×C₆₁	C₂×C₄
cosocle	C₂²×Dic₆₁	C₂²
hypercenter	C₂²×C₄	#6923
hypocenter	C₆₁	#324
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
11. C₂×C₄×C₆₁	C₈	semidirect
31. C₆₁	#324	semidirect
46. C₂×C₄	#92420	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#92420	2
C₂×C₄⋊Dic₆₁	2
#324	61