Subgroups — #30595306

Order 15616.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840601	C₂	1	2	yes	maximal
2	C₃₂×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#16840601	—	2	2	no	—
4	C₂×#14519683	C₂²	1	1	yes	characteristic
5	C₁₆×Dic₆₁	C₄	1	1	yes	characteristic
6	C₂×C₃₂×C₆₁	C₄	1	1	yes	characteristic
7	C₂×#11758185	C₈	1	1	yes	characteristic
8	C₈×Dic₆₁	C₈	1	1	yes	characteristic
9	#14519683	D₄	1	1	yes	characteristic
10	#14519683	Q₈	1	1	yes	characteristic
11	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₆₁	—	2	1	no	—
13	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
14	C₄×Dic₆₁	C₁₆	1	1	yes	characteristic
15	C₂×#92418	C₁₆	1	1	yes	characteristic
16	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₆₁	M₁₆	1	1	yes	characteristic
18	#11758185	—	2	1	no	—
19	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
20	C₂×Dic₆₁	C₃₂	1	1	yes	characteristic
21	C₂×Dic₆₁	C₃₂	1	1	yes	characteristic
22	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₆₁	M₃₂	1	1	yes	characteristic
24	#92418	—	2	1	no	—
25	C₄⋊C₆₄	—	61	1	no	maximal, Sylow
26	C₂²×C₆₁	C₂×C₃₂	1	1	yes	characteristic
27	C₄×C₆₁	C₄⋊C₁₆	1	1	yes	characteristic
28	C₄×C₆₁	M₆₄	1	1	yes	characteristic
29	Dic₆₁	—	2	1	no	—
30	Dic₆₁	C₆₄	1	2	yes	complemented
31	C₂×C₆₄	—	61	2	no	—
32	C₄×C₃₂	—	61	1	no	—
33	C₂×C₆₁	C₄⋊C₃₂	1	1	yes	characteristic
34	C₂×C₆₁	C₂×C₆₄	1	1	yes	characteristic
35	C₂×C₆₁	M₁₂₈	1	1	yes	characteristic
36	C₂×C₃₂	—	61	1	no	—
37	C₂×C₃₂	#6923	1	1	yes	characteristic
38	C₄×C₁₆	—	61	1	no	—
39	C₆₄	—	122	2	no	—
40	C₆₁	C₄⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₁₆	—	61	1	no	—
42	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
43	C₄×C₈	—	61	1	no	—
44	C₃₂	—	2	1	no	—
45	C₃₂	—	61	1	no	—
46	C₃₂	—	61	1	no	—
47	C₂×C₈	—	61	1	no	—
48	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
49	C₄²	—	61	1	no	—
50	C₁₆	—	122	1	no	—
51	C₁₆	C₄⋊#6923	1	1	yes	characteristic
52	C₁₆	#11758217	1	1	yes	characteristic
53	C₂×C₄	—	61	1	no	—
54	C₂×C₄	—	61	1	no	—
55	C₂×C₄	D₆₁⋊C₁₆	1	1	yes	characteristic
56	C₈	—	122	1	no	—
57	C₈	#14519844	1	1	yes	characteristic
58	C₈	#14519852	1	1	yes	characteristic
59	C₂²	D₆₁⋊C₃₂	1	1	yes	characteristic
60	C₄	—	61	2	no	—
61	C₄	—	122	1	no	—
62	C₄	#16840683	1	1	yes	characteristic
63	C₄	#16840695	1	1	yes	characteristic
64	C₂	#21460977	1	1	yes	characteristic
65	C₂	D₆₁⋊C₆₄	1	1	yes	characteristic
66	C₂	#21460989	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×#6923
commutator	C₂×C₆₁	C₂×C₆₄
Fitting	C₂×C₃₂×C₆₁	C₄
socle, prime radical	C₂²×C₆₁	C₂×C₃₂
cosocle	C₂×#14519683	C₂²
hypercenter	C₂×C₃₂	#6923
hypocenter	C₆₁	C₄⋊C₆₄
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
30. Dic₆₁	C₆₄	semidirect
40. C₆₁	C₄⋊C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840601	2
C₃₂×Dic₆₁	2
C₄⋊C₆₄	61