Subgroups — #28334667

Order 13696.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20292457	C₂	1	1	yes	maximal, characteristic
2	#20292455	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	#16812045	—	2	1	no	—
5	#16812043	—	2	1	no	—
6	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
7	C₃₂×C₁₀₇	C₄	1	2	yes	complemented
8	#13133802	—	4	1	no	—
9	C₄⋊Dic₁₀₇	—	4	1	no	—
10	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
11	C₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₀₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
14	C₂×Dic₁₀₇	—	8	1	no	—
15	C₈×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
16	C₈×C₁₀₇	Q₁₆	1	1	yes	characteristic
17	C₁₀₇⋊C₈	—	8	1	no	—
18	C₂²×C₁₀₇	SD₃₂	1	1	yes	characteristic
19	C₄×C₁₀₇	#107	1	1	yes	characteristic
20	C₄×C₁₀₇	SD₃₂	1	1	yes	characteristic
21	Dic₁₀₇	—	16	1	no	—
22	C₂×C₁₀₇	—	2	1	no	—
23	C₂×C₁₀₇	#367	1	1	yes	characteristic
24	#1426	—	107	1	no	maximal, Sylow
25	C₁₀₇	#1426	1	1	yes	characteristic, complemented, Sylow
26	M₆₄	D₁₀₇	1	1	yes	characteristic
27	#368	—	107	1	no	—
28	#367	—	107	1	no	—
29	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
30	#108	—	214	1	no	—
31	#107	—	214	1	no	—
32	C₃₂	Dic₁₀₇	1	2	yes	complemented
33	M₁₆	—	428	1	no	—
34	C₂×C₈	D₄₂₈	1	1	yes	characteristic
35	C₄⋊C₄	—	428	1	no	—
36	C₁₆	Dic₂₁₄	1	1	yes	characteristic
37	C₁₆	C₂×Dic₁₀₇	1	1	yes	characteristic
38	C₂×C₄	—	856	1	no	—
39	C₂×C₄	D₈₅₆	1	1	yes	characteristic
40	C₈	—	856	1	no	—
41	C₈	C₄⋊Dic₁₀₇	1	1	yes	characteristic
42	C₈	Dic₄₂₈	1	1	yes	characteristic
43	C₂²	#16812054	1	1	yes	characteristic
44	C₄	—	1712	1	no	—
45	C₄	#16812043	1	1	yes	characteristic
46	C₄	#16812054	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#20292455	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20292455
commutator	C₁₆×C₁₀₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₇
Fitting	M₆₄×C₁₀₇	C₂
socle	C₂×C₁₀₇	#367
prime radical	C₂²×C₁₀₇	SD₃₂
cosocle	C₂×C₁₆×C₁₀₇	C₂²
hypercenter	M₆₄	D₁₀₇
hypocenter	C₁₀₇	#1426
LCS	G ▷ C₁₆×C₁₀₇ ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₁₀₇	C₄	semidirect
25. C₁₀₇	#1426	semidirect
32. C₃₂	Dic₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20292457	2
#20292455	2
M₆₄×C₁₀₇	2
#1426	107