Subgroups — #28334649

Order 13696.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293059	C₂	1	1	yes	maximal, characteristic
2	#20292455	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812131	—	2	1	no	—
5	Dic₈₅₆	C₄	1	1	yes	characteristic, complemented
6	#16812054	—	2	1	no	—
7	D₁₇₁₂	C₄	1	1	yes	characteristic, complemented
8	#16812043	—	2	1	no	—
9	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₀₇	—	2	1	no	—
11	#13133818	—	4	1	no	—
12	Dic₄₂₈	—	2	1	no	—
13	#13133807	—	4	1	no	—
14	D₈₅₆	—	2	1	no	—
15	C₄⋊Dic₁₀₇	—	4	1	no	—
16	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₀₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
20	C₂²⋊D₁₀₇	—	8	1	no	—
21	C₂×Dic₁₀₇	—	8	1	no	—
22	D₄₂₈	—	4	1	no	—
23	C₄×D₁₀₇	—	8	1	no	—
24	Dic₂₁₄	—	4	1	no	—
25	C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₀₇	D₁₆	1	1	yes	characteristic
28	C₂×D₁₀₇	—	8	1	no	—
29	C₄×C₁₀₇	#102	1	1	yes	characteristic
30	C₄×C₁₀₇	SD₃₂	1	1	yes	characteristic
31	Dic₁₀₇	—	8	1	no	—
32	Dic₁₀₇	—	16	1	no	—
33	C₂×C₁₀₇	—	2	1	no	—
34	C₂×C₁₀₇	#357	1	1	yes	characteristic
35	D₁₀₇	—	16	1	no	—
36	#1418	—	107	1	no	maximal, Sylow
37	C₁₀₇	#1418	1	1	yes	characteristic, complemented, Sylow
38	#508	—	107	1	no	—
39	M₆₄	D₁₀₇	1	1	yes	characteristic, complemented
40	#367	—	107	1	no	—
41	#135	—	214	1	no	—
42	Q₃₂	—	107	1	no	—
43	SD₃₂	—	214	1	no	—
44	D₁₆	—	107	1	no	—
45	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
46	#107	—	214	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	428	1	no	—
49	Q₁₆	—	214	1	no	—
50	SD₁₆	—	428	1	no	—
51	D₈	—	214	1	no	—
52	C₂×C₈	D₄₂₈	1	1	yes	characteristic
53	C₄⋊C₄	—	428	1	no	—
54	C₁₆	C₄×D₁₀₇	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₀₇	1	1	yes	characteristic
56	Q₈	—	428	1	no	—
57	D₄	—	856	1	no	—
58	D₄	—	428	1	no	—
59	C₂×C₄	—	856	1	no	—
60	C₂×C₄	—	856	1	no	—
61	C₂×C₄	D₈₅₆	1	1	yes	characteristic
62	C₈	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
63	C₈	#13133807	1	1	yes	characteristic
64	C₂²	—	856	1	no	—
65	C₂²	D₁₇₁₂	1	1	yes	characteristic
66	C₄	—	856	1	no	—
67	C₄	—	1712	1	no	—
68	C₄	#16812031	1	1	yes	characteristic
69	C₄	#16812054	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	1712	1	no	—
72	C₂	#20292429	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20292429
commutator	C₁₆×C₁₀₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₇
Fitting	M₆₄×C₁₀₇	C₂
socle	C₂×C₁₀₇	#357
prime radical	#20293059	C₂
cosocle	C₂×C₁₆×C₁₀₇	C₂²
hypercenter	M₆₄	D₁₀₇
hypocenter	C₁₀₇	#1418
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
2. #20292455	C₂	semidirect
3. M₆₄×C₁₀₇	C₂	semidirect
5. Dic₈₅₆	C₄	semidirect
7. D₁₇₁₂	C₄	semidirect
37. C₁₀₇	#1418	semidirect
39. M₆₄	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293059	2
#20292455	2
M₆₄×C₁₀₇	2
#1418	107