Subgroups — #28334521

Order 13696.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293029	C₂	1	1	yes	maximal, characteristic
2	#20292448	C₂	1	1	yes	maximal, characteristic
3	#364×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	#16812101	C₄	1	2	yes	—
5	#16812049	—	2	1	no	—
6	#16812043	—	2	1	no	—
7	#16812041	—	2	1	no	—
8	M₃₂×C₁₀₇	—	2	1	no	—
9	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₀₇	—	2	2	no	—
11	Dic₁₀₇⋊C₄	—	4	2	no	—
12	C₁₀₇⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
15	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₀₇	—	4	1	no	—
17	C₂×C₄×C₁₀₇	—	2	1	no	—
18	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×Dic₁₀₇	—	4	2	no	—
20	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
21	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
22	C₈×C₁₀₇	Q₁₆	1	1	yes	characteristic
23	C₈×C₁₀₇	Q₁₆	1	1	yes	characteristic
24	C₂²×C₁₀₇	#100	1	1	yes	characteristic
25	C₄×C₁₀₇	—	4	1	no	—
26	C₄×C₁₀₇	#103	1	1	yes	characteristic
27	C₄×C₁₀₇	#103	1	1	yes	characteristic
28	Dic₁₀₇	—	8	2	no	—
29	C₂×C₁₀₇	—	2	1	no	—
30	C₂×C₁₀₇	#333	1	1	yes	characteristic
31	#1366	—	107	1	no	maximal, Sylow
32	C₁₀₇	#1366	1	1	yes	characteristic, complemented, Sylow
33	#499	—	107	1	no	—
34	#364	—	107	1	no	—
35	#364	D₁₀₇	1	1	yes	characteristic
36	#125	—	107	2	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	214	1	no	—
39	#107	—	214	1	no	—
40	#106	—	214	1	no	—
41	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
42	C₂×C₈	D₄₂₈	1	1	yes	characteristic
43	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
44	C₄⋊C₄	—	428	2	no	—
45	C₄⋊C₄	—	214	2	no	—
46	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
47	C₁₆	—	4	1	no	—
48	C₁₆	—	428	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	428	2	no	—
51	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
52	C₈	#13133806	1	1	yes	characteristic
53	C₈	#13133807	1	1	yes	characteristic
54	C₈	#13133808	1	1	yes	characteristic
55	C₈	Dic₄₂₈	1	1	yes	characteristic
56	C₂²	#16812025	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	856	2	no	—
59	C₄	#16812032	1	1	yes	characteristic
60	C₄	#16812034	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#20292375	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20292375
commutator	C₄²×C₁₀₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₇
Fitting	#364×C₁₀₇	C₂
socle	C₂×C₁₀₇	#333
prime radical	C₂²×C₁₀₇	#100
cosocle	C₄×C₈×C₁₀₇	C₂²
hypercenter	#364	D₁₀₇
hypocenter	C₁₀₇	#1366
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
32. C₁₀₇	#1366	semidirect

Maximal subgroups

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

subgroup	index
#20293029	2
#20292448	2
#364×C₁₀₇	2
#1366	107