Subgroups — #20292377

Order 6848.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₀₇⋊C₈	—	2	1	no	—
5	C₂×C₁₀₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₁₀₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₀₇	—	2	1	no	—
8	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
11	C₂×C₄×C₁₀₇	Q₈	1	1	yes	characteristic
12	C₈×C₁₀₇	—	4	1	no	—
13	C₁₀₇⋊C₈	—	2	1	no	—
14	C₁₀₇⋊C₈	—	4	1	no	—
15	C₁₀₇⋊C₈	C₈	1	2	yes	complemented
16	C₂²×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₁₀₇	—	2	1	no	—
18	C₄×C₁₀₇	C₂×C₈	1	1	yes	characteristic
19	C₄×C₁₀₇	M₁₆	1	1	yes	characteristic
20	C₄×C₁₀₇	SD₁₆	1	1	yes	characteristic
21	C₄×C₁₀₇	SD₁₆	1	1	yes	characteristic
22	C₂×C₁₀₇	C₄⋊C₈	1	1	yes	characteristic
23	C₂×C₁₀₇	#106	1	1	yes	characteristic
24	C₂×C₁₀₇	#108	1	1	yes	characteristic
25	C₁₀₇	#334	1	1	yes	characteristic, complemented, Sylow
26	#334	—	107	1	no	maximal, Sylow
27	C₄⋊C₈	—	107	1	no	—
28	C₄⋊C₈	D₁₀₇	1	1	yes	characteristic
29	C₄×C₈	—	107	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	107	1	no	—
32	C₂×C₈	—	107	1	no	—
33	C₂×C₈	—	214	1	no	—
34	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
35	C₂×C₄	Dic₂₁₄	1	1	yes	characteristic
36	C₂×C₄	C₄×D₁₀₇	1	1	yes	characteristic
37	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
38	C₈	—	4	1	no	—
39	C₈	—	107	2	no	—
40	C₈	—	428	1	no	—
41	C₈	—	214	1	no	—
42	C₂²	Dic₁₀₇⋊C₄	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₁₀₇	1	1	yes	characteristic
45	C₄	#13133801	1	1	yes	characteristic
46	C₄	#13133805	1	1	yes	characteristic
47	C₄	#13133806	1	1	yes	characteristic
48	C₂	Dic₁₀₇⋊C₈	1	1	yes	characteristic
49	C₂	#16812040	1	1	yes	characteristic
50	C₂	#16812044	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₀₇
commutator	C₄×C₁₀₇	C₂×C₈
Frattini	C₄²	C₂×D₁₀₇
Fitting	C₄⋊C₈×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₄⋊C₄
hypercenter	C₄⋊C₈	D₁₀₇
hypocenter	C₁₀₇	#334
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
15. C₁₀₇⋊C₈	C₈	semidirect
25. C₁₀₇	#334	semidirect