Subgroups — #20292357

Order 6848.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₂₁₄	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×Dic₂₁₄	C₄	1	1	yes	characteristic
5	C₂×C₁₀₇⋊C₈	—	2	1	no	—
6	C₄⋊Dic₁₀₇	C₄	1	1	yes	characteristic
7	Dic₁₀₇⋊C₄	—	2	1	no	—
8	C₄×Dic₁₀₇	—	2	1	no	—
9	C₂×C₈×C₁₀₇	—	2	1	no	—
10	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
13	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
14	C₂×Dic₁₀₇	—	2	1	no	—
15	C₂×Dic₁₀₇	—	2	1	no	—
16	Dic₂₁₄	—	2	1	no	—
17	Dic₂₁₄	C₈	1	2	yes	complemented
18	C₈×C₁₀₇	—	4	1	no	—
19	C₁₀₇⋊C₈	—	2	2	no	—
20	C₂²×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₀₇	—	2	1	no	—
22	C₄×C₁₀₇	C₂×C₈	1	1	yes	characteristic
23	C₄×C₁₀₇	M₁₆	1	1	yes	characteristic
24	C₄×C₁₀₇	SD₁₆	1	1	yes	characteristic
25	C₄×C₁₀₇	Q₁₆	1	1	yes	characteristic
26	Dic₁₀₇	—	2	2	no	—
27	Dic₁₀₇	—	4	1	no	—
28	C₂×C₁₀₇	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₁₀₇	#103	1	1	yes	characteristic
30	C₂×C₁₀₇	#104	1	1	yes	characteristic
31	C₁₀₇	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	107	1	no	maximal, Sylow
33	C₄×Q₈	—	107	1	no	—
34	C₄⋊C₈	D₁₀₇	1	1	yes	characteristic
35	C₄×C₈	—	107	1	no	—
36	C₂×Q₈	—	107	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	214	1	no	—
39	C₄⋊C₄	—	107	1	no	—
40	C₄⋊C₄	—	214	1	no	—
41	C₄²	—	214	1	no	—
42	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
43	Q₈	—	107	2	no	—
44	Q₈	—	214	1	no	—
45	C₂×C₄	—	214	1	no	—
46	C₂×C₄	—	214	1	no	—
47	C₂×C₄	C₄×D₁₀₇	1	1	yes	characteristic
48	C₂×C₄	D₄₂₈	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	214	2	no	—
52	C₂²	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	428	1	no	—
55	C₄	—	214	2	no	—
56	C₄	C₈×D₁₀₇	1	1	yes	characteristic
57	C₄	#13133801	1	1	yes	characteristic
58	C₄	#13133806	1	1	yes	characteristic
59	C₄	#13133808	1	1	yes	characteristic
60	C₂	(C₂×D₁₀₇)⋊C₈	1	1	yes	characteristic
61	C₂	#16812032	1	1	yes	characteristic
62	C₂	#16812035	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	C₂²⋊C₄
hypercenter	C₄⋊C₈	D₁₀₇
hypocenter	C₁₀₇	Q₈⋊C₈
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
17. Dic₂₁₄	C₈	semidirect
31. C₁₀₇	Q₈⋊C₈	semidirect