Subgroups — #20292356

Order 6848.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄₂₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₂₈	C₄	1	1	yes	characteristic
5	C₂×C₄×D₁₀₇	—	2	1	no	—
6	C₂×C₁₀₇⋊C₈	—	2	1	no	—
7	C₄⋊Dic₁₀₇	C₄	1	1	yes	characteristic
8	(C₂×D₁₀₇)⋊C₄	—	2	1	no	—
9	C₂×C₈×C₁₀₇	—	2	1	no	—
10	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂²×D₁₀₇	—	2	1	no	—
12	C₂×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
14	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
15	C₂×Dic₁₀₇	—	2	1	no	—
16	D₄₂₈	—	2	1	no	—
17	D₄₂₈	C₈	1	2	yes	complemented
18	C₄×D₁₀₇	—	4	1	no	—
19	C₄×D₁₀₇	—	4	1	no	—
20	C₈×C₁₀₇	—	2	2	no	—
21	C₁₀₇⋊C₈	—	4	1	no	—
22	C₂²×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₁₀₇	—	2	2	no	—
24	C₂×D₁₀₇	—	4	1	no	—
25	C₂×D₁₀₇	—	4	1	no	—
26	C₄×C₁₀₇	—	2	1	no	—
27	C₄×C₁₀₇	C₂×C₈	1	1	yes	characteristic
28	C₄×C₁₀₇	M₁₆	1	1	yes	characteristic
29	C₄×C₁₀₇	D₈	1	1	yes	characteristic
30	C₄×C₁₀₇	SD₁₆	1	1	yes	characteristic
31	Dic₁₀₇	—	4	1	no	—
32	C₂×C₁₀₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₂×C₁₀₇	#102	1	1	yes	characteristic
34	C₂×C₁₀₇	#104	1	1	yes	characteristic
35	D₁₀₇	—	4	2	no	—
36	C₁₀₇	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
37	D₄⋊C₈	—	107	1	no	maximal, Sylow
38	C₄×D₄	—	107	1	no	—
39	C₄⋊C₈	—	107	1	no	—
40	C₄×C₈	D₁₀₇	1	1	yes	characteristic, complemented
41	C₂×D₄	—	107	1	no	—
42	C₂²×C₄	—	214	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	214	1	no	—
45	C₄⋊C₄	—	107	1	no	—
46	C₂²⋊C₄	—	214	1	no	—
47	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
48	C₂³	—	214	1	no	—
49	D₄	—	107	2	no	—
50	D₄	—	214	1	no	—
51	C₂×C₄	—	428	1	no	—
52	C₂×C₄	—	428	1	no	—
53	C₂×C₄	—	214	1	no	—
54	C₂×C₄	C₄×D₁₀₇	1	1	yes	characteristic
55	C₂×C₄	D₄₂₈	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₈	—	428	1	no	—
59	C₂²	—	428	1	no	—
60	C₂²	—	428	1	no	—
61	C₂²	—	214	2	no	—
62	C₂²	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	428	1	no	—
65	C₄	C₈×D₁₀₇	1	1	yes	characteristic
66	C₄	#13133801	1	1	yes	characteristic
67	C₄	D₈₅₆	1	1	yes	characteristic
68	C₄	#13133807	1	1	yes	characteristic
69	C₂	—	428	2	no	—
70	C₂	(C₂×D₁₀₇)⋊C₈	1	1	yes	characteristic
71	C₂	#16812031	1	1	yes	characteristic
72	C₂	#16812037	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄₂₈
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	D₄⋊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
2. C₄⋊(C₁₀₇⋊C₈)	C₂	semidirect
3. C₄×C₈×C₁₀₇	C₂	semidirect
17. D₄₂₈	C₈	semidirect
36. C₁₀₇	D₄⋊C₈	semidirect
40. C₄×C₈	D₁₀₇	semidirect