Subgroups — #20292665

Order 6848.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812068	C₂	1	2	yes	maximal
2	#16812043	C₂	1	2	yes	maximal, complemented
3	#16812041	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₁₀₇	C₂²	1	4	yes	—
6	C₂²⋊Dic₁₀₇	—	2	2	no	—
7	C₄×Dic₁₀₇	—	2	2	no	—
8	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₁₀₇	C₄	1	4	yes	complemented
10	C₂×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂³×C₁₀₇	D₄	1	1	yes	characteristic
13	C₂×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₁₀₇	Q₈	1	2	yes	—
16	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
17	C₂×Dic₁₀₇	—	2	4	no	—
18	C₈×C₁₀₇	C₂×C₄	1	2	yes	—
19	C₈×C₁₀₇	C₂×C₄	1	2	yes	—
20	C₂²×C₁₀₇	—	2	2	no	—
21	C₂²×C₁₀₇	C₄⋊C₄	1	2	yes	—
22	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
23	C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
24	C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
26	Dic₁₀₇	—	4	4	no	—
27	C₂×C₁₀₇	—	2	2	no	—
28	C₂×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₁₀₇	#135	1	2	yes	—
30	C₁₀₇	#427	1	1	yes	characteristic, complemented, Sylow
31	#427	—	107	1	no	maximal, Sylow
32	C₂²×C₈	D₁₀₇	1	1	yes	characteristic
33	#117	—	107	2	no	—
34	#107	—	107	2	no	—
35	#106	—	107	2	no	—
36	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
37	C₂×C₈	Dic₁₀₇	1	4	yes	complemented
38	C₂×C₈	C₂×D₁₀₇	1	1	yes	characteristic
39	C₂×C₈	C₂×D₁₀₇	1	1	yes	characteristic
40	C₄⋊C₄	—	107	4	no	—
41	C₂²⋊C₄	—	214	2	no	—
42	C₄²	—	214	2	no	—
43	C₂³	D₄₂₈	1	1	yes	characteristic
44	C₂×C₄	—	214	4	no	—
45	C₂×C₄	Dic₂₁₄	1	2	yes	—
46	C₂×C₄	D₄₂₈	1	1	yes	characteristic
47	C₂×C₄	C₂×Dic₁₀₇	1	2	yes	—
48	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
49	C₈	C₂×Dic₁₀₇	1	2	yes	—
50	C₈	C₂×Dic₁₀₇	1	2	yes	—
51	C₂²	—	2	2	no	—
52	C₂²	C₄⋊Dic₁₀₇	1	2	yes	—
53	C₂²	C₂×D₄₂₈	1	1	yes	characteristic
54	C₄	—	428	4	no	—
55	C₄	C₄⋊Dic₁₀₇	1	2	yes	—
56	C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
57	C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
60	C₂	#16812131	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄₂₈
commutator	C₄×C₁₀₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	C₂²×C₈×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₄
hypercenter	C₂²×C₈	D₁₀₇
hypocenter	C₁₀₇	#427
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #16812043	C₂	semidirect
3. #16812041	C₂	semidirect
9. C₂×C₈×C₁₀₇	C₄	semidirect
30. C₁₀₇	#427	semidirect
37. C₂×C₈	Dic₁₀₇	semidirect