Subgroups — #20292745

Order 6848.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812131	C₂	1	1	yes	maximal, characteristic
2	#16812115	C₂	1	2	yes	maximal, complemented
3	#16812045	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812037	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133818	C₂²	1	2	yes	—
7	Dic₄₂₈	C₄	1	1	yes	characteristic, complemented
8	#13133807	C₄	1	2	yes	complemented
9	D₈₅₆	C₄	1	1	yes	characteristic, complemented
10	#13133802	C₂²	1	2	yes	—
11	#13133801	—	2	2	no	—
12	C₈×D₁₀₇	—	2	2	no	—
13	C₂×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₀₇	—	2	1	no	—
17	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
18	C₂²⋊D₁₀₇	—	2	2	no	—
19	D₄₂₈	C₂×C₄	1	2	yes	—
20	C₄×D₁₀₇	—	2	2	no	—
21	Dic₂₁₄	C₂×C₄	1	2	yes	—
22	C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
23	C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
24	C₈×C₁₀₇	D₄	1	1	yes	characteristic
25	C₈×C₁₀₇	D₄	1	1	yes	characteristic
26	C₁₀₇⋊C₈	—	2	2	no	—
27	C₂²×C₁₀₇	#41	1	1	yes	characteristic
28	C₂×D₁₀₇	—	2	2	no	—
29	C₄×C₁₀₇	—	2	2	no	—
30	C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
32	Dic₁₀₇	—	2	2	no	—
33	C₂×C₁₀₇	—	2	1	no	—
34	C₂×C₁₀₇	C₄×D₄	1	1	yes	characteristic
35	D₁₀₇	—	4	2	no	—
36	C₁₀₇	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	107	1	no	maximal, Sylow
38	#135	—	107	1	no	—
39	#131	—	107	2	no	—
40	#108	—	107	1	no	—
41	#104	—	107	2	no	—
42	C₄×C₈	D₁₀₇	1	1	yes	characteristic, complemented
43	#41	—	107	2	no	—
44	Q₁₆	—	107	1	no	—
45	SD₁₆	—	107	2	no	—
46	D₈	—	107	1	no	—
47	M₁₆	—	107	2	no	—
48	M₁₆	—	214	2	no	—
49	C₂×C₈	—	214	2	no	—
50	C₂×C₈	C₂×D₁₀₇	1	1	yes	characteristic
51	C₂×C₈	C₂×D₁₀₇	1	1	yes	characteristic
52	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
53	Q₈	—	107	2	no	—
54	D₄	—	107	2	no	—
55	D₄	—	214	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	214	2	no	—
58	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
59	C₈	—	214	2	no	—
60	C₈	C₄×D₁₀₇	1	1	yes	characteristic
61	C₈	C₄×D₁₀₇	1	1	yes	characteristic
62	C₈	D₄₂₈	1	1	yes	characteristic
63	C₈	D₄₂₈	1	1	yes	characteristic
64	C₂²	—	214	2	no	—
65	C₂²	#13133818	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	214	2	no	—
68	C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
69	C₄	C₂×D₄₂₈	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	428	2	no	—
72	C₂	C₄×D₄₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄₂₈
commutator	C₄×C₁₀₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	C₄×C₈×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₄×D₄
hypercenter	C₄×C₈	D₁₀₇
hypocenter	C₁₀₇	#443
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #16812115	C₂	semidirect
3. #16812045	C₂	semidirect
4. #16812037	C₂	semidirect
5. C₄×C₈×C₁₀₇	C₂	semidirect
7. Dic₄₂₈	C₄	semidirect
8. #13133807	C₄	semidirect
9. D₈₅₆	C₄	semidirect
36. C₁₀₇	#443	semidirect
42. C₄×C₈	D₁₀₇	semidirect