Subgroups — #20293444

Order 6848.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812156	C₂	1	2	yes	maximal, complemented
2	#16812142	C₂	1	6	yes	maximal, complemented
3	#16812131	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₄₂₈	C₂	1	3	yes	maximal, complemented
5	#131×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133818	C₂²	1	6	yes	—
7	#13133817	—	2	6	no	—
8	C₂×Dic₂₁₄	C₂²	1	6	yes	—
9	Q₈×D₁₀₇	—	2	2	no	—
10	Dic₄₂₈	C₂²	1	3	yes	—
11	Dic₄₂₈	C₂²	1	6	yes	—
12	#13133807	C₂²	1	6	yes	—
13	D₈₅₆	C₂²	1	1	yes	characteristic
14	#41×C₁₀₇	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₀₇	C₂²	1	3	yes	—
16	C₂×C₈×C₁₀₇	C₂²	1	3	yes	—
17	Q₈×C₁₀₇	D₄	1	1	yes	characteristic
18	D₄×C₁₀₇	D₄	1	3	yes	—
19	C₂×C₄×C₁₀₇	C₂³	1	3	yes	—
20	C₂²⋊D₁₀₇	—	2	6	no	—
21	C₂×Dic₁₀₇	—	2	6	no	—
22	D₄₂₈	C₂³	1	2	yes	—
23	C₄×D₁₀₇	—	2	6	no	—
24	Dic₂₁₄	—	2	6	no	—
25	Dic₂₁₄	C₂³	1	6	yes	—
26	C₈×C₁₀₇	C₂³	1	1	yes	characteristic
27	C₈×C₁₀₇	C₂³	1	3	yes	—
28	C₂²×C₁₀₇	C₂×D₄	1	3	yes	—
29	C₂×D₁₀₇	—	2	2	no	—
30	C₄×C₁₀₇	C₂×D₄	1	3	yes	—
31	C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
32	Dic₁₀₇	—	2	6	no	—
33	C₂×C₁₀₇	—	2	3	no	—
34	C₂×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
35	D₁₀₇	—	4	2	no	—
36	C₁₀₇	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	107	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	107	2	no	—
39	#137	—	107	6	no	—
40	#135	—	107	3	no	—
41	C₂×Q₁₆	—	107	3	no	—
42	#131	D₁₀₇	1	1	yes	characteristic, complemented
43	#41	—	107	6	no	—
44	#41	—	214	6	no	—
45	#41	C₂×D₁₀₇	1	1	yes	characteristic
46	C₂×Q₈	—	107	6	no	—
47	C₂×Q₈	—	214	2	no	—
48	Q₁₆	—	107	3	no	—
49	Q₁₆	—	107	6	no	—
50	SD₁₆	—	107	6	no	—
51	D₈	—	107	1	no	—
52	M₁₆	C₂×D₁₀₇	1	3	yes	—
53	C₂×C₈	C₂×D₁₀₇	1	3	yes	—
54	Q₈	—	107	6	no	—
55	Q₈	—	214	6	no	—
56	Q₈	D₄₂₈	1	1	yes	characteristic
57	D₄	—	107	2	no	—
58	D₄	—	214	6	no	—
59	D₄	D₄₂₈	1	3	yes	—
60	C₂×C₄	—	214	6	no	—
61	C₂×C₄	—	214	6	no	—
62	C₂×C₄	C₂²×D₁₀₇	1	3	yes	—
63	C₈	C₂²×D₁₀₇	1	1	yes	characteristic
64	C₈	C₂²×D₁₀₇	1	3	yes	—
65	C₂²	—	214	2	no	—
66	C₂²	C₂×D₄₂₈	1	3	yes	—
67	C₄	—	214	6	no	—
68	C₄	C₂×D₄₂₈	1	3	yes	—
69	C₄	C₂³×D₁₀₇	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	428	2	no	—
72	C₂	C₂²×D₄₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄₂₈
commutator	C₄×C₁₀₇	C₂⁴
Frattini	C₄	C₂³×D₁₀₇
Fitting	#131×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₂²×D₄
hypercenter	#131	D₁₀₇
hypocenter	C₁₀₇	#578
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #16812156	C₂	semidirect
2. #16812142	C₂	semidirect
3. #16812131	C₂	semidirect
4. C₂×Dic₄₂₈	C₂	semidirect
5. #131×C₁₀₇	C₂	semidirect
36. C₁₀₇	#578	semidirect
42. #131	D₁₀₇	semidirect