Subgroups — #20292809

Order 6848.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812157	C₂	1	1	yes	maximal, characteristic
2	#16812140	C₂	1	2	yes	maximal, complemented
3	#16812037	C₂	1	2	yes	maximal, complemented
4	#16812028	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133819	—	2	2	no	—
7	#13133818	—	2	2	no	—
8	#13133818	C₂²	1	2	yes	—
9	Q₈×D₁₀₇	—	2	2	no	—
10	#13133808	—	2	2	no	—
11	#13133805	—	2	2	no	—
12	#13133802	C₂²	1	2	yes	—
13	C₂×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₁₀₇	—	2	2	no	—
16	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
17	Q₈×C₁₀₇	—	2	2	no	—
18	Q₈×C₁₀₇	—	2	2	no	—
19	C₂×C₄×C₁₀₇	—	2	1	no	—
20	C₂×C₄×C₁₀₇	—	2	1	no	—
21	C₂×C₄×C₁₀₇	D₄	1	2	yes	—
22	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
23	C₂²⋊D₁₀₇	—	2	2	no	—
24	D₄₂₈	—	2	2	no	—
25	D₄₂₈	D₄	1	2	yes	—
26	C₄×D₁₀₇	—	2	4	no	—
27	C₄×D₁₀₇	—	2	2	no	—
28	Dic₂₁₄	—	2	2	no	—
29	Dic₂₁₄	D₄	1	2	yes	—
30	C₁₀₇⋊C₈	—	2	2	no	—
31	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
32	C₂×D₁₀₇	—	2	2	no	—
33	C₄×C₁₀₇	—	2	2	no	—
34	C₄×C₁₀₇	—	2	2	no	—
35	C₄×C₁₀₇	—	4	1	no	—
36	C₄×C₁₀₇	C₂×D₄	1	2	yes	—
37	Dic₁₀₇	—	2	2	no	—
38	C₂×C₁₀₇	—	2	1	no	—
39	C₂×C₁₀₇	#120	1	1	yes	characteristic
40	D₁₀₇	—	4	2	no	—
41	C₁₀₇	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	107	1	no	maximal, Sylow
43	2₋¹⁺⁴	—	107	1	no	—
44	#137	—	107	2	no	—
45	C₄⋊Q₈	D₁₀₇	1	1	yes	characteristic, complemented
46	#104	—	107	2	no	—
47	#101	—	107	1	no	—
48	#41	—	107	2	no	—
49	#41	—	214	2	no	—
50	#41	—	214	2	no	—
51	C₂×Q₈	—	214	2	no	—
52	C₂×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
53	C₂×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
54	Q₁₆	—	214	2	no	—
55	SD₁₆	—	214	2	no	—
56	M₁₆	—	107	2	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	107	2	no	—
62	Q₈	—	214	2	no	—
63	D₄	—	107	2	no	—
64	D₄	—	214	2	no	—
65	D₄	—	214	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	214	4	no	—
69	C₂×C₄	—	214	2	no	—
70	C₂×C₄	C₂²⋊D₁₀₇	1	2	yes	—
71	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
72	C₈	—	214	2	no	—
73	C₂²	—	214	2	no	—
74	C₂²	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	214	2	no	—
79	C₄	D₄×D₁₀₇	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	428	2	no	—
82	C₂	#16812077	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16812077
commutator	C₂×C₄×C₁₀₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	C₄⋊Q₈×C₁₀₇	C₂
socle	C₂×C₁₀₇	#120
hypercenter	C₄⋊Q₈	D₁₀₇
hypocenter	C₁₀₇	#456
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #16812140	C₂	semidirect
3. #16812037	C₂	semidirect
4. #16812028	C₂	semidirect
5. C₄⋊Q₈×C₁₀₇	C₂	semidirect
41. C₁₀₇	#456	semidirect
45. C₄⋊Q₈	D₁₀₇	semidirect