Subgroups — #20293033

Order 6848.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812101	C₂	1	1	yes	maximal, characteristic
2	#16812042	C₂	1	2	yes	maximal
3	#16812040	C₂	1	2	yes	maximal
4	#16812020	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₀₇⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
9	Dic₁₀₇⋊C₄	—	2	1	no	—
10	Dic₁₀₇⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₁₀₇	—	2	1	no	—
12	C₄⋊C₄×C₁₀₇	—	2	1	no	—
13	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
14	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
15	Q₈×C₁₀₇	—	4	2	no	—
16	C₂×C₄×C₁₀₇	—	2	2	no	—
17	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₁₀₇	—	2	1	no	—
21	C₂×Dic₁₀₇	—	2	1	no	—
22	C₁₀₇⋊C₈	Q₈	1	4	yes	complemented
23	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₀₇	—	2	1	no	—
25	C₄×C₁₀₇	—	2	1	no	—
26	C₄×C₁₀₇	—	4	2	no	—
27	C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
28	Dic₁₀₇	—	4	1	no	—
29	Dic₁₀₇	—	4	1	no	—
30	C₂×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₁₀₇	#137	1	1	yes	characteristic
33	C₁₀₇	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	107	1	no	maximal, Sylow
35	C₄⋊Q₈	D₁₀₇	1	1	yes	characteristic
36	#125	—	107	1	no	—
37	#107	—	107	2	no	—
38	#106	—	107	2	no	—
39	#97	—	107	1	no	—
40	C₂×Q₈	—	2	1	no	—
41	C₂×C₈	—	107	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	107	1	no	—
44	C₄⋊C₄	—	107	1	no	—
45	C₄⋊C₄	—	214	1	no	—
46	C₄⋊C₄	—	214	1	no	—
47	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
48	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	214	1	no	—
52	C₂×C₄	—	214	1	no	—
53	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
56	C₈	—	107	4	no	—
57	C₂²	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	—	428	1	no	—
62	C₄	—	428	1	no	—
63	C₄	Q₈×D₁₀₇	1	2	yes	—
64	C₂	#16812107	1	1	yes	characteristic
65	C₂	#16812135	1	1	yes	characteristic
66	C₂	#16812140	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	C₄⋊Q₈×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₀₇
hypocenter	C₁₀₇	#501
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
22. C₁₀₇⋊C₈	Q₈	semidirect
33. C₁₀₇	#501	semidirect