Subgroups — #20293030

Order 6848.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812109	C₂	1	1	yes	maximal, characteristic
2	#16812042	C₂	1	4	yes	maximal
3	C₄×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₁₄	—	2	1	no	—
6	C₂×C₁₀₇⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₁₀₇	—	2	1	no	—
8	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₀₇	—	2	1	no	—
10	C₄⋊C₄×C₁₀₇	—	2	1	no	—
11	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
12	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
13	Q₈×C₁₀₇	—	4	2	no	—
14	C₂×C₄×C₁₀₇	—	2	2	no	—
15	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₁₀₇	—	2	2	no	—
19	Dic₂₁₄	—	4	2	no	—
20	C₁₀₇⋊C₈	Q₈	1	4	yes	complemented
21	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
22	C₄×C₁₀₇	—	4	2	no	—
23	C₄×C₁₀₇	D₈	1	2	yes	—
24	C₄×C₁₀₇	Q₁₆	1	2	yes	—
25	C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
26	Dic₁₀₇	—	4	2	no	—
27	C₂×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₁₀₇	C₂×D₈	1	1	yes	characteristic
29	C₂×C₁₀₇	C₂×Q₁₆	1	1	yes	characteristic
30	C₁₀₇	#500	1	1	yes	characteristic, complemented, Sylow
31	#500	—	107	1	no	maximal, Sylow
32	C₄⋊Q₈	—	107	1	no	—
33	C₄⋊Q₈	D₁₀₇	1	1	yes	characteristic
34	#107	—	107	4	no	—
35	C₄×C₈	—	107	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	214	1	no	—
38	C₂×C₈	—	107	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	107	2	no	—
41	C₄⋊C₄	—	214	1	no	—
42	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
43	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	428	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	214	2	no	—
48	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
51	C₈	—	107	4	no	—
52	C₂²	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	428	2	no	—
55	C₄	#13133803	1	2	yes	—
56	C₄	#13133808	1	2	yes	—
57	C₄	Q₈×D₁₀₇	1	2	yes	—
58	C₂	#16812107	1	1	yes	characteristic
59	C₂	C₂×#13133803	1	1	yes	characteristic
60	C₂	C₂×#13133808	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	#500
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
20. C₁₀₇⋊C₈	Q₈	semidirect
30. C₁₀₇	#500	semidirect