Subgroups — #20293001

Order 6848.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133805	C₂	1	4	yes	maximal, complemented
2	#16812106	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄₂₈	—	2	2	no	—
6	C₂×D₄₂₈	C₂²	1	2	yes	—
7	#13133805	—	2	8	no	—
8	C₂×C₁₀₇⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₀₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₀₇	—	2	2	no	—
11	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂²×D₁₀₇	—	2	2	no	—
13	Q₈×C₁₀₇	—	2	4	no	—
14	C₂×C₄×C₁₀₇	—	2	2	no	—
15	C₂×C₄×C₁₀₇	D₄	1	2	yes	—
16	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
17	D₄₂₈	—	2	4	no	—
18	D₄₂₈	—	4	4	no	—
19	C₁₀₇⋊C₈	D₄	1	4	yes	complemented
20	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
21	C₂×D₁₀₇	—	4	2	no	—
22	C₂×D₁₀₇	—	4	4	no	—
23	C₄×C₁₀₇	—	4	2	no	—
24	C₄×C₁₀₇	SD₁₆	1	4	yes	complemented
25	C₄×C₁₀₇	C₂×D₄	1	2	yes	—
26	C₂×C₁₀₇	#127	1	1	yes	characteristic
27	C₂×C₁₀₇	C₂×SD₁₆	1	2	yes	—
28	D₁₀₇	—	8	2	no	—
29	C₁₀₇	#492	1	1	yes	characteristic, complemented, Sylow
30	#492	—	107	1	no	maximal, Sylow
31	C₂×SD₁₆	—	107	4	no	—
32	C₄⋊Q₈	D₁₀₇	1	1	yes	characteristic, complemented
33	#127	—	107	1	no	—
34	C₄×C₈	—	107	1	no	—
35	C₂×Q₈	C₂×D₁₀₇	1	2	yes	—
36	C₂×D₄	—	107	2	no	—
37	C₂×D₄	—	214	2	no	—
38	SD₁₆	—	214	8	no	—
39	C₂×C₈	—	107	2	no	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
42	C₂³	—	214	2	no	—
43	Q₈	—	2	4	no	—
44	D₄	—	428	4	no	—
45	D₄	—	214	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₂²⋊D₁₀₇	1	2	yes	—
48	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
49	C₈	—	107	4	no	—
50	C₂²	—	428	2	no	—
51	C₂²	—	428	4	no	—
52	C₂²	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	#13133805	1	4	yes	complemented
55	C₄	D₄×D₁₀₇	1	2	yes	—
56	C₂	—	856	2	no	—
57	C₂	#16812105	1	1	yes	characteristic
58	C₂	C₂×#13133805	1	2	yes	—

	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	#492
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
1. C₂×#13133805	C₂	semidirect
3. C₄×C₁₀₇⋊C₈	C₂	semidirect
4. C₄⋊Q₈×C₁₀₇	C₂	semidirect
19. C₁₀₇⋊C₈	D₄	semidirect
24. C₄×C₁₀₇	SD₁₆	semidirect
29. C₁₀₇	#492	semidirect
32. C₄⋊Q₈	D₁₀₇	semidirect
54. C₄	#13133805	semidirect