Subgroups — #20298972

Order 7232.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268634	C₂	1	4	yes	maximal, complemented
2	#16813619	C₂	1	1	yes	maximal, characteristic
3	C₄×#11744751	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄₅₂	—	2	2	no	—
6	C₂×D₄₅₂	C₂²	1	2	yes	—
7	#14268634	—	2	8	no	—
8	C₂×#11744751	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₁₃	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₁₃	—	2	2	no	—
11	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂²×D₁₁₃	—	2	2	no	—
13	Q₈×C₁₁₃	—	2	4	no	—
14	C₂×C₄×C₁₁₃	—	2	2	no	—
15	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
16	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
17	D₄₅₂	—	2	4	no	—
18	D₄₅₂	—	4	4	no	—
19	#11744751	D₄	1	4	yes	complemented
20	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
21	C₂×D₁₁₃	—	4	2	no	—
22	C₂×D₁₁₃	—	4	4	no	—
23	C₄×C₁₁₃	—	4	2	no	—
24	C₄×C₁₁₃	SD₁₆	1	4	yes	complemented
25	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
26	C₂×C₁₁₃	#127	1	1	yes	characteristic
27	C₂×C₁₁₃	C₂×SD₁₆	1	2	yes	—
28	D₁₁₃	—	8	2	no	—
29	C₁₁₃	#492	1	1	yes	characteristic, complemented, Sylow
30	#492	—	113	1	no	maximal, Sylow
31	C₂×SD₁₆	—	113	4	no	—
32	C₄⋊Q₈	D₁₁₃	1	1	yes	characteristic, complemented
33	#127	—	113	1	no	—
34	C₄×C₈	—	113	1	no	—
35	C₂×Q₈	C₂×D₁₁₃	1	2	yes	—
36	C₂×D₄	—	113	2	no	—
37	C₂×D₄	—	226	2	no	—
38	SD₁₆	—	226	8	no	—
39	C₂×C₈	—	113	2	no	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
42	C₂³	—	226	2	no	—
43	Q₈	—	2	4	no	—
44	D₄	—	452	4	no	—
45	D₄	—	226	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₂²⋊D₁₁₃	1	2	yes	—
48	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
49	C₈	—	113	4	no	—
50	C₂²	—	452	2	no	—
51	C₂²	—	452	4	no	—
52	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	#14268634	1	4	yes	complemented
55	C₄	D₄×D₁₁₃	1	2	yes	—
56	C₂	—	904	2	no	—
57	C₂	#16813618	1	1	yes	characteristic
58	C₂	C₂×#14268634	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	C₄⋊Q₈×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	#492
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14268634	C₂	semidirect
3. C₄×#11744751	C₂	semidirect
4. C₄⋊Q₈×C₁₁₃	C₂	semidirect
19. #11744751	D₄	semidirect
24. C₄×C₁₁₃	SD₁₆	semidirect
29. C₁₁₃	#492	semidirect
32. C₄⋊Q₈	D₁₁₃	semidirect
54. C₄	#14268634	semidirect