Subgroups — #20298977

Order 7232.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268637	C₂	1	4	yes	maximal
2	#16813622	C₂	1	1	yes	maximal, characteristic
3	C₄×#11744751	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₂₆	C₂²	1	2	yes	—
6	#14268637	—	2	8	no	—
7	C₂×#11744751	C₂²	1	2	yes	—
8	C₄⋊Dic₁₁₃	—	2	2	no	—
9	C₂×Q₈×C₁₁₃	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₁₃	—	2	2	no	—
11	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
12	Q₈×C₁₁₃	—	2	4	no	—
13	C₂×C₄×C₁₁₃	—	2	2	no	—
14	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
15	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
16	C₂×Dic₁₁₃	—	2	2	no	—
17	Dic₂₂₆	—	2	4	no	—
18	#11744751	D₄	1	4	yes	—
19	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
20	C₄×C₁₁₃	—	4	2	no	—
21	C₄×C₁₁₃	Q₁₆	1	4	yes	complemented
22	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
23	Dic₁₁₃	—	4	2	no	—
24	C₂×C₁₁₃	#127	1	1	yes	characteristic
25	C₂×C₁₁₃	C₂×Q₁₆	1	2	yes	—
26	C₁₁₃	#494	1	1	yes	characteristic, complemented, Sylow
27	#494	—	113	1	no	maximal, Sylow
28	C₂×Q₁₆	—	113	4	no	—
29	C₄⋊Q₈	—	113	1	no	—
30	C₄⋊Q₈	D₁₁₃	1	1	yes	characteristic
31	C₄×C₈	—	113	1	no	—
32	C₂×Q₈	—	113	2	no	—
33	C₂×Q₈	C₂×D₁₁₃	1	2	yes	—
34	Q₁₆	—	226	8	no	—
35	C₂×C₈	—	113	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	226	2	no	—
38	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	226	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	226	2	no	—
43	C₂×C₄	C₂²⋊D₁₁₃	1	2	yes	—
44	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
45	C₈	—	113	4	no	—
46	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	452	2	no	—
49	C₄	#14268637	1	4	yes	complemented
50	C₄	D₄×D₁₁₃	1	2	yes	—
51	C₂	#16813618	1	1	yes	characteristic
52	C₂	C₂×#14268637	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	#494
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
21. C₄×C₁₁₃	Q₁₆	semidirect
26. C₁₁₃	#494	semidirect
49. C₄	#14268637	semidirect