Subgroups — #20298880

Order 7232.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic
2	#16813553	C₂	1	2	yes	maximal
3	C₄⋊#11744751	C₂	1	1	yes	maximal, characteristic
4	#16813545	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
7	C₂×#11744751	C₂²	1	2	yes	—
8	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
9	#14268626	—	2	1	no	—
10	C₄×Dic₁₁₃	—	2	1	no	—
11	C₂×Q₈×C₁₁₃	—	2	1	no	—
12	C₄⋊C₄×C₁₁₃	—	2	1	no	—
13	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
14	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
15	Q₈×C₁₁₃	—	4	2	no	—
16	C₂×C₄×C₁₁₃	—	2	2	no	—
17	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
20	C₂×Dic₁₁₃	—	2	1	no	—
21	C₂×Dic₁₁₃	—	2	1	no	—
22	Dic₂₂₆	—	2	1	no	—
23	Dic₂₂₆	Q₈	1	2	yes	complemented
24	#11744751	—	2	2	no	—
25	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
26	C₄×C₁₁₃	—	2	1	no	—
27	C₄×C₁₁₃	—	4	2	no	—
28	C₄×C₁₁₃	SD₁₆	1	2	yes	—
29	C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
30	C₄×C₁₁₃	#41	1	1	yes	characteristic
31	Dic₁₁₃	—	2	2	no	—
32	Dic₁₁₃	—	4	1	no	—
33	C₂×C₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
35	C₂×C₁₁₃	#137	1	1	yes	characteristic
36	C₁₁₃	#475	1	1	yes	characteristic, complemented, Sylow
37	#475	—	113	1	no	maximal, Sylow
38	C₄⋊Q₈	D₁₁₃	1	1	yes	characteristic
39	C₄×Q₈	—	113	1	no	—
40	#106	—	113	2	no	—
41	C₄⋊C₈	—	113	1	no	—
42	#103	—	113	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	—	113	1	no	—
45	C₂×C₈	—	113	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	113	1	no	—
48	C₄⋊C₄	—	226	1	no	—
49	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
50	C₄²	—	226	1	no	—
51	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
52	Q₈	—	4	2	no	—
53	Q₈	—	113	2	no	—
54	Q₈	—	226	1	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	226	1	no	—
57	C₂×C₄	—	226	1	no	—
58	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
61	C₈	—	226	2	no	—
62	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	2	no	—
65	C₄	—	452	1	no	—
66	C₄	—	226	2	no	—
67	C₄	#14268635	1	2	yes	—
68	C₄	Q₈×D₁₁₃	1	1	yes	characteristic
69	C₄	#14268648	1	1	yes	characteristic
70	C₂	#16813598	1	1	yes	characteristic
71	C₂	C₂×#14268635	1	1	yes	characteristic
72	C₂	#16813653	1	1	yes	characteristic

	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	#475
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
23. Dic₂₂₆	Q₈	semidirect
36. C₁₁₃	#475	semidirect