Subgroups — #20298705

Order 7232.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₁₃	C₂	1	2	yes	maximal
2	#16813554	C₂	1	1	yes	maximal, characteristic
3	#16813546	C₂	1	2	yes	maximal
4	#16813532	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11744751	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₁₁₃	—	2	2	no	—
8	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
9	C₄×Dic₁₁₃	—	2	2	no	—
10	C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₁₁₃	C₂²	1	2	yes	—
12	Q₁₆×C₁₁₃	C₄	1	4	yes	complemented
13	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
14	Q₈×C₁₁₃	—	2	2	no	—
15	Q₈×C₁₁₃	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₁₁₃	—	2	2	no	—
17	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₁₁₃	—	2	2	no	—
19	C₂×Dic₁₁₃	D₄	1	2	yes	—
20	C₈×C₁₁₃	C₂×C₄	1	2	yes	—
21	#11744751	—	2	1	no	—
22	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
23	C₄×C₁₁₃	—	2	4	no	—
24	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₁₁₃	#41	1	1	yes	characteristic
26	Dic₁₁₃	—	2	2	no	—
27	Dic₁₁₃	—	4	2	no	—
28	C₂×C₁₁₃	C₄×D₄	1	1	yes	characteristic
29	C₂×C₁₁₃	#137	1	2	yes	—
30	C₁₁₃	#441	1	1	yes	characteristic, complemented, Sylow
31	#441	—	113	1	no	maximal, Sylow
32	C₂×Q₁₆	D₁₁₃	1	1	yes	characteristic
33	C₄×Q₈	—	113	2	no	—
34	#106	—	113	1	no	—
35	#103	—	113	2	no	—
36	#97	—	113	1	no	—
37	C₂×Q₈	C₂×D₁₁₃	1	2	yes	—
38	Q₁₆	Dic₁₁₃	1	4	yes	complemented
39	C₂×C₈	—	113	1	no	—
40	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
41	C₄⋊C₄	—	113	2	no	—
42	C₄⋊C₄	—	226	2	no	—
43	C₄²	—	113	1	no	—
44	C₄²	—	226	2	no	—
45	Q₈	—	2	2	no	—
46	Q₈	C₂×Dic₁₁₃	1	4	yes	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	113	2	no	—
49	C₂×C₄	—	226	2	no	—
50	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
51	C₈	—	226	1	no	—
52	C₈	C₂×Dic₁₁₃	1	2	yes	—
53	C₂²	D₄×D₁₁₃	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	—	452	2	no	—
56	C₄	—	226	2	no	—
57	C₄	C₂²×Dic₁₁₃	1	1	yes	characteristic
58	C₄	#14268646	1	1	yes	characteristic
59	C₂	D₄×Dic₁₁₃	1	1	yes	characteristic
60	C₂	#16813654	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₁₃
commutator	C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	C₂×Q₁₆×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	C₂×Q₁₆	D₁₁₃
hypocenter	C₁₁₃	#441
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
12. Q₁₆×C₁₁₃	C₄	semidirect
30. C₁₁₃	#441	semidirect
38. Q₁₆	Dic₁₁₃	semidirect