Subgroups — #20298720

Order 7232.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813629	C₂	1	2	yes	maximal, complemented
2	#16813558	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813549	C₂	1	2	yes	maximal, complemented
4	#16813532	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	#14268631	—	2	2	no	—
7	#14268631	C₂²	1	2	yes	—
8	C₂×#11744751	—	2	2	no	—
9	C₂×#11744751	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
11	#41×C₁₁₃	C₂²	1	2	yes	—
12	Q₁₆×C₁₁₃	C₄	1	1	yes	characteristic, complemented
13	SD₁₆×C₁₁₃	C₄	1	2	yes	complemented
14	D₈×C₁₁₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
16	Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
17	D₄×C₁₁₃	—	2	2	no	—
18	D₄×C₁₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₁₃	—	2	2	no	—
20	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₁₃	—	2	1	no	—
22	C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
23	C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
24	#11744751	—	2	2	no	—
25	#11744751	D₄	1	2	yes	—
26	C₂²×C₁₁₃	—	2	2	no	—
27	C₂²×C₁₁₃	#41	1	1	yes	characteristic
28	C₄×C₁₁₃	—	2	2	no	—
29	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
31	Dic₁₁₃	—	4	1	no	—
32	C₂×C₁₁₃	—	2	1	no	—
33	C₂×C₁₁₃	—	4	2	no	—
34	C₂×C₁₁₃	C₄×D₄	1	1	yes	characteristic
35	C₁₁₃	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	113	1	no	maximal, Sylow
37	#135	D₁₁₃	1	1	yes	characteristic
38	#131	—	113	2	no	—
39	#108	—	113	1	no	—
40	#104	—	113	2	no	—
41	#97	—	113	1	no	—
42	#41	C₂×D₁₁₃	1	2	yes	—
43	Q₁₆	Dic₁₁₃	1	1	yes	characteristic, complemented
44	SD₁₆	Dic₁₁₃	1	2	yes	complemented
45	D₈	Dic₁₁₃	1	1	yes	characteristic, complemented
46	M₁₆	—	113	2	no	—
47	M₁₆	—	226	2	no	—
48	C₂×C₈	—	113	1	no	—
49	C₂×C₈	—	226	2	no	—
50	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
51	C₄²	—	113	1	no	—
52	Q₈	C₂×Dic₁₁₃	1	2	yes	—
53	D₄	—	2	2	no	—
54	D₄	C₂×Dic₁₁₃	1	2	yes	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	226	1	no	—
57	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
58	C₈	—	113	2	no	—
59	C₈	—	226	2	no	—
60	C₈	C₂×Dic₁₁₃	1	1	yes	characteristic
61	C₈	C₂×Dic₁₁₃	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	#14268646	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	452	1	no	—
66	C₄	C₂²×Dic₁₁₃	1	1	yes	characteristic
67	C₄	D₄×D₁₁₃	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	D₄×Dic₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₁₁₃
commutator	C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#135×C₁₁₃	C₂
socle	C₂×C₁₁₃	C₄×D₄
hypercenter	#135	D₁₁₃
hypocenter	C₁₁₃	#444
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #16813629	C₂	semidirect
2. #16813558	C₂	semidirect
3. #16813549	C₂	semidirect
4. #16813532	C₂	semidirect
12. Q₁₆×C₁₁₃	C₄	semidirect
13. SD₁₆×C₁₁₃	C₄	semidirect
14. D₈×C₁₁₃	C₄	semidirect
35. C₁₁₃	#444	semidirect
43. Q₁₆	Dic₁₁₃	semidirect
44. SD₁₆	Dic₁₁₃	semidirect
45. D₈	Dic₁₁₃	semidirect