Subgroups — #20298641

Order 7232.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268631	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
3	#16813555	C₂	1	2	yes	maximal, complemented
4	#16813553	C₂	1	2	yes	maximal, complemented
5	#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₃	—	2	1	no	—
7	#14268631	C₄	1	4	yes	complemented
8	C₂×#11744751	C₂²	1	2	yes	—
9	C₄⋊Dic₁₁₃	—	2	1	no	—
10	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₁₃	—	2	1	no	—
15	C₄²×C₁₁₃	—	2	1	no	—
16	C₂³×C₁₁₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₁₃	—	2	1	no	—
18	C₂×C₄×C₁₁₃	—	2	1	no	—
19	C₂×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₁₃	Q₈	1	2	yes	—
22	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
23	C₂×Dic₁₁₃	—	2	2	no	—
24	C₂×Dic₁₁₃	—	4	2	no	—
25	#11744751	C₂×C₄	1	4	yes	complemented
26	C₂²×C₁₁₃	—	2	1	no	—
27	C₂²×C₁₁₃	—	2	1	no	—
28	C₂²×C₁₁₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
30	C₄×C₁₁₃	—	4	1	no	—
31	C₄×C₁₁₃	—	4	1	no	—
32	C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
33	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
35	Dic₁₁₃	—	4	2	no	—
36	C₂×C₁₁₃	—	2	2	no	—
37	C₂×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₁₁₃	#137	1	1	yes	characteristic
40	C₁₁₃	#428	1	1	yes	characteristic, complemented, Sylow
41	#428	—	113	1	no	maximal, Sylow
42	C₂×M₁₆	—	113	1	no	—
43	#117	D₁₁₃	1	1	yes	characteristic
44	C₂×C₄⋊C₄	—	113	1	no	—
45	#107	—	113	2	no	—
46	#106	—	113	2	no	—
47	C₂²×C₄	—	226	1	no	—
48	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
49	M₁₆	—	113	4	no	—
50	C₂×C₈	—	113	2	no	—
51	C₄⋊C₄	—	113	2	no	—
52	C₄⋊C₄	—	226	1	no	—
53	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
54	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
55	C₂²⋊C₄	—	2	1	no	—
56	C₄²	—	2	1	no	—
57	C₂³	C₂²⋊D₁₁₃	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	452	2	no	—
61	C₂×C₄	—	226	2	no	—
62	C₂×C₄	Dic₂₂₆	1	2	yes	—
63	C₂×C₄	C₄×D₁₁₃	1	2	yes	—
64	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
66	C₈	—	113	4	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	#14268626	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	452	2	no	—
74	C₄	#14268626	1	2	yes	—
75	C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
76	C₄	C₂×Dic₂₂₆	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×#14268626	1	1	yes	characteristic
79	C₂	#16813647	1	1	yes	characteristic
80	C₂	#16813652	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₁₃
commutator	C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#117×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#117	D₁₁₃
hypocenter	C₁₁₃	#428
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #16813555	C₂	semidirect
4. #16813553	C₂	semidirect
7. #14268631	C₄	semidirect
25. #11744751	C₂×C₄	semidirect
40. C₁₁₃	#428	semidirect