Subgroups — #20298626

Order 7232.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11744751	C₂	1	1	yes	maximal, characteristic
2	#16813581	C₂	1	1	yes	maximal, characteristic
3	#16813551	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11744751	—	2	2	no	—
6	C₂×#11744751	C₂²	1	2	yes	—
7	C₄⋊Dic₁₁₃	C₄	1	2	yes	—
8	C₂²⋊Dic₁₁₃	C₄	1	2	yes	—
9	C₄×Dic₁₁₃	C₂²	1	2	yes	—
10	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
11	M₁₆×C₁₁₃	—	2	2	no	—
12	C₂×C₈×C₁₁₃	C₂²	1	2	yes	—
13	C₂³×C₁₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
16	C₂×C₄×C₁₁₃	Q₈	1	2	yes	—
17	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₁₁₃	C₂×C₄	1	4	yes	—
19	C₈×C₁₁₃	—	2	2	no	—
20	#11744751	—	2	2	no	—
21	C₂²×C₁₁₃	—	2	2	no	—
22	C₂²×C₁₁₃	C₄⋊C₄	1	2	yes	—
23	C₂²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
25	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
26	C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
27	Dic₁₁₃	—	2	4	no	—
28	C₂×C₁₁₃	—	2	2	no	—
29	C₂×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₁₁₃	#131	1	2	yes	—
31	C₁₁₃	#424	1	1	yes	characteristic, complemented, Sylow
32	#424	—	113	1	no	maximal, Sylow
33	C₂×M₁₆	D₁₁₃	1	1	yes	characteristic
34	C₂²×C₈	—	113	1	no	—
35	#117	—	113	1	no	—
36	C₄⋊C₈	—	113	4	no	—
37	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
38	M₁₆	—	2	2	no	—
39	C₂×C₈	—	113	2	no	—
40	C₂×C₈	—	226	2	no	—
41	C₂×C₈	C₂×D₁₁₃	1	2	yes	—
42	C₄⋊C₄	—	113	2	no	—
43	C₂²⋊C₄	—	113	2	no	—
44	C₄²	—	113	2	no	—
45	C₂³	C₄×D₁₁₃	1	1	yes	characteristic
46	C₂×C₄	—	113	4	no	—
47	C₂×C₄	Dic₂₂₆	1	2	yes	—
48	C₂×C₄	C₄×D₁₁₃	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₁₃	1	2	yes	—
50	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	226	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	#14268626	1	2	yes	—
55	C₂²	C₂×C₄×D₁₁₃	1	1	yes	characteristic
56	C₄	—	226	4	no	—
57	C₄	#14268626	1	2	yes	—
58	C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
59	C₄	C₂×Dic₂₂₆	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×#14268626	1	1	yes	characteristic
62	C₂	#16813630	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂²×C₄
Fitting	C₂×M₁₆×C₁₁₃	C₂
hypercenter	C₂×M₁₆	D₁₁₃
hypocenter	C₁₁₃	#424
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16813551	C₂	semidirect
31. C₁₁₃	#424	semidirect