Subgroups — #28405608

Order 14464.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299461	C₂	1	2	yes	maximal
2	C₁₆×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11744753	—	2	2	no	—
4	C₂×#14268622	C₄	1	1	yes	characteristic
5	C₈×Dic₁₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₁₁₃	C₄	1	1	yes	characteristic
7	C₂×#11744753	—	4	2	no	—
8	C₂×#11744751	Q₈	1	1	yes	characteristic
9	C₄×Dic₁₁₃	D₄	1	1	yes	characteristic
10	#14268622	C₈	1	2	yes	complemented
11	C₂×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₁₃	—	2	1	no	—
13	C₂×C₄×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
14	C₂×Dic₁₁₃	D₈	1	1	yes	characteristic
15	C₂×Dic₁₁₃	Q₁₆	1	1	yes	characteristic
16	#11744753	—	8	2	no	—
17	C₈×C₁₁₃	C₂×C₈	1	1	yes	characteristic
18	C₈×C₁₁₃	M₁₆	1	1	yes	characteristic
19	#11744751	—	2	1	no	—
20	C₂²×C₁₁₃	#107	1	1	yes	characteristic
21	C₄×C₁₁₃	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₁₁₃	#108	1	1	yes	characteristic
23	Dic₁₁₃	—	2	1	no	—
24	Dic₁₁₃	SD₃₂	1	2	yes	—
25	C₂×C₁₁₃	#335	1	1	yes	characteristic
26	C₂×C₁₁₃	#367	1	1	yes	characteristic
27	C₂×C₁₁₃	#368	1	1	yes	characteristic
28	#1372	—	113	1	no	maximal, Sylow
29	C₁₁₃	#1372	1	1	yes	characteristic, complemented, Sylow
30	C₄×C₁₆	—	113	1	no	—
31	#335	—	113	2	no	—
32	C₂×C₁₆	—	113	1	no	—
33	C₂×C₁₆	#90673	1	1	yes	characteristic
34	C₄⋊C₈	—	226	2	no	—
35	C₄×C₈	—	113	1	no	—
36	C₂×C₈	—	113	1	no	—
37	C₂×C₈	—	452	2	no	—
38	C₂×C₈	C₂×#90673	1	1	yes	characteristic
39	C₄²	—	113	1	no	—
40	C₁₆	—	2	1	no	—
41	C₁₆	—	113	2	no	—
42	C₂×C₄	—	113	1	no	—
43	C₂×C₄	—	113	1	no	—
44	C₂×C₄	C₄⋊#90673	1	1	yes	characteristic
45	C₈	—	904	2	no	—
46	C₈	—	226	1	no	—
47	C₈	#14268655	1	1	yes	characteristic
48	C₈	#14268656	1	1	yes	characteristic
49	C₂²	#16813691	1	1	yes	characteristic
50	C₄	—	113	2	no	—
51	C₄	—	226	1	no	—
52	C₄	#16813689	1	1	yes	characteristic
53	C₄	#16813692	1	1	yes	characteristic
54	C₂	#20299462	1	1	yes	characteristic
55	C₂	#20299525	1	1	yes	characteristic
56	C₂	#20299526	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16813691
commutator	C₈×C₁₁₃	C₂×C₈
Frattini	C₂×C₈	C₂×#90673
Fitting	C₂×C₁₆×C₁₁₃	C₄
socle, prime radical	C₂²×C₁₁₃	#107
cosocle	C₈×Dic₁₁₃	C₂²
hypercenter	C₂×C₁₆	#90673
hypocenter	C₁₁₃	#1372
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
10. #14268622	C₈	semidirect
29. C₁₁₃	#1372	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20299461	2
C₁₆×Dic₁₁₃	2
#1372	113