Subgroups — #28388780

Order 14464.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298428	C₂	1	1	yes	maximal, characteristic
2	#20298426	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	#16813558	—	2	1	no	—
5	#16813556	—	2	1	no	—
6	C₂×C₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
7	C₃₂×C₁₁₃	C₄	1	2	yes	complemented
8	#14268631	—	4	1	no	—
9	C₄⋊Dic₁₁₃	—	4	1	no	—
10	C₂×C₈×C₁₁₃	D₄	1	1	yes	characteristic
11	C₁₆×C₁₁₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₁₃	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₁₁₃	D₈	1	1	yes	characteristic
14	C₂×Dic₁₁₃	—	8	1	no	—
15	C₈×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
16	C₈×C₁₁₃	Q₁₆	1	1	yes	characteristic
17	#11744751	—	8	1	no	—
18	C₂²×C₁₁₃	SD₃₂	1	1	yes	characteristic
19	C₄×C₁₁₃	#107	1	1	yes	characteristic
20	C₄×C₁₁₃	SD₃₂	1	1	yes	characteristic
21	Dic₁₁₃	—	16	1	no	—
22	C₂×C₁₁₃	—	2	1	no	—
23	C₂×C₁₁₃	#367	1	1	yes	characteristic
24	#1426	—	113	1	no	maximal, Sylow
25	C₁₁₃	#1426	1	1	yes	characteristic, complemented, Sylow
26	M₆₄	D₁₁₃	1	1	yes	characteristic
27	#368	—	113	1	no	—
28	#367	—	113	1	no	—
29	C₂×C₁₆	C₂×D₁₁₃	1	1	yes	characteristic
30	#108	—	226	1	no	—
31	#107	—	226	1	no	—
32	C₃₂	Dic₁₁₃	1	2	yes	complemented
33	M₁₆	—	452	1	no	—
34	C₂×C₈	D₄₅₂	1	1	yes	characteristic
35	C₄⋊C₄	—	452	1	no	—
36	C₁₆	Dic₂₂₆	1	1	yes	characteristic
37	C₁₆	C₂×Dic₁₁₃	1	1	yes	characteristic
38	C₂×C₄	—	904	1	no	—
39	C₂×C₄	D₉₀₄	1	1	yes	characteristic
40	C₈	—	904	1	no	—
41	C₈	C₄⋊Dic₁₁₃	1	1	yes	characteristic
42	C₈	Dic₄₅₂	1	1	yes	characteristic
43	C₂²	#16813567	1	1	yes	characteristic
44	C₄	—	1808	1	no	—
45	C₄	#16813556	1	1	yes	characteristic
46	C₄	#16813567	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#20298426	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20298426
commutator	C₁₆×C₁₁₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₁₃
Fitting	M₆₄×C₁₁₃	C₂
socle	C₂×C₁₁₃	#367
prime radical	C₂²×C₁₁₃	SD₃₂
cosocle	C₂×C₁₆×C₁₁₃	C₂²
hypercenter	M₆₄	D₁₁₃
hypocenter	C₁₁₃	#1426
LCS	G ▷ C₁₆×C₁₁₃ ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₁₁₃	C₄	semidirect
25. C₁₁₃	#1426	semidirect
32. C₃₂	Dic₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20298428	2
#20298426	2
M₆₄×C₁₁₃	2
#1426	113