Subgroups — #28388771

Order 14464.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298432	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14268622	C₄	1	1	yes	characteristic
5	C₈×Dic₁₁₃	C₄	1	1	yes	characteristic
6	#16813528	—	2	1	no	—
7	C₂×C₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
8	C₃₂×C₁₁₃	—	2	1	no	—
9	C₂×#11744751	C₈	1	1	yes	characteristic
10	C₄×Dic₁₁₃	C₈	1	1	yes	characteristic
11	#14268622	—	2	1	no	—
12	C₂×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₁₃	D₄	1	1	yes	characteristic
14	C₁₆×C₁₁₃	Q₈	1	1	yes	characteristic
15	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
16	C₂×Dic₁₁₃	—	2	1	no	—
17	C₈×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₁₁₃	M₁₆	1	1	yes	characteristic
19	#11744751	C₁₆	1	2	yes	—
20	C₂²×C₁₁₃	M₃₂	1	1	yes	characteristic
21	C₄×C₁₁₃	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₁₁₃	C₂×C₁₆	1	1	yes	characteristic
23	Dic₁₁₃	—	2	2	no	—
24	C₂×C₁₁₃	—	2	1	no	—
25	C₂×C₁₁₃	C₄⋊C₁₆	1	1	yes	characteristic
26	#1422	—	113	1	no	maximal, Sylow
27	C₁₁₃	#1422	1	1	yes	characteristic, complemented, Sylow
28	M₆₄	—	113	1	no	—
29	M₆₄	D₁₁₃	1	1	yes	characteristic
30	C₄×C₁₆	—	113	1	no	—
31	C₂×C₁₆	—	113	1	no	—
32	C₂×C₁₆	C₂×D₁₁₃	1	1	yes	characteristic
33	C₄×C₈	—	113	1	no	—
34	C₃₂	—	2	1	no	—
35	C₃₂	—	226	1	no	—
36	C₂×C₈	—	113	1	no	—
37	C₂×C₈	C₄×D₁₁₃	1	1	yes	characteristic
38	C₄²	—	113	1	no	—
39	C₁₆	—	226	1	no	—
40	C₁₆	Dic₂₂₆	1	1	yes	characteristic
41	C₁₆	C₂²⋊D₁₁₃	1	1	yes	characteristic
42	C₂×C₄	—	226	1	no	—
43	C₂×C₄	C₈×D₁₁₃	1	1	yes	characteristic
44	C₈	—	113	2	no	—
45	C₈	#14268626	1	1	yes	characteristic
46	C₈	#14268630	1	1	yes	characteristic
47	C₂²	#16813561	1	1	yes	characteristic
48	C₄	—	226	2	no	—
49	C₄	#16813551	1	1	yes	characteristic
50	C₄	C₁₆×D₁₁₃	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#20298416	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₁₃
commutator	C₄×C₁₁₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₁₃
Fitting	M₆₄×C₁₁₃	C₂
socle	C₂×C₁₁₃	C₄⋊C₁₆
prime radical	C₂²×C₁₁₃	M₃₂
cosocle	C₂×C₁₆×C₁₁₃	C₂²
hypercenter	M₆₄	D₁₁₃
hypocenter	C₁₁₃	#1422
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
27. C₁₁₃	#1422	semidirect

Maximal subgroups

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

subgroup	index
#20298432	2
C₁₆×Dic₁₁₃	2
M₆₄×C₁₁₃	2
#1422	113