Subgroups — #41011609

Order 27008.

38 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₁₁⋊C₆₄	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₂₁₁	C₄	1	1	yes	characteristic
4	C₃₂×C₂₁₁	C₄	1	1	yes	characteristic
5	C₃₂×C₂₁₁	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₂₁₁	C₈	1	1	yes	characteristic
7	C₁₆×C₂₁₁	C₈	1	1	yes	characteristic
8	C₁₆×C₂₁₁	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₂₁₁	C₁₆	1	1	yes	characteristic
10	C₈×C₂₁₁	C₁₆	1	1	yes	characteristic
11	C₈×C₂₁₁	C₂×C₈	1	1	yes	characteristic
12	C₂²×C₂₁₁	C₃₂	1	1	yes	characteristic
13	C₄×C₂₁₁	C₃₂	1	1	yes	characteristic
14	C₄×C₂₁₁	C₂×C₁₆	1	1	yes	characteristic
15	C₂×C₂₁₁	—	2	1	no	—
16	C₂×C₂₁₁	C₂×C₃₂	1	1	yes	characteristic
17	C₂₁₁	M₁₂₈	1	1	yes	characteristic, complemented, Sylow
18	M₁₂₈	—	211	1	no	maximal, Sylow
19	C₂×C₃₂	D₂₁₁	1	1	yes	characteristic
20	C₆₄	—	211	2	no	—
21	C₂×C₁₆	Dic₂₁₁	1	1	yes	characteristic
22	C₃₂	Dic₂₁₁	1	1	yes	characteristic
23	C₃₂	C₂×D₂₁₁	1	1	yes	characteristic
24	C₂×C₈	C₂₁₁⋊C₈	1	1	yes	characteristic
25	C₁₆	C₂₁₁⋊C₈	1	1	yes	characteristic
26	C₁₆	C₂×Dic₂₁₁	1	1	yes	characteristic
27	C₂×C₄	C₂₁₁⋊C₁₆	1	1	yes	characteristic
28	C₈	C₂₁₁⋊C₁₆	1	1	yes	characteristic
29	C₈	C₂×C₂₁₁⋊C₈	1	1	yes	characteristic
30	C₂²	C₂₁₁⋊C₃₂	1	1	yes	characteristic
31	C₄	C₂₁₁⋊C₃₂	1	1	yes	characteristic
32	C₄	C₂×C₂₁₁⋊C₁₆	1	1	yes	characteristic
33	C₂	—	2	1	no	—
34	C₂	C₂×C₂₁₁⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂×D₂₁₁
commutator, socle	C₂×C₂₁₁	C₂×C₃₂
Fitting	C₂×C₃₂×C₂₁₁	C₂
prime radical	C₂²×C₂₁₁	C₃₂
cosocle	C₃₂×C₂₁₁	C₂²
hypercenter	C₂×C₃₂	D₂₁₁
hypocenter	C₂₁₁	M₁₂₈
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂₁₁⋊C₆₄	C₂	semidirect
17. C₂₁₁	M₁₂₈	semidirect

Maximal subgroups

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

subgroup	index
C₂₁₁⋊C₆₄	2
C₂×C₃₂×C₂₁₁	2
M₁₂₈	211