Subgroups — C₄⋊(C₂₁₁⋊C₁₆) #27212965

Order 13504.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₁₁⋊C₁₆	C₂	1	2	yes	maximal
2	C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂₁₁⋊C₁₆	—	2	2	no	—
4	C₂×C₈×C₂₁₁	C₄	1	1	yes	characteristic
5	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
6	C₄²×C₂₁₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₂₁₁	C₈	1	1	yes	characteristic
8	C₂×C₄×C₂₁₁	C₈	1	1	yes	characteristic
9	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
10	C₈×C₂₁₁	—	2	1	no	—
11	C₈×C₂₁₁	D₄	1	1	yes	characteristic
12	C₈×C₂₁₁	Q₈	1	1	yes	characteristic
13	C₂²×C₂₁₁	C₂×C₈	1	1	yes	characteristic
14	C₄×C₂₁₁	—	2	1	no	—
15	C₄×C₂₁₁	C₁₆	1	2	yes	complemented
16	C₄×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₂₁₁	M₁₆	1	1	yes	characteristic
18	C₂×C₂₁₁	C₄⋊C₈	1	1	yes	characteristic
19	C₂×C₂₁₁	C₂×C₁₆	1	1	yes	characteristic
20	C₂×C₂₁₁	M₃₂	1	1	yes	characteristic
21	C₂₁₁	C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
22	C₄⋊C₁₆	—	211	1	no	maximal, Sylow
23	C₂×C₁₆	—	211	2	no	—
24	C₄×C₈	D₂₁₁	1	1	yes	characteristic
25	C₂×C₈	Dic₂₁₁	1	1	yes	characteristic
26	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
27	C₄²	Dic₂₁₁	1	1	yes	characteristic
28	C₁₆	—	422	2	no	—
29	C₂×C₄	C₂₁₁⋊C₈	1	1	yes	characteristic
30	C₂×C₄	C₂₁₁⋊C₈	1	1	yes	characteristic
31	C₂×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
32	C₈	—	2	1	no	—
33	C₈	D₈₄₄	1	1	yes	characteristic
34	C₈	Dic₄₂₂	1	1	yes	characteristic
35	C₂²	C₂×C₂₁₁⋊C₈	1	1	yes	characteristic
36	C₄	—	2	1	no	—
37	C₄	C₂₁₁⋊C₁₆	1	2	yes	complemented
38	C₄	C₄⋊Dic₂₁₁	1	1	yes	characteristic
39	C₄	#16809946	1	1	yes	characteristic
40	C₂	C₄⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
41	C₂	C₂×C₂₁₁⋊C₁₆	1	1	yes	characteristic
42	C₂	#20269917	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₂₁₁
commutator	C₂×C₂₁₁	C₂×C₁₆
Fitting	C₄×C₈×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂×C₈
cosocle	C₂×C₈×C₂₁₁	C₂²
hypercenter	C₄×C₈	D₂₁₁
hypocenter	C₂₁₁	C₄⋊C₁₆
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
15. C₄×C₂₁₁	C₁₆	semidirect
21. C₂₁₁	C₄⋊C₁₆	semidirect
37. C₄	C₂₁₁⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₁₁⋊C₁₆	2
C₄×C₈×C₂₁₁	2
C₄⋊C₁₆	211

Subgroups — C4⋊(C211⋊C16) #27212965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊(C₂₁₁⋊C₁₆) #27212965