Subgroups — #27212928

Order 13504.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269917	C₂	1	2	yes	maximal, complemented
2	C₂×M₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂₁₁⋊C₁₆	—	2	2	no	—
4	C₂²×C₄×C₂₁₁	C₄	1	1	yes	characteristic
5	M₁₆×C₂₁₁	—	2	2	no	—
6	C₂×C₈×C₂₁₁	C₄	1	1	yes	characteristic
7	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₂³×C₂₁₁	C₈	1	1	yes	characteristic
9	C₂×C₄×C₂₁₁	—	2	1	no	—
10	C₂×C₄×C₂₁₁	—	2	1	no	—
11	C₂×C₄×C₂₁₁	C₈	1	1	yes	characteristic
12	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
13	C₈×C₂₁₁	—	2	1	no	—
14	C₈×C₂₁₁	D₄	1	2	yes	—
15	C₂²×C₂₁₁	—	2	1	no	—
16	C₂²×C₂₁₁	—	4	1	no	—
17	C₂²×C₂₁₁	C₂×C₈	1	1	yes	characteristic
18	C₄×C₂₁₁	—	2	1	no	—
19	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₂₁₁	M₁₆	1	1	yes	characteristic
21	C₂×C₂₁₁	—	2	1	no	—
22	C₂×C₂₁₁	—	4	1	no	—
23	C₂×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
24	C₂₁₁	#349	1	1	yes	characteristic, complemented, Sylow
25	#349	—	211	1	no	maximal, Sylow
26	C₂×M₁₆	D₂₁₁	1	1	yes	characteristic
27	M₃₂	—	211	2	no	—
28	C₂²×C₄	Dic₂₁₁	1	1	yes	characteristic
29	M₁₆	—	2	2	no	—
30	C₂×C₈	Dic₂₁₁	1	1	yes	characteristic
31	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
32	C₁₆	—	422	2	no	—
33	C₂³	C₂₁₁⋊C₈	1	1	yes	characteristic
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	C₂₁₁⋊C₈	1	1	yes	characteristic
37	C₂×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	C₂²⋊D₂₁₁	1	2	yes	—
40	C₂²	—	2	1	no	—
41	C₂²	—	4	1	no	—
42	C₂²	C₂×C₂₁₁⋊C₈	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
45	C₄	#16809946	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	4	1	no	—
48	C₂	C₂²⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊Dic₂₁₁
commutator	C₂²×C₂₁₁	C₂×C₈
Frattini	C₂×C₈	C₂×D₂₁₁
Fitting	C₂×M₁₆×C₂₁₁	C₂
socle	C₂×C₂₁₁	C₂²⋊C₈
prime radical	C₂³×C₂₁₁	C₈
cosocle	C₂×C₈×C₂₁₁	C₂²
hypercenter	C₂×M₁₆	D₂₁₁
hypocenter	C₂₁₁	#349
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. #20269917	C₂	semidirect
24. C₂₁₁	#349	semidirect

Maximal subgroups

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

subgroup	index
#20269917	2
C₂×M₁₆×C₂₁₁	2
#349	211