Subgroups — #27212921

Order 13504.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₁₁⋊C₈	C₄	1	1	yes	characteristic
5	C₄×Dic₂₁₁	C₄	1	1	yes	characteristic
6	C₂₁₁⋊C₁₆	C₄	1	2	yes	complemented
7	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₁₆×C₂₁₁	C₄	1	2	yes	complemented
9	C₂×Dic₂₁₁	C₈	1	2	yes	—
10	C₂₁₁⋊C₈	C₈	1	2	yes	—
11	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
12	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
13	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₂₁₁	C₂×C₈	1	1	yes	characteristic
15	C₄×C₂₁₁	C₄²	1	1	yes	characteristic
16	C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
17	Dic₂₁₁	—	2	2	no	—
18	C₂×C₂₁₁	C₄×C₈	1	1	yes	characteristic
19	C₂×C₂₁₁	M₃₂	1	2	yes	—
20	C₂₁₁	#346	1	1	yes	characteristic, complemented, Sylow
21	#346	—	211	1	no	maximal, Sylow
22	C₂×C₁₆	—	211	1	no	—
23	C₂×C₁₆	D₂₁₁	1	1	yes	characteristic
24	C₄×C₈	—	211	1	no	—
25	C₂×C₈	—	211	1	no	—
26	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
27	C₄²	—	211	1	no	—
28	C₁₆	—	211	2	no	—
29	C₁₆	Dic₂₁₁	1	2	yes	complemented
30	C₂×C₄	—	211	2	no	—
31	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
32	C₈	—	211	2	no	—
33	C₈	C₄×D₂₁₁	1	1	yes	characteristic
34	C₈	C₂×Dic₂₁₁	1	1	yes	characteristic
35	C₂²	C₈×D₂₁₁	1	1	yes	characteristic
36	C₄	—	422	2	no	—
37	C₄	C₄×Dic₂₁₁	1	1	yes	characteristic
38	C₄	C₈×D₂₁₁	1	1	yes	characteristic
39	C₂	C₈×Dic₂₁₁	1	1	yes	characteristic
40	C₂	#20269916	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₂₁₁
commutator	C₂×C₂₁₁	C₄×C₈
Fitting	C₂×C₁₆×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂×C₈
cosocle	C₂×C₈×C₂₁₁	C₂²
hypercenter	C₂×C₁₆	D₂₁₁
hypocenter	C₂₁₁	#346
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
6. C₂₁₁⋊C₁₆	C₄	semidirect
8. C₁₆×C₂₁₁	C₄	semidirect
20. C₂₁₁	#346	semidirect
29. C₁₆	Dic₂₁₁	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
C₈×Dic₂₁₁	2
C₂×C₁₆×C₂₁₁	2
#346	211