Subgroups — #27213016

Order 13504.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₁₁	C₂	1	6	yes	maximal
2	C₂×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₂₁₁	C₂²	1	4	yes	—
4	C₄⋊Dic₂₁₁	—	2	12	no	—
5	C₂²×C₄×C₂₁₁	C₂²	1	3	yes	—
6	C₄²×C₂₁₁	C₄	1	4	yes	complemented
7	C₂×Dic₂₁₁	—	2	12	no	—
8	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
9	C₂×C₄×C₂₁₁	C₂×C₄	1	6	yes	—
10	C₂×C₄×C₂₁₁	D₄	1	6	yes	—
11	C₂×C₄×C₂₁₁	Q₈	1	6	yes	—
12	C₂²×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
13	C₂²×C₂₁₁	C₂×D₄	1	3	yes	—
14	C₂²×C₂₁₁	C₂×Q₈	1	3	yes	—
15	C₄×C₂₁₁	C₄⋊C₄	1	12	yes	complemented
16	Dic₂₁₁	—	4	4	no	—
17	C₂×C₂₁₁	C₂×C₄⋊C₄	1	3	yes	—
18	C₂×C₂₁₁	#127	1	1	yes	characteristic
19	C₂×C₂₁₁	C₄⋊Q₈	1	3	yes	—
20	C₂₁₁	#384	1	1	yes	characteristic, complemented, Sylow
21	#384	—	211	1	no	maximal, Sylow
22	C₂×C₄⋊C₄	—	211	6	no	—
23	C₂×C₄²	D₂₁₁	1	1	yes	characteristic
24	C₂²×C₄	—	211	4	no	—
25	C₂²×C₄	C₂×D₂₁₁	1	3	yes	—
26	C₄⋊C₄	—	422	12	no	—
27	C₄²	Dic₂₁₁	1	4	yes	complemented
28	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
29	C₂×C₄	—	422	12	no	—
30	C₂×C₄	C₂×Dic₂₁₁	1	6	yes	—
31	C₂×C₄	D₈₄₄	1	6	yes	—
32	C₂×C₄	Dic₄₂₂	1	6	yes	—
33	C₂²	C₂²×Dic₂₁₁	1	1	yes	characteristic
34	C₂²	C₂×D₈₄₄	1	3	yes	—
35	C₂²	C₂×Dic₄₂₂	1	3	yes	—
36	C₄	—	844	4	no	—
37	C₄	C₄⋊Dic₂₁₁	1	12	yes	complemented
38	C₂	C₂×C₄⋊Dic₂₁₁	1	3	yes	—
39	C₂	#20269974	1	1	yes	characteristic
40	C₂	#20269977	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₁₁
commutator	C₂²×C₂₁₁	C₂²×C₄
Fitting	C₂×C₄²×C₂₁₁	C₂
socle, prime radical, cosocle	C₂³×C₂₁₁	C₂³
hypercenter	C₂×C₄²	D₂₁₁
hypocenter	C₂₁₁	#384
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. C₄²×C₂₁₁	C₄	semidirect
15. C₄×C₂₁₁	C₄⋊C₄	semidirect
20. C₂₁₁	#384	semidirect
27. C₄²	Dic₂₁₁	semidirect
37. C₄	C₄⋊Dic₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₁₁	2
C₂×C₄²×C₂₁₁	2
#384	211