Subgroups — #27213012

Order 13504.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₁₁	C₂	1	2	yes	maximal
2	C₂₁₁⋊#95	C₂	1	4	yes	maximal
3	C₂×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₁₁	C₂²	1	4	yes	—
5	C₄⋊Dic₂₁₁	—	2	4	no	—
6	C₂²×C₄×C₂₁₁	C₂²	1	2	yes	—
7	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₄²×C₂₁₁	C₄	1	4	yes	complemented
9	C₂×Dic₂₁₁	—	2	4	no	—
10	C₂×Dic₂₁₁	—	2	8	no	—
11	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
12	C₂×C₄×C₂₁₁	—	2	4	no	—
13	C₂×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
14	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
16	C₂×C₄×C₂₁₁	Q₈	1	2	yes	—
17	C₂²×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
18	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
19	C₂²×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
20	C₂²×C₂₁₁	#41	1	4	yes	—
21	C₄×C₂₁₁	—	2	4	no	—
22	C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
23	Dic₂₁₁	—	4	4	no	—
24	C₂×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₂₁₁	#117	1	2	yes	—
26	C₂×C₂₁₁	#124	1	2	yes	—
27	C₂×C₂₁₁	#125	1	2	yes	—
28	C₂₁₁	#382	1	1	yes	characteristic, complemented, Sylow
29	#382	—	211	1	no	maximal, Sylow
30	C₂×C₄⋊C₄	—	211	2	no	—
31	C₂×C₄²	D₂₁₁	1	1	yes	characteristic
32	#95	—	211	4	no	—
33	C₂²×C₄	—	211	4	no	—
34	C₂²×C₄	C₂×D₂₁₁	1	2	yes	—
35	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
36	C₄⋊C₄	—	422	4	no	—
37	C₄²	Dic₂₁₁	1	4	yes	complemented
38	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
39	C₂×C₄	—	2	4	no	—
40	C₂×C₄	—	422	4	no	—
41	C₂×C₄	—	422	8	no	—
42	C₂×C₄	C₂×Dic₂₁₁	1	4	yes	—
43	C₂×C₄	C₂×Dic₂₁₁	1	2	yes	—
44	C₂×C₄	D₈₄₄	1	2	yes	—
45	C₂×C₄	Dic₄₂₂	1	2	yes	—
46	C₂²	C₂²×Dic₂₁₁	1	1	yes	characteristic
47	C₂²	C₂×D₈₄₄	1	1	yes	characteristic
48	C₂²	C₂×Dic₄₂₂	1	1	yes	characteristic
49	C₂²	#16809962	1	4	yes	—
50	C₄	—	2	4	no	—
51	C₄	—	844	4	no	—
52	C₄	C₄⋊Dic₂₁₁	1	4	yes	—
53	C₂	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
54	C₂	#20269936	1	2	yes	—
55	C₂	#20269965	1	2	yes	—
56	C₂	#20269969	1	2	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	#382
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
8. C₄²×C₂₁₁	C₄	semidirect
28. C₂₁₁	#382	semidirect
37. C₄²	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
C₂₁₁⋊#95	2
C₂×C₄²×C₂₁₁	2
#382	211