Subgroups — #27213624

Order 13504.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₁₁	C₂	1	8	yes	maximal, complemented
2	#20269936	C₂	1	6	yes	maximal
3	C₂²×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₂₁₁	C₂²	1	12	yes	—
5	C₂²⋊Dic₂₁₁	C₂²	1	4	yes	—
6	C₄×Dic₂₁₁	C₂²	1	12	yes	—
7	C₂×Q₈×C₂₁₁	C₄	1	8	yes	complemented
8	C₂×Q₈×C₂₁₁	C₂²	1	4	yes	—
9	C₂²×C₄×C₂₁₁	C₂²	1	3	yes	—
10	C₂×Dic₂₁₁	C₂³	1	8	yes	—
11	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
12	Q₈×C₂₁₁	C₂×C₄	1	16	yes	—
13	C₂×C₄×C₂₁₁	C₂×C₄	1	12	yes	—
14	C₂×C₄×C₂₁₁	C₂³	1	6	yes	—
15	C₂²×C₂₁₁	—	2	2	no	—
16	C₂²×C₂₁₁	C₂²×C₄	1	2	yes	—
17	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
18	C₄×C₂₁₁	C₂²×C₄	1	12	yes	—
19	Dic₂₁₁	—	2	8	no	—
20	C₂×C₂₁₁	—	2	2	no	—
21	C₂×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
22	C₂×C₂₁₁	2₋¹⁺⁴	1	2	yes	—
23	C₂₁₁	#519	1	1	yes	characteristic, complemented, Sylow
24	#519	—	211	1	no	maximal, Sylow
25	C₂²×Q₈	D₂₁₁	1	1	yes	characteristic
26	C₄×Q₈	—	211	8	no	—
27	#117	—	211	6	no	—
28	C₂×Q₈	Dic₂₁₁	1	8	yes	complemented
29	C₂×Q₈	C₂×D₂₁₁	1	4	yes	—
30	C₂²×C₄	C₂×D₂₁₁	1	3	yes	—
31	C₄⋊C₄	—	211	12	no	—
32	C₂²⋊C₄	—	211	4	no	—
33	C₄²	—	211	12	no	—
34	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
35	Q₈	C₂×Dic₂₁₁	1	16	yes	—
36	C₂×C₄	—	211	8	no	—
37	C₂×C₄	C₂×Dic₂₁₁	1	12	yes	—
38	C₂×C₄	C₂²×D₂₁₁	1	6	yes	—
39	C₂²	—	2	2	no	—
40	C₂²	C₂²×Dic₂₁₁	1	2	yes	—
41	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
42	C₄	—	422	8	no	—
43	C₄	C₂²×Dic₂₁₁	1	12	yes	—
44	C₂	—	2	2	no	—
45	C₂	C₂³×Dic₂₁₁	1	1	yes	characteristic
46	C₂	#20270025	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator	C₂×C₂₁₁	C₂³×C₄
Fitting	C₂²×Q₈×C₂₁₁	C₂
socle, cosocle	C₂²×C₂₁₁	C₂⁴
prime radical	C₂³×C₂₁₁	C₂³
hypercenter	C₂²×Q₈	D₂₁₁
hypocenter	C₂₁₁	#519
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₁₁	C₂	semidirect
7. C₂×Q₈×C₂₁₁	C₄	semidirect
23. C₂₁₁	#519	semidirect
28. C₂×Q₈	Dic₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₁₁	2
#20269936	2
C₂²×Q₈×C₂₁₁	2
#519	211