Subgroups — #27213556

Order 13504.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269977	C₂	1	1	yes	maximal, characteristic
2	#20269910	C₂	1	2	yes	maximal
3	#20269908	C₂	1	2	yes	maximal
4	#20269888	C₂	1	1	yes	maximal, characteristic
5	#125×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	—	2	1	no	—
7	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₁₁	—	2	1	no	—
9	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
10	C₄⋊C₄×C₂₁₁	—	2	1	no	—
11	C₄⋊C₄×C₂₁₁	—	2	1	no	—
12	C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	Dic₄₂₂	—	4	2	no	—
16	C₂×Dic₂₁₁	—	2	2	no	—
17	C₂₁₁⋊C₈	Q₈	1	4	yes	complemented
18	C₂×C₄×C₂₁₁	—	2	1	no	—
19	C₂×C₄×C₂₁₁	—	2	1	no	—
20	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
23	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
24	C₄×C₂₁₁	—	2	1	no	—
25	C₄×C₂₁₁	—	2	1	no	—
26	C₄×C₂₁₁	—	4	1	no	—
27	C₄×C₂₁₁	—	4	1	no	—
28	C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
29	Dic₂₁₁	—	4	2	no	—
30	C₂×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₂₁₁	#137	1	1	yes	characteristic
33	C₂₁₁	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	211	1	no	maximal, Sylow
35	C₄⋊Q₈	—	211	1	no	—
36	#125	D₂₁₁	1	1	yes	characteristic
37	#107	—	211	2	no	—
38	#106	—	211	2	no	—
39	#97	—	211	1	no	—
40	C₂×Q₈	—	422	1	no	—
41	C₂×C₈	—	211	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	422	1	no	—
45	C₄⋊C₄	—	211	2	no	—
46	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
47	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
48	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
49	Q₈	—	844	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	422	2	no	—
53	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
56	C₈	—	211	4	no	—
57	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	844	2	no	—
63	C₄	Q₈×D₂₁₁	1	2	yes	—
64	C₂	#20269975	1	1	yes	characteristic
65	C₂	#20270002	1	1	yes	characteristic
66	C₂	#20270007	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#125×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂×D₄
hypercenter	#125	D₂₁₁
hypocenter	C₂₁₁	#501
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
17. C₂₁₁⋊C₈	Q₈	semidirect
33. C₂₁₁	#501	semidirect

Maximal subgroups

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

subgroup	index
#20269977	2
#20269910	2
#20269908	2
#20269888	2
#125×C₂₁₁	2
#501	211