Subgroups — #27213554

Order 13504.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269976	C₂	1	1	yes	maximal, characteristic
2	#20269968	C₂	1	1	yes	maximal, characteristic
3	#20269910	C₂	1	1	yes	maximal, characteristic
4	#20269909	C₂	1	1	yes	maximal, characteristic
5	#20269908	C₂	1	1	yes	maximal, characteristic
6	#20269887	C₂	1	1	yes	maximal, characteristic
7	#107×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₂₂	—	2	1	no	—
9	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₁₁	—	2	1	no	—
11	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
13	Dic₂₁₁⋊C₄	—	2	1	no	—
14	Dic₂₁₁⋊C₄	—	2	1	no	—
15	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
19	Dic₄₂₂	—	4	2	no	—
20	C₂×Dic₂₁₁	—	2	1	no	—
21	C₂×Dic₂₁₁	—	2	1	no	—
22	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
23	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
24	C₂₁₁⋊C₈	Q₈	1	2	yes	complemented
25	C₂×C₄×C₂₁₁	—	2	1	no	—
26	C₂×C₄×C₂₁₁	—	2	1	no	—
27	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₈×C₂₁₁	Q₈	1	2	yes	complemented
29	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₁₁	—	4	1	no	—
31	C₄×C₂₁₁	—	4	1	no	—
32	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
34	Dic₂₁₁	—	2	1	no	—
35	Dic₂₁₁	—	2	1	no	—
36	Dic₂₁₁	—	4	1	no	—
37	Dic₂₁₁	—	4	1	no	—
38	C₂×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₂×C₂₁₁	#137	1	1	yes	characteristic
41	C₂₁₁	#501	1	1	yes	characteristic, complemented, Sylow
42	#501	—	211	1	no	maximal, Sylow
43	C₄⋊Q₈	—	211	1	no	—
44	#125	—	211	1	no	—
45	#107	—	211	1	no	—
46	#107	D₂₁₁	1	1	yes	characteristic
47	#106	—	211	1	no	—
48	#106	—	211	1	no	—
49	#97	—	211	1	no	—
50	C₂×Q₈	—	422	1	no	—
51	C₂×C₈	—	211	1	no	—
52	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
53	C₄⋊C₄	—	422	1	no	—
54	C₄⋊C₄	—	422	1	no	—
55	C₄⋊C₄	—	422	1	no	—
56	C₄⋊C₄	—	211	1	no	—
57	C₄⋊C₄	—	211	1	no	—
58	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
60	C₄²	—	211	1	no	—
61	Q₈	—	844	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	422	1	no	—
65	C₂×C₄	—	422	1	no	—
66	C₂×C₄	—	211	1	no	—
67	C₂×C₄	—	211	1	no	—
68	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
69	C₈	—	211	2	no	—
70	C₈	Dic₄₂₂	1	2	yes	complemented
71	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	844	1	no	—
75	C₄	—	844	1	no	—
76	C₄	—	422	1	no	—
77	C₄	—	422	1	no	—
78	C₄	Q₈×D₂₁₁	1	1	yes	characteristic
79	C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
80	C₂	#20269976	1	1	yes	characteristic
81	C₂	#20270001	1	1	yes	characteristic
82	C₂	#20270009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#107×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂×D₄
hypercenter	#107	D₂₁₁
hypocenter	C₂₁₁	#501
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
24. C₂₁₁⋊C₈	Q₈	semidirect
28. C₈×C₂₁₁	Q₈	semidirect
41. C₂₁₁	#501	semidirect
70. C₈	Dic₄₂₂	semidirect

Maximal subgroups

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

subgroup	index
#20269976	2
#20269968	2
#20269910	2
#20269909	2
#20269908	2
#20269887	2
#107×C₂₁₁	2
#501	211