Subgroups — #27213439

Order 13504.

96 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	Q₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#20269911	C₂	1	1	yes	maximal, characteristic
4	#20269910	C₂	1	1	yes	maximal, characteristic
5	Dic₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#20269901	C₂	1	1	yes	maximal, characteristic
7	#103×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	C₄×Dic₂₁₁	—	2	1	no	—
15	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×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	—	2	1	no	—
25	Q₈×C₂₁₁	—	2	1	no	—
26	Q₈×C₂₁₁	Q₈	1	2	yes	complemented
27	C₂×C₄×C₂₁₁	—	2	1	no	—
28	C₂×C₄×C₂₁₁	—	2	1	no	—
29	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₈×C₂₁₁	—	2	1	no	—
31	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₁₁	—	2	2	no	—
33	C₄×C₂₁₁	—	4	1	no	—
34	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₂₁₁	#41	1	1	yes	characteristic
36	Dic₂₁₁	—	2	1	no	—
37	Dic₂₁₁	—	4	1	no	—
38	Dic₂₁₁	—	4	1	no	—
39	Dic₂₁₁	Q₁₆	1	2	yes	—
40	C₂×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₂₁₁	#477	1	1	yes	characteristic, complemented, Sylow
44	#477	—	211	1	no	maximal, Sylow
45	C₄⋊Q₈	—	211	1	no	—
46	C₄×Q₈	—	211	1	no	—
47	#107	—	211	1	no	—
48	#107	—	211	1	no	—
49	C₄⋊C₈	—	211	1	no	—
50	#103	—	211	1	no	—
51	#103	D₂₁₁	1	1	yes	characteristic
52	C₂×Q₈	—	422	1	no	—
53	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
54	C₂×C₈	—	211	1	no	—
55	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
56	C₄⋊C₄	—	422	1	no	—
57	C₄⋊C₄	—	422	1	no	—
58	C₄⋊C₄	—	211	1	no	—
59	C₄⋊C₄	—	211	1	no	—
60	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
61	C₄²	—	422	1	no	—
62	C₄²	—	211	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	844	2	no	—
65	Q₈	Dic₄₂₂	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	422	1	no	—
69	C₂×C₄	—	422	1	no	—
70	C₂×C₄	—	211	1	no	—
71	C₂×C₄	—	211	1	no	—
72	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	422	1	no	—
75	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	844	1	no	—
79	C₄	—	844	1	no	—
80	C₄	—	422	1	no	—
81	C₄	—	211	2	no	—
82	C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
83	C₄	#16809961	1	1	yes	characteristic
84	C₂	#20269954	1	1	yes	characteristic
85	C₂	Q₁₆×D₂₁₁	1	1	yes	characteristic
86	C₂	#20270004	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#103×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂×D₄
hypercenter	#103	D₂₁₁
hypocenter	C₂₁₁	#477
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₂₁₁	Q₈	semidirect
43. C₂₁₁	#477	semidirect
65. Q₈	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
Q₈×Dic₂₁₁	2
#20269911	2
#20269910	2
Dic₂₁₁⋊C₈	2
#20269901	2
#103×C₂₁₁	2
#477	211