Subgroups — #27213513

Order 13504.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269975	C₂	1	1	yes	maximal, characteristic
2	#20269968	C₂	1	1	yes	maximal, characteristic
3	#20269902	C₂	1	1	yes	maximal, characteristic
4	#20269901	C₂	1	1	yes	maximal, characteristic
5	#20269900	C₂	1	1	yes	maximal, characteristic
6	#20269887	C₂	1	1	yes	maximal, characteristic
7	#103×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₂₂	C₂²	1	1	yes	characteristic
9	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₁₁	—	2	1	no	—
11	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
12	Dic₂₁₁⋊C₄	—	2	1	no	—
13	Dic₂₁₁⋊C₄	—	2	1	no	—
14	Dic₂₁₁⋊C₄	—	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	—	2	1	no	—
20	Dic₄₂₂	—	2	1	no	—
21	C₂×Dic₂₁₁	—	2	1	no	—
22	C₂×Dic₂₁₁	—	2	1	no	—
23	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
24	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
25	C₂₁₁⋊C₈	—	2	1	no	—
26	Q₈×C₂₁₁	—	2	1	no	—
27	Q₈×C₂₁₁	—	2	1	no	—
28	C₂×C₄×C₂₁₁	—	2	1	no	—
29	C₂×C₄×C₂₁₁	—	2	1	no	—
30	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₈×C₂₁₁	—	2	1	no	—
32	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₁₁	—	4	1	no	—
34	C₄×C₂₁₁	—	4	1	no	—
35	C₄×C₂₁₁	#41	1	1	yes	characteristic
36	C₄×C₂₁₁	#41	1	1	yes	characteristic
37	Dic₂₁₁	—	2	1	no	—
38	Dic₂₁₁	—	2	1	no	—
39	Dic₂₁₁	—	4	1	no	—
40	Dic₂₁₁	—	4	1	no	—
41	C₂×C₂₁₁	#124	1	1	yes	characteristic
42	C₂×C₂₁₁	#137	1	1	yes	characteristic
43	C₂×C₂₁₁	#137	1	1	yes	characteristic
44	C₂₁₁	#491	1	1	yes	characteristic, complemented, Sylow
45	#491	—	211	1	no	maximal, Sylow
46	C₄⋊Q₈	—	211	1	no	—
47	#125	—	211	1	no	—
48	#103	—	211	1	no	—
49	#103	—	211	1	no	—
50	#103	—	211	1	no	—
51	#103	D₂₁₁	1	1	yes	characteristic
52	#97	—	211	1	no	—
53	C₂×Q₈	—	211	1	no	—
54	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
55	C₂×C₈	—	211	1	no	—
56	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
57	C₄⋊C₄	—	422	1	no	—
58	C₄⋊C₄	—	422	1	no	—
59	C₄⋊C₄	—	422	1	no	—
60	C₄⋊C₄	—	422	1	no	—
61	C₄⋊C₄	—	211	1	no	—
62	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
63	C₄²	—	211	1	no	—
64	Q₈	—	2	1	no	—
65	Q₈	—	2	1	no	—
66	Q₈	—	422	1	no	—
67	Q₈	—	422	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	422	1	no	—
71	C₂×C₄	—	422	1	no	—
72	C₂×C₄	—	211	1	no	—
73	C₂×C₄	—	211	1	no	—
74	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	—	422	1	no	—
77	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
78	C₄	—	4	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	844	1	no	—
81	C₄	—	844	1	no	—
82	C₄	—	422	1	no	—
83	C₄	—	422	1	no	—
84	C₄	#16809961	1	1	yes	characteristic
85	C₄	#16809962	1	1	yes	characteristic
86	C₂	#20269963	1	1	yes	characteristic
87	C₂	#20270006	1	1	yes	characteristic
88	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	#103×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂×D₄
hypercenter	#103	D₂₁₁
hypocenter	C₂₁₁	#491
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
44. C₂₁₁	#491	semidirect

Maximal subgroups

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

subgroup	index
#20269975	2
#20269968	2
#20269902	2
#20269901	2
#20269900	2
#20269887	2
#103×C₂₁₁	2
#491	211