Subgroups — #11760917

Order 1056.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63143	C₂	1	2	yes	maximal
2	C₄⋊Dic₃₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₁×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
5	Dic₃⋊Dic₁₁	C₂	1	2	yes	maximal
6	#66922	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×Dic₃₃	C₂²	1	2	yes	—
9	C₂×C₁₁×Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×Dic₁₁	C₂²	1	2	yes	—
12	#63143	—	2	4	no	—
13	C₂×Dic₂₂	—	3	2	no	—
14	C₄²×C₁₁	—	3	1	no	—
15	C₄⋊Dic₁₁	—	3	2	no	—
16	C₄⋊Dic₁₁	—	3	1	no	—
17	C₄⋊Dic₁₁	D₃	1	1	yes	characteristic
18	C₂²×C₃×C₁₁	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₁₁	Q₈	1	2	yes	complemented
20	Dic₃₃	—	2	2	no	—
21	C₃×Dic₁₁	—	2	2	no	—
22	C₁₁×Dic₃	D₄	1	2	yes	—
23	C₁₁×Dic₃	Q₈	1	2	yes	complemented
24	#888	—	11	1	no	maximal, Hall
25	C₂×C₄×C₁₁	—	3	1	no	—
26	C₂×C₄×C₁₁	—	3	1	no	—
27	C₂×C₄×C₁₁	C₂×D₃	1	1	yes	characteristic
28	C₂×Dic₁₁	—	3	2	no	—
29	C₂×Dic₁₁	C₂×D₃	1	2	yes	—
30	Dic₂₂	—	6	4	no	—
31	C₂×C₃×C₁₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃×C₁₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₃×C₁₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×Dic₆	—	11	2	no	—
35	C₃×C₄⋊C₄	—	11	1	no	—
36	C₄⋊Dic₃	—	11	1	no	—
37	Dic₃⋊C₄	—	11	2	no	—
38	C₄×Dic₃	D₁₁	1	1	yes	characteristic
39	C₂²×C₁₁	C₂²×D₃	1	1	yes	characteristic
40	C₄×C₁₁	—	3	2	no	—
41	C₄×C₁₁	—	3	2	no	—
42	C₄×C₁₁	Dic₆	1	2	yes	complemented
43	Dic₁₁	—	2	2	no	—
44	Dic₁₁	—	6	2	no	—
45	C₃×C₁₁	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
46	C₄⋊Q₈	—	33	1	no	Sylow
47	C₂×C₃×C₄	—	11	2	no	—
48	C₂×C₃×C₄	C₂×D₁₁	1	1	yes	characteristic
49	C₂×Dic₃	—	11	2	no	—
50	C₂×Dic₃	C₂×D₁₁	1	1	yes	characteristic
51	C₂×Dic₃	C₂×D₁₁	1	1	yes	characteristic
52	Dic₆	—	22	4	no	—
53	C₂×C₁₁	C₂×Dic₆	1	1	yes	characteristic
54	C₂×C₁₁	D₃×D₄	1	1	yes	characteristic
55	C₂×C₁₁	D₃×Q₈	1	1	yes	characteristic
56	C₂×Q₈	—	33	2	no	—
57	C₄⋊C₄	—	11	1	no	—
58	C₄⋊C₄	—	33	2	no	—
59	C₄⋊C₄	—	33	1	no	—
60	C₄²	—	3	1	no	—
61	C₂²×C₃	C₂²×D₁₁	1	1	yes	characteristic
62	C₃×C₄	—	22	2	no	—
63	C₃×C₄	Dic₂₂	1	2	yes	complemented
64	Dic₃	—	22	2	no	—
65	Dic₃	Dic₂₂	1	2	yes	complemented
66	Dic₃	D₄₄	1	2	yes	—
67	C₁₁	#888	1	1	yes	characteristic, complemented, Sylow
68	Q₈	—	66	4	no	—
69	C₂×C₄	—	3	1	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₄	—	11	2	no	—
72	C₂×C₄	—	33	2	no	—
73	C₂×C₄	D₃×D₁₁	1	1	yes	characteristic
74	C₂×C₃	C₂×Dic₂₂	1	1	yes	characteristic
75	C₂×C₃	C₂×Dic₂₂	1	1	yes	characteristic
76	C₂×C₃	C₂×D₄₄	1	1	yes	characteristic
77	C₂²	C₂×D₃×D₁₁	1	1	yes	characteristic
78	C₄	—	3	2	no	—
79	C₄	—	3	2	no	—
80	C₄	—	22	2	no	—
81	C₄	—	66	2	no	—
82	C₄	#63143	1	2	yes	complemented
83	C₃	#66922	1	1	yes	characteristic, complemented, Sylow
84	C₂	D₃×Dic₂₂	1	1	yes	characteristic
85	C₂	D₃×D₄₄	1	1	yes	characteristic
86	C₂	C₂×#63143	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×D₁₁
commutator, socle, prime radical, cosocle	C₂²×C₃×C₁₁	C₂³
Fitting	C₂×C₃×C₄×C₁₁	C₂²
hypercenter	C₂×C₄	D₃×D₁₁
hypocenter	C₃×C₁₁	C₄⋊Q₈
LCS	G ▷ C₂²×C₃×C₁₁ ▷ C₃×C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
19. C₃×C₄×C₁₁	Q₈	semidirect
23. C₁₁×Dic₃	Q₈	semidirect
42. C₄×C₁₁	Dic₆	semidirect
45. C₃×C₁₁	C₄⋊Q₈	semidirect
63. C₃×C₄	Dic₂₂	semidirect
65. Dic₃	Dic₂₂	semidirect
67. C₁₁	#888	semidirect
82. C₄	#63143	semidirect
83. C₃	#66922	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63143	2
C₄⋊Dic₃₃	2
C₄×C₁₁×Dic₃	2
C₃×C₄⋊Dic₁₁	2
Dic₃⋊Dic₁₁	2
#66922	3
#888	11