Subgroups — #13068225

Order 1408.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#500	C₂	1	1	yes	maximal, characteristic
2	#10622706	C₂	1	2	yes	maximal
3	#10622694	C₂	1	2	yes	maximal
4	#10622295	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	#66922	C₂²	1	2	yes	—
8	C₁₁×#107	—	2	2	no	—
9	C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	#66899	—	2	4	no	—
11	#66882	—	2	2	no	—
12	#66874	—	2	4	no	—
13	C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₁₁	—	2	2	no	—
16	C₂×Dic₂₂	—	2	2	no	—
17	C₂×C₈×C₁₁	D₄	1	2	yes	—
18	C₁₁×C₄⋊C₄	—	2	2	no	—
19	C₁₁×C₄⋊C₄	—	2	2	no	—
20	C₄²×C₁₁	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₁₁	—	2	2	no	—
22	C₄⋊Dic₁₁	—	2	2	no	—
23	C₂×C₁₁⋊C₈	D₄	1	2	yes	—
24	C₂×C₁₁⋊C₈	D₄	1	2	yes	—
25	#1710	—	11	1	no	maximal, Sylow
26	Q₈×C₁₁	—	4	4	no	—
27	C₂×C₄×C₁₁	—	4	2	no	—
28	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₁₁	—	4	2	no	—
32	Dic₂₂	—	4	4	no	—
33	C₈×C₁₁	Q₁₆	1	4	yes	—
34	C₁₁⋊C₈	—	2	2	no	—
35	C₁₁⋊C₈	—	2	2	no	—
36	#500	—	11	1	no	—
37	#500	D₁₁	1	1	yes	characteristic
38	#487	—	11	2	no	—
39	#487	—	11	2	no	—
40	C₈²	—	11	1	no	—
41	C₂²×C₁₁	#127	1	1	yes	characteristic
42	C₄×C₁₁	—	8	2	no	—
43	C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
44	C₄×C₁₁	#135	1	2	yes	—
45	C₄×C₁₁	#135	1	2	yes	—
46	Dic₁₁	—	8	2	no	—
47	C₄⋊Q₈	—	11	2	no	—
48	C₄⋊Q₈	C₂×D₁₁	1	2	yes	—
49	#107	—	2	2	no	—
50	#107	—	22	2	no	—
51	#103	—	22	4	no	—
52	#103	—	22	4	no	—
53	C₄×C₈	—	11	1	no	—
54	C₄×C₈	—	11	1	no	—
55	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
56	C₂×C₁₁	#494	1	1	yes	characteristic
57	C₂×C₁₁	#495	1	1	yes	characteristic
58	C₂×C₁₁	#495	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	22	2	no	—
61	C₂×C₈	—	11	2	no	—
62	C₂×C₈	—	11	2	no	—
63	C₂×C₈	C₂²⋊D₁₁	1	2	yes	—
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	22	2	no	—
67	C₄⋊C₄	—	22	2	no	—
68	C₄²	C₂²×D₁₁	1	1	yes	characteristic
69	C₁₁	#1710	1	1	yes	characteristic, complemented, Sylow
70	Q₈	—	4	4	no	—
71	Q₈	—	44	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	44	2	no	—
74	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
75	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
76	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
77	C₈	—	22	2	no	—
78	C₈	—	22	2	no	—
79	C₈	#4326	1	4	yes	—
80	C₂²	#66993	1	1	yes	characteristic
81	C₄	—	8	2	no	—
82	C₄	—	88	2	no	—
83	C₄	#66965	1	2	yes	—
84	C₄	#66972	1	2	yes	—
85	C₄	C₂×#4326	1	2	yes	—
86	C₂	#10622705	1	1	yes	characteristic
87	C₂	#10622768	1	1	yes	characteristic
88	C₂	#10622804	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#66993
commutator, cosocle	C₄²×C₁₁	C₂³
Frattini	C₄²	C₂²×D₁₁
Fitting	C₁₁×#500	C₂
socle, prime radical	C₂²×C₁₁	#127
hypercenter	#500	D₁₁
hypocenter	C₁₁	#1710
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
69. C₁₁	#1710	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×#500	2
#10622706	2
#10622694	2
#10622295	2
C₈×C₁₁⋊C₈	2
#1710	11