Subgroups — #13068214

Order 1408.

180 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	#10622694	C₂	1	2	yes	maximal
3	#10622692	C₂	1	2	yes	maximal
4	#10622312	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	C₂×Dic₄₄	—	2	2	no	—
8	#66922	C₂²	1	2	yes	—
9	C₁₁×#107	—	2	2	no	—
10	C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	#66899	—	2	4	no	—
12	#66872	—	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	Dic₄₄	—	4	4	no	—
26	#1707	—	11	1	no	maximal, Sylow
27	Q₈×C₁₁	—	4	4	no	—
28	C₂×C₄×C₁₁	—	4	2	no	—
29	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₁₁	—	4	2	no	—
33	Dic₂₂	—	4	4	no	—
34	C₈×C₁₁	SD₁₆	1	4	yes	—
35	C₁₁⋊C₈	—	2	2	no	—
36	C₁₁⋊C₈	Q₁₆	1	4	yes	complemented
37	#500	D₁₁	1	1	yes	characteristic
38	#498	—	11	2	no	—
39	#494	—	11	1	no	—
40	#487	—	11	2	no	—
41	C₈²	—	11	1	no	—
42	C₂²×C₁₁	#127	1	1	yes	characteristic
43	C₄×C₁₁	—	8	2	no	—
44	C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
46	C₄×C₁₁	#135	1	2	yes	—
47	Dic₁₁	—	8	2	no	—
48	C₂×Q₁₆	—	22	2	no	—
49	C₄⋊Q₈	—	11	2	no	—
50	C₄⋊Q₈	C₂×D₁₁	1	2	yes	—
51	#107	—	2	2	no	—
52	#106	—	22	4	no	—
53	#103	—	22	4	no	—
54	C₄×C₈	—	11	1	no	—
55	C₄×C₈	—	11	1	no	—
56	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
57	C₂×C₁₁	#492	1	1	yes	characteristic
58	C₂×C₁₁	#494	1	1	yes	characteristic
59	C₂×C₁₁	#495	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	22	2	no	—
62	Q₁₆	—	44	4	no	—
63	C₂×C₈	—	11	2	no	—
64	C₂×C₈	—	11	2	no	—
65	C₂×C₈	C₂²⋊D₁₁	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	22	2	no	—
69	C₄⋊C₄	—	22	2	no	—
70	C₄²	C₂²×D₁₁	1	1	yes	characteristic
71	C₁₁	#1707	1	1	yes	characteristic, complemented, Sylow
72	Q₈	—	4	4	no	—
73	Q₈	—	44	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	44	2	no	—
76	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
77	C₂×C₄	D₄×D₁₁	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
79	C₈	—	11	4	no	—
80	C₈	—	22	2	no	—
81	C₈	#4325	1	4	yes	—
82	C₂²	#66993	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	88	2	no	—
85	C₄	#66965	1	2	yes	—
86	C₄	Q₁₆×D₁₁	1	2	yes	—
87	C₄	C₂×#4325	1	2	yes	—
88	C₂	#10622698	1	1	yes	characteristic
89	C₂	#10622768	1	1	yes	characteristic
90	C₂	#10622794	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	#1707
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
36. C₁₁⋊C₈	Q₁₆	semidirect
71. C₁₁	#1707	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
#10622694	2
#10622692	2
#10622312	2
C₈×C₁₁⋊C₈	2
#1707	11