Subgroups — #13183350

Order 1760.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#711	C₂	1	1	yes	maximal, characteristic
2	#11724979	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₁⋊#170	C₂	1	2	yes	maximal, complemented
4	#11724975	C₂	1	1	yes	maximal, characteristic, complemented
5	D₅⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×D₄×C₁₁	C₄	1	1	yes	characteristic
7	C₁₁×C₂²⋊D₅	C₄	1	2	yes	—
8	C₂×C₁₁×Dic₅	C₂²	1	2	yes	—
9	C₄×D₅×C₁₁	C₂²	1	1	yes	characteristic
10	C₁₁×Dic₁₀	C₄	1	1	yes	characteristic
11	C₁₁⋊#170	C₂²	1	1	yes	characteristic
12	C₁₁⋊#170	C₂²	1	1	yes	characteristic
13	C₁₁⋊#170	C₂²	1	2	yes	—
14	#67001	—	5	1	no	maximal, Hall
15	C₂²×C₅×C₁₁	C₂×C₄	1	2	yes	—
16	C₂×D₅×C₁₁	C₂×C₄	1	1	yes	characteristic
17	C₄×C₅×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₁₁×Dic₅	C₂×C₄	1	2	yes	—
19	C₁₁×Dic₅	C₂³	1	1	yes	characteristic
20	C₁₁×#41	—	5	1	no	—
21	#4318	—	5	1	no	—
22	#4318	—	5	2	no	—
23	C₂×C₁₁⋊C₈	—	5	1	no	—
24	C₂×C₁₁⋊C₈	—	5	2	no	—
25	#4131	—	11	1	no	maximal, Hall
26	C₂×C₅×C₁₁	—	2	2	no	—
27	C₂×C₅×C₁₁	C₂²×C₄	1	1	yes	characteristic
28	D₅×C₁₁	—	2	1	no	—
29	Q₈×C₁₁	—	5	1	no	—
30	D₄×C₁₁	—	5	2	no	—
31	D₄×C₁₁	AGL(1,𝔽₅)	1	1	yes	characteristic
32	C₂×C₄×C₁₁	—	5	2	no	—
33	C₂×C₄×C₁₁	—	5	1	no	—
34	C₁₁⋊C₈	—	5	1	no	—
35	C₁₁⋊C₈	—	5	1	no	—
36	C₁₁⋊C₈	—	5	2	no	—
37	#711	D₁₁	1	1	yes	characteristic
38	#704	—	11	2	no	—
39	C₂×#170	—	11	2	no	—
40	#700	—	11	1	no	—
41	D₅⋊C₈	—	11	1	no	—
42	C₅×C₁₁	#131	1	1	yes	characteristic, complemented, Hall
43	C₂²×C₁₁	—	5	1	no	—
44	C₂²×C₁₁	C₂×AGL(1,𝔽₅)	1	2	yes	—
45	C₄×C₁₁	—	5	1	no	—
46	C₄×C₁₁	—	5	2	no	—
47	C₄×C₁₁	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
48	C₅×D₄	Dic₁₁	1	1	yes	characteristic
49	C₂²⋊D₅	Dic₁₁	1	2	yes	—
50	C₂×Dic₅	C₂×D₁₁	1	2	yes	—
51	C₄×D₅	C₂×D₁₁	1	1	yes	characteristic
52	Dic₁₀	Dic₁₁	1	1	yes	characteristic
53	#170	—	11	1	no	—
54	#170	—	11	1	no	—
55	#170	—	11	2	no	—
56	#131	—	55	1	no	Sylow
57	C₂×C₁₁	—	2	2	no	—
58	C₂×C₁₁	—	10	1	no	—
59	C₂×C₁₁	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₂²×C₅	C₂×Dic₁₁	1	2	yes	—
61	C₂×D₅	C₂×Dic₁₁	1	1	yes	characteristic
62	C₄×C₅	C₂×Dic₁₁	1	1	yes	characteristic
63	Dic₅	C₂×Dic₁₁	1	2	yes	—
64	Dic₅	C₂²×D₁₁	1	1	yes	characteristic
65	#41	—	5	1	no	—
66	M₁₆	—	55	1	no	—
67	M₁₆	—	55	2	no	—
68	C₂×C₈	—	55	1	no	—
69	C₂×C₈	—	55	2	no	—
70	C₁₁	#4131	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₅	—	2	2	no	—
72	C₂×C₅	C₂²×Dic₁₁	1	1	yes	characteristic
73	D₅	—	2	1	no	—
74	Q₈	—	5	1	no	—
75	D₄	—	5	2	no	—
76	D₄	C₁₁⋊AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₂×C₄	—	5	2	no	—
78	C₂×C₄	—	5	1	no	—
79	C₈	—	55	1	no	—
80	C₈	—	55	1	no	—
81	C₈	—	55	2	no	—
82	C₅	#67001	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	5	1	no	—
84	C₂²	C₂×C₁₁⋊AGL(1,𝔽₅)	1	2	yes	—
85	C₄	—	5	1	no	—
86	C₄	—	5	2	no	—
87	C₄	C₂×C₁₁⋊AGL(1,𝔽₅)	1	1	yes	characteristic
88	C₂	—	2	2	no	—
89	C₂	—	10	1	no	—
90	C₂	C₂²×C₁₁⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×C₁₁⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₅×C₁₁	C₂²×C₄
Fitting	C₅×D₄×C₁₁	C₄
prime radical	C₁₁×#711	C₂
cosocle	C₁₁×Dic₅	C₂³
hypercenter	D₄	C₁₁⋊AGL(1,𝔽₅)
hypocenter	C₅×C₁₁	#131
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
2. #11724979	C₂	semidirect
3. C₂×C₁₁⋊#170	C₂	semidirect
4. #11724975	C₂	semidirect
5. D₅⋊(C₁₁⋊C₈)	C₂	semidirect
42. C₅×C₁₁	#131	semidirect
70. C₁₁	#4131	semidirect
82. C₅	#67001	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×#711	2
#11724979	2
C₂×C₁₁⋊#170	2
#11724975	2
D₅⋊(C₁₁⋊C₈)	2
#67001	5
#4131	11