Subgroups — #117×C₁₁⋊C₅ #13182589

Order 1760.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₁⋊C₅	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×C₁₁⋊C₅	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₄×C₁₁⋊C₅	C₂	1	2	yes	maximal
4	C₄²×C₁₁⋊C₅	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₁⋊C₅	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₁₁⋊C₅	C₄	1	4	yes	complemented
7	C₂×C₄×C₁₁⋊C₅	C₂²	1	4	yes	—
8	C₂×C₄×C₁₁⋊C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₁⋊C₅	C₂²	1	1	yes	characteristic
10	C₁₁×#117	C₅	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂²×C₁₁⋊C₅	—	2	2	no	—
12	C₂²×C₁₁⋊C₅	C₂×C₄	1	2	yes	—
13	C₂²×C₁₁⋊C₅	C₂³	1	1	yes	characteristic
14	C₄×C₁₁⋊C₅	—	2	4	no	—
15	C₄×C₁₁⋊C₅	C₂×C₄	1	2	yes	—
16	C₄×C₁₁⋊C₅	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₁₁	C₂×C₅	1	1	yes	characteristic
18	C₁₁×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
19	C₁₁×C₂²⋊C₄	C₂×C₅	1	2	yes	—
20	C₄²×C₁₁	C₂×C₅	1	2	yes	complemented
21	C₅×#117	—	11	1	no	maximal, Hall
22	C₂×C₁₁⋊C₅	—	2	2	no	—
23	C₂×C₁₁⋊C₅	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₁₁⋊C₅	#41	1	2	yes	—
25	C₂³×C₁₁	C₂²×C₅	1	1	yes	characteristic
26	C₂×C₄×C₁₁	C₄×C₅	1	4	yes	complemented
27	C₂×C₄×C₁₁	C₂²×C₅	1	4	yes	—
28	C₂×C₄×C₁₁	C₂²×C₅	1	1	yes	characteristic
29	C₂×C₄×C₁₁	C₂²×C₅	1	1	yes	characteristic
30	C₂²×C₄×C₅	—	11	1	no	—
31	C₅×C₄⋊C₄	—	11	2	no	—
32	C₅×C₂²⋊C₄	—	11	2	no	—
33	C₄²×C₅	—	11	2	no	—
34	C₁₁⋊C₅	#117	1	1	yes	characteristic, direct product factor, Hall
35	C₂²×C₁₁	—	2	2	no	—
36	C₂²×C₁₁	C₂×C₄×C₅	1	2	yes	—
37	C₂²×C₁₁	C₂³×C₅	1	1	yes	characteristic
38	C₄×C₁₁	—	2	4	no	—
39	C₄×C₁₁	C₂×C₄×C₅	1	2	yes	—
40	C₄×C₁₁	C₂×C₄×C₅	1	2	yes	—
41	C₂³×C₅	—	11	1	no	—
42	C₂×C₄×C₅	—	11	4	no	—
43	C₂×C₄×C₅	—	11	1	no	—
44	C₂×C₄×C₅	—	11	4	no	—
45	C₂×C₄×C₅	—	11	1	no	—
46	#117	C₁₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×C₁₁	—	2	2	no	—
48	C₂×C₁₁	C₂²×C₄×C₅	1	1	yes	characteristic
49	C₂×C₁₁	C₅×#41	1	2	yes	—
50	C₂²×C₅	—	11	2	no	—
51	C₂²×C₅	—	11	1	no	—
52	C₂²×C₅	—	22	2	no	—
53	C₄×C₅	—	11	2	no	—
54	C₄×C₅	—	11	2	no	—
55	C₄×C₅	—	22	4	no	—
56	C₂²×C₄	C₂×C₁₁⋊C₅	1	1	yes	characteristic
57	C₄⋊C₄	C₂×C₁₁⋊C₅	1	2	yes	complemented
58	C₂²⋊C₄	C₂×C₁₁⋊C₅	1	2	yes	—
59	C₄²	C₂×C₁₁⋊C₅	1	2	yes	complemented
60	C₁₁	C₅×#117	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₅	—	11	1	no	—
62	C₂×C₅	—	11	2	no	—
63	C₂×C₅	—	22	2	no	—
64	C₂³	C₂²×C₁₁⋊C₅	1	1	yes	characteristic
65	C₂×C₄	C₄×C₁₁⋊C₅	1	4	yes	complemented
66	C₂×C₄	C₂²×C₁₁⋊C₅	1	4	yes	—
67	C₂×C₄	C₂²×C₁₁⋊C₅	1	1	yes	characteristic
68	C₂×C₄	C₂²×C₁₁⋊C₅	1	1	yes	characteristic
69	C₅	—	11	1	no	Sylow
70	C₂²	—	2	2	no	—
71	C₂²	C₂×C₄×C₁₁⋊C₅	1	2	yes	—
72	C₂²	C₂³×C₁₁⋊C₅	1	1	yes	characteristic
73	C₄	—	2	4	no	—
74	C₄	C₂×C₄×C₁₁⋊C₅	1	2	yes	—
75	C₄	C₂×C₄×C₁₁⋊C₅	1	2	yes	—
76	C₂	—	2	2	no	—
77	C₂	C₂²×C₄×C₁₁⋊C₅	1	1	yes	characteristic
78	C₂	#41×C₁₁⋊C₅	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₁₁⋊C₅
commutator	C₂×C₁₁	C₂²×C₄×C₅
Frattini	C₂²	C₂³×C₁₁⋊C₅
Fitting	C₁₁×#117	C₅
socle, cosocle	C₂²×C₁₁	C₂³×C₅
prime radical	C₂³×C₁₁⋊C₅	C₂²
hypercenter	#117	C₁₁⋊C₅
hypocenter	C₁₁	C₅×#117
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×C₁₁⋊C₅	C₂	semidirect
4. C₄²×C₁₁⋊C₅	C₂	semidirect
6. C₂×C₄×C₁₁⋊C₅	C₄	semidirect
10. C₁₁×#117	C₅	semidirect
18. C₁₁×C₄⋊C₄	C₂×C₅	semidirect
20. C₄²×C₁₁	C₂×C₅	semidirect
26. C₂×C₄×C₁₁	C₄×C₅	semidirect
34. C₁₁⋊C₅	#117	direct (N × H)
46. #117	C₁₁⋊C₅	direct (N × H)
57. C₄⋊C₄	C₂×C₁₁⋊C₅	semidirect
59. C₄²	C₂×C₁₁⋊C₅	semidirect
60. C₁₁	C₅×#117	semidirect
65. C₂×C₄	C₄×C₁₁⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₁₁⋊C₅	2
C₄⋊C₄×C₁₁⋊C₅	2
C₂²⋊C₄×C₁₁⋊C₅	2
C₄²×C₁₁⋊C₅	2
C₁₁×#117	5
C₅×#117	11

Subgroups — #117×C11⋊C5 #13182589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #117×C₁₁⋊C₅ #13182589