Subgroups — #13083328

Order 1408.

1424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#585	C₂	1	1	yes	maximal, characteristic
2	#10623350	C₂	1	15	yes	maximal, complemented
3	C₂×#67001	C₂	1	15	yes	maximal, complemented
4	C₁₁×2₋¹⁺⁴	C₄	1	6	yes	—
5	C₁₁×H(5,𝔽₂)	C₄	1	10	yes	—
6	C₂×C₁₁×#41	C₂²	1	15	yes	—
7	#67001	C₂²	1	20	yes	complemented
8	#67001	C₂²	1	60	yes	complemented
9	C₂×#4318	C₂²	1	45	yes	complemented
10	C₂²×C₁₁⋊C₈	C₂²	1	15	yes	complemented
11	C₁₁×#41	C₂×C₄	1	60	yes	—
12	C₁₁×#41	C₂³	1	20	yes	—
13	C₂×Q₈×C₁₁	C₂×C₄	1	15	yes	—
14	C₂×D₄×C₁₁	C₂×C₄	1	45	yes	—
15	C₂²×C₄×C₁₁	C₂³	1	15	yes	—
16	#4318	C₂³	1	60	yes	—
17	C₂×C₁₁⋊C₈	C₂³	1	15	yes	—
18	C₂×C₁₁⋊C₈	C₂³	1	45	yes	—
19	#3573	—	11	1	no	maximal, Sylow
20	C₂³×C₁₁	C₂²×C₄	1	15	yes	—
21	Q₈×C₁₁	C₂²×C₄	1	20	yes	—
22	D₄×C₁₁	C₂²×C₄	1	60	yes	—
23	C₂×C₄×C₁₁	C₂²×C₄	1	45	yes	—
24	C₂×C₄×C₁₁	C₂⁴	1	15	yes	—
25	C₁₁⋊C₈	C₂⁴	1	1	yes	characteristic
26	C₁₁⋊C₈	C₂⁴	1	15	yes	—
27	#585	D₁₁	1	1	yes	characteristic
28	#568	—	11	15	no	—
29	C₂×#131	—	11	15	no	—
30	C₂²×C₁₁	—	4	15	no	—
31	C₂²×C₁₁	C₂³×C₄	1	15	yes	—
32	C₄×C₁₁	C₂³×C₄	1	15	yes	—
33	C₄×C₁₁	C₂⁵	1	1	yes	characteristic
34	2₋¹⁺⁴	Dic₁₁	1	6	yes	—
35	H(5,𝔽₂)	Dic₁₁	1	10	yes	—
36	C₂×#41	C₂×D₁₁	1	15	yes	—
37	#131	—	11	20	no	—
38	#131	—	11	60	no	—
39	C₂×M₁₆	—	11	45	no	—
40	C₂²×C₈	—	11	15	no	—
41	C₂×C₁₁	—	2	15	no	—
42	C₂×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
43	#41	C₂×Dic₁₁	1	60	yes	—
44	#41	C₂²×D₁₁	1	20	yes	—
45	C₂×Q₈	C₂×Dic₁₁	1	15	yes	—
46	C₂×D₄	C₂×Dic₁₁	1	45	yes	—
47	C₂²×C₄	C₂²×D₁₁	1	15	yes	—
48	M₁₆	—	11	60	no	—
49	C₂×C₈	—	11	15	no	—
50	C₂×C₈	—	11	45	no	—
51	C₁₁	#3573	1	1	yes	characteristic, complemented, Sylow
52	C₂³	C₂²×Dic₁₁	1	15	yes	—
53	Q₈	C₂²×Dic₁₁	1	20	yes	—
54	D₄	C₂²×Dic₁₁	1	60	yes	—
55	C₂×C₄	C₂²×Dic₁₁	1	45	yes	—
56	C₂×C₄	C₂³×D₁₁	1	15	yes	—
57	C₈	—	11	1	no	—
58	C₈	—	11	15	no	—
59	C₂²	—	4	15	no	—
60	C₂²	C₂³×Dic₁₁	1	15	yes	—
61	C₄	C₂³×Dic₁₁	1	15	yes	—
62	C₄	C₂⁴×D₁₁	1	1	yes	characteristic
63	C₂	—	2	15	no	—
64	C₂	C₂⁴×Dic₁₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×D₁₁
commutator, socle	C₂×C₁₁	C₂⁴×C₄
Fitting, prime radical	C₁₁×#585	C₂
cosocle	C₄×C₁₁	C₂⁵
hypercenter	#585	D₁₁
hypocenter	C₁₁	#3573
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
2. #10623350	C₂	semidirect
3. C₂×#67001	C₂	semidirect
7. #67001	C₂²	semidirect
8. #67001	C₂²	semidirect
9. C₂×#4318	C₂²	semidirect
10. C₂²×C₁₁⋊C₈	C₂²	semidirect
51. C₁₁	#3573	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×#585	2
#10623350	2
C₂×#67001	2
#3573	11