Subgroups — #72248917

Order 3520.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈₈⋊C₅	C₂	1	4	yes	maximal, complemented
2	#66927⋊C₅	C₂	1	2	yes	maximal, complemented
3	C₄×C₈×C₁₁⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄⋊AGL(1,𝔽₁₁)	—	2	4	no	—
5	C₂×C₄⋊AGL(1,𝔽₁₁)	C₂²	1	4	yes	—
6	C₂×C₈×C₁₁⋊C₅	C₂²	1	2	yes	—
7	C₄²×C₁₁⋊C₅	C₂²	1	1	yes	characteristic
8	D₈₈⋊C₅	—	2	8	no	—
9	#10622308	C₅	1	1	yes	maximal, characteristic, complemented, Hall
10	C₂²×AGL(1,𝔽₁₁)	—	2	4	no	—
11	C₂×C₄×C₁₁⋊C₅	D₄	1	2	yes	—
12	C₂×C₄×C₁₁⋊C₅	C₂³	1	1	yes	characteristic
13	C₄⋊AGL(1,𝔽₁₁)	—	2	8	no	—
14	C₄⋊AGL(1,𝔽₁₁)	—	4	8	no	—
15	C₈×C₁₁⋊C₅	D₄	1	4	yes	complemented
16	C₂×D₈₈	C₂×C₅	1	4	yes	complemented
17	#66927	C₂×C₅	1	2	yes	complemented
18	C₄×C₈×C₁₁	C₂×C₅	1	1	yes	characteristic, complemented
19	C₅×#493	—	11	1	no	maximal, Hall
20	C₂²×C₁₁⋊C₅	C₂×D₄	1	1	yes	characteristic
21	C₂×AGL(1,𝔽₁₁)	—	4	4	no	—
22	C₂×AGL(1,𝔽₁₁)	—	4	8	no	—
23	C₄×C₁₁⋊C₅	D₈	1	4	yes	complemented
24	C₄×C₁₁⋊C₅	C₂×D₄	1	2	yes	—
25	C₂×D₄₄	—	2	4	no	—
26	C₂×D₄₄	C₂²×C₅	1	4	yes	—
27	C₂×C₈×C₁₁	C₂²×C₅	1	2	yes	—
28	C₄²×C₁₁	C₂²×C₅	1	1	yes	characteristic
29	D₈₈	—	2	8	no	—
30	C₂×C₅×D₈	—	11	4	no	—
31	C₅×#127	—	11	2	no	—
32	C₄×C₅×C₈	—	11	1	no	—
33	C₂×C₁₁⋊C₅	#127	1	1	yes	characteristic
34	C₂×C₁₁⋊C₅	C₂×D₈	1	2	yes	—
35	AGL(1,𝔽₁₁)	—	8	4	no	—
36	C₂²×D₁₁	—	2	4	no	—
37	C₂×C₄×C₁₁	C₅×D₄	1	2	yes	—
38	C₂×C₄×C₁₁	C₂³×C₅	1	1	yes	characteristic
39	D₄₄	—	2	8	no	—
40	D₄₄	—	4	8	no	—
41	C₈×C₁₁	C₅×D₄	1	4	yes	complemented
42	C₂×C₅×D₄	—	11	4	no	—
43	C₂×C₅×D₄	—	22	4	no	—
44	C₅×D₈	—	22	8	no	—
45	C₂×C₅×C₈	—	11	2	no	—
46	C₄²×C₅	—	11	1	no	—
47	#493	—	11	1	no	Sylow
48	C₁₁⋊C₅	#493	1	1	yes	characteristic, complemented, Hall
49	C₂²×C₁₁	C₂×C₅×D₄	1	1	yes	characteristic
50	C₂×D₁₁	—	4	4	no	—
51	C₂×D₁₁	—	4	8	no	—
52	C₄×C₁₁	C₅×D₈	1	4	yes	complemented
53	C₄×C₁₁	C₂×C₅×D₄	1	2	yes	—
54	C₂³×C₅	—	22	4	no	—
55	C₅×D₄	—	22	8	no	—
56	C₅×D₄	—	44	8	no	—
57	C₂×C₄×C₅	—	11	2	no	—
58	C₂×C₄×C₅	—	11	1	no	—
59	C₅×C₈	—	11	4	no	—
60	C₂×D₈	—	11	4	no	—
61	#127	—	11	2	no	—
62	C₄×C₈	AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented
63	C₂×C₁₁	C₅×#127	1	1	yes	characteristic
64	C₂×C₁₁	C₂×C₅×D₈	1	2	yes	—
65	D₁₁	—	8	4	no	—
66	C₂²×C₅	—	11	1	no	—
67	C₂²×C₅	—	44	4	no	—
68	C₂²×C₅	—	44	8	no	—
69	C₄×C₅	—	11	4	no	—
70	C₄×C₅	—	11	2	no	—
71	C₂×D₄	—	11	4	no	—
72	C₂×D₄	—	22	4	no	—
73	D₈	—	22	8	no	—
74	C₂×C₈	C₂×AGL(1,𝔽₁₁)	1	2	yes	—
75	C₄²	C₂×AGL(1,𝔽₁₁)	1	1	yes	characteristic
76	C₁₁	C₅×#493	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₅	—	11	1	no	—
78	C₂×C₅	—	11	2	no	—
79	C₂×C₅	—	88	4	no	—
80	C₂³	—	22	4	no	—
81	D₄	—	22	8	no	—
82	D₄	—	44	8	no	—
83	C₂×C₄	C₄⋊AGL(1,𝔽₁₁)	1	2	yes	—
84	C₂×C₄	C₂²×AGL(1,𝔽₁₁)	1	1	yes	characteristic
85	C₈	C₄⋊AGL(1,𝔽₁₁)	1	4	yes	complemented
86	C₅	—	11	1	no	Sylow
87	C₂²	—	44	4	no	—
88	C₂²	—	44	8	no	—
89	C₂²	C₂×C₄⋊AGL(1,𝔽₁₁)	1	1	yes	characteristic
90	C₄	D₈₈⋊C₅	1	4	yes	complemented
91	C₄	C₂×C₄⋊AGL(1,𝔽₁₁)	1	2	yes	—
92	C₂	—	88	4	no	—
93	C₂	#66927⋊C₅	1	1	yes	characteristic
94	C₂	C₂×D₈₈⋊C₅	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,𝔽₁₁)
commutator, cosocle	C₂×C₄×C₁₁	C₂³×C₅
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₁₁)
Fitting	C₄×C₈×C₁₁	C₂×C₅
socle	C₂²×C₁₁	C₂×C₅×D₄
hypercenter	C₄×C₈	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#493
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₈₈⋊C₅	C₂	semidirect
2. #66927⋊C₅	C₂	semidirect
3. C₄×C₈×C₁₁⋊C₅	C₂	semidirect
9. #10622308	C₅	semidirect
15. C₈×C₁₁⋊C₅	D₄	semidirect
16. C₂×D₈₈	C₂×C₅	semidirect
17. #66927	C₂×C₅	semidirect
18. C₄×C₈×C₁₁	C₂×C₅	semidirect
23. C₄×C₁₁⋊C₅	D₈	semidirect
41. C₈×C₁₁	C₅×D₄	semidirect
48. C₁₁⋊C₅	#493	semidirect
52. C₄×C₁₁	C₅×D₈	semidirect
62. C₄×C₈	AGL(1,𝔽₁₁)	semidirect
76. C₁₁	C₅×#493	semidirect
85. C₈	C₄⋊AGL(1,𝔽₁₁)	semidirect
90. C₄	D₈₈⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈₈⋊C₅	2
#66927⋊C₅	2
C₄×C₈×C₁₁⋊C₅	2
#10622308	5
C₅×#493	11