Subgroups — C₂×C₁₁⋊#10587717 #26046524

Order 12012.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁⋊#10587717	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₁₁×#63271	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×D₁₄₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₁×#63271	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×D₁₄₃	C₂×C₃	1	2	yes	complemented
6	C₂×C₇×C₁₁×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₁₁⋊(D₁₃⋊C₃)	—	7	1	no	maximal, Hall
8	C₂×#10587717	—	11	1	no	maximal, Hall
9	C₇×C₁₁×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×ΓL(1,𝔽₈)×D₁₁	—	13	1	no	maximal, Hall
11	C₂×C₁₁×C₁₃⋊C₃	—	7	1	no	—
12	C₁₁⋊(D₁₃⋊C₃)	—	7	2	no	—
13	C₂×D₁₄₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
14	C₂×#63271	D₁₁	1	1	yes	characteristic, complemented
15	#10587717	—	11	2	no	—
16	C₂×C₁₁×ΓL(1,𝔽₈)	—	13	1	no	—
17	ΓL(1,𝔽₈)×D₁₁	—	13	2	no	—
18	C₁₁×C₁₃⋊C₃	—	7	1	no	Hall
19	C₂×C₇×D₁₃	—	11	1	no	Hall
20	C₂×C₇×D₁₁	—	13	1	no	Hall
21	C₂×C₁₁×C₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	D₁₄₃	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
23	#63271	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
24	C₁₁×ΓL(1,𝔽₈)	—	13	1	no	Hall
25	C₂×C₇×C₁₃	C₃×D₁₁	1	1	yes	characteristic, complemented
26	C₇×D₁₃	—	11	2	no	—
27	C₂×D₁₃⋊C₃	—	77	1	no	Hall
28	C₂×C₇×C₁₁	D₁₃⋊C₃	1	1	yes	characteristic, complemented
29	C₇×D₁₁	—	13	2	no	—
30	C₁₁×C₁₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
31	C₂×C₃×D₁₁	—	91	1	no	Hall
32	C₇×C₁₃	C₂×C₃×D₁₁	1	1	yes	characteristic, complemented, Hall
33	C₂²×ΓL(1,𝔽₈)	—	143	1	no	Hall
34	C₂×C₁₃⋊C₃	—	7	1	no	—
35	D₁₃⋊C₃	—	77	2	no	—
36	C₇×C₁₁	C₂×D₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₁₁	—	91	1	no	—
38	C₃×D₁₁	—	91	2	no	—
39	C₂×D₁₃	—	11	1	no	Hall
40	C₂×D₁₁	—	13	1	no	Hall
41	C₂×ΓL(1,𝔽₈)	—	13	1	no	—
42	C₂×ΓL(1,𝔽₈)	—	143	2	no	—
43	C₁₃⋊C₃	—	7	1	no	Hall
44	C₃×C₁₁	—	91	1	no	Hall
45	C₂²×C₇	—	143	1	no	Hall
46	C₂×C₁₃	ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, complemented
47	D₁₃	—	11	2	no	—
48	C₂×C₁₁	#10587717	1	1	yes	characteristic, complemented
49	D₁₁	—	13	2	no	—
50	ΓL(1,𝔽₈)	—	13	1	no	Hall
51	C₂×C₇	—	143	2	no	—
52	C₂×C₇	C₁₁⋊(D₁₃⋊C₃)	1	1	yes	characteristic, complemented
53	C₁₃	C₂×ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₃	—	1001	1	no	Hall
55	C₁₁	C₂×#10587717	1	1	yes	characteristic, complemented, Sylow
56	C₇	C₂×C₁₁⋊(D₁₃⋊C₃)	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃	—	91	1	no	—
58	C₂×C₃	—	1001	2	no	—
59	C₂²	—	143	1	no	Sylow
60	C₃	—	91	1	no	Sylow
61	C₂	—	143	2	no	—
62	C₂	C₁₁⋊#10587717	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₁₁⋊#10587717
commutator, cosocle, hypocenter	C₇×C₁₁×C₁₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₁×C₁₃	C₂×C₃
LCS	G ▷ C₇×C₁₁×C₁₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₁₁⋊#10587717	C₂	direct (N × H)
2. C₂×C₁₁×#63271	C₂	semidirect
3. C₂×C₇×D₁₄₃	C₃	semidirect
4. C₁₁×#63271	C₂²	semidirect
5. C₇×D₁₄₃	C₂×C₃	semidirect
6. C₂×C₇×C₁₁×C₁₃	C₂×C₃	semidirect
9. C₇×C₁₁×C₁₃	C₂²×C₃	semidirect
13. C₂×D₁₄₃	ΓL(1,𝔽₈)	semidirect
14. C₂×#63271	D₁₁	semidirect
21. C₂×C₁₁×C₁₃	C₂×ΓL(1,𝔽₈)	semidirect
22. D₁₄₃	C₂×ΓL(1,𝔽₈)	semidirect
23. #63271	C₂×D₁₁	semidirect
25. C₂×C₇×C₁₃	C₃×D₁₁	semidirect
28. C₂×C₇×C₁₁	D₁₃⋊C₃	semidirect
30. C₁₁×C₁₃	C₂²×ΓL(1,𝔽₈)	semidirect
32. C₇×C₁₃	C₂×C₃×D₁₁	semidirect
36. C₇×C₁₁	C₂×D₁₃⋊C₃	semidirect
46. C₂×C₁₃	ΓL(1,𝔽₈)×D₁₁	semidirect
48. C₂×C₁₁	#10587717	semidirect
52. C₂×C₇	C₁₁⋊(D₁₃⋊C₃)	semidirect
53. C₁₃	C₂×ΓL(1,𝔽₈)×D₁₁	semidirect
55. C₁₁	C₂×#10587717	semidirect
56. C₇	C₂×C₁₁⋊(D₁₃⋊C₃)	semidirect
62. C₂	C₁₁⋊#10587717	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₁⋊#10587717	2
C₂×C₁₁×#63271	2
C₂×C₇×D₁₄₃	3
C₂×C₁₁⋊(D₁₃⋊C₃)	7
C₂×#10587717	11
C₂×ΓL(1,𝔽₈)×D₁₁	13

Subgroups — C2×C11⋊#10587717 #26046524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁⋊#10587717 #26046524