Subgroups — C₂²⋊ΓL(1,𝔽₈)×C₁₁²⋊C₃ #44479136

Order 30492.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₁₁²⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₁₁²×C₂²⋊ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₇⋊(C₁₁²⋊C₃))	C₃	1	2	yes	maximal, complemented
4	ΓL(1,𝔽₈)×C₁₁²⋊C₃	—	4	1	no	maximal, Hall
5	C₂×C₇×C₁₁²⋊C₃	—	3	1	no	—
6	A₄×C₁₁²⋊C₃	—	7	1	no	maximal, Hall
7	C₂²×C₇×C₁₁²	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₇⋊(C₁₁²⋊C₃)	—	4	2	no	—
9	C₁₁²×ΓL(1,𝔽₈)	—	4	1	no	—
10	C₇×C₁₁²⋊C₃	A₄	1	1	yes	characteristic, complemented
11	C₂×C₇×C₁₁²	—	3	1	no	—
12	C₂²⋊(C₁₁²⋊C₃)	—	7	2	no	—
13	C₁₁²×A₄	—	7	1	no	—
14	C₂²×C₁₁²⋊C₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
15	C₃×C₁₁²⋊C₃	—	28	1	no	Hall
16	C₁₁×C₂²⋊ΓL(1,𝔽₈)	—	3	4	no	—
17	C₇×C₁₁²	C₃×A₄	1	1	yes	characteristic, complemented, Hall
18	C₂×C₁₁²⋊C₃	—	3	1	no	—
19	C₂²×C₁₁²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
20	C₃×C₁₁²	—	28	1	no	—
21	C₁₁²⋊C₃	—	28	2	no	—
22	C₁₁²⋊C₃	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
23	C₂²×C₇×C₁₁	—	3	4	no	—
24	C₃×C₂²⋊ΓL(1,𝔽₈)	—	121	1	no	maximal, Hall
25	C₂×C₁₁²	—	3	1	no	—
26	C₁₁×ΓL(1,𝔽₈)	—	12	4	no	—
27	C₂×C₇×C₁₁	—	9	4	no	—
28	C₁₁×A₄	—	21	4	no	—
29	C₁₁²	C₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
30	C₂²×C₃×C₇	—	121	1	no	—
31	C₂²⋊ΓL(1,𝔽₈)	—	121	2	no	—
32	C₂²⋊ΓL(1,𝔽₈)	C₁₁²⋊C₃	1	1	yes	characteristic, direct product factor
33	C₇×C₁₁	—	3	4	no	—
34	C₃×ΓL(1,𝔽₈)	—	484	1	no	Hall
35	C₂²×C₁₁	—	3	4	no	—
36	C₂×C₃×C₇	—	363	1	no	—
37	C₃×A₄	—	847	1	no	Hall
38	C₃×C₁₁	—	84	4	no	—
39	C₂²×C₇	C₃×C₁₁²⋊C₃	1	1	yes	characteristic, complemented, Hall
40	C₂×C₁₁	—	9	4	no	—
41	C₃×C₇	—	121	1	no	—
42	ΓL(1,𝔽₈)	—	4	1	no	—
43	ΓL(1,𝔽₈)	—	484	2	no	—
44	C₂×C₇	—	3	1	no	—
45	C₂²×C₃	—	121	1	no	—
46	A₄	—	7	1	no	—
47	A₄	—	847	2	no	—
48	C₁₁	—	3	4	no	—
49	C₃²	—	3388	1	no	Sylow
50	C₇	A₄×C₁₁²⋊C₃	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃	—	363	1	no	—
52	C₂²	ΓL(1,𝔽₈)×C₁₁²⋊C₃	1	1	yes	characteristic, complemented, Sylow
53	C₃	—	28	1	no	—
54	C₃	—	3388	2	no	—
55	C₃	—	121	1	no	—
56	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₇×C₁₁²	C₃²
LCS	G ▷ C₂²×C₇×C₁₁²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₁₁²⋊C₃	C₃	semidirect
2. C₁₁²×C₂²⋊ΓL(1,𝔽₈)	C₃	semidirect
3. C₂²⋊(C₇⋊(C₁₁²⋊C₃))	C₃	semidirect
7. C₂²×C₇×C₁₁²	C₃²	semidirect
10. C₇×C₁₁²⋊C₃	A₄	semidirect
14. C₂²×C₁₁²⋊C₃	ΓL(1,𝔽₈)	semidirect
17. C₇×C₁₁²	C₃×A₄	semidirect
19. C₂²×C₁₁²	C₃×ΓL(1,𝔽₈)	semidirect
22. C₁₁²⋊C₃	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
29. C₁₁²	C₃×C₂²⋊ΓL(1,𝔽₈)	semidirect
32. C₂²⋊ΓL(1,𝔽₈)	C₁₁²⋊C₃	direct (N × H)
39. C₂²×C₇	C₃×C₁₁²⋊C₃	semidirect
50. C₇	A₄×C₁₁²⋊C₃	semidirect
52. C₂²	ΓL(1,𝔽₈)×C₁₁²⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₁₁²⋊C₃	3
C₁₁²×C₂²⋊ΓL(1,𝔽₈)	3
C₂²⋊(C₇⋊(C₁₁²⋊C₃))	3
ΓL(1,𝔽₈)×C₁₁²⋊C₃	4
A₄×C₁₁²⋊C₃	7
C₃×C₂²⋊ΓL(1,𝔽₈)	121

Subgroups — C22⋊ΓL(1,𝔽8)×C112⋊C3 #44479136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊ΓL(1,𝔽₈)×C₁₁²⋊C₃ #44479136