Subgroups — ΓL(1,𝔽₈)×C₄²⋊C₃ #11759071

Order 1008.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇⋊(C₄²⋊C₃)	C₃	1	2	yes	maximal, complemented
2	C₇×C₄²⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₄²×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
4	A₄×ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₂×C₄×ΓL(1,𝔽₈)	—	3	1	no	—
6	C₃×C₄²⋊C₃	—	7	1	no	maximal, Hall
7	C₄²×C₇	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₂²⋊ΓL(1,𝔽₈)	—	4	2	no	—
9	C₇×A₄	—	4	1	no	—
10	C₂²×ΓL(1,𝔽₈)	A₄	1	1	yes	characteristic
11	C₄×ΓL(1,𝔽₈)	—	3	2	no	—
12	C₃×ΓL(1,𝔽₈)	—	16	1	no	Hall
13	C₂×C₄×C₇	—	3	1	no	—
14	C₃×C₄²	—	7	1	no	—
15	C₄²⋊C₃	—	7	2	no	—
16	C₄²⋊C₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
17	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
18	C₃×A₄	—	28	1	no	—
19	C₂²×C₇	C₃×A₄	1	1	yes	characteristic
20	C₄×C₇	—	3	2	no	—
21	C₂×C₃×C₄	—	21	1	no	—
22	C₃×C₇	—	16	1	no	—
23	ΓL(1,𝔽₈)	—	16	2	no	—
24	ΓL(1,𝔽₈)	C₄²⋊C₃	1	1	yes	characteristic, direct product factor
25	C₄²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₇	—	3	1	no	—
27	C₂²×C₃	—	7	1	no	—
28	A₄	—	4	1	no	—
29	A₄	—	28	2	no	—
30	C₃×C₄	—	21	2	no	—
31	C₃²	—	112	1	no	Sylow
32	C₂×C₄	—	3	1	no	—
33	C₇	C₃×C₄²⋊C₃	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃	—	21	1	no	—
35	C₂²	A₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
36	C₄	—	3	2	no	—
37	C₃	—	7	1	no	—
38	C₃	—	16	1	no	—
39	C₃	—	112	2	no	—
40	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄²×C₇	C₃²
Frattini	C₂²	A₄×ΓL(1,𝔽₈)
socle	C₂²×C₇	C₃×A₄
LCS	G ▷ C₄²×C₇
UCS	C₁

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

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