Subgroups — #127⋊ΓL(1,𝔽₈) #10621176

Order 672.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇⋊(C₄²⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#127	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₂×C₇×D₄	—	3	1	no	—
5	C₂×C₇×D₄	—	3	1	no	—
6	C₄²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
7	#127⋊C₃	—	7	1	no	maximal, Hall
8	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
9	C₂³×C₇	—	3	1	no	—
10	C₂³×C₇	A₄	1	1	yes	characteristic
11	C₇×D₄	—	6	2	no	—
12	C₇×D₄	—	6	2	no	—
13	C₂×C₄×C₇	—	3	1	no	—
14	C₄²⋊C₃	—	7	1	no	—
15	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
16	#127	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
17	C₂²×C₇	—	6	1	no	—
18	C₂²×C₇	—	6	1	no	—
19	C₂²×C₇	—	6	2	no	—
20	C₂²×C₇	C₂×A₄	1	1	yes	characteristic
21	C₄×C₇	—	3	2	no	—
22	C₂×A₄	—	28	1	no	—
23	ΓL(1,𝔽₈)	—	16	1	no	Hall
24	C₂×D₄	—	3	1	no	—
25	C₂×D₄	—	3	1	no	—
26	C₄²	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
27	C₂×C₇	—	3	1	no	—
28	C₂×C₇	—	4	1	no	—
29	C₂×C₇	—	12	1	no	—
30	A₄	—	28	1	no	—
31	C₂³	—	3	1	no	—
32	C₂³	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
33	D₄	—	6	2	no	—
34	D₄	—	6	2	no	—
35	C₂×C₄	—	3	1	no	—
36	C₇	#127⋊C₃	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃	—	112	1	no	—
38	C₂²	—	6	1	no	—
39	C₂²	—	6	1	no	—
40	C₂²	—	6	2	no	—
41	C₂²	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
42	C₄	—	3	2	no	—
43	C₃	—	112	1	no	Sylow
44	C₂	—	3	1	no	—
45	C₂	—	4	1	no	—
46	C₂	—	12	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₄²×C₇	C₂×C₃
Frattini	C₂²	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×#127	C₃
socle	C₂²×C₇	C₂×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.