Subgroups — ΓL(1,𝔽₈)²×C₆₁ #41002018

Order 26901.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3861	C₃	1	1	yes	maximal, characteristic, complemented
2	C₇×ΓL(1,𝔽₈)×C₆₁	C₃	1	2	yes	maximal, complemented
3	C₆₁×#3862	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×ΓL(1,𝔽₈)×C₆₁	—	7	2	no	maximal
5	C₇²×C₆₁	C₃²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₇×C₆₁	—	7	2	no	—
7	ΓL(1,𝔽₈)×C₆₁	—	7	2	no	—
8	ΓL(1,𝔽₈)×C₆₁	—	7	2	no	—
9	ΓL(1,𝔽₈)×C₆₁	—	21	2	no	—
10	ΓL(1,𝔽₈)×C₆₁	ΓL(1,𝔽₈)	1	2	yes	direct product factor
11	C₃²×C₆₁	—	49	1	no	Hall
12	ΓL(1,𝔽₈)²	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₇×C₆₁	—	3	2	no	—
14	C₇×C₆₁	C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
15	C₃×C₆₁	—	7	2	no	—
16	C₃×C₆₁	—	49	1	no	—
17	C₃×C₆₁	—	49	1	no	—
18	#3862	C₃×C₆₁	1	1	yes	characteristic, complemented
19	#3861	C₃×C₆₁	1	1	yes	characteristic, complemented
20	C₇×ΓL(1,𝔽₈)	C₃×C₆₁	1	2	yes	complemented
21	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
22	C₆₁	ΓL(1,𝔽₈)²	1	1	yes	characteristic, direct product factor, Sylow
23	C₇²	C₃²×C₆₁	1	1	yes	characteristic, complemented, Sylow
24	C₃×C₇	—	7	2	no	—
25	ΓL(1,𝔽₈)	—	7	2	no	—
26	ΓL(1,𝔽₈)	—	7	2	no	—
27	ΓL(1,𝔽₈)	—	21	2	no	—
28	ΓL(1,𝔽₈)	ΓL(1,𝔽₈)×C₆₁	1	2	yes	direct product factor
29	C₃²	—	49	1	no	Sylow
30	C₇	—	3	2	no	—
31	C₇	C₃×ΓL(1,𝔽₈)×C₆₁	1	2	yes	complemented
32	C₃	—	7	2	no	—
33	C₃	—	49	1	no	—
34	C₃	—	49	1	no	—

	subgroup	G/H
center, hypercenter	C₆₁	ΓL(1,𝔽₈)²
commutator, cosocle, hypocenter	C₇²	C₃²×C₆₁
Frattini	C₁	G
Fitting, socle	C₇²×C₆₁	C₃²
LCS	G ▷ C₇²
UCS	C₁ ◁ C₆₁

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.