Subgroups — C₂×AΣL(1,𝔽₂₇) #4192

Order 162.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	AΣL(1,𝔽₂₇)	C₂	1	1	yes	maximal, characteristic, direct product factor, Sylow
2	C₂×C₃³	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
4	C₂×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³	C₂×C₃	1	1	yes	characteristic, complemented
6	3₋¹⁺²	C₂×C₃	1	2	yes	complemented
7	H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₃²	—	3	1	no	—
9	C₂×C₃²	—	3	3	no	—
10	C₂×C₃²	—	3	1	no	—
11	C₂×C₃²	C₃²	1	1	yes	characteristic
12	C₂×C₉	—	3	2	no	—
13	C₃²	—	3	3	no	—
14	C₃²	—	3	1	no	—
15	C₃²	—	3	1	no	—
16	C₃²	C₂×C₃²	1	1	yes	characteristic
17	C₉	—	3	2	no	—
18	C₂×C₃	—	3	3	no	—
19	C₂×C₃	—	3	1	no	—
20	C₂×C₃	—	9	1	no	—
21	C₂×C₃	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃	—	3	3	no	—
23	C₃	—	3	1	no	—
24	C₃	—	9	1	no	—
25	C₃	C₂×H(3,𝔽₃)	1	1	yes	characteristic
26	C₂	AΣL(1,𝔽₂₇)	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, socle	C₂×C₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂×C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃² ◁ G

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.