Subgroups — C₂₃×#10624562 #30661069

Order 16767.

2138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂₃×H(3,𝔽₃)	C₃	1	13	yes	maximal, complemented
2	C₃⁴×C₂₃	C₃²	1	13	yes	—
3	C₃×C₂₃×H(3,𝔽₃)	—	3	156	no	—
4	#10624562	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₃³×C₂₃	—	3	156	no	—
6	C₃³×C₂₃	H(3,𝔽₃)	1	39	yes	complemented
7	C₃³×C₂₃	C₃³	1	1	yes	characteristic
8	C₂₃×H(3,𝔽₃)	—	3	351	no	—
9	C₃²×H(3,𝔽₃)	C₃×C₂₃	1	13	yes	complemented
10	C₃²×C₂₃	—	3	156	no	—
11	C₃²×C₂₃	—	9	117	no	—
12	C₃²×C₂₃	C₃×H(3,𝔽₃)	1	13	yes	—
13	C₃⁴	C₃²×C₂₃	1	13	yes	—
14	C₃×H(3,𝔽₃)	—	3	156	no	—
15	C₃×C₂₃	—	9	39	no	—
16	C₃×C₂₃	C₃²⋊H(3,𝔽₃)	1	13	yes	—
17	C₃³	—	3	156	no	—
18	C₃³	C₂₃×H(3,𝔽₃)	1	39	yes	complemented
19	C₃³	C₃³×C₂₃	1	1	yes	characteristic
20	H(3,𝔽₃)	—	3	351	no	—
21	C₂₃	#10624562	1	1	yes	characteristic, direct product factor, Sylow
22	C₃²	—	3	156	no	—
23	C₃²	—	9	117	no	—
24	C₃²	C₃×C₂₃×H(3,𝔽₃)	1	13	yes	—
25	C₃	—	9	39	no	—
26	C₃	C₂₃×C₃²⋊H(3,𝔽₃)	1	13	yes	—

	subgroup	G/H
center, socle	C₃³×C₂₃	C₃³
commutator, Frattini, cosocle	C₃³	C₃³×C₂₃
LCS	G ▷ C₃³ ▷ C₁
UCS	C₁ ◁ C₃³×C₂₃ ◁ 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.

subgroup	index
C₃²×C₂₃×H(3,𝔽₃)	3
#10624562	23