Subgroups — C₂₃×#10624486 #30660993

Order 16767.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₃×#733	C₃	1	1	yes	maximal, characteristic
2	C₂₃×#6895	C₃	1	2	yes	maximal
3	C₂₃×#6867	C₃	1	1	yes	maximal, characteristic
4	C₃×C₂₃×3₋¹⁺²	—	3	1	no	—
5	C₃²×C₉×C₂₃	C₃²	1	1	yes	characteristic
6	C₂₃×#733	—	3	3	no	—
7	C₂₃×C₉⋊C₉	—	3	6	no	—
8	C₂₃×C₃²⋊C₉	—	3	2	no	—
9	C₂₃×C₃²⋊C₉	—	3	1	no	—
10	#10624486	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃³×C₂₃	H(3,𝔽₃)	1	1	yes	characteristic
12	C₂₃×3₋¹⁺²	—	3	3	no	—
13	C₂₃×3₋¹⁺²	—	9	2	no	—
14	C₃×C₉×C₂₃	—	3	6	no	—
15	C₃×C₉×C₂₃	—	3	2	no	—
16	C₃×C₉×C₂₃	—	3	1	no	—
17	C₃×C₉×C₂₃	—	3	1	no	—
18	C₃×C₉×C₂₃	—	9	1	no	—
19	C₃×C₉×C₂₃	—	9	1	no	—
20	C₃×#733	C₃×C₂₃	1	1	yes	characteristic
21	#6895	C₃×C₂₃	1	2	yes	—
22	#6867	C₃×C₂₃	1	1	yes	characteristic
23	C₃²×C₂₃	—	9	1	no	—
24	C₃²×C₂₃	AΣL(1,𝔽₂₇)	1	2	yes	—
25	C₃²×C₂₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
26	C₃²×C₂₃	#731	1	1	yes	characteristic
27	C₉×C₂₃	—	9	6	no	—
28	C₉×C₂₃	—	9	1	no	—
29	C₉×C₂₃	—	9	3	no	—
30	C₉×C₂₃	—	27	1	no	—
31	C₃×3₋¹⁺²	—	3	1	no	—
32	C₃²×C₉	C₃²×C₂₃	1	1	yes	characteristic
33	#733	—	3	3	no	—
34	C₉⋊C₉	—	3	6	no	—
35	C₃²⋊C₉	—	3	2	no	—
36	C₃²⋊C₉	—	3	1	no	—
37	C₃×C₂₃	—	3	2	no	—
38	C₃×C₂₃	—	3	1	no	—
39	C₃×C₂₃	—	3	1	no	—
40	C₃×C₂₃	#6857	1	1	yes	characteristic
41	C₃³	C₂₃×H(3,𝔽₃)	1	1	yes	characteristic
42	3₋¹⁺²	—	3	3	no	—
43	3₋¹⁺²	—	9	2	no	—
44	C₃×C₉	—	3	6	no	—
45	C₃×C₉	—	3	2	no	—
46	C₃×C₉	—	3	1	no	—
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	9	1	no	—
49	C₃×C₉	—	9	1	no	—
50	C₂₃	#10624486	1	1	yes	characteristic, direct product factor, Sylow
51	C₃²	—	9	1	no	—
52	C₃²	C₂₃×AΣL(1,𝔽₂₇)	1	2	yes	—
53	C₃²	C₂₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
54	C₃²	C₂₃×#731	1	1	yes	characteristic
55	C₉	—	9	6	no	—
56	C₉	—	9	1	no	—
57	C₉	—	9	3	no	—
58	C₉	—	27	1	no	—
59	C₃	—	3	2	no	—
60	C₃	—	3	1	no	—
61	C₃	—	3	1	no	—
62	C₃	C₂₃×#6857	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₂₃	#6857
commutator, Frattini, cosocle	C₃²×C₉	C₃²×C₂₃
prime radical	C₃³×C₂₃	H(3,𝔽₃)
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₂₃ ◁ C₃³×C₂₃ ◁ C₃²×C₉×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. #10624486	C₂₃	direct (N × H)
50. C₂₃	#10624486	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₂₃×#733	3
C₂₃×#6895	3
C₂₃×#6867	3
#10624486	23

Subgroups — C23×#10624486 #30660993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10624486 #30660993