Group #23744867 — subgroups

Subgroups — C₉×C₁₃×AΣL(1,𝔽₂₇) #23744867

Order 9477.

472 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₉×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic
3	C₉×C₁₃×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
4	C₉×C₁₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₁₃×3₋¹⁺²⋊C₉	C₃	1	4	yes	maximal, complemented
6	C₁₃×H(3,𝔽₃)⋊C₉	C₃	1	2	yes	maximal, complemented
7	C₁₃×#6866	C₃	1	2	yes	maximal, complemented
8	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
9	C₃×C₁₃×3₋¹⁺²	C₃²	1	2	yes	—
10	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃²×C₉×C₁₃	—	3	2	no	—
12	C₃²×C₉×C₁₃	—	3	1	no	—
13	C₃²×C₉×C₁₃	—	3	9	no	—
14	C₃²×C₉×C₁₃	—	3	1	no	—
15	C₃²×C₉×C₁₃	C₃²	1	2	yes	—
16	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
17	C₁₃×AΣL(1,𝔽₂₇)	C₉	1	9	yes	direct product factor
18	C₁₃×C₉⋊C₉	—	3	4	no	—
19	C₁₃×C₃²⋊C₉	—	3	2	no	—
20	C₁₃×C₃²⋊C₉	C₃²	1	4	yes	—
21	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₉²×C₁₃	—	3	2	no	—
23	C₉×AΣL(1,𝔽₂₇)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₃³×C₁₃	—	3	3	no	—
25	C₃³×C₁₃	—	3	6	no	—
26	C₃³×C₁₃	—	3	1	no	—
27	C₃³×C₁₃	—	3	2	no	—
28	C₃³×C₁₃	—	3	1	no	—
29	C₃³×C₁₃	C₃×C₉	1	3	yes	complemented
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₁₃×3₋¹⁺²	—	3	4	no	—
32	C₁₃×3₋¹⁺²	C₃×C₉	1	6	yes	complemented
33	C₁₃×H(3,𝔽₃)	—	3	2	no	—
34	C₁₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	complemented
35	C₃×C₉×C₁₃	—	3	18	no	—
36	C₃×C₉×C₁₃	—	3	2	no	—
37	C₃×C₉×C₁₃	—	3	2	no	—
38	C₃×C₉×C₁₃	—	3	9	no	—
39	C₃×C₉×C₁₃	—	3	3	no	—
40	C₃×C₉×C₁₃	—	3	6	no	—
41	C₃×C₉×C₁₃	—	3	4	no	—
42	C₃×C₉×C₁₃	—	3	2	no	—
43	C₃×C₉×C₁₃	—	9	3	no	—
44	C₃×C₉×C₁₃	H(3,𝔽₃)	1	2	yes	—
45	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
46	C₃³×C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
47	C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	1	1	yes	characteristic
48	C₉×3₋¹⁺²	C₃×C₁₃	1	2	yes	complemented
49	C₉×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic, complemented
50	3₋¹⁺²⋊C₉	C₃×C₁₃	1	4	yes	complemented
51	H(3,𝔽₃)⋊C₉	C₃×C₁₃	1	2	yes	complemented
52	#6866	C₃×C₁₃	1	2	yes	complemented
53	C₃²×C₁₃	—	3	9	no	—
54	C₃²×C₁₃	—	3	18	no	—
55	C₃²×C₁₃	—	3	3	no	—
56	C₃²×C₁₃	—	3	3	no	—
57	C₃²×C₁₃	—	3	1	no	—
58	C₃²×C₁₃	—	3	6	no	—
59	C₃²×C₁₃	—	3	2	no	—
60	C₃²×C₁₃	—	3	3	no	—
61	C₃²×C₁₃	—	9	1	no	—
62	C₃²×C₁₃	—	9	2	no	—
63	C₃²×C₁₃	C₃²×C₉	1	1	yes	characteristic
64	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃²×C₁₃	#737	1	2	yes	—
66	C₉×C₁₃	—	3	6	no	—
67	C₉×C₁₃	—	3	6	no	—
68	C₉×C₁₃	—	3	2	no	—
69	C₉×C₁₃	—	9	2	no	—
70	C₉×C₁₃	—	9	4	no	—
71	C₉×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	direct product factor
72	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
73	C₃×3₋¹⁺²	C₃²×C₁₃	1	2	yes	—
74	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
75	C₃²×C₉	—	3	2	no	—
76	C₃²×C₉	—	3	1	no	—
77	C₃²×C₉	—	3	9	no	—
78	C₃²×C₉	—	3	1	no	—
79	C₃²×C₉	C₃²×C₁₃	1	2	yes	—
80	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
81	AΣL(1,𝔽₂₇)	C₉×C₁₃	1	9	yes	direct product factor
82	C₉⋊C₉	—	3	4	no	—
83	C₃²⋊C₉	—	3	2	no	—
84	C₃²⋊C₉	C₃²×C₁₃	1	4	yes	—
85	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
86	C₉²	—	3	2	no	—
87	C₃×C₁₃	—	3	9	no	—
88	C₃×C₁₃	—	3	1	no	—
89	C₃×C₁₃	—	3	2	no	—
90	C₃×C₁₃	—	9	3	no	—
91	C₃×C₁₃	C₉×H(3,𝔽₃)	1	1	yes	characteristic
92	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
93	C₃×C₁₃	#6908	1	2	yes	—
94	C₃³	—	3	3	no	—
95	C₃³	—	3	6	no	—
96	C₃³	—	3	1	no	—
97	C₃³	—	3	2	no	—
98	C₃³	—	3	1	no	—
99	C₃³	C₃×C₉×C₁₃	1	3	yes	complemented
100	C₃³	C₃³×C₁₃	1	1	yes	characteristic
101	3₋¹⁺²	—	3	4	no	—
102	3₋¹⁺²	C₃×C₉×C₁₃	1	6	yes	complemented
103	H(3,𝔽₃)	—	3	2	no	—
104	H(3,𝔽₃)	C₃×C₉×C₁₃	1	3	yes	complemented
105	C₃×C₉	—	3	18	no	—
106	C₃×C₉	—	3	2	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	3	9	no	—
109	C₃×C₉	—	3	3	no	—
110	C₃×C₉	—	3	6	no	—
111	C₃×C₉	—	3	4	no	—
112	C₃×C₉	—	3	2	no	—
113	C₃×C₉	—	9	3	no	—
114	C₃×C₉	C₁₃×H(3,𝔽₃)	1	2	yes	—
115	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
116	C₁₃	C₉×AΣL(1,𝔽₂₇)	1	1	yes	characteristic, direct product factor, Sylow
117	C₃²	—	3	9	no	—
118	C₃²	—	3	18	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	3	no	—
122	C₃²	—	3	2	no	—
123	C₃²	—	3	6	no	—
124	C₃²	—	3	3	no	—
125	C₃²	—	9	1	no	—
126	C₃²	—	9	2	no	—
127	C₃²	C₃²×C₉×C₁₃	1	1	yes	characteristic
128	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
129	C₃²	C₁₃×#737	1	2	yes	—
130	C₉	—	3	6	no	—
131	C₉	—	3	2	no	—
132	C₉	—	3	6	no	—
133	C₉	—	9	2	no	—
134	C₉	—	9	4	no	—
135	C₉	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	direct product factor
136	C₃	—	3	1	no	—
137	C₃	—	3	2	no	—
138	C₃	—	3	9	no	—
139	C₃	—	9	3	no	—
140	C₃	C₉×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
141	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
142	C₃	C₁₃×#6908	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×C₉×C₁₃

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

yes

maximal, characteristic

C₉×C₁₃×3₋¹⁺²

C₃

yes

maximal, complemented

C₉×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₁₃×3₋¹⁺²⋊C₉

C₃

yes

maximal, complemented

C₁₃×H(3,𝔽₃)⋊C₉

C₃

yes

maximal, complemented

C₁₃×#6866

C₃

yes

maximal, complemented

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×AΣL(1,𝔽₂₇)

C₉

yes

direct product factor

C₁₃×C₉⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₉²×C₁₃

—

C₉×AΣL(1,𝔽₂₇)

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

C₃×C₉

yes

complemented

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

C₃×C₉

yes

complemented

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃×C₉

yes

complemented

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₉

C₃×C₁₃

yes

characteristic, complemented

C₃×AΣL(1,𝔽₂₇)

C₃×C₁₃

yes

characteristic

C₉×3₋¹⁺²

C₃×C₁₃

yes

complemented

C₉×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic, complemented

3₋¹⁺²⋊C₉

C₃×C₁₃

yes

complemented

H(3,𝔽₃)⋊C₉

C₃×C₁₃

yes

complemented

#6866

C₃×C₁₃

yes

complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃²×C₉

yes

characteristic

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

#737

yes

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

AΣL(1,𝔽₂₇)

yes

direct product factor

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

characteristic

AΣL(1,𝔽₂₇)

C₉×C₁₃

yes

direct product factor

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₉²

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₉×H(3,𝔽₃)

yes

characteristic

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₁₃

#6908

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₉×C₁₃

yes

complemented

100

C₃³

C₃³×C₁₃

yes

characteristic

101

3₋¹⁺²

—

102

3₋¹⁺²

C₃×C₉×C₁₃

yes

complemented

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

C₃×C₉×C₁₃

yes

complemented

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

115

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

116

C₁₃

C₉×AΣL(1,𝔽₂₇)

yes

characteristic, direct product factor, Sylow

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

C₃²×C₉×C₁₃

yes

characteristic

128

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

129

C₃²

C₁₃×#737

yes

—

130

C₉

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

C₁₃×AΣL(1,𝔽₂₇)

yes

direct product factor

136

C₃

—

137

C₃

—

138

C₃

—

139

C₃

—

140

C₃

C₉×C₁₃×H(3,𝔽₃)

yes

characteristic

141

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

142

C₃

C₁₃×#6908

yes

—

	subgroup	G/H
center	C₃×C₉×C₁₃	H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₁₃
Frattini, cosocle	C₃³	C₃³×C₁₃
socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
prime radical	C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

subgroup

G/H

center

C₃×C₉×C₁₃

H(3,𝔽₃)

commutator

C₃²

C₃²×C₉×C₁₃

Frattini, cosocle

C₃³

C₃³×C₁₃

socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

prime radical

C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₉×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃³×C₉×C₁₃	C₃	semidirect
3. C₉×C₁₃×3₋¹⁺²	C₃	semidirect
4. C₉×C₁₃×H(3,𝔽₃)	C₃	semidirect
5. C₁₃×3₋¹⁺²⋊C₉	C₃	semidirect
6. C₁₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
7. C₁₃×#6866	C₃	semidirect
17. C₁₃×AΣL(1,𝔽₂₇)	C₉	direct (N × H)
23. C₉×AΣL(1,𝔽₂₇)	C₁₃	direct (N × H)
29. C₃³×C₁₃	C₃×C₉	semidirect
32. C₁₃×3₋¹⁺²	C₃×C₉	semidirect
34. C₁₃×H(3,𝔽₃)	C₃×C₉	semidirect
46. C₃³×C₉	C₃×C₁₃	semidirect
48. C₉×3₋¹⁺²	C₃×C₁₃	semidirect
49. C₉×H(3,𝔽₃)	C₃×C₁₃	semidirect
50. 3₋¹⁺²⋊C₉	C₃×C₁₃	semidirect
51. H(3,𝔽₃)⋊C₉	C₃×C₁₃	semidirect
52. #6866	C₃×C₁₃	semidirect
71. C₉×C₁₃	AΣL(1,𝔽₂₇)	direct (N × H)
81. AΣL(1,𝔽₂₇)	C₉×C₁₃	direct (N × H)
99. C₃³	C₃×C₉×C₁₃	semidirect
102. 3₋¹⁺²	C₃×C₉×C₁₃	semidirect
104. H(3,𝔽₃)	C₃×C₉×C₁₃	semidirect
116. C₁₃	C₉×AΣL(1,𝔽₂₇)	direct (N × H)
135. C₉	C₁₃×AΣL(1,𝔽₂₇)	direct (N × H)

H ≅ G/N

type

1. C₃³×C₉×C₁₃

C₃

semidirect

3. C₉×C₁₃×3₋¹⁺²

C₃

semidirect

4. C₉×C₁₃×H(3,𝔽₃)

C₃

semidirect

5. C₁₃×3₋¹⁺²⋊C₉

C₃

semidirect

6. C₁₃×H(3,𝔽₃)⋊C₉

C₃

semidirect

7. C₁₃×#6866

C₃

semidirect

17. C₁₃×AΣL(1,𝔽₂₇)

C₉

direct (N × H)

23. C₉×AΣL(1,𝔽₂₇)

C₁₃

direct (N × H)

29. C₃³×C₁₃

C₃×C₉

semidirect

32. C₁₃×3₋¹⁺²

C₃×C₉

semidirect

34. C₁₃×H(3,𝔽₃)

C₃×C₉

semidirect

46. C₃³×C₉

C₃×C₁₃

semidirect

48. C₉×3₋¹⁺²

C₃×C₁₃

semidirect

49. C₉×H(3,𝔽₃)

C₃×C₁₃

semidirect

50. 3₋¹⁺²⋊C₉

C₃×C₁₃

semidirect

51. H(3,𝔽₃)⋊C₉

C₃×C₁₃

semidirect

52. #6866

C₃×C₁₃

semidirect

71. C₉×C₁₃

AΣL(1,𝔽₂₇)

direct (N × H)

81. AΣL(1,𝔽₂₇)

C₉×C₁₃

direct (N × H)

99. C₃³

C₃×C₉×C₁₃

semidirect

102. 3₋¹⁺²

C₃×C₉×C₁₃

semidirect

104. H(3,𝔽₃)

C₃×C₉×C₁₃

semidirect

116. C₁₃

C₉×AΣL(1,𝔽₂₇)

direct (N × H)

135. C₉

C₁₃×AΣL(1,𝔽₂₇)

direct (N × H)

Subgroups — C₉×C₁₃×AΣL(1,𝔽₂₇) #23744867

Conjugacy classes of subgroups

Complements

Subgroups — C9×C13×AΣL(1,𝔽27) #23744867

Conjugacy classes of subgroups

Complements

Subgroups — C₉×C₁₃×AΣL(1,𝔽₂₇) #23744867