Group #13146771 — subgroups

Subgroups — C₉×#5407 #13146771

Order 1728.

209 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 39 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	C₉×#986	C₂	1	2	yes	maximal, complemented
2	C₉×#131⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5407	C₃	1	1	yes	maximal, characteristic
4	C₉×#472	—	3	1	no	maximal
5	C₉×#41⋊C₃	C₂²	1	1	yes	characteristic
6	C₉×GL(2,𝔽₃)	—	2	2	no	—
7	C₉×D₂₄	—	4	1	no	maximal
8	C₃×#986	C₂×C₃	1	2	yes	—
9	C₃×#131⋊C₃	C₂×C₃	1	1	yes	characteristic
10	C₉×AGL(1,ℤ/8)	—	3	2	no	—
11	C₂×C₉×D₈	—	3	1	no	—
12	C₉×#131	D₃	1	1	yes	characteristic, complemented
13	#131⋊C₉	—	2	1	no	—
14	C₉×#108	—	3	1	no	—
15	C₉×#100	—	3	2	no	—
16	C₉×D₁₂	—	4	2	no	—
17	C₉×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
18	C₃×C₈×C₉	—	4	1	no	—
19	#5407	C₉	1	1	yes	characteristic, direct product factor
20	C₃×#472	—	3	1	no	—
21	C₃×#41⋊C₃	C₂²×C₃	1	1	yes	characteristic
22	C₃×GL(2,𝔽₃)	—	2	2	no	—
23	C₃×D₂₄	—	4	1	no	—
24	C₉×#41	C₂×D₃	1	1	yes	characteristic
25	C₂×D₄×C₉	—	3	2	no	—
26	#41⋊C₉	—	2	1	no	—
27	C₉×SD₁₆	—	6	2	no	—
28	C₉×D₈	—	6	1	no	—
29	C₉×D₈	—	6	2	no	—
30	C₉×D₈	—	6	1	no	—
31	C₉×M₁₆	—	3	1	no	—
32	C₉×M₁₆	—	3	2	no	—
33	C₂×C₈×C₉	—	3	1	no	—
34	C₂×D₃×C₉	—	8	2	no	—
35	C₃×C₄×C₉	—	4	1	no	—
36	#986	C₂×C₉	1	2	yes	complemented
37	C₃×AGL(1,ℤ/8)	—	3	2	no	—
38	C₂×C₃×D₈	—	3	1	no	—
39	C₃×#131	C₃×D₃	1	1	yes	characteristic
40	#131⋊C₃	—	2	1	no	—
41	#131⋊C₃	C₂×C₉	1	1	yes	characteristic, complemented
42	C₃×#108	—	3	1	no	—
43	C₃×#100	—	3	2	no	—
44	C₃×D₁₂	—	4	2	no	—
45	C₃×SL(2,𝔽₃)	C₃×D₄	1	1	yes	characteristic
46	C₂³×C₉	—	6	2	no	—
47	C₃²×C₈	—	4	1	no	—
48	Q₈×C₉	D₁₂	1	1	yes	characteristic
49	D₄×C₉	—	3	1	no	—
50	D₄×C₉	—	6	2	no	—
51	D₄×C₉	—	6	2	no	—
52	C₂×C₄×C₉	—	3	1	no	—
53	Q₈⋊C₉	—	2	1	no	—
54	C₈×C₉	—	3	1	no	—
55	C₈×C₉	—	6	2	no	—
56	C₈×C₉	—	8	1	no	—
57	C₈×C₉	S₄	1	1	yes	characteristic
58	#472	—	3	1	no	Sylow
59	C₂×C₃×C₉	—	4	1	no	—
60	D₃×C₉	—	16	2	no	—
61	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
62	C₂×C₃×D₄	—	3	2	no	—
63	#41⋊C₃	—	2	1	no	—
64	#41⋊C₃	C₂²×C₉	1	1	yes	characteristic
65	GL(2,𝔽₃)	—	2	2	no	—
66	C₃×SD₁₆	—	6	2	no	—
67	C₃×D₈	—	6	1	no	—
68	C₃×D₈	—	6	2	no	—
69	C₃×D₈	—	6	1	no	—
70	C₃×M₁₆	—	3	1	no	—
71	C₃×M₁₆	—	3	2	no	—
72	C₂×C₃×C₈	—	3	1	no	—
73	D₂₄	—	4	1	no	—
74	C₂×C₃×D₃	—	8	2	no	—
75	C₃²×C₄	—	4	1	no	—
76	C₂²×C₉	—	3	1	no	—
77	C₂²×C₉	—	12	2	no	—
78	C₂²×C₉	—	24	2	no	—
79	C₄×C₉	—	3	1	no	—
80	C₄×C₉	—	8	1	no	—
81	C₄×C₉	C₂×S₄	1	1	yes	characteristic
82	AGL(1,ℤ/8)	—	3	2	no	—
83	C₂×D₈	—	3	1	no	—
84	#131	D₃×C₉	1	1	yes	characteristic, complemented
85	#108	—	3	1	no	—
86	#100	—	3	2	no	—
87	C₃×C₉	—	4	1	no	Sylow
88	C₂³×C₃	—	6	2	no	—
89	C₃×Q₈	C₃×D₁₂	1	1	yes	characteristic
90	C₃×D₄	—	3	1	no	—
91	C₃×D₄	—	6	2	no	—
92	C₃×D₄	—	6	2	no	—
93	C₂×C₃×C₄	—	3	1	no	—
94	D₁₂	—	4	2	no	—
95	SL(2,𝔽₃)	—	2	1	no	—
96	SL(2,𝔽₃)	D₄×C₉	1	1	yes	characteristic
97	C₃×C₈	—	3	1	no	—
98	C₃×C₈	—	4	1	no	—
99	C₃×C₈	—	6	2	no	—
100	C₃×C₈	—	8	1	no	—
101	C₃×C₈	C₃×S₄	1	1	yes	characteristic
102	C₂×C₃²	—	4	1	no	—
103	C₃×D₃	—	16	2	no	—
104	C₂×C₉	—	6	1	no	—
105	C₂×C₉	—	8	1	no	—
106	C₂×C₉	—	24	2	no	—
107	C₂×C₉	C₄⋊S₄	1	1	yes	characteristic
108	#41	C₂×D₃×C₉	1	1	yes	characteristic
109	C₂×D₄	—	3	2	no	—
110	SD₁₆	—	6	2	no	—
111	D₈	—	6	1	no	—
112	D₈	—	6	2	no	—
113	D₈	—	6	1	no	—
114	M₁₆	—	3	1	no	—
115	M₁₆	—	3	2	no	—
116	C₂×C₈	—	3	1	no	—
117	C₂²×C₃	—	3	1	no	—
118	C₂²×C₃	—	12	2	no	—
119	C₂²×C₃	—	24	2	no	—
120	C₂×D₃	—	8	2	no	—
121	C₃×C₄	—	3	1	no	—
122	C₃×C₄	—	4	1	no	—
123	C₃×C₄	—	8	1	no	—
124	C₃×C₄	C₂×C₃×S₄	1	1	yes	characteristic
125	C₃²	—	4	1	no	—
126	C₉	—	8	1	no	—
127	C₉	#5407	1	1	yes	characteristic, direct product factor
128	C₂³	—	6	2	no	—
129	Q₈	C₉×D₁₂	1	1	yes	characteristic
130	D₄	—	3	1	no	—
131	D₄	—	6	2	no	—
132	D₄	—	6	2	no	—
133	C₂×C₄	—	3	1	no	—
134	C₈	—	3	1	no	—
135	C₈	—	6	2	no	—
136	C₈	C₉×S₄	1	1	yes	characteristic
137	C₂×C₃	—	4	1	no	—
138	C₂×C₃	—	6	1	no	—
139	C₂×C₃	—	8	1	no	—
140	C₂×C₃	—	24	2	no	—
141	C₂×C₃	C₃×C₄⋊S₄	1	1	yes	characteristic
142	D₃	—	16	2	no	—
143	C₂²	—	3	1	no	—
144	C₂²	—	12	2	no	—
145	C₂²	—	24	2	no	—
146	C₄	—	3	1	no	—
147	C₄	C₂×C₉×S₄	1	1	yes	characteristic
148	C₃	—	4	1	no	—
149	C₃	—	8	1	no	—
150	C₃	C₃×#5407	1	1	yes	characteristic
151	C₂	—	6	1	no	—
152	C₂	—	24	2	no	—
153	C₂	C₉×C₄⋊S₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#986

C₂

yes

maximal, complemented

C₉×#131⋊C₃

C₂

yes

maximal, characteristic, complemented

C₃×#5407

C₃

yes

maximal, characteristic

C₉×#472

—

maximal

C₉×#41⋊C₃

C₂²

yes

characteristic

C₉×GL(2,𝔽₃)

—

C₉×D₂₄

—

maximal

C₃×#986

C₂×C₃

yes

—

C₃×#131⋊C₃

C₂×C₃

yes

characteristic

C₉×AGL(1,ℤ/8)

—

C₂×C₉×D₈

—

C₉×#131

D₃

yes

characteristic, complemented

#131⋊C₉

—

C₉×#108

—

C₉×#100

—

C₉×D₁₂

—

C₉×SL(2,𝔽₃)

D₄

yes

characteristic

C₃×C₈×C₉

—

#5407

C₉

yes

characteristic, direct product factor

C₃×#472

—

C₃×#41⋊C₃

C₂²×C₃

yes

characteristic

C₃×GL(2,𝔽₃)

—

C₃×D₂₄

—

C₉×#41

C₂×D₃

yes

characteristic

C₂×D₄×C₉

—

#41⋊C₉

—

C₉×SD₁₆

—

C₉×D₈

—

C₉×D₈

—

C₉×D₈

—

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×D₃×C₉

—

C₃×C₄×C₉

—

#986

C₂×C₉

yes

complemented

C₃×AGL(1,ℤ/8)

—

C₂×C₃×D₈

—

C₃×#131

C₃×D₃

yes

characteristic

#131⋊C₃

—

#131⋊C₃

C₂×C₉

yes

characteristic, complemented

C₃×#108

—

C₃×#100

—

C₃×D₁₂

—

C₃×SL(2,𝔽₃)

C₃×D₄

yes

characteristic

C₂³×C₉

—

C₃²×C₈

—

Q₈×C₉

D₁₂

yes

characteristic

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

Q₈⋊C₉

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

S₄

yes

characteristic

#472

—

Sylow

C₂×C₃×C₉

—

D₃×C₉

—

C₃×#41

C₂×C₃×D₃

yes

characteristic

C₂×C₃×D₄

—

#41⋊C₃

—

#41⋊C₃

C₂²×C₉

yes

characteristic

GL(2,𝔽₃)

—

C₃×SD₁₆

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

D₂₄

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×S₄

yes

characteristic

AGL(1,ℤ/8)

—

C₂×D₈

—

#131

D₃×C₉

yes

characteristic, complemented

#108

—

#100

—

C₃×C₉

—

Sylow

C₂³×C₃

—

C₃×Q₈

C₃×D₁₂

yes

characteristic

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

D₁₂

—

SL(2,𝔽₃)

—

SL(2,𝔽₃)

D₄×C₉

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

100

C₃×C₈

—

101

C₃×C₈

C₃×S₄

yes

characteristic

102

C₂×C₃²

—

103

C₃×D₃

—

104

C₂×C₉

—

105

C₂×C₉

—

106

C₂×C₉

—

107

C₂×C₉

C₄⋊S₄

yes

characteristic

108

#41

C₂×D₃×C₉

yes

characteristic

109

C₂×D₄

—

110

SD₁₆

—

111

D₈

—

112

D₈

—

113

D₈

—

114

M₁₆

—

115

M₁₆

—

116

C₂×C₈

—

117

C₂²×C₃

—

118

C₂²×C₃

—

119

C₂²×C₃

—

120

C₂×D₃

—

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

C₂×C₃×S₄

yes

characteristic

125

C₃²

—

126

C₉

—

127

C₉

#5407

yes

characteristic, direct product factor

128

C₂³

—

129

Q₈

C₉×D₁₂

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₈

—

135

C₈

—

136

C₈

C₉×S₄

yes

characteristic

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

C₃×C₄⋊S₄

yes

characteristic

142

D₃

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₄

—

147

C₄

C₂×C₉×S₄

yes

characteristic

148

C₃

—

149

C₃

—

150

C₃

C₃×#5407

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

C₉×C₄⋊S₄

yes

characteristic

	subgroup	G/H
center	C₂×C₉	C₄⋊S₄
commutator	#41⋊C₃	C₂²×C₉
Frattini	C₃×C₄	C₂×C₃×S₄
Fitting	C₉×#131	D₃
socle	C₂×C₃	C₃×C₄⋊S₄
prime radical	C₃×#5407	C₃
cosocle	C₃×#41⋊C₃	C₂²×C₃
hypercenter	C₈×C₉	S₄
hypocenter	SL(2,𝔽₃)	D₄×C₉
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ C₈×C₉

subgroup

G/H

center

C₂×C₉

C₄⋊S₄

commutator

#41⋊C₃

C₂²×C₉

Frattini

C₃×C₄

C₂×C₃×S₄

Fitting

C₉×#131

D₃

socle

C₂×C₃

C₃×C₄⋊S₄

prime radical

C₃×#5407

C₃

cosocle

C₃×#41⋊C₃

C₂²×C₃

hypercenter

C₈×C₉

S₄

hypocenter

SL(2,𝔽₃)

D₄×C₉

LCS

G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ C₈×C₉

N	H ≅ G/N	type
1. C₉×#986	C₂	semidirect
2. C₉×#131⋊C₃	C₂	semidirect
12. C₉×#131	D₃	semidirect
19. #5407	C₉	direct (N × H)
36. #986	C₂×C₉	semidirect
41. #131⋊C₃	C₂×C₉	semidirect
84. #131	D₃×C₉	semidirect
127. C₉	#5407	direct (N × H)

H ≅ G/N

type

1. C₉×#986

C₂

semidirect

2. C₉×#131⋊C₃

C₂

semidirect

12. C₉×#131

D₃

semidirect

19. #5407

C₉

direct (N × H)

36. #986

C₂×C₉

semidirect

41. #131⋊C₃

C₂×C₉

semidirect

84. #131

D₃×C₉

semidirect

127. C₉

#5407

direct (N × H)

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

subgroup	index
C₉×#986	2
C₉×#131⋊C₃	2
C₃×#5407	3
C₉×#472	3
C₉×D₂₄	4

subgroup

index

C₉×#986

C₉×#131⋊C₃

C₃×#5407

C₉×#472

C₉×D₂₄

Subgroups — C₉×#5407 #13146771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#5407 #13146771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5407 #13146771