Group #11776883 — subgroups

Subgroups — #11776883

Order 1152.

218 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 66 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	C₂×C₄×AGL(1,𝔽₉)	C₂	1	1	yes	maximal, characteristic
2	#10593539	C₂	1	1	yes	maximal, characteristic
3	#10593328	C₂	1	1	yes	maximal, characteristic
4	AGL(1,𝔽₉)⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂²×AGL(1,𝔽₉)	C₄	1	2	yes	—
6	C₂×C₄×#157	C₂²	1	1	yes	characteristic
7	#64283	—	2	1	no	—
8	C₄×AGL(1,𝔽₉)	C₄	1	1	yes	characteristic
9	C₄×AGL(1,𝔽₉)	C₄	1	1	yes	characteristic
10	C₄×AGL(1,𝔽₉)	C₂²	1	2	yes	—
11	#63818	—	2	1	no	—
12	#63744	—	2	1	no	—
13	#63744	C₂²	1	2	yes	—
14	#63732	—	2	1	no	—
15	#63732	C₂²	1	2	yes	—
16	C₂²×#157	C₂×C₄	1	1	yes	characteristic
17	C₂²×#157	Q₈	1	1	yes	characteristic
18	C₂×AGL(1,𝔽₉)	—	2	4	no	—
19	C₂×AGL(1,𝔽₉)	C₂×C₄	1	2	yes	—
20	C₂×AGL(1,𝔽₉)	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
22	#3792	—	4	1	no	—
23	C₄×#157	C₂×C₄	1	1	yes	characteristic
24	C₄×#157	D₄	1	1	yes	characteristic
25	C₄×#157	Q₈	1	1	yes	characteristic
26	C₄×#157	C₂³	1	1	yes	characteristic
27	#3788	—	4	1	no	—
28	#3787	—	2	2	no	—
29	C₃×#208	—	4	1	no	—
30	#3712	—	4	1	no	—
31	#3710	—	2	2	no	—
32	#1566	—	9	1	no	maximal, Sylow
33	D₃×M₁₆	—	12	1	no	—
34	C₂²×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×#157	—	2	2	no	—
36	C₂×#157	—	2	2	no	—
37	C₂×#157	C₄⋊C₄	1	1	yes	characteristic
38	C₂×#157	C₄⋊C₄	1	1	yes	characteristic
39	C₂×#157	C₂²×C₄	1	1	yes	characteristic
40	C₂×#157	C₂×Q₈	1	1	yes	characteristic
41	AGL(1,𝔽₉)	—	2	2	no	—
42	AGL(1,𝔽₉)	M₁₆	1	4	yes	complemented
43	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
44	C₂×C₃⋊Dic₃	SD₁₆	1	1	yes	characteristic
45	C₄×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₃⋊D₃	SD₁₆	1	1	yes	characteristic
47	C₄×C₃⋊D₃	SD₁₆	1	1	yes	characteristic
48	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
49	#622	—	4	2	no	—
50	C₃×C₃⋊C₈	—	4	2	no	—
51	#423	—	9	1	no	—
52	#423	—	9	1	no	—
53	C₂×C₄×C₈	—	9	1	no	—
54	#334	—	9	4	no	—
55	C₂×C₄×D₃	—	12	1	no	—
56	C₃×M₁₆	—	12	1	no	—
57	#208	—	4	1	no	—
58	ΓL(1,𝔽₂₅)	—	12	2	no	—
59	D₃×C₈	—	12	2	no	—
60	C₂²×C₃²	#106	1	1	yes	characteristic
61	C₂×C₃⋊D₃	—	2	1	no	—
62	C₂×C₃⋊D₃	—	2	1	no	—
63	C₂×C₃⋊D₃	#106	1	1	yes	characteristic
64	C₂×C₃⋊D₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	#157	—	2	1	no	—
66	#157	—	2	1	no	—
67	#157	#108	1	2	yes	—
68	#157	C₂×M₁₆	1	2	yes	—
69	C₃²×C₄	#106	1	1	yes	characteristic
70	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
71	C₃⋊Dic₃	#106	1	1	yes	characteristic
72	C₃⋊Dic₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×M₁₆	—	18	1	no	—
74	C₂×M₁₆	—	18	1	no	—
75	C₂²×C₈	—	9	2	no	—
76	C₂×C₄²	—	9	1	no	—
77	C₄⋊C₈	—	9	2	no	—
78	C₄⋊C₈	—	9	2	no	—
79	C₄⋊C₈	—	18	1	no	—
80	C₄⋊C₈	—	18	1	no	—
81	C₄×C₈	—	9	2	no	—
82	C₄×C₈	—	9	1	no	—
83	C₄×C₈	—	9	1	no	—
84	C₂²×D₃	—	12	1	no	—
85	C₂×C₃×C₄	—	4	1	no	—
86	C₂×Dic₃	—	12	1	no	—
87	C₄×D₃	—	12	1	no	—
88	C₄×D₃	—	12	1	no	—
89	C₄×D₃	—	12	1	no	—
90	C₄×D₃	—	12	1	no	—
91	C₃×C₈	—	12	2	no	—
92	C₃⋊C₈	—	4	2	no	—
93	C₂×C₃²	—	2	1	no	—
94	C₂×C₃²	C₂×#106	1	1	yes	characteristic
95	C₃⋊D₃	—	2	1	no	—
96	C₃⋊D₃	#423	1	1	yes	characteristic
97	C₃⋊D₃	C₂×#108	1	1	yes	characteristic
98	C₂²×C₄	—	9	1	no	—
99	C₂²×C₄	—	9	1	no	—
100	C₂²×C₄	—	9	1	no	—
101	M₁₆	—	12	1	no	—
102	M₁₆	—	36	1	no	—
103	M₁₆	—	36	1	no	—
104	M₁₆	—	36	1	no	—
105	C₂×C₈	—	9	2	no	—
106	C₂×C₈	—	9	2	no	—
107	C₂×C₈	—	18	4	no	—
108	C₂×C₈	—	18	2	no	—
109	C₂×C₈	—	18	2	no	—
110	C₄²	—	9	1	no	—
111	C₄²	—	9	1	no	—
112	C₄²	—	9	1	no	—
113	C₄²	—	9	1	no	—
114	C₂²×C₃	—	4	1	no	—
115	C₂×D₃	—	12	1	no	—
116	C₂×D₃	—	12	1	no	—
117	C₂×D₃	—	24	1	no	—
118	C₂×D₃	—	24	1	no	—
119	C₃×C₄	—	4	1	no	—
120	C₃×C₄	—	4	1	no	—
121	Dic₃	—	12	1	no	—
122	Dic₃	—	12	1	no	—
123	C₃²	#1566	1	1	yes	characteristic, complemented, Sylow
124	C₂³	—	9	1	no	—
125	C₂×C₄	—	9	1	no	—
126	C₂×C₄	—	9	1	no	—
127	C₂×C₄	—	9	1	no	—
128	C₂×C₄	—	9	1	no	—
129	C₂×C₄	—	9	1	no	—
130	C₂×C₄	—	9	1	no	—
131	C₂×C₄	—	9	1	no	—
132	C₂×C₄	—	9	1	no	—
133	C₂×C₄	—	9	1	no	—
134	C₂×C₄	—	18	2	no	—
135	C₂×C₄	—	18	2	no	—
136	C₂×C₄	AΓL(1,𝔽₉)	1	1	yes	characteristic
137	C₈	—	9	4	no	—
138	C₈	—	12	2	no	—
139	C₈	—	18	2	no	—
140	C₈	—	36	2	no	—
141	C₂×C₃	—	4	1	no	—
142	C₂×C₃	—	8	1	no	—
143	D₃	—	12	1	no	—
144	D₃	—	12	1	no	—
145	D₃	—	24	1	no	—
146	C₂²	—	9	1	no	—
147	C₂²	—	9	1	no	—
148	C₂²	—	18	1	no	—
149	C₂²	—	18	1	no	—
150	C₂²	AGL(1,𝔽₉)⋊C₄	1	1	yes	characteristic
151	C₄	—	9	2	no	—
152	C₄	—	9	2	no	—
153	C₄	—	9	1	no	—
154	C₄	—	9	1	no	—
155	C₄	—	18	1	no	—
156	C₄	—	18	1	no	—
157	C₄	AGL(1,𝔽₉)⋊C₄	1	1	yes	characteristic
158	C₄	C₂×AΓL(1,𝔽₉)	1	1	yes	characteristic
159	C₃	—	4	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	9	1	no	—
162	C₂	—	9	1	no	—
163	C₂	—	18	1	no	—
164	C₂	C₂×AGL(1,𝔽₉)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×AGL(1,𝔽₉)

C₂

yes

maximal, characteristic

#10593539

C₂

yes

maximal, characteristic

#10593328

C₂

yes

maximal, characteristic

AGL(1,𝔽₉)⋊C₈

C₂

yes

maximal, complemented

C₂²×AGL(1,𝔽₉)

C₄

yes

—

C₂×C₄×#157

C₂²

yes

characteristic

#64283

—

C₄×AGL(1,𝔽₉)

C₄

yes

characteristic

C₄×AGL(1,𝔽₉)

C₄

yes

characteristic

C₄×AGL(1,𝔽₉)

C₂²

yes

—

#63818

—

#63744

—

#63744

C₂²

yes

—

#63732

—

#63732

C₂²

yes

—

C₂²×#157

C₂×C₄

yes

characteristic

C₂²×#157

Q₈

yes

characteristic

C₂×AGL(1,𝔽₉)

—

C₂×AGL(1,𝔽₉)

C₂×C₄

yes

—

C₂×AGL(1,𝔽₉)

C₂×C₄

yes

—

C₂×C₄×C₃⋊D₃

D₄

yes

characteristic

#3792

—

C₄×#157

C₂×C₄

yes

characteristic

C₄×#157

D₄

yes

characteristic

C₄×#157

Q₈

yes

characteristic

C₄×#157

C₂³

yes

—

—

—

—

—

—

D₃×M₁₆

—

C₂²×C₃⋊D₃

C₄⋊C₄

yes

—

—

C₄⋊C₄

yes

characteristic

C₂×#157

C₄⋊C₄

yes

characteristic

C₂×#157

C₂²×C₄

yes

characteristic

C₂×#157

C₂×Q₈

yes

characteristic

AGL(1,𝔽₉)

—

AGL(1,𝔽₉)

M₁₆

yes

complemented

C₂×C₃²×C₄

SD₁₆

yes

characteristic

C₂×C₃⋊Dic₃

SD₁₆

yes

characteristic

C₄×C₃⋊D₃

C₄⋊C₄

yes

characteristic

C₄×C₃⋊D₃

SD₁₆

yes

characteristic

C₄×C₃⋊D₃

SD₁₆

yes

characteristic

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

#622

—

C₃×C₃⋊C₈

—

#423

—

#423

—

C₂×C₄×C₈

—

#334

—

C₂×C₄×D₃

—

C₃×M₁₆

—

#208

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₂²×C₃²

#106

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

#106

yes

characteristic

C₂×C₃⋊D₃

C₂×C₄⋊C₄

yes

—

—

yes

—

#157

C₂×M₁₆

yes

—

C₃²×C₄

#106

yes

characteristic

C₃²×C₄

C₂×SD₁₆

yes

characteristic

C₃⋊Dic₃

#106

yes

characteristic

C₃⋊Dic₃

C₂×SD₁₆

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

C₂×#106

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

yes

C₃⋊D₃

yes

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₂²×C₃

—

115

C₂×D₃

—

116

C₂×D₃

—

117

C₂×D₃

—

118

C₂×D₃

—

119

C₃×C₄

—

120

C₃×C₄

—

121

Dic₃

—

122

Dic₃

—

123

C₃²

#1566

yes

characteristic, complemented, Sylow

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

AΓL(1,𝔽₉)

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂×C₃

—

142

C₂×C₃

—

143

D₃

—

144

D₃

—

145

D₃

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

AGL(1,𝔽₉)⋊C₄

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

AGL(1,𝔽₉)⋊C₄

yes

characteristic

158

C₄

C₂×AΓL(1,𝔽₉)

yes

characteristic

159

C₃

—

160

C₂

—

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

C₂×AGL(1,𝔽₉)⋊C₄

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×AΓL(1,𝔽₉)
commutator	C₂×#157	C₂²×C₄
Fitting	C₂×C₃²×C₄	SD₁₆
socle	C₂×C₃²	C₂×#106
prime radical	C₂²×C₃⋊D₃	C₄⋊C₄
cosocle	C₄×#157	C₂³
hypercenter	C₂×C₄	AΓL(1,𝔽₉)
hypocenter	C₃²	#1566
LCS	G ▷ C₂×#157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

subgroup

G/H

C₄

C₂²×C₄

C₂×C₃²×C₄

SD₁₆

socle

C₂×C₃²

C₂×#106

prime radical

C₂²×C₃⋊D₃

C₄⋊C₄

C₂³

C₂×C₄

C₃²

G ▷ C₂×#157 ▷ C₃⋊D₃ ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂×C₄

N	H ≅ G/N	type
4. AGL(1,𝔽₉)⋊C₈	C₂	semidirect
42. AGL(1,𝔽₉)	M₁₆	semidirect
123. C₃²	#1566	semidirect

H ≅ G/N

type

4. AGL(1,𝔽₉)⋊C₈

C₂

semidirect

42. AGL(1,𝔽₉)

M₁₆

semidirect

123. C₃²

#1566

semidirect

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

subgroup	index
C₂×C₄×AGL(1,𝔽₉)	2
#10593539	2
#10593328	2
AGL(1,𝔽₉)⋊C₈	2
#1566	9

subgroup

index

C₂×C₄×AGL(1,𝔽₉)

Subgroups — #11776883

Conjugacy classes of subgroups

Complements

Maximal subgroups