Group #11779322 — subgroups

Subgroups — #11779322

Order 1152.

327 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 72 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	#10589815	C₂	1	1	yes	maximal, characteristic
2	C₉×#492	C₂	1	1	yes	maximal, characteristic
3	#10588764	C₂	1	2	yes	maximal, complemented
4	#10588221	C₂	1	1	yes	maximal, characteristic
5	#10588133	C₂	1	2	yes	maximal, complemented
6	#71270	—	3	1	no	maximal
7	C₂×C₉×SD₁₆	—	2	2	no	—
8	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₉×#127	C₂²	1	1	yes	characteristic
10	#63498	—	2	2	no	—
11	#63497	—	2	2	no	—
12	D₄×Dic₉	—	2	2	no	—
13	C₄×Dic₁₈	C₂²	1	2	yes	—
14	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
15	#63368	—	2	2	no	—
16	#63366	—	2	2	no	—
17	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
18	#5713	—	3	1	no	—
19	C₃×#492	D₃	1	1	yes	characteristic
20	#5091	—	3	2	no	—
21	#4544	—	3	1	no	—
22	#4456	—	3	2	no	—
23	C₂×Q₈×C₉	—	2	1	no	—
24	C₂×D₄×C₉	—	2	2	no	—
25	C₂×D₄×C₉	D₄	1	2	yes	—
26	C₂²×Dic₉	—	2	2	no	—
27	C₂×Dic₁₈	D₄	1	2	yes	—
28	C₉×SD₁₆	—	4	4	no	—
29	C₂×C₈×C₉	—	2	1	no	—
30	C₉×C₄⋊C₄	—	2	2	no	—
31	C₄²×C₉	C₂³	1	1	yes	characteristic
32	#3676	—	2	4	no	—
33	#3676	—	2	2	no	—
34	C₄⋊Dic₉	D₄	1	2	yes	—
35	Dic₉⋊C₄	—	2	2	no	—
36	C₄×Dic₉	—	2	2	no	—
37	C₂×C₉⋊C₈	—	2	2	no	—
38	#1630	—	9	1	no	Sylow
39	C₂×C₃×SD₁₆	—	2	2	no	—
40	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
41	C₃×#127	C₂×D₃	1	1	yes	characteristic
42	#936	—	6	2	no	—
43	#935	—	6	2	no	—
44	D₄×Dic₃	—	6	2	no	—
45	C₄×Dic₆	—	3	2	no	—
46	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
47	#810	—	6	2	no	—
48	#807	—	6	2	no	—
49	C₄⋊(C₃⋊C₈)	—	3	2	no	—
50	C₂³×C₉	—	2	2	no	—
51	Q₈×C₉	—	4	2	no	—
52	D₄×C₉	—	4	2	no	—
53	D₄×C₉	—	4	2	no	—
54	D₄×C₉	—	4	2	no	—
55	C₂×C₄×C₉	—	4	1	no	—
56	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₉	—	2	2	no	—
59	C₂×Dic₉	—	2	2	no	—
60	C₂×Dic₉	—	8	2	no	—
61	Dic₁₈	—	2	2	no	—
62	Dic₁₈	—	2	2	no	—
63	C₈×C₉	—	2	2	no	—
64	C₉⋊C₈	—	4	2	no	—
65	#559	—	9	1	no	—
66	#492	D₉	1	1	yes	characteristic
67	#477	—	9	2	no	—
68	#331	—	9	1	no	—
69	Q₈⋊C₈	—	9	2	no	—
70	C₂×C₃×Q₈	—	2	1	no	—
71	C₂×C₃×D₄	—	2	2	no	—
72	C₂×C₃×D₄	#67	1	2	yes	—
73	C₂²×Dic₃	—	6	2	no	—
74	C₂×Dic₆	—	3	2	no	—
75	C₃×SD₁₆	—	4	4	no	—
76	C₂×C₃×C₈	—	2	1	no	—
77	C₃×C₄⋊C₄	—	2	2	no	—
78	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
79	#217	—	6	4	no	—
80	#217	—	6	2	no	—
81	C₄⋊Dic₃	—	3	2	no	—
82	Dic₃⋊C₄	—	6	2	no	—
83	C₄×Dic₃	—	6	2	no	—
84	C₂×C₃⋊C₈	—	6	2	no	—
85	C₂²×C₉	—	4	4	no	—
86	C₂²×C₉	—	4	2	no	—
87	C₂²×C₉	#120	1	1	yes	characteristic
88	C₄×C₉	—	2	1	no	—
89	C₄×C₉	—	8	1	no	—
90	C₄×C₉	#135	1	2	yes	—
91	C₄×C₉	#137	1	2	yes	—
92	Dic₉	—	4	2	no	—
93	Dic₉	—	4	2	no	—
94	C₂×SD₁₆	—	2	2	no	—
95	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
96	#127	C₂×D₉	1	1	yes	characteristic
97	#124	—	18	2	no	—
98	#123	—	18	2	no	—
99	C₄×Q₈	—	9	2	no	—
100	C₄×D₄	—	18	2	no	—
101	#107	—	18	2	no	—
102	C₄⋊C₈	—	9	2	no	—
103	#103	—	18	2	no	—
104	C₄×C₈	C₂×D₉	1	1	yes	characteristic
105	C₂³×C₃	—	2	2	no	—
106	C₃×Q₈	—	4	2	no	—
107	C₃×D₄	—	4	2	no	—
108	C₃×D₄	—	4	2	no	—
109	C₃×D₄	—	4	2	no	—
110	C₂×C₃×C₄	—	4	1	no	—
111	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
112	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
113	C₂×Dic₃	—	6	2	no	—
114	C₂×Dic₃	—	6	2	no	—
115	C₂×Dic₃	—	24	2	no	—
116	Dic₆	—	6	2	no	—
117	Dic₆	—	6	2	no	—
118	C₃×C₈	—	2	2	no	—
119	C₃⋊C₈	—	12	2	no	—
120	C₂×C₉	—	8	2	no	—
121	C₂×C₉	#452	1	2	yes	—
122	C₂×C₉	#453	1	1	yes	characteristic
123	C₂×Q₈	—	2	1	no	—
124	C₂×Q₈	—	9	2	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	#611	1	2	yes	—
127	C₂²×C₄	—	18	2	no	—
128	SD₁₆	—	4	4	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	18	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	9	2	no	—
133	C₄⋊C₄	—	18	2	no	—
134	C₂²⋊C₄	—	18	4	no	—
135	C₂²⋊C₄	—	18	2	no	—
136	C₄²	—	18	2	no	—
137	C₄²	C₂²×D₉	1	1	yes	characteristic
138	C₂²×C₃	—	4	4	no	—
139	C₂²×C₃	—	4	2	no	—
140	C₂²×C₃	#937	1	1	yes	characteristic
141	C₃×C₄	—	2	1	no	—
142	C₃×C₄	—	8	1	no	—
143	C₃×C₄	#915	1	2	yes	—
144	C₃×C₄	#916	1	2	yes	—
145	Dic₃	—	12	2	no	—
146	Dic₃	—	12	2	no	—
147	C₉	#1630	1	1	yes	characteristic, complemented, Sylow
148	C₂³	—	2	2	no	—
149	Q₈	—	4	2	no	—
150	Q₈	—	18	2	no	—
151	Q₈	—	18	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	4	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	18	2	no	—
157	C₂×C₄	—	18	2	no	—
158	C₂×C₄	—	72	2	no	—
159	C₂×C₄	D₄×D₉	1	2	yes	—
160	C₂×C₄	C₂×#611	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	36	2	no	—
163	C₂×C₃	—	8	2	no	—
164	C₂×C₃	#5077	1	1	yes	characteristic
165	C₂×C₃	#5173	1	2	yes	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	4	no	—
168	C₂²	#63499	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	8	1	no	—
171	C₄	—	36	2	no	—
172	C₄	—	36	2	no	—
173	C₄	#63477	1	2	yes	—
174	C₄	#63478	1	2	yes	—
175	C₃	#71270	1	1	yes	characteristic
176	C₂	—	8	2	no	—
177	C₂	#10588757	1	1	yes	characteristic
178	C₂	#10588853	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10589815

C₂

yes

maximal, characteristic

C₉×#492

C₂

yes

maximal, characteristic

#10588764

C₂

yes

maximal, complemented

#10588221

C₂

yes

maximal, characteristic

#10588133

C₂

yes

maximal, complemented

#71270

—

maximal

C₂×C₉×SD₁₆

—

C₉×C₄⋊Q₈

C₂²

yes

characteristic

C₉×#127

C₂²

yes

characteristic

#63498

—

#63497

—

D₄×Dic₉

—

C₄×Dic₁₈

C₂²

yes

—

C₄×C₈×C₉

C₂²

yes

characteristic

#63368

—

#63366

—

C₄⋊(C₉⋊C₈)

C₂²

yes

—

#5713

—

C₃×#492

D₃

yes

—

—

—

C₂×Q₈×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

D₄

yes

—

C₂²×Dic₉

—

C₂×Dic₁₈

D₄

yes

—

C₉×SD₁₆

—

C₂×C₈×C₉

—

C₉×C₄⋊C₄

—

C₄²×C₉

C₂³

yes

characteristic

#3676

—

#3676

—

C₄⋊Dic₉

D₄

yes

—

Dic₉⋊C₄

—

C₄×Dic₉

—

C₂×C₉⋊C₈

—

#1630

—

Sylow

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

C₂×D₃

yes

characteristic

C₃×#127

C₂×D₃

yes

characteristic

#936

—

#935

—

D₄×Dic₃

—

C₄×Dic₆

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

#810

—

#807

—

C₄⋊(C₃⋊C₈)

—

C₂³×C₉

—

Q₈×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

—

C₉⋊C₈

—

#559

—

#492

D₉

yes

characteristic

#477

—

#331

—

Q₈⋊C₈

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

#67

yes

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₃×SD₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

characteristic

#217

—

#217

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

#120

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#135

yes

—

C₄×C₉

#137

yes

—

Dic₉

—

Dic₉

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₉

yes

characteristic

#127

C₂×D₉

yes

characteristic

#124

—

#123

—

C₄×Q₈

—

100

C₄×D₄

—

101

#107

—

102

C₄⋊C₈

—

103

#103

—

104

C₄×C₈

C₂×D₉

yes

characteristic

105

C₂³×C₃

—

106

C₃×Q₈

—

107

C₃×D₄

—

108

C₃×D₄

—

109

C₃×D₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

D₃×D₄

yes

—

112

C₂×C₃×C₄

C₂×#67

yes

characteristic

113

C₂×Dic₃

—

114

C₂×Dic₃

—

115

C₂×Dic₃

—

116

Dic₆

—

117

Dic₆

—

118

C₃×C₈

—

119

C₃⋊C₈

—

120

C₂×C₉

—

121

C₂×C₉

#452

yes

—

122

C₂×C₉

#453

yes

characteristic

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×D₄

—

126

C₂×D₄

#611

yes

—

127

C₂²×C₄

—

128

SD₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

C₂²×D₉

yes

characteristic

138

C₂²×C₃

—

139

C₂²×C₃

—

140

C₂²×C₃

#937

yes

characteristic

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

#915

yes

—

144

C₃×C₄

#916

yes

—

145

Dic₃

—

146

Dic₃

—

147

C₉

#1630

yes

characteristic, complemented, Sylow

148

C₂³

—

149

Q₈

—

150

Q₈

—

151

Q₈

—

152

D₄

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

D₄×D₉

yes

—

160

C₂×C₄

C₂×#611

yes

characteristic

161

C₈

—

162

C₈

—

163

C₂×C₃

—

164

C₂×C₃

#5077

yes

characteristic

165

C₂×C₃

#5173

yes

—

166

C₂²

—

167

C₂²

—

168

C₂²

#63499

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#63477

yes

—

174

C₄

#63478

yes

—

175

C₃

#71270

yes

characteristic

176

C₂

—

177

C₂

#10588757

yes

characteristic

178

C₂

#10588853

yes

—

	subgroup	G/H
center	C₂²	#63499
commutator, cosocle	C₄²×C₉	C₂³
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#492	C₂
socle	C₂²×C₃	#937
prime radical	C₃×#127	C₂×D₃
hypercenter	#492	D₉
hypocenter	C₉	#1630
LCS	G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#63499

commutator, cosocle

C₄²×C₉

C₂³

Frattini

C₃×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂×D₃

D₉

C₉

G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
3. #10588764	C₂	semidirect
5. #10588133	C₂	semidirect
147. C₉	#1630	semidirect

H ≅ G/N

type

3. #10588764

C₂

semidirect

5. #10588133

C₂

semidirect

147. C₉

#1630

semidirect

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

subgroup	index
#10589815	2
C₉×#492	2
#10588764	2
#10588221	2
#10588133	2
#71270	3

subgroup

index

Subgroups — #11779322

Conjugacy classes of subgroups

Complements

Maximal subgroups