Group #11771122 — subgroups

Subgroups — #11771122

Order 1152.

225 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 94 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×#431	C₂	1	1	yes	maximal, characteristic
2	#10588604	C₂	1	1	yes	maximal, characteristic
3	C₄×#3667	C₂	1	1	yes	maximal, characteristic
4	#67743	—	3	1	no	maximal
5	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
6	C₂×#3667	—	2	1	no	—
7	C₂×#3667	C₄	1	2	yes	—
8	C₉×C₄⋊C₈	C₄	1	2	yes	—
9	C₉×C₂²⋊C₈	—	2	1	no	—
10	C₄×C₈×C₉	—	2	1	no	—
11	C₄⋊(C₉⋊C₈)	—	2	1	no	—
12	C₄⋊(C₉⋊C₈)	—	2	1	no	—
13	#63362	C₄	1	2	yes	—
14	C₄×C₉⋊C₈	—	2	1	no	—
15	C₃×#431	D₃	1	1	yes	characteristic
16	#4924	—	3	1	no	—
17	C₄×#208	—	3	1	no	—
18	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₉	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₉	—	2	2	no	—
22	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₉	D₄	1	1	yes	characteristic
25	C₄²×C₉	D₄	1	1	yes	characteristic
26	#3667	—	2	4	no	—
27	#3667	—	4	2	no	—
28	C₂×C₉⋊C₈	—	2	2	no	—
29	C₂×C₉⋊C₈	—	2	2	no	—
30	#1299	—	9	1	no	Sylow
31	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
32	C₂×#208	—	3	2	no	—
33	C₂×#208	—	6	1	no	—
34	C₃×C₄⋊C₈	Dic₃	1	2	yes	—
35	C₃×C₂²⋊C₈	—	2	1	no	—
36	C₃×C₄×C₈	—	2	1	no	—
37	C₄⋊(C₃⋊C₈)	—	6	1	no	—
38	C₄⋊(C₃⋊C₈)	—	6	1	no	—
39	#803	—	3	2	no	—
40	C₄×C₃⋊C₈	—	6	1	no	—
41	C₂³×C₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₉	—	2	1	no	—
43	C₂×C₄×C₉	—	2	1	no	—
44	C₂×C₄×C₉	—	2	2	no	—
45	C₂×C₄×C₉	—	2	2	no	—
46	C₂×C₄×C₉	C₄²	1	1	yes	characteristic
47	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
52	C₈×C₉	—	4	2	no	—
53	C₉⋊C₈	—	2	4	no	—
54	C₉⋊C₈	—	4	2	no	—
55	#431	D₉	1	1	yes	characteristic
56	#423	—	9	1	no	—
57	C₄×M₁₆	—	9	1	no	—
58	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
60	C₂²×C₃×C₄	#67	1	1	yes	characteristic
61	C₂×C₃×C₈	—	2	2	no	—
62	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
63	C₃×C₄²	D₁₂	1	1	yes	characteristic
64	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
65	C₃×C₄²	#67	1	1	yes	characteristic
66	#208	—	6	4	no	—
67	#208	—	12	2	no	—
68	C₂×C₃⋊C₈	—	6	2	no	—
69	C₂×C₃⋊C₈	—	6	2	no	—
70	C₂²×C₉	—	2	1	no	—
71	C₂²×C₉	—	2	1	no	—
72	C₂²×C₉	—	2	1	no	—
73	C₂²×C₉	#95	1	1	yes	characteristic
74	C₄×C₉	—	2	1	no	—
75	C₄×C₉	—	2	1	no	—
76	C₄×C₉	—	2	1	no	—
77	C₄×C₉	#100	1	1	yes	characteristic
78	C₄×C₉	#101	1	1	yes	characteristic
79	C₄×C₉	#104	1	2	yes	—
80	C₄×C₉	#108	1	2	yes	—
81	C₂×M₁₆	—	9	2	no	—
82	C₂×M₁₆	—	18	1	no	—
83	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
84	C₄⋊C₈	—	18	1	no	—
85	C₄⋊C₈	—	18	1	no	—
86	C₄⋊C₈	Dic₉	1	2	yes	—
87	C₂²⋊C₈	—	2	1	no	—
88	#97	—	9	2	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	18	1	no	—
91	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	2	1	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
97	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
98	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
99	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
100	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
101	C₂×C₃×C₄	#217	1	1	yes	characteristic
102	C₃×C₈	—	4	2	no	—
103	C₃⋊C₈	—	6	4	no	—
104	C₃⋊C₈	—	12	2	no	—
105	C₂×C₉	—	4	1	no	—
106	C₂×C₉	#339	1	1	yes	characteristic
107	C₂×C₉	#341	1	1	yes	characteristic
108	C₂×C₉	#343	1	1	yes	characteristic
109	C₂²×C₄	Dic₁₈	1	1	yes	characteristic
110	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
111	C₂²×C₄	#611	1	1	yes	characteristic
112	M₁₆	—	18	4	no	—
113	M₁₆	—	36	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	18	2	no	—
116	C₂×C₈	—	18	2	no	—
117	C₄²	C₄×D₉	1	1	yes	characteristic
118	C₄²	D₃₆	1	1	yes	characteristic
119	C₄²	C₂×Dic₉	1	1	yes	characteristic
120	C₄²	#611	1	1	yes	characteristic
121	C₂²×C₃	—	2	1	no	—
122	C₂²×C₃	—	2	1	no	—
123	C₂²×C₃	—	2	1	no	—
124	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
125	C₃×C₄	—	2	1	no	—
126	C₃×C₄	—	2	1	no	—
127	C₃×C₄	—	2	1	no	—
128	C₃×C₄	#819	1	2	yes	—
129	C₃×C₄	#823	1	1	yes	characteristic
130	C₃×C₄	#824	1	1	yes	characteristic
131	C₃×C₄	#837	1	2	yes	—
132	C₉	#1299	1	1	yes	characteristic, complemented, Sylow
133	C₂³	Dic₉⋊C₄	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₄×Dic₉	1	1	yes	characteristic
139	C₂×C₄	Dic₉⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₄⋊Dic₉	1	1	yes	characteristic
141	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
142	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
143	C₂×C₄	#3676	1	1	yes	characteristic
144	C₈	—	4	2	no	—
145	C₈	—	18	4	no	—
146	C₈	—	36	2	no	—
147	C₂×C₃	—	4	1	no	—
148	C₂×C₃	#4532	1	1	yes	characteristic
149	C₂×C₃	#4552	1	1	yes	characteristic
150	C₂×C₃	#4554	1	1	yes	characteristic
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	C₉⋊#95	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#63372	1	2	yes	—
159	C₄	#63382	1	1	yes	characteristic
160	C₄	#63383	1	1	yes	characteristic
161	C₄	#63396	1	2	yes	—
162	C₃	#67743	1	1	yes	characteristic
163	C₂	—	4	1	no	—
164	C₂	#10588208	1	1	yes	characteristic
165	C₂	#10588227	1	1	yes	characteristic
166	C₂	#10588234	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#431

C₂

yes

maximal, characteristic

#10588604

C₂

yes

maximal, characteristic

C₄×#3667

C₂

yes

maximal, characteristic

#67743

—

maximal

C₂×C₄²×C₉

C₂²

yes

characteristic

C₂×#3667

—

C₂×#3667

C₄

yes

—

C₉×C₄⋊C₈

C₄

yes

—

C₉×C₂²⋊C₈

—

C₄×C₈×C₉

—

C₄⋊(C₉⋊C₈)

—

C₄⋊(C₉⋊C₈)

—

#63362

C₄

yes

—

C₄×C₉⋊C₈

—

C₃×#431

D₃

yes

characteristic

#4924

—

C₄×#208

—

C₂²×C₄×C₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₉

D₄

yes

characteristic

C₂²×C₄×C₉

Q₈

yes

characteristic

C₂×C₈×C₉

—

C₄²×C₉

C₂×C₄

yes

characteristic

C₄²×C₉

C₂×C₄

yes

characteristic

C₄²×C₉

D₄

yes

characteristic

C₄²×C₉

D₄

yes

characteristic

#3667

—

#3667

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

#1299

—

Sylow

C₂×C₃×C₄²

C₂×D₃

yes

characteristic

C₂×#208

—

C₂×#208

—

C₃×C₄⋊C₈

Dic₃

yes

—

C₃×C₂²⋊C₈

—

C₃×C₄×C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₄×C₃⋊C₈

—

C₂³×C₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₄²

yes

characteristic

C₂×C₄×C₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉

C₄⋊C₄

yes

characteristic

C₈×C₉

—

C₉⋊C₈

—

C₉⋊C₈

—

#431

D₉

yes

characteristic

#423

—

C₄×M₁₆

—

C₂²×C₃×C₄

Dic₆

yes

characteristic

C₂²×C₃×C₄

C₄×D₃

yes

characteristic

C₂²×C₃×C₄

#67

yes

characteristic

C₂×C₃×C₈

—

C₃×C₄²

C₄×D₃

yes

characteristic

C₃×C₄²

D₁₂

yes

characteristic

C₃×C₄²

C₂×Dic₃

yes

C₃×C₄²

yes

—

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

#95

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

yes

C₄×C₉

yes

C₄×C₉

yes

—

C₄×C₉

#108

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄²

C₂×D₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₉

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

Dic₃⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×Dic₃

yes

characteristic

C₂×C₃×C₄

Dic₃⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊Dic₃

yes

characteristic

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

characteristic

100

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

characteristic

101

C₂×C₃×C₄

#217

yes

characteristic

102

C₃×C₈

—

103

C₃⋊C₈

—

104

C₃⋊C₈

—

105

C₂×C₉

—

106

C₂×C₉

#339

yes

characteristic

107

C₂×C₉

#341

yes

characteristic

108

C₂×C₉

#343

yes

characteristic

109

C₂²×C₄

Dic₁₈

yes

characteristic

110

C₂²×C₄

C₄×D₉

yes

characteristic

111

C₂²×C₄

#611

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄²

C₄×D₉

yes

characteristic

118

C₄²

D₃₆

yes

characteristic

119

C₄²

C₂×Dic₉

yes

characteristic

120

C₄²

#611

yes

characteristic

121

C₂²×C₃

—

122

C₂²×C₃

—

123

C₂²×C₃

—

124

C₂²×C₃

C₃⋊#95

yes

characteristic

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

—

128

C₃×C₄

#819

yes

—

129

C₃×C₄

#823

yes

characteristic

130

C₃×C₄

#824

yes

characteristic

131

C₃×C₄

#837

yes

—

132

C₉

#1299

yes

characteristic, complemented, Sylow

133

C₂³

Dic₉⋊C₄

yes

characteristic

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₄×Dic₉

yes

characteristic

139

C₂×C₄

Dic₉⋊C₄

yes

characteristic

140

C₂×C₄

C₄⋊Dic₉

yes

characteristic

141

C₂×C₄

(C₂×D₉)⋊C₄

yes

characteristic

142

C₂×C₄

(C₂×D₉)⋊C₄

yes

characteristic

143

C₂×C₄

#3676

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₂×C₃

—

148

C₂×C₃

#4532

yes

characteristic

149

C₂×C₃

#4552

yes

characteristic

150

C₂×C₃

#4554

yes

characteristic

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

C₉⋊#95

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#63372

yes

—

159

C₄

#63382

yes

characteristic

160

C₄

#63383

yes

characteristic

161

C₄

#63396

yes

—

162

C₃

#67743

yes

characteristic

163

C₂

—

164

C₂

#10588208

yes

characteristic

165

C₂

#10588227

yes

characteristic

166

C₂

#10588234

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₉)⋊C₄
commutator	C₂×C₄×C₉	C₄²
Frattini	C₂×C₃×C₄²	C₂×D₃
Fitting	C₉×#431	C₂
socle	C₂²×C₃	C₃⋊#95
prime radical	C₂³×C₃	Dic₃⋊C₄
cosocle	C₂×C₄²×C₉	C₂²
hypercenter	#431	D₉
hypocenter	C₉	#1299
LCS	G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₉)⋊C₄

commutator

C₂×C₄×C₉

C₄²

Frattini

C₂×C₃×C₄²

C₂×D₃

C₂

C₂²×C₃

C₂³×C₃

Dic₃⋊C₄

cosocle

C₂×C₄²×C₉

C₂²

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
132. C₉	#1299	semidirect

H ≅ G/N

type

132. C₉

#1299

semidirect

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

subgroup	index
C₉×#431	2
#10588604	2
C₄×#3667	2
#67743	3

subgroup

index

Subgroups — #11771122

Conjugacy classes of subgroups

Complements

Maximal subgroups