Group #11784221 — subgroups

Subgroups — #11784221

Order 1152.

390 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 174 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×C₉×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₉⋊C₈	C₂	1	2	yes	maximal
3	#10588612	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#10588228	C₂	1	2	yes	maximal
6	#69832	—	3	1	no	maximal
7	C₂²×C₈×C₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
11	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
12	C₉×#97	—	2	2	no	—
13	C₉⋊#95	C₄	1	4	yes	—
14	Dic₉⋊C₈	C₄	1	8	yes	complemented
15	C₄×C₉⋊C₈	—	2	2	no	—
16	C₂×C₃×#97	D₃	1	1	yes	characteristic
17	C₂×Dic₃⋊C₈	—	3	2	no	—
18	#4932	—	3	1	no	—
19	C₂×C₄×C₃⋊C₈	—	3	1	no	—
20	#4550	—	3	2	no	—
21	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₉	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₉	—	2	2	no	—
26	C₂×C₈×C₉	—	2	2	no	—
27	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
28	C₄²×C₉	D₄	1	2	yes	—
29	C₄²×C₉	Q₈	1	2	yes	—
30	C₄×Dic₉	—	2	2	no	—
31	C₄×Dic₉	C₂×C₄	1	4	yes	—
32	C₂×C₉⋊C₈	—	2	2	no	—
33	C₂×C₉⋊C₈	—	2	2	no	—
34	C₂×C₉⋊C₈	C₂×C₄	1	4	yes	—
35	#1768	—	9	1	no	Sylow
36	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
37	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
38	C₂×C₄×Dic₃	—	3	1	no	—
39	C₂×C₄×Dic₃	—	3	1	no	—
40	C₂²×C₃⋊C₈	—	3	2	no	—
41	C₃×#97	—	2	2	no	—
42	C₃⋊#95	—	3	4	no	—
43	Dic₃⋊C₈	—	3	8	no	—
44	C₄×C₃⋊C₈	—	6	2	no	—
45	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₉	—	2	4	no	—
47	C₂×C₄×C₉	C₄⋊C₄	1	4	yes	—
48	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₉	#41	1	2	yes	—
52	C₂×Dic₉	—	2	4	no	—
53	C₂×Dic₉	—	2	2	no	—
54	C₂×Dic₉	—	2	4	no	—
55	C₂×Dic₉	C₄²	1	4	yes	—
56	C₈×C₉	—	2	4	no	—
57	C₉⋊C₈	—	2	4	no	—
58	C₂×C₄⋊C₈	—	9	2	no	—
59	C₂×#97	D₉	1	1	yes	characteristic
60	C₂×C₄×C₈	—	9	1	no	—
61	#376	—	9	1	no	—
62	#336	—	9	2	no	—
63	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
64	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
65	C₂²×Dic₃	—	3	2	no	—
66	C₂²×Dic₃	—	3	2	no	—
67	C₂×C₃×C₈	—	2	2	no	—
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
70	C₃×C₄²	Dic₆	1	2	yes	—
71	C₃×C₄²	#67	1	2	yes	—
72	C₄×Dic₃	—	3	4	no	—
73	C₄×Dic₃	—	6	2	no	—
74	C₂×C₃⋊C₈	—	3	4	no	—
75	C₂×C₃⋊C₈	—	6	2	no	—
76	C₂×C₃⋊C₈	—	6	2	no	—
77	C₂²×C₉	C₂×C₄²	1	1	yes	characteristic
78	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₉	#117	1	1	yes	characteristic
80	C₂²×C₉	#131	1	4	yes	—
81	C₄×C₉	—	2	4	no	—
82	C₄×C₉	C₄×D₄	1	2	yes	—
83	C₄×C₉	C₄×Q₈	1	2	yes	—
84	Dic₉	—	2	4	no	—
85	Dic₉	—	4	2	no	—
86	C₂²×C₈	—	9	2	no	—
87	C₂²×C₈	C₂×D₉	1	2	yes	—
88	C₂×C₄²	—	9	1	no	—
89	C₂×C₄²	—	9	1	no	—
90	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
91	C₄⋊C₈	—	9	8	no	—
92	#97	—	2	2	no	—
93	C₄×C₈	—	18	2	no	—
94	#95	—	9	4	no	—
95	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
96	C₂×C₃×C₄	—	2	4	no	—
97	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
98	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
100	C₂×C₃×C₄	#235	1	2	yes	—
101	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
102	C₂×Dic₃	—	3	4	no	—
103	C₂×Dic₃	—	6	4	no	—
104	C₂×Dic₃	—	6	2	no	—
105	C₂×Dic₃	—	6	4	no	—
106	C₃×C₈	—	2	4	no	—
107	C₃⋊C₈	—	6	4	no	—
108	C₂×C₉	C₄×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₉	#405	1	2	yes	—
110	C₂×C₉	#424	1	2	yes	—
111	C₂×C₉	#433	1	2	yes	—
112	C₂²×C₄	—	9	2	no	—
113	C₂²×C₄	—	9	2	no	—
114	C₂²×C₄	C₄×D₉	1	2	yes	—
115	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	9	4	no	—
119	C₂×C₈	—	18	2	no	—
120	C₂×C₈	—	18	2	no	—
121	C₂×C₈	C₄×D₉	1	4	yes	—
122	C₄²	—	9	4	no	—
123	C₄²	—	18	2	no	—
124	C₄²	Dic₁₈	1	2	yes	—
125	C₄²	#611	1	2	yes	—
126	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
127	C₂²×C₃	#872	1	1	yes	characteristic
128	C₂²×C₃	#907	1	4	yes	—
129	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
130	C₃×C₄	—	2	4	no	—
131	C₃×C₄	C₄×Dic₆	1	2	yes	—
132	C₃×C₄	C₄×#67	1	2	yes	—
133	Dic₃	—	6	4	no	—
134	Dic₃	—	12	2	no	—
135	C₉	#1768	1	1	yes	characteristic, complemented, Sylow
136	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	9	4	no	—
139	C₂×C₄	—	18	4	no	—
140	C₂×C₄	—	18	2	no	—
141	C₂×C₄	—	18	4	no	—
142	C₂×C₄	Dic₉⋊C₄	1	4	yes	—
143	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
145	C₂×C₄	#3697	1	2	yes	—
146	C₂×C₄	C₂×#611	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₈	—	18	4	no	—
149	C₂×C₃	#4708	1	2	yes	—
150	C₂×C₃	#4709	1	2	yes	—
151	C₂×C₃	C₄×Dic₃⋊C₄	1	1	yes	characteristic
152	C₂×C₃	#5119	1	2	yes	—
153	C₂²	C₄²×D₉	1	1	yes	characteristic
154	C₂²	#63434	1	1	yes	characteristic
155	C₂²	#63469	1	4	yes	—
156	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	18	4	no	—
159	C₄	—	36	2	no	—
160	C₄	C₄×Dic₁₈	1	2	yes	—
161	C₄	C₄×#611	1	2	yes	—
162	C₃	#69832	1	1	yes	characteristic
163	C₂	#10588389	1	2	yes	—
164	C₂	#10588390	1	2	yes	—
165	C₂	C₄×Dic₉⋊C₄	1	1	yes	characteristic
166	C₂	#10588803	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×#97

C₂

yes

maximal, characteristic

C₂×Dic₉⋊C₈

C₂

yes

maximal

#10588612

C₂

yes

maximal, characteristic

C₂×C₄×C₉⋊C₈

C₂

yes

maximal, characteristic

#10588228

C₂

yes

maximal

#69832

—

maximal

C₂²×C₈×C₉

C₂²

yes

—

C₂×C₄²×C₉

C₂²

yes

characteristic

C₂×C₄×Dic₉

C₂²

yes

characteristic

C₂×C₄×Dic₉

C₂²

yes

characteristic

C₂²×C₉⋊C₈

C₂²

yes

—

C₉×#97

—

C₉⋊#95

C₄

yes

—

Dic₉⋊C₈

C₄

yes

complemented

C₄×C₉⋊C₈

—

C₂×C₃×#97

D₃

yes

characteristic

C₂×Dic₃⋊C₈

—

#4932

—

C₂×C₄×C₃⋊C₈

—

#4550

—

C₂²×C₄×C₉

C₂×C₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂²×Dic₉

C₂×C₄

yes

—

C₂²×Dic₉

C₂×C₄

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂×C₄

yes

—

C₄²×C₉

D₄

yes

—

C₄²×C₉

Q₈

yes

—

C₄×Dic₉

—

C₄×Dic₉

C₂×C₄

yes

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

C₂×C₄

yes

—

#1768

—

Sylow

C₂²×C₃×C₈

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

characteristic

C₂×C₄×Dic₃

—

C₂×C₄×Dic₃

—

C₂²×C₃⋊C₈

—

C₃×#97

—

C₃⋊#95

—

Dic₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂³×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

C₄⋊C₄

yes

—

C₂×C₄×C₉

C₂²×C₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₉

#41

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

C₄²

yes

—

C₈×C₉

—

C₉⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

D₉

yes

characteristic

C₂×C₄×C₈

—

#376

—

#336

—

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₃×C₄²

Dic₆

yes

—

C₃×C₄²

#67

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

C₂×C₄²

yes

characteristic

C₂²×C₉

C₂×C₄⋊C₄

yes

C₂²×C₉

yes

C₂²×C₉

yes

—

C₄×C₉

—

C₄×C₉

C₄×D₄

yes

—

C₄×C₉

C₄×Q₈

yes

—

Dic₉

—

Dic₉

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉

yes

characteristic

C₄⋊C₈

—

#97

—

C₄×C₈

—

#95

—

C₂³×C₃

C₂×C₄×D₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

C₂×Dic₆

yes

characteristic

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

100

C₂×C₃×C₄

#235

yes

—

101

C₂×C₃×C₄

C₂×#67

yes

characteristic

102

C₂×Dic₃

—

103

C₂×Dic₃

—

104

C₂×Dic₃

—

105

C₂×Dic₃

—

106

C₃×C₈

—

107

C₃⋊C₈

—

108

C₂×C₉

C₄×C₄⋊C₄

yes

characteristic

109

C₂×C₉

#405

yes

—

110

C₂×C₉

#424

yes

—

111

C₂×C₉

#433

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₄×D₉

yes

—

115

C₂²×C₄

C₂²×D₉

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₄×D₉

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

Dic₁₈

yes

—

125

C₄²

#611

yes

—

126

C₂²×C₃

C₄²×D₃

yes

characteristic

127

C₂²×C₃

#872

yes

characteristic

128

C₂²×C₃

#907

yes

—

129

C₂²×C₃

C₂×Dic₃⋊C₄

yes

characteristic

130

C₃×C₄

—

131

C₃×C₄

C₄×Dic₆

yes

—

132

C₃×C₄

C₄×#67

yes

—

133

Dic₃

—

134

Dic₃

—

135

C₉

#1768

yes

characteristic, complemented, Sylow

136

C₂³

C₂×C₄×D₉

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

Dic₉⋊C₄

yes

—

143

C₂×C₄

C₂×Dic₁₈

yes

characteristic

144

C₂×C₄

C₂×C₄×D₉

yes

—

145

C₂×C₄

#3697

yes

—

146

C₂×C₄

C₂×#611

yes

characteristic

147

C₈

—

148

C₈

—

149

C₂×C₃

#4708

yes

—

150

C₂×C₃

#4709

yes

—

151

C₂×C₃

C₄×Dic₃⋊C₄

yes

characteristic

152

C₂×C₃

#5119

yes

—

153

C₂²

C₄²×D₉

yes

characteristic

154

C₂²

#63434

yes

characteristic

155

C₂²

#63469

yes

—

156

C₂²

C₂×Dic₉⋊C₄

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

C₄×Dic₁₈

yes

—

161

C₄

C₄×#611

yes

—

162

C₃

#69832

yes

characteristic

163

C₂

#10588389

yes

—

164

C₂

#10588390

yes

—

165

C₂

C₄×Dic₉⋊C₄

yes

characteristic

166

C₂

#10588803

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#97	C₂
socle, prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#97	D₉
hypocenter	C₉	#1768
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center

C₂²×C₄

C₂²×D₉

commutator

C₂²×C₉

C₂×C₄²

Frattini

C₂²×C₃×C₄

C₂²×D₃

Fitting

C₂×C₉×#97

C₂

socle, prime radical

C₂³×C₃

C₂×C₄×D₃

cosocle

C₂²×C₄×C₉

C₂³

hypercenter

C₂×#97

D₉

C₉

G ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
14. Dic₉⋊C₈	C₄	semidirect
135. C₉	#1768	semidirect

H ≅ G/N

type

14. Dic₉⋊C₈

C₄

semidirect

135. C₉

#1768

semidirect

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

subgroup	index
C₂×C₉×#97	2
C₂×Dic₉⋊C₈	2
#10588612	2
C₂×C₄×C₉⋊C₈	2
#10588228	2
#69832	3

subgroup

index

C₂×C₉×#97

C₂×Dic₉⋊C₈

#10588612

C₂×C₄×C₉⋊C₈

#10588228

#69832

Subgroups — #11784221

Conjugacy classes of subgroups

Complements

Maximal subgroups