Group #11781949 — subgroups

Subgroups — #11781949

Order 1152.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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×#500	C₂	1	1	yes	maximal, characteristic
2	#10588763	C₂	1	2	yes	maximal
3	#10588359	C₂	1	1	yes	maximal, characteristic
4	#10588224	C₂	1	2	yes	maximal
5	#10588134	C₂	1	1	yes	maximal, characteristic
6	#71379	—	3	1	no	maximal
7	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#3674	—	2	2	no	—
9	C₄×Dic₁₈	C₂²	1	1	yes	characteristic
10	C₉×#107	—	2	1	no	—
11	C₉×#107	—	2	1	no	—
12	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
13	#63395	—	2	2	no	—
14	#63375	—	2	1	no	—
15	Dic₉⋊C₈	—	2	1	no	—
16	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
18	C₃×#500	D₃	1	1	yes	characteristic
19	#5090	—	3	2	no	—
20	#4679	—	3	1	no	—
21	#4547	—	3	2	no	—
22	#4457	—	3	1	no	—
23	C₂×Q₈×C₉	—	2	2	no	—
24	C₂×Dic₁₈	D₄	1	1	yes	characteristic
25	C₂×C₈×C₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₉	D₄	1	1	yes	characteristic
27	C₉×C₄⋊C₄	—	2	2	no	—
28	C₉×C₄⋊C₄	—	2	2	no	—
29	C₄²×C₉	C₂³	1	1	yes	characteristic
30	#3674	—	4	4	no	—
31	C₄⋊Dic₉	D₄	1	1	yes	characteristic
32	Dic₉⋊C₄	—	2	1	no	—
33	C₄×Dic₉	—	2	1	no	—
34	C₂×C₉⋊C₈	—	2	2	no	—
35	C₂×C₉⋊C₈	—	2	1	no	—
36	#1694	—	9	1	no	Sylow
37	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
38	C₂×#216	—	6	2	no	—
39	C₄×Dic₆	—	3	1	no	—
40	C₃×#107	—	2	1	no	—
41	C₃×#107	—	2	1	no	—
42	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
43	#835	—	6	2	no	—
44	#815	—	6	1	no	—
45	Dic₃⋊C₈	—	6	1	no	—
46	C₄⋊(C₃⋊C₈)	—	3	1	no	—
47	C₄⋊(C₃⋊C₈)	—	3	2	no	—
48	Q₈×C₉	—	4	2	no	—
49	Q₈×C₉	—	4	2	no	—
50	C₂×C₄×C₉	—	4	2	no	—
51	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₉	#41	1	1	yes	characteristic
54	C₂×Dic₉	—	2	1	no	—
55	C₂×Dic₉	—	2	1	no	—
56	Dic₁₈	—	2	2	no	—
57	C₈×C₉	—	2	1	no	—
58	C₈×C₉	Q₁₆	1	2	yes	complemented
59	C₉⋊C₈	—	4	2	no	—
60	C₉⋊C₈	—	4	1	no	—
61	#500	D₉	1	1	yes	characteristic
62	#462	—	9	2	no	—
63	#446	—	9	1	no	—
64	#334	—	9	1	no	—
65	#332	—	9	2	no	—
66	C₂×C₃×Q₈	—	2	2	no	—
67	C₂×Dic₆	—	3	1	no	—
68	C₂×C₃×C₈	#67	1	1	yes	characteristic
69	C₂×C₃×C₈	#67	1	1	yes	characteristic
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
73	#216	—	12	4	no	—
74	C₄⋊Dic₃	—	3	1	no	—
75	Dic₃⋊C₄	—	6	1	no	—
76	C₄×Dic₃	—	6	1	no	—
77	C₂×C₃⋊C₈	—	6	2	no	—
78	C₂×C₃⋊C₈	—	6	1	no	—
79	C₂²×C₉	#121	1	1	yes	characteristic
80	C₄×C₉	—	2	1	no	—
81	C₄×C₉	—	8	2	no	—
82	C₄×C₉	C₂×Q₁₆	1	1	yes	characteristic
83	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
84	C₄×C₉	#137	1	1	yes	characteristic
85	Dic₉	—	4	1	no	—
86	Dic₉	—	4	1	no	—
87	C₂×Q₁₆	—	18	2	no	—
88	C₄⋊Q₈	C₂×D₉	1	2	yes	—
89	C₄×Q₈	—	9	1	no	—
90	#107	—	2	1	no	—
91	#107	—	2	1	no	—
92	C₄⋊C₈	—	9	1	no	—
93	C₄⋊C₈	—	9	2	no	—
94	C₄⋊C₈	—	18	1	no	—
95	#103	—	18	2	no	—
96	#97	—	18	1	no	—
97	C₄×C₈	C₂×D₉	1	1	yes	characteristic
98	C₃×Q₈	—	4	2	no	—
99	C₃×Q₈	—	4	2	no	—
100	C₂×C₃×C₄	—	4	2	no	—
101	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
102	C₂×C₃×C₄	#237	1	1	yes	characteristic
103	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
104	C₂×Dic₃	—	6	1	no	—
105	C₂×Dic₃	—	6	1	no	—
106	Dic₆	—	6	2	no	—
107	C₃×C₈	—	2	1	no	—
108	C₃×C₈	#216	1	2	yes	—
109	C₃⋊C₈	—	12	2	no	—
110	C₃⋊C₈	—	12	1	no	—
111	C₂×C₉	#462	1	1	yes	characteristic
112	C₂×C₉	#469	1	1	yes	characteristic
113	C₂×C₉	#473	1	1	yes	characteristic
114	C₂×Q₈	—	2	2	no	—
115	C₂×Q₈	—	9	1	no	—
116	Q₁₆	—	36	4	no	—
117	C₂×C₈	—	18	2	no	—
118	C₂×C₈	—	18	1	no	—
119	C₂×C₈	#611	1	1	yes	characteristic
120	C₂×C₈	#611	1	1	yes	characteristic
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	9	1	no	—
124	C₄⋊C₄	—	18	1	no	—
125	C₄²	—	18	1	no	—
126	C₄²	C₂²×D₉	1	1	yes	characteristic
127	C₂²×C₃	#938	1	1	yes	characteristic
128	C₃×C₄	—	2	1	no	—
129	C₃×C₄	—	8	2	no	—
130	C₃×C₄	#911	1	2	yes	—
131	C₃×C₄	#942	1	1	yes	characteristic
132	C₃×C₄	C₂×#216	1	1	yes	characteristic
133	Dic₃	—	12	1	no	—
134	Dic₃	—	12	1	no	—
135	C₉	#1694	1	1	yes	characteristic, complemented, Sylow
136	Q₈	—	4	2	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	18	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	18	1	no	—
141	C₂×C₄	—	18	1	no	—
142	C₂×C₄	D₄×D₉	1	1	yes	characteristic
143	C₂×C₄	#3699	1	1	yes	characteristic
144	C₂×C₄	C₂×#611	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	36	1	no	—
147	C₈	—	36	2	no	—
148	C₈	#3674	1	2	yes	complemented
149	C₂×C₃	#5090	1	1	yes	characteristic
150	C₂×C₃	#5159	1	1	yes	characteristic
151	C₂×C₃	#5192	1	1	yes	characteristic
152	C₂²	#63500	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	8	2	no	—
155	C₄	—	36	1	no	—
156	C₄	—	36	1	no	—
157	C₄	#63473	1	2	yes	—
158	C₄	C₂×#3674	1	1	yes	characteristic
159	C₄	#63505	1	1	yes	characteristic
160	C₃	#71379	1	1	yes	characteristic
161	C₂	#10588763	1	1	yes	characteristic
162	C₂	#10588839	1	1	yes	characteristic
163	C₂	#10588872	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#500

C₂

yes

maximal, characteristic

#10588763

C₂

yes

maximal

#10588359

C₂

yes

maximal, characteristic

#10588224

C₂

yes

maximal

#10588134

C₂

yes

maximal, characteristic

#71379

—

maximal

C₉×C₄⋊Q₈

C₂²

yes

—

C₂×#3674

—

C₄×Dic₁₈

C₂²

yes

characteristic

C₉×#107

—

C₉×#107

—

C₄×C₈×C₉

C₂²

yes

characteristic

#63395

—

#63375

—

Dic₉⋊C₈

—

C₄⋊(C₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₉⋊C₈)

C₂²

yes

—

C₃×#500

D₃

yes

—

—

—

—

C₂×Q₈×C₉

—

C₂×Dic₁₈

D₄

yes

characteristic

C₂×C₈×C₉

D₄

yes

characteristic

C₂×C₈×C₉

D₄

yes

characteristic

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

C₂³

yes

characteristic

#3674

—

C₄⋊Dic₉

D₄

yes

characteristic

Dic₉⋊C₄

—

C₄×Dic₉

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

#1694

—

Sylow

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₂×#216

—

C₄×Dic₆

—

C₃×#107

—

C₃×#107

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

#835

—

#815

—

Dic₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

#41

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

Q₁₆

yes

complemented

C₉⋊C₈

—

C₉⋊C₈

—

#500

D₉

yes

—

—

—

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

yes

C₂×C₃×C₈

yes

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

characteristic

#216

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

#121

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×Q₁₆

yes

characteristic

C₄×C₉

AGL(1,ℤ/8)

yes

—

C₄×C₉

#137

yes

characteristic

Dic₉

—

Dic₉

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₉

yes

—

C₄×Q₈

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

C₂×D₉

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

D₃×D₄

yes

characteristic

102

C₂×C₃×C₄

#237

yes

characteristic

103

C₂×C₃×C₄

C₂×#67

yes

characteristic

104

C₂×Dic₃

—

105

C₂×Dic₃

—

106

Dic₆

—

107

C₃×C₈

—

108

C₃×C₈

#216

yes

—

109

C₃⋊C₈

—

110

C₃⋊C₈

—

111

C₂×C₉

#462

yes

characteristic

112

C₂×C₉

#469

yes

characteristic

113

C₂×C₉

#473

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

116

Q₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

#611

yes

characteristic

120

C₂×C₈

#611

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

—

126

C₄²

C₂²×D₉

yes

characteristic

127

C₂²×C₃

#938

yes

characteristic

128

C₃×C₄

—

129

C₃×C₄

—

130

C₃×C₄

#911

yes

—

131

C₃×C₄

#942

yes

characteristic

132

C₃×C₄

C₂×#216

yes

characteristic

133

Dic₃

—

134

Dic₃

—

135

C₉

#1694

yes

characteristic, complemented, Sylow

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

D₄×D₉

yes

characteristic

143

C₂×C₄

#3699

yes

characteristic

144

C₂×C₄

C₂×#611

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

#3674

yes

complemented

149

C₂×C₃

#5090

yes

characteristic

150

C₂×C₃

#5159

yes

characteristic

151

C₂×C₃

#5192

yes

characteristic

152

C₂²

#63500

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#63473

yes

—

158

C₄

C₂×#3674

yes

characteristic

159

C₄

#63505

yes

characteristic

160

C₃

#71379

yes

characteristic

161

C₂

#10588763

yes

characteristic

162

C₂

#10588839

yes

characteristic

163

C₂

#10588872

yes

characteristic

	subgroup	G/H
center	C₂²	#63500
commutator, cosocle	C₄²×C₉	C₂³
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#500	C₂
socle, prime radical	C₂²×C₃	#938
hypercenter	#500	D₉
hypocenter	C₉	#1694
LCS	G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#63500

commutator, cosocle

C₄²×C₉

C₂³

Frattini

C₃×C₄²

C₂²×D₃

C₂

C₂²×C₃

D₉

C₉

G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
58. C₈×C₉	Q₁₆	semidirect
135. C₉	#1694	semidirect
148. C₈	#3674	semidirect

H ≅ G/N

type

58. C₈×C₉

Q₁₆

semidirect

135. C₉

#1694

semidirect

148. C₈

#3674

semidirect

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

subgroup	index
C₉×#500	2
#10588763	2
#10588359	2
#10588224	2
#10588134	2
#71379	3

subgroup

index

Subgroups — #11781949

Conjugacy classes of subgroups

Complements

Maximal subgroups