Group #11773603 — subgroups

Subgroups — #11773603

Order 1152.

326 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₄×#3788	C₂	1	1	yes	maximal, characteristic
2	C₄×#3787	C₂	1	1	yes	maximal, characteristic
3	#10592532	C₂	1	1	yes	maximal, characteristic
4	#10590169	C₂	1	2	yes	maximal, complemented
5	#10590168	C₂	1	2	yes	maximal, complemented
6	C₂×#3788	C₄	1	2	yes	—
7	C₂×#3787	C₄	1	2	yes	—
8	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
9	#63777	C₄	1	2	yes	—
10	C₄×#622	C₄	1	2	yes	—
11	C₄×#622	C₂²	1	1	yes	characteristic
12	C₄×#622	C₂²	1	1	yes	characteristic
13	C₄×#622	C₂²	1	2	yes	—
14	C₃²×C₄⋊C₈	C₄	1	1	yes	characteristic
15	#63647	C₄	1	2	yes	—
16	C₈×C₃⋊Dic₃	C₂²	1	2	yes	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	1	yes	characteristic
18	#4833	—	6	2	no	—
19	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#622	C₂×C₄	1	2	yes	—
23	C₂×#622	C₂×C₄	1	4	yes	—
24	C₂×#622	C₂×C₄	1	2	yes	—
25	#3788	—	2	4	no	—
26	#3787	—	2	4	no	—
27	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
28	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
32	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
33	#1451	—	9	1	no	maximal, Sylow
34	C₄²×D₃	—	6	2	no	—
35	C₃×C₄⋊C₈	—	2	2	no	—
36	(C₂×D₃)⋊C₈	—	6	4	no	—
37	C₈×Dic₃	—	6	4	no	—
38	C₄⋊(C₃⋊C₈)	—	6	2	no	—
39	C₂²×C₃⋊D₃	C₄²	1	1	yes	characteristic
40	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
41	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
42	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₃⋊Dic₃	C₄²	1	1	yes	characteristic
44	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
46	C₄×C₃⋊D₃	—	2	1	no	—
47	C₄×C₃⋊D₃	—	2	1	no	—
48	C₄×C₃⋊D₃	—	2	2	no	—
49	C₄×C₃⋊D₃	M₁₆	1	4	yes	—
50	#622	—	2	2	no	—
51	#622	—	2	4	no	—
52	#622	M₁₆	1	4	yes	complemented
53	C₃²×C₈	—	2	2	no	—
54	C₃⋊(C₃⋊C₈)	—	2	2	no	—
55	#431	—	9	1	no	—
56	C₄×M₁₆	—	9	1	no	—
57	C₂×C₄×C₈	—	9	1	no	—
58	#322	—	9	2	no	—
59	#322	—	9	2	no	—
60	C₂×C₄×D₃	—	6	2	no	—
61	C₂×C₄×D₃	—	6	2	no	—
62	C₂×C₄×D₃	—	6	2	no	—
63	C₂×C₃×C₈	—	2	4	no	—
64	C₃×C₄²	—	2	2	no	—
65	C₄×Dic₃	—	6	2	no	—
66	C₄×Dic₃	—	6	2	no	—
67	C₄×Dic₃	—	6	2	no	—
68	C₂×C₃⋊C₈	—	6	4	no	—
69	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₃⋊D₃	—	2	1	no	—
71	C₂×C₃⋊D₃	—	2	1	no	—
72	C₂×C₃⋊D₃	#97	1	2	yes	—
73	C₃²×C₄	—	2	1	no	—
74	C₃²×C₄	#97	1	2	yes	—
75	C₃²×C₄	#131	1	1	yes	characteristic
76	C₃²×C₄	#131	1	1	yes	characteristic
77	C₃⋊Dic₃	—	2	1	no	—
78	C₃⋊Dic₃	C₂×M₁₆	1	2	yes	—
79	C₃⋊Dic₃	C₂×M₁₆	1	2	yes	—
80	C₂×M₁₆	—	9	2	no	—
81	C₂²×C₈	—	9	2	no	—
82	C₂×C₄²	—	9	1	no	—
83	C₄⋊C₈	—	9	1	no	—
84	C₄⋊C₈	#157	1	1	yes	characteristic
85	C₂²⋊C₈	—	9	2	no	—
86	#97	—	9	2	no	—
87	C₄×C₈	—	9	2	no	—
88	C₄×C₈	—	9	1	no	—
89	C₄×C₈	—	9	1	no	—
90	C₄×C₈	—	9	2	no	—
91	C₄×C₈	—	9	2	no	—
92	C₂²×D₃	—	6	2	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×Dic₃	—	6	2	no	—
97	C₂×Dic₃	—	6	2	no	—
98	C₂×Dic₃	—	6	2	no	—
99	C₄×D₃	—	6	8	no	—
100	C₄×D₃	—	12	4	no	—
101	C₄×D₃	—	12	2	no	—
102	C₄×D₃	—	12	2	no	—
103	C₃×C₈	—	4	4	no	—
104	C₃⋊C₈	—	12	4	no	—
105	C₂×C₃²	C₂×#97	1	1	yes	characteristic
106	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
107	C₂×C₃²	#405	1	1	yes	characteristic
108	C₃⋊D₃	—	2	2	no	—
109	C₂²×C₄	—	9	1	no	—
110	C₂²×C₄	—	9	1	no	—
111	C₂²×C₄	—	9	1	no	—
112	M₁₆	—	18	4	no	—
113	C₂×C₈	—	9	2	no	—
114	C₂×C₈	—	9	2	no	—
115	C₂×C₈	—	9	4	no	—
116	C₂×C₈	—	9	2	no	—
117	C₂×C₈	—	18	4	no	—
118	C₂×C₈	C₂×#157	1	2	yes	—
119	C₄²	—	9	1	no	—
120	C₄²	—	9	1	no	—
121	C₄²	—	9	1	no	—
122	C₄²	C₂×#157	1	1	yes	characteristic
123	C₂²×C₃	—	2	2	no	—
124	C₂×D₃	—	6	4	no	—
125	C₂×D₃	—	12	2	no	—
126	C₂×D₃	—	12	2	no	—
127	C₃×C₄	—	2	4	no	—
128	C₃×C₄	—	2	2	no	—
129	C₃×C₄	—	2	2	no	—
130	C₃×C₄	—	4	2	no	—
131	Dic₃	—	6	4	no	—
132	Dic₃	—	6	4	no	—
133	Dic₃	—	12	2	no	—
134	C₃²	#1451	1	1	yes	characteristic, complemented, Sylow
135	C₂³	—	9	1	no	—
136	C₂×C₄	—	9	1	no	—
137	C₂×C₄	—	9	1	no	—
138	C₂×C₄	—	9	4	no	—
139	C₂×C₄	—	9	1	no	—
140	C₂×C₄	—	18	2	no	—
141	C₂×C₄	—	18	1	no	—
142	C₂×C₄	—	18	1	no	—
143	C₂×C₄	C₄×#157	1	1	yes	characteristic
144	C₂×C₄	C₄×#157	1	1	yes	characteristic
145	C₂×C₄	C₂²×#157	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	9	4	no	—
148	C₈	—	18	2	no	—
149	C₈	—	18	4	no	—
150	C₈	—	18	2	no	—
151	C₂×C₃	—	2	2	no	—
152	C₂×C₃	—	2	2	no	—
153	C₂×C₃	—	2	2	no	—
154	D₃	—	12	4	no	—
155	C₂²	—	9	2	no	—
156	C₂²	—	18	1	no	—
157	C₂²	—	18	1	no	—
158	C₂²	C₂×C₄×#157	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	9	2	no	—
161	C₄	—	9	2	no	—
162	C₄	—	18	1	no	—
163	C₄	#63767	1	2	yes	—
164	C₄	#64287	1	1	yes	characteristic
165	C₄	#64289	1	1	yes	characteristic
166	C₃	—	2	2	no	—
167	C₂	—	18	2	no	—
168	C₂	#10593748	1	1	yes	characteristic
169	C₂	C₂×#63767	1	1	yes	characteristic
170	C₂	M₁₆×#157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#3788

C₂

yes

maximal, characteristic

C₄×#3787

C₂

yes

maximal, characteristic

#10592532

C₂

yes

maximal, characteristic

#10590169

C₂

yes

maximal, complemented

#10590168

C₂

yes

maximal, complemented

C₂×#3788

C₄

yes

—

C₂×#3787

C₄

yes

—

C₄²×C₃⋊D₃

C₂²

yes

characteristic

#63777

C₄

yes

—

C₄×#622

C₄

yes

—

C₄×#622

C₂²

yes

characteristic

C₄×#622

C₂²

yes

characteristic

C₄×#622

C₂²

yes

—

C₃²×C₄⋊C₈

C₄

yes

characteristic

#63647

C₄

yes

—

C₈×C₃⋊Dic₃

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊C₈))

C₄

yes

characteristic

#4833

—

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

characteristic

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

characteristic

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

characteristic

C₂×#622

C₂×C₄

yes

—

C₂×#622

C₂×C₄

yes

—

C₂×#622

C₂×C₄

yes

—

#3788

—

#3787

—

C₂×C₃²×C₈

C₂×C₄

yes

—

C₃²×C₄²

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

#1451

—

maximal, Sylow

C₄²×D₃

—

C₃×C₄⋊C₈

—

(C₂×D₃)⋊C₈

—

C₈×Dic₃

—

C₄⋊(C₃⋊C₈)

—

C₂²×C₃⋊D₃

C₄²

yes

characteristic

C₂×C₃²×C₄

C₄²

yes

characteristic

C₂×C₃²×C₄

C₄²

yes

characteristic

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₄²

yes

characteristic

C₂×C₃⋊Dic₃

C₂²×C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂²×C₄

yes

characteristic

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

M₁₆

yes

—

#622

—

#622

—

#622

M₁₆

yes

complemented

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

#322

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×C₄²

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

#97

yes

—

C₃²×C₄

—

C₃²×C₄

#97

yes

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

—

C₃⋊Dic₃

C₂×M₁₆

yes

—

C₃⋊Dic₃

C₂×M₁₆

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

#157

yes

characteristic

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

100

C₄×D₃

—

101

C₄×D₃

—

102

C₄×D₃

—

103

C₃×C₈

—

104

C₃⋊C₈

—

105

C₂×C₃²

C₂×#97

yes

characteristic

106

C₂×C₃²

C₄×M₁₆

yes

characteristic

107

C₂×C₃²

#405

yes

characteristic

108

C₃⋊D₃

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂×#157

yes

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×#157

yes

characteristic

123

C₂²×C₃

—

124

C₂×D₃

—

125

C₂×D₃

—

126

C₂×D₃

—

127

C₃×C₄

—

128

C₃×C₄

—

129

C₃×C₄

—

130

C₃×C₄

—

131

Dic₃

—

132

Dic₃

—

133

Dic₃

—

134

C₃²

#1451

yes

characteristic, complemented, Sylow

135

C₂³

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

C₄×#157

yes

characteristic

144

C₂×C₄

C₄×#157

yes

characteristic

145

C₂×C₄

C₂²×#157

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₂×C₃

—

152

C₂×C₃

—

153

C₂×C₃

—

154

D₃

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₄×#157

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#63767

yes

—

164

C₄

#64287

yes

characteristic

165

C₄

#64289

yes

characteristic

166

C₃

—

167

C₂

—

168

C₂

#10593748

yes

characteristic

169

C₂

C₂×#63767

yes

characteristic

170

C₂

M₁₆×#157

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#157
commutator, socle	C₂²×C₃²	C₂×C₄²
Fitting	C₃²×C₄⋊C₈	C₄
prime radical	C₂²×C₃⋊D₃	C₄²
cosocle	C₄×C₃⋊Dic₃	C₂³
hypercenter	C₄⋊C₈	#157
hypocenter	C₃²	#1451
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#157

commutator, socle

C₂²×C₃²

C₂×C₄²

Fitting

C₃²×C₄⋊C₈

C₄

prime radical

C₂²×C₃⋊D₃

C₄²

cosocle

C₄×C₃⋊Dic₃

C₂³

C₄⋊C₈

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
4. #10590169	C₂	semidirect
5. #10590168	C₂	semidirect
52. #622	M₁₆	semidirect
134. C₃²	#1451	semidirect

H ≅ G/N

type

4. #10590169

C₂

semidirect

5. #10590168

C₂

semidirect

52. #622

M₁₆

semidirect

134. C₃²

#1451

semidirect

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

subgroup	index
C₄×#3788	2
C₄×#3787	2
#10592532	2
#10590169	2
#10590168	2
#1451	9

subgroup

index

Subgroups — #11773603

Conjugacy classes of subgroups

Complements

Maximal subgroups