Group #11790085 — subgroups

Subgroups — #11790085

Order 1152.

286 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 77 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×#63767	C₂	1	1	yes	maximal, characteristic
2	C₄×#3777	C₂	1	1	yes	maximal, characteristic
3	C₂×#63744	C₂	1	1	yes	maximal, characteristic
4	C₂×#63744	C₂	1	2	yes	maximal
5	#10590181	C₂	1	2	yes	maximal
6	C₂×C₄×#157	C₂²	1	2	yes	—
7	C₂×C₄×#157	C₂²	1	1	yes	characteristic
8	C₂×#3787	C₂²	1	2	yes	—
9	C₂×#3787	C₂²	1	1	yes	characteristic
10	C₂×#3777	C₄	1	2	yes	—
11	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	#63767	—	2	2	no	—
13	#63747	C₄	1	1	yes	characteristic
14	#63747	C₄	1	1	yes	characteristic
15	#63744	—	2	2	no	—
16	#63744	—	2	4	no	—
17	C₂²×#157	C₂×C₄	1	2	yes	—
18	C₂²×#157	C₂×C₄	1	1	yes	characteristic
19	C₂²×#157	C₂×C₄	1	1	yes	characteristic
20	C₂²×#157	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
22	C₂×#622	—	2	1	no	—
23	C₂×#622	—	2	2	no	—
24	C₄×#157	—	2	2	no	—
25	C₄×#157	—	2	2	no	—
26	C₄×#157	D₄	1	2	yes	—
27	C₄×#157	Q₈	1	2	yes	—
28	#3787	—	2	1	no	—
29	#3787	—	2	2	no	—
30	#3787	—	2	2	no	—
31	#3787	D₄	1	2	yes	—
32	#3777	—	2	4	no	—
33	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
34	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic
35	C₈×C₃⋊D₃	—	2	1	no	—
36	C₈×C₃⋊D₃	—	2	1	no	—
37	#1940	—	9	1	no	maximal, Sylow
38	C₂×D₃×C₈	—	12	1	no	—
39	C₂²×C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×#157	—	2	2	no	—
41	C₂×#157	—	2	2	no	—
42	C₂×#157	—	2	2	no	—
43	C₂×#157	—	2	2	no	—
44	C₂×#157	—	2	2	no	—
45	C₂×#157	—	2	2	no	—
46	C₂×#157	—	2	2	no	—
47	C₂×#157	C₄⋊C₄	1	2	yes	—
48	C₂×#157	C₄⋊C₄	1	2	yes	—
49	C₂×#157	M₁₆	1	4	yes	—
50	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₃⋊Dic₃	#41	1	1	yes	characteristic
52	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
53	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
54	C₄×C₃⋊D₃	C₂×Q₈	1	1	yes	characteristic
55	C₄×C₃⋊D₃	#41	1	1	yes	characteristic
56	#622	—	2	2	no	—
57	#622	—	4	2	no	—
58	C₃²×C₈	—	2	1	no	—
59	C₃⋊(C₃⋊C₈)	—	2	1	no	—
60	C₂×C₄⋊C₈	—	9	1	no	—
61	C₂×C₄⋊C₈	—	9	2	no	—
62	C₂×#97	—	9	1	no	—
63	C₄×C₄⋊C₄	—	9	1	no	—
64	#336	—	9	2	no	—
65	C₂×C₄×D₃	—	12	1	no	—
66	C₂×C₃×C₈	—	4	1	no	—
67	C₂×C₃⋊C₈	—	12	1	no	—
68	D₃×C₈	—	12	2	no	—
69	D₃×C₈	—	12	2	no	—
70	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₃⋊D₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃⋊D₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₃⋊D₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₃⋊D₃	#131	1	2	yes	—
75	#157	—	2	2	no	—
76	#157	—	2	2	no	—
77	#157	—	2	4	no	—
78	#157	—	4	2	no	—
79	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
80	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
81	C₃⋊Dic₃	#121	1	1	yes	characteristic
82	C₃⋊Dic₃	#125	1	1	yes	characteristic
83	C₂²×C₈	—	9	1	no	—
84	C₂²×C₈	—	9	1	no	—
85	C₂²×C₈	—	9	2	no	—
86	C₂×C₄⋊C₄	—	9	2	no	—
87	C₂×C₄²	—	9	1	no	—
88	C₂×C₄²	—	9	2	no	—
89	C₄⋊C₈	—	18	2	no	—
90	C₄⋊C₈	—	18	4	no	—
91	#97	—	18	2	no	—
92	#95	—	9	1	no	—
93	#95	—	9	1	no	—
94	C₂²×D₃	—	12	1	no	—
95	C₂×C₃×C₄	—	4	1	no	—
96	C₂×Dic₃	—	12	1	no	—
97	C₄×D₃	—	12	1	no	—
98	C₄×D₃	—	12	1	no	—
99	C₄×D₃	—	12	1	no	—
100	C₄×D₃	—	12	1	no	—
101	C₃×C₈	—	4	2	no	—
102	C₃⋊C₈	—	12	2	no	—
103	C₂×C₃²	#389	1	1	yes	characteristic
104	C₂×C₃²	#446	1	2	yes	—
105	C₃⋊D₃	#423	1	1	yes	characteristic
106	C₃⋊D₃	#424	1	1	yes	characteristic
107	C₃⋊D₃	#435	1	2	yes	—
108	C₂²×C₄	—	9	1	no	—
109	C₂²×C₄	—	9	1	no	—
110	C₂²×C₄	—	9	2	no	—
111	C₂²×C₄	—	9	1	no	—
112	C₂²×C₄	—	9	2	no	—
113	C₂×C₈	—	9	2	no	—
114	C₂×C₈	—	9	1	no	—
115	C₂×C₈	—	18	1	no	—
116	C₂×C₈	—	18	1	no	—
117	C₂×C₈	—	18	1	no	—
118	C₂×C₈	—	18	1	no	—
119	C₂×C₈	—	18	2	no	—
120	C₂×C₈	—	18	2	no	—
121	C₂×C₈	—	18	2	no	—
122	C₂×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
123	C₄⋊C₄	—	18	4	no	—
124	C₄²	—	9	2	no	—
125	C₄²	—	9	2	no	—
126	C₄²	—	18	2	no	—
127	C₄²	—	18	2	no	—
128	C₂²×C₃	—	4	1	no	—
129	C₂×D₃	—	12	1	no	—
130	C₂×D₃	—	12	1	no	—
131	C₂×D₃	—	12	2	no	—
132	C₂×D₃	—	12	2	no	—
133	C₃×C₄	—	4	1	no	—
134	C₃×C₄	—	4	1	no	—
135	Dic₃	—	12	1	no	—
136	Dic₃	—	12	1	no	—
137	C₃²	#1940	1	1	yes	characteristic, complemented, Sylow
138	C₂³	—	9	1	no	—
139	C₂×C₄	—	9	1	no	—
140	C₂×C₄	—	9	1	no	—
141	C₂×C₄	—	9	1	no	—
142	C₂×C₄	—	9	1	no	—
143	C₂×C₄	—	9	1	no	—
144	C₂×C₄	—	9	2	no	—
145	C₂×C₄	—	9	2	no	—
146	C₂×C₄	—	9	4	no	—
147	C₂×C₄	—	18	2	no	—
148	C₂×C₄	—	18	2	no	—
149	C₂×C₄	—	18	2	no	—
150	C₂×C₄	—	18	2	no	—
151	C₂×C₄	—	18	2	no	—
152	C₂×C₄	—	18	2	no	—
153	C₂×C₄	—	18	2	no	—
154	C₂×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
155	C₈	—	2	1	no	—
156	C₈	—	18	2	no	—
157	C₈	—	18	1	no	—
158	C₈	—	36	2	no	—
159	C₂×C₃	—	4	1	no	—
160	C₂×C₃	—	4	2	no	—
161	D₃	—	12	1	no	—
162	D₃	—	12	1	no	—
163	D₃	—	12	2	no	—
164	C₂²	—	9	1	no	—
165	C₂²	—	9	1	no	—
166	C₂²	—	9	2	no	—
167	C₂²	—	9	2	no	—
168	C₂²	C₄×PSU(3,𝔽₂)	1	1	yes	characteristic
169	C₄	—	9	1	no	—
170	C₄	—	9	1	no	—
171	C₄	—	18	4	no	—
172	C₄	—	18	2	no	—
173	C₄	—	18	2	no	—
174	C₄	—	36	2	no	—
175	C₄	C₄⋊PSU(3,𝔽₂)	1	1	yes	characteristic
176	C₄	C₂²⋊PSU(3,𝔽₂)	1	1	yes	characteristic
177	C₃	—	4	1	no	—
178	C₂	—	9	2	no	—
179	C₂	—	9	1	no	—
180	C₂	—	9	1	no	—
181	C₂	#10593532	1	2	yes	—
182	C₂	#10593548	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63767

C₂

yes

maximal, characteristic

C₄×#3777

C₂

yes

maximal, characteristic

C₂×#63744

C₂

yes

maximal, characteristic

C₂×#63744

C₂

yes

maximal

#10590181

C₂

yes

maximal

C₂×C₄×#157

C₂²

yes

—

C₂×C₄×#157

C₂²

yes

characteristic

C₂×#3787

C₂²

yes

—

C₂×#3787

C₂²

yes

characteristic

C₂×#3777

C₄

yes

—

C₂×C₈×C₃⋊D₃

C₂²

yes

characteristic

#63767

—

#63747

C₄

yes

characteristic

#63747

C₄

yes

characteristic

#63744

—

#63744

—

C₂²×#157

C₂×C₄

yes

—

C₂²×#157

C₂×C₄

yes

characteristic

C₂²×#157

C₂×C₄

yes

characteristic

C₂²×#157

C₂×C₄

yes

—

C₂×C₄×C₃⋊D₃

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×#157

Q₈

yes

—

—

—

—

D₄

yes

—

#3777

—

C₂×C₃²×C₈

Q₈

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

Q₈

yes

characteristic

C₈×C₃⋊D₃

—

C₈×C₃⋊D₃

—

#1940

—

maximal, Sylow

C₂×D₃×C₈

—

C₂²×C₃⋊D₃

C₂²×C₄

yes

—

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#157

C₄⋊C₄

yes

—

C₂×#157

M₁₆

yes

—

C₂×C₃²×C₄

C₂×Q₈

yes

characteristic

C₂×C₃⋊Dic₃

#41

yes

characteristic

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

C₄×C₃⋊D₃

C₂×Q₈

yes

C₄×C₃⋊D₃

yes

—

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄×C₄⋊C₄

—

#336

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

C₂²×C₃²

C₄×Q₈

yes

characteristic

C₂×C₃⋊D₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃⋊D₃

C₄×D₄

yes

characteristic

C₂×C₃⋊D₃

C₂×M₁₆

yes

—

C₂×C₃⋊D₃

#131

yes

—

—

—

—

—

C₃²×C₄

C₂²⋊Q₈

yes

characteristic

C₃²×C₄

C₄⋊Q₈

yes

C₃⋊Dic₃

yes

C₃⋊Dic₃

yes

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

—

#95

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

100

C₄×D₃

—

101

C₃×C₈

—

102

C₃⋊C₈

—

103

C₂×C₃²

#389

yes

characteristic

104

C₂×C₃²

#446

yes

—

105

C₃⋊D₃

#423

yes

characteristic

106

C₃⋊D₃

#424

yes

characteristic

107

C₃⋊D₃

#435

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

PSU(3,𝔽₂)

yes

characteristic

123

C₄⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₂²×C₃

—

129

C₂×D₃

—

130

C₂×D₃

—

131

C₂×D₃

—

132

C₂×D₃

—

133

C₃×C₄

—

134

C₃×C₄

—

135

Dic₃

—

136

Dic₃

—

137

C₃²

#1940

yes

characteristic, complemented, Sylow

138

C₂³

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₂×C₃

—

160

C₂×C₃

—

161

D₃

—

162

D₃

—

163

D₃

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₄×PSU(3,𝔽₂)

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

C₄⋊PSU(3,𝔽₂)

yes

characteristic

176

C₄

C₂²⋊PSU(3,𝔽₂)

yes

characteristic

177

C₃

—

178

C₂

—

179

C₂

—

180

C₂

—

181

C₂

#10593532

yes

—

182

C₂

#10593548

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×PSU(3,𝔽₂)
commutator, prime radical	C₂²×C₃⋊D₃	C₂²×C₄
Fitting	C₂×C₃²×C₈	Q₈
socle	C₂²×C₃²	C₄×Q₈
cosocle	C₂×C₄×C₃⋊D₃	C₂³
hypercenter	C₂×C₈	PSU(3,𝔽₂)
hypocenter	C₃²	#1940
LCS	G ▷ C₂²×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂×PSU(3,𝔽₂)

commutator, prime radical

C₂²×C₃⋊D₃

C₂²×C₄

Fitting

C₂×C₃²×C₈

Q₈

socle

C₂²×C₃²

C₄×Q₈

cosocle

C₂×C₄×C₃⋊D₃

C₂³

C₂×C₈

C₃²

G ▷ C₂²×C₃⋊D₃ ▷ C₃²

C₁ ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
137. C₃²	#1940	semidirect

H ≅ G/N

type

137. C₃²

#1940

semidirect

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

subgroup	index
C₂×#63767	2
C₄×#3777	2
C₂×#63744	2
#10590181	2
#1940	9

subgroup

index

Subgroups — #11790085

Conjugacy classes of subgroups

Complements

Maximal subgroups