Group #140787 — subgroups

Subgroups — #140787

Order 512.

317 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 88 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	#33463	C₂	1	1	yes	maximal, characteristic
2	#13197	C₂	1	2	yes	maximal, complemented
3	#12002	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1335	C₂	1	2	yes	maximal, complemented
5	#11402	C₂	1	1	yes	maximal, characteristic, complemented
6	#3414	C₄	1	1	yes	characteristic
7	C₂×#443	C₂²	1	2	yes	—
8	#2179	—	2	2	no	—
9	#2178	—	2	2	no	—
10	#2161	C₄	1	1	yes	characteristic
11	C₂×#364	C₂²	1	2	yes	—
12	#2143	C₄	1	2	yes	—
13	#2117	—	2	2	no	—
14	#2115	—	2	2	no	—
15	C₄×M₃₂	C₂²	1	1	yes	characteristic
16	C₄×M₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1369	—	2	2	no	—
19	#1335	—	2	2	no	—
20	#1335	—	2	2	no	—
21	#1312	—	2	2	no	—
22	C₂×#135	C₂×C₄	1	2	yes	—
23	C₂×#131	—	2	2	no	—
24	#510	—	2	2	no	—
25	#509	—	2	2	no	—
26	#508	—	2	2	no	—
27	C₂×M₃₂	—	2	2	no	—
28	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
29	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
30	C₂×M₃₂	D₄	1	1	yes	characteristic
31	C₂×M₃₂	D₄	1	1	yes	characteristic
32	#443	—	2	2	no	—
33	#443	—	2	2	no	—
34	#443	—	2	2	no	—
35	#443	—	2	2	no	—
36	C₂×#108	C₂×C₄	1	2	yes	—
37	C₂×#104	—	2	2	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#368	—	2	2	no	—
41	#364	—	2	2	no	—
42	#364	—	2	2	no	—
43	#359	—	2	4	no	—
44	#350	—	4	4	no	—
45	#349	—	4	4	no	—
46	#346	—	2	1	no	—
47	#346	D₄	1	2	yes	—
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	D₄	1	2	yes	—
50	C₈²	—	2	1	no	—
51	C₈²	D₄	1	1	yes	characteristic
52	C₈²	D₄	1	1	yes	characteristic
53	C₂×#41	—	2	2	no	—
54	#135	—	2	2	no	—
55	#135	—	2	2	no	—
56	#135	—	2	2	no	—
57	#135	—	2	2	no	—
58	C₂×Q₁₆	M₁₆	1	2	yes	—
59	C₂×SD₁₆	—	2	2	no	—
60	C₂×D₈	M₁₆	1	2	yes	—
61	#131	—	4	2	no	—
62	#131	—	4	2	no	—
63	#131	—	4	2	no	—
64	#131	—	4	2	no	—
65	C₂×M₁₆	—	2	2	no	—
66	C₂×M₁₆	—	4	2	no	—
67	C₂²×C₈	—	2	2	no	—
68	C₂²×C₈	—	4	2	no	—
69	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄²	#41	1	1	yes	characteristic
72	Q₃₂	—	2	2	no	—
73	SD₃₂	—	2	4	no	—
74	D₁₆	—	2	2	no	—
75	M₃₂	—	2	2	no	—
76	M₃₂	—	2	2	no	—
77	M₃₂	—	2	2	no	—
78	M₃₂	—	2	2	no	—
79	M₃₂	—	4	2	no	—
80	M₃₂	—	4	2	no	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	4	2	no	—
84	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
85	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
86	#108	—	2	2	no	—
87	#108	—	2	2	no	—
88	#104	—	4	2	no	—
89	#104	—	4	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	C₂×D₄	1	1	yes	characteristic
94	C₄×C₈	C₂×D₄	1	1	yes	characteristic
95	C₄×C₈	C₂×D₄	1	1	yes	characteristic
96	C₄×C₈	#41	1	1	yes	characteristic
97	#41	—	4	2	no	—
98	#41	—	4	2	no	—
99	#41	—	4	2	no	—
100	#41	—	4	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	4	2	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	4	2	no	—
106	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
107	Q₁₆	—	2	2	no	—
108	Q₁₆	—	2	2	no	—
109	SD₁₆	—	4	2	no	—
110	SD₁₆	—	4	2	no	—
111	D₈	—	2	2	no	—
112	D₈	—	2	2	no	—
113	M₁₆	—	4	2	no	—
114	M₁₆	—	4	2	no	—
115	M₁₆	—	8	2	no	—
116	M₁₆	—	8	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	4	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	8	2	no	—
122	C₂×C₈	—	8	2	no	—
123	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₄×D₄	1	1	yes	characteristic
125	C₂×C₈	#121	1	1	yes	characteristic
126	C₂×C₈	#127	1	1	yes	characteristic
127	C₂×C₈	C₂×M₁₆	1	2	yes	—
128	C₂×C₈	#131	1	2	yes	—
129	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
131	C₄²	—	2	1	no	—
132	C₄²	#121	1	1	yes	characteristic
133	C₄²	#124	1	1	yes	characteristic
134	C₁₆	—	2	2	no	—
135	C₁₆	—	2	2	no	—
136	C₁₆	—	4	1	no	—
137	C₁₆	—	4	1	no	—
138	C₁₆	—	8	2	no	—
139	C₂³	—	4	2	no	—
140	C₂³	#443	1	1	yes	characteristic
141	Q₈	—	4	2	no	—
142	Q₈	—	4	2	no	—
143	D₄	—	4	2	no	—
144	D₄	—	4	2	no	—
145	D₄	—	8	2	no	—
146	D₄	—	8	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	#390	1	1	yes	characteristic
154	C₂×C₄	#435	1	2	yes	—
155	C₂×C₄	#442	1	2	yes	—
156	C₂×C₄	#443	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₈	—	4	2	no	—
159	C₈	—	8	2	no	—
160	C₈	#407	1	1	yes	characteristic
161	C₈	#408	1	1	yes	characteristic
162	C₈	#435	1	2	yes	—
163	C₈	#469	1	1	yes	characteristic
164	C₈	#469	1	1	yes	characteristic
165	C₈	#496	1	2	yes	—
166	C₂²	—	4	1	no	—
167	C₂²	—	8	2	no	—
168	C₂²	—	16	2	no	—
169	C₂²	#1586	1	2	yes	—
170	C₂²	#1974	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	4	1	no	—
173	C₄	—	8	2	no	—
174	C₄	#1586	1	2	yes	—
175	C₄	#1955	1	1	yes	characteristic
176	C₄	#1972	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	2	no	—
179	C₂	—	16	2	no	—
180	C₂	#9665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33463

C₂

yes

maximal, characteristic

#13197

C₂

yes

maximal, complemented

#12002

C₂

yes

maximal, characteristic, complemented

C₂×#1335

C₂

yes

maximal, complemented

#11402

C₂

yes

maximal, characteristic, complemented

#3414

C₄

yes

characteristic

C₂×#443

C₂²

yes

—

#2179

—

#2178

—

#2161

C₄

yes

characteristic

C₂×#364

C₂²

yes

—

#2143

C₄

yes

—

#2117

—

#2115

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₈²

C₂²

yes

—

—

—

—

C₂×C₄

yes

—

—

—

—

—

C₂×M₃₂

—

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

D₄

yes

characteristic

C₂×M₃₂

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#104

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

D₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₈²

—

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×#41

—

—

—

—

—

C₂×Q₁₆

M₁₆

yes

—

C₂×SD₁₆

—

C₂×D₈

M₁₆

yes

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄²

#41

yes

characteristic

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

—

—

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₄×D₄

yes

characteristic

107

Q₁₆

—

108

Q₁₆

—

109

SD₁₆

—

110

SD₁₆

—

111

D₈

—

112

D₈

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

124

C₂×C₈

C₄×D₄

yes

characteristic

125

C₂×C₈

#121

yes

characteristic

126

C₂×C₈

#127

yes

characteristic

127

C₂×C₈

C₂×M₁₆

yes

—

128

C₂×C₈

#131

yes

—

129

C₂×C₈

AGL(1,ℤ/8)

yes

—

130

C₂×C₈

AGL(1,ℤ/8)

yes

—

131

C₄²

—

132

C₄²

#121

yes

characteristic

133

C₄²

#124

yes

characteristic

134

C₁₆

—

135

C₁₆

—

136

C₁₆

—

137

C₁₆

—

138

C₁₆

—

139

C₂³

—

140

C₂³

#443

yes

characteristic

141

Q₈

—

142

Q₈

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#390

yes

characteristic

154

C₂×C₄

#435

yes

—

155

C₂×C₄

#442

yes

—

156

C₂×C₄

#443

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

#407

yes

characteristic

161

C₈

#408

yes

characteristic

162

C₈

#435

yes

—

163

C₈

#469

yes

characteristic

164

C₈

#469

yes

characteristic

165

C₈

#496

yes

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

#1586

yes

—

170

C₂²

#1974

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#1586

yes

—

175

C₄

#1955

yes

characteristic

176

C₄

#1972

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

—

180

C₂

#9665

yes

characteristic

	subgroup	G/H
center	C₈	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9665
prime radical	#3414	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#469

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₄

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13197	C₂	semidirect
3. #12002	C₂	semidirect
4. C₂×#1335	C₂	semidirect
5. #11402	C₂	semidirect

H ≅ G/N

type

2. #13197

C₂

semidirect

3. #12002

C₂

semidirect

4. C₂×#1335

C₂

semidirect

5. #11402

C₂

semidirect

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

subgroup	index
#33463	2
#13197	2
#12002	2
C₂×#1335	2
#11402	2

subgroup

index

Subgroups — #140787

Conjugacy classes of subgroups

Complements

Maximal subgroups