Group #140776 — subgroups

Subgroups — #140776

Order 512.

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

normal subgroups only

This group has 311 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	#33463	C₂	1	1	yes	maximal, characteristic
2	#13197	C₂	1	2	yes	maximal, complemented
3	#12011	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1336	C₂	1	2	yes	maximal, complemented
5	#11404	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×#322	C₂²	1	1	yes	characteristic
18	#1373	—	2	2	no	—
19	#1336	—	2	4	no	—
20	#1313	—	2	2	no	—
21	C₂×#135	C₂×C₄	1	2	yes	—
22	C₂×#131	—	2	2	no	—
23	#510	—	2	2	no	—
24	#509	—	2	2	no	—
25	#508	—	2	2	no	—
26	C₂×M₃₂	—	2	2	no	—
27	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
28	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
29	C₂×M₃₂	D₄	1	2	yes	—
30	#443	—	2	2	no	—
31	#443	—	2	2	no	—
32	#443	—	2	2	no	—
33	#443	—	2	2	no	—
34	C₂×#108	C₂×C₄	1	2	yes	—
35	C₂×#104	—	2	2	no	—
36	C₂×C₄×C₈	—	2	1	no	—
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	#368	—	2	2	no	—
39	#364	—	2	2	no	—
40	#364	—	2	2	no	—
41	#359	—	2	4	no	—
42	#350	—	4	4	no	—
43	#349	—	4	4	no	—
44	#346	—	2	1	no	—
45	#346	D₄	1	2	yes	—
46	C₄×C₁₆	—	2	1	no	—
47	C₄×C₁₆	D₄	1	2	yes	—
48	#322	—	2	1	no	—
49	#322	D₄	1	2	yes	—
50	C₂×#41	—	2	2	no	—
51	#135	—	2	2	no	—
52	#135	—	2	2	no	—
53	#135	—	2	2	no	—
54	#135	—	2	2	no	—
55	C₂×Q₁₆	M₁₆	1	2	yes	—
56	C₂×SD₁₆	—	2	2	no	—
57	C₂×D₈	M₁₆	1	2	yes	—
58	#131	—	4	2	no	—
59	#131	—	4	2	no	—
60	#131	—	4	2	no	—
61	#131	—	4	2	no	—
62	C₂×M₁₆	—	2	2	no	—
63	C₂×M₁₆	—	4	2	no	—
64	C₂²×C₈	—	4	1	no	—
65	C₂²×C₈	—	4	2	no	—
66	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄²	#41	1	1	yes	characteristic
69	Q₃₂	—	2	2	no	—
70	SD₃₂	—	2	4	no	—
71	D₁₆	—	2	2	no	—
72	M₃₂	—	2	2	no	—
73	M₃₂	—	2	2	no	—
74	M₃₂	—	2	2	no	—
75	M₃₂	—	4	2	no	—
76	M₃₂	—	4	2	no	—
77	M₃₂	D₈	1	4	yes	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	4	2	no	—
80	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
81	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
82	#108	—	2	2	no	—
83	#108	—	2	2	no	—
84	#104	—	4	2	no	—
85	#104	—	4	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	4	1	no	—
88	C₄×C₈	C₂×D₄	1	1	yes	characteristic
89	C₄×C₈	C₂×D₄	1	1	yes	characteristic
90	C₄×C₈	C₂×D₄	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	#41	—	4	2	no	—
93	#41	—	4	2	no	—
94	#41	—	4	2	no	—
95	#41	—	4	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×D₄	—	2	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂²×C₄	—	2	1	no	—
100	C₂²×C₄	—	4	2	no	—
101	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
102	Q₁₆	—	2	2	no	—
103	Q₁₆	—	2	2	no	—
104	SD₁₆	—	4	2	no	—
105	SD₁₆	—	4	2	no	—
106	D₈	—	2	2	no	—
107	D₈	—	2	2	no	—
108	M₁₆	—	4	2	no	—
109	M₁₆	—	4	2	no	—
110	M₁₆	—	8	2	no	—
111	M₁₆	—	8	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	8	2	no	—
115	C₂×C₈	—	8	2	no	—
116	C₂×C₈	—	8	2	no	—
117	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₄×D₄	1	1	yes	characteristic
119	C₂×C₈	#121	1	1	yes	characteristic
120	C₂×C₈	#127	1	1	yes	characteristic
121	C₂×C₈	C₂×M₁₆	1	2	yes	—
122	C₂×C₈	#131	1	2	yes	—
123	C₂×C₈	C₂×D₈	1	2	yes	—
124	C₂×C₈	#135	1	2	yes	—
125	C₄²	—	2	1	no	—
126	C₄²	#121	1	1	yes	characteristic
127	C₄²	#124	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₁₆	—	2	2	no	—
130	C₁₆	—	2	4	no	—
131	C₁₆	—	8	2	no	—
132	C₂³	—	4	2	no	—
133	C₂³	#444	1	1	yes	characteristic
134	Q₈	—	4	2	no	—
135	Q₈	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	4	2	no	—
138	D₄	—	8	2	no	—
139	D₄	—	8	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	#390	1	1	yes	characteristic
147	C₂×C₄	#435	1	2	yes	—
148	C₂×C₄	C₄×D₈	1	2	yes	—
149	C₂×C₄	#444	1	1	yes	characteristic
150	C₈	—	8	2	no	—
151	C₈	—	8	2	no	—
152	C₈	#407	1	1	yes	characteristic
153	C₈	#408	1	1	yes	characteristic
154	C₈	#435	1	2	yes	—
155	C₈	#466	1	2	yes	—
156	C₈	#493	1	1	yes	characteristic
157	C₈	#495	1	1	yes	characteristic
158	C₂²	—	4	1	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	16	2	no	—
161	C₂²	#1581	1	2	yes	—
162	C₂²	#1975	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	4	1	no	—
165	C₄	—	8	2	no	—
166	C₄	#1581	1	2	yes	—
167	C₄	#1955	1	1	yes	characteristic
168	C₄	#1970	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	2	no	—
171	C₂	—	16	2	no	—
172	C₂	#9663	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33463

C₂

yes

maximal, characteristic

#13197

C₂

yes

maximal, complemented

#12011

C₂

yes

maximal, characteristic, complemented

C₂×#1336

C₂

yes

maximal, complemented

#11404

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×#322

C₂²

yes

—

—

—

C₂×C₄

yes

—

—

—

—

—

C₂×M₃₂

—

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

C₂×C₄

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

—

#322

—

#322

D₄

yes

—

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

D₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₄

yes

characteristic

102

Q₁₆

—

103

Q₁₆

—

104

SD₁₆

—

105

SD₁₆

—

106

D₈

—

107

D₈

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

118

C₂×C₈

C₄×D₄

yes

characteristic

119

C₂×C₈

#121

yes

characteristic

120

C₂×C₈

#127

yes

characteristic

121

C₂×C₈

C₂×M₁₆

yes

—

122

C₂×C₈

#131

yes

—

123

C₂×C₈

C₂×D₈

yes

—

124

C₂×C₈

#135

yes

—

125

C₄²

—

126

C₄²

#121

yes

characteristic

127

C₄²

#124

yes

characteristic

128

C₁₆

—

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

132

C₂³

—

133

C₂³

#444

yes

characteristic

134

Q₈

—

135

Q₈

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#390

yes

characteristic

147

C₂×C₄

#435

yes

—

148

C₂×C₄

C₄×D₈

yes

—

149

C₂×C₄

#444

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

#407

yes

characteristic

153

C₈

#408

yes

characteristic

154

C₈

#435

yes

—

155

C₈

#466

yes

—

156

C₈

#493

yes

characteristic

157

C₈

#495

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

#1581

yes

—

162

C₂²

#1975

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#1581

yes

—

167

C₄

#1955

yes

characteristic

168

C₄

#1970

yes

characteristic

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

#9663

yes

characteristic

	subgroup	G/H
center	C₄	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9663
prime radical	#3414	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#1970

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₄

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13197	C₂	semidirect
3. #12011	C₂	semidirect
4. C₂×#1336	C₂	semidirect
5. #11404	C₂	semidirect

H ≅ G/N

type

2. #13197

C₂

semidirect

3. #12011

C₂

semidirect

4. C₂×#1336

C₂

semidirect

5. #11404

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
#12011	2
C₂×#1336	2
#11404	2

subgroup

index

Subgroups — #140776

Conjugacy classes of subgroups

Complements

Maximal subgroups