Group #153409 — subgroups

Subgroups — #153409

Order 512.

196 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 66 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	#13638	C₂	1	1	yes	maximal, characteristic
2	#13636	C₂	1	1	yes	maximal, characteristic
3	C₂×#1420	C₂	1	1	yes	maximal, characteristic
4	#13618	C₂	1	1	yes	maximal, characteristic
5	#13244	C₂	1	1	yes	maximal, characteristic
6	#12911	C₂	1	1	yes	maximal, characteristic
7	#7469	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	1	yes	characteristic
12	C₂×#368	C₂²	1	1	yes	characteristic
13	#2158	C₂²	1	1	yes	characteristic
14	#2155	—	2	1	no	—
15	C₂×#362	—	2	1	no	—
16	#2108	C₂²	1	1	yes	characteristic
17	#1979	—	2	1	no	—
18	#1833	—	2	1	no	—
19	#1426	—	2	2	no	—
20	#1425	—	2	2	no	—
21	#1420	—	2	2	no	—
22	#1416	—	2	2	no	—
23	C₂²×Q₁₆	—	2	1	no	—
24	C₂×Q₃₂	—	2	1	no	—
25	C₂×Q₃₂	—	2	1	no	—
26	C₂×Q₃₂	—	2	1	no	—
27	C₂×Q₃₂	—	2	1	no	—
28	C₂×Q₃₂	—	2	1	no	—
29	C₂×Q₃₂	Q₈	1	2	yes	—
30	C₂×M₃₂	D₄	1	1	yes	characteristic
31	C₂²×C₁₆	C₂³	1	1	yes	characteristic
32	C₂×#108	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	#427	—	2	1	no	—
35	#419	—	4	1	no	—
36	C₂×#101	—	4	1	no	—
37	#405	D₄	1	1	yes	characteristic
38	M₆₄	—	2	1	no	—
39	M₆₄	—	2	1	no	—
40	M₆₄	—	2	1	no	—
41	M₆₄	D₄	1	2	yes	—
42	C₂×C₃₂	—	2	1	no	—
43	C₂×C₃₂	D₄	1	2	yes	—
44	#368	—	2	1	no	—
45	#368	—	2	1	no	—
46	#368	—	2	1	no	—
47	#368	—	2	1	no	—
48	#367	—	2	1	no	—
49	#366	—	2	1	no	—
50	#365	—	4	2	no	—
51	#362	—	4	2	no	—
52	#346	—	2	1	no	—
53	C₄×C₁₆	—	2	1	no	—
54	#344	—	4	1	no	—
55	H(3,ℤ/4)	—	4	1	no	—
56	C₂²×Q₈	—	4	1	no	—
57	C₂×Q₁₆	—	2	2	no	—
58	C₂×Q₁₆	—	4	1	no	—
59	C₂×Q₁₆	—	4	1	no	—
60	C₂×Q₁₆	—	4	1	no	—
61	C₂×Q₁₆	—	4	1	no	—
62	C₂×Q₁₆	—	4	1	no	—
63	C₂×M₁₆	—	4	1	no	—
64	C₂×M₁₆	—	4	1	no	—
65	C₂×M₁₆	D₈	1	1	yes	characteristic
66	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
67	#117	—	4	1	no	—
68	#117	D₈	1	1	yes	characteristic
69	Q₃₂	—	2	2	no	—
70	Q₃₂	—	2	2	no	—
71	Q₃₂	—	4	1	no	—
72	Q₃₂	—	4	1	no	—
73	M₃₂	—	2	1	no	—
74	M₃₂	D₈	1	2	yes	—
75	C₂×C₁₆	—	2	1	no	—
76	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
77	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
78	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₁₆	#41	1	1	yes	characteristic
80	C₂×C₁₆	#41	1	1	yes	characteristic
81	C₂×C₁₆	#41	1	1	yes	characteristic
82	#108	—	4	1	no	—
83	#108	—	4	1	no	—
84	#108	—	4	1	no	—
85	#108	—	4	1	no	—
86	#107	—	4	1	no	—
87	#106	—	4	1	no	—
88	#103	—	8	2	no	—
89	#101	—	8	2	no	—
90	#97	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₃₂	—	2	2	no	—
93	C₃₂	—	2	2	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	8	1	no	—
96	C₂×Q₈	—	8	1	no	—
97	C₂×Q₈	—	8	1	no	—
98	C₂×Q₈	—	8	1	no	—
99	C₂×Q₈	—	8	1	no	—
100	C₂²×C₄	—	8	1	no	—
101	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
102	Q₁₆	—	4	2	no	—
103	Q₁₆	—	4	2	no	—
104	Q₁₆	—	8	1	no	—
105	Q₁₆	—	8	1	no	—
106	M₁₆	—	2	1	no	—
107	M₁₆	—	2	1	no	—
108	M₁₆	—	8	1	no	—
109	M₁₆	—	8	1	no	—
110	M₁₆	—	8	1	no	—
111	M₁₆	—	8	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	8	1	no	—
114	C₂×C₈	—	8	1	no	—
115	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₈	#123	1	1	yes	characteristic
117	C₂×C₈	#127	1	1	yes	characteristic
118	C₂×C₈	C₂×D₈	1	1	yes	characteristic
119	C₂×C₈	#137	1	1	yes	characteristic
120	C₂×C₈	#137	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	8	1	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	8	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	8	1	no	—
127	C₁₆	—	2	2	no	—
128	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
129	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
130	C₁₆	#123	1	1	yes	characteristic
131	C₁₆	#123	1	1	yes	characteristic
132	C₂³	#508	1	1	yes	characteristic
133	Q₈	—	8	2	no	—
134	Q₈	—	8	2	no	—
135	Q₈	—	16	1	no	—
136	Q₈	—	16	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	16	1	no	—
140	C₂×C₄	—	16	1	no	—
141	C₂×C₄	—	16	1	no	—
142	C₂×C₄	#474	1	1	yes	characteristic
143	C₂×C₄	#480	1	1	yes	characteristic
144	C₂×C₄	#493	1	1	yes	characteristic
145	C₂×C₄	#508	1	1	yes	characteristic
146	C₂×C₄	#509	1	1	yes	characteristic
147	C₂×C₄	#509	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	—	16	1	no	—
150	C₈	—	16	1	no	—
151	C₈	#399	1	1	yes	characteristic
152	C₈	#474	1	1	yes	characteristic
153	C₈	#480	1	1	yes	characteristic
154	C₈	#497	1	1	yes	characteristic
155	C₂²	—	4	1	no	—
156	C₂²	#2228	1	1	yes	characteristic
157	C₂²	#2234	1	1	yes	characteristic
158	C₂²	#2249	1	1	yes	characteristic
159	C₄	—	8	1	no	—
160	C₄	—	16	2	no	—
161	C₄	—	32	1	no	—
162	C₄	#2093	1	1	yes	characteristic
163	C₄	#2228	1	1	yes	characteristic
164	C₄	#2234	1	1	yes	characteristic
165	C₄	#2250	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#13528	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13638

C₂

yes

maximal, characteristic

#13636

C₂

yes

maximal, characteristic

C₂×#1420

C₂

yes

maximal, characteristic

#13618

C₂

yes

maximal, characteristic

#13244

C₂

yes

maximal, characteristic

#12911

C₂

yes

maximal, characteristic

#7469

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×#368

C₂²

yes

characteristic

C₂×#368

C₂²

yes

characteristic

#2158

C₂²

yes

—

—

C₂²

yes

—

—

—

—

—

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

Q₈

yes

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

D₄

yes

characteristic

M₆₄

—

M₆₄

—

M₆₄

—

M₆₄

D₄

yes

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

—

—

—

—

—

—

—

—

—

C₄×C₁₆

—

#344

—

H(3,ℤ/4)

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

#117

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

M₃₂

—

M₃₂

D₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂×D₈

yes

characteristic

102

Q₁₆

—

103

Q₁₆

—

104

Q₁₆

—

105

Q₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂²⋊Q₈

yes

characteristic

116

C₂×C₈

#123

yes

characteristic

117

C₂×C₈

#127

yes

characteristic

118

C₂×C₈

C₂×D₈

yes

characteristic

119

C₂×C₈

#137

yes

characteristic

120

C₂×C₈

#137

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₁₆

—

128

C₁₆

C₂²⋊Q₈

yes

characteristic

129

C₁₆

C₂²⋊Q₈

yes

characteristic

130

C₁₆

#123

yes

characteristic

131

C₁₆

#123

yes

characteristic

132

C₂³

#508

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#474

yes

characteristic

143

C₂×C₄

#480

yes

characteristic

144

C₂×C₄

#493

yes

characteristic

145

C₂×C₄

#508

yes

characteristic

146

C₂×C₄

#509

yes

characteristic

147

C₂×C₄

#509

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

#399

yes

characteristic

152

C₈

#474

yes

characteristic

153

C₈

#480

yes

characteristic

154

C₈

#497

yes

characteristic

155

C₂²

—

156

C₂²

#2228

yes

characteristic

157

C₂²

#2234

yes

characteristic

158

C₂²

#2249

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#2093

yes

characteristic

163

C₄

#2228

yes

characteristic

164

C₄

#2234

yes

characteristic

165

C₄

#2250

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

#13528

yes

characteristic

	subgroup	G/H
center, socle	C₂	#13528
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

#13528

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

#508

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#13638	2
#13636	2
C₂×#1420	2
#13618	2
#13244	2
#12911	2
#7469	2

subgroup

index

Subgroups — #153409

Conjugacy classes of subgroups

Maximal subgroups