Group #153398 — subgroups

Subgroups — #153398

Order 512.

249 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 84 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×#1424	C₂	1	2	yes	maximal, complemented
2	C₂×#1415	C₂	1	2	yes	maximal
3	#13235	C₂	1	1	yes	maximal, characteristic
4	#12905	C₂	1	1	yes	maximal, characteristic, complemented
5	#7465	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	2	yes	—
8	C₂×#366	—	2	1	no	—
9	C₂×#366	C₂²	1	2	yes	—
10	C₂×#366	C₂²	1	1	yes	characteristic
11	C₂×#357	—	2	1	no	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1970	—	2	1	no	—
14	#1845	—	2	1	no	—
15	#1424	—	2	4	no	—
16	#1415	—	2	4	no	—
17	C₂²×D₈	—	2	1	no	—
18	C₂×D₁₆	—	2	2	no	—
19	C₂×D₁₆	—	2	2	no	—
20	C₂×D₁₆	—	2	1	no	—
21	C₂×D₁₆	Q₈	1	2	yes	—
22	C₂²×C₁₆	D₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂³	1	1	yes	characteristic
24	C₂×#107	—	2	2	no	—
25	C₂×#107	—	2	1	no	—
26	C₂×#107	—	4	1	no	—
27	C₂×#102	—	4	1	no	—
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	#390	—	4	1	no	—
30	#384	—	4	1	no	—
31	C₂×C₃₂	—	2	2	no	—
32	C₂×C₃₂	—	2	2	no	—
33	C₂×C₃₂	D₄	1	4	yes	—
34	#366	—	2	2	no	—
35	#366	—	2	2	no	—
36	#366	—	2	2	no	—
37	#366	—	4	2	no	—
38	#357	—	4	2	no	—
39	C₄×C₁₆	—	2	2	no	—
40	C₂²×D₄	—	4	1	no	—
41	C₂×D₈	—	2	2	no	—
42	C₂×D₈	—	4	1	no	—
43	C₂×D₈	—	4	1	no	—
44	C₂×D₈	—	4	1	no	—
45	C₂×D₈	—	4	1	no	—
46	C₂×D₈	—	4	1	no	—
47	C₂²×C₈	D₈	1	1	yes	characteristic
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄	—	4	2	no	—
50	C₂×C₄⋊C₄	—	4	1	no	—
51	C₂×C₄⋊C₄	—	8	2	no	—
52	C₂×C₂²⋊C₄	—	8	2	no	—
53	C₂×C₄²	D₈	1	1	yes	characteristic
54	D₁₆	—	2	4	no	—
55	D₁₆	—	4	2	no	—
56	C₂×C₁₆	—	2	1	no	—
57	C₂×C₁₆	—	2	1	no	—
58	C₂×C₁₆	D₈	1	2	yes	—
59	C₂×C₁₆	C₂×D₄	1	2	yes	—
60	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₁₆	#41	1	2	yes	—
62	C₂×C₁₆	#41	1	1	yes	characteristic
63	#107	—	4	2	no	—
64	#107	—	4	1	no	—
65	#107	—	4	1	no	—
66	#107	—	4	2	no	—
67	#107	—	8	2	no	—
68	#102	—	8	2	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₃₂	—	2	4	no	—
72	C₂⁴	—	8	1	no	—
73	C₂×D₄	—	4	2	no	—
74	C₂×D₄	—	8	1	no	—
75	C₂×D₄	—	8	1	no	—
76	C₂×D₄	—	8	1	no	—
77	C₂×D₄	—	8	1	no	—
78	C₂×D₄	—	8	1	no	—
79	C₂²×C₄	—	8	1	no	—
80	C₂²×C₄	—	8	2	no	—
81	C₂²×C₄	D₁₆	1	2	yes	—
82	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
83	D₈	—	4	2	no	—
84	D₈	—	4	2	no	—
85	D₈	—	8	1	no	—
86	D₈	—	8	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	D₁₆	1	2	yes	—
90	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₈	#123	1	1	yes	characteristic
92	C₂×C₈	#127	1	1	yes	characteristic
93	C₂×C₈	C₂×D₈	1	1	yes	characteristic
94	C₂×C₈	#137	1	1	yes	characteristic
95	C₂×C₈	#137	1	1	yes	characteristic
96	C₄⋊C₄	—	8	1	no	—
97	C₄⋊C₄	—	8	2	no	—
98	C₄⋊C₄	—	8	1	no	—
99	C₄⋊C₄	—	8	2	no	—
100	C₄⋊C₄	—	16	4	no	—
101	C₂²⋊C₄	—	16	4	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	1	no	—
104	C₁₆	—	2	2	no	—
105	C₁₆	C₂²⋊Q₈	1	2	yes	—
106	C₁₆	#123	1	2	yes	—
107	C₂³	—	8	2	no	—
108	C₂³	—	16	1	no	—
109	C₂³	—	16	1	no	—
110	C₂³	—	16	2	no	—
111	C₂³	—	16	2	no	—
112	C₂³	C₂×D₁₆	1	1	yes	characteristic
113	D₄	—	8	2	no	—
114	D₄	—	8	2	no	—
115	D₄	—	16	1	no	—
116	D₄	—	16	1	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	16	1	no	—
120	C₂×C₄	—	16	2	no	—
121	C₂×C₄	—	16	2	no	—
122	C₂×C₄	—	16	4	no	—
123	C₂×C₄	SD₆₄	1	4	yes	—
124	C₂×C₄	#474	1	1	yes	characteristic
125	C₂×C₄	#480	1	1	yes	characteristic
126	C₂×C₄	#493	1	1	yes	characteristic
127	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
128	C₂×C₄	#510	1	1	yes	characteristic
129	C₂×C₄	#510	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	#399	1	1	yes	characteristic
132	C₈	#474	1	1	yes	characteristic
133	C₈	#480	1	1	yes	characteristic
134	C₈	#497	1	1	yes	characteristic
135	C₂²	—	16	4	no	—
136	C₂²	—	16	2	no	—
137	C₂²	—	32	2	no	—
138	C₂²	—	32	1	no	—
139	C₂²	—	32	1	no	—
140	C₂²	#2224	1	1	yes	characteristic
141	C₂²	#2232	1	1	yes	characteristic
142	C₂²	#2246	1	1	yes	characteristic
143	C₂²	C₂×SD₆₄	1	2	yes	—
144	C₂²	#2263	1	2	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	32	1	no	—
147	C₄	—	32	2	no	—
148	C₄	#2093	1	1	yes	characteristic
149	C₄	#2224	1	1	yes	characteristic
150	C₄	#2232	1	1	yes	characteristic
151	C₄	#2251	1	1	yes	characteristic
152	C₂	—	32	2	no	—
153	C₂	#13526	1	1	yes	characteristic
154	C₂	#13705	1	2	yes	—
155	C₂	#13712	1	2	yes	—
156	C₂	#13727	1	1	yes	characteristic
157	C₂	#13729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1424

C₂

yes

maximal, complemented

C₂×#1415

C₂

yes

maximal

#13235

C₂

yes

maximal, characteristic

#12905

C₂

yes

maximal, characteristic, complemented

#7465

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₆

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

—

C₂×#366

—

C₂×#366

C₂²

yes

—

C₂×#366

C₂²

yes

characteristic

C₂×#357

—

C₂×C₄×C₁₆

C₂²

yes

—

—

—

—

C₂²×D₈

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

Q₈

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#390

—

#384

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

D₁₆

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

D₈

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

C₂²⋊Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×D₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₁₆

—

105

C₁₆

C₂²⋊Q₈

yes

—

106

C₁₆

#123

yes

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

C₂×D₁₆

yes

characteristic

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

SD₆₄

yes

—

124

C₂×C₄

#474

yes

characteristic

125

C₂×C₄

#480

yes

characteristic

126

C₂×C₄

#493

yes

characteristic

127

C₂×C₄

C₂×D₁₆

yes

characteristic

128

C₂×C₄

#510

yes

characteristic

129

C₂×C₄

#510

yes

characteristic

130

C₈

—

131

C₈

#399

yes

characteristic

132

C₈

#474

yes

characteristic

133

C₈

#480

yes

characteristic

134

C₈

#497

yes

characteristic

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#2224

yes

characteristic

141

C₂²

#2232

yes

characteristic

142

C₂²

#2246

yes

characteristic

143

C₂²

C₂×SD₆₄

yes

—

144

C₂²

#2263

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#2093

yes

characteristic

149

C₄

#2224

yes

characteristic

150

C₄

#2232

yes

characteristic

151

C₄

#2251

yes

characteristic

152

C₂

—

153

C₂

#13526

yes

characteristic

154

C₂

#13705

yes

—

155

C₂

#13712

yes

—

156

C₂

#13727

yes

characteristic

157

C₂

#13729

yes

characteristic

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂³

C₂×D₁₆

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂²×D₁₆

C₂²

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#1424	C₂	semidirect
4. #12905	C₂	semidirect
5. #7465	C₂	semidirect

H ≅ G/N

type

1. C₂×#1424

C₂

semidirect

4. #12905

C₂

semidirect

5. #7465

C₂

semidirect

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

subgroup	index
C₂×#1424	2
C₂×#1415	2
#13235	2
#12905	2
#7465	2

subgroup

index

Subgroups — #153398

Conjugacy classes of subgroups

Complements

Maximal subgroups