Group #10506 — subgroups

Subgroups — #10506

Order 256.

187 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 57 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	#3230	C₂	1	1	yes	maximal, characteristic
2	#1997	C₂	1	1	yes	maximal, characteristic
3	#1987	C₂	1	1	yes	maximal, characteristic
4	#1986	C₂	1	1	yes	maximal, characteristic
5	#1565	C₂	1	1	yes	maximal, characteristic
6	#1550	C₂	1	1	yes	maximal, characteristic
7	#1303	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#500	—	2	1	no	—
11	#487	—	2	1	no	—
12	#484	—	2	2	no	—
13	#477	—	2	1	no	—
14	#477	—	2	1	no	—
15	#431	C₂²	1	1	yes	characteristic
16	#423	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×M₁₆	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	#343	—	2	1	no	—
21	#343	—	2	1	no	—
22	#340	—	2	2	no	—
23	#339	—	2	1	no	—
24	#339	—	2	1	no	—
25	#335	—	2	1	no	—
26	#334	—	2	1	no	—
27	#333	—	2	1	no	—
28	#332	—	2	1	no	—
29	C₂²×Q₈	—	2	1	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	D₄	1	1	yes	characteristic
32	C₂×M₁₆	D₄	1	1	yes	characteristic
33	C₂²×C₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	C₄⋊Q₈	—	2	1	no	—
37	C₄⋊Q₈	—	2	1	no	—
38	C₄⋊Q₈	D₄	1	1	yes	characteristic
39	C₄⋊Q₈	D₄	1	1	yes	characteristic
40	#125	—	2	1	no	—
41	C₂²⋊Q₈	—	2	1	no	—
42	C₂²⋊Q₈	—	2	1	no	—
43	C₄×Q₈	—	2	1	no	—
44	C₄×Q₈	—	2	1	no	—
45	#117	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄⋊C₄	—	2	1	no	—
48	C₂×C₄⋊C₄	D₄	1	2	yes	—
49	C₂×C₄²	—	2	1	no	—
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	#107	—	2	2	no	—
52	#107	—	2	2	no	—
53	C₄⋊C₈	—	2	1	no	—
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	—	2	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	D₄	1	1	yes	characteristic
58	C₄⋊C₈	D₄	1	1	yes	characteristic
59	#103	—	2	2	no	—
60	#103	—	2	2	no	—
61	C₂²⋊C₈	—	2	1	no	—
62	#97	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	#95	—	2	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	2	1	no	—
68	C₂×Q₈	—	4	1	no	—
69	C₂×Q₈	—	4	1	no	—
70	C₂×Q₈	—	8	2	no	—
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	—	4	1	no	—
73	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄	#41	1	1	yes	characteristic
76	M₁₆	—	2	1	no	—
77	M₁₆	—	2	1	no	—
78	M₁₆	—	2	1	no	—
79	M₁₆	—	4	1	no	—
80	M₁₆	—	4	1	no	—
81	M₁₆	Q₁₆	1	2	yes	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	4	1	no	—
97	C₄⋊C₄	—	4	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₂²⋊C₄	—	4	1	no	—
102	C₄²	—	4	1	no	—
103	C₄²	—	4	1	no	—
104	C₄²	—	4	1	no	—
105	C₄²	C₂×D₄	1	1	yes	characteristic
106	C₄²	C₂×D₄	1	1	yes	characteristic
107	C₄²	#41	1	1	yes	characteristic
108	C₄²	#41	1	1	yes	characteristic
109	C₂³	#121	1	1	yes	characteristic
110	Q₈	—	4	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	8	1	no	—
113	Q₈	—	8	2	no	—
114	Q₈	—	8	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	#120	1	1	yes	characteristic
126	C₂×C₄	#121	1	1	yes	characteristic
127	C₂×C₄	#121	1	1	yes	characteristic
128	C₂×C₄	#123	1	1	yes	characteristic
129	C₂×C₄	#123	1	1	yes	characteristic
130	C₂×C₄	#124	1	1	yes	characteristic
131	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×C₄	#135	1	1	yes	characteristic
133	C₂×C₄	#137	1	1	yes	characteristic
134	C₂×C₄	#137	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	4	1	no	—
137	C₈	—	4	1	no	—
138	C₈	—	4	1	no	—
139	C₈	—	8	1	no	—
140	C₈	—	8	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#394	1	1	yes	characteristic
144	C₂²	#462	1	1	yes	characteristic
145	C₂²	#464	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	8	2	no	—
150	C₄	—	8	2	no	—
151	C₄	—	8	1	no	—
152	C₄	—	8	1	no	—
153	C₄	#454	1	1	yes	characteristic
154	C₄	#456	1	1	yes	characteristic
155	C₄	#462	1	1	yes	characteristic
156	C₄	#464	1	1	yes	characteristic
157	C₄	#467	1	1	yes	characteristic
158	C₄	#470	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	#2038	1	1	yes	characteristic
162	C₂	#2048	1	1	yes	characteristic
163	C₂	#2056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3230

C₂

yes

maximal, characteristic

#1997

C₂

yes

maximal, characteristic

#1987

C₂

yes

maximal, characteristic

#1986

C₂

yes

maximal, characteristic

#1565

C₂

yes

maximal, characteristic

#1550

C₂

yes

maximal, characteristic

#1303

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₄×M₁₆

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

characteristic

C₄⋊Q₈

D₄

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

#103

—

#103

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

Q₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×D₄

yes

characteristic

106

C₄²

C₂×D₄

yes

characteristic

107

C₄²

#41

yes

characteristic

108

C₄²

#41

yes

characteristic

109

C₂³

#121

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#120

yes

characteristic

126

C₂×C₄

#121

yes

characteristic

127

C₂×C₄

#121

yes

characteristic

128

C₂×C₄

#123

yes

characteristic

129

C₂×C₄

#123

yes

characteristic

130

C₂×C₄

#124

yes

characteristic

131

C₂×C₄

C₂×Q₁₆

yes

characteristic

132

C₂×C₄

#135

yes

characteristic

133

C₂×C₄

#137

yes

characteristic

134

C₂×C₄

#137

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

#394

yes

characteristic

144

C₂²

#462

yes

characteristic

145

C₂²

#464

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#454

yes

characteristic

154

C₄

#456

yes

characteristic

155

C₄

#462

yes

characteristic

156

C₄

#464

yes

characteristic

157

C₄

#467

yes

characteristic

158

C₄

#470

yes

characteristic

159

C₂

—

160

C₂

—

161

C₂

#2038

yes

characteristic

162

C₂

#2048

yes

characteristic

163

C₂

#2056

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³

prime radical

C₂³

#121

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#3230	2
#1997	2
#1987	2
#1986	2
#1565	2
#1550	2
#1303	2

subgroup

index

Subgroups — #10506

Conjugacy classes of subgroups

Maximal subgroups