Group #10165 — subgroups

Subgroups — #10165

Order 256.

238 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 65 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	#3165	C₂	1	1	yes	maximal, characteristic
3	C₂×#500	C₂	1	1	yes	maximal, characteristic
4	#1997	C₂	1	1	yes	maximal, characteristic
5	#1548	C₂	1	1	yes	maximal, characteristic
6	#1493	C₂	1	1	yes	maximal, characteristic
7	#1278	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	#501	—	2	2	no	—
12	#500	—	2	2	no	—
13	#500	—	2	1	no	—
14	#500	—	2	1	no	—
15	#500	—	2	2	no	—
16	#498	—	2	1	no	—
17	#487	—	2	1	no	—
18	#484	—	2	2	no	—
19	#477	—	2	2	no	—
20	#431	C₂²	1	1	yes	characteristic
21	#428	—	2	2	no	—
22	C₂×#107	—	2	2	no	—
23	#423	C₂²	1	1	yes	characteristic
24	C₄×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	#343	—	2	2	no	—
27	#333	—	2	2	no	—
28	Q₈⋊C₈	—	2	2	no	—
29	C₂²×Q₈	—	2	1	no	—
30	C₂²×Q₈	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	D₄	1	2	yes	—
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	—	2	1	no	—
39	C₄⋊Q₈	—	2	1	no	—
40	C₄⋊Q₈	—	2	1	no	—
41	C₄⋊Q₈	D₄	1	2	yes	—
42	C₄⋊Q₈	D₄	1	1	yes	characteristic
43	C₄⋊Q₈	D₄	1	1	yes	characteristic
44	#125	—	2	1	no	—
45	C₂²⋊Q₈	—	2	1	no	—
46	C₂²⋊Q₈	—	2	1	no	—
47	C₄×Q₈	—	2	1	no	—
48	C₄×Q₈	—	2	1	no	—
49	#117	—	2	1	no	—
50	C₂×C₄⋊C₄	—	2	1	no	—
51	C₂×C₄⋊C₄	—	2	1	no	—
52	C₂×C₄⋊C₄	—	2	1	no	—
53	C₂×C₄⋊C₄	D₄	1	2	yes	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#107	—	2	8	no	—
56	#107	—	2	4	no	—
57	#107	—	2	4	no	—
58	#106	—	2	4	no	—
59	C₄⋊C₈	—	2	1	no	—
60	C₄⋊C₈	—	2	1	no	—
61	C₄⋊C₈	D₄	1	2	yes	—
62	#103	—	2	4	no	—
63	C₂²⋊C₈	—	2	1	no	—
64	#97	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	D₄	1	2	yes	—
69	C₂×Q₈	—	2	1	no	—
70	C₂×Q₈	—	2	1	no	—
71	C₂×Q₈	—	2	2	no	—
72	C₂×Q₈	—	4	1	no	—
73	C₂×Q₈	—	4	1	no	—
74	C₂×Q₈	—	4	1	no	—
75	C₂×Q₈	—	8	2	no	—
76	C₂×Q₈	—	8	2	no	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	—	4	1	no	—
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄	#41	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	4	2	no	—
84	M₁₆	Q₁₆	1	4	yes	—
85	C₂×C₈	—	2	4	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	4	1	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₂²⋊C₄	—	4	1	no	—
105	C₄²	—	4	1	no	—
106	C₄²	C₂×D₄	1	1	yes	characteristic
107	C₄²	C₂×D₄	1	1	yes	characteristic
108	C₄²	C₂×D₄	1	1	yes	characteristic
109	C₄²	C₂×D₄	1	1	yes	characteristic
110	C₂³	#121	1	1	yes	characteristic
111	Q₈	—	4	2	no	—
112	Q₈	—	4	2	no	—
113	Q₈	—	8	2	no	—
114	Q₈	—	8	2	no	—
115	Q₈	—	8	2	no	—
116	Q₈	—	8	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	8	2	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	#120	1	1	yes	characteristic
126	C₂×C₄	#120	1	1	yes	characteristic
127	C₂×C₄	#120	1	1	yes	characteristic
128	C₂×C₄	#121	1	1	yes	characteristic
129	C₂×C₄	#121	1	1	yes	characteristic
130	C₂×C₄	#127	1	1	yes	characteristic
131	C₂×C₄	C₂×Q₁₆	1	2	yes	—
132	C₂×C₄	#137	1	2	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	2	4	no	—
135	C₈	—	4	2	no	—
136	C₈	—	8	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	#392	1	1	yes	characteristic
140	C₂²	#462	1	2	yes	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	8	1	no	—
144	C₄	—	8	2	no	—
145	C₄	—	8	2	no	—
146	C₄	—	8	1	no	—
147	C₄	#451	1	2	yes	—
148	C₄	#456	1	2	yes	—
149	C₄	#472	1	2	yes	—
150	C₄	#494	1	1	yes	characteristic
151	C₄	#497	1	1	yes	characteristic
152	C₂	—	2	2	no	—
153	C₂	#2005	1	1	yes	characteristic
154	C₂	#2016	1	1	yes	characteristic
155	C₂	#2018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3230

C₂

yes

maximal, characteristic

#3165

C₂

yes

maximal, characteristic

C₂×#500

C₂

yes

maximal, characteristic

#1997

C₂

yes

maximal, characteristic

#1548

C₂

yes

maximal, characteristic

#1493

C₂

yes

maximal, characteristic

#1278

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#343

—

#333

—

Q₈⋊C₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

—

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

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄²

C₂³

yes

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

#103

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

Q₁₆

yes

—

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

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₂×D₄

yes

characteristic

107

C₄²

C₂×D₄

yes

characteristic

108

C₄²

C₂×D₄

yes

characteristic

109

C₄²

C₂×D₄

yes

characteristic

110

C₂³

#121

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

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

#120

yes

characteristic

127

C₂×C₄

#120

yes

characteristic

128

C₂×C₄

#121

yes

characteristic

129

C₂×C₄

#121

yes

characteristic

130

C₂×C₄

#127

yes

characteristic

131

C₂×C₄

C₂×Q₁₆

yes

—

132

C₂×C₄

#137

yes

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₂²

—

138

C₂²

—

139

C₂²

#392

yes

characteristic

140

C₂²

#462

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#451

yes

—

148

C₄

#456

yes

—

149

C₄

#472

yes

—

150

C₄

#494

yes

characteristic

151

C₄

#497

yes

characteristic

152

C₂

—

153

C₂

#2005

yes

characteristic

154

C₂

#2016

yes

characteristic

155

C₂

#2018

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#392
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

#392

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
#3165	2
C₂×#500	2
#1997	2
#1548	2
#1493	2
#1278	2

subgroup

index

Subgroups — #10165

Conjugacy classes of subgroups

Maximal subgroups