Group #10658 — subgroups

Subgroups — #10658

Order 256.

167 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 71 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	#1948	C₂	1	1	yes	maximal, characteristic
2	#1947	C₂	1	1	yes	maximal, characteristic
3	#1939	C₂	1	1	yes	maximal, characteristic
4	#1834	C₂	1	1	yes	maximal, characteristic
5	#1296	C₂	1	1	yes	maximal, characteristic
6	#1296	C₂	1	1	yes	maximal, characteristic
7	#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×#107	—	2	1	no	—
9	C₂×#106	—	2	1	no	—
10	C₂×C₄⋊C₈	—	2	1	no	—
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#384	C₂²	1	1	yes	characteristic
16	#384	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	#340	—	2	1	no	—
19	#340	—	2	1	no	—
20	#340	—	2	1	no	—
21	#340	—	2	1	no	—
22	#336	—	2	1	no	—
23	C₂²×C₈	—	2	1	no	—
24	C₂²×C₈	—	2	1	no	—
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄	—	2	1	no	—
29	C₂×C₄⋊C₄	—	2	1	no	—
30	C₂×C₄⋊C₄	—	2	1	no	—
31	C₂×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄⋊C₄	—	2	1	no	—
34	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
36	C₂×C₄²	—	2	1	no	—
37	C₂×C₄²	C₂³	1	1	yes	characteristic
38	#107	—	4	2	no	—
39	#106	—	4	2	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	2	1	no	—
42	C₄⋊C₈	—	2	1	no	—
43	C₄⋊C₈	—	2	1	no	—
44	C₄⋊C₈	—	4	2	no	—
45	C₄⋊C₈	Q₈	1	2	yes	—
46	C₄⋊C₈	Q₈	1	2	yes	—
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	—	2	1	no	—
49	#95	—	2	1	no	—
50	C₂²×C₄	—	2	1	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	—	4	1	no	—
53	C₂²×C₄	—	4	1	no	—
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄	#41	1	1	yes	characteristic
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	—	4	1	no	—
68	C₂×C₈	—	4	1	no	—
69	C₂×C₈	—	4	1	no	—
70	C₂×C₈	—	4	1	no	—
71	C₂×C₈	—	4	1	no	—
72	C₄⋊C₄	—	2	1	no	—
73	C₄⋊C₄	—	2	1	no	—
74	C₄⋊C₄	—	2	1	no	—
75	C₄⋊C₄	—	4	1	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₄⋊C₄	—	4	1	no	—
78	C₄⋊C₄	—	4	1	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄⋊C₄	—	4	1	no	—
81	C₄⋊C₄	—	4	1	no	—
82	C₄⋊C₄	—	4	1	no	—
83	C₄⋊C₄	—	8	1	no	—
84	C₄⋊C₄	—	8	1	no	—
85	C₄⋊C₄	SD₁₆	1	2	yes	—
86	C₄²	—	4	1	no	—
87	C₄²	—	4	1	no	—
88	C₄²	C₂×Q₈	1	1	yes	characteristic
89	C₄²	C₂×Q₈	1	1	yes	characteristic
90	C₄²	#41	1	1	yes	characteristic
91	C₄²	#41	1	1	yes	characteristic
92	C₂³	#121	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	8	1	no	—
102	C₂×C₄	—	8	1	no	—
103	C₂×C₄	—	8	1	no	—
104	C₂×C₄	—	8	1	no	—
105	C₂×C₄	—	8	1	no	—
106	C₂×C₄	—	8	1	no	—
107	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄	#123	1	1	yes	characteristic
110	C₂×C₄	#125	1	1	yes	characteristic
111	C₂×C₄	#125	1	1	yes	characteristic
112	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
113	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄	#135	1	1	yes	characteristic
115	C₂×C₄	#135	1	1	yes	characteristic
116	C₂×C₄	#135	1	1	yes	characteristic
117	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄	#137	1	1	yes	characteristic
120	C₂×C₄	#137	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	4	2	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	8	1	no	—
126	C₂²	#398	1	1	yes	characteristic
127	C₂²	#463	1	1	yes	characteristic
128	C₂²	#464	1	1	yes	characteristic
129	C₂²	#465	1	1	yes	characteristic
130	C₂²	#468	1	1	yes	characteristic
131	C₂²	#472	1	1	yes	characteristic
132	C₂²	#473	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	2	no	—
136	C₄	—	8	1	no	—
137	C₄	—	16	1	no	—
138	C₄	—	16	1	no	—
139	C₄	#475	1	1	yes	characteristic
140	C₄	#476	1	1	yes	characteristic
141	C₄	#478	1	1	yes	characteristic
142	C₄	#479	1	1	yes	characteristic
143	C₄	#482	1	1	yes	characteristic
144	C₄	#485	1	1	yes	characteristic
145	C₄	#499	1	1	yes	characteristic
146	C₄	#501	1	1	yes	characteristic
147	C₂	#1678	1	1	yes	characteristic
148	C₂	#1679	1	1	yes	characteristic
149	C₂	#1697	1	1	yes	characteristic
150	C₂	#1700	1	1	yes	characteristic
151	C₂	#2085	1	1	yes	characteristic
152	C₂	#2088	1	1	yes	characteristic
153	C₂	#2089	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1948

C₂

yes

maximal, characteristic

#1947

C₂

yes

maximal, characteristic

#1939

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

#1296

C₂

yes

maximal, characteristic

#1296

C₂

yes

maximal, characteristic

#1277

C₂

yes

maximal, characteristic

C₂×#107

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Q₈

yes

—

C₄⋊C₈

Q₈

yes

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

SD₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₂×Q₈

yes

characteristic

C₄²

C₂×Q₈

yes

C₄²

yes

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²⋊Q₈

yes

characteristic

108

C₂×C₄

C₂²⋊Q₈

yes

characteristic

109

C₂×C₄

#123

yes

characteristic

110

C₂×C₄

#125

yes

characteristic

111

C₂×C₄

#125

yes

characteristic

112

C₂×C₄

C₄⋊Q₈

yes

characteristic

113

C₂×C₄

C₂×SD₁₆

yes

characteristic

114

C₂×C₄

#135

yes

characteristic

115

C₂×C₄

#135

yes

characteristic

116

C₂×C₄

#135

yes

characteristic

117

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

118

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

119

C₂×C₄

#137

yes

characteristic

120

C₂×C₄

#137

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₂²

#398

yes

characteristic

127

C₂²

#463

yes

characteristic

128

C₂²

#464

yes

characteristic

129

C₂²

#465

yes

characteristic

130

C₂²

#468

yes

characteristic

131

C₂²

#472

yes

characteristic

132

C₂²

#473

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#475

yes

characteristic

140

C₄

#476

yes

characteristic

141

C₄

#478

yes

characteristic

142

C₄

#479

yes

characteristic

143

C₄

#482

yes

characteristic

144

C₄

#485

yes

characteristic

145

C₄

#499

yes

characteristic

146

C₄

#501

yes

characteristic

147

C₂

#1678

yes

characteristic

148

C₂

#1679

yes

characteristic

149

C₂

#1697

yes

characteristic

150

C₂

#1700

yes

characteristic

151

C₂

#2085

yes

characteristic

152

C₂

#2088

yes

characteristic

153

C₂

#2089

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³

#121

commutator, Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ 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
#1948	2
#1947	2
#1939	2
#1834	2
#1296	2
#1277	2

subgroup

index

Subgroups — #10658

Conjugacy classes of subgroups

Maximal subgroups