Group #10556 — subgroups

Subgroups — #10556

Order 256.

182 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 59 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	#3165	C₂	1	1	yes	maximal, characteristic
2	#1950	C₂	1	1	yes	maximal, characteristic
3	#1550	C₂	1	1	yes	maximal, characteristic
4	#1548	C₂	1	1	yes	maximal, characteristic
5	C₂×#332	C₂	1	1	yes	maximal, characteristic
6	#1481	C₂	1	1	yes	maximal, characteristic
7	#1283	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#501	—	2	2	no	—
11	#500	—	2	1	no	—
12	#498	—	2	1	no	—
13	#431	C₂²	1	1	yes	characteristic
14	C₂×#108	—	2	1	no	—
15	#428	—	2	1	no	—
16	#428	—	2	1	no	—
17	#423	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×M₁₆	C₂²	1	1	yes	characteristic
21	#341	—	2	1	no	—
22	#333	—	2	2	no	—
23	#333	—	2	1	no	—
24	#332	—	2	1	no	—
25	#332	—	2	2	no	—
26	Q₈⋊C₈	—	2	2	no	—
27	C₂²×Q₈	—	2	1	no	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	D₄	1	1	yes	characteristic
30	C₂×M₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₈	—	2	1	no	—
32	C₂²×C₈	—	2	1	no	—
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	C₄⋊Q₈	D₄	1	2	yes	—
37	C₄⋊Q₈	D₄	1	1	yes	characteristic
38	C₄⋊Q₈	D₄	1	1	yes	characteristic
39	#125	—	2	1	no	—
40	C₂²⋊Q₈	—	2	1	no	—
41	C₂²⋊Q₈	—	2	1	no	—
42	C₄×Q₈	—	2	1	no	—
43	C₄×Q₈	—	2	1	no	—
44	#117	—	2	1	no	—
45	C₂×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄²	C₂³	1	1	yes	characteristic
48	#108	—	4	2	no	—
49	#107	—	2	2	no	—
50	#107	—	2	2	no	—
51	#106	—	2	2	no	—
52	#106	—	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	—	2	1	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	D₄	1	2	yes	—
60	C₂²⋊C₈	—	2	1	no	—
61	#97	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₂×Q₈	—	2	1	no	—
65	C₂×Q₈	—	2	1	no	—
66	C₂×Q₈	—	2	2	no	—
67	C₂×Q₈	—	4	1	no	—
68	C₂×Q₈	—	8	1	no	—
69	C₂×Q₈	—	8	1	no	—
70	C₂²×C₄	—	4	1	no	—
71	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄	#41	1	1	yes	characteristic
73	C₂²×C₄	#41	1	1	yes	characteristic
74	M₁₆	—	2	1	no	—
75	M₁₆	—	2	1	no	—
76	M₁₆	—	4	1	no	—
77	M₁₆	—	4	1	no	—
78	M₁₆	SD₁₆	1	2	yes	—
79	M₁₆	Q₁₆	1	2	yes	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	1	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	8	1	no	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄⋊C₄	—	4	1	no	—
97	C₄⋊C₄	—	4	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₂²⋊C₄	—	4	1	no	—
100	C₄²	—	4	1	no	—
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₄²	C₂×D₄	1	1	yes	characteristic
103	C₄²	C₂×D₄	1	1	yes	characteristic
104	C₄²	#41	1	1	yes	characteristic
105	C₂³	#124	1	1	yes	characteristic
106	Q₈	—	4	1	no	—
107	Q₈	—	4	1	no	—
108	Q₈	—	8	1	no	—
109	Q₈	—	8	2	no	—
110	Q₈	—	8	1	no	—
111	Q₈	—	8	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	#120	1	1	yes	characteristic
120	C₂×C₄	#121	1	1	yes	characteristic
121	C₂×C₄	#121	1	1	yes	characteristic
122	C₂×C₄	#121	1	1	yes	characteristic
123	C₂×C₄	#123	1	1	yes	characteristic
124	C₂×C₄	#123	1	1	yes	characteristic
125	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄	#137	1	1	yes	characteristic
128	C₂×C₄	#137	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	2	2	no	—
131	C₈	—	4	2	no	—
132	C₈	—	8	1	no	—
133	C₈	—	8	1	no	—
134	C₈	—	8	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	#394	1	1	yes	characteristic
138	C₂²	#487	1	1	yes	characteristic
139	C₂²	#491	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	8	1	no	—
144	C₄	—	8	2	no	—
145	C₄	—	8	1	no	—
146	C₄	#448	1	1	yes	characteristic
147	C₄	#451	1	1	yes	characteristic
148	C₄	#461	1	1	yes	characteristic
149	C₄	#462	1	1	yes	characteristic
150	C₄	#471	1	1	yes	characteristic
151	C₄	#472	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	2	1	no	—
154	C₂	#2043	1	1	yes	characteristic
155	C₂	#2049	1	1	yes	characteristic
156	C₂	#2051	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3165

C₂

yes

maximal, characteristic

#1950

C₂

yes

maximal, characteristic

#1550

C₂

yes

maximal, characteristic

#1548

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal, characteristic

#1481

C₂

yes

maximal, characteristic

#1283

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

—

—

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₄×M₁₆

C₂²

yes

—

—

—

—

—

Q₈⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

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×C₄²

C₂³

yes

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

SD₁₆

yes

—

M₁₆

Q₁₆

yes

—

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

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

C₂×D₄

yes

characteristic

102

C₄²

C₂×D₄

yes

characteristic

103

C₄²

C₂×D₄

yes

characteristic

104

C₄²

#41

yes

characteristic

105

C₂³

#124

yes

characteristic

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#120

yes

characteristic

120

C₂×C₄

#121

yes

characteristic

121

C₂×C₄

#121

yes

characteristic

122

C₂×C₄

#121

yes

characteristic

123

C₂×C₄

#123

yes

characteristic

124

C₂×C₄

#123

yes

characteristic

125

C₂×C₄

C₂×SD₁₆

yes

characteristic

126

C₂×C₄

C₂×Q₁₆

yes

characteristic

127

C₂×C₄

#137

yes

characteristic

128

C₂×C₄

#137

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

#394

yes

characteristic

138

C₂²

#487

yes

characteristic

139

C₂²

#491

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#448

yes

characteristic

147

C₄

#451

yes

characteristic

148

C₄

#461

yes

characteristic

149

C₄

#462

yes

characteristic

150

C₄

#471

yes

characteristic

151

C₄

#472

yes

characteristic

152

C₂

—

153

C₂

—

154

C₂

#2043

yes

characteristic

155

C₂

#2049

yes

characteristic

156

C₂

#2051

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#124
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

#124

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
#3165	2
#1950	2
#1550	2
#1548	2
C₂×#332	2
#1481	2
#1283	2

subgroup

index

Subgroups — #10556

Conjugacy classes of subgroups

Maximal subgroups