Group #25005934 — subgroups

Subgroups — #25005934

Order 11008.

342 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 155 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	#19136394	C₂	1	2	yes	maximal
2	#19132535	C₂	1	4	yes	maximal
3	C₂×C₈×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
5	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
6	C₄×C₄₃⋊C₁₆	—	2	2	no	—
7	C₂×C₄×Q₈×C₄₃	C₄	1	1	yes	characteristic
8	C₈×Q₈×C₄₃	—	2	2	no	—
9	C₈×Q₈×C₄₃	—	2	2	no	—
10	C₂×C₄⋊C₈×C₄₃	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₄₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄₃⋊C₁₆	—	2	4	no	—
15	C₂²×Q₈×C₄₃	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
19	C₄×Q₈×C₄₃	—	2	2	no	—
20	C₄×Q₈×C₄₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₄₃	C₈	1	2	yes	—
22	C₂×C₄⋊C₄×C₄₃	C₈	1	1	yes	characteristic
23	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₈×C₄₃	—	2	4	no	—
26	C₄⋊C₈×C₄₃	—	2	2	no	—
27	C₄×C₈×C₄₃	—	2	2	no	—
28	C₄×C₈×C₄₃	—	2	2	no	—
29	C₄×C₈×C₄₃	D₄	1	2	yes	—
30	C₄×C₈×C₄₃	D₄	1	2	yes	—
31	C₂×Q₈×C₄₃	—	2	2	no	—
32	C₂×Q₈×C₄₃	—	2	2	no	—
33	C₂×Q₈×C₄₃	C₁₆	1	4	yes	complemented
34	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
35	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₃	—	2	2	no	—
40	C₂×C₈×C₄₃	—	2	2	no	—
41	C₂×C₈×C₄₃	—	4	1	no	—
42	C₂×C₈×C₄₃	—	4	2	no	—
43	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
46	C₄⋊C₄×C₄₃	—	2	4	no	—
47	C₄⋊C₄×C₄₃	—	2	2	no	—
48	C₄²×C₄₃	—	2	2	no	—
49	C₄²×C₄₃	—	2	2	no	—
50	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₄₃	M₁₆	1	2	yes	—
52	C₄₃⋊C₁₆	—	4	4	no	—
53	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
54	Q₈×C₄₃	—	2	8	no	—
55	C₂×C₄×C₄₃	—	2	2	no	—
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	—	2	1	no	—
58	C₂×C₄×C₄₃	—	2	1	no	—
59	C₂×C₄×C₄₃	—	2	2	no	—
60	C₂×C₄×C₄₃	—	4	1	no	—
61	C₂×C₄×C₄₃	—	4	2	no	—
62	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₄₃	C₂×C₁₆	1	4	yes	—
65	C₂×C₄×C₄₃	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₄₃	M₃₂	1	2	yes	—
67	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
69	C₈×C₄₃	—	2	2	no	—
70	C₈×C₄₃	—	2	1	no	—
71	C₈×C₄₃	—	4	2	no	—
72	C₈×C₄₃	#101	1	2	yes	—
73	#11493	—	43	1	no	maximal, Sylow
74	C₂²×C₄₃	—	2	1	no	—
75	C₂²×C₄₃	—	2	1	no	—
76	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂²×C₄₃	C₂²×C₁₆	1	1	yes	characteristic
78	C₂²×C₄₃	C₂×M₃₂	1	1	yes	characteristic
79	C₄×C₄₃	—	2	4	no	—
80	C₄×C₄₃	—	2	2	no	—
81	C₄×C₄₃	—	2	1	no	—
82	C₄×C₄₃	—	4	2	no	—
83	C₄×C₄₃	C₂²⋊C₁₆	1	4	yes	—
84	C₄×C₄₃	#410	1	1	yes	characteristic
85	C₄×C₄₃	C₂×#101	1	1	yes	characteristic
86	C₂×C₈×Q₈	D₄₃	1	1	yes	characteristic
87	#2162	—	43	2	no	—
88	#1322	—	43	4	no	—
89	C₂×C₄₃	—	2	1	no	—
90	C₂×C₄₃	—	2	1	no	—
91	C₂×C₄₃	#1461	1	1	yes	characteristic
92	C₂×C₄₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
93	C₂×C₄₃	#2115	1	1	yes	characteristic
94	C₂×C₄×Q₈	Dic₄₃	1	1	yes	characteristic
95	C₈×Q₈	—	2	2	no	—
96	C₈×Q₈	—	2	2	no	—
97	C₂×C₄⋊C₈	Dic₄₃	1	1	yes	characteristic
98	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
99	C₂×C₄×C₈	Dic₄₃	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
101	C₄⋊C₁₆	—	86	2	no	—
102	C₂²⋊C₁₆	—	43	4	no	—
103	C₄×C₁₆	—	86	2	no	—
104	C₄₃	#11493	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	C₄₃⋊C₈	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
108	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	C₂×C₄⋊C₄	C₄₃⋊C₈	1	2	yes	—
112	C₂×C₄⋊C₄	C₄₃⋊C₈	1	1	yes	characteristic
113	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
114	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
115	C₂×C₁₆	—	86	4	no	—
116	C₄⋊C₈	—	2	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
121	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	C₄₃⋊C₁₆	1	4	yes	complemented
125	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
126	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
127	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
129	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
135	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
136	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	#10621877	1	2	yes	—
142	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
143	C₁₆	—	172	4	no	—
144	C₂³	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
154	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
155	C₂×C₄	C₂×C₄₃⋊C₁₆	1	4	yes	—
156	C₂×C₄	C₂×C₄₃⋊C₁₆	1	2	yes	—
157	C₂×C₄	#13063265	1	2	yes	—
158	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
159	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	1	no	—
162	C₈	—	4	2	no	—
163	C₈	#13063244	1	2	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
167	C₂²	C₂²×C₄₃⋊C₁₆	1	1	yes	characteristic
168	C₂²	C₂×#13063265	1	1	yes	characteristic
169	C₄	—	2	4	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	C₂²⋊(C₄₃⋊C₁₆)	1	4	yes	—
174	C₄	#14582873	1	1	yes	characteristic
175	C₄	C₂×#13063244	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#19132919	1	1	yes	characteristic
179	C₂	C₂×C₂²⋊(C₄₃⋊C₁₆)	1	1	yes	characteristic
180	C₂	#19136216	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136394

C₂

yes

maximal

#19132535

C₂

yes

maximal

C₂×C₈×Q₈×C₄₃

C₂

yes

maximal, characteristic

C₄⋊(C₄₃⋊C₁₆)

—

C₂²⋊(C₄₃⋊C₁₆)

C₂²

yes

—

C₄×C₄₃⋊C₁₆

—

C₂×C₄×Q₈×C₄₃

C₄

yes

characteristic

C₈×Q₈×C₄₃

—

C₈×Q₈×C₄₃

—

C₂×C₄⋊C₈×C₄₃

C₄

yes

characteristic

C₂×C₄⋊C₈×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄₃⋊C₁₆

—

C₂²×Q₈×C₄₃

C₈

yes

characteristic

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₈

yes

—

C₂×C₄⋊C₄×C₄₃

C₈

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

C₁₆

yes

complemented

C₂²×C₄×C₄₃

C₂×C₈

yes

—

C₂²×C₄×C₄₃

C₂×C₈

yes

—

C₂²×C₄×C₄₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

M₁₆

yes

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂²×C₈

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₈

yes

—

C₂×C₄×C₄₃

C₂²⋊C₈

yes

—

C₂×C₄×C₄₃

C₂×C₁₆

yes

—

C₂×C₄×C₄₃

C₂×C₁₆

yes

—

C₂×C₄×C₄₃

M₃₂

yes

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×M₁₆

yes

characteristic

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#101

yes

—

#11493

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₄₃

C₂²×C₁₆

yes

characteristic

C₂²×C₄₃

C₂×M₃₂

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊C₁₆

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₂×C₈×Q₈

D₄₃

yes

characteristic

#2162

—

#1322

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#1461

yes

characteristic

C₂×C₄₃

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₄₃

#2115

yes

characteristic

C₂×C₄×Q₈

Dic₄₃

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

Dic₄₃

yes

characteristic

C₂×C₄⋊C₈

C₂×D₄₃

yes

—

C₂×C₄×C₈

Dic₄₃

yes

—

100

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

101

C₄⋊C₁₆

—

102

C₂²⋊C₁₆

—

103

C₄×C₁₆

—

104

C₄₃

#11493

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

C₄₃⋊C₈

yes

characteristic

106

C₂²×C₈

C₂×Dic₄₃

yes

characteristic

107

C₂²×C₈

C₂×Dic₄₃

yes

—

108

C₂²×C₈

C₂²×D₄₃

yes

characteristic

109

C₄×Q₈

—

110

C₄×Q₈

—

111

C₂×C₄⋊C₄

C₄₃⋊C₈

yes

—

112

C₂×C₄⋊C₄

C₄₃⋊C₈

yes

characteristic

113

C₂×C₄²

C₂×Dic₄₃

yes

—

114

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

115

C₂×C₁₆

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₂²⋊D₄₃

yes

—

121

C₄×C₈

C₂²⋊D₄₃

yes

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

C₄₃⋊C₁₆

yes

complemented

125

C₂²×C₄

C₂×C₄₃⋊C₈

yes

—

126

C₂²×C₄

C₂×C₄₃⋊C₈

yes

—

127

C₂²×C₄

C₂×C₄₃⋊C₈

yes

characteristic

128

C₂²×C₄

C₂×C₄₃⋊C₈

yes

characteristic

129

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

C₂²⋊Dic₄₃

yes

characteristic

135

C₂×C₈

C₂²⋊Dic₄₃

yes

characteristic

136

C₂×C₈

C₂×C₂²⋊D₄₃

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

—

140

C₄²

—

141

C₄²

#10621877

yes

—

142

C₄²

C₂²⋊Dic₄₃

yes

—

143

C₁₆

—

172

—

144

C₂³

C₂²×C₄₃⋊C₈

yes

characteristic

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂²⋊(C₄₃⋊C₈)

yes

—

154

C₂×C₄

C₂²⋊(C₄₃⋊C₈)

yes

—

155

C₂×C₄

C₂×C₄₃⋊C₁₆

yes

—

156

C₂×C₄

C₂×C₄₃⋊C₁₆

yes

—

157

C₂×C₄

#13063265

yes

—

158

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

159

C₂×C₄

C₂×#10621877

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

#13063244

yes

—

164

C₂²

—

165

C₂²

—

166

C₂²

C₂×C₂²⋊(C₄₃⋊C₈)

yes

characteristic

167

C₂²

C₂²×C₄₃⋊C₁₆

yes

characteristic

168

C₂²

C₂×#13063265

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

C₂²⋊(C₄₃⋊C₁₆)

yes

—

174

C₄

#14582873

yes

characteristic

175

C₄

C₂×#13063244

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#19132919

yes

characteristic

179

C₂

C₂×C₂²⋊(C₄₃⋊C₁₆)

yes

characteristic

180

C₂

#19136216

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₄₃
commutator	C₂²×C₄₃	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₈×Q₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₄₃	C₂²×C₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×C₈×Q₈	D₄₃
hypocenter	C₄₃	#11493
LCS	G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₄₃

commutator

C₂²×C₄₃

C₂²×C₁₆

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂×C₈×Q₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₄₃

C₂²×C₈

cosocle

C₂²×C₈×C₄₃

C₂³

hypercenter

C₂×C₈×Q₈

D₄₃

hypocenter

C₄₃

#11493

LCS

G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

N	H ≅ G/N	type
33. C₂×Q₈×C₄₃	C₁₆	semidirect
104. C₄₃	#11493	semidirect
124. C₂×Q₈	C₄₃⋊C₁₆	semidirect

H ≅ G/N

type

33. C₂×Q₈×C₄₃

C₁₆

semidirect

104. C₄₃

#11493

semidirect

124. C₂×Q₈

C₄₃⋊C₁₆

semidirect

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

subgroup	index
#19136394	2
#19132535	2
C₂×C₈×Q₈×C₄₃	2
#11493	43

subgroup

index

#19136394

#19132535

C₂×C₈×Q₈×C₄₃

#11493

Subgroups — #25005934

Conjugacy classes of subgroups

Complements

Maximal subgroups