Group #24983851 — subgroups

Subgroups — #24983851

Order 11008.

216 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₄×#13063261	C₂	1	1	yes	maximal, characteristic
2	#19133354	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#2162	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063261	—	2	1	no	—
5	#14582923	—	2	1	no	—
6	C₄×#10621877	—	2	1	no	—
7	#14582664	C₄	1	2	yes	—
8	#14582652	—	2	1	no	—
9	#14582652	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
11	C₄₃×C₄⋊C₁₆	C₄	1	2	yes	—
12	C₄₃×C₂²⋊C₁₆	—	2	1	no	—
13	C₄×C₁₆×C₄₃	—	2	1	no	—
14	C₂×#10621877	—	2	2	no	—
15	C₂×#10621877	—	4	1	no	—
16	#13063261	—	2	4	no	—
17	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
18	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
19	#13063236	—	4	1	no	—
20	C₄×C₄₃⋊C₈	—	4	1	no	—
21	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₄₃	—	2	2	no	—
25	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₂×C₈×C₄₃	—	4	1	no	—
34	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₄₃	D₈	1	1	yes	characteristic
41	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
42	#10621877	—	2	4	no	—
43	#10621877	—	4	2	no	—
44	#10621877	—	8	2	no	—
45	C₂×C₄₃⋊C₈	—	4	2	no	—
46	C₂×C₄₃⋊C₈	—	4	2	no	—
47	C₁₆×C₄₃	—	4	2	no	—
48	C₂³×C₄₃	#103	1	1	yes	characteristic
49	C₂×C₄×C₄₃	—	2	1	no	—
50	C₂×C₄×C₄₃	—	2	1	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
60	C₈×C₄₃	—	2	1	no	—
61	C₈×C₄₃	—	2	1	no	—
62	C₈×C₄₃	—	2	1	no	—
63	C₈×C₄₃	#104	1	2	yes	—
64	C₄₃⋊C₈	—	4	4	no	—
65	C₄₃⋊C₈	—	8	2	no	—
66	#7298	—	43	1	no	maximal, Sylow
67	C₂²×C₄₃	—	2	1	no	—
68	C₂²×C₄₃	—	2	1	no	—
69	C₂²×C₄₃	—	2	1	no	—
70	C₂²×C₄₃	#340	1	1	yes	characteristic
71	C₄×C₄₃	—	2	1	no	—
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	—	2	1	no	—
74	C₄×C₄₃	#339	1	1	yes	characteristic
75	C₄×C₄₃	#344	1	1	yes	characteristic
76	C₄×C₄₃	#361	1	1	yes	characteristic
77	C₄×C₄₃	#362	1	1	yes	characteristic
78	C₄×C₄₃	#368	1	2	yes	—
79	#2162	D₄₃	1	1	yes	characteristic
80	C₄×#108	—	43	1	no	—
81	#1567	—	43	1	no	—
82	C₂×C₄₃	—	4	1	no	—
83	C₂×C₄₃	#1303	1	1	yes	characteristic
84	C₂×C₄₃	#1381	1	1	yes	characteristic
85	C₂×C₄₃	#1387	1	1	yes	characteristic
86	C₂×#108	—	86	1	no	—
87	#423	—	86	1	no	—
88	C₄×M₁₆	—	86	1	no	—
89	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
90	C₄⋊C₁₆	Dic₄₃	1	2	yes	—
91	C₂²⋊C₁₆	—	2	1	no	—
92	C₄×C₁₆	—	2	1	no	—
93	#341	—	43	2	no	—
94	#335	—	43	2	no	—
95	#335	—	86	1	no	—
96	C₄₃	#7298	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	—	86	2	no	—
98	C₂×M₁₆	—	172	1	no	—
99	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
101	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
102	C₂×C₁₆	—	2	2	no	—
103	#108	—	86	4	no	—
104	C₄⋊C₈	—	86	2	no	—
105	C₄⋊C₈	—	172	1	no	—
106	#97	—	172	1	no	—
107	C₄×C₈	—	172	1	no	—
108	C₄×C₈	Dic₈₆	1	1	yes	characteristic
109	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
112	C₂²×C₄	#10621873	1	1	yes	characteristic
113	C₂²×C₄	Dic₁₇₂	1	1	yes	characteristic
114	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
115	M₁₆	—	86	4	no	—
116	M₁₆	—	344	2	no	—
117	M₁₆	—	172	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	172	2	no	—
121	C₂×C₈	—	172	2	no	—
122	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
123	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
124	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
126	C₄²	#10621873	1	1	yes	characteristic
127	C₄²	D₃₄₄	1	1	yes	characteristic
128	C₄²	C₄⋊Dic₄₃	1	1	yes	characteristic
129	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
130	C₁₆	—	4	2	no	—
131	C₂³	#13063250	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
138	C₂×C₄	#13063247	1	1	yes	characteristic
139	C₂×C₄	#13063247	1	1	yes	characteristic
140	C₂×C₄	#13063250	1	1	yes	characteristic
141	C₂×C₄	#13063257	1	1	yes	characteristic
142	C₂×C₄	#13063259	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	344	2	no	—
147	C₈	—	172	4	no	—
148	C₈	#13063252	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#14582662	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#14582658	1	1	yes	characteristic
157	C₄	#14582671	1	1	yes	characteristic
158	C₄	#14582713	1	1	yes	characteristic
159	C₄	#14582716	1	1	yes	characteristic
160	C₄	#14582729	1	2	yes	—
161	C₂	—	4	1	no	—
162	C₂	#19132497	1	1	yes	characteristic
163	C₂	#19132689	1	1	yes	characteristic
164	C₂	#19132704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13063261

C₂

yes

maximal, characteristic

#19133354

C₂

yes

maximal, characteristic

C₄₃×#2162

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#14582652

—

#14582652

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×C₄⋊C₁₆

C₄

yes

—

C₄₃×C₂²⋊C₁₆

—

C₄×C₁₆×C₄₃

—

C₂×#10621877

—

C₂×#10621877

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

Q₈

yes

characteristic

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

SD₁₆

yes

characteristic

C₂²×C₄×C₄₃

Q₁₆

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄²

yes

characteristic

C₂×C₈×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

characteristic

C₄²×C₄₃

C₂²⋊C₄

yes

characteristic

C₄²×C₄₃

C₄⋊C₄

yes

characteristic

C₄²×C₄₃

D₈

yes

characteristic

C₄²×C₄₃

SD₁₆

yes

—

—

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₂³×C₄₃

#103

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#104

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#7298

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#340

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

#2162

D₄₃

yes

characteristic

C₄×#108

—

#1567

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄⋊C₁₆

Dic₄₃

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#341

—

#335

—

#335

—

C₄₃

#7298

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

172

—

C₂²×C₈

C₄×D₄₃

yes

characteristic

100

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

101

C₂×C₄²

D₁₇₂

yes

characteristic

102

C₂×C₁₆

—

103

#108

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

172

—

106

#97

—

172

—

107

C₄×C₈

—

172

—

108

C₄×C₈

Dic₈₆

yes

characteristic

109

C₄×C₈

C₄×D₄₃

yes

characteristic

110

C₄×C₈

C₂×Dic₄₃

yes

characteristic

111

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

112

C₂²×C₄

#10621873

yes

characteristic

113

C₂²×C₄

Dic₁₇₂

yes

characteristic

114

C₂²×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

115

M₁₆

—

116

M₁₆

—

344

—

117

M₁₆

—

172

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

172

—

121

C₂×C₈

—

172

—

122

C₂×C₈

C₄×Dic₄₃

yes

characteristic

123

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

124

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

125

C₂×C₈

C₂²⋊Dic₄₃

yes

characteristic

126

C₄²

#10621873

yes

characteristic

127

C₄²

D₃₄₄

yes

characteristic

128

C₄²

C₄⋊Dic₄₃

yes

characteristic

129

C₄²

(C₂×D₄₃)⋊C₄

yes

characteristic

130

C₁₆

—

131

C₂³

#13063250

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₄₃⋊#95

yes

characteristic

138

C₂×C₄

#13063247

yes

characteristic

139

C₂×C₄

#13063247

yes

characteristic

140

C₂×C₄

#13063250

yes

characteristic

141

C₂×C₄

#13063257

yes

characteristic

142

C₂×C₄

#13063259

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

344

—

147

C₈

—

172

—

148

C₈

#13063252

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

#14582662

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#14582658

yes

characteristic

157

C₄

#14582671

yes

characteristic

158

C₄

#14582713

yes

characteristic

159

C₄

#14582716

yes

characteristic

160

C₄

#14582729

yes

—

161

C₂

—

162

C₂

#19132497

yes

characteristic

163

C₂

#19132689

yes

characteristic

164

C₂

#19132704

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063247
commutator	C₂×C₈×C₄₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₄₃×#2162	C₂
socle	C₂²×C₄₃	#340
prime radical	C₂³×C₄₃	#103
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	#2162	D₄₃
hypocenter	C₄₃	#7298
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2162

subgroup

G/H

center

C₂×C₄

#13063247

commutator

C₂×C₈×C₄₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₄₃

Fitting

C₄₃×#2162

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄×C₈×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2162

N	H ≅ G/N	type
96. C₄₃	#7298	semidirect

H ≅ G/N

type

96. C₄₃

#7298

semidirect

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

subgroup	index
C₄×#13063261	2
#19133354	2
C₄₃×#2162	2
#7298	43

subgroup

index

C₄×#13063261

#19133354

C₄₃×#2162

#7298

Subgroups — #24983851

Conjugacy classes of subgroups

Complements

Maximal subgroups