Group #24983498 — subgroups

Subgroups — #24983498

Order 11008.

216 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 103 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	1	yes	maximal, characteristic
2	C₄×#13063265	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#1451	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063265	—	2	1	no	—
5	C₄⋊(C₄₃⋊C₁₆)	C₄	1	2	yes	—
6	C₂²⋊(C₄₃⋊C₁₆)	—	2	1	no	—
7	#14582675	C₄	1	2	yes	—
8	C₄×C₄₃⋊C₁₆	—	2	1	no	—
9	C₄×C₄₃⋊C₁₆	—	2	1	no	—
10	C₄₃×#431	C₄	1	1	yes	characteristic
11	C₄×M₁₆×C₄₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₄₃×#322	—	2	1	no	—
14	C₄₃×#322	—	2	1	no	—
15	#13063265	—	2	4	no	—
16	C₂×C₄₃⋊C₁₆	—	2	2	no	—
17	C₂×C₄₃⋊C₁₆	—	2	2	no	—
18	C₂×M₁₆×C₄₃	C₈	1	2	yes	—
19	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₈×C₄₃	C₈	1	2	yes	—
23	C₂²⋊C₈×C₄₃	—	2	1	no	—
24	#97×C₄₃	—	2	1	no	—
25	C₄×C₈×C₄₃	—	2	1	no	—
26	C₄×C₈×C₄₃	—	2	1	no	—
27	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	M₁₆	1	1	yes	characteristic
34	M₁₆×C₄₃	—	4	2	no	—
35	C₂×C₈×C₄₃	—	2	2	no	—
36	C₂×C₈×C₄₃	—	2	2	no	—
37	C₂×C₈×C₄₃	—	2	2	no	—
38	C₂×C₈×C₄₃	—	2	2	no	—
39	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₄₃	C₄²	1	1	yes	characteristic
44	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₄₃	M₁₆	1	1	yes	characteristic
47	C₄₃⋊C₁₆	—	2	4	no	—
48	C₄₃⋊C₁₆	—	4	2	no	—
49	C₂³×C₄₃	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₄₃	—	2	1	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#97	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₄₃	—	2	1	no	—
61	C₈×C₄₃	—	2	1	no	—
62	C₈×C₄₃	—	4	2	no	—
63	C₈×C₄₃	—	4	2	no	—
64	C₈×C₄₃	#104	1	2	yes	—
65	C₈×C₄₃	#108	1	2	yes	—
66	#7166	—	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	#336	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	#341	1	1	yes	characteristic
76	C₄×C₄₃	#349	1	2	yes	—
77	C₄×C₄₃	#350	1	2	yes	—
78	#2162	—	43	1	no	—
79	C₄×M₃₂	—	43	1	no	—
80	#1451	D₄₃	1	1	yes	characteristic
81	C₂×C₄₃	—	4	1	no	—
82	C₂×C₄₃	#1278	1	1	yes	characteristic
83	C₂×C₄₃	#1377	1	1	yes	characteristic
84	C₂×C₄₃	#1379	1	1	yes	characteristic
85	C₂×M₃₂	—	86	1	no	—
86	#431	Dic₄₃	1	1	yes	characteristic
87	C₄×M₁₆	Dic₄₃	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
89	C₄⋊C₁₆	—	43	2	no	—
90	C₂²⋊C₁₆	—	86	1	no	—
91	#346	—	43	2	no	—
92	C₄×C₁₆	—	86	1	no	—
93	C₄×C₁₆	—	86	1	no	—
94	#322	—	2	1	no	—
95	#322	—	2	1	no	—
96	C₄₃	#7166	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	C₄₃⋊C₈	1	2	yes	—
98	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
99	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
100	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
101	M₃₂	—	86	4	no	—
102	C₂×C₁₆	—	86	2	no	—
103	C₂×C₁₆	—	86	2	no	—
104	C₄⋊C₈	C₄₃⋊C₈	1	2	yes	—
105	C₂²⋊C₈	—	2	1	no	—
106	#97	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
110	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
114	C₂²×C₄	#10621877	1	1	yes	characteristic
115	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
122	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
123	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
124	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
125	C₄²	C₂×C₄₃⋊C₈	1	1	yes	characteristic
126	C₄²	#10621877	1	1	yes	characteristic
127	C₄²	C₄×Dic₄₃	1	1	yes	characteristic
128	C₄²	C₂²⋊Dic₄₃	1	1	yes	characteristic
129	C₁₆	—	86	4	no	—
130	C₁₆	—	172	2	no	—
131	C₂³	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
137	C₂×C₄	C₄×C₄₃⋊C₈	1	1	yes	characteristic
138	C₂×C₄	#13063236	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
141	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	4	2	no	—
145	C₈	—	4	2	no	—
146	C₈	#13063251	1	2	yes	—
147	C₈	#13063260	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#14582654	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#14582660	1	1	yes	characteristic
156	C₄	#14582663	1	1	yes	characteristic
157	C₄	#14582681	1	2	yes	—
158	C₄	#14582683	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#19132433	1	1	yes	characteristic
161	C₂	#19132679	1	1	yes	characteristic
162	C₂	#19132684	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136394

C₂

yes

maximal, characteristic

C₄×#13063265

C₂

yes

maximal, characteristic

C₄₃×#1451

C₂

yes

maximal, characteristic

C₂×#13063265

—

C₄⋊(C₄₃⋊C₁₆)

C₄

yes

—

C₂²⋊(C₄₃⋊C₁₆)

—

#14582675

C₄

yes

—

C₄×C₄₃⋊C₁₆

—

C₄×C₄₃⋊C₁₆

—

C₄₃×#431

C₄

yes

characteristic

C₄×M₁₆×C₄₃

C₄

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#322

—

C₄₃×#322

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₂×M₁₆×C₄₃

C₈

yes

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

Q₈

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₃

C₈

yes

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₃

M₁₆

yes

characteristic

M₁₆×C₄₃

—

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

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

characteristic

C₄²×C₄₃

C₄²

yes

characteristic

C₄²×C₄₃

C₂²⋊C₄

yes

characteristic

C₄²×C₄₃

C₂×C₈

yes

characteristic

C₄²×C₄₃

M₁₆

yes

characteristic

C₄₃⋊C₁₆

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#95

yes

characteristic

C₂×C₄×C₄₃

C₄×C₈

yes

characteristic

C₂×C₄×C₄₃

#97

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₈

yes

characteristic

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#104

yes

—

C₈×C₄₃

#108

yes

—

#7166

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#336

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₄×C₄₃

#350

yes

—

#2162

—

C₄×M₃₂

—

#1451

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×M₃₂

—

#431

Dic₄₃

yes

characteristic

C₄×M₁₆

Dic₄₃

yes

characteristic

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

—

#322

—

C₄₃

#7166

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₄₃⋊C₈

yes

—

C₂²×C₈

Dic₈₆

yes

characteristic

C₂²×C₈

D₁₇₂

yes

characteristic

100

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

101

M₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₄⋊C₈

C₄₃⋊C₈

yes

—

105

C₂²⋊C₈

—

106

#97

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₄×D₄₃

yes

characteristic

110

C₄×C₈

C₄×D₄₃

yes

characteristic

111

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

112

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

113

C₂²×C₄

C₂×C₄₃⋊C₈

yes

characteristic

114

C₂²×C₄

#10621877

yes

characteristic

115

C₂²×C₄

C₄⋊Dic₄₃

yes

characteristic

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

122

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

123

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

characteristic

124

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

characteristic

125

C₄²

C₂×C₄₃⋊C₈

yes

characteristic

126

C₄²

#10621877

yes

characteristic

127

C₄²

C₄×Dic₄₃

yes

characteristic

128

C₄²

C₂²⋊Dic₄₃

yes

characteristic

129

C₁₆

—

130

C₁₆

—

172

—

131

C₂³

C₄⋊(C₄₃⋊C₈)

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₄₃⋊#95

yes

characteristic

137

C₂×C₄

C₄×C₄₃⋊C₈

yes

characteristic

138

C₂×C₄

#13063236

yes

characteristic

139

C₂×C₄

C₂²⋊(C₄₃⋊C₈)

yes

characteristic

140

C₂×C₄

C₂²⋊(C₄₃⋊C₈)

yes

characteristic

141

C₂×C₄

C₄⋊(C₄₃⋊C₈)

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

#13063251

yes

—

147

C₈

#13063260

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#14582654

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#14582660

yes

characteristic

156

C₄

#14582663

yes

characteristic

157

C₄

#14582681

yes

—

158

C₄

#14582683

yes

—

159

C₂

—

160

C₂

#19132433

yes

characteristic

161

C₂

#19132679

yes

characteristic

162

C₂

#19132684

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₄₃×#1451	C₂
socle	C₂²×C₄₃	#336
prime radical	C₂³×C₄₃	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	#1451	D₄₃
hypocenter	C₄₃	#7166
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1451

subgroup

G/H

center

C₄²

C₂²⋊Dic₄₃

commutator

C₂×C₄×C₄₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₃

Fitting

C₄₃×#1451

C₂

socle

C₂²×C₄₃

#336

prime radical

C₂³×C₄₃

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1451

N	H ≅ G/N	type
96. C₄₃	#7166	semidirect

H ≅ G/N

type

96. C₄₃

#7166

semidirect

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

subgroup	index
#19136394	2
C₄×#13063265	2
C₄₃×#1451	2
#7166	43

subgroup

index

#19136394

C₄×#13063265

C₄₃×#1451

#7166

Subgroups — #24983498

Conjugacy classes of subgroups

Complements

Maximal subgroups