Group #25012015 — subgroups

Subgroups — #25012015

Order 11008.

306 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₂×#14582729	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582674	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₄₃	C₂	1	2	yes	maximal
4	#19132688	C₂	1	2	yes	maximal
5	C₂×C₄₃×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13063261	C₂²	1	2	yes	—
8	M₁₆×Dic₄₃	—	2	2	no	—
9	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
10	#14582729	C₄	1	4	yes	complemented
11	#14582674	C₄	1	4	yes	—
12	#14582663	—	2	2	no	—
13	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₄₃	C₂²	1	2	yes	—
15	C₄₃×#368	C₄	1	4	yes	complemented
16	C₂×#10621877	—	2	2	no	—
17	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
19	C₂×C₄₃⋊C₁₆	—	2	1	no	—
20	C₂×C₄₃⋊C₁₆	—	2	1	no	—
21	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
22	#13063261	—	2	2	no	—
23	#13063261	C₂×C₄	1	4	yes	—
24	#13063235	—	4	2	no	—
25	C₈×Dic₄₃	—	4	2	no	—
26	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
27	C₈×Dic₄₃	Q₈	1	2	yes	—
28	C₂×M₁₆×C₄₃	—	2	2	no	—
29	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
33	#108×C₄₃	—	2	2	no	—
34	#108×C₄₃	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₄₃	Q₁₆	1	2	yes	—
37	M₁₆×C₄₃	—	2	4	no	—
38	M₁₆×C₄₃	—	4	2	no	—
39	C₂×C₈×C₄₃	—	4	2	no	—
40	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
46	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
47	C₄×Dic₄₃	D₈	1	2	yes	—
48	#10621877	—	2	4	no	—
49	#10621877	—	4	2	no	—
50	C₂×C₄₃⋊C₈	—	2	1	no	—
51	C₂×C₄₃⋊C₈	—	2	1	no	—
52	C₂×C₄₃⋊C₈	—	4	2	no	—
53	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
54	C₁₆×C₄₃	C₄²	1	4	yes	—
55	C₄₃⋊C₁₆	—	2	2	no	—
56	C₂³×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
63	C₂×Dic₄₃	—	4	2	no	—
64	C₂×Dic₄₃	#107	1	4	yes	—
65	C₈×C₄₃	—	4	4	no	—
66	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₄₃	#117	1	1	yes	characteristic
68	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
69	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
70	C₄₃⋊C₈	—	2	2	no	—
71	C₄₃⋊C₈	—	4	4	no	—
72	#12676	—	43	1	no	maximal, Sylow
73	C₂²×C₄₃	—	2	2	no	—
74	C₂²×C₄₃	C₂×#107	1	1	yes	characteristic
75	C₂²×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
76	C₂²×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
77	C₄×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₄₃	#427	1	1	yes	characteristic
79	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
80	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
81	Dic₄₃	—	2	4	no	—
82	C₂×#368	—	43	1	no	—
83	C₂×#368	D₄₃	1	1	yes	characteristic
84	C₂×#346	—	43	1	no	—
85	C₄×#108	—	43	2	no	—
86	#1381	—	43	2	no	—
87	C₂×C₄₃	—	2	1	no	—
88	C₂×C₄₃	—	2	1	no	—
89	C₂×C₄₃	C₄×#107	1	1	yes	characteristic
90	C₂×C₄₃	#2161	1	2	yes	—
91	C₂²×C₁₆	—	43	1	no	—
92	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
93	C₂×#108	—	43	2	no	—
94	C₂×#108	C₂×D₄₃	1	2	yes	—
95	C₄×M₁₆	—	86	2	no	—
96	C₂×C₄×C₈	—	43	1	no	—
97	#368	—	43	4	no	—
98	#368	Dic₄₃	1	4	yes	complemented
99	#346	—	43	4	no	—
100	#341	—	86	2	no	—
101	C₄₃	#12676	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	86	2	no	—
104	C₂²×C₈	—	43	1	no	—
105	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
106	C₂×C₄²	—	43	1	no	—
107	C₂×C₁₆	—	43	2	no	—
108	C₂×C₁₆	—	86	1	no	—
109	C₂×C₁₆	—	86	1	no	—
110	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
111	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₄₃	1	2	yes	—
113	#108	—	2	2	no	—
114	#108	—	43	4	no	—
115	#108	—	86	2	no	—
116	#108	C₂×Dic₄₃	1	4	yes	—
117	#97	—	172	2	no	—
118	C₄×C₈	—	43	2	no	—
119	C₄×C₈	—	43	2	no	—
120	C₄×C₈	—	172	2	no	—
121	C₂²×C₄	—	43	2	no	—
122	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	86	4	no	—
126	M₁₆	—	172	2	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	43	2	no	—
129	C₂×C₈	—	86	1	no	—
130	C₂×C₈	—	86	1	no	—
131	C₂×C₈	—	172	2	no	—
132	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
134	C₂×C₈	#10621900	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
136	C₂×C₈	#10621902	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
138	C₄²	—	43	2	no	—
139	C₄²	—	43	2	no	—
140	C₁₆	—	86	2	no	—
141	C₁₆	C₄×Dic₄₃	1	4	yes	—
142	C₂³	Q₁₆×D₄₃	1	1	yes	characteristic
143	C₂×C₄	—	43	4	no	—
144	C₂×C₄	—	172	2	no	—
145	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
146	C₂×C₄	#13063287	1	1	yes	characteristic
147	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
148	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
149	C₂×C₄	#13063344	1	1	yes	characteristic
150	C₂×C₄	#13063348	1	1	yes	characteristic
151	C₈	—	4	4	no	—
152	C₈	—	86	2	no	—
153	C₈	—	172	4	no	—
154	C₈	C₂×C₄×Dic₄₃	1	1	yes	characteristic
155	C₈	#13063283	1	1	yes	characteristic
156	C₈	#13063290	1	1	yes	characteristic
157	C₈	Q₈×Dic₄₃	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	#107×D₄₃	1	1	yes	characteristic
160	C₂²	#14582993	1	1	yes	characteristic
161	C₂²	Q₁₆×Dic₄₃	1	1	yes	characteristic
162	C₄	—	86	4	no	—
163	C₄	C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
164	C₄	#14582936	1	1	yes	characteristic
165	C₄	#14582981	1	1	yes	characteristic
166	C₄	D₈×Dic₄₃	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#107×Dic₄₃	1	1	yes	characteristic
170	C₂	#19136388	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582729

C₂

yes

maximal, characteristic

C₂×#14582674

C₂

yes

maximal, characteristic

#108×Dic₄₃

C₂

yes

maximal

#19132688

C₂

yes

maximal

C₂×C₄₃×#368

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₁₆

C₂²

yes

characteristic

C₂×#13063261

C₂²

yes

—

M₁₆×Dic₄₃

—

C₂×C₈×Dic₄₃

C₂²

yes

characteristic

#14582729

C₄

yes

complemented

#14582674

C₄

yes

—

#14582663

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×#108×C₄₃

C₂²

yes

—

C₄₃×#368

C₄

yes

complemented

C₂×#10621877

—

C₂²×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

#13063261

—

#13063261

C₂×C₄

yes

—

#13063235

—

C₈×Dic₄₃

—

C₈×Dic₄₃

C₂×C₄

yes

—

C₈×Dic₄₃

Q₈

yes

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₁₆×C₄₃

C₂×C₄

yes

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

—

#108×C₄₃

—

#108×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

Q₁₆

yes

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

C₂×C₈×C₄₃

yes

C₂×C₈×C₄₃

yes

C₄×Dic₄₃

C₄⋊C₄

yes

—

C₄×Dic₄₃

D₈

yes

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₄⋊C₄

yes

—

C₁₆×C₄₃

C₄²

yes

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

#107

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₂×C₄²

yes

characteristic

C₈×C₄₃

#117

yes

characteristic

C₈×C₄₃

C₄×Q₈

yes

characteristic

C₈×C₄₃

C₄×Q₈

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#12676

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×#107

yes

characteristic

C₂²×C₄₃

C₄×Q₁₆

yes

characteristic

C₂²×C₄₃

C₄×Q₁₆

yes

characteristic

C₄×C₄₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₄₃

#427

yes

characteristic

C₄×C₄₃

C₄×D₈

yes

characteristic

C₄×C₄₃

C₄×D₈

yes

characteristic

Dic₄₃

—

C₂×#368

—

C₂×#368

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₄×#107

yes

characteristic

C₂×C₄₃

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₄₃

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#368

—

#368

Dic₄₃

yes

complemented

#346

—

100

#341

—

101

C₄₃

#12676

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

—

105

C₂²×C₈

C₂²×D₄₃

yes

characteristic

106

C₂×C₄²

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

C₄×D₄₃

yes

—

111

C₂×C₁₆

C₄×D₄₃

yes

—

112

C₂×C₁₆

C₂×Dic₄₃

yes

—

113

#108

—

114

#108

—

115

#108

—

116

#108

C₂×Dic₄₃

yes

—

117

#97

—

172

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

172

—

121

C₂²×C₄

—

122

C₂²×C₄

D₄×D₄₃

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

172

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

172

—

132

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

133

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

134

C₂×C₈

#10621900

yes

characteristic

135

C₂×C₈

Q₈×D₄₃

yes

characteristic

136

C₂×C₈

#10621902

yes

characteristic

137

C₂×C₈

C₂²×Dic₄₃

yes

characteristic

138

C₄²

—

139

C₄²

—

140

C₁₆

—

141

C₁₆

C₄×Dic₄₃

yes

—

142

C₂³

Q₁₆×D₄₃

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

172

—

145

C₂×C₄

C₄⋊C₄×D₄₃

yes

characteristic

146

C₂×C₄

#13063287

yes

characteristic

147

C₂×C₄

D₄×Dic₄₃

yes

characteristic

148

C₂×C₄

D₈×D₄₃

yes

characteristic

149

C₂×C₄

#13063344

yes

characteristic

150

C₂×C₄

#13063348

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

172

—

154

C₈

C₂×C₄×Dic₄₃

yes

characteristic

155

C₈

#13063283

yes

characteristic

156

C₈

#13063290

yes

characteristic

157

C₈

Q₈×Dic₄₃

yes

characteristic

158

C₂²

—

159

C₂²

#107×D₄₃

yes

characteristic

160

C₂²

#14582993

yes

characteristic

161

C₂²

Q₁₆×Dic₄₃

yes

characteristic

162

C₄

—

163

C₄

C₄⋊C₄×Dic₄₃

yes

characteristic

164

C₄

#14582936

yes

characteristic

165

C₄

#14582981

yes

characteristic

166

C₄

D₈×Dic₄₃

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#107×Dic₄₃

yes

characteristic

170

C₂

#19136388

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#368	C₂
socle	C₂²×C₄₃	C₂×#107
prime radical	C₂³×C₄₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#368	D₄₃
hypocenter	C₄₃	#12676
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₄₃

commutator

C₈×C₄₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂×C₄₃×#368

C₂

socle

C₂²×C₄₃

C₂×#107

prime radical

C₂³×C₄₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
10. #14582729	C₄	semidirect
15. C₄₃×#368	C₄	semidirect
98. #368	Dic₄₃	semidirect
101. C₄₃	#12676	semidirect

H ≅ G/N

type

10. #14582729

C₄

semidirect

15. C₄₃×#368

C₄

semidirect

98. #368

Dic₄₃

semidirect

101. C₄₃

#12676

semidirect

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

subgroup	index
C₂×#14582729	2
C₂×#14582674	2
#108×Dic₄₃	2
#19132688	2
C₂×C₄₃×#368	2
#12676	43

subgroup

index

#108×Dic₄₃

C₂×C₄₃×#368

Subgroups — #25012015

Conjugacy classes of subgroups

Complements

Maximal subgroups