Group #25012782 — subgroups

Subgroups — #25012782

Order 11008.

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

normal subgroups only

This group has 374 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	2	yes	maximal, complemented
2	#19134633	C₂	1	2	yes	maximal
3	#19132689	C₂	1	2	yes	maximal, complemented
4	C₂²×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063261	C₂²	1	4	yes	—
6	#14582863	—	2	2	no	—
7	#14582729	—	2	4	no	—
8	#14582664	—	2	2	no	—
9	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₄₃	—	2	4	no	—
11	C₂×M₃₂×C₄₃	C₄	1	4	yes	—
12	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#10621877	—	2	4	no	—
15	#13063261	—	2	4	no	—
16	#13063261	—	2	4	no	—
17	C₂²⋊(C₄₃⋊C₈)	—	4	4	no	—
18	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₄₃	—	2	2	no	—
20	C₂²×C₈×C₄₃	—	2	2	no	—
21	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₃	Q₈	1	2	yes	—
24	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
25	M₃₂×C₄₃	—	2	8	no	—
26	C₂×C₁₆×C₄₃	—	2	2	no	—
27	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
30	C₂⁴×C₄₃	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	—	2	2	no	—
32	C₂²×C₄×C₄₃	—	2	2	no	—
33	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₄₃	D₈	1	2	yes	—
35	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₄₃	—	2	2	no	—
38	C₂×C₈×C₄₃	—	2	2	no	—
39	C₂×C₈×C₄₃	—	2	2	no	—
40	C₂×C₈×C₄₃	—	2	4	no	—
41	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₄₃	#41	1	2	yes	—
46	#10621877	—	4	4	no	—
47	#10621877	—	4	4	no	—
48	C₂×C₄₃⋊C₈	—	4	4	no	—
49	C₁₆×C₄₃	—	2	2	no	—
50	C₁₆×C₄₃	C₂²⋊C₄	1	4	yes	—
51	C₂³×C₄₃	—	2	2	no	—
52	C₂³×C₄₃	—	4	2	no	—
53	C₂³×C₄₃	#107	1	2	yes	—
54	C₂³×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₄₃	—	2	2	no	—
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	—	2	2	no	—
58	C₂×C₄×C₄₃	—	2	4	no	—
59	C₂×C₄×C₄₃	#107	1	2	yes	—
60	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#121	1	2	yes	—
62	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#135	1	2	yes	—
64	C₈×C₄₃	—	2	2	no	—
65	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₄₃	#117	1	1	yes	characteristic
67	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
68	C₄₃⋊C₈	—	8	4	no	—
69	#12838	—	43	1	no	maximal, Sylow
70	C₂²×C₄₃	—	2	2	no	—
71	C₂²×C₄₃	—	2	2	no	—
72	C₂²×C₄₃	—	2	4	no	—
73	C₂²×C₄₃	—	4	4	no	—
74	C₂²×C₄₃	C₂×#107	1	1	yes	characteristic
75	C₂²×C₄₃	#467	1	2	yes	—
76	C₄×C₄₃	—	2	2	no	—
77	C₄×C₄₃	#380	1	1	yes	characteristic
78	C₄×C₄₃	#427	1	1	yes	characteristic
79	C₄×C₄₃	#466	1	2	yes	—
80	C₂²×M₃₂	D₄₃	1	1	yes	characteristic
81	C₂×#368	—	43	2	no	—
82	#1810	—	43	2	no	—
83	#1381	—	43	2	no	—
84	C₂×C₄₃	—	2	2	no	—
85	C₂×C₄₃	—	4	2	no	—
86	C₂×C₄₃	#1808	1	1	yes	characteristic
87	C₂×C₄₃	#2161	1	2	yes	—
88	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
89	C₂×M₃₂	—	2	4	no	—
90	C₂×M₃₂	Dic₄₃	1	4	yes	—
91	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
92	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
93	C₂×#108	—	43	4	no	—
94	#407	—	86	2	no	—
95	#368	—	86	4	no	—
96	#341	—	86	2	no	—
97	C₄₃	#12838	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
99	C₂×M₁₆	—	86	4	no	—
100	C₂²×C₈	—	2	2	no	—
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	Dic₈₆	1	2	yes	—
103	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
104	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
105	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
106	M₃₂	—	2	8	no	—
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	C₂×Dic₄₃	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₄₃	1	2	yes	—
110	C₂×C₁₆	C₂²⋊D₄₃	1	4	yes	—
111	#108	—	86	4	no	—
112	#108	—	86	4	no	—
113	C₂²⋊C₈	—	172	4	no	—
114	C₂⁴	Dic₁₇₂	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	D₃₄₄	1	2	yes	—
118	C₂²×C₄	Dic₁₇₂	1	1	yes	characteristic
119	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
120	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
121	M₁₆	—	172	4	no	—
122	M₁₆	—	172	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	172	4	no	—
128	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
129	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
130	C₂×C₈	#10621898	1	2	yes	—
131	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
132	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
133	C₁₆	—	2	2	no	—
134	C₁₆	C₂²⋊Dic₄₃	1	4	yes	—
135	C₂³	—	2	2	no	—
136	C₂³	—	4	2	no	—
137	C₂³	#13063259	1	2	yes	—
138	C₂³	C₂×Dic₁₇₂	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	#13063259	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
145	C₂×C₄	#13063299	1	2	yes	—
146	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
147	C₂×C₄	#13063347	1	2	yes	—
148	C₈	—	2	2	no	—
149	C₈	—	344	4	no	—
150	C₈	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
151	C₈	#13063284	1	1	yes	characteristic
152	C₈	#13063304	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×#13063259	1	1	yes	characteristic
158	C₂²	#14583140	1	2	yes	—
159	C₄	—	2	2	no	—
160	C₄	#14582759	1	1	yes	characteristic
161	C₄	#14582937	1	1	yes	characteristic
162	C₄	#14583135	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	4	2	no	—
165	C₂	#19134624	1	1	yes	characteristic
166	C₂	#19136390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582729

C₂

yes

maximal, complemented

#19134633

C₂

yes

maximal

#19132689

C₂

yes

maximal, complemented

C₂²×M₃₂×C₄₃

C₂

yes

maximal, characteristic

C₂×#13063261

C₂²

yes

—

#14582863

—

#14582729

—

#14582664

—

C₂³×C₈×C₄₃

C₂²

yes

characteristic

C₂×M₃₂×C₄₃

—

C₂×M₃₂×C₄₃

C₄

yes

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₂²×C₁₆×C₄₃

C₂²

yes

—

—

—

C₂²⋊(C₄₃⋊C₈)

—

C₂³×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

—

C₂²×C₈×C₄₃

—

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

Q₈

yes

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

M₃₂×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₂⁴×C₄₃

Q₁₆

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₄⋊C₄

yes

—

C₂²×C₄×C₄₃

D₈

yes

—

C₂²×C₄×C₄₃

Q₁₆

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₁₆×C₄₃

C₂²⋊C₄

yes

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

#107

yes

—

C₂³×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#107

yes

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₄₃

#117

yes

characteristic

C₈×C₄₃

C₂²⋊Q₈

yes

—

C₄₃⋊C₈

—

#12838

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×#107

yes

characteristic

C₂²×C₄₃

#467

yes

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂²×M₃₂

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#1808

yes

characteristic

C₂×C₄₃

#2161

yes

—

C₂³×C₈

C₂×D₄₃

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

Dic₄₃

yes

—

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

C₂²×C₁₆

C₂×D₄₃

yes

—

—

—

—

C₄₃

#12838

yes

characteristic, complemented, Sylow

C₂³×C₄

D₁₇₂

yes

characteristic

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

Dic₈₆

yes

—

103

C₂²×C₈

D₁₇₂

yes

characteristic

104

C₂²×C₈

C₂×Dic₄₃

yes

—

105

C₂²×C₈

C₂²×D₄₃

yes

characteristic

106

M₃₂

—

107

C₂×C₁₆

—

108

C₂×C₁₆

C₂×Dic₄₃

yes

—

109

C₂×C₁₆

C₂×Dic₄₃

yes

—

110

C₂×C₁₆

C₂²⋊D₄₃

yes

—

111

#108

—

112

#108

—

113

C₂²⋊C₈

—

172

—

114

C₂⁴

Dic₁₇₂

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

D₃₄₄

yes

—

118

C₂²×C₄

Dic₁₇₂

yes

characteristic

119

C₂²×C₄

C₄⋊Dic₄₃

yes

—

120

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

121

M₁₆

—

172

—

122

M₁₆

—

172

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

172

—

128

C₂×C₈

C₄⋊Dic₄₃

yes

—

129

C₂×C₈

C₂×Dic₈₆

yes

characteristic

130

C₂×C₈

#10621898

yes

—

131

C₂×C₈

C₂²×Dic₄₃

yes

characteristic

132

C₂×C₈

C₂×C₂²⋊D₄₃

yes

—

133

C₁₆

—

134

C₁₆

C₂²⋊Dic₄₃

yes

—

135

C₂³

—

136

C₂³

—

137

C₂³

#13063259

yes

—

138

C₂³

C₂×Dic₁₇₂

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

#13063259

yes

—

144

C₂×C₄

C₂×C₄⋊Dic₄₃

yes

characteristic

145

C₂×C₄

#13063299

yes

—

146

C₂×C₄

C₂×D₃₄₄

yes

characteristic

147

C₂×C₄

#13063347

yes

—

148

C₈

—

149

C₈

—

344

—

150

C₈

C₂×C₂²⋊Dic₄₃

yes

characteristic

151

C₈

#13063284

yes

characteristic

152

C₈

#13063304

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×#13063259

yes

characteristic

158

C₂²

#14583140

yes

—

159

C₄

—

160

C₄

#14582759

yes

characteristic

161

C₄

#14582937

yes

characteristic

162

C₄

#14583135

yes

—

163

C₂

—

164

C₂

—

165

C₂

#19134624

yes

characteristic

166

C₂

#19136390

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂²×M₃₂×C₄₃	C₂
socle	C₂²×C₄₃	C₂×#107
prime radical	C₂⁴×C₄₃	Q₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂²×M₃₂	D₄₃
hypocenter	C₄₃	#12838
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

subgroup

G/H

center

C₂×C₄

C₂×D₃₄₄

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂²×M₃₂×C₄₃

C₂

socle

C₂²×C₄₃

C₂×#107

prime radical

C₂⁴×C₄₃

Q₁₆

cosocle

C₂²×C₈×C₄₃

C₂³

hypercenter

C₂²×M₃₂

D₄₃

hypocenter

C₄₃

#12838

LCS

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

N	H ≅ G/N	type
1. C₂×#14582729	C₂	semidirect
3. #19132689	C₂	semidirect
97. C₄₃	#12838	semidirect

H ≅ G/N

type

1. C₂×#14582729

C₂

semidirect

3. #19132689

C₂

semidirect

97. C₄₃

#12838

semidirect

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

subgroup	index
C₂×#14582729	2
#19134633	2
#19132689	2
C₂²×M₃₂×C₄₃	2
#12838	43

subgroup

index

C₂²×M₃₂×C₄₃

Subgroups — #25012782

Conjugacy classes of subgroups

Complements

Maximal subgroups