Group #19144246 — subgroups

Subgroups — #19144246

Order 5504.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13063331	C₂	1	1	yes	maximal, characteristic
2	#14583473	C₂	1	2	yes	maximal
3	#14583154	C₂	1	4	yes	maximal, complemented
4	#14583137	C₂	1	4	yes	maximal, complemented
5	#14582937	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582908	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621898	C₂²	1	1	yes	characteristic
9	#13063331	—	2	4	no	—
10	C₂×#10621877	C₂²	1	1	yes	characteristic
11	C₂×#10621872	C₂²	1	2	yes	—
12	C₂×C₈×D₄₃	C₂²	1	2	yes	—
13	#13063323	—	2	2	no	—
14	#13063314	—	2	2	no	—
15	#13063308	—	2	4	no	—
16	#13063301	C₂²	1	8	yes	—
17	#13063284	C₂²	1	2	yes	—
18	#13063259	C₂²	1	2	yes	—
19	#13063257	C₂²	1	2	yes	—
20	#13063249	C₂²	1	8	yes	—
21	C₂²×Q₈×C₄₃	C₂²	1	2	yes	—
22	C₂×Q₁₆×C₄₃	—	2	4	no	—
23	C₂×Q₁₆×C₄₃	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₄₃	C₂²	1	2	yes	—
25	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₄₃	—	2	2	no	—
27	C₂×Q₈×C₄₃	—	2	4	no	—
28	C₂×Q₈×C₄₃	—	2	4	no	—
29	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₄₃	—	2	2	no	—
31	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×C₂²⋊D₄₃	D₄	1	2	yes	—
33	#10621898	—	2	2	no	—
34	#10621898	—	2	2	no	—
35	C₂×D₁₇₂	D₄	1	1	yes	characteristic
36	C₂×C₄×D₄₃	C₂³	1	2	yes	—
37	C₂×Dic₈₆	D₄	1	1	yes	characteristic
38	Q₁₆×C₄₃	—	2	4	no	—
39	Q₁₆×C₄₃	—	2	4	no	—
40	C₂×C₈×C₄₃	D₄	1	4	yes	—
41	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₄₃	—	2	2	no	—
44	(C₂×D₄₃)⋊C₄	—	2	8	no	—
45	C₄⋊Dic₄₃	C₂³	1	4	yes	—
46	Dic₄₃⋊C₄	—	2	8	no	—
47	C₄×Dic₄₃	—	2	2	no	—
48	#10621877	—	2	2	no	—
49	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
50	#10621872	—	2	4	no	—
51	C₈×D₄₃	—	2	4	no	—
52	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×D₄₃	C₂×D₄	1	2	yes	—
54	Q₈×C₄₃	—	2	4	no	—
55	Q₈×C₄₃	—	2	4	no	—
56	Q₈×C₄₃	—	4	4	no	—
57	C₂×C₄×C₄₃	—	2	4	no	—
58	C₂×C₄×C₄₃	—	4	4	no	—
59	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₃	#41	1	2	yes	—
62	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
63	C₂²⋊D₄₃	—	2	4	no	—
64	C₂×Dic₄₃	—	2	4	no	—
65	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
66	D₁₇₂	—	2	2	no	—
67	C₄×D₄₃	—	2	2	no	—
68	C₄×D₄₃	—	2	2	no	—
69	Dic₈₆	—	2	2	no	—
70	C₈×C₄₃	C₂×D₄	1	2	yes	—
71	C₈×C₄₃	C₂×D₄	1	2	yes	—
72	C₄₃⋊C₈	—	2	2	no	—
73	C₂²×C₄₃	—	2	2	no	—
74	C₂²×C₄₃	#121	1	2	yes	—
75	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×D₄₃	—	2	2	no	—
77	C₂×D₄₃	—	2	4	no	—
78	C₄×C₄₃	—	4	4	no	—
79	C₄×C₄₃	#121	1	2	yes	—
80	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
81	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
82	Dic₄₃	—	2	2	no	—
83	Dic₄₃	—	4	4	no	—
84	#3059	—	43	1	no	maximal, Sylow
85	C₂×C₄₃	—	2	2	no	—
86	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
87	C₂×C₄₃	#578	1	2	yes	—
88	D₄₃	—	4	2	no	—
89	C₂²×Q₁₆	D₄₃	1	1	yes	characteristic, complemented
90	C₂×#131	—	43	1	no	—
91	#536	—	43	2	no	—
92	#470	—	43	4	no	—
93	#467	—	43	4	no	—
94	#427	—	43	1	no	—
95	#419	—	43	2	no	—
96	C₄₃	#3059	1	1	yes	characteristic, complemented, Sylow
97	C₂×#41	—	43	1	no	—
98	C₂²×Q₈	C₂×D₄₃	1	2	yes	—
99	C₂×Q₁₆	—	2	4	no	—
100	C₂×Q₁₆	C₂×D₄₃	1	2	yes	—
101	C₂×Q₁₆	C₂×D₄₃	1	2	yes	—
102	#131	—	86	4	no	—
103	C₂×M₁₆	—	43	1	no	—
104	C₂×M₁₆	—	43	2	no	—
105	C₂²×C₈	—	43	2	no	—
106	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
107	C₄⋊Q₈	—	86	2	no	—
108	#124	—	86	2	no	—
109	#123	—	86	4	no	—
110	C₂²⋊Q₈	—	43	8	no	—
111	#117	—	43	2	no	—
112	#107	—	43	2	no	—
113	#106	—	43	2	no	—
114	#103	—	43	8	no	—
115	#41	—	86	2	no	—
116	#41	—	86	2	no	—
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	43	1	no	—
121	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
122	C₂×D₄	—	43	2	no	—
123	C₂×D₄	—	43	1	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	43	2	no	—
126	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
127	Q₁₆	—	2	4	no	—
128	Q₁₆	—	2	4	no	—
129	M₁₆	—	86	2	no	—
130	M₁₆	—	86	4	no	—
131	C₂×C₈	—	43	2	no	—
132	C₂×C₈	—	86	4	no	—
133	C₂×C₈	C₂²⋊D₄₃	1	4	yes	—
134	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
136	C₄⋊C₄	—	43	4	no	—
137	C₄⋊C₄	—	86	8	no	—
138	C₂²⋊C₄	—	86	8	no	—
139	C₂²⋊C₄	—	86	2	no	—
140	C₄²	—	86	2	no	—
141	C₂³	—	43	2	no	—
142	C₂³	D₄×D₄₃	1	1	yes	characteristic
143	Q₈	—	2	4	no	—
144	Q₈	—	2	4	no	—
145	Q₈	—	4	4	no	—
146	Q₈	—	86	2	no	—
147	D₄	—	86	4	no	—
148	D₄	—	86	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	—	43	2	no	—
152	C₂×C₄	—	86	2	no	—
153	C₂×C₄	—	86	2	no	—
154	C₂×C₄	—	86	4	no	—
155	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
156	C₂×C₄	#10621900	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
158	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
159	C₈	—	86	2	no	—
160	C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
161	C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	86	2	no	—
164	C₂²	—	86	4	no	—
165	C₂²	#13063296	1	2	yes	—
166	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
167	C₄	—	4	4	no	—
168	C₄	—	86	2	no	—
169	C₄	—	172	4	no	—
170	C₄	#13063296	1	2	yes	—
171	C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
172	C₄	C₂×#10621900	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	172	2	no	—
175	C₂	C₂×#13063296	1	1	yes	characteristic
176	C₂	#14583712	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063331

C₂

yes

maximal, characteristic

#14583473

C₂

yes

maximal

#14583154

C₂

yes

maximal, complemented

#14583137

C₂

yes

maximal, complemented

#14582937

C₂

yes

maximal, characteristic, complemented

#14582908

C₂

yes

maximal, complemented

C₂²×Q₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₂²

yes

characteristic

#13063331

—

C₂×#10621877

C₂²

yes

characteristic

C₂×#10621872

C₂²

yes

—

C₂×C₈×D₄₃

C₂²

yes

—

—

—

—

C₂²

yes

—

#13063284

C₂²

yes

—

#13063259

C₂²

yes

—

#13063257

C₂²

yes

—

#13063249

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

C₂²

yes

—

C₂×Q₁₆×C₄₃

C₂²

yes

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

D₄

yes

—

#10621898

—

#10621898

—

C₂×D₁₇₂

D₄

yes

characteristic

C₂×C₄×D₄₃

C₂³

yes

—

C₂×Dic₈₆

D₄

yes

characteristic

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₂²⋊Dic₄₃

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

C₂³

yes

—

#10621872

—

C₈×D₄₃

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

C₂×D₄

yes

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

C₂×D₄

yes

—

C₈×C₄₃

C₂×D₄

yes

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

#121

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#121

yes

—

C₄×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

C₂×#41

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#3059

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂×#121

yes

characteristic

C₂×C₄₃

#578

yes

—

D₄₃

—

C₂²×Q₁₆

D₄₃

yes

characteristic, complemented

—

—

—

—

—

—

C₄₃

#3059

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

C₂×D₄₃

yes

—

C₂×Q₁₆

—

100

C₂×Q₁₆

C₂×D₄₃

yes

—

101

C₂×Q₁₆

C₂×D₄₃

yes

—

102

#131

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×D₄₃

yes

characteristic

107

C₄⋊Q₈

—

108

#124

—

109

#123

—

110

C₂²⋊Q₈

—

111

#117

—

112

#107

—

113

#106

—

114

#103

—

115

#41

—

116

#41

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

C₂²×D₄₃

yes

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²×D₄₃

yes

characteristic

127

Q₁₆

—

128

Q₁₆

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

C₂²⋊D₄₃

yes

—

134

C₂×C₈

C₂²×D₄₃

yes

characteristic

135

C₂×C₈

C₂²×D₄₃

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

—

142

C₂³

D₄×D₄₃

yes

characteristic

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

D₄×D₄₃

yes

characteristic

156

C₂×C₄

#10621900

yes

—

157

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

158

C₂×C₄

C₂³×D₄₃

yes

characteristic

159

C₈

—

160

C₈

C₂×C₂²⋊D₄₃

yes

—

161

C₈

C₂×C₂²⋊D₄₃

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#13063296

yes

—

166

C₂²

C₂×D₄×D₄₃

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

172

—

170

C₄

#13063296

yes

—

171

C₄

C₂²×C₂²⋊D₄₃

yes

characteristic

172

C₄

C₂×#10621900

yes

characteristic

173

C₂

—

174

C₂

—

172

—

175

C₂

C₂×#13063296

yes

characteristic

176

C₂

#14583712

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂²×Q₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₂²×Q₁₆	D₄₃
hypocenter	C₄₃	#3059
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₂²×Q₁₆×C₄₃

C₂

socle

C₂²×C₄₃

C₂²×D₄

hypercenter

C₂²×Q₁₆

D₄₃

hypocenter

C₄₃

#3059

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
3. #14583154	C₂	semidirect
4. #14583137	C₂	semidirect
5. #14582937	C₂	semidirect
6. #14582908	C₂	semidirect
7. C₂²×Q₁₆×C₄₃	C₂	semidirect
89. C₂²×Q₁₆	D₄₃	semidirect
96. C₄₃	#3059	semidirect

H ≅ G/N

type

3. #14583154

C₂

semidirect

4. #14583137

C₂

semidirect

5. #14582937

C₂

semidirect

6. #14582908

C₂

semidirect

7. C₂²×Q₁₆×C₄₃

C₂

semidirect

89. C₂²×Q₁₆

D₄₃

semidirect

96. C₄₃

#3059

semidirect

Subgroups — #19144246

Conjugacy classes of subgroups

Complements