Group #19135142 — subgroups

Subgroups — #19135142

Order 5504.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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	#14583463	C₂	1	1	yes	maximal, characteristic
2	#14582883	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13063241	C₂	1	2	yes	maximal, complemented
4	#14582671	C₂	1	2	yes	maximal, complemented
5	C₄₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
7	C₂×#10621877	C₂²	1	2	yes	—
8	#13063321	—	2	1	no	—
9	#13063314	—	2	1	no	—
10	#13063299	—	2	2	no	—
11	#13063293	C₄	1	4	yes	—
12	#13063284	C₂²	1	1	yes	characteristic
13	#13063244	—	2	1	no	—
14	#13063242	—	2	1	no	—
15	#13063241	—	2	4	no	—
16	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
17	#131×C₄₃	—	2	4	no	—
18	C₂×M₁₆×C₄₃	—	2	2	no	—
19	C₂×M₁₆×C₄₃	C₂²	1	2	yes	—
20	C₂³×D₄₃	C₂×C₄	1	2	yes	—
21	#41×C₄₃	—	2	2	no	—
22	#41×C₄₃	—	2	2	no	—
23	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
24	C₂×D₄×C₄₃	C₂×C₄	1	2	yes	—
25	C₂×D₄×C₄₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	D₄	1	2	yes	—
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₄₃	—	2	4	no	—
29	C₂×D₁₇₂	—	2	2	no	—
30	C₂×D₁₇₂	—	2	1	no	—
31	C₂×D₁₇₂	—	2	1	no	—
32	C₂×D₁₇₂	—	2	2	no	—
33	M₁₆×C₄₃	—	2	4	no	—
34	M₁₆×C₄₃	—	2	2	no	—
35	M₁₆×C₄₃	D₄	1	4	yes	complemented
36	C₂×C₈×C₄₃	—	2	4	no	—
37	C₂×C₈×C₄₃	—	2	2	no	—
38	C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
39	(C₂×D₄₃)⋊C₄	—	2	4	no	—
40	C₄⋊Dic₄₃	—	2	1	no	—
41	C₄×Dic₄₃	—	2	1	no	—
42	#10621877	—	2	2	no	—
43	#10621877	—	2	2	no	—
44	C₂×C₄₃⋊C₈	—	2	2	no	—
45	C₂³×C₄₃	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₄₃	—	2	2	no	—
48	C₂²×D₄₃	—	2	4	no	—
49	C₂²×D₄₃	—	8	2	no	—
50	Q₈×C₄₃	—	2	2	no	—
51	D₄×C₄₃	—	2	4	no	—
52	D₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#41	1	2	yes	—
60	C₂²⋊D₄₃	—	8	4	no	—
61	C₂×Dic₄₃	—	2	2	no	—
62	D₁₇₂	—	4	1	no	—
63	D₁₇₂	—	4	2	no	—
64	D₁₇₂	—	4	1	no	—
65	C₈×C₄₃	—	2	4	no	—
66	C₄₃⋊C₈	—	4	2	no	—
67	C₂²×C₄₃	—	2	2	no	—
68	C₂²×C₄₃	—	4	1	no	—
69	C₂²×C₄₃	—	4	2	no	—
70	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄₃	C₄×D₄	1	2	yes	—
72	C₂×D₄₃	—	4	4	no	—
73	C₂×D₄₃	—	4	2	no	—
74	C₂×D₄₃	—	8	4	no	—
75	C₄×C₄₃	—	2	2	no	—
76	C₄×C₄₃	#120	1	1	yes	characteristic
77	C₄×C₄₃	#121	1	2	yes	—
78	C₄×C₄₃	#123	1	1	yes	characteristic
79	Dic₄₃	—	4	2	no	—
80	#1914	—	43	1	no	maximal, Sylow
81	C₂×C₄₃	—	2	1	no	—
82	C₂×C₄₃	—	2	2	no	—
83	C₂×C₄₃	—	4	2	no	—
84	C₂×C₄₃	#386	1	1	yes	characteristic
85	D₄₃	—	8	2	no	—
86	#568	D₄₃	1	1	yes	characteristic, complemented
87	#535	—	43	1	no	—
88	#413	—	43	1	no	—
89	C₂×#100	—	43	2	no	—
90	#344	—	43	2	no	—
91	C₄₃	#1914	1	1	yes	characteristic, complemented, Sylow
92	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
93	C₂²×D₄	—	43	1	no	—
94	#131	—	2	4	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	43	2	no	—
97	C₂×M₁₆	C₂×D₄₃	1	2	yes	—
98	#127	—	86	1	no	—
99	#124	—	86	1	no	—
100	#121	—	86	2	no	—
101	#120	—	43	4	no	—
102	#117	—	43	1	no	—
103	#101	—	86	1	no	—
104	#100	—	86	1	no	—
105	#100	—	86	4	no	—
106	C₂⁴	—	43	2	no	—
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	C₂×Q₈	D₁₇₂	1	1	yes	characteristic
110	C₂×D₄	—	86	4	no	—
111	C₂×D₄	—	86	1	no	—
112	C₂×D₄	—	86	2	no	—
113	C₂×D₄	—	86	2	no	—
114	C₂×D₄	—	86	1	no	—
115	C₂×D₄	C₄×D₄₃	1	2	yes	—
116	C₂×D₄	D₁₇₂	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
118	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
119	M₁₆	—	2	4	no	—
120	M₁₆	—	2	2	no	—
121	M₁₆	—	86	2	no	—
122	M₁₆	—	86	2	no	—
123	M₁₆	C₂²⋊D₄₃	1	4	yes	complemented
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	86	2	no	—
127	C₄⋊C₄	—	86	1	no	—
128	C₂²⋊C₄	—	43	2	no	—
129	C₂²⋊C₄	—	86	4	no	—
130	C₄²	—	86	1	no	—
131	C₂³	—	86	4	no	—
132	C₂³	—	86	2	no	—
133	C₂³	—	344	2	no	—
134	C₂³	(C₂×D₄₃)⋊C₄	1	2	yes	—
135	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
136	Q₈	—	2	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	D₄	—	344	4	no	—
140	D₄	—	172	1	no	—
141	D₄	—	172	2	no	—
142	D₄	—	172	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	86	2	no	—
146	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
147	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
148	C₂×C₄	#10621898	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₈	—	172	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	344	4	no	—
157	C₂²	—	172	4	no	—
158	C₂²	—	172	2	no	—
159	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
160	C₂²	C₄×C₂²⋊D₄₃	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	—	172	2	no	—
163	C₄	#13063294	1	1	yes	characteristic
164	C₄	#13063299	1	2	yes	—
165	C₄	#13063308	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	—	4	2	no	—
169	C₂	—	344	2	no	—
170	C₂	#14582778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583463

C₂

yes

maximal, characteristic

#14582883

C₂

yes

maximal, characteristic, complemented

C₂×#13063241

C₂

yes

maximal, complemented

#14582671

C₂

yes

maximal, complemented

C₄₃×#568

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

characteristic

C₂×#10621877

C₂²

yes

—

—

—

—

C₄

yes

—

#13063284

C₂²

yes

—

—

—

C₂×#41×C₄₃

C₂²

yes

characteristic

#131×C₄₃

—

C₂×M₁₆×C₄₃

—

C₂×M₁₆×C₄₃

C₂²

yes

—

C₂³×D₄₃

C₂×C₄

yes

—

#41×C₄₃

—

#41×C₄₃

—

C₂×Q₈×C₄₃

D₄

yes

characteristic

C₂×D₄×C₄₃

C₂×C₄

yes

—

C₂×D₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

M₁₆×C₄₃

D₄

yes

complemented

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂²⋊Dic₄₃

C₂×C₄

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₄×D₄

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#120

yes

characteristic

C₄×C₄₃

#121

yes

—

C₄×C₄₃

#123

yes

characteristic

Dic₄₃

—

#1914

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#386

yes

characteristic

D₄₃

—

#568

D₄₃

yes

characteristic, complemented

—

—

—

—

C₄₃

#1914

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₃

yes

characteristic

C₂²×D₄

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₃

yes

—

#127

—

#124

—

100

#121

—

101

#120

—

102

#117

—

103

#101

—

104

#100

—

105

#100

—

106

C₂⁴

—

107

#41

—

108

#41

—

109

C₂×Q₈

D₁₇₂

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

C₄×D₄₃

yes

—

116

C₂×D₄

D₁₇₂

yes

characteristic

117

C₂²×C₄

C₂²⋊D₄₃

yes

—

118

C₂²×C₄

C₂²×D₄₃

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

122

M₁₆

—

123

M₁₆

C₂²⋊D₄₃

yes

complemented

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

130

C₄²

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

344

—

134

C₂³

(C₂×D₄₃)⋊C₄

yes

—

135

C₂³

C₂×C₄×D₄₃

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

344

—

140

D₄

—

172

—

141

D₄

—

172

—

142

D₄

—

172

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

147

C₂×C₄

C₂×D₁₇₂

yes

characteristic

148

C₂×C₄

#10621898

yes

—

149

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

150

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

151

C₈

—

152

C₈

—

172

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

344

—

157

C₂²

—

172

—

158

C₂²

—

172

—

159

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

160

C₂²

C₄×C₂²⋊D₄₃

yes

—

161

C₄

—

162

C₄

—

172

—

163

C₄

#13063294

yes

characteristic

164

C₄

#13063299

yes

—

165

C₄

#13063308

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

344

—

170

C₂

#14582778

yes

characteristic

	subgroup	G/H
center	C₂	#14582778
commutator	C₂³×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#568	C₂
socle	C₂×C₄₃	#386
hypercenter	#568	D₄₃
hypocenter	C₄₃	#1914
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#14582778

commutator

C₂³×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#568

C₂

C₂×C₄₃

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
2. #14582883	C₂	semidirect
3. C₂×#13063241	C₂	semidirect
4. #14582671	C₂	semidirect
5. C₄₃×#568	C₂	semidirect
35. M₁₆×C₄₃	D₄	semidirect
86. #568	D₄₃	semidirect
91. C₄₃	#1914	semidirect
123. M₁₆	C₂²⋊D₄₃	semidirect

H ≅ G/N

type

2. #14582883

C₂

semidirect

3. C₂×#13063241

C₂

semidirect

4. #14582671

C₂

semidirect

5. C₄₃×#568

C₂

semidirect

35. M₁₆×C₄₃

D₄

semidirect

86. #568

D₄₃

semidirect

91. C₄₃

#1914

semidirect

123. M₁₆

C₂²⋊D₄₃

semidirect

Subgroups — #19135142

Conjugacy classes of subgroups

Complements