Group #19135819 — subgroups

Subgroups — #19135819

Order 5504.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14583480	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063253	C₂	1	2	yes	maximal, complemented
3	#14582898	C₂	1	2	yes	maximal, complemented
4	#14582668	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10621898	C₂²	1	2	yes	—
7	C₂×#10621877	C₂²	1	2	yes	—
8	#13063305	—	2	2	no	—
9	#13063300	—	2	2	no	—
10	#13063299	—	2	2	no	—
11	#13063253	—	2	4	no	—
12	#13063248	—	2	2	no	—
13	#13063245	—	2	2	no	—
14	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
15	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₄₃	—	2	1	no	—
19	C₂²×C₄×C₄₃	—	2	1	no	—
20	C₂²×C₄×C₄₃	D₄	1	2	yes	—
21	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₄₃	—	2	2	no	—
23	#10621898	—	2	2	no	—
24	#10621898	—	2	2	no	—
25	#10621898	—	2	2	no	—
26	#10621898	—	2	2	no	—
27	C₂×D₁₇₂	—	2	2	no	—
28	C₂×D₁₇₂	D₄	1	2	yes	—
29	C₂×C₄×D₄₃	—	2	2	no	—
30	C₂×Dic₈₆	D₄	1	2	yes	—
31	C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₄⋊C₄×C₄₃	—	2	2	no	—
33	C₄⋊C₄×C₄₃	—	4	4	no	—
34	C₄²×C₄₃	—	2	1	no	—
35	C₄²×C₄₃	D₄	1	2	yes	complemented
36	(C₂×D₄₃)⋊C₄	—	2	4	no	—
37	C₄⋊Dic₄₃	—	2	2	no	—
38	#10621877	—	2	2	no	—
39	#10621877	—	2	2	no	—
40	C₂×C₄₃⋊C₈	—	2	2	no	—
41	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	4	2	no	—
48	C₂×C₄×C₄₃	—	4	2	no	—
49	C₂×C₄×C₄₃	—	4	1	no	—
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
53	C₂×C₄×C₄₃	#41	1	2	yes	—
54	C₂²⋊D₄₃	—	4	2	no	—
55	C₂²⋊D₄₃	—	4	2	no	—
56	C₂×Dic₄₃	—	2	2	no	—
57	D₁₇₂	—	2	2	no	—
58	D₁₇₂	—	2	2	no	—
59	D₁₇₂	—	8	2	no	—
60	C₄×D₄₃	—	4	2	no	—
61	C₄×D₄₃	—	4	2	no	—
62	Dic₈₆	—	2	2	no	—
63	Dic₈₆	—	2	2	no	—
64	C₄₃⋊C₈	—	4	2	no	—
65	C₂²×C₄₃	—	2	1	no	—
66	C₂²×C₄₃	—	2	1	no	—
67	C₂²×C₄₃	#120	1	1	yes	characteristic
68	C₂²×C₄₃	#121	1	1	yes	characteristic
69	C₂²×C₄₃	#124	1	1	yes	characteristic
70	C₂×D₄₃	—	4	2	no	—
71	C₂×D₄₃	—	4	2	no	—
72	C₂×D₄₃	—	4	2	no	—
73	C₄×C₄₃	—	2	4	no	—
74	C₄×C₄₃	—	4	2	no	—
75	C₄×C₄₃	—	8	1	no	—
76	C₄×C₄₃	#121	1	2	yes	—
77	C₄×C₄₃	#123	1	2	yes	—
78	Dic₄₃	—	4	2	no	—
79	#2039	—	43	1	no	maximal, Sylow
80	C₂×C₄₃	—	2	1	no	—
81	C₂×C₄₃	—	2	1	no	—
82	C₂×C₄₃	#394	1	1	yes	characteristic
83	C₂×C₄₃	#453	1	1	yes	characteristic
84	C₂×C₄₃	#456	1	1	yes	characteristic
85	D₄₃	—	8	2	no	—
86	#537	—	43	1	no	—
87	C₂×#104	—	43	2	no	—
88	#417	—	43	2	no	—
89	#384	D₄₃	1	1	yes	characteristic, complemented
90	#343	—	43	1	no	—
91	C₄₃	#2039	1	1	yes	characteristic, complemented, Sylow
92	C₂×#41	—	43	2	no	—
93	C₂×M₁₆	—	43	2	no	—
94	C₂²⋊Q₈	—	86	2	no	—
95	#121	—	86	2	no	—
96	#121	—	86	2	no	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
101	#104	—	86	4	no	—
102	#103	—	86	2	no	—
103	#102	—	86	2	no	—
104	#41	—	86	2	no	—
105	#41	—	86	2	no	—
106	#41	—	86	2	no	—
107	#41	—	86	2	no	—
108	C₂×Q₈	—	43	2	no	—
109	C₂×D₄	—	43	2	no	—
110	C₂×D₄	—	86	2	no	—
111	C₂×D₄	—	86	2	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	86	2	no	—
115	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
116	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
117	M₁₆	—	86	2	no	—
118	M₁₆	—	86	2	no	—
119	C₂×C₈	—	86	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄⋊C₄	—	86	2	no	—
124	C₂²⋊C₄	—	86	4	no	—
125	C₄²	—	2	1	no	—
126	C₄²	D₁₇₂	1	2	yes	complemented
127	C₂³	—	86	2	no	—
128	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
129	Q₈	—	86	2	no	—
130	Q₈	—	86	2	no	—
131	D₄	—	86	2	no	—
132	D₄	—	86	2	no	—
133	D₄	—	344	2	no	—
134	D₄	—	172	2	no	—
135	D₄	—	172	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	86	2	no	—
144	C₂×C₄	—	172	2	no	—
145	C₂×C₄	—	172	2	no	—
146	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
147	C₂×C₄	#10621898	1	1	yes	characteristic
148	C₂×C₄	D₄×D₄₃	1	2	yes	—
149	C₂×C₄	#10621902	1	2	yes	—
150	C₈	—	172	2	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	172	2	no	—
154	C₂²	—	172	2	no	—
155	C₂²	—	172	2	no	—
156	C₂²	#13063293	1	1	yes	characteristic
157	C₂²	#13063299	1	1	yes	characteristic
158	C₂²	#13063314	1	1	yes	characteristic
159	C₄	—	2	4	no	—
160	C₄	—	4	2	no	—
161	C₄	—	8	1	no	—
162	C₄	—	172	2	no	—
163	C₄	#13063300	1	2	yes	—
164	C₄	#13063310	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	344	2	no	—
168	C₂	#14582816	1	1	yes	characteristic
169	C₂	#14583066	1	1	yes	characteristic
170	C₂	#14583081	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583480

C₂

yes

maximal, characteristic

C₂×#13063253

C₂

yes

maximal, complemented

#14582898

C₂

yes

maximal, complemented

#14582668

C₂

yes

maximal, characteristic, complemented

C₄₃×#384

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₂²

yes

—

C₂×#10621877

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

—

—

—

—

—

C₂×D₁₇₂

—

C₂×D₁₇₂

D₄

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₄

yes

complemented

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×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

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂²⋊D₄₃

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#121

yes

—

C₄×C₄₃

#123

yes

—

Dic₄₃

—

#2039

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

—

—

—

D₄₃

yes

characteristic, complemented

#343

—

C₄₃

#2039

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂²⋊Q₈

—

#121

—

#121

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

100

C₂×C₄²

C₂×D₄₃

yes

characteristic

101

#104

—

102

#103

—

103

#102

—

104

#41

—

105

#41

—

106

#41

—

107

#41

—

108

C₂×Q₈

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂²⋊D₄₃

yes

—

116

C₂²×C₄

C₂²×D₄₃

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

D₁₇₂

yes

complemented

127

C₂³

—

128

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

344

—

134

D₄

—

172

—

135

D₄

—

172

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

172

—

145

C₂×C₄

—

172

—

146

C₂×C₄

C₂×D₁₇₂

yes

characteristic

147

C₂×C₄

#10621898

yes

characteristic

148

C₂×C₄

D₄×D₄₃

yes

—

149

C₂×C₄

#10621902

yes

—

150

C₈

—

172

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

172

—

154

C₂²

—

172

—

155

C₂²

—

172

—

156

C₂²

#13063293

yes

characteristic

157

C₂²

#13063299

yes

characteristic

158

C₂²

#13063314

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

172

—

163

C₄

#13063300

yes

—

164

C₄

#13063310

yes

—

165

C₂

—

166

C₂

—

167

C₂

—

344

—

168

C₂

#14582816

yes

characteristic

169

C₂

#14583066

yes

characteristic

170

C₂

#14583081

yes

characteristic

	subgroup	G/H
center	C₂²	#13063293
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#384	C₂
socle	C₂²×C₄₃	#120
hypercenter	#384	D₄₃
hypocenter	C₄₃	#2039
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #384

subgroup

G/H

center

C₂²

#13063293

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#384

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #384

N	H ≅ G/N	type
2. C₂×#13063253	C₂	semidirect
3. #14582898	C₂	semidirect
4. #14582668	C₂	semidirect
5. C₄₃×#384	C₂	semidirect
35. C₄²×C₄₃	D₄	semidirect
89. #384	D₄₃	semidirect
91. C₄₃	#2039	semidirect
126. C₄²	D₁₇₂	semidirect

H ≅ G/N

type

2. C₂×#13063253

C₂

semidirect

3. #14582898

C₂

semidirect

4. #14582668

C₂

semidirect

5. C₄₃×#384

C₂

semidirect

35. C₄²×C₄₃

D₄

semidirect

89. #384

D₄₃

semidirect

91. C₄₃

#2039

semidirect

126. C₄²

D₁₇₂

semidirect

Subgroups — #19135819

Conjugacy classes of subgroups

Complements