Group #19135388 — subgroups

Subgroups — #19135388

Order 5504.

396 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₄₃)⋊C₈	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₇₂	C₄	1	1	yes	characteristic
6	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
7	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
8	C₄×D₁₇₂	—	2	2	no	—
9	C₄×D₁₇₂	—	2	2	no	—
10	C₂×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
11	C₂×(C₂×D₄₃)⋊C₄	C₄	1	2	yes	—
12	(C₂×D₄₃)⋊C₈	—	2	8	no	—
13	C₄×C₄₃⋊C₈	—	2	2	no	—
14	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₃	—	2	2	no	—
17	C₂³×D₄₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₄₃	C₂×C₄	1	2	yes	—
22	C₂×D₁₇₂	—	2	2	no	—
23	C₂×D₁₇₂	—	2	2	no	—
24	C₂×D₁₇₂	C₈	1	4	yes	complemented
25	C₂×C₄×D₄₃	—	2	4	no	—
26	C₂×C₄×D₄₃	—	2	2	no	—
27	C₂×C₄×D₄₃	—	2	2	no	—
28	C₂×C₄×D₄₃	—	2	4	no	—
29	C₂×C₈×C₄₃	—	2	2	no	—
30	C₂×C₈×C₄₃	—	2	4	no	—
31	C₄²×C₄₃	D₄	1	2	yes	—
32	C₄²×C₄₃	D₄	1	2	yes	—
33	(C₂×D₄₃)⋊C₄	—	2	4	no	—
34	C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄₃⋊C₈	—	2	2	no	—
36	C₂×C₄₃⋊C₈	—	2	2	no	—
37	C₂×C₄₃⋊C₈	D₄	1	4	yes	complemented
38	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×D₄₃	—	2	2	no	—
40	C₂²×D₄₃	—	2	4	no	—
41	C₂²×D₄₃	—	2	2	no	—
42	C₂²×D₄₃	—	2	4	no	—
43	C₂²×D₄₃	C₂×C₈	1	4	yes	—
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
48	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	#41	1	2	yes	—
53	C₂×Dic₄₃	—	2	2	no	—
54	C₂×Dic₄₃	—	2	4	no	—
55	D₁₇₂	—	2	8	no	—
56	C₄×D₄₃	—	4	4	no	—
57	C₄×D₄₃	—	4	2	no	—
58	C₄×D₄₃	—	4	2	no	—
59	C₈×C₄₃	—	4	2	no	—
60	C₄₃⋊C₈	—	2	4	no	—
61	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄₃	C₄×D₄	1	2	yes	—
63	C₂²×C₄₃	C₂²×C₈	1	1	yes	characteristic
64	C₂²×C₄₃	C₂×M₁₆	1	1	yes	characteristic
65	C₂²×C₄₃	#131	1	2	yes	—
66	C₂×D₄₃	—	2	4	no	—
67	C₂×D₄₃	—	2	8	no	—
68	C₂×D₄₃	—	4	2	no	—
69	C₂×D₄₃	—	4	2	no	—
70	C₂×D₄₃	—	4	4	no	—
71	C₄×C₄₃	—	2	2	no	—
72	C₄×C₄₃	C₂²⋊C₈	1	4	yes	complemented
73	C₄×C₄₃	#121	1	2	yes	—
74	C₄×C₄₃	#124	1	1	yes	characteristic
75	C₄×C₄₃	#127	1	1	yes	characteristic
76	Dic₄₃	—	4	2	no	—
77	#1954	—	43	1	no	maximal, Sylow
78	C₂×C₄₃	#390	1	1	yes	characteristic
79	C₂×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄₃	#408	1	1	yes	characteristic
81	C₂×C₄₃	C₈×D₄	1	2	yes	—
82	C₂×C₄₃	#436	1	2	yes	—
83	D₄₃	—	4	4	no	—
84	C₂×C₄×D₄	—	43	1	no	—
85	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic, complemented
86	C₂×C₂²⋊C₈	—	43	4	no	—
87	C₂×C₄×C₈	—	43	1	no	—
88	C₄₃	#1954	1	1	yes	characteristic, complemented, Sylow
89	C₂²×D₄	—	43	1	no	—
90	C₂³×C₄	—	43	2	no	—
91	C₂²×C₈	—	43	2	no	—
92	C₂²×C₈	C₂×D₄₃	1	2	yes	—
93	C₄×D₄	—	86	2	no	—
94	C₄×D₄	—	86	2	no	—
95	C₂×C₄⋊C₄	—	43	1	no	—
96	C₂×C₂²⋊C₄	—	43	2	no	—
97	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
98	C₄⋊C₈	—	2	2	no	—
99	C₂²⋊C₈	—	86	8	no	—
100	C₄×C₈	—	86	2	no	—
101	C₂⁴	—	43	2	no	—
102	C₂×D₄	—	43	4	no	—
103	C₂×D₄	—	86	2	no	—
104	C₂×D₄	—	86	2	no	—
105	C₂²×C₄	—	43	2	no	—
106	C₂²×C₄	—	86	2	no	—
107	C₂²×C₄	—	86	4	no	—
108	C₂²×C₄	—	86	4	no	—
109	C₂²×C₄	—	86	2	no	—
110	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
111	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	43	4	no	—
116	C₂×C₈	—	86	2	no	—
117	C₂×C₈	—	86	2	no	—
118	C₄⋊C₄	—	86	2	no	—
119	C₂²⋊C₄	—	86	4	no	—
120	C₄²	D₁₇₂	1	2	yes	—
121	C₄²	C₂²⋊D₄₃	1	2	yes	—
122	C₂³	—	43	4	no	—
123	C₂³	—	86	2	no	—
124	C₂³	—	86	4	no	—
125	C₂³	—	86	2	no	—
126	C₂³	—	86	4	no	—
127	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
128	D₄	—	86	8	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	86	2	no	—
131	C₂×C₄	—	86	4	no	—
132	C₂×C₄	—	172	4	no	—
133	C₂×C₄	—	172	2	no	—
134	C₂×C₄	—	172	2	no	—
135	C₂×C₄	C₈×D₄₃	1	2	yes	—
136	C₂×C₄	#10621872	1	2	yes	—
137	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
138	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
139	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
140	C₂×C₄	D₄×D₄₃	1	2	yes	—
141	C₂×C₄	#10621902	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
143	C₈	—	4	2	no	—
144	C₈	—	86	4	no	—
145	C₂²	—	86	4	no	—
146	C₂²	—	86	8	no	—
147	C₂²	—	172	2	no	—
148	C₂²	—	172	2	no	—
149	C₂²	—	172	4	no	—
150	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
151	C₂²	#13063287	1	2	yes	—
152	C₂²	C₂×C₈×D₄₃	1	1	yes	characteristic
153	C₂²	C₂×#10621872	1	1	yes	characteristic
154	C₂²	#13063333	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	172	2	no	—
157	C₄	(C₂×D₄₃)⋊C₈	1	4	yes	complemented
158	C₄	#13063300	1	2	yes	—
159	C₄	#13063314	1	1	yes	characteristic
160	C₄	#13063321	1	1	yes	characteristic
161	C₂	—	172	4	no	—
162	C₂	#14582800	1	1	yes	characteristic
163	C₂	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
164	C₂	#14582867	1	1	yes	characteristic
165	C₂	#14582964	1	2	yes	—
166	C₂	#14582976	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₁₇₂

C₂

yes

maximal, characteristic

C₂×(C₂×D₄₃)⋊C₈

C₂

yes

maximal, complemented

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₄

yes

characteristic

C₂²×C₄×D₄₃

C₂²

yes

—

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₄×D₁₇₂

—

C₄×D₁₇₂

—

C₂×C₄⋊Dic₄₃

C₄

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

C₄

yes

—

(C₂×D₄₃)⋊C₈

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂³×D₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂×C₄

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₈

yes

complemented

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

D₄

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

complemented

C₂³×C₄₃

C₂²×C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

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

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₄×D₄

yes

—

C₂²×C₄₃

C₂²×C₈

yes

characteristic

C₂²×C₄₃

C₂×M₁₆

yes

characteristic

C₂²×C₄₃

#131

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊C₈

yes

complemented

C₄×C₄₃

#121

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₄₃

#408

yes

characteristic

C₂×C₄₃

C₈×D₄

yes

—

C₂×C₄₃

#436

yes

—

D₄₃

—

C₂×C₄×D₄

—

C₂×C₄⋊C₈

D₄₃

yes

characteristic, complemented

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

C₄₃

#1954

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₃

yes

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

100

C₄×C₈

—

101

C₂⁴

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₄₃

yes

characteristic

111

C₂²×C₄

C₄×D₄₃

yes

characteristic

112

C₂²×C₄

C₂²×D₄₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

D₁₇₂

yes

—

121

C₄²

C₂²⋊D₄₃

yes

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂×C₄×D₄₃

yes

characteristic

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

172

—

133

C₂×C₄

—

172

—

134

C₂×C₄

—

172

—

135

C₂×C₄

C₈×D₄₃

yes

—

136

C₂×C₄

#10621872

yes

—

137

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

138

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

139

C₂×C₄

C₂×D₁₇₂

yes

characteristic

140

C₂×C₄

D₄×D₄₃

yes

—

141

C₂×C₄

#10621902

yes

—

142

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

172

—

148

C₂²

—

172

—

149

C₂²

—

172

—

150

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

151

C₂²

#13063287

yes

—

152

C₂²

C₂×C₈×D₄₃

yes

characteristic

153

C₂²

C₂×#10621872

yes

characteristic

154

C₂²

#13063333

yes

—

155

C₄

—

156

C₄

—

172

—

157

C₄

(C₂×D₄₃)⋊C₈

yes

complemented

158

C₄

#13063300

yes

—

159

C₄

#13063314

yes

characteristic

160

C₄

#13063321

yes

characteristic

161

C₂

—

172

—

162

C₂

#14582800

yes

characteristic

163

C₂

C₂×(C₂×D₄₃)⋊C₈

yes

characteristic

164

C₂

#14582867

yes

characteristic

165

C₂

#14582964

yes

—

166

C₂

#14582976

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#1954
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₃

commutator

C₂²×C₄₃

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. C₂×(C₂×D₄₃)⋊C₈	C₂	semidirect
3. C₂×C₄×C₄₃⋊C₈	C₂	semidirect
4. C₂×C₄⋊C₈×C₄₃	C₂	semidirect
24. C₂×D₁₇₂	C₈	semidirect
37. C₂×C₄₃⋊C₈	D₄	semidirect
72. C₄×C₄₃	C₂²⋊C₈	semidirect
85. C₂×C₄⋊C₈	D₄₃	semidirect
88. C₄₃	#1954	semidirect
157. C₄	(C₂×D₄₃)⋊C₈	semidirect

H ≅ G/N

type

2. C₂×(C₂×D₄₃)⋊C₈

C₂

semidirect

3. C₂×C₄×C₄₃⋊C₈

C₂

semidirect

4. C₂×C₄⋊C₈×C₄₃

C₂

semidirect

24. C₂×D₁₇₂

C₈

semidirect

37. C₂×C₄₃⋊C₈

D₄

semidirect

72. C₄×C₄₃

C₂²⋊C₈

semidirect

85. C₂×C₄⋊C₈

D₄₃

semidirect

88. C₄₃

#1954

semidirect

157. C₄

(C₂×D₄₃)⋊C₈

semidirect

Subgroups — #19135388

Conjugacy classes of subgroups

Complements