Group #19132947 — subgroups

Subgroups — #19132947

Order 5504.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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	#14582956	C₂	1	1	yes	maximal, characteristic, complemented
2	#14582952	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582623	C₂	1	4	yes	maximal, complemented
4	C₂×#124×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
6	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
9	#13063236	—	2	1	no	—
10	C₄×C₄₃⋊C₈	—	2	1	no	—
11	C₂²×Q₈×C₄₃	C₄	1	1	yes	characteristic
12	C₂²×D₄×C₄₃	C₄	1	1	yes	characteristic
13	#124×C₄₃	—	2	1	no	—
14	#124×C₄₃	—	2	1	no	—
15	#124×C₄₃	—	2	1	no	—
16	#124×C₄₃	—	2	1	no	—
17	C₂×C₂²⋊C₄×C₄₃	C₄	1	2	yes	—
18	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄₃	C₂×C₄	1	2	yes	—
21	C₂×Q₈×C₄₃	—	2	1	no	—
22	C₂×Q₈×C₄₃	—	2	1	no	—
23	C₂×Q₈×C₄₃	—	4	2	no	—
24	C₂×D₄×C₄₃	—	2	1	no	—
25	C₂×D₄×C₄₃	—	2	1	no	—
26	C₂×D₄×C₄₃	—	4	2	no	—
27	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂²⋊C₄×C₄₃	—	2	2	no	—
32	C₂²⋊C₄×C₄₃	—	2	2	no	—
33	C₂²⋊C₄×C₄₃	—	2	2	no	—
34	C₂²⋊C₄×C₄₃	—	2	2	no	—
35	C₄²×C₄₃	D₄	1	1	yes	characteristic
36	C₄²×C₄₃	D₄	1	1	yes	characteristic
37	C₄²×C₄₃	D₄	1	1	yes	characteristic
38	C₄²×C₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄₃⋊C₈	—	2	2	no	—
40	C₂×C₄₃⋊C₈	—	2	2	no	—
41	C₂³×C₄₃	—	2	2	no	—
42	C₂³×C₄₃	—	2	4	no	—
43	C₂³×C₄₃	—	4	2	no	—
44	C₂³×C₄₃	—	4	2	no	—
45	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
46	Q₈×C₄₃	—	4	2	no	—
47	Q₈×C₄₃	—	4	2	no	—
48	D₄×C₄₃	—	4	2	no	—
49	D₄×C₄₃	—	4	2	no	—
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	—	4	1	no	—
54	C₂×C₄×C₄₃	—	4	2	no	—
55	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₄₃⋊C₈	—	4	2	no	—
62	C₄₃⋊C₈	—	4	2	no	—
63	C₂²×C₄₃	—	2	1	no	—
64	C₂²×C₄₃	—	2	1	no	—
65	C₂²×C₄₃	—	4	2	no	—
66	C₂²×C₄₃	—	4	2	no	—
67	C₂²×C₄₃	—	4	2	no	—
68	C₂²×C₄₃	—	8	4	no	—
69	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄₃	#131	1	2	yes	—
71	C₄×C₄₃	—	2	1	no	—
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	—	2	1	no	—
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	—	4	2	no	—
77	C₄×C₄₃	#100	1	2	yes	—
78	#1469	—	43	1	no	maximal, Sylow
79	C₂×C₄₃	—	2	2	no	—
80	C₂×C₄₃	—	8	2	no	—
81	C₂×C₄₃	#408	1	1	yes	characteristic
82	C₂×C₄₃	#410	1	1	yes	characteristic
83	C₂×C₄₃	C₂×#100	1	1	yes	characteristic
84	C₂×#124	D₄₃	1	1	yes	characteristic
85	#432	—	43	1	no	—
86	#431	—	43	1	no	—
87	#323	—	43	4	no	—
88	C₄₃	#1469	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	Dic₄₃	1	1	yes	characteristic
90	C₂²×D₄	Dic₄₃	1	1	yes	characteristic
91	#124	—	2	1	no	—
92	#124	—	2	1	no	—
93	#124	—	2	1	no	—
94	#124	—	2	1	no	—
95	C₂×C₂²⋊C₄	Dic₄₃	1	2	yes	—
96	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
97	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
98	C₄⋊C₈	—	86	1	no	—
99	C₄⋊C₈	—	86	1	no	—
100	C₂²⋊C₈	—	43	2	no	—
101	C₂²⋊C₈	—	43	2	no	—
102	#97	—	86	1	no	—
103	C₄×C₈	—	86	1	no	—
104	C₂⁴	C₂×Dic₄₃	1	2	yes	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	4	2	no	—
111	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
114	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
115	C₂×C₈	—	86	2	no	—
116	C₂×C₈	—	86	2	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
122	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
123	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
124	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	—	2	4	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	4	2	no	—
129	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
130	Q₈	—	4	2	no	—
131	Q₈	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
145	C₈	—	172	2	no	—
146	C₈	—	172	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	8	4	no	—
153	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
154	C₂²	#13063332	1	2	yes	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	4	2	no	—
161	C₄	#13063242	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	—	8	2	no	—
164	C₂	#14582866	1	1	yes	characteristic
165	C₂	#14582873	1	1	yes	characteristic
166	C₂	C₂×#13063242	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14582956

C₂

yes

maximal, characteristic, complemented

#14582952

C₂

yes

maximal, characteristic, complemented

#14582623

C₂

yes

maximal, complemented

C₂×#124×C₄₃

C₂

yes

maximal, characteristic

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₃⋊C₈)

C₂²

yes

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₂²×Q₈×C₄₃

C₄

yes

characteristic

C₂²×D₄×C₄₃

C₄

yes

characteristic

#124×C₄₃

—

#124×C₄₃

—

#124×C₄₃

—

#124×C₄₃

—

C₂×C₂²⋊C₄×C₄₃

C₄

yes

—

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂⁴×C₄₃

C₂×C₄

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

#131

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#100

yes

—

#1469

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

yes

—

—

—

C₄₃

#1469

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₄₃

yes

characteristic

C₂²×D₄

Dic₄₃

yes

—

—

—

—

C₂×C₂²⋊C₄

Dic₄₃

yes

—

C₂×C₂²⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

C₄×C₈

—

104

C₂⁴

C₂×Dic₄₃

yes

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

112

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

113

C₂²×C₄

C₂×Dic₄₃

yes

—

114

C₂²×C₄

C₂²×D₄₃

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₂²⋊D₄₃

yes

characteristic

122

C₄²

C₂²⋊D₄₃

yes

characteristic

123

C₄²

C₂²⋊D₄₃

yes

characteristic

124

C₄²

C₂²⋊D₄₃

yes

characteristic

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂²×Dic₄₃

yes

characteristic

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

140

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

141

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

142

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

145

C₈

—

172

—

146

C₈

—

172

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₂×C₂²⋊Dic₄₃

yes

characteristic

154

C₂²

#13063332

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#13063242

yes

—

162

C₂

—

163

C₂

—

164

C₂

#14582866

yes

characteristic

165

C₂

#14582873

yes

characteristic

166

C₂

C₂×#13063242

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₄₃
commutator	C₂³×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#124×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	C₂×#124	D₄₃
hypocenter	C₄₃	#1469
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₄₃

commutator

C₂³×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#124×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #14582956	C₂	semidirect
2. #14582952	C₂	semidirect
3. #14582623	C₂	semidirect
88. C₄₃	#1469	semidirect

H ≅ G/N

type

1. #14582956

C₂

semidirect

2. #14582952

C₂

semidirect

3. #14582623

C₂

semidirect

88. C₄₃

#1469

semidirect

Subgroups — #19132947

Conjugacy classes of subgroups

Complements