Group #21424130 — subgroups

Subgroups — #21424130

Order 7552.

316 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×#11745659	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273659	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837259	C₂	1	2	yes	maximal
4	#16837164	C₂	1	2	yes	maximal, complemented
5	C₂×#107×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₉	C₂²	1	1	yes	characteristic
7	C₂×#11745659	—	2	2	no	—
8	C₂×#11745659	—	2	2	no	—
9	C₂×#11745659	C₄	1	4	yes	complemented
10	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
12	C₂×(C₂×D₅₉)⋊C₄	—	2	2	no	—
13	#14273659	—	2	2	no	—
14	C₅₉⋊#95	—	2	2	no	—
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
17	#107×C₅₉	—	2	2	no	—
18	C₂³×D₅₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	—	2	2	no	—
20	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₉	—	2	2	no	—
22	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
23	C₂×C₄×D₅₉	—	2	1	no	—
24	C₂×C₄×D₅₉	—	2	1	no	—
25	C₂×C₄×D₅₉	—	2	1	no	—
26	C₂×C₄×D₅₉	—	2	1	no	—
27	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₅₉	Q₈	1	2	yes	—
29	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₅₉	D₄	1	2	yes	—
31	C₂×C₈×C₅₉	D₄	1	2	yes	—
32	C₄⋊C₄×C₅₉	—	2	2	no	—
33	C₄⋊C₄×C₅₉	—	2	2	no	—
34	(C₂×D₅₉)⋊C₄	—	4	4	no	—
35	C₄⋊Dic₅₉	—	2	2	no	—
36	C₄⋊Dic₅₉	—	2	2	no	—
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
40	#11745659	—	2	8	no	—
41	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₅₉	—	2	1	no	—
43	C₂²×D₅₉	—	2	1	no	—
44	C₂²×D₅₉	—	2	2	no	—
45	C₂²×D₅₉	—	2	2	no	—
46	C₂²×D₅₉	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₅₉	—	4	2	no	—
48	C₂×C₄×C₅₉	—	4	2	no	—
49	C₂×C₄×C₅₉	—	4	2	no	—
50	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
56	C₂×Dic₅₉	—	2	1	no	—
57	C₂×Dic₅₉	—	2	1	no	—
58	C₂×Dic₅₉	—	4	2	no	—
59	C₂×Dic₅₉	—	4	2	no	—
60	C₂×Dic₅₉	—	4	2	no	—
61	C₂×Dic₅₉	C₄⋊C₄	1	2	yes	—
62	C₄×D₅₉	—	2	2	no	—
63	C₄×D₅₉	—	2	2	no	—
64	C₄×D₅₉	—	2	2	no	—
65	C₄×D₅₉	—	2	2	no	—
66	C₈×C₅₉	C₂²⋊C₄	1	4	yes	complemented
67	C₅₉⋊C₈	—	2	2	no	—
68	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₅₉	#121	1	1	yes	characteristic
70	C₂²×C₅₉	#121	1	1	yes	characteristic
71	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
72	C₂²×C₅₉	#137	1	2	yes	—
73	C₂×D₅₉	—	2	2	no	—
74	C₂×D₅₉	—	2	4	no	—
75	C₂×D₅₉	—	4	2	no	—
76	C₂×D₅₉	—	4	1	no	—
77	C₂×D₅₉	—	4	1	no	—
78	C₄×C₅₉	—	8	2	no	—
79	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₄×C₅₉	#117	1	1	yes	characteristic
81	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
83	Dic₅₉	—	2	2	no	—
84	Dic₅₉	—	8	2	no	—
85	#1809	—	59	1	no	maximal, Sylow
86	C₂×C₅₉	#380	1	1	yes	characteristic
87	C₂×C₅₉	#428	1	2	yes	—
88	C₂×C₅₉	#468	1	2	yes	—
89	C₂×C₅₉	#469	1	1	yes	characteristic
90	C₂×C₅₉	#470	1	1	yes	characteristic
91	D₅₉	—	4	2	no	—
92	C₂²×M₁₆	—	59	1	no	—
93	C₂×#107	D₅₉	1	1	yes	characteristic, complemented
94	C₂×#106	—	59	1	no	—
95	#380	—	59	2	no	—
96	#340	—	59	2	no	—
97	C₅₉	#1809	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	—	59	1	no	—
99	C₂×M₁₆	—	59	4	no	—
100	C₂×M₁₆	—	118	2	no	—
101	C₂×M₁₆	—	118	2	no	—
102	C₂²×C₈	—	59	1	no	—
103	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	59	2	no	—
105	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
106	C₂×C₂²⋊C₄	—	118	2	no	—
107	#107	—	2	2	no	—
108	#106	—	118	2	no	—
109	#95	—	118	2	no	—
110	C₂⁴	—	59	1	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	59	1	no	—
113	C₂²×C₄	—	59	2	no	—
114	C₂²×C₄	—	59	2	no	—
115	C₂²×C₄	—	118	1	no	—
116	C₂²×C₄	—	118	1	no	—
117	C₂²×C₄	—	118	1	no	—
118	C₂²×C₄	—	118	1	no	—
119	C₂²×C₄	—	118	2	no	—
120	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
121	M₁₆	—	118	8	no	—
122	C₂×C₈	—	59	2	no	—
123	C₂×C₈	—	118	1	no	—
124	C₂×C₈	—	118	1	no	—
125	C₂×C₈	C₄×D₅₉	1	2	yes	—
126	C₂×C₈	D₂₃₆	1	2	yes	—
127	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	118	2	no	—
131	C₄⋊C₄	—	118	2	no	—
132	C₂²⋊C₄	—	236	4	no	—
133	C₂³	—	59	2	no	—
134	C₂³	—	118	1	no	—
135	C₂³	—	118	1	no	—
136	C₂³	—	118	2	no	—
137	C₂³	—	118	2	no	—
138	C₂³	D₄×D₅₉	1	1	yes	characteristic
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	59	2	no	—
143	C₂×C₄	—	118	1	no	—
144	C₂×C₄	—	118	1	no	—
145	C₂×C₄	—	118	2	no	—
146	C₂×C₄	—	118	2	no	—
147	C₂×C₄	—	118	2	no	—
148	C₂×C₄	—	118	2	no	—
149	C₂×C₄	—	236	2	no	—
150	C₂×C₄	—	236	2	no	—
151	C₂×C₄	—	236	2	no	—
152	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
153	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
154	C₂×C₄	#11745687	1	1	yes	characteristic
155	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
156	C₂×C₄	#11745689	1	1	yes	characteristic
157	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
158	C₈	—	118	2	no	—
159	C₈	(C₂×D₅₉)⋊C₄	1	4	yes	complemented
160	C₂²	—	118	4	no	—
161	C₂²	—	118	2	no	—
162	C₂²	—	236	2	no	—
163	C₂²	—	236	1	no	—
164	C₂²	—	236	1	no	—
165	C₂²	C₄⋊C₄×D₅₉	1	1	yes	characteristic
166	C₂²	#14273698	1	1	yes	characteristic
167	C₂²	#14273702	1	1	yes	characteristic
168	C₂²	#14273751	1	2	yes	—
169	C₂²	#14273759	1	2	yes	—
170	C₄	—	8	2	no	—
171	C₄	—	118	2	no	—
172	C₄	—	472	2	no	—
173	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
174	C₄	#14273685	1	1	yes	characteristic
175	C₄	#14273703	1	1	yes	characteristic
176	C₄	#14273707	1	1	yes	characteristic
177	C₂	—	236	2	no	—
178	C₂	#16837259	1	1	yes	characteristic
179	C₂	#16837443	1	2	yes	—
180	C₂	#16837649	1	2	yes	—
181	C₂	#16837652	1	1	yes	characteristic
182	C₂	#16837657	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745659

C₂

yes

maximal, characteristic

C₂×#14273659

C₂

yes

maximal, characteristic, complemented

#16837259

C₂

yes

maximal

#16837164

C₂

yes

maximal, complemented

C₂×#107×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₉

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×(C₂×D₅₉)⋊C₄

—

#14273659

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

#107×C₅₉

—

C₂³×D₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

C₂×C₄

yes

—

C₂×C₄×D₅₉

Q₈

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

#11745659

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₄⋊C₄

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#137

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#468

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

C₂²×M₁₆

—

C₂×#107

D₅₉

yes

characteristic, complemented

C₂×#106

—

#380

—

#340

—

C₅₉

#1809

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

100

C₂×M₁₆

—

118

—

101

C₂×M₁₆

—

118

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂×D₅₉

yes

characteristic

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

106

C₂×C₂²⋊C₄

—

118

—

107

#107

—

108

#106

—

118

—

109

#95

—

118

—

110

C₂⁴

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

118

—

116

C₂²×C₄

—

118

—

117

C₂²×C₄

—

118

—

118

C₂²×C₄

—

118

—

119

C₂²×C₄

—

118

—

120

C₂²×C₄

C₂²×D₅₉

yes

characteristic

121

M₁₆

—

118

—

122

C₂×C₈

—

123

C₂×C₈

—

118

—

124

C₂×C₈

—

118

—

125

C₂×C₈

C₄×D₅₉

yes

—

126

C₂×C₈

D₂₃₆

yes

—

127

C₂×C₈

C₂²⋊D₅₉

yes

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

118

—

131

C₄⋊C₄

—

118

—

132

C₂²⋊C₄

—

236

—

133

C₂³

—

134

C₂³

—

118

—

135

C₂³

—

118

—

136

C₂³

—

118

—

137

C₂³

—

118

—

138

C₂³

D₄×D₅₉

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

118

—

144

C₂×C₄

—

118

—

145

C₂×C₄

—

118

—

146

C₂×C₄

—

118

—

147

C₂×C₄

—

118

—

148

C₂×C₄

—

118

—

149

C₂×C₄

—

236

—

150

C₂×C₄

—

236

—

151

C₂×C₄

—

236

—

152

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

153

C₂×C₄

C₂×D₂₃₆

yes

characteristic

154

C₂×C₄

#11745687

yes

characteristic

155

C₂×C₄

Q₈×D₅₉

yes

characteristic

156

C₂×C₄

#11745689

yes

characteristic

157

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

158

C₈

—

118

—

159

C₈

(C₂×D₅₉)⋊C₄

yes

complemented

160

C₂²

—

118

—

161

C₂²

—

118

—

162

C₂²

—

236

—

163

C₂²

—

236

—

164

C₂²

—

236

—

165

C₂²

C₄⋊C₄×D₅₉

yes

characteristic

166

C₂²

#14273698

yes

characteristic

167

C₂²

#14273702

yes

characteristic

168

C₂²

#14273751

yes

—

169

C₂²

#14273759

yes

—

170

C₄

—

171

C₄

—

118

—

172

C₄

—

472

—

173

C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

174

C₄

#14273685

yes

characteristic

175

C₄

#14273703

yes

characteristic

176

C₄

#14273707

yes

characteristic

177

C₂

—

236

—

178

C₂

#16837259

yes

characteristic

179

C₂

#16837443

yes

—

180

C₂

#16837649

yes

—

181

C₂

#16837652

yes

characteristic

182

C₂

#16837657

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#107×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#107	D₅₉
hypocenter	C₅₉	#1809
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#107×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. C₂×#14273659	C₂	semidirect
4. #16837164	C₂	semidirect
5. C₂×#107×C₅₉	C₂	semidirect
9. C₂×#11745659	C₄	semidirect
66. C₈×C₅₉	C₂²⋊C₄	semidirect
93. C₂×#107	D₅₉	semidirect
97. C₅₉	#1809	semidirect
159. C₈	(C₂×D₅₉)⋊C₄	semidirect

H ≅ G/N

type

2. C₂×#14273659

C₂

semidirect

4. #16837164

C₂

semidirect

5. C₂×#107×C₅₉

C₂

semidirect

9. C₂×#11745659

C₄

semidirect

66. C₈×C₅₉

C₂²⋊C₄

semidirect

93. C₂×#107

D₅₉

semidirect

97. C₅₉

#1809

semidirect

159. C₈

(C₂×D₅₉)⋊C₄

semidirect

Subgroups — #21424130

Conjugacy classes of subgroups

Complements