Group #21423971 — subgroups

Subgroups — #21423971

Order 7552.

272 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 141 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⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273638	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273638	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#16837157	C₂	1	1	yes	maximal, characteristic
6	#16837157	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊(C₅₉⋊C₈)	C₄	1	4	yes	complemented
13	#14273638	—	2	2	no	—
14	#14273638	C₄	1	4	yes	complemented
15	C₄×C₅₉⋊C₈	—	2	2	no	—
16	C₂×C₄⋊C₄×C₅₉	C₄	1	2	yes	—
17	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
20	#95×C₅₉	C₄	1	2	yes	—
21	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₉	—	2	4	no	—
27	C₄²×C₅₉	—	2	1	no	—
28	C₄²×C₅₉	—	2	1	no	—
29	C₄²×C₅₉	—	2	1	no	—
30	C₄²×C₅₉	—	2	1	no	—
31	C₄²×C₅₉	C₂×C₄	1	2	yes	—
32	C₄²×C₅₉	C₂×C₄	1	2	yes	—
33	C₂×C₅₉⋊C₈	—	2	1	no	—
34	C₂×C₅₉⋊C₈	—	2	1	no	—
35	C₂×C₅₉⋊C₈	—	2	1	no	—
36	C₂×C₅₉⋊C₈	—	2	1	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	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
44	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
45	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	2	no	—
48	C₂×C₄×C₅₉	—	2	2	no	—
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	C₄²	1	2	yes	—
54	C₂×C₄×C₅₉	C₄²	1	2	yes	—
55	C₂×C₄×C₅₉	M₁₆	1	4	yes	—
56	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
62	C₅₉⋊C₈	—	2	2	no	—
63	C₅₉⋊C₈	—	2	2	no	—
64	C₅₉⋊C₈	—	2	2	no	—
65	C₅₉⋊C₈	C₄⋊C₄	1	4	yes	complemented
66	C₂²×C₅₉	C₂×C₄²	1	1	yes	characteristic
67	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
68	C₂²×C₅₉	C₄×Q₈	1	1	yes	characteristic
69	C₂²×C₅₉	C₂×M₁₆	1	2	yes	—
70	C₂²×C₅₉	#131	1	2	yes	—
71	C₄×C₅₉	—	2	2	no	—
72	C₄×C₅₉	—	2	2	no	—
73	C₄×C₅₉	—	2	4	no	—
74	C₄×C₅₉	—	4	2	no	—
75	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₅₉	#117	1	1	yes	characteristic
77	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
78	C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
79	#1771	—	59	1	no	maximal, Sylow
80	C₂×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₅₉	C₄×M₁₆	1	1	yes	characteristic
82	C₂×C₅₉	#405	1	1	yes	characteristic
83	C₂×C₅₉	#435	1	2	yes	—
84	C₂×C₅₉	#446	1	2	yes	—
85	C₂×C₄⋊C₈	—	59	1	no	—
86	C₂×#97	—	59	1	no	—
87	C₂×#97	—	59	1	no	—
88	C₂×C₄×C₈	—	59	1	no	—
89	C₄×C₄⋊C₄	D₅₉	1	1	yes	characteristic
90	#336	—	59	1	no	—
91	#336	—	59	1	no	—
92	C₅₉	#1771	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	—	59	1	no	—
94	C₂²×C₈	—	59	1	no	—
95	C₂²×C₈	—	59	1	no	—
96	C₂²×C₈	—	59	1	no	—
97	C₂×C₄⋊C₄	Dic₅₉	1	2	yes	—
98	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
101	C₄⋊C₈	—	59	4	no	—
102	#97	—	59	4	no	—
103	#97	—	118	2	no	—
104	C₄×C₈	—	118	2	no	—
105	#95	Dic₅₉	1	2	yes	—
106	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
107	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
108	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
109	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
111	C₂×C₈	—	59	2	no	—
112	C₂×C₈	—	59	2	no	—
113	C₂×C₈	—	59	2	no	—
114	C₂×C₈	—	59	2	no	—
115	C₂×C₈	—	59	2	no	—
116	C₂×C₈	—	59	2	no	—
117	C₂×C₈	—	118	1	no	—
118	C₂×C₈	—	118	1	no	—
119	C₂×C₈	—	118	1	no	—
120	C₂×C₈	—	118	1	no	—
121	C₂×C₈	—	118	1	no	—
122	C₂×C₈	—	118	1	no	—
123	C₄⋊C₄	—	2	4	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	C₄×D₅₉	1	2	yes	—
129	C₄²	C₄×D₅₉	1	2	yes	—
130	C₂³	C₂²×Dic₅₉	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	#11745664	1	4	yes	—
139	C₂×C₄	C₄×Dic₅₉	1	2	yes	—
140	C₂×C₄	C₄×Dic₅₉	1	2	yes	—
141	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
144	C₂×C₄	#11745687	1	1	yes	characteristic
145	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
146	C₂×C₄	#11745689	1	1	yes	characteristic
147	C₈	—	59	4	no	—
148	C₈	—	118	2	no	—
149	C₈	—	118	2	no	—
150	C₈	—	118	2	no	—
151	C₂²	C₂×C₄×Dic₅₉	1	1	yes	characteristic
152	C₂²	D₄×Dic₅₉	1	1	yes	characteristic
153	C₂²	Q₈×Dic₅₉	1	1	yes	characteristic
154	C₂²	C₂×#11745664	1	2	yes	—
155	C₂²	#14273734	1	2	yes	—
156	C₄	—	2	4	no	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	2	no	—
160	C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
161	C₄	#14273685	1	1	yes	characteristic
162	C₄	#14273689	1	1	yes	characteristic
163	C₄	#14273692	1	1	yes	characteristic
164	C₂	C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
165	C₂	C₄×#11745664	1	1	yes	characteristic
166	C₂	#16837358	1	1	yes	characteristic
167	C₂	#16837474	1	2	yes	—
168	C₂	#16837533	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14273638

C₂

yes

maximal, characteristic

C₂×#14273638

C₂

yes

maximal, characteristic

C₂×C₄×C₅₉⋊C₈

C₂

yes

maximal, characteristic

#16837157

C₂

yes

maximal, characteristic

#16837157

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₄

yes

complemented

#14273638

—

#14273638

C₄

yes

complemented

C₄×C₅₉⋊C₈

—

C₂×C₄⋊C₄×C₅₉

C₄

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#95×C₅₉

C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄²

yes

—

C₂×C₄×C₅₉

C₄²

yes

—

C₂×C₄×C₅₉

M₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₅₉

C₂×C₄²

yes

characteristic

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂²×C₅₉

C₄×Q₈

yes

characteristic

C₂²×C₅₉

C₂×M₁₆

yes

—

C₂²×C₅₉

#131

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₄×Q₈

yes

characteristic

#1771

—

maximal, Sylow

C₂×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₉

C₄×M₁₆

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#446

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

D₅₉

yes

characteristic

#336

—

#336

—

C₅₉

#1771

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₂×C₄²

C₂×D₅₉

yes

characteristic

100

C₂×C₄²

C₂×D₅₉

yes

characteristic

101

C₄⋊C₈

—

102

#97

—

103

#97

—

118

—

104

C₄×C₈

—

118

—

105

#95

Dic₅₉

yes

—

106

C₂²×C₄

C₂×Dic₅₉

yes

—

107

C₂²×C₄

C₂×Dic₅₉

yes

—

108

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

109

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

110

C₂²×C₄

C₂²×D₅₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

—

118

C₂×C₈

—

118

—

119

C₂×C₈

—

118

—

120

C₂×C₈

—

118

—

121

C₂×C₈

—

118

—

122

C₂×C₈

—

118

—

123

C₄⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₄×D₅₉

yes

—

129

C₄²

C₄×D₅₉

yes

—

130

C₂³

C₂²×Dic₅₉

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#11745664

yes

—

139

C₂×C₄

C₄×Dic₅₉

yes

—

140

C₂×C₄

C₄×Dic₅₉

yes

—

141

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

142

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

143

C₂×C₄

D₄×D₅₉

yes

characteristic

144

C₂×C₄

#11745687

yes

characteristic

145

C₂×C₄

Q₈×D₅₉

yes

characteristic

146

C₂×C₄

#11745689

yes

characteristic

147

C₈

—

148

C₈

—

118

—

149

C₈

—

118

—

150

C₈

—

118

—

151

C₂²

C₂×C₄×Dic₅₉

yes

characteristic

152

C₂²

D₄×Dic₅₉

yes

characteristic

153

C₂²

Q₈×Dic₅₉

yes

characteristic

154

C₂²

C₂×#11745664

yes

—

155

C₂²

#14273734

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

C₄⋊C₄×D₅₉

yes

characteristic

161

C₄

#14273685

yes

characteristic

162

C₄

#14273689

yes

characteristic

163

C₄

#14273692

yes

characteristic

164

C₂

C₄⋊C₄×Dic₅₉

yes

characteristic

165

C₂

C₄×#11745664

yes

characteristic

166

C₂

#16837358

yes

characteristic

167

C₂

#16837474

yes

—

168

C₂

#16837533

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂×C₄²
Fitting	C₄×C₄⋊C₄×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₄
hypercenter	C₄×C₄⋊C₄	D₅₉
hypocenter	C₅₉	#1771
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₉

commutator

C₂²×C₅₉

C₂×C₄²

Fitting

C₄×C₄⋊C₄×C₅₉

C₂

socle

C₂³×C₅₉

C₂²×C₄

hypercenter

C₄×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
12. C₄⋊(C₅₉⋊C₈)	C₄	semidirect
14. #14273638	C₄	semidirect
65. C₅₉⋊C₈	C₄⋊C₄	semidirect
92. C₅₉	#1771	semidirect

H ≅ G/N

type

12. C₄⋊(C₅₉⋊C₈)

C₄

semidirect

14. #14273638

C₄

semidirect

65. C₅₉⋊C₈

C₄⋊C₄

semidirect

92. C₅₉

#1771

semidirect

Subgroups — #21423971

Conjugacy classes of subgroups

Complements