Group #21424812 — subgroups

Subgroups — #21424812

Order 7552.

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

normal subgroups only

This group has 260 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×Dic₅₉⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837156	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₉	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₉	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
11	Dic₅₉⋊C₈	—	2	4	no	—
12	C₄×C₅₉⋊C₈	—	2	2	no	—
13	C₅₉⋊#95	C₄	1	2	yes	—
14	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₉	—	2	2	no	—
17	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
21	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₉	—	2	2	no	—
23	C₂×C₈×C₅₉	—	2	2	no	—
24	C₂×C₈×C₅₉	—	2	2	no	—
25	C₄²×C₅₉	D₄	1	2	yes	—
26	C₄²×C₅₉	Q₈	1	2	yes	—
27	C₄⋊Dic₅₉	—	2	2	no	—
28	C₄⋊Dic₅₉	C₈	1	4	yes	complemented
29	C₄×Dic₅₉	—	2	2	no	—
30	C₄×Dic₅₉	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	2	1	no	—
32	C₂×C₅₉⋊C₈	—	2	1	no	—
33	C₂×C₅₉⋊C₈	—	2	1	no	—
34	C₂×C₅₉⋊C₈	—	2	1	no	—
35	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
36	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
37	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	—	2	1	no	—
39	C₂×C₄×C₅₉	—	2	1	no	—
40	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₉	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₅₉	M₁₆	1	2	yes	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
49	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	2	no	—
51	C₂×Dic₅₉	—	2	2	no	—
52	C₂×Dic₅₉	—	2	2	no	—
53	C₂×Dic₅₉	—	2	2	no	—
54	C₂×Dic₅₉	—	2	2	no	—
55	C₂×Dic₅₉	C₂×C₈	1	4	yes	—
56	C₈×C₅₉	—	4	2	no	—
57	C₅₉⋊C₈	—	2	2	no	—
58	C₅₉⋊C₈	—	2	2	no	—
59	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₅₉	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₅₉	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₅₉	#131	1	1	yes	characteristic
65	C₂²×C₅₉	#131	1	1	yes	characteristic
66	C₄×C₅₉	—	2	2	no	—
67	C₄×C₅₉	C₄⋊C₈	1	4	yes	complemented
68	C₄×C₅₉	#121	1	1	yes	characteristic
69	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
70	C₄×C₅₉	#125	1	1	yes	characteristic
71	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
72	Dic₅₉	—	2	4	no	—
73	Dic₅₉	—	4	2	no	—
74	#1939	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	#389	1	1	yes	characteristic
76	C₂×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₅₉	#424	1	1	yes	characteristic
78	C₂×C₅₉	C₈×D₄	1	1	yes	characteristic
79	C₂×C₅₉	#436	1	1	yes	characteristic
80	C₂×C₅₉	C₈×Q₈	1	1	yes	characteristic
81	C₂×C₅₉	#446	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	59	2	no	—
83	C₂×C₄⋊C₈	D₅₉	1	1	yes	characteristic
84	C₂×C₄×C₈	—	59	1	no	—
85	C₄×C₄⋊C₄	—	59	1	no	—
86	#336	—	59	2	no	—
87	C₅₉	#1939	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	59	1	no	—
89	C₂²×C₈	—	59	1	no	—
90	C₂²×C₈	C₂×D₅₉	1	2	yes	—
91	C₂×C₄⋊C₄	—	59	1	no	—
92	C₂×C₄⋊C₄	—	59	1	no	—
93	C₂×C₄²	—	59	2	no	—
94	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	118	4	no	—
97	C₄×C₈	—	118	2	no	—
98	#95	—	59	2	no	—
99	C₂²×C₄	—	59	2	no	—
100	C₂²×C₄	—	59	2	no	—
101	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
102	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	59	2	no	—
108	C₂×C₈	—	59	2	no	—
109	C₂×C₈	—	118	1	no	—
110	C₂×C₈	—	118	1	no	—
111	C₂×C₈	—	118	1	no	—
112	C₂×C₈	—	118	1	no	—
113	C₄⋊C₄	—	59	4	no	—
114	C₄⋊C₄	—	118	2	no	—
115	C₄²	—	118	2	no	—
116	C₄²	—	118	2	no	—
117	C₄²	Dic₁₁₈	1	2	yes	—
118	C₄²	C₂²⋊D₅₉	1	2	yes	—
119	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	59	4	no	—
123	C₂×C₄	—	118	2	no	—
124	C₂×C₄	—	118	2	no	—
125	C₂×C₄	—	118	2	no	—
126	C₂×C₄	—	118	2	no	—
127	C₂×C₄	—	118	2	no	—
128	C₂×C₄	C₈×D₅₉	1	2	yes	—
129	C₂×C₄	#11745659	1	2	yes	—
130	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
131	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
132	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
133	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
134	C₂×C₄	#11745687	1	1	yes	characteristic
135	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
136	C₂×C₄	#11745689	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	118	2	no	—
140	C₈	—	118	2	no	—
141	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
142	C₂²	#14273689	1	1	yes	characteristic
143	C₂²	#14273692	1	1	yes	characteristic
144	C₂²	C₂×C₈×D₅₉	1	1	yes	characteristic
145	C₂²	C₂×#11745659	1	1	yes	characteristic
146	C₂²	#14273735	1	1	yes	characteristic
147	C₂²	#14273735	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	118	4	no	—
150	C₄	—	236	2	no	—
151	C₄	Dic₅₉⋊C₈	1	4	yes	complemented
152	C₄	#14273698	1	1	yes	characteristic
153	C₄	#14273703	1	1	yes	characteristic
154	C₄	#14273718	1	1	yes	characteristic
155	C₄	#14273726	1	1	yes	characteristic
156	C₂	#16837298	1	1	yes	characteristic
157	C₂	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
158	C₂	#16837430	1	1	yes	characteristic
159	C₂	#16837467	1	1	yes	characteristic
160	C₂	#16837479	1	1	yes	characteristic
161	C₂	#16837526	1	1	yes	characteristic
162	C₂	#16837529	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₅₉⋊C₈

C₂

yes

maximal

C₂×C₄×C₅₉⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

#16837156

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₄

yes

characteristic

C₂×C₄⋊Dic₅₉

C₄

yes

characteristic

C₂×C₄×Dic₅₉

C₂²

yes

—

Dic₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₅₉⋊#95

C₄

yes

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₈

yes

complemented

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

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

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×C₈

yes

—

C₂×C₄×C₅₉

M₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

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

C₂×C₈

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂²×C₅₉

C₄×Q₈

yes

characteristic

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₂²×C₅₉

C₂×M₁₆

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

C₄⋊C₈

yes

complemented

C₄×C₅₉

#121

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#125

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₂×C₅₉

#424

yes

characteristic

C₂×C₅₉

C₈×D₄

yes

characteristic

C₂×C₅₉

#436

yes

characteristic

C₂×C₅₉

C₈×Q₈

yes

characteristic

C₂×C₅₉

#446

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₅₉

yes

characteristic

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#336

—

C₅₉

#1939

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄×C₈

—

118

—

#95

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₅₉

yes

characteristic

102

C₂²×C₄

C₄×D₅₉

yes

characteristic

103

C₂²×C₄

C₂²×D₅₉

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

118

—

110

C₂×C₈

—

118

—

111

C₂×C₈

—

118

—

112

C₂×C₈

—

118

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

118

—

115

C₄²

—

118

—

116

C₄²

—

118

—

117

C₄²

Dic₁₁₈

yes

—

118

C₄²

C₂²⋊D₅₉

yes

—

119

C₂³

C₂×C₄×D₅₉

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

118

—

124

C₂×C₄

—

118

—

125

C₂×C₄

—

118

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

118

—

128

C₂×C₄

C₈×D₅₉

yes

—

129

C₂×C₄

#11745659

yes

—

130

C₂×C₄

Dic₅₉⋊C₄

yes

—

131

C₂×C₄

Dic₅₉⋊C₄

yes

—

132

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

133

C₂×C₄

D₄×D₅₉

yes

characteristic

134

C₂×C₄

#11745687

yes

characteristic

135

C₂×C₄

Q₈×D₅₉

yes

characteristic

136

C₂×C₄

#11745689

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

138

C₈

—

139

C₈

—

118

—

140

C₈

—

118

—

141

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

142

C₂²

#14273689

yes

characteristic

143

C₂²

#14273692

yes

characteristic

144

C₂²

C₂×C₈×D₅₉

yes

characteristic

145

C₂²

C₂×#11745659

yes

characteristic

146

C₂²

#14273735

yes

characteristic

147

C₂²

#14273735

yes

characteristic

148

C₄

—

149

C₄

—

118

—

150

C₄

—

236

—

151

C₄

Dic₅₉⋊C₈

yes

complemented

152

C₄

#14273698

yes

characteristic

153

C₄

#14273703

yes

characteristic

154

C₄

#14273718

yes

characteristic

155

C₄

#14273726

yes

characteristic

156

C₂

#16837298

yes

characteristic

157

C₂

C₂×Dic₅₉⋊C₈

yes

characteristic

158

C₂

#16837430

yes

characteristic

159

C₂

#16837467

yes

characteristic

160

C₂

#16837479

yes

characteristic

161

C₂

#16837526

yes

characteristic

162

C₂

#16837529

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#1939
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₉

commutator

C₂²×C₅₉

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
28. C₄⋊Dic₅₉	C₈	semidirect
67. C₄×C₅₉	C₄⋊C₈	semidirect
87. C₅₉	#1939	semidirect
151. C₄	Dic₅₉⋊C₈	semidirect

H ≅ G/N

type

28. C₄⋊Dic₅₉

C₈

semidirect

67. C₄×C₅₉

C₄⋊C₈

semidirect

87. C₅₉

#1939

semidirect

151. C₄

Dic₅₉⋊C₈

semidirect

Subgroups — #21424812

Conjugacy classes of subgroups

Complements