Group #28509212 — subgroups

Subgroups — #28509212

Order 15104.

202 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 101 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×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
2	Dic₅₉⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₅₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	C₃₂×Dic₅₉	C₂	1	2	yes	maximal, complemented
5	C₅₉×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
8	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	—
9	C₁₆×Dic₅₉	C₄	1	1	yes	characteristic
10	C₁₆×Dic₅₉	C₄	1	1	yes	characteristic
11	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
12	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₅₉⋊C₈	C₈	1	1	yes	characteristic
16	C₂×C₄×Dic₅₉	C₈	1	1	yes	characteristic
17	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₅₉	C₈	1	1	yes	characteristic
21	C₈×Dic₅₉	C₈	1	1	yes	characteristic
22	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
24	C₅₉⋊C₃₂	—	2	2	no	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	—	2	2	no	—
27	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
29	C₃₂×C₅₉	—	2	2	no	—
30	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
31	C₂²×Dic₅₉	C₁₆	1	1	yes	characteristic
32	C₂²×Dic₅₉	C₁₆	1	1	yes	characteristic
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	—	2	1	no	—
35	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₄×Dic₅₉	C₁₆	1	1	yes	characteristic
38	C₄×Dic₅₉	C₁₆	1	1	yes	characteristic
39	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
40	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
41	C₂×C₅₉⋊C₈	—	2	2	no	—
42	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
43	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
44	C₁₆×C₅₉	—	2	1	no	—
45	C₁₆×C₅₉	#41	1	2	yes	—
46	C₅₉⋊C₁₆	—	2	1	no	—
47	C₅₉⋊C₁₆	M₁₆	1	2	yes	—
48	C₂³×C₅₉	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₅₉	—	2	1	no	—
50	C₂×C₄×C₅₉	—	2	1	no	—
51	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
53	C₂×Dic₅₉	—	2	2	no	—
54	C₂×Dic₅₉	C₃₂	1	4	yes	complemented
55	C₂×Dic₅₉	C₂×C₁₆	1	1	yes	characteristic
56	C₂×Dic₅₉	C₂×C₁₆	1	1	yes	characteristic
57	C₂×Dic₅₉	C₂×C₁₆	1	1	yes	characteristic
58	C₂×Dic₅₉	C₂×C₁₆	1	1	yes	characteristic
59	C₈×C₅₉	—	2	1	no	—
60	C₈×C₅₉	#117	1	1	yes	characteristic
61	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
62	C₅₉⋊C₈	—	2	1	no	—
63	C₅₉⋊C₈	M₃₂	1	2	yes	—
64	#13639	—	59	1	no	maximal, Sylow
65	C₂²×C₅₉	—	2	1	no	—
66	C₂²×C₅₉	—	2	1	no	—
67	C₂²×C₅₉	C₂×C₃₂	1	2	yes	—
68	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
69	C₄×C₅₉	—	2	1	no	—
70	C₄×C₅₉	#431	1	1	yes	characteristic
71	C₄×C₅₉	C₂×M₃₂	1	1	yes	characteristic
72	Dic₅₉	—	2	1	no	—
73	Dic₅₉	C₂×C₃₂	1	2	yes	—
74	Dic₅₉	C₂×C₃₂	1	2	yes	—
75	Dic₅₉	M₆₄	1	2	yes	—
76	C₂×C₄×C₁₆	—	59	1	no	—
77	C₄⋊C₃₂	—	59	2	no	—
78	C₂²⋊C₃₂	—	59	1	no	—
79	C₂²⋊C₃₂	D₅₉	1	1	yes	characteristic
80	C₄×C₃₂	—	59	2	no	—
81	C₂×C₅₉	—	2	2	no	—
82	C₂×C₅₉	#2162	1	1	yes	characteristic
83	C₂×C₅₉	C₂²×C₃₂	1	1	yes	characteristic
84	C₂×C₅₉	C₂×M₆₄	1	1	yes	characteristic
85	C₂²×C₁₆	—	59	1	no	—
86	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
87	C₂×C₄×C₈	—	59	1	no	—
88	C₂×C₃₂	—	59	2	no	—
89	C₂×C₃₂	C₂×D₅₉	1	2	yes	—
90	C₄×C₁₆	—	59	1	no	—
91	C₄×C₁₆	—	59	1	no	—
92	C₄×C₁₆	—	59	1	no	—
93	C₄×C₁₆	—	59	1	no	—
94	C₅₉	#13639	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	59	1	no	—
96	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
97	C₂×C₄²	—	59	1	no	—
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	59	1	no	—
100	C₂×C₁₆	—	59	1	no	—
101	C₂×C₁₆	—	118	2	no	—
102	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
103	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
104	C₄×C₈	—	59	1	no	—
105	C₄×C₈	—	59	1	no	—
106	C₄×C₈	—	59	1	no	—
107	C₄×C₈	—	59	1	no	—
108	C₃₂	—	2	2	no	—
109	C₃₂	—	118	2	no	—
110	C₂²×C₄	—	59	1	no	—
111	C₂²×C₄	—	59	1	no	—
112	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	59	1	no	—
116	C₂×C₈	—	59	1	no	—
117	C₂×C₈	—	118	2	no	—
118	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
119	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
120	C₄²	—	59	1	no	—
121	C₄²	—	59	1	no	—
122	C₄²	—	59	1	no	—
123	C₄²	—	59	1	no	—
124	C₁₆	—	2	1	no	—
125	C₁₆	—	59	2	no	—
126	C₁₆	—	118	1	no	—
127	C₁₆	#11745687	1	2	yes	—
128	C₂³	C₁₆×D₅₉	1	1	yes	characteristic
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	59	4	no	—
132	C₂×C₄	—	59	1	no	—
133	C₂×C₄	—	59	1	no	—
134	C₂×C₄	—	59	1	no	—
135	C₂×C₄	—	59	1	no	—
136	C₂×C₄	—	118	2	no	—
137	C₂×C₄	C₁₆×D₅₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	59	2	no	—
141	C₈	—	118	1	no	—
142	C₈	#14273683	1	1	yes	characteristic
143	C₈	M₁₆×D₅₉	1	1	yes	characteristic
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	C₃₂×D₅₉	1	2	yes	—
147	C₂²	C₂×C₁₆×D₅₉	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	59	2	no	—
150	C₄	—	59	2	no	—
151	C₄	—	59	2	no	—
152	C₄	—	118	1	no	—
153	C₄	#16837452	1	1	yes	characteristic
154	C₄	M₃₂×D₅₉	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#21425892	1	1	yes	characteristic
157	C₂	C₂×C₃₂×D₅₉	1	1	yes	characteristic
158	C₂	M₆₄×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×Dic₅₉

C₂

yes

maximal, characteristic

Dic₅₉⋊C₃₂

C₂

yes

maximal, complemented

C₂²⋊(C₅₉⋊C₃₂)

C₂

yes

maximal, characteristic

C₃₂×Dic₅₉

C₂

yes

maximal, complemented

C₅₉×C₂²⋊C₃₂

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₄

yes

characteristic

C₂×C₈×Dic₅₉

C₄

yes

characteristic

C₂×C₅₉⋊C₃₂

C₂²

yes

—

C₁₆×Dic₅₉

C₄

yes

characteristic

C₁₆×Dic₅₉

C₄

yes

characteristic

C₁₆×Dic₅₉

C₂²

yes

characteristic

C₁₆×Dic₅₉

C₂²

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₃₂×C₅₉

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₈

yes

characteristic

C₂×C₄×Dic₅₉

C₈

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₈×Dic₅₉

C₈

yes

characteristic

C₈×Dic₅₉

C₈

yes

characteristic

C₈×Dic₅₉

C₂×C₄

yes

characteristic

C₈×Dic₅₉

C₂×C₄

yes

characteristic

C₅₉⋊C₃₂

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₃₂×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂²×Dic₅₉

C₁₆

yes

characteristic

C₂²×Dic₅₉

C₁₆

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₄×Dic₅₉

C₁₆

yes

characteristic

C₄×Dic₅₉

C₁₆

yes

characteristic

C₄×Dic₅₉

C₂×C₈

yes

characteristic

C₄×Dic₅₉

C₂×C₈

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₈

yes

characteristic

C₁₆×C₅₉

—

C₁₆×C₅₉

#41

yes

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

M₁₆

yes

—

C₂³×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

C₃₂

yes

complemented

C₂×Dic₅₉

C₂×C₁₆

yes

characteristic

C₂×Dic₅₉

C₂×C₁₆

yes

characteristic

C₂×Dic₅₉

C₂×C₁₆

yes

characteristic

C₂×Dic₅₉

C₂×C₁₆

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

#117

yes

characteristic

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

M₃₂

yes

—

#13639

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₃₂

yes

—

C₂²×C₅₉

C₂²×C₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#431

yes

characteristic

C₄×C₅₉

C₂×M₃₂

yes

characteristic

Dic₅₉

—

Dic₅₉

C₂×C₃₂

yes

—

Dic₅₉

C₂×C₃₂

yes

—

Dic₅₉

M₆₄

yes

—

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

C₂²⋊C₃₂

D₅₉

yes

characteristic

C₄×C₃₂

—

C₂×C₅₉

—

C₂×C₅₉

#2162

yes

characteristic

C₂×C₅₉

C₂²×C₃₂

yes

characteristic

C₂×C₅₉

C₂×M₆₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

C₂×C₄×C₈

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₉

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₅₉

#13639

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

118

—

102

C₂×C₁₆

C₄×D₅₉

yes

characteristic

103

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₃₂

—

109

C₃₂

—

118

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₈×D₅₉

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

—

118

C₂×C₈

C₈×D₅₉

yes

characteristic

119

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₁₆

—

125

C₁₆

—

126

C₁₆

—

118

—

127

C₁₆

#11745687

yes

—

128

C₂³

C₁₆×D₅₉

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

118

—

137

C₂×C₄

C₁₆×D₅₉

yes

characteristic

138

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

118

—

142

C₈

#14273683

yes

characteristic

143

C₈

M₁₆×D₅₉

yes

characteristic

144

C₂²

—

145

C₂²

—

146

C₂²

C₃₂×D₅₉

yes

—

147

C₂²

C₂×C₁₆×D₅₉

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

118

—

153

C₄

#16837452

yes

characteristic

154

C₄

M₃₂×D₅₉

yes

characteristic

155

C₂

—

156

C₂

#21425892

yes

characteristic

157

C₂

C₂×C₃₂×D₅₉

yes

characteristic

158

C₂

M₆₄×D₅₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₉
commutator	C₂×C₅₉	C₂²×C₃₂
Fitting	C₅₉×C₂²⋊C₃₂	C₂
socle	C₂²×C₅₉	C₂²×C₁₆
prime radical	C₂³×C₅₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	C₂²⋊C₃₂	D₅₉
hypocenter	C₅₉	#13639
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²⋊C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₉

commutator

C₂×C₅₉

C₂²×C₃₂

Fitting

C₅₉×C₂²⋊C₃₂

C₂

socle

C₂²×C₅₉

C₂²×C₁₆

prime radical

C₂³×C₅₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₅₉

C₂³

hypercenter

C₂²⋊C₃₂

D₅₉

C₅₉

G ▷ C₂×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₁₆ ◁ C₂²⋊C₃₂

N	H ≅ G/N	type
2. Dic₅₉⋊C₃₂	C₂	semidirect
4. C₃₂×Dic₅₉	C₂	semidirect
54. C₂×Dic₅₉	C₃₂	semidirect
94. C₅₉	#13639	semidirect

H ≅ G/N

type

2. Dic₅₉⋊C₃₂

C₂

semidirect

4. C₃₂×Dic₅₉

C₂

semidirect

54. C₂×Dic₅₉

C₃₂

semidirect

94. C₅₉

#13639

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₆×Dic₅₉	2
Dic₅₉⋊C₃₂	2
C₂²⋊(C₅₉⋊C₃₂)	2
C₃₂×Dic₅₉	2
C₅₉×C₂²⋊C₃₂	2
#13639	59

subgroup

index

C₂×C₁₆×Dic₅₉

Dic₅₉⋊C₃₂

C₂²⋊(C₅₉⋊C₃₂)

C₃₂×Dic₅₉

C₅₉×C₂²⋊C₃₂

#13639

Subgroups — #28509212

Conjugacy classes of subgroups

Complements

Maximal subgroups