Group #28504903 — subgroups

Subgroups — #28504903

Order 15104.

454 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 171 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×C₂²⋊(C₅₉⋊C₁₆)	C₂	1	2	yes	maximal
2	#21422174	C₂	1	4	yes	maximal, complemented
3	C₂×C₅₉×#405	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅₉⋊C₁₆	C₂²	1	4	yes	—
5	C₂²⋊(C₅₉⋊C₁₆)	—	2	4	no	—
6	C₂²×M₁₆×C₅₉	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₂×#117×C₅₉	C₄	1	1	yes	characteristic
9	C₅₉×#405	—	2	4	no	—
10	C₂×#97×C₅₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
12	C₂×C₅₉⋊C₁₆	—	2	4	no	—
13	C₂×C₅₉⋊C₁₆	—	2	8	no	—
14	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
15	C₂×M₁₆×C₅₉	—	2	2	no	—
16	C₂×M₁₆×C₅₉	—	2	4	no	—
17	C₂²×C₈×C₅₉	—	2	4	no	—
18	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₉	D₄	1	4	yes	—
21	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	#117×C₅₉	—	2	2	no	—
23	#117×C₅₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₅₉	C₈	1	2	yes	—
25	C₂×C₂²⋊C₄×C₅₉	C₈	1	2	yes	—
26	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
27	#97×C₅₉	—	2	4	no	—
28	C₄×C₈×C₅₉	—	2	4	no	—
29	C₂⁴×C₅₉	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	—	2	2	no	—
31	C₂²×C₄×C₅₉	—	2	2	no	—
32	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₅₉	C₂×C₈	1	4	yes	—
34	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
36	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	M₁₆×C₅₉	—	2	8	no	—
38	C₂×C₈×C₅₉	—	2	2	no	—
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	—	2	2	no	—
41	C₂×C₈×C₅₉	—	2	8	no	—
42	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
44	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₅₉	#41	1	4	yes	—
46	C₄⋊C₄×C₅₉	—	2	4	no	—
47	C₂²⋊C₄×C₅₉	—	2	4	no	—
48	C₄²×C₅₉	—	2	2	no	—
49	C₄²×C₅₉	—	2	2	no	—
50	C₅₉⋊C₁₆	—	4	4	no	—
51	C₂³×C₅₉	—	2	2	no	—
52	C₂³×C₅₉	—	2	4	no	—
53	C₂³×C₅₉	C₂²⋊C₈	1	2	yes	—
54	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	2	2	no	—
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	2	4	no	—
59	C₂×C₄×C₅₉	—	2	4	no	—
60	C₂×C₄×C₅₉	—	2	8	no	—
61	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#117	1	2	yes	—
64	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₅₉	—	2	2	no	—
67	C₈×C₅₉	—	2	4	no	—
68	C₈×C₅₉	#123	1	4	yes	—
69	#12845	—	59	1	no	maximal, Sylow
70	C₂²×C₅₉	—	2	2	no	—
71	C₂²×C₅₉	—	2	4	no	—
72	C₂²×C₅₉	—	4	4	no	—
73	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₅₉	#431	1	2	yes	—
75	C₂²×C₅₉	#504	1	4	yes	—
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	4	4	no	—
78	C₄×C₅₉	#381	1	1	yes	characteristic
79	C₄×C₅₉	#407	1	1	yes	characteristic
80	C₄×C₅₉	#432	1	2	yes	—
81	C₂×#405	D₅₉	1	1	yes	characteristic
82	C₂×C₂²⋊C₁₆	—	59	2	no	—
83	#1374	—	59	4	no	—
84	C₂×C₅₉	—	4	2	no	—
85	C₂×C₅₉	#1817	1	1	yes	characteristic
86	C₂×C₅₉	#2113	1	2	yes	—
87	C₂×C₅₉	#2164	1	4	yes	—
88	C₂²×M₁₆	Dic₅₉	1	1	yes	characteristic
89	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
90	C₂×#117	Dic₅₉	1	1	yes	characteristic
91	C₂²×C₁₆	—	59	4	no	—
92	#405	—	2	4	no	—
93	C₂×#97	Dic₅₉	1	2	yes	—
94	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
95	C₂²⋊C₁₆	—	118	4	no	—
96	C₅₉	#12845	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	4	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
102	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
103	C₂²×C₈	C₂²⋊D₅₉	1	4	yes	—
104	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄⋊C₄	C₅₉⋊C₈	1	2	yes	—
108	C₂×C₂²⋊C₄	C₅₉⋊C₈	1	2	yes	—
109	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
110	C₂×C₁₆	—	118	8	no	—
111	C₂×C₁₆	—	118	4	no	—
112	#97	—	2	4	no	—
113	C₄×C₈	—	2	4	no	—
114	C₂⁴	C₂×C₅₉⋊C₈	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂×C₅₉⋊C₈	1	4	yes	—
118	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
119	C₂²×C₄	#11745664	1	2	yes	—
120	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
121	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	#11745664	1	4	yes	—
128	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
129	C₂×C₈	#11745685	1	4	yes	—
130	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	236	4	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	4	no	—
138	C₂³	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
139	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
148	C₂×C₄	#14273686	1	2	yes	—
149	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
150	C₂×C₄	C₂×#11745664	1	2	yes	—
151	C₈	—	2	2	no	—
152	C₈	—	2	4	no	—
153	C₈	#14273710	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
158	C₂²	#16837455	1	2	yes	—
159	C₂²	#16837819	1	4	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#16837265	1	1	yes	characteristic
163	C₄	#16837366	1	1	yes	characteristic
164	C₄	#16837459	1	2	yes	—
165	C₂	—	4	2	no	—
166	C₂	#21424164	1	1	yes	characteristic
167	C₂	#21425710	1	2	yes	—
168	C₂	#21425903	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₅₉⋊C₁₆)

C₂

yes

maximal

#21422174

C₂

yes

maximal, complemented

C₂×C₅₉×#405

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

—

C₂²⋊(C₅₉⋊C₁₆)

—

C₂²×M₁₆×C₅₉

C₄

yes

characteristic

C₂³×C₈×C₅₉

C₂²

yes

characteristic

C₂×#117×C₅₉

C₄

yes

characteristic

C₅₉×#405

—

C₂×#97×C₅₉

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂³×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

—

#117×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₈

yes

—

C₂×C₂²⋊C₄×C₅₉

C₈

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₂⁴×C₅₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₈

yes

—

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₉

M₁₆

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

M₁₆

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂²⋊C₈

yes

—

C₂³×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#123

yes

—

#12845

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₉

#431

yes

—

C₂²×C₅₉

#504

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×#405

D₅₉

yes

characteristic

C₂×C₂²⋊C₁₆

—

#1374

—

C₂×C₅₉

—

C₂×C₅₉

#1817

yes

characteristic

C₂×C₅₉

#2113

yes

—

C₂×C₅₉

#2164

yes

—

C₂²×M₁₆

Dic₅₉

yes

characteristic

C₂³×C₈

C₂×D₅₉

yes

characteristic

C₂×#117

Dic₅₉

yes

characteristic

C₂²×C₁₆

—

#405

—

C₂×#97

Dic₅₉

yes

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₂²⋊C₁₆

—

118

—

C₅₉

#12845

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×Dic₅₉

yes

—

102

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

103

C₂²×C₈

C₂²⋊D₅₉

yes

—

104

C₂²×C₈

C₂²×D₅₉

yes

characteristic

105

#117

—

106

#117

—

107

C₂×C₄⋊C₄

C₅₉⋊C₈

yes

—

108

C₂×C₂²⋊C₄

C₅₉⋊C₈

yes

—

109

C₂×C₄²

C₂×Dic₅₉

yes

—

110

C₂×C₁₆

—

118

—

111

C₂×C₁₆

—

118

—

112

#97

—

113

C₄×C₈

—

114

C₂⁴

C₂×C₅₉⋊C₈

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂×C₅₉⋊C₈

yes

—

118

C₂²×C₄

C₂×C₅₉⋊C₈

yes

characteristic

119

C₂²×C₄

#11745664

yes

—

120

C₂²×C₄

C₂²⋊Dic₅₉

yes

—

121

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

#11745664

yes

—

128

C₂×C₈

C₂²⋊Dic₅₉

yes

—

129

C₂×C₈

#11745685

yes

—

130

C₂×C₈

C₂×C₂²⋊D₅₉

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

236

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊(C₅₉⋊C₈)

yes

—

139

C₂³

C₂²×C₅₉⋊C₈

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊(C₅₉⋊C₈)

yes

—

147

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

148

C₂×C₄

#14273686

yes

—

149

C₂×C₄

C₂×#11745664

yes

characteristic

150

C₂×C₄

C₂×#11745664

yes

—

151

C₈

—

152

C₈

—

153

C₈

#14273710

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂²⋊(C₅₉⋊C₈)

yes

characteristic

158

C₂²

#16837455

yes

—

159

C₂²

#16837819

yes

—

160

C₄

—

161

C₄

—

162

C₄

#16837265

yes

characteristic

163

C₄

#16837366

yes

characteristic

164

C₄

#16837459

yes

—

165

C₂

—

166

C₂

#21424164

yes

characteristic

167

C₂

#21425710

yes

—

168

C₂

#21425903

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator, socle	C₂³×C₅₉	C₂²×C₈
Fitting	C₂×C₅₉×#405	C₂
prime radical	C₂⁴×C₅₉	C₂×C₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#405	D₅₉
hypocenter	C₅₉	#12845
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₅₉

commutator, socle

C₂³×C₅₉

C₂²×C₈

Fitting

C₂×C₅₉×#405

C₂

prime radical

C₂⁴×C₅₉

C₂×C₈

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₈ ◁ C₂×#405

N	H ≅ G/N	type
2. #21422174	C₂	semidirect
96. C₅₉	#12845	semidirect

H ≅ G/N

type

2. #21422174

C₂

semidirect

96. C₅₉

#12845

semidirect

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

subgroup	index
C₂×C₂²⋊(C₅₉⋊C₁₆)	2
#21422174	2
C₂×C₅₉×#405	2
#12845	59

subgroup

index

C₂×C₂²⋊(C₅₉⋊C₁₆)

#21422174

C₂×C₅₉×#405

#12845

Subgroups — #28504903

Conjugacy classes of subgroups

Complements

Maximal subgroups