Group #28475539 — subgroups

Subgroups — #28475539

Order 15104.

210 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 91 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	#21425899	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837124	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2163	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
6	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
7	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
8	#16837178	C₄	1	2	yes	—
9	#16837124	—	2	1	no	—
10	#16837124	—	2	1	no	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×C₄⋊C₁₆	—	2	1	no	—
13	C₅₉×C₂²⋊C₁₆	—	2	1	no	—
14	C₅₉×#346	C₄	1	2	yes	—
15	C₂²×C₅₉⋊C₈	—	2	1	no	—
16	C₂²×C₅₉⋊C₈	C₈	1	2	yes	—
17	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	C₄×C₅₉⋊C₈	—	2	1	no	—
19	C₄×C₅₉⋊C₈	—	2	1	no	—
20	C₄×C₅₉⋊C₈	—	2	1	no	—
21	C₄×C₅₉⋊C₈	C₈	1	2	yes	—
22	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₉	—	2	2	no	—
26	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₅₉	—	4	1	no	—
34	C₂×C₈×C₅₉	—	4	1	no	—
35	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₅₉	C₄²	1	1	yes	characteristic
40	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
43	C₂×C₅₉⋊C₈	—	2	4	no	—
44	C₂×C₅₉⋊C₈	—	2	2	no	—
45	C₂×C₅₉⋊C₈	—	2	2	no	—
46	C₂×C₅₉⋊C₈	—	4	2	no	—
47	C₁₆×C₅₉	—	4	2	no	—
48	C₅₉⋊C₁₆	—	4	2	no	—
49	C₂³×C₅₉	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	2	1	no	—
52	C₂×C₄×C₅₉	—	2	1	no	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#97	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₅₉	—	2	1	no	—
61	C₈×C₅₉	—	2	1	no	—
62	C₈×C₅₉	—	2	1	no	—
63	C₈×C₅₉	—	2	1	no	—
64	C₅₉⋊C₈	—	2	4	no	—
65	C₅₉⋊C₈	—	4	2	no	—
66	#7146	—	59	1	no	maximal, Sylow
67	C₂²×C₅₉	—	2	1	no	—
68	C₂²×C₅₉	—	2	1	no	—
69	C₂²×C₅₉	#336	1	1	yes	characteristic
70	C₂²×C₅₉	#364	1	2	yes	—
71	C₄×C₅₉	—	2	1	no	—
72	C₄×C₅₉	—	2	1	no	—
73	C₄×C₅₉	—	2	1	no	—
74	C₄×C₅₉	—	2	1	no	—
75	C₄×C₅₉	H(3,ℤ/4)	1	1	yes	characteristic
76	C₄×C₅₉	#338	1	1	yes	characteristic
77	C₄×C₅₉	#349	1	2	yes	—
78	#2163	—	59	1	no	—
79	#2163	D₅₉	1	1	yes	characteristic
80	C₂×#322	—	59	1	no	—
81	C₂×C₅₉	—	2	2	no	—
82	C₂×C₅₉	#1275	1	1	yes	characteristic
83	C₂×C₅₉	#1376	1	1	yes	characteristic
84	C₂×C₅₉	#1377	1	1	yes	characteristic
85	C₂×C₄×C₈	—	59	1	no	—
86	C₂×C₄×C₈	—	59	1	no	—
87	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
88	C₄⋊C₁₆	—	2	1	no	—
89	C₄⋊C₁₆	—	118	1	no	—
90	C₂²⋊C₁₆	—	2	1	no	—
91	C₂²⋊C₁₆	—	118	1	no	—
92	#346	—	59	2	no	—
93	#346	Dic₅₉	1	2	yes	—
94	#322	—	118	1	no	—
95	#322	—	118	1	no	—
96	C₅₉	#7146	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₈	—	59	2	no	—
98	C₂²×C₈	—	118	1	no	—
99	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
101	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	118	2	no	—
104	C₄×C₈	—	59	2	no	—
105	C₄×C₈	—	118	1	no	—
106	C₄×C₈	—	118	1	no	—
107	C₄×C₈	—	118	1	no	—
108	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
109	C₄×C₈	D₂₃₆	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
112	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
113	C₂²×C₄	#11745659	1	1	yes	characteristic
114	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	118	4	no	—
118	C₂×C₈	—	118	2	no	—
119	C₂×C₈	—	118	2	no	—
120	C₂×C₈	—	236	2	no	—
121	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
122	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
123	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
125	C₄²	C₈×D₅₉	1	1	yes	characteristic
126	C₄²	#11745659	1	1	yes	characteristic
127	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
128	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
129	C₁₆	—	4	2	no	—
130	C₁₆	—	236	2	no	—
131	C₂³	Dic₅₉⋊C₈	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
137	C₂×C₄	C₈×Dic₅₉	1	1	yes	characteristic
138	C₂×C₄	#14273637	1	1	yes	characteristic
139	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
140	C₂×C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
141	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	118	4	no	—
147	C₈	—	236	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#16837156	1	1	yes	characteristic
151	C₂²	#16837222	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#16837158	1	1	yes	characteristic
157	C₄	C₅₉⋊#338	1	1	yes	characteristic
158	C₄	#16837183	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#21421926	1	1	yes	characteristic
161	C₂	#21422177	1	1	yes	characteristic
162	C₂	#21422179	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425899

C₂

yes

maximal, characteristic

C₂×#16837124

C₂

yes

maximal, characteristic

C₅₉×#2163

C₂

yes

maximal, characteristic

C₂×C₄×C₅₉⋊C₈

C₄

yes

characteristic

C₂×C₄×C₅₉⋊C₈

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₁₆)

—

C₂²⋊(C₅₉⋊C₁₆)

—

#16837178

C₄

yes

—

#16837124

—

#16837124

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₅₉×C₂²⋊C₁₆

—

C₅₉×#346

C₄

yes

—

C₂²×C₅₉⋊C₈

—

C₂²×C₅₉⋊C₈

C₈

yes

—

C₂×C₅₉⋊C₁₆

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₈

yes

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

Q₈

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₉

M₁₆

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₄²×C₅₉

C₄²

yes

characteristic

C₄²×C₅₉

C₂²⋊C₄

yes

characteristic

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄²×C₅₉

M₁₆

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#95

yes

characteristic

C₂×C₄×C₅₉

C₄×C₈

yes

characteristic

C₂×C₄×C₅₉

#97

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#7146

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#336

yes

characteristic

C₂²×C₅₉

#364

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#2163

—

#2163

D₅₉

yes

characteristic

C₂×#322

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

118

—

#346

—

#346

Dic₅₉

yes

—

#322

—

118

—

#322

—

118

—

C₅₉

#7146

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

118

—

C₂²×C₈

Dic₁₁₈

yes

characteristic

100

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

101

C₂×C₄²

C₄×D₅₉

yes

characteristic

102

C₂×C₁₆

—

103

C₂×C₁₆

—

118

—

104

C₄×C₈

—

105

C₄×C₈

—

118

—

106

C₄×C₈

—

118

—

107

C₄×C₈

—

118

—

108

C₄×C₈

C₄×D₅₉

yes

characteristic

109

C₄×C₈

D₂₃₆

yes

characteristic

110

C₄×C₈

C₂×Dic₅₉

yes

characteristic

111

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

112

C₂²×C₄

C₈×D₅₉

yes

characteristic

113

C₂²×C₄

#11745659

yes

characteristic

114

C₂²×C₄

Dic₅₉⋊C₄

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

—

118

C₂×C₈

—

118

—

119

C₂×C₈

—

118

—

120

C₂×C₈

—

236

—

121

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

122

C₂×C₈

C₄⋊Dic₅₉

yes

characteristic

123

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

characteristic

124

C₂×C₈

C₂²⋊Dic₅₉

yes

characteristic

125

C₄²

C₈×D₅₉

yes

characteristic

126

C₄²

#11745659

yes

characteristic

127

C₄²

C₄×Dic₅₉

yes

characteristic

128

C₄²

(C₂×D₅₉)⋊C₄

yes

characteristic

129

C₁₆

—

130

C₁₆

—

236

—

131

C₂³

Dic₅₉⋊C₈

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₅₉⋊#95

yes

characteristic

137

C₂×C₄

C₈×Dic₅₉

yes

characteristic

138

C₂×C₄

#14273637

yes

characteristic

139

C₂×C₄

(C₂×D₅₉)⋊C₈

yes

characteristic

140

C₂×C₄

(C₂×D₅₉)⋊C₈

yes

characteristic

141

C₂×C₄

Dic₅₉⋊C₈

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

118

—

147

C₈

—

236

—

148

C₂²

—

149

C₂²

—

150

C₂²

#16837156

yes

characteristic

151

C₂²

#16837222

yes

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#16837158

yes

characteristic

157

C₄

C₅₉⋊#338

yes

characteristic

158

C₄

#16837183

yes

—

159

C₂

—

160

C₂

#21421926

yes

characteristic

161

C₂

#21422177

yes

characteristic

162

C₂

#21422179

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₅₉⋊#95
commutator	C₂×C₄×C₅₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₅₉×#2163	C₂
socle	C₂²×C₅₉	#336
prime radical	C₂³×C₅₉	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	#2163	D₅₉
hypocenter	C₅₉	#7146
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂×C₄

C₅₉⋊#95

commutator

C₂×C₄×C₅₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₅₉

Fitting

C₅₉×#2163

C₂

socle

C₂²×C₅₉

#336

prime radical

C₂³×C₅₉

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
96. C₅₉	#7146	semidirect

H ≅ G/N

type

96. C₅₉

#7146

semidirect

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

subgroup	index
#21425899	2
C₂×#16837124	2
C₅₉×#2163	2
#7146	59

subgroup

index

#21425899

C₂×#16837124

C₅₉×#2163

#7146

Subgroups — #28475539

Conjugacy classes of subgroups

Complements

Maximal subgroups