Group #28507219 — subgroups

Subgroups — #28507219

Order 15104.

342 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 143 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	#21433030	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#21422186	C₂	1	4	yes	maximal
4	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
5	#16837959	C₄	1	1	yes	characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
7	#16837532	—	2	2	no	—
8	C₈×Dic₁₁₈	—	2	2	no	—
9	#16837428	C₄	1	2	yes	—
10	#16837426	C₄	1	1	yes	characteristic
11	#16837357	C₂²	1	2	yes	—
12	C₄×C₅₉⋊C₁₆	—	2	2	no	—
13	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×C₄⋊C₁₆	—	2	2	no	—
16	C₂×#11745664	C₂×C₄	1	2	yes	—
17	#14273727	C₈	1	2	yes	—
18	#14273719	C₈	1	2	yes	—
19	#14273706	—	2	2	no	—
20	C₄×Dic₁₁₈	—	2	2	no	—
21	#14273686	C₂×C₄	1	2	yes	—
22	C₂×C₅₉⋊C₁₆	—	2	2	no	—
23	C₂×C₅₉⋊C₁₆	—	2	2	no	—
24	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	complemented
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	Dic₅₉⋊C₈	—	2	4	no	—
27	#14273637	—	2	2	no	—
28	C₈×Dic₅₉	—	2	2	no	—
29	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₅₉	—	2	4	no	—
33	C₂×C₁₆×C₅₉	—	2	2	no	—
34	C₄×C₈×C₅₉	D₄	1	2	yes	—
35	C₄×C₈×C₅₉	D₄	1	2	yes	—
36	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₁₁₈	—	2	2	no	—
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₅₉	#41	1	2	yes	—
45	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
48	C₂²⋊Dic₅₉	—	2	2	no	—
49	C₄⋊Dic₅₉	—	2	2	no	—
50	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	—
51	Dic₅₉⋊C₄	—	2	4	no	—
52	C₄×Dic₅₉	—	2	2	no	—
53	#11745664	—	2	4	no	—
54	C₂×C₅₉⋊C₈	—	2	2	no	—
55	C₁₆×C₅₉	—	4	2	no	—
56	C₅₉⋊C₁₆	—	2	4	no	—
57	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅₉	—	2	4	no	—
59	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#131	1	2	yes	—
65	C₂×Dic₅₉	—	2	4	no	—
66	Dic₁₁₈	—	4	4	no	—
67	C₈×C₅₉	—	2	2	no	—
68	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₅₉	#124	1	1	yes	characteristic
70	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
71	C₅₉⋊C₈	—	4	2	no	—
72	#13266	—	59	1	no	maximal, Sylow
73	C₂²×C₅₉	—	2	2	no	—
74	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₂²×C₅₉	#446	1	2	yes	—
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	#350	1	4	yes	—
78	C₄×C₅₉	#391	1	1	yes	characteristic
79	C₄×C₅₉	#408	1	1	yes	characteristic
80	C₄×C₅₉	C₈×Q₈	1	2	yes	—
81	Dic₅₉	—	4	4	no	—
82	#2960	—	59	1	no	—
83	#2150	D₅₉	1	1	yes	characteristic
84	C₂×C₄×C₁₆	—	59	1	no	—
85	#1379	—	59	4	no	—
86	C₂×C₅₉	—	2	2	no	—
87	C₂×C₅₉	#1984	1	1	yes	characteristic
88	C₂×C₅₉	C₂×#350	1	2	yes	—
89	#533	—	59	1	no	—
90	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
91	C₂²×C₁₆	—	59	2	no	—
92	#446	—	118	2	no	—
93	C₈×Q₈	—	118	2	no	—
94	#424	—	59	2	no	—
95	#423	—	59	1	no	—
96	#405	—	59	2	no	—
97	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
98	C₄⋊C₁₆	—	2	2	no	—
99	C₄×C₁₆	—	118	2	no	—
100	C₅₉	#13266	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	59	2	no	—
102	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
104	C₄⋊Q₈	—	59	2	no	—
105	#125	—	59	2	no	—
106	C₂²⋊Q₈	—	118	2	no	—
107	C₄×Q₈	—	118	2	no	—
108	#117	—	59	2	no	—
109	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
110	M₃₂	—	2	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	59	4	no	—
113	C₂×C₁₆	—	118	2	no	—
114	C₂×C₁₆	—	118	2	no	—
115	C₄⋊C₈	—	118	4	no	—
116	C₄⋊C₈	—	118	2	no	—
117	#97	—	118	2	no	—
118	C₄×C₈	—	118	2	no	—
119	C₄×C₈	D₂₃₆	1	2	yes	—
120	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
121	C₂×Q₈	—	118	2	no	—
122	C₂²×C₄	#11745659	1	2	yes	—
123	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
124	M₁₆	—	118	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	118	2	no	—
127	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
129	C₂×C₈	#11745687	1	2	yes	—
130	C₂×C₈	Q₈×D₅₉	1	2	yes	—
131	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
132	C₄⋊C₄	—	59	4	no	—
133	C₄⋊C₄	—	118	4	no	—
134	C₄⋊C₄	—	118	2	no	—
135	C₂²⋊C₄	—	118	2	no	—
136	C₄²	—	118	2	no	—
137	C₄²	C₈×D₅₉	1	1	yes	characteristic
138	C₄²	C₈×D₅₉	1	1	yes	characteristic
139	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
140	C₁₆	—	4	2	no	—
141	C₁₆	—	118	4	no	—
142	C₂³	C₂×#11745659	1	1	yes	characteristic
143	Q₈	—	236	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	118	4	no	—
146	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
147	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
148	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#14273692	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
151	C₂×C₄	#14273735	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	236	2	no	—
154	C₈	#14273707	1	2	yes	—
155	C₈	#14273714	1	1	yes	characteristic
156	C₈	#14273725	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
159	C₂²	#16837529	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	236	4	no	—
162	C₄	#16837187	1	4	yes	—
163	C₄	#16837307	1	1	yes	characteristic
164	C₄	#16837370	1	1	yes	characteristic
165	C₄	#16837526	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#21425031	1	1	yes	characteristic
168	C₂	C₂×#16837187	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21433030

C₂

yes

maximal, characteristic

C₂×C₄×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic

#21422186

C₂

yes

maximal

C₅₉×#2150

C₂

yes

maximal, characteristic

#16837959

C₄

yes

characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

—

#16837532

—

C₈×Dic₁₁₈

—

#16837428

C₄

yes

—

#16837426

C₄

yes

characteristic

#16837357

C₂²

yes

—

C₄×C₅₉⋊C₁₆

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₂×#11745664

C₂×C₄

yes

—

#14273727

C₈

yes

—

#14273719

C₈

yes

—

#14273706

—

C₄×Dic₁₁₈

—

#14273686

C₂×C₄

yes

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

Q₈

yes

complemented

C₄⋊(C₅₉⋊C₈)

—

Dic₅₉⋊C₈

—

#14273637

—

C₈×Dic₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

M₁₆

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₈

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₈

yes

—

C₂×C₄×C₅₉

C₂²⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×Q₈

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

#131

yes

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊Q₈

yes

—

C₈×C₅₉

#124

yes

characteristic

C₈×C₅₉

C₄⋊Q₈

yes

characteristic

C₅₉⋊C₈

—

#13266

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₉

#446

yes

—

C₄×C₅₉

—

C₄×C₅₉

#350

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₈×Q₈

yes

—

Dic₅₉

—

#2960

—

#2150

D₅₉

yes

characteristic

C₂×C₄×C₁₆

—

#1379

—

C₂×C₅₉

—

C₂×C₅₉

#1984

yes

characteristic

C₂×C₅₉

C₂×#350

yes

—

#533

—

C₂×M₃₂

C₂×D₅₉

yes

—

C₂²×C₁₆

—

#446

—

118

—

C₈×Q₈

—

118

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₁₆

—

C₄×C₁₆

—

118

—

100

C₅₉

#13266

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

102

C₂²×C₈

C₄×D₅₉

yes

characteristic

103

C₂²×C₈

C₂²×D₅₉

yes

characteristic

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

118

—

107

C₄×Q₈

—

118

—

108

#117

—

109

C₂×C₄²

C₄×D₅₉

yes

characteristic

110

M₃₂

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

118

—

114

C₂×C₁₆

—

118

—

115

C₄⋊C₈

—

118

—

116

C₄⋊C₈

—

118

—

117

#97

—

118

—

118

C₄×C₈

—

118

—

119

C₄×C₈

D₂₃₆

yes

—

120

C₄×C₈

C₂²⋊D₅₉

yes

—

121

C₂×Q₈

—

118

—

122

C₂²×C₄

#11745659

yes

—

123

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

124

M₁₆

—

118

—

125

C₂×C₈

—

126

C₂×C₈

—

118

—

127

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

128

C₂×C₈

C₂×D₂₃₆

yes

characteristic

129

C₂×C₈

#11745687

yes

—

130

C₂×C₈

Q₈×D₅₉

yes

—

131

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

118

—

134

C₄⋊C₄

—

118

—

135

C₂²⋊C₄

—

118

—

136

C₄²

—

118

—

137

C₄²

C₈×D₅₉

yes

characteristic

138

C₄²

C₈×D₅₉

yes

characteristic

139

C₄²

(C₂×D₅₉)⋊C₄

yes

—

140

C₁₆

—

141

C₁₆

—

118

—

142

C₂³

C₂×#11745659

yes

characteristic

143

Q₈

—

236

—

144

C₂×C₄

—

145

C₂×C₄

—

118

—

146

C₂×C₄

(C₂×D₅₉)⋊C₈

yes

—

147

C₂×C₄

(C₂×D₅₉)⋊C₈

yes

—

148

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

149

C₂×C₄

#14273692

yes

—

150

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

151

C₂×C₄

#14273735

yes

—

152

C₈

—

153

C₈

—

236

—

154

C₈

#14273707

yes

—

155

C₈

#14273714

yes

characteristic

156

C₈

#14273725

yes

characteristic

157

C₂²

—

158

C₂²

C₂×(C₂×D₅₉)⋊C₈

yes

characteristic

159

C₂²

#16837529

yes

—

160

C₄

—

161

C₄

—

236

—

162

C₄

#16837187

yes

—

163

C₄

#16837307

yes

characteristic

164

C₄

#16837370

yes

characteristic

165

C₄

#16837526

yes

—

166

C₂

—

167

C₂

#21425031

yes

characteristic

168

C₂

C₂×#16837187

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#13266
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#2150

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₈

prime radical

C₂³×C₅₉

C₂×M₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
24. C₂×C₅₉⋊C₁₆	Q₈	semidirect
100. C₅₉	#13266	semidirect

H ≅ G/N

type

24. C₂×C₅₉⋊C₁₆

Q₈

semidirect

100. C₅₉

#13266

semidirect

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

subgroup	index
#21433030	2
C₂×C₄×C₅₉⋊C₁₆	2
#21422186	2
C₅₉×#2150	2
#13266	59

subgroup

index

#21433030

C₂×C₄×C₅₉⋊C₁₆

#21422186

C₅₉×#2150

#13266

Subgroups — #28507219

Conjugacy classes of subgroups

Complements

Maximal subgroups