Group #28488153 — subgroups

Subgroups — #28488153

Order 15104.

478 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 159 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	#21424268	C₂	1	1	yes	maximal, characteristic
2	#21422768	C₂	1	4	yes	maximal
3	#21422379	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic
5	#16837455	C₂²	1	2	yes	—
6	C₂×#14273663	—	2	2	no	—
7	#16837426	C₂²	1	2	yes	—
8	#16837148	—	2	8	no	—
9	#16837123	—	2	1	no	—
10	C₈×C₅₉⋊C₈	—	2	1	no	—
11	C₂²×Q₁₆×C₅₉	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
13	C₅₉×#494	—	2	1	no	—
14	C₅₉×#494	—	2	1	no	—
15	C₅₉×#494	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745664	—	2	2	no	—
18	#14273663	—	4	4	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
20	C₄⋊(C₅₉⋊C₈)	D₄	1	4	yes	—
21	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
22	C₄×C₅₉⋊C₈	—	2	2	no	—
23	C₂²×Q₈×C₅₉	—	2	2	no	—
24	C₂×Q₁₆×C₅₉	—	2	4	no	—
25	C₂×Q₁₆×C₅₉	—	2	4	no	—
26	C₂×Q₁₆×C₅₉	—	4	4	no	—
27	C₂×Q₁₆×C₅₉	—	4	4	no	—
28	C₂²×C₈×C₅₉	D₄	1	2	yes	—
29	C₄⋊Q₈×C₅₉	—	2	4	no	—
30	C₄⋊Q₈×C₅₉	—	2	2	no	—
31	C₄⋊Q₈×C₅₉	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
33	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
36	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₅₉	—	2	4	no	—
38	C₂×Q₈×C₅₉	—	4	2	no	—
39	C₂×Q₈×C₅₉	—	4	4	no	—
40	C₂×Q₈×C₅₉	—	4	4	no	—
41	C₂²×C₄×C₅₉	—	4	2	no	—
42	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	#41	1	2	yes	—
44	Q₁₆×C₅₉	—	4	8	no	—
45	Q₁₆×C₅₉	—	4	8	no	—
46	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₉	D₈	1	4	yes	—
49	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
50	C₄⋊C₄×C₅₉	—	2	8	no	—
51	C₄⋊C₄×C₅₉	—	4	4	no	—
52	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₉	C₂×D₄	1	2	yes	—
55	#11745664	—	4	4	no	—
56	C₂×C₅₉⋊C₈	—	2	4	no	—
57	C₂×C₅₉⋊C₈	—	4	2	no	—
58	C₂³×C₅₉	#124	1	1	yes	characteristic
59	Q₈×C₅₉	—	4	8	no	—
60	Q₈×C₅₉	—	8	2	no	—
61	Q₈×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	—	2	4	no	—
63	C₂×C₄×C₅₉	—	4	4	no	—
64	C₂×C₄×C₅₉	—	8	4	no	—
65	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₅₉	#121	1	2	yes	—
68	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
71	C₈×C₅₉	#102	1	4	yes	—
72	C₈×C₅₉	#102	1	4	yes	—
73	C₅₉⋊C₈	—	4	4	no	—
74	C₅₉⋊C₈	—	8	2	no	—
75	#9772	—	59	1	no	maximal, Sylow
76	C₂²×C₅₉	—	2	2	no	—
77	C₂²×C₅₉	#390	1	1	yes	characteristic
78	C₂²×C₅₉	#486	1	2	yes	—
79	C₄×C₅₉	—	2	2	no	—
80	C₄×C₅₉	—	8	4	no	—
81	C₄×C₅₉	C₂×#102	1	2	yes	—
82	C₄×C₅₉	#473	1	4	yes	—
83	C₄×C₅₉	#492	1	1	yes	characteristic
84	C₄×C₅₉	#493	1	1	yes	characteristic
85	C₂×#494	D₅₉	1	1	yes	characteristic
86	#1848	—	59	1	no	—
87	#1546	—	59	4	no	—
88	#1450	—	59	1	no	—
89	C₂×C₅₉	—	2	2	no	—
90	C₂×C₅₉	#1957	1	1	yes	characteristic
91	C₂×C₅₉	#1979	1	2	yes	—
92	C₂²×Q₁₆	—	2	2	no	—
93	C₂×C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
94	#494	—	2	1	no	—
95	#494	—	2	1	no	—
96	#494	Dic₅₉	1	4	yes	—
97	#431	—	59	2	no	—
98	C₂×#108	—	118	2	no	—
99	#423	—	59	2	no	—
100	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
101	#332	—	118	8	no	—
102	#322	—	118	1	no	—
103	C₈²	—	118	1	no	—
104	C₅₉	#9772	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	2	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂×Q₁₆	—	4	4	no	—
110	C₂×M₁₆	—	118	2	no	—
111	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₂×Dic₅₉	1	4	yes	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
117	#108	—	236	4	no	—
118	C₄⋊C₈	—	59	4	no	—
119	C₄⋊C₈	—	118	4	no	—
120	C₂²⋊C₈	—	118	2	no	—
121	C₄×C₈	—	118	2	no	—
122	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
123	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	#11745687	1	2	yes	—
131	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
132	Q₁₆	—	4	8	no	—
133	Q₁₆	—	4	8	no	—
134	M₁₆	—	236	4	no	—
135	C₂×C₈	—	118	4	no	—
136	C₂×C₈	—	236	2	no	—
137	C₂×C₈	#11745669	1	4	yes	—
138	C₂×C₈	#11745670	1	4	yes	—
139	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
140	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	D₄×D₅₉	1	2	yes	—
144	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
145	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
146	C₂³	#14273714	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
154	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
155	C₂×C₄	#14273698	1	2	yes	—
156	C₂×C₄	#14273723	1	1	yes	characteristic
157	C₂×C₄	C₂×#11745669	1	2	yes	—
158	C₂×C₄	C₂×#11745670	1	2	yes	—
159	C₈	—	472	2	no	—
160	C₈	—	236	4	no	—
161	C₈	#14273648	1	4	yes	—
162	C₈	#14273648	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	#16837305	1	1	yes	characteristic
165	C₂²	#16837748	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	8	4	no	—
168	C₄	C₂×#14273648	1	2	yes	—
169	C₄	#16837674	1	4	yes	—
170	C₄	#16837774	1	1	yes	characteristic
171	C₄	#16837778	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	#21424903	1	1	yes	characteristic
174	C₂	#21425012	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424268

C₂

yes

maximal, characteristic

#21422768

C₂

yes

maximal

#21422379

C₂

yes

maximal, characteristic

C₂×C₅₉×#494

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

C₂×#14273663

—

#16837426

C₂²

yes

—

#16837148

—

#16837123

—

C₈×C₅₉⋊C₈

—

C₂²×Q₁₆×C₅₉

—

C₂×C₄⋊Q₈×C₅₉

C₂²

yes

—

C₅₉×#494

—

C₅₉×#494

—

C₅₉×#494

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂²×Q₈×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#124

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₈×C₅₉

#102

yes

—

C₈×C₅₉

#102

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9772

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#486

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#102

yes

—

C₄×C₅₉

#473

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

#1957

yes

characteristic

C₂×C₅₉

#1979

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₅₉

yes

—

#494

—

#494

—

#494

Dic₅₉

yes

—

#431

—

C₂×#108

—

118

—

#423

—

100

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

101

#332

—

118

—

102

#322

—

118

—

103

C₈²

—

118

—

104

C₅₉

#9772

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×M₁₆

—

118

—

111

C₂²×C₈

C₂²⋊D₅₉

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×Dic₅₉

yes

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²×D₅₉

yes

characteristic

117

#108

—

236

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

118

—

120

C₂²⋊C₈

—

118

—

121

C₄×C₈

—

118

—

122

C₄×C₈

C₂×Dic₅₉

yes

—

123

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

124

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

#11745687

yes

—

131

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

132

Q₁₆

—

133

Q₁₆

—

134

M₁₆

—

236

—

135

C₂×C₈

—

118

—

136

C₂×C₈

—

236

—

137

C₂×C₈

#11745669

yes

—

138

C₂×C₈

#11745670

yes

—

139

C₂×C₈

C₂²⋊Dic₅₉

yes

—

140

C₂×C₈

C₂²⋊Dic₅₉

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

D₄×D₅₉

yes

—

144

C₄²

C₂²×Dic₅₉

yes

characteristic

145

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

146

C₂³

#14273714

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

154

C₂×C₄

D₄×Dic₅₉

yes

—

155

C₂×C₄

#14273698

yes

—

156

C₂×C₄

#14273723

yes

characteristic

157

C₂×C₄

C₂×#11745669

yes

—

158

C₂×C₄

C₂×#11745670

yes

—

159

C₈

—

472

—

160

C₈

—

236

—

161

C₈

#14273648

yes

—

162

C₈

#14273648

yes

—

163

C₂²

—

164

C₂²

#16837305

yes

characteristic

165

C₂²

#16837748

yes

—

166

C₄

—

167

C₄

—

168

C₄

C₂×#14273648

yes

—

169

C₄

#16837674

yes

—

170

C₄

#16837774

yes

characteristic

171

C₄

#16837778

yes

characteristic

172

C₂

—

173

C₂

#21424903

yes

characteristic

174

C₂

#21425012

yes

—

	subgroup	G/H
center	C₂²	#16837305
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#494	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂³×C₅₉	#124
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#494	D₅₉
hypocenter	C₅₉	#9772
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#16837305

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₅₉×#494

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
104. C₅₉	#9772	semidirect

H ≅ G/N

type

104. C₅₉

#9772

semidirect

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

subgroup	index
#21424268	2
#21422768	2
#21422379	2
C₂×C₅₉×#494	2
#9772	59

subgroup

index

C₂×C₅₉×#494

Subgroups — #28488153

Conjugacy classes of subgroups

Complements

Maximal subgroups