Group #28481684 — subgroups

Subgroups — #28481684

Order 15104.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#21433026	C₂	1	1	yes	maximal, characteristic
2	#21422857	C₂	1	2	yes	maximal
3	#21422733	C₂	1	2	yes	maximal
4	#21422007	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1307	C₂	1	1	yes	maximal, characteristic
6	#16837959	C₂²	1	1	yes	characteristic
7	#16837529	—	2	2	no	—
8	#16837526	—	2	2	no	—
9	#16837459	C₂²	1	2	yes	—
10	#16837430	C₄	1	2	yes	—
11	#16837360	C₄	1	2	yes	—
12	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
13	#16837154	—	2	2	no	—
14	#16837152	—	2	2	no	—
15	#16837143	—	2	4	no	—
16	C₅₉×#432	C₂²	1	2	yes	—
17	C₅₉×#423	C₂²	1	1	yes	characteristic
18	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
20	#14273727	Q₈	1	2	yes	—
21	#14273719	D₄	1	2	yes	—
22	#14273706	—	2	2	no	—
23	C₄×Dic₁₁₈	—	2	2	no	—
24	#14273686	C₂×C₄	1	2	yes	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	Dic₅₉⋊C₈	—	2	4	no	—
27	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
28	#14273638	—	2	2	no	—
29	#14273637	—	2	2	no	—
30	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
31	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
32	C₈×Dic₅₉	—	2	2	no	—
33	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₉	—	2	2	no	—
36	C₄⋊C₈×C₅₉	—	2	2	no	—
37	C₂²⋊C₈×C₅₉	—	2	2	no	—
38	#97×C₅₉	—	2	2	no	—
39	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	#41	1	2	yes	—
41	C₂×Dic₁₁₈	—	2	2	no	—
42	M₁₆×C₅₉	—	4	2	no	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	4	2	no	—
45	C₄²×C₅₉	C₂×D₄	1	2	yes	—
46	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₂²⋊Dic₅₉	—	2	2	no	—
49	C₄⋊Dic₅₉	—	2	2	no	—
50	C₄⋊Dic₅₉	C₄⋊C₄	1	4	yes	—
51	Dic₅₉⋊C₄	—	2	4	no	—
52	C₄×Dic₅₉	—	2	2	no	—
53	C₂×C₅₉⋊C₈	—	2	2	no	—
54	C₂×C₅₉⋊C₈	—	4	1	no	—
55	C₂×C₅₉⋊C₈	—	4	2	no	—
56	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
58	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂³×C₅₉	#117	1	1	yes	characteristic
60	C₂×C₄×C₅₉	—	2	1	no	—
61	C₂×C₄×C₅₉	—	2	1	no	—
62	C₂×C₄×C₅₉	—	4	2	no	—
63	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#121	1	2	yes	—
66	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
67	C₂×Dic₅₉	—	2	4	no	—
68	Dic₁₁₈	—	4	4	no	—
69	C₈×C₅₉	—	4	2	no	—
70	C₈×C₅₉	—	8	2	no	—
71	C₅₉⋊C₈	—	2	1	no	—
72	C₅₉⋊C₈	—	2	2	no	—
73	C₅₉⋊C₈	—	8	2	no	—
74	C₅₉⋊C₈	#101	1	2	yes	—
75	#8631	—	59	1	no	maximal, Sylow
76	C₂²×C₅₉	—	2	1	no	—
77	C₂²×C₅₉	—	2	2	no	—
78	C₂²×C₅₉	#380	1	1	yes	characteristic
79	C₄×C₅₉	—	2	1	no	—
80	C₄×C₅₉	—	2	2	no	—
81	C₄×C₅₉	—	2	2	no	—
82	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
83	C₄×C₅₉	#413	1	1	yes	characteristic
84	Dic₅₉	—	4	4	no	—
85	#2960	—	59	1	no	—
86	#1568	—	59	2	no	—
87	#1537	—	59	2	no	—
88	#1307	—	59	1	no	—
89	#1307	D₅₉	1	1	yes	characteristic
90	C₂×C₅₉	—	4	1	no	—
91	C₂×C₅₉	#1801	1	1	yes	characteristic
92	C₂×C₅₉	#1814	1	2	yes	—
93	#533	—	59	1	no	—
94	#446	—	118	2	no	—
95	C₈×Q₈	—	118	2	no	—
96	#432	—	59	2	no	—
97	#432	C₂×D₅₉	1	2	yes	—
98	#424	—	59	2	no	—
99	#423	C₂×D₅₉	1	1	yes	characteristic
100	#405	—	59	2	no	—
101	C₂×C₄×C₈	—	59	1	no	—
102	#335	—	118	2	no	—
103	#334	—	118	2	no	—
104	#330	—	118	4	no	—
105	C₅₉	#8631	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₁₆	C₄×D₅₉	1	2	yes	—
107	C₂²×C₈	—	59	1	no	—
108	C₂²×C₈	—	59	1	no	—
109	C₄⋊Q₈	—	59	2	no	—
110	#125	—	59	2	no	—
111	C₂²⋊Q₈	—	118	2	no	—
112	C₄×Q₈	—	118	2	no	—
113	#117	—	59	2	no	—
114	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	118	4	no	—
118	C₄⋊C₈	—	118	2	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	118	2	no	—
121	#97	—	2	2	no	—
122	#97	—	118	2	no	—
123	#97	—	118	2	no	—
124	C₄×C₈	—	59	2	no	—
125	C₄×C₈	—	59	2	no	—
126	C₄×C₈	—	118	2	no	—
127	C₂×Q₈	—	118	2	no	—
128	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
129	C₂²×C₄	#11745685	1	2	yes	—
130	M₁₆	—	4	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	59	1	no	—
134	C₂×C₈	—	59	1	no	—
135	C₂×C₈	—	59	2	no	—
136	C₂×C₈	—	118	2	no	—
137	C₂×C₈	—	236	1	no	—
138	C₂×C₈	—	236	2	no	—
139	C₄⋊C₄	—	59	4	no	—
140	C₄⋊C₄	—	118	4	no	—
141	C₄⋊C₄	—	118	2	no	—
142	C₂²⋊C₄	—	118	2	no	—
143	C₄²	—	118	2	no	—
144	C₄²	D₄×D₅₉	1	2	yes	—
145	C₄²	D₄×D₅₉	1	1	yes	characteristic
146	C₄²	Q₈×D₅₉	1	1	yes	characteristic
147	C₂³	#14273684	1	1	yes	characteristic
148	Q₈	—	236	4	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	118	4	no	—
153	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
154	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
155	C₂×C₄	#14273700	1	2	yes	—
156	C₂×C₄	#14273705	1	2	yes	—
157	C₈	—	4	2	no	—
158	C₈	—	8	2	no	—
159	C₈	—	59	2	no	—
160	C₈	—	118	1	no	—
161	C₈	—	118	2	no	—
162	C₈	—	472	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#16837261	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	236	4	no	—
170	C₄	#101×D₅₉	1	1	yes	characteristic
171	C₄	#16837383	1	1	yes	characteristic
172	C₂	—	4	1	no	—
173	C₂	#21424091	1	1	yes	characteristic
174	C₂	#21424149	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21433026

C₂

yes

maximal, characteristic

#21422857

C₂

yes

maximal

#21422733

C₂

yes

maximal

#21422007

C₂

yes

maximal, characteristic

C₅₉×#1307

C₂

yes

maximal, characteristic

#16837959

C₂²

yes

—

—

C₂²

yes

—

#16837430

C₄

yes

—

#16837360

C₄

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

—

—

C₅₉×#432

C₂²

yes

—

C₅₉×#423

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

#14273727

Q₈

yes

—

#14273719

D₄

yes

—

#14273706

—

C₄×Dic₁₁₈

—

#14273686

C₂×C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

Dic₅₉⋊C₈

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

#14273637

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₄⋊C₄

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂×C₅₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂³×C₅₉

#117

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

#101

yes

—

#8631

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#380

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

#1801

yes

characteristic

C₂×C₅₉

#1814

yes

—

#533

—

#446

—

118

—

C₈×Q₈

—

118

—

#432

—

#432

C₂×D₅₉

yes

—

#424

—

#423

C₂×D₅₉

yes

characteristic

100

#405

—

101

C₂×C₄×C₈

—

102

#335

—

118

—

103

#334

—

118

—

104

#330

—

118

—

105

C₅₉

#8631

yes

characteristic, complemented, Sylow

106

C₂×M₁₆

C₄×D₅₉

yes

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₄⋊Q₈

—

110

#125

—

111

C₂²⋊Q₈

—

118

—

112

C₄×Q₈

—

118

—

113

#117

—

114

C₂×C₄²

C₂²×D₅₉

yes

characteristic

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

—

118

C₄⋊C₈

—

118

—

119

C₂²⋊C₈

—

120

C₂²⋊C₈

—

118

—

121

#97

—

122

#97

—

118

—

123

#97

—

118

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

118

—

127

C₂×Q₈

—

118

—

128

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

129

C₂²×C₄

#11745685

yes

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

118

—

137

C₂×C₈

—

236

—

138

C₂×C₈

—

236

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

118

—

141

C₄⋊C₄

—

118

—

142

C₂²⋊C₄

—

118

—

143

C₄²

—

118

—

144

C₄²

D₄×D₅₉

yes

—

145

C₄²

D₄×D₅₉

yes

characteristic

146

C₄²

Q₈×D₅₉

yes

characteristic

147

C₂³

#14273684

yes

characteristic

148

Q₈

—

236

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

118

—

153

C₂×C₄

C₂²⋊C₄×D₅₉

yes

characteristic

154

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

155

C₂×C₄

#14273700

yes

—

156

C₂×C₄

#14273705

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

118

—

161

C₈

—

118

—

162

C₈

—

472

—

163

C₂²

—

164

C₂²

—

165

C₂²

#16837261

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

236

—

170

C₄

#101×D₅₉

yes

characteristic

171

C₄

#16837383

yes

characteristic

172

C₂

—

173

C₂

#21424091

yes

characteristic

174

C₂

#21424149

yes

—

	subgroup	G/H
center	C₂²	#16837261
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1307	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1307	D₅₉
hypocenter	C₅₉	#8631
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#16837261

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1307

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

N	H ≅ G/N	type
105. C₅₉	#8631	semidirect

H ≅ G/N

type

105. C₅₉

#8631

semidirect

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

subgroup	index
#21433026	2
#21422857	2
#21422733	2
#21422007	2
C₅₉×#1307	2
#8631	59

subgroup

index

C₅₉×#1307

Subgroups — #28481684

Conjugacy classes of subgroups

Complements

Maximal subgroups