Group #28482391 — subgroups

Subgroups — #28482391

Order 15104.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#21424234	C₂	1	1	yes	maximal, characteristic
2	#21422388	C₂	1	1	yes	maximal, characteristic
3	#21421954	C₂	1	2	yes	maximal
4	#21421954	C₂	1	2	yes	maximal
5	C₅₉×#1846	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	#16837426	C₂²	1	1	yes	characteristic
8	#16837426	C₂²	1	1	yes	characteristic
9	#16837167	—	2	1	no	—
10	#16837167	—	2	1	no	—
11	#16837124	C₄	1	4	yes	complemented
12	C₂×#107×C₅₉	—	2	1	no	—
13	C₂×#106×C₅₉	—	2	1	no	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#382	C₂²	1	2	yes	—
16	C₂×#11745664	—	2	1	no	—
17	C₂×#11745664	—	2	1	no	—
18	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
20	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
21	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
22	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
26	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
27	#107×C₅₉	—	4	2	no	—
28	#106×C₅₉	—	4	2	no	—
29	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₉	Q₈	1	2	yes	—
31	#95×C₅₉	—	2	4	no	—
32	C₂²×C₄×C₅₉	—	4	2	no	—
33	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	#41	1	2	yes	—
35	C₂×C₈×C₅₉	—	2	1	no	—
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₂×C₈×C₅₉	—	2	1	no	—
39	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
41	C₄⋊C₄×C₅₉	—	4	2	no	—
42	C₄⋊C₄×C₅₉	—	4	2	no	—
43	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₅₉	#41	1	2	yes	—
46	#11745664	—	4	1	no	—
47	#11745664	—	4	1	no	—
48	#11745664	—	4	1	no	—
49	#11745664	—	4	1	no	—
50	C₂×C₅₉⋊C₈	—	2	4	no	—
51	C₂×C₅₉⋊C₈	—	4	1	no	—
52	C₂×C₅₉⋊C₈	—	4	1	no	—
53	C₂³×C₅₉	#124	1	1	yes	characteristic
54	C₂×C₄×C₅₉	—	4	2	no	—
55	C₂×C₄×C₅₉	—	8	4	no	—
56	C₂×C₄×C₅₉	—	8	2	no	—
57	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#117	1	2	yes	—
59	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#125	1	2	yes	—
61	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
65	C₈×C₅₉	—	2	2	no	—
66	C₈×C₅₉	—	2	2	no	—
67	C₅₉⋊C₈	—	4	4	no	—
68	C₅₉⋊C₈	—	8	1	no	—
69	C₅₉⋊C₈	—	8	1	no	—
70	#8795	—	59	1	no	maximal, Sylow
71	C₂²×C₅₉	—	2	2	no	—
72	C₂²×C₅₉	#382	1	1	yes	characteristic
73	C₂²×C₅₉	#490	1	1	yes	characteristic
74	C₂²×C₅₉	#491	1	1	yes	characteristic
75	C₄×C₅₉	—	2	2	no	—
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	16	2	no	—
78	C₄×C₅₉	#428	1	1	yes	characteristic
79	C₄×C₅₉	#428	1	1	yes	characteristic
80	C₄×C₅₉	#489	1	1	yes	characteristic
81	C₄×C₅₉	#489	1	1	yes	characteristic
82	#1846	D₅₉	1	1	yes	characteristic
83	#1837	—	59	1	no	—
84	#1452	—	59	1	no	—
85	#1286	—	59	2	no	—
86	#1286	—	59	2	no	—
87	C₂×C₅₉	—	2	1	no	—
88	C₂×C₅₉	—	2	1	no	—
89	C₂×C₅₉	#1833	1	2	yes	—
90	C₂×C₅₉	#1836	1	1	yes	characteristic
91	#431	—	59	2	no	—
92	C₂×#107	—	2	1	no	—
93	C₂×#106	—	2	1	no	—
94	#423	—	59	1	no	—
95	#423	—	59	1	no	—
96	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
97	#382	C₂×D₅₉	1	2	yes	—
98	#341	—	118	1	no	—
99	#341	—	118	1	no	—
100	#322	—	59	4	no	—
101	C₅₉	#8795	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	118	1	no	—
103	C₂×M₁₆	—	118	1	no	—
104	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
108	#107	—	4	2	no	—
109	#106	—	4	2	no	—
110	C₄⋊C₈	—	118	2	no	—
111	C₄⋊C₈	—	118	2	no	—
112	C₄⋊C₈	—	118	2	no	—
113	C₂²⋊C₈	—	118	2	no	—
114	C₄×C₈	—	59	4	no	—
115	C₄×C₈	Dic₁₁₈	1	2	yes	—
116	C₄×C₈	C₄×D₅₉	1	2	yes	—
117	#95	—	2	4	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	#11745687	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
121	M₁₆	—	236	1	no	—
122	M₁₆	—	236	1	no	—
123	M₁₆	—	236	1	no	—
124	M₁₆	—	236	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	118	4	no	—
130	C₂×C₈	—	236	1	no	—
131	C₂×C₈	—	236	1	no	—
132	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
133	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
138	C₄²	#11745689	1	2	yes	—
139	C₂³	#14273714	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#14273685	1	2	yes	—
145	C₂×C₄	#14273716	1	1	yes	characteristic
146	C₂×C₄	#14273718	1	2	yes	—
147	C₂×C₄	#14273753	1	1	yes	characteristic
148	C₂×C₄	#14273753	1	1	yes	characteristic
149	C₂×C₄	#14273758	1	1	yes	characteristic
150	C₂×C₄	#14273758	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	472	1	no	—
154	C₈	—	472	1	no	—
155	C₈	—	236	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#16837267	1	1	yes	characteristic
158	C₂²	#16837766	1	1	yes	characteristic
159	C₂²	#16837770	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	16	2	no	—
163	C₄	#16837442	1	1	yes	characteristic
164	C₄	#16837442	1	1	yes	characteristic
165	C₄	#16837764	1	1	yes	characteristic
166	C₄	#16837764	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#21424219	1	2	yes	—
170	C₂	#21424231	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424234

C₂

yes

maximal, characteristic

#21422388

C₂

yes

maximal, characteristic

#21421954

C₂

yes

maximal

#21421954

C₂

yes

maximal

C₅₉×#1846

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

#16837426

C₂²

yes

characteristic

#16837426

C₂²

yes

—

—

C₄

yes

complemented

C₂×#107×C₅₉

—

C₂×#106×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#382

C₂²

yes

—

C₂×#11745664

—

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#107×C₅₉

—

#106×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

#41

yes

—

—

—

—

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

#125

yes

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#8795

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1833

yes

—

C₂×C₅₉

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#382

C₂×D₅₉

yes

—

#341

—

118

—

#341

—

118

—

100

#322

—

101

C₅₉

#8795

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

118

—

103

C₂×M₁₆

—

118

—

104

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

105

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₅₉

yes

characteristic

108

#107

—

109

#106

—

110

C₄⋊C₈

—

118

—

111

C₄⋊C₈

—

118

—

112

C₄⋊C₈

—

118

—

113

C₂²⋊C₈

—

118

—

114

C₄×C₈

—

115

C₄×C₈

Dic₁₁₈

yes

—

116

C₄×C₈

C₄×D₅₉

yes

—

117

#95

—

118

C₂²×C₄

—

119

C₂²×C₄

#11745687

yes

—

120

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

121

M₁₆

—

236

—

122

M₁₆

—

236

—

123

M₁₆

—

236

—

124

M₁₆

—

236

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

118

—

130

C₂×C₈

—

236

—

131

C₂×C₈

—

236

—

132

C₂×C₈

Dic₅₉⋊C₄

yes

—

133

C₂×C₈

Dic₅₉⋊C₄

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

C₂×Dic₁₁₈

yes

characteristic

137

C₄²

C₂×C₄×D₅₉

yes

characteristic

138

C₄²

#11745689

yes

—

139

C₂³

#14273714

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

144

C₂×C₄

#14273685

yes

—

145

C₂×C₄

#14273716

yes

characteristic

146

C₂×C₄

#14273718

yes

—

147

C₂×C₄

#14273753

yes

characteristic

148

C₂×C₄

#14273753

yes

characteristic

149

C₂×C₄

#14273758

yes

characteristic

150

C₂×C₄

#14273758

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

472

—

154

C₈

—

472

—

155

C₈

—

236

—

156

C₂²

—

157

C₂²

#16837267

yes

characteristic

158

C₂²

#16837766

yes

characteristic

159

C₂²

#16837770

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#16837442

yes

characteristic

164

C₄

#16837442

yes

characteristic

165

C₄

#16837764

yes

characteristic

166

C₄

#16837764

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#21424219

yes

—

170

C₂

#21424231

yes

characteristic

	subgroup	G/H
center	C₂²	#16837267
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1846	C₂
socle	C₂²×C₅₉	#382
prime radical	C₂³×C₅₉	#124
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1846	D₅₉
hypocenter	C₅₉	#8795
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

subgroup

G/H

center

C₂²

#16837267

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1846

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

N	H ≅ G/N	type
11. #16837124	C₄	semidirect
101. C₅₉	#8795	semidirect

H ≅ G/N

type

11. #16837124

C₄

semidirect

101. C₅₉

#8795

semidirect

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

subgroup	index
#21424234	2
#21422388	2
#21421954	2
C₅₉×#1846	2
#8795	59

subgroup

index

C₅₉×#1846

Subgroups — #28482391

Conjugacy classes of subgroups

Complements

Maximal subgroups