Group #28737310 — subgroups

Subgroups — #28737310

Order 15104.

712 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 127 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	#21425612	C₂	1	3	yes	maximal, complemented
2	#21425495	C₂	1	2	yes	maximal, complemented
3	#21425493	C₂	1	6	yes	maximal, complemented
4	#21424016	C₂	1	3	yes	maximal
5	C₅₉×#2091	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₂₃₆	C₂²	1	1	yes	characteristic
7	#16837451	C₂²	1	3	yes	complemented
8	#16837450	C₂²	1	6	yes	complemented
9	C₂×#14273643	C₂²	1	6	yes	—
10	C₂×#14273643	C₂²	1	6	yes	—
11	#16837366	C₂²	1	3	yes	—
12	C₅₉⋊#338	C₂²	1	3	yes	complemented
13	C₅₉×#430	C₂²	1	3	yes	complemented
14	C₅₉×#407	C₂²	1	3	yes	—
15	C₅₉×#338	C₂²	1	1	yes	characteristic, complemented
16	C₂⁴×D₅₉	Q₈	1	2	yes	—
17	C₂²×D₂₃₆	—	4	3	no	—
18	C₂²×D₂₃₆	—	4	3	no	—
19	C₂²×D₂₃₆	Q₈	1	2	yes	—
20	C₂²×D₂₃₆	C₂³	1	2	yes	—
21	C₂×#11745664	C₂³	1	6	yes	—
22	#14273663	—	4	3	no	—
23	#14273662	—	4	6	no	—
24	#14273643	—	4	6	no	—
25	#14273643	—	4	6	no	—
26	C₂²⋊(C₅₉⋊C₈)	—	4	3	no	—
27	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₅₉	C₂³	1	3	yes	—
29	C₂×M₁₆×C₅₉	C₂³	1	3	yes	—
30	#108×C₅₉	—	4	3	no	—
31	C₂²⋊C₈×C₅₉	—	4	3	no	—
32	C₂⁴×C₅₉	C₂×Q₈	1	1	yes	characteristic
33	C₂³×D₅₉	—	4	6	no	—
34	C₂³×D₅₉	—	8	4	no	—
35	C₂³×D₅₉	C₂×Q₈	1	4	yes	—
36	C₂²×C₄×C₅₉	—	2	3	no	—
37	C₂²×C₄×C₅₉	—	2	3	no	—
38	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
40	C₂×D₂₃₆	—	2	6	no	—
41	C₂×D₂₃₆	—	2	6	no	—
42	C₂×D₂₃₆	—	4	6	no	—
43	C₂×D₂₃₆	—	4	4	no	—
44	C₂×D₂₃₆	—	4	12	no	—
45	M₁₆×C₅₉	—	2	6	no	—
46	M₁₆×C₅₉	—	2	6	no	—
47	C₂×C₈×C₅₉	—	2	3	no	—
48	C₂×C₈×C₅₉	—	2	3	no	—
49	#11745664	—	2	6	no	—
50	#11745664	—	2	6	no	—
51	C₂×C₅₉⋊C₈	—	2	6	no	—
52	C₂³×C₅₉	—	2	3	no	—
53	C₂³×C₅₉	—	4	2	no	—
54	C₂³×C₅₉	C₂²×Q₈	1	1	yes	characteristic
55	C₂²×D₅₉	—	2	12	no	—
56	C₂²×D₅₉	—	4	8	no	—
57	C₂²×D₅₉	—	8	12	no	—
58	C₂²×D₅₉	—	8	12	no	—
59	C₂²×D₅₉	—	8	4	no	—
60	C₂×C₄×C₅₉	—	2	3	no	—
61	C₂×C₄×C₅₉	—	4	1	no	—
62	C₂×C₄×C₅₉	—	4	3	no	—
63	C₂×C₄×C₅₉	H(5,𝔽₂)	1	3	yes	—
64	C₂×C₄×C₅₉	H(5,𝔽₂)	1	3	yes	—
65	D₂₃₆	—	4	8	no	—
66	D₂₃₆	—	4	2	no	—
67	D₂₃₆	—	4	6	no	—
68	C₈×C₅₉	—	4	3	no	—
69	C₈×C₅₉	—	4	3	no	—
70	C₅₉⋊C₈	—	4	6	no	—
71	#32875	—	59	1	no	maximal, Sylow
72	C₂²×C₅₉	—	2	6	no	—
73	C₂²×C₅₉	—	4	1	no	—
74	C₂²×C₅₉	—	4	3	no	—
75	C₂²×C₅₉	—	4	1	no	—
76	C₂²×C₅₉	#543	1	3	yes	—
77	C₂×D₅₉	—	4	4	no	—
78	C₂×D₅₉	—	8	16	no	—
79	C₂×D₅₉	—	8	12	no	—
80	C₄×C₅₉	—	4	1	no	—
81	C₄×C₅₉	#560	1	1	yes	characteristic
82	C₄×C₅₉	#560	1	3	yes	—
83	#2091	—	59	3	no	—
84	#2091	D₅₉	1	1	yes	characteristic, complemented
85	#2069	—	59	6	no	—
86	#2069	—	59	2	no	—
87	#1783	—	59	3	no	—
88	C₂×C₅₉	—	2	3	no	—
89	C₂×C₅₉	—	4	2	no	—
90	C₂×C₅₉	#2842	1	1	yes	characteristic
91	D₅₉	—	8	4	no	—
92	C₂³×D₄	—	59	1	no	—
93	#430	—	59	6	no	—
94	#430	—	59	3	no	—
95	#430	C₂×D₅₉	1	3	yes	complemented
96	C₂×#100	—	59	6	no	—
97	C₂×#100	—	59	6	no	—
98	#407	—	59	3	no	—
99	#407	C₂×D₅₉	1	3	yes	—
100	#338	—	59	3	no	—
101	#338	C₂×D₅₉	1	1	yes	characteristic, complemented
102	C₅₉	#32875	1	1	yes	characteristic, complemented, Sylow
103	C₂⁵	—	59	2	no	—
104	C₂²×D₄	—	59	2	no	—
105	C₂²×D₄	—	59	2	no	—
106	C₂²×D₄	—	236	3	no	—
107	C₂²×D₄	—	236	3	no	—
108	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
109	C₂×M₁₆	—	59	6	no	—
110	C₂×M₁₆	C₂²×D₅₉	1	3	yes	—
111	C₂×M₁₆	C₂²×D₅₉	1	3	yes	—
112	#108	—	4	3	no	—
113	#108	—	236	3	no	—
114	#108	—	236	6	no	—
115	#100	—	236	6	no	—
116	#100	—	236	6	no	—
117	C₂²⋊C₈	—	4	3	no	—
118	C₂²⋊C₈	—	236	3	no	—
119	C₂⁴	—	59	4	no	—
120	C₂⁴	—	472	4	no	—
121	C₂⁴	—	236	6	no	—
122	C₂⁴	Q₈×D₅₉	1	1	yes	characteristic
123	C₂×D₄	—	118	6	no	—
124	C₂×D₄	—	118	6	no	—
125	C₂×D₄	—	236	6	no	—
126	C₂×D₄	—	236	4	no	—
127	C₂×D₄	—	236	12	no	—
128	C₂²×C₄	—	2	3	no	—
129	C₂²×C₄	—	2	3	no	—
130	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
131	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
132	M₁₆	—	2	6	no	—
133	M₁₆	—	2	6	no	—
134	M₁₆	—	118	6	no	—
135	M₁₆	—	118	6	no	—
136	C₂×C₈	—	2	3	no	—
137	C₂×C₈	—	2	3	no	—
138	C₂×C₈	—	118	6	no	—
139	C₂³	—	2	3	no	—
140	C₂³	—	4	2	no	—
141	C₂³	—	118	12	no	—
142	C₂³	—	472	12	no	—
143	C₂³	—	472	4	no	—
144	C₂³	—	472	12	no	—
145	C₂³	—	236	8	no	—
146	C₂³	C₂×Q₈×D₅₉	1	1	yes	characteristic
147	D₄	—	236	8	no	—
148	D₄	—	236	2	no	—
149	D₄	—	236	6	no	—
150	C₂×C₄	—	2	3	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	3	no	—
153	C₂×C₄	#14273772	1	3	yes	—
154	C₂×C₄	#14273773	1	3	yes	—
155	C₈	—	4	3	no	—
156	C₈	—	4	3	no	—
157	C₈	—	236	6	no	—
158	C₂²	—	2	6	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	4	3	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	472	12	no	—
163	C₂²	—	472	16	no	—
164	C₂²	—	236	4	no	—
165	C₂²	#16838027	1	3	yes	—
166	C₄	—	4	1	no	—
167	C₄	#16838139	1	1	yes	characteristic
168	C₄	#16838142	1	3	yes	—
169	C₂	—	2	3	no	—
170	C₂	—	4	2	no	—
171	C₂	—	472	4	no	—
172	C₂	#21432361	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425612

C₂

yes

maximal, complemented

#21425495

C₂

yes

maximal, complemented

#21425493

C₂

yes

maximal, complemented

#21424016

C₂

yes

maximal

C₅₉×#2091

C₂

yes

maximal, characteristic, complemented

C₂³×D₂₃₆

C₂²

yes

characteristic

#16837451

C₂²

yes

complemented

#16837450

C₂²

yes

complemented

C₂×#14273643

C₂²

yes

—

C₂×#14273643

C₂²

yes

—

#16837366

C₂²

yes

—

C₅₉⋊#338

C₂²

yes

complemented

C₅₉×#430

C₂²

yes

complemented

C₅₉×#407

C₂²

yes

—

C₅₉×#338

C₂²

yes

characteristic, complemented

C₂⁴×D₅₉

Q₈

yes

—

C₂²×D₂₃₆

—

C₂²×D₂₃₆

—

C₂²×D₂₃₆

Q₈

yes

—

C₂²×D₂₃₆

C₂³

yes

—

C₂×#11745664

C₂³

yes

—

—

—

—

—

C₂²⋊(C₅₉⋊C₈)

—

C₂³×C₄×C₅₉

C₂³

yes

characteristic

C₂×M₁₆×C₅₉

C₂³

yes

—

C₂×M₁₆×C₅₉

C₂³

yes

—

#108×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₂⁴×C₅₉

C₂×Q₈

yes

characteristic

C₂³×D₅₉

—

C₂³×D₅₉

—

C₂³×D₅₉

C₂×Q₈

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂²×Q₈

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

H(5,𝔽₂)

yes

—

C₂×C₄×C₅₉

H(5,𝔽₂)

yes

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

#32875

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#543

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#560

yes

characteristic

C₄×C₅₉

#560

yes

—

#2091

—

#2091

D₅₉

yes

characteristic, complemented

#2069

—

#2069

—

#1783

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#2842

yes

characteristic

D₅₉

—

C₂³×D₄

—

#430

—

#430

—

#430

C₂×D₅₉

yes

—

—

—

C₂×D₅₉

yes

—

100

#338

—

101

#338

C₂×D₅₉

yes

characteristic, complemented

102

C₅₉

#32875

yes

characteristic, complemented, Sylow

103

C₂⁵

—

104

C₂²×D₄

—

105

C₂²×D₄

—

106

C₂²×D₄

—

236

—

107

C₂²×D₄

—

236

—

108

C₂³×C₄

C₂²×D₅₉

yes

characteristic

109

C₂×M₁₆

—

110

C₂×M₁₆

C₂²×D₅₉

yes

—

111

C₂×M₁₆

C₂²×D₅₉

yes

—

112

#108

—

113

#108

—

236

—

114

#108

—

236

—

115

#100

—

236

—

116

#100

—

236

—

117

C₂²⋊C₈

—

118

C₂²⋊C₈

—

236

—

119

C₂⁴

—

120

C₂⁴

—

472

—

121

C₂⁴

—

236

—

122

C₂⁴

Q₈×D₅₉

yes

characteristic

123

C₂×D₄

—

118

—

124

C₂×D₄

—

118

—

125

C₂×D₄

—

236

—

126

C₂×D₄

—

236

—

127

C₂×D₄

—

236

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

Q₈×D₅₉

yes

characteristic

131

C₂²×C₄

C₂³×D₅₉

yes

characteristic

132

M₁₆

—

133

M₁₆

—

134

M₁₆

—

118

—

135

M₁₆

—

118

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

118

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

118

—

142

C₂³

—

472

—

143

C₂³

—

472

—

144

C₂³

—

472

—

145

C₂³

—

236

—

146

C₂³

C₂×Q₈×D₅₉

yes

characteristic

147

D₄

—

236

—

148

D₄

—

236

—

149

D₄

—

236

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#14273772

yes

—

154

C₂×C₄

#14273773

yes

—

155

C₈

—

156

C₈

—

157

C₈

—

236

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

472

—

163

C₂²

—

472

—

164

C₂²

—

236

—

165

C₂²

#16838027

yes

—

166

C₄

—

167

C₄

#16838139

yes

characteristic

168

C₄

#16838142

yes

—

169

C₂

—

170

C₂

—

171

C₂

—

472

—

172

C₂

#21432361

yes

characteristic

	subgroup	G/H
center	C₂	#21432361
commutator, cosocle	C₂²×C₄×C₅₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₅₉×#2091	C₂
socle	C₂×C₅₉	#2842
prime radical	C₂³×D₂₃₆	C₂²
hypercenter	#2091	D₅₉
hypocenter	C₅₉	#32875
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2091

subgroup

G/H

center

C₂

#21432361

commutator, cosocle

C₂²×C₄×C₅₉

C₂⁴

Frattini

C₂²×C₄

C₂³×D₅₉

Fitting

C₅₉×#2091

C₂

socle

C₂×C₅₉

#2842

prime radical

C₂³×D₂₃₆

C₂²

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2091

N	H ≅ G/N	type
1. #21425612	C₂	semidirect
2. #21425495	C₂	semidirect
3. #21425493	C₂	semidirect
5. C₅₉×#2091	C₂	semidirect
7. #16837451	C₂²	semidirect
8. #16837450	C₂²	semidirect
12. C₅₉⋊#338	C₂²	semidirect
13. C₅₉×#430	C₂²	semidirect
15. C₅₉×#338	C₂²	semidirect
84. #2091	D₅₉	semidirect
95. #430	C₂×D₅₉	semidirect
101. #338	C₂×D₅₉	semidirect
102. C₅₉	#32875	semidirect

H ≅ G/N

type

1. #21425612

C₂

semidirect

2. #21425495

C₂

semidirect

3. #21425493

C₂

semidirect

5. C₅₉×#2091

C₂

semidirect

7. #16837451

C₂²

semidirect

8. #16837450

C₂²

semidirect

12. C₅₉⋊#338

C₂²

semidirect

13. C₅₉×#430

C₂²

semidirect

15. C₅₉×#338

C₂²

semidirect

84. #2091

D₅₉

semidirect

95. #430

C₂×D₅₉

semidirect

101. #338

C₂×D₅₉

semidirect

102. C₅₉

#32875

semidirect

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

subgroup	index
#21425612	2
#21425495	2
#21425493	2
#21424016	2
C₅₉×#2091	2
#32875	59

subgroup

index

C₅₉×#2091

Subgroups — #28737310

Conjugacy classes of subgroups

Complements

Maximal subgroups