Group #28475608 — subgroups

Subgroups — #28475608

Order 15104.

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

normal subgroups only

This group has 246 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	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#21422844	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
5	#16837455	—	2	1	no	—
6	C₄×C₅₉⋊C₁₆	—	2	1	no	—
7	C₄×C₅₉⋊C₁₆	C₄	1	2	yes	—
8	#16837155	C₄	1	2	yes	—
9	#16837153	—	2	1	no	—
10	C₂×#108×C₅₉	C₄	1	2	yes	—
11	C₄×M₁₆×C₅₉	—	2	1	no	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#341	—	2	1	no	—
14	C₂×C₅₉⋊C₁₆	—	2	4	no	—
15	C₂×C₅₉⋊C₁₆	—	2	2	no	—
16	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
17	C₂²⋊(C₅₉⋊C₈)	—	4	1	no	—
18	C₄×C₅₉⋊C₈	—	4	1	no	—
19	C₂×M₁₆×C₅₉	—	2	2	no	—
20	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
23	#108×C₅₉	—	2	2	no	—
24	#108×C₅₉	C₈	1	4	yes	complemented
25	#97×C₅₉	—	4	1	no	—
26	C₄×C₈×C₅₉	—	4	1	no	—
27	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
34	M₁₆×C₅₉	—	2	4	no	—
35	M₁₆×C₅₉	—	4	2	no	—
36	C₂×C₈×C₅₉	—	2	2	no	—
37	C₂×C₈×C₅₉	—	4	2	no	—
38	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
44	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₅₉	D₈	1	1	yes	characteristic
47	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
48	C₂×C₅₉⋊C₈	—	4	2	no	—
49	C₅₉⋊C₁₆	—	2	4	no	—
50	C₂³×C₅₉	#103	1	1	yes	characteristic
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#104	1	2	yes	—
59	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
61	C₈×C₅₉	—	2	1	no	—
62	C₈×C₅₉	—	2	1	no	—
63	C₈×C₅₉	—	4	4	no	—
64	C₈×C₅₉	C₄×C₈	1	1	yes	characteristic
65	C₈×C₅₉	#97	1	1	yes	characteristic
66	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
67	C₅₉⋊C₈	—	8	2	no	—
68	#7172	—	59	1	no	maximal, Sylow
69	C₂²×C₅₉	—	2	1	no	—
70	C₂²×C₅₉	—	2	1	no	—
71	C₂²×C₅₉	Q₈⋊C₈	1	2	yes	—
72	C₂²×C₅₉	#340	1	1	yes	characteristic
73	C₄×C₅₉	—	2	1	no	—
74	C₄×C₅₉	—	2	1	no	—
75	C₄×C₅₉	D₄⋊C₈	1	2	yes	—
76	C₄×C₅₉	#336	1	1	yes	characteristic
77	C₄×C₅₉	#339	1	1	yes	characteristic
78	C₄×C₅₉	#359	1	2	yes	—
79	C₄×C₅₉	#368	1	2	yes	—
80	C₂×C₄×C₁₆	—	59	1	no	—
81	C₄×#108	D₅₉	1	1	yes	characteristic
82	#1564	—	59	1	no	—
83	C₂×C₅₉	—	2	2	no	—
84	C₂×C₅₉	#1279	1	1	yes	characteristic
85	C₂×C₅₉	#1381	1	1	yes	characteristic
86	C₂×C₅₉	#1382	1	1	yes	characteristic
87	C₂²×C₁₆	—	118	1	no	—
88	#431	—	118	1	no	—
89	C₂×#108	Dic₅₉	1	2	yes	—
90	C₄×M₁₆	—	2	1	no	—
91	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
92	C₄×C₁₆	—	59	2	no	—
93	C₄×C₁₆	—	118	1	no	—
94	#341	—	2	1	no	—
95	#335	—	59	2	no	—
96	#334	—	118	1	no	—
97	C₅₉	#7172	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	2	2	no	—
99	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
102	C₂×C₁₆	—	118	4	no	—
103	C₂×C₁₆	—	118	2	no	—
104	#108	—	2	2	no	—
105	#108	C₅₉⋊C₈	1	4	yes	complemented
106	C₄⋊C₈	—	118	2	no	—
107	C₂²⋊C₈	—	236	1	no	—
108	#97	—	4	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	—	236	1	no	—
111	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
112	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
113	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
114	C₄×C₈	D₂₃₆	1	1	yes	characteristic
115	C₂²×C₄	#11745671	1	1	yes	characteristic
116	C₂²×C₄	#11745672	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	M₁₆	—	4	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	236	2	no	—
123	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
124	C₂×C₈	#11745664	1	2	yes	—
125	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
126	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
128	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
129	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
130	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
131	C₄²	#11745669	1	1	yes	characteristic
132	C₄²	#11745670	1	1	yes	characteristic
133	C₁₆	—	118	4	no	—
134	C₂³	#14273651	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
139	C₂×C₄	#14273647	1	1	yes	characteristic
140	C₂×C₄	#14273648	1	1	yes	characteristic
141	C₂×C₄	#14273650	1	1	yes	characteristic
142	C₂×C₄	#14273654	1	2	yes	—
143	C₂×C₄	#14273658	1	1	yes	characteristic
144	C₂×C₄	#14273660	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	4	4	no	—
148	C₈	—	472	2	no	—
149	C₈	C₄×C₅₉⋊C₈	1	1	yes	characteristic
150	C₈	#14273638	1	1	yes	characteristic
151	C₈	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	Q₈⋊(C₅₉⋊C₈)	1	2	yes	—
155	C₂²	#16837164	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	D₄⋊(C₅₉⋊C₈)	1	2	yes	—
159	C₄	#16837157	1	1	yes	characteristic
160	C₄	#16837161	1	1	yes	characteristic
161	C₄	#16837208	1	2	yes	—
162	C₄	#16837231	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#21421937	1	1	yes	characteristic
165	C₂	#21422189	1	1	yes	characteristic
166	C₂	#21422193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic

#21422844

C₂

yes

maximal, characteristic

C₄×#108×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

—

#16837455

—

C₄×C₅₉⋊C₁₆

—

C₄×C₅₉⋊C₁₆

C₄

yes

—

#16837155

C₄

yes

—

#16837153

—

C₂×#108×C₅₉

C₄

yes

—

C₄×M₁₆×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#341

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

#108×C₅₉

—

#108×C₅₉

C₈

yes

complemented

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₉

SD₁₆

yes

characteristic

C₂²×C₄×C₅₉

Q₁₆

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄²

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₈

yes

—

C₂×C₈×C₅₉

M₁₆

yes

—

C₄²×C₅₉

C₂²⋊C₄

yes

characteristic

C₄²×C₅₉

C₄⋊C₄

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

SD₁₆

yes

characteristic

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

#103

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#95

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₈

yes

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₄×C₈

yes

characteristic

C₈×C₅₉

#97

yes

characteristic

C₈×C₅₉

C₄⋊C₈

yes

—

C₅₉⋊C₈

—

#7172

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

Q₈⋊C₈

yes

—

C₂²×C₅₉

#340

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

D₄⋊C₈

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₄×C₅₉

#368

yes

—

C₂×C₄×C₁₆

—

C₄×#108

D₅₉

yes

characteristic

#1564

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂²×C₁₆

—

118

—

#431

—

118

—

C₂×#108

Dic₅₉

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

118

—

#341

—

#335

—

#334

—

118

—

C₅₉

#7172

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

100

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

101

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

102

C₂×C₁₆

—

118

—

103

C₂×C₁₆

—

118

—

104

#108

—

105

#108

C₅₉⋊C₈

yes

complemented

106

C₄⋊C₈

—

118

—

107

C₂²⋊C₈

—

236

—

108

#97

—

109

C₄×C₈

—

110

C₄×C₈

—

236

—

111

C₄×C₈

Dic₁₁₈

yes

characteristic

112

C₄×C₈

C₄×D₅₉

yes

characteristic

113

C₄×C₈

C₄×D₅₉

yes

characteristic

114

C₄×C₈

D₂₃₆

yes

characteristic

115

C₂²×C₄

#11745671

yes

characteristic

116

C₂²×C₄

#11745672

yes

characteristic

117

C₂²×C₄

C₂²⋊Dic₅₉

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

236

—

123

C₂×C₈

C₂×C₅₉⋊C₈

yes

—

124

C₂×C₈

#11745664

yes

—

125

C₂×C₈

C₄×Dic₅₉

yes

characteristic

126

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

127

C₂×C₈

C₄⋊Dic₅₉

yes

characteristic

128

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

characteristic

129

C₄²

Dic₅₉⋊C₄

yes

characteristic

130

C₄²

(C₂×D₅₉)⋊C₄

yes

characteristic

131

C₄²

#11745669

yes

characteristic

132

C₄²

#11745670

yes

characteristic

133

C₁₆

—

118

—

134

C₂³

#14273651

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₅₉⋊#95

yes

characteristic

138

C₂×C₄

C₂²⋊(C₅₉⋊C₈)

yes

—

139

C₂×C₄

#14273647

yes

characteristic

140

C₂×C₄

#14273648

yes

characteristic

141

C₂×C₄

#14273650

yes

characteristic

142

C₂×C₄

#14273654

yes

—

143

C₂×C₄

#14273658

yes

characteristic

144

C₂×C₄

#14273660

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

472

—

149

C₈

C₄×C₅₉⋊C₈

yes

characteristic

150

C₈

#14273638

yes

characteristic

151

C₈

C₄⋊(C₅₉⋊C₈)

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

Q₈⋊(C₅₉⋊C₈)

yes

—

155

C₂²

#16837164

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

D₄⋊(C₅₉⋊C₈)

yes

—

159

C₄

#16837157

yes

characteristic

160

C₄

#16837161

yes

characteristic

161

C₄

#16837208

yes

—

162

C₄

#16837231

yes

—

163

C₂

—

164

C₂

#21421937

yes

characteristic

165

C₂

#21422189

yes

characteristic

166

C₂

#21422193

yes

characteristic

	subgroup	G/H
center	C₄²	#11745669
commutator	C₈×C₅₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₄×#108×C₅₉	C₂
socle	C₂²×C₅₉	#340
prime radical	C₂³×C₅₉	#103
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	C₄×#108	D₅₉
hypocenter	C₅₉	#7172
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₄²

#11745669

commutator

C₈×C₅₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₅₉

Fitting

C₄×#108×C₅₉

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄×C₈×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
24. #108×C₅₉	C₈	semidirect
97. C₅₉	#7172	semidirect
105. #108	C₅₉⋊C₈	semidirect

H ≅ G/N

type

24. #108×C₅₉

C₈

semidirect

97. C₅₉

#7172

semidirect

105. #108

C₅₉⋊C₈

semidirect

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

subgroup	index
C₂×C₄×C₅₉⋊C₁₆	2
#21422844	2
C₄×#108×C₅₉	2
#7172	59

subgroup

index

C₂×C₄×C₅₉⋊C₁₆

#21422844

C₄×#108×C₅₉

#7172

Subgroups — #28475608

Conjugacy classes of subgroups

Complements

Maximal subgroups