Group #28505038 — subgroups

Subgroups — #28505038

Order 15104.

322 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 151 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	#21423782	C₂	1	1	yes	maximal, characteristic
3	#21422173	C₂	1	4	yes	maximal
4	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14273638	C₄	1	1	yes	characteristic
7	C₂×#14273637	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
9	#16837248	C₄	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
11	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
16	C₂×C₅₉⋊C₁₆	—	2	2	no	—
17	C₂×C₅₉⋊C₁₆	—	2	4	no	—
18	#14273638	—	2	2	no	—
19	#14273637	—	2	4	no	—
20	C₈×Dic₅₉	—	2	2	no	—
21	C₈×Dic₅₉	—	2	2	no	—
22	C₅₉⋊#95	C₈	1	2	yes	—
23	C₅₉⋊#95	C₈	1	2	yes	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₉	—	2	2	no	—
28	C₂×C₁₆×C₅₉	—	2	4	no	—
29	C₄×C₈×C₅₉	D₄	1	2	yes	—
30	C₄×C₈×C₅₉	Q₈	1	2	yes	—
31	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₉	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₅₉	—	2	2	no	—
36	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₅₉	#41	1	2	yes	—
40	C₂×C₈×C₅₉	#41	1	2	yes	—
41	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
42	C₄²×C₅₉	M₁₆	1	2	yes	—
43	C₄×Dic₅₉	—	2	2	no	—
44	C₄×Dic₅₉	—	2	2	no	—
45	C₂×C₅₉⋊C₈	—	2	2	no	—
46	C₂×C₅₉⋊C₈	—	2	2	no	—
47	C₂×C₅₉⋊C₈	M₁₆	1	4	yes	—
48	C₁₆×C₅₉	—	4	2	no	—
49	C₅₉⋊C₁₆	—	4	2	no	—
50	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#117	1	2	yes	—
57	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
59	C₂×Dic₅₉	—	2	4	no	—
60	C₂×Dic₅₉	—	2	4	no	—
61	C₂×Dic₅₉	—	2	4	no	—
62	C₈×C₅₉	—	2	2	no	—
63	C₈×C₅₉	#124	1	1	yes	characteristic
64	C₈×C₅₉	#124	1	1	yes	characteristic
65	C₈×C₅₉	#125	1	2	yes	—
66	C₅₉⋊C₈	—	2	4	no	—
67	#12880	—	59	1	no	maximal, Sylow
68	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₅₉	#431	1	2	yes	—
70	C₂²×C₅₉	#504	1	2	yes	—
71	C₂²×C₅₉	#504	1	2	yes	—
72	C₄×C₅₉	—	2	2	no	—
73	C₄×C₅₉	—	2	2	no	—
74	C₄×C₅₉	#382	1	1	yes	characteristic
75	C₄×C₅₉	#423	1	1	yes	characteristic
76	C₄×C₅₉	#432	1	2	yes	—
77	Dic₅₉	—	4	4	no	—
78	C₂×C₄⋊C₁₆	—	59	1	no	—
79	C₂×C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
80	#1726	—	59	1	no	—
81	#1374	—	59	4	no	—
82	C₂×C₅₉	#1831	1	1	yes	characteristic
83	C₂×C₅₉	#2151	1	1	yes	characteristic
84	C₂×C₅₉	#2151	1	1	yes	characteristic
85	C₂×C₅₉	#2164	1	4	yes	—
86	C₂²×C₁₆	—	59	2	no	—
87	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
88	C₂×#97	—	59	2	no	—
89	C₂×#97	—	59	1	no	—
90	C₂×C₄×C₈	—	59	2	no	—
91	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
92	#376	—	59	1	no	—
93	C₄⋊C₁₆	—	2	2	no	—
94	C₄⋊C₁₆	—	118	2	no	—
95	C₅₉	#12880	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	59	2	no	—
97	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
99	C₂×C₄²	—	59	2	no	—
100	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	118	4	no	—
104	C₂×C₁₆	—	118	2	no	—
105	#97	—	118	4	no	—
106	#97	—	118	2	no	—
107	C₄×C₈	—	118	2	no	—
108	C₄×C₈	—	118	2	no	—
109	C₄×C₈	Dic₁₁₈	1	2	yes	—
110	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
111	#95	—	59	2	no	—
112	#95	—	59	2	no	—
113	C₂²×C₄	—	59	4	no	—
114	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
115	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
116	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	59	4	no	—
119	C₂×C₈	—	118	2	no	—
120	C₂×C₈	—	118	2	no	—
121	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
122	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
123	C₂×C₈	#11745687	1	2	yes	—
124	C₂×C₈	#11745689	1	2	yes	—
125	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
126	C₄²	—	118	2	no	—
127	C₄²	—	118	2	no	—
128	C₄²	#11745659	1	2	yes	—
129	C₄²	Dic₅₉⋊C₄	1	2	yes	—
130	C₁₆	—	4	2	no	—
131	C₁₆	—	236	2	no	—
132	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	118	4	no	—
136	C₂×C₄	—	118	4	no	—
137	C₂×C₄	—	118	4	no	—
138	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
139	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
140	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
141	C₂×C₄	#14273685	1	2	yes	—
142	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
143	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	118	4	no	—
146	C₈	#14273714	1	1	yes	characteristic
147	C₈	#14273716	1	1	yes	characteristic
148	C₈	#14273718	1	2	yes	—
149	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
150	C₂²	#16837454	1	2	yes	—
151	C₂²	#16837818	1	2	yes	—
152	C₂²	#16837820	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	236	4	no	—
156	C₄	#16837267	1	1	yes	characteristic
157	C₄	#16837425	1	1	yes	characteristic
158	C₄	#16837458	1	2	yes	—
159	C₂	#21424212	1	1	yes	characteristic
160	C₂	#21425854	1	1	yes	characteristic
161	C₂	#21425856	1	1	yes	characteristic
162	C₂	#21425902	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₅₉⋊C₁₆)

C₂

yes

maximal, characteristic

#21423782

C₂

yes

maximal, characteristic

#21422173

C₂

yes

maximal

C₂×C₅₉×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

—

C₂×#14273638

C₄

yes

characteristic

C₂×#14273637

C₄

yes

—

C₂×C₈×Dic₅₉

C₂²

yes

—

#16837248

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₁₆)

—

C₂²×C₁₆×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

#14273638

—

#14273637

—

C₈×Dic₅₉

—

C₈×Dic₅₉

—

C₅₉⋊#95

C₈

yes

—

C₅₉⋊#95

C₈

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×Dic₅₉

C₂×C₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

M₁₆

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

M₁₆

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₈×C₅₉

yes

—

C₅₉⋊C₈

—

#12880

—

maximal, Sylow

C₂²×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₉

#431

yes

—

C₂²×C₅₉

#504

yes

—

C₂²×C₅₉

#504

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

D₅₉

yes

characteristic

#1726

—

#1374

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₉

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

C₅₉

#12880

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₄×D₅₉

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

118

—

104

C₂×C₁₆

—

118

—

105

#97

—

118

—

106

#97

—

118

—

107

C₄×C₈

—

118

—

108

C₄×C₈

—

118

—

109

C₄×C₈

Dic₁₁₈

yes

—

110

C₄×C₈

C₂²⋊D₅₉

yes

—

111

#95

—

112

#95

—

113

C₂²×C₄

—

114

C₂²×C₄

C₈×D₅₉

yes

characteristic

115

C₂²×C₄

C₈×D₅₉

yes

characteristic

116

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

118

—

120

C₂×C₈

—

118

—

121

C₂×C₈

Dic₅₉⋊C₄

yes

—

122

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

123

C₂×C₈

#11745687

yes

—

124

C₂×C₈

#11745689

yes

—

125

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

126

C₄²

—

118

—

127

C₄²

—

118

—

128

C₄²

#11745659

yes

—

129

C₄²

Dic₅₉⋊C₄

yes

—

130

C₁₆

—

131

C₁₆

—

236

—

132

C₂³

C₂×C₈×D₅₉

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

118

—

136

C₂×C₄

—

118

—

137

C₂×C₄

—

118

—

138

C₂×C₄

Dic₅₉⋊C₈

yes

—

139

C₂×C₄

Dic₅₉⋊C₈

yes

—

140

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

141

C₂×C₄

#14273685

yes

—

142

C₂×C₄

M₁₆×D₅₉

yes

—

143

C₂×C₄

C₂×#11745659

yes

characteristic

144

C₈

—

145

C₈

—

118

—

146

C₈

#14273714

yes

characteristic

147

C₈

#14273716

yes

characteristic

148

C₈

#14273718

yes

—

149

C₂²

C₂×Dic₅₉⋊C₈

yes

characteristic

150

C₂²

#16837454

yes

—

151

C₂²

#16837818

yes

—

152

C₂²

#16837820

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

236

—

156

C₄

#16837267

yes

characteristic

157

C₄

#16837425

yes

characteristic

158

C₄

#16837458

yes

—

159

C₂

#21424212

yes

characteristic

160

C₂

#21425854

yes

characteristic

161

C₂

#21425856

yes

characteristic

162

C₂

#21425902

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator, socle, prime radical	C₂³×C₅₉	C₂²×C₈
Fitting	C₂×C₅₉×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#12880
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₅₉

commutator, socle, prime radical

C₂³×C₅₉

C₂²×C₈

Fitting

C₂×C₅₉×C₄⋊C₁₆

C₂

cosocle

C₂²×C₈×C₅₉

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
95. C₅₉	#12880	semidirect

H ≅ G/N

type

95. C₅₉

#12880

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
#21423782	2
#21422173	2
C₂×C₅₉×C₄⋊C₁₆	2
#12880	59

subgroup

index

C₂×C₄⋊(C₅₉⋊C₁₆)

#21423782

#21422173

C₂×C₅₉×C₄⋊C₁₆

#12880

Subgroups — #28505038

Conjugacy classes of subgroups

Complements

Maximal subgroups