Group #28504870 — subgroups

Subgroups — #28504870

Order 15104.

394 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16837232	C₂	1	2	yes	maximal, complemented
2	#21424134	C₂	1	2	yes	maximal
3	#21422190	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273663	C₂²	1	4	yes	—
6	#16837366	—	2	2	no	—
7	#16837232	—	2	4	no	—
8	#16837167	—	2	2	no	—
9	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₉	—	2	4	no	—
11	C₂²×C₁₆×C₅₉	C₄	1	4	yes	—
12	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#11745664	—	2	4	no	—
15	#14273663	—	2	4	no	—
16	#14273663	—	2	4	no	—
17	C₂²⋊(C₅₉⋊C₈)	—	4	4	no	—
18	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₉	—	2	2	no	—
20	C₂²×C₈×C₅₉	—	2	2	no	—
21	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	Q₈	1	2	yes	—
24	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₂×C₁₆×C₅₉	—	2	2	no	—
26	C₂×C₁₆×C₅₉	—	2	8	no	—
27	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₅₉	D₄	1	4	yes	—
30	C₂⁴×C₅₉	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	—	2	2	no	—
32	C₂²×C₄×C₅₉	—	2	2	no	—
33	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅₉	D₈	1	2	yes	—
35	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	—	2	2	no	—
38	C₂×C₈×C₅₉	—	2	2	no	—
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	—	2	4	no	—
41	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₅₉	#41	1	2	yes	—
46	#11745664	—	4	4	no	—
47	#11745664	—	4	4	no	—
48	C₂×C₅₉⋊C₈	—	4	4	no	—
49	C₁₆×C₅₉	—	2	2	no	—
50	C₁₆×C₅₉	C₂²⋊C₄	1	4	yes	—
51	C₂³×C₅₉	—	2	4	no	—
52	C₂³×C₅₉	—	2	2	no	—
53	C₂³×C₅₉	#107	1	2	yes	—
54	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	2	2	no	—
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	2	4	no	—
59	C₂×C₄×C₅₉	#107	1	2	yes	—
60	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#121	1	2	yes	—
62	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#135	1	2	yes	—
64	C₈×C₅₉	—	2	2	no	—
65	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₅₉	#117	1	1	yes	characteristic
67	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
68	C₅₉⋊C₈	—	8	4	no	—
69	#12837	—	59	1	no	maximal, Sylow
70	C₂²×C₅₉	—	2	8	no	—
71	C₂²×C₅₉	—	2	2	no	—
72	C₂²×C₅₉	—	2	2	no	—
73	C₂²×C₅₉	—	2	2	no	—
74	C₂²×C₅₉	#368	1	4	yes	complemented
75	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
76	C₂²×C₅₉	#467	1	2	yes	—
77	C₄×C₅₉	—	2	2	no	—
78	C₄×C₅₉	#380	1	1	yes	characteristic
79	C₄×C₅₉	#427	1	1	yes	characteristic
80	C₄×C₅₉	#466	1	2	yes	—
81	C₂³×C₁₆	D₅₉	1	1	yes	characteristic
82	C₂×#368	—	59	2	no	—
83	#1810	—	59	2	no	—
84	#1381	—	59	2	no	—
85	C₂×C₅₉	—	2	4	no	—
86	C₂×C₅₉	—	2	2	no	—
87	C₂×C₅₉	#1808	1	1	yes	characteristic
88	C₂×C₅₉	C₂×#368	1	2	yes	—
89	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
90	C₂²×C₁₆	—	2	4	no	—
91	C₂²×C₁₆	Dic₅₉	1	4	yes	—
92	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
93	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
94	C₂×#108	—	59	4	no	—
95	#407	—	118	2	no	—
96	#368	—	118	4	no	—
97	#341	—	118	2	no	—
98	C₅₉	#12837	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
100	C₂×M₁₆	—	118	4	no	—
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	Dic₁₁₈	1	2	yes	—
104	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
106	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	2	8	no	—
109	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
110	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
111	C₂×C₁₆	C₂²⋊D₅₉	1	4	yes	—
112	#108	—	118	4	no	—
113	#108	—	118	4	no	—
114	C₂²⋊C₈	—	236	4	no	—
115	C₂⁴	Dic₂₃₆	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	D₄₇₂	1	2	yes	—
119	C₂²×C₄	Dic₂₃₆	1	1	yes	characteristic
120	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
121	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
122	M₁₆	—	236	4	no	—
123	M₁₆	—	236	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	236	4	no	—
129	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
130	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
131	C₂×C₈	#11745685	1	2	yes	—
132	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
133	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
134	C₁₆	—	2	2	no	—
135	C₁₆	C₂²⋊Dic₅₉	1	4	yes	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	4	no	—
138	C₂³	#14273661	1	2	yes	—
139	C₂³	C₂×Dic₂₃₆	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	#14273661	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
146	C₂×C₄	#14273701	1	2	yes	—
147	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
148	C₂×C₄	#14273749	1	2	yes	—
149	C₈	—	2	2	no	—
150	C₈	—	472	4	no	—
151	C₈	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
152	C₈	#14273686	1	1	yes	characteristic
153	C₈	#14273706	1	2	yes	—
154	C₂²	—	2	8	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	#16837232	1	4	yes	complemented
159	C₂²	C₂×#14273661	1	1	yes	characteristic
160	C₂²	#16837643	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	#16837262	1	1	yes	characteristic
163	C₄	#16837440	1	1	yes	characteristic
164	C₄	#16837638	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	—	2	4	no	—
167	C₂	#21424125	1	1	yes	characteristic
168	C₂	C₂×#16837232	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837232

C₂

yes

maximal, complemented

#21424134

C₂

yes

maximal

#21422190

C₂

yes

maximal, complemented

C₂³×C₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×#14273663

C₂²

yes

—

#16837366

—

#16837232

—

#16837167

—

C₂³×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₁₆×C₅₉

—

C₂²×C₁₆×C₅₉

C₄

yes

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

—

—

—

C₂²⋊(C₅₉⋊C₈)

—

C₂³×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

Q₈

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

D₄

yes

—

C₂⁴×C₅₉

Q₁₆

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₄⋊C₄

yes

—

C₂²×C₄×C₅₉

D₈

yes

—

C₂²×C₄×C₅₉

Q₁₆

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

C₂²⋊C₄

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

#107

yes

—

C₂³×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#107

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₉

#117

yes

characteristic

C₈×C₅₉

C₂²⋊Q₈

yes

—

C₅₉⋊C₈

—

#12837

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂³×C₁₆

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1808

yes

characteristic

C₂×C₅₉

C₂×#368

yes

—

C₂³×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

Dic₅₉

yes

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

C₂×#108

—

#407

—

118

—

#368

—

118

—

#341

—

118

—

C₅₉

#12837

yes

characteristic, complemented, Sylow

C₂³×C₄

D₂₃₆

yes

characteristic

100

C₂×M₁₆

—

118

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

Dic₁₁₈

yes

—

104

C₂²×C₈

D₂₃₆

yes

characteristic

105

C₂²×C₈

C₂×Dic₅₉

yes

—

106

C₂²×C₈

C₂²×D₅₉

yes

characteristic

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₂×Dic₅₉

yes

—

110

C₂×C₁₆

C₂×Dic₅₉

yes

—

111

C₂×C₁₆

C₂²⋊D₅₉

yes

—

112

#108

—

118

—

113

#108

—

118

—

114

C₂²⋊C₈

—

236

—

115

C₂⁴

Dic₂₃₆

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

D₄₇₂

yes

—

119

C₂²×C₄

Dic₂₃₆

yes

characteristic

120

C₂²×C₄

C₄⋊Dic₅₉

yes

—

121

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

122

M₁₆

—

236

—

123

M₁₆

—

236

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

236

—

129

C₂×C₈

C₄⋊Dic₅₉

yes

—

130

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

131

C₂×C₈

#11745685

yes

—

132

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

133

C₂×C₈

C₂×C₂²⋊D₅₉

yes

—

134

C₁₆

—

135

C₁₆

C₂²⋊Dic₅₉

yes

—

136

C₂³

—

137

C₂³

—

138

C₂³

#14273661

yes

—

139

C₂³

C₂×Dic₂₃₆

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#14273661

yes

—

145

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

146

C₂×C₄

#14273701

yes

—

147

C₂×C₄

C₂×D₄₇₂

yes

characteristic

148

C₂×C₄

#14273749

yes

—

149

C₈

—

150

C₈

—

472

—

151

C₈

C₂×C₂²⋊Dic₅₉

yes

characteristic

152

C₈

#14273686

yes

characteristic

153

C₈

#14273706

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#16837232

yes

complemented

159

C₂²

C₂×#14273661

yes

characteristic

160

C₂²

#16837643

yes

—

161

C₄

—

162

C₄

#16837262

yes

characteristic

163

C₄

#16837440

yes

characteristic

164

C₄

#16837638

yes

—

165

C₂

—

166

C₂

—

167

C₂

#21424125

yes

characteristic

168

C₂

C₂×#16837232

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂³×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂⁴×C₅₉	Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂³×C₁₆	D₅₉
hypocenter	C₅₉	#12837
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂³×C₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂×#107

prime radical

C₂⁴×C₅₉

Q₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

hypercenter

C₂³×C₁₆

D₅₉

hypocenter

C₅₉

#12837

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

N	H ≅ G/N	type
1. C₂×#16837232	C₂	semidirect
3. #21422190	C₂	semidirect
74. C₂²×C₅₉	#368	semidirect
98. C₅₉	#12837	semidirect
158. C₂²	#16837232	semidirect

H ≅ G/N

type

1. C₂×#16837232

C₂

semidirect

3. #21422190

C₂

semidirect

74. C₂²×C₅₉

#368

semidirect

98. C₅₉

#12837

semidirect

158. C₂²

#16837232

semidirect

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

subgroup	index
C₂×#16837232	2
#21424134	2
#21422190	2
C₂³×C₁₆×C₅₉	2
#12837	59

subgroup

index

C₂³×C₁₆×C₅₉

Subgroups — #28504870

Conjugacy classes of subgroups

Complements

Maximal subgroups