Group #28507286 — subgroups

Subgroups — #28507286

Order 15104.

366 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16837804	C₂	1	1	yes	maximal, characteristic
2	#21424260	C₂	1	1	yes	maximal, characteristic
3	#21422188	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273719	—	2	1	no	—
6	#16837804	—	2	2	no	—
7	#16837804	C₄	1	4	yes	complemented
8	C₂×#14273661	C₂²	1	2	yes	—
9	C₂×#14273661	C₂²	1	2	yes	—
10	C₂×#14273659	—	2	1	no	—
11	C₂×#14273659	—	2	1	no	—
12	#16837268	—	2	1	no	—
13	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₅₉	—	2	2	no	—
16	#14273719	—	2	4	no	—
17	#14273719	—	4	2	no	—
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×Dic₅₉⋊C₄	—	4	2	no	—
21	#14273661	—	2	2	no	—
22	#14273661	—	2	4	no	—
23	#14273661	C₂×C₄	1	4	yes	—
24	#14273659	—	2	4	no	—
25	#14273659	—	4	2	no	—
26	C₅₉⋊#95	—	4	2	no	—
27	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₉	—	2	2	no	—
31	C₂×C₁₆×C₅₉	—	2	2	no	—
32	C₂×C₁₆×C₅₉	Q₈	1	4	yes	—
33	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₉	D₄	1	2	yes	—
35	C₂²×C₄×C₅₉	D₈	1	2	yes	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₉	—	4	2	no	—
38	C₂²×Dic₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₅₉	#41	1	2	yes	—
46	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₅₉	SD₁₆	1	2	yes	—
48	C₄⋊Dic₅₉	—	2	4	no	—
49	C₄⋊Dic₅₉	—	4	2	no	—
50	C₄⋊Dic₅₉	—	4	2	no	—
51	C₄⋊Dic₅₉	—	4	2	no	—
52	Dic₅₉⋊C₄	—	4	8	no	—
53	C₁₆×C₅₉	—	2	4	no	—
54	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	2	4	no	—
56	C₂×C₄×C₅₉	#102	1	4	yes	—
57	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₅₉	—	4	4	no	—
64	C₂×Dic₅₉	—	8	4	no	—
65	C₂×Dic₅₉	—	8	2	no	—
66	C₂×Dic₅₉	—	8	4	no	—
67	C₈×C₅₉	#103	1	4	yes	—
68	C₈×C₅₉	C₄×Q₈	1	2	yes	—
69	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
70	#13281	—	59	1	no	maximal, Sylow
71	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
72	C₂²×C₅₉	#486	1	1	yes	characteristic
73	C₂²×C₅₉	#500	1	1	yes	characteristic
74	C₂²×C₅₉	#508	1	4	yes	—
75	C₄×C₅₉	—	2	4	no	—
76	C₄×C₅₉	#391	1	1	yes	characteristic
77	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
78	C₄×C₅₉	#487	1	1	yes	characteristic
79	C₄×C₅₉	#498	1	1	yes	characteristic
80	Dic₅₉	—	8	4	no	—
81	Dic₅₉	—	16	2	no	—
82	C₂×#499	—	59	1	no	—
83	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
84	#1846	—	59	1	no	—
85	#1380	—	59	4	no	—
86	C₂×C₅₉	#1990	1	1	yes	characteristic
87	C₂×C₅₉	#2138	1	2	yes	—
88	C₂×C₅₉	#2242	1	2	yes	—
89	C₂×C₅₉	#2253	1	2	yes	—
90	C₂×#125	—	118	1	no	—
91	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
92	#499	—	59	4	no	—
93	#499	—	118	2	no	—
94	C₂×#107	—	59	2	no	—
95	C₂×#107	—	59	2	no	—
96	C₂×#106	—	118	1	no	—
97	C₂×#106	—	118	1	no	—
98	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
99	#382	—	118	1	no	—
100	C₄×C₁₆	—	2	2	no	—
101	C₅₉	#13281	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
104	#125	—	118	4	no	—
105	#125	—	236	2	no	—
106	C₂×C₄⋊C₄	—	118	2	no	—
107	C₂×C₄⋊C₄	—	118	2	no	—
108	C₂×C₄⋊C₄	—	236	2	no	—
109	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	Dic₁₁₈	1	4	yes	—
113	#107	—	59	4	no	—
114	#107	—	118	2	no	—
115	#107	—	118	4	no	—
116	#106	—	118	4	no	—
117	#106	—	236	2	no	—
118	C₄×C₈	C₄×D₅₉	1	2	yes	—
119	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
120	#95	—	236	2	no	—
121	C₂²×C₄	—	236	2	no	—
122	C₂²×C₄	—	236	2	no	—
123	C₂²×C₄	D₄₇₂	1	2	yes	—
124	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
125	C₂×C₈	#11745660	1	2	yes	—
126	C₂×C₈	Dic₂₃₆	1	2	yes	—
127	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
130	C₂×C₈	#11745685	1	2	yes	—
131	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
132	C₄⋊C₄	—	118	4	no	—
133	C₄⋊C₄	—	236	2	no	—
134	C₄⋊C₄	—	236	2	no	—
135	C₄⋊C₄	—	236	8	no	—
136	C₄⋊C₄	—	236	2	no	—
137	C₄²	#11745660	1	2	yes	—
138	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	472	2	no	—
143	C₂×C₄	—	472	4	no	—
144	C₂×C₄	—	472	4	no	—
145	C₂×C₄	—	236	4	no	—
146	C₂×C₄	#14273649	1	4	yes	—
147	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#14273715	1	1	yes	characteristic
149	C₂×C₄	#14273727	1	1	yes	characteristic
150	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
151	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
152	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
153	C₈	#14273652	1	4	yes	—
154	C₈	C₄×Dic₁₁₈	1	2	yes	—
155	C₈	#14273706	1	2	yes	—
156	C₂²	C₂×#14273649	1	1	yes	characteristic
157	C₂²	#16837749	1	1	yes	characteristic
158	C₂²	#16837807	1	1	yes	characteristic
159	C₂²	#16837834	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	944	2	no	—
162	C₄	—	472	4	no	—
163	C₄	#16837308	1	1	yes	characteristic
164	C₄	C₂×#14273652	1	1	yes	characteristic
165	C₄	#16837753	1	1	yes	characteristic
166	C₄	#16837800	1	1	yes	characteristic
167	C₂	#21425057	1	1	yes	characteristic
168	C₂	#21425804	1	2	yes	—
169	C₂	#21426328	1	2	yes	—
170	C₂	#21426375	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837804

C₂

yes

maximal, characteristic

#21424260

C₂

yes

maximal, characteristic

#21422188

C₂

yes

maximal

C₂×C₄×C₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×#14273719

—

#16837804

—

#16837804

C₄

yes

complemented

C₂×#14273661

C₂²

yes

—

C₂×#14273661

C₂²

yes

—

C₂×#14273659

—

C₂×#14273659

—

#16837268

—

C₂²×C₁₆×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₁₆×C₅₉

—

#14273719

—

#14273719

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×Dic₅₉⋊C₄

—

#14273661

—

#14273661

—

#14273661

C₂×C₄

yes

—

#14273659

—

#14273659

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

Q₈

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

D₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

SD₁₆

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₁₆×C₅₉

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

#103

yes

—

C₈×C₅₉

C₄×Q₈

yes

—

C₈×C₅₉

C₂²⋊Q₈

yes

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

C₂×#499

—

C₂×C₄×C₁₆

D₅₉

yes

characteristic

#1846

—

#1380

—

C₂×C₅₉

#1990

yes

characteristic

C₂×C₅₉

#2138

yes

—

C₂×C₅₉

#2242

yes

—

C₂×C₅₉

#2253

yes

—

C₂×#125

—

118

—

C₂²×C₁₆

C₂×D₅₉

yes

—

#499

—

#499

—

118

—

C₂×#107

—

C₂×#107

—

C₂×#106

—

118

—

C₂×#106

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#382

—

118

—

100

C₄×C₁₆

—

101

C₅₉

#13281

yes

characteristic, complemented, Sylow

102

C₂²×C₈

D₂₃₆

yes

characteristic

103

C₂²×C₈

C₂²×D₅₉

yes

characteristic

104

#125

—

118

—

105

#125

—

236

—

106

C₂×C₄⋊C₄

—

118

—

107

C₂×C₄⋊C₄

—

118

—

108

C₂×C₄⋊C₄

—

236

—

109

C₂×C₄²

D₂₃₆

yes

characteristic

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

Dic₁₁₈

yes

—

113

#107

—

114

#107

—

118

—

115

#107

—

118

—

116

#106

—

118

—

117

#106

—

236

—

118

C₄×C₈

C₄×D₅₉

yes

—

119

C₄×C₈

C₂²⋊D₅₉

yes

—

120

#95

—

236

—

121

C₂²×C₄

—

236

—

122

C₂²×C₄

—

236

—

123

C₂²×C₄

D₄₇₂

yes

—

124

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

125

C₂×C₈

#11745660

yes

—

126

C₂×C₈

Dic₂₃₆

yes

—

127

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

128

C₂×C₈

C₂×Dic₁₁₈

yes

—

129

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

130

C₂×C₈

#11745685

yes

—

131

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

132

C₄⋊C₄

—

118

—

133

C₄⋊C₄

—

236

—

134

C₄⋊C₄

—

236

—

135

C₄⋊C₄

—

236

—

136

C₄⋊C₄

—

236

—

137

C₄²

#11745660

yes

—

138

C₄²

(C₂×D₅₉)⋊C₄

yes

—

139

C₁₆

—

140

C₂³

C₂×D₄₇₂

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

472

—

143

C₂×C₄

—

472

—

144

C₂×C₄

—

472

—

145

C₂×C₄

—

236

—

146

C₂×C₄

#14273649

yes

—

147

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

148

C₂×C₄

#14273715

yes

characteristic

149

C₂×C₄

#14273727

yes

characteristic

150

C₂×C₄

C₂×#11745660

yes

characteristic

151

C₂×C₄

C₂×#11745660

yes

characteristic

152

C₂×C₄

C₂×Dic₂₃₆

yes

characteristic

153

C₈

#14273652

yes

—

154

C₈

C₄×Dic₁₁₈

yes

—

155

C₈

#14273706

yes

—

156

C₂²

C₂×#14273649

yes

characteristic

157

C₂²

#16837749

yes

characteristic

158

C₂²

#16837807

yes

characteristic

159

C₂²

#16837834

yes

—

160

C₄

—

161

C₄

—

944

—

162

C₄

—

472

—

163

C₄

#16837308

yes

characteristic

164

C₄

C₂×#14273652

yes

characteristic

165

C₄

#16837753

yes

characteristic

166

C₄

#16837800

yes

characteristic

167

C₂

#21425057

yes

characteristic

168

C₂

#21425804

yes

—

169

C₂

#21426328

yes

—

170

C₂

#21426375

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#13281
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₄×C₁₆×C₅₉

C₂

socle, prime radical

C₂³×C₅₉

C₂×D₈

cosocle

C₂²×C₈×C₅₉

C₂³

hypercenter

C₂×C₄×C₁₆

D₅₉

hypocenter

C₅₉

#13281

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
7. #16837804	C₄	semidirect
101. C₅₉	#13281	semidirect

H ≅ G/N

type

7. #16837804

C₄

semidirect

101. C₅₉

#13281

semidirect

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

subgroup	index
C₂×#16837804	2
#21424260	2
#21422188	2
C₂×C₄×C₁₆×C₅₉	2
#13281	59

subgroup

index

C₂×C₄×C₁₆×C₅₉

Subgroups — #28507286

Conjugacy classes of subgroups

Complements

Maximal subgroups