Group #28509284 — subgroups

Subgroups — #28509284

Order 15104.

188 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 65 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×#16837242	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#21425954	C₂	1	1	yes	maximal, characteristic
4	#21422297	C₂	1	1	yes	maximal, characteristic
5	#21422278	C₂	1	1	yes	maximal, characteristic
6	#21422229	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₅₉	—	2	1	no	—
10	#16837496	—	2	1	no	—
11	#16837242	C₄	1	4	yes	complemented
12	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
13	#16837228	C₂²	1	1	yes	characteristic
14	#16837206	—	2	1	no	—
15	#16837205	—	2	1	no	—
16	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#366	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₃₆	—	2	1	no	—
21	Q₈×Dic₅₉	—	4	1	no	—
22	#14273692	—	4	1	no	—
23	Dic₄₇₂	—	2	1	no	—
24	Dic₄₇₂	C₂×C₄	1	2	yes	—
25	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
26	#14273661	—	2	1	no	—
27	#14273651	—	4	1	no	—
28	#14273650	—	4	1	no	—
29	C₈×Dic₅₉	D₄	1	1	yes	characteristic
30	C₅₉⋊C₃₂	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₅₉	—	2	1	no	—
32	Q₃₂×C₅₉	—	2	1	no	—
33	Q₃₂×C₅₉	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
35	#107×C₅₉	—	2	1	no	—
36	C₃₂×C₅₉	—	2	1	no	—
37	C₂×Q₈×C₅₉	—	4	1	no	—
38	C₂×Dic₁₁₈	—	4	1	no	—
39	Q₁₆×C₅₉	—	2	2	no	—
40	Q₁₆×C₅₉	—	4	1	no	—
41	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₅₉	—	4	1	no	—
43	C₄⋊Dic₅₉	—	4	1	no	—
44	C₄⋊Dic₅₉	—	8	1	no	—
45	Dic₅₉⋊C₄	—	8	1	no	—
46	C₄×Dic₅₉	—	8	1	no	—
47	C₄×Dic₅₉	—	8	1	no	—
48	C₄×Dic₅₉	D₈	1	1	yes	characteristic
49	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
50	Dic₂₃₆	—	2	2	no	—
51	Dic₂₃₆	—	4	1	no	—
52	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₅₉	#41	1	1	yes	characteristic
54	C₅₉⋊C₁₆	—	2	1	no	—
55	Q₈×C₅₉	—	4	2	no	—
56	Q₈×C₅₉	—	8	1	no	—
57	C₂×C₄×C₅₉	—	8	1	no	—
58	C₂×C₄×C₅₉	—	8	1	no	—
59	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₅₉	—	8	1	no	—
61	C₂×Dic₅₉	—	8	1	no	—
62	C₂×Dic₅₉	D₁₆	1	1	yes	characteristic
63	C₂×Dic₅₉	D₁₆	1	1	yes	characteristic
64	Dic₁₁₈	—	4	2	no	—
65	Dic₁₁₈	—	8	1	no	—
66	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
67	C₈×C₅₉	#135	1	1	yes	characteristic
68	C₅₉⋊C₈	—	2	1	no	—
69	#13653	—	59	1	no	maximal, Sylow
70	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
71	C₄×C₅₉	—	8	2	no	—
72	C₄×C₅₉	—	16	1	no	—
73	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
74	C₄×C₅₉	#508	1	1	yes	characteristic
75	Dic₅₉	—	2	1	no	—
76	Dic₅₉	—	2	1	no	—
77	Dic₅₉	—	8	2	no	—
78	Dic₅₉	—	16	1	no	—
79	C₂×Q₆₄	—	59	1	no	—
80	C₄×Q₃₂	—	59	1	no	—
81	C₄×Q₃₂	—	59	1	no	—
82	#1424	—	59	1	no	—
83	#1416	—	59	1	no	—
84	#1416	D₅₉	1	1	yes	characteristic
85	#1397	—	59	1	no	—
86	C₂×C₅₉	C₄×D₁₆	1	1	yes	characteristic
87	C₂×C₅₉	#2264	1	1	yes	characteristic
88	C₂×C₅₉	#2264	1	1	yes	characteristic
89	C₂×Q₃₂	—	59	1	no	—
90	C₂×Q₃₂	C₂×D₅₉	1	1	yes	characteristic
91	C₄×Q₁₆	—	118	1	no	—
92	C₄×Q₁₆	—	118	1	no	—
93	Q₆₄	—	59	4	no	—
94	C₂×C₃₂	—	59	1	no	—
95	C₂×C₃₂	C₂×D₅₉	1	1	yes	characteristic
96	#366	—	59	1	no	—
97	#366	C₂×D₅₉	1	1	yes	characteristic
98	#358	—	118	1	no	—
99	#358	—	118	1	no	—
100	C₄×C₁₆	—	59	1	no	—
101	C₅₉	#13653	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₁₆	—	2	1	no	—
103	C₂×Q₁₆	—	118	1	no	—
104	C₄×Q₈	—	236	1	no	—
105	C₄×Q₈	—	236	1	no	—
106	Q₃₂	—	2	1	no	—
107	Q₃₂	—	59	2	no	—
108	Q₃₂	—	118	1	no	—
109	Q₃₂	C₄×D₅₉	1	2	yes	—
110	C₂×C₁₆	—	59	1	no	—
111	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
112	#107	—	2	1	no	—
113	#107	—	118	1	no	—
114	#103	—	236	1	no	—
115	#103	—	236	1	no	—
116	C₄×C₈	—	59	1	no	—
117	C₃₂	—	2	1	no	—
118	C₃₂	—	59	2	no	—
119	C₂×Q₈	—	4	1	no	—
120	C₂×Q₈	—	236	1	no	—
121	Q₁₆	—	2	2	no	—
122	Q₁₆	—	4	1	no	—
123	Q₁₆	—	118	2	no	—
124	Q₁₆	—	236	1	no	—
125	C₂×C₈	—	59	1	no	—
126	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	472	1	no	—
129	C₄⋊C₄	—	472	1	no	—
130	C₄⋊C₄	—	236	1	no	—
131	C₄²	—	59	1	no	—
132	C₄²	—	472	1	no	—
133	C₄²	—	472	1	no	—
134	C₁₆	—	118	1	no	—
135	C₁₆	C₂×C₄×D₅₉	1	1	yes	characteristic
136	C₁₆	#11745687	1	1	yes	characteristic
137	Q₈	—	4	2	no	—
138	Q₈	—	8	1	no	—
139	Q₈	—	472	1	no	—
140	Q₈	—	236	2	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	59	1	no	—
144	C₂×C₄	—	59	1	no	—
145	C₂×C₄	—	472	1	no	—
146	C₂×C₄	—	472	1	no	—
147	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
148	C₈	—	118	1	no	—
149	C₈	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
150	C₈	#14273748	1	1	yes	characteristic
151	C₂²	D₁₆×D₅₉	1	1	yes	characteristic
152	C₄	—	8	2	no	—
153	C₄	—	16	1	no	—
154	C₄	—	118	1	no	—
155	C₄	—	118	1	no	—
156	C₄	—	944	1	no	—
157	C₄	—	472	2	no	—
158	C₄	#16837482	1	1	yes	characteristic
159	C₄	#16837833	1	1	yes	characteristic
160	C₂	#21425940	1	1	yes	characteristic
161	C₂	#21426412	1	1	yes	characteristic
162	C₂	#21426415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837242

C₂

yes

maximal, characteristic

Q₃₂×Dic₅₉

C₂

yes

maximal, characteristic

#21425954

C₂

yes

maximal, characteristic

#21422297

C₂

yes

maximal, characteristic

#21422278

C₂

yes

maximal, characteristic

#21422229

C₂

yes

maximal, characteristic

C₅₉×#1416

C₂

yes

maximal, characteristic

C₂×Dic₄₇₂

C₂²

yes

characteristic

Q₁₆×Dic₅₉

—

#16837496

—

#16837242

C₄

yes

complemented

C₂×C₅₉⋊C₃₂

C₂²

yes

characteristic

#16837228

C₂²

yes

characteristic

#16837206

—

#16837205

—

C₁₆×Dic₅₉

C₂²

yes

characteristic

C₂×Q₃₂×C₅₉

C₂²

yes

characteristic

C₂×C₃₂×C₅₉

C₂²

yes

characteristic

C₅₉×#366

C₂²

yes

characteristic

C₂×Dic₂₃₆

—

Q₈×Dic₅₉

—

#14273692

—

Dic₄₇₂

—

Dic₄₇₂

C₂×C₄

yes

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

—

—

C₈×Dic₅₉

D₄

yes

characteristic

C₅₉⋊C₃₂

C₂×C₄

yes

—

C₂×Q₁₆×C₅₉

—

Q₃₂×C₅₉

—

Q₃₂×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

#107×C₅₉

—

C₃₂×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₈

yes

characteristic

C₂×C₅₉⋊C₈

D₈

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

C₂²×C₄

yes

characteristic

C₁₆×C₅₉

#41

yes

characteristic

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₁₆

yes

characteristic

C₂×Dic₅₉

D₁₆

yes

characteristic

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₄×D₄

yes

characteristic

C₈×C₅₉

#135

yes

characteristic

C₅₉⋊C₈

—

#13653

—

maximal, Sylow

C₂²×C₅₉

C₂×D₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₈

yes

characteristic

C₄×C₅₉

#508

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

C₂×Q₆₄

—

C₄×Q₃₂

—

C₄×Q₃₂

—

#1424

—

#1416

—

#1416

D₅₉

yes

characteristic

#1397

—

C₂×C₅₉

C₄×D₁₆

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₅₉

yes

characteristic

C₄×Q₁₆

—

118

—

C₄×Q₁₆

—

118

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₉

yes

characteristic

#366

—

#366

C₂×D₅₉

yes

characteristic

#358

—

118

—

#358

—

118

—

100

C₄×C₁₆

—

101

C₅₉

#13653

yes

characteristic, complemented, Sylow

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

118

—

104

C₄×Q₈

—

236

—

105

C₄×Q₈

—

236

—

106

Q₃₂

—

107

Q₃₂

—

108

Q₃₂

—

118

—

109

Q₃₂

C₄×D₅₉

yes

—

110

C₂×C₁₆

—

111

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

112

#107

—

113

#107

—

118

—

114

#103

—

236

—

115

#103

—

236

—

116

C₄×C₈

—

117

C₃₂

—

118

C₃₂

—

119

C₂×Q₈

—

120

C₂×Q₈

—

236

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

118

—

124

Q₁₆

—

236

—

125

C₂×C₈

—

126

C₂×C₈

D₄×D₅₉

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

472

—

129

C₄⋊C₄

—

472

—

130

C₄⋊C₄

—

236

—

131

C₄²

—

132

C₄²

—

472

—

133

C₄²

—

472

—

134

C₁₆

—

118

—

135

C₁₆

C₂×C₄×D₅₉

yes

characteristic

136

C₁₆

#11745687

yes

characteristic

137

Q₈

—

138

Q₈

—

139

Q₈

—

472

—

140

Q₈

—

236

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

472

—

146

C₂×C₄

—

472

—

147

C₂×C₄

D₈×D₅₉

yes

characteristic

148

C₈

—

118

—

149

C₈

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

150

C₈

#14273748

yes

characteristic

151

C₂²

D₁₆×D₅₉

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

118

—

155

C₄

—

118

—

156

C₄

—

944

—

157

C₄

—

472

—

158

C₄

#16837482

yes

characteristic

159

C₄

#16837833

yes

characteristic

160

C₂

#21425940

yes

characteristic

161

C₂

#21426412

yes

characteristic

162

C₂

#21426415

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₅₉
commutator	C₁₆×C₅₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₅₉×#1416	C₂
socle, prime radical	C₂²×C₅₉	C₂×D₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	#1416	D₅₉
hypocenter	C₅₉	#13653
LCS	G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₅₉

commutator

C₁₆×C₅₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₅₉×#1416

C₂

socle, prime radical

C₂²×C₅₉

C₂×D₁₆

cosocle

C₂×C₁₆×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
11. #16837242	C₄	semidirect
101. C₅₉	#13653	semidirect

H ≅ G/N

type

11. #16837242

C₄

semidirect

101. C₅₉

#13653

semidirect

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

subgroup	index
C₂×#16837242	2
Q₃₂×Dic₅₉	2
#21425954	2
#21422297	2
#21422278	2
#21422229	2
C₅₉×#1416	2
#13653	59

subgroup

index

Q₃₂×Dic₅₉

C₅₉×#1416

Subgroups — #28509284

Conjugacy classes of subgroups

Complements

Maximal subgroups