Group #28503019 — subgroups

Subgroups — #28503019

Order 15104.

212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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	#21426371	C₂	1	1	yes	maximal, characteristic, complemented
2	#21423495	C₂	1	2	yes	maximal
3	#21422168	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422153	C₂	1	2	yes	maximal, complemented
5	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837807	C₂²	1	2	yes	—
7	#16837692	—	2	2	no	—
8	#16837228	—	2	1	no	—
9	#16837228	—	2	1	no	—
10	#16837155	C₂²	1	2	yes	—
11	#16837150	—	2	2	no	—
12	C₂²×C₁₆×C₅₉	—	2	1	no	—
13	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅₉×C₂²⋊C₁₆	—	2	1	no	—
16	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	#14273727	—	2	2	no	—
18	#14273661	—	2	2	no	—
19	#14273661	—	2	2	no	—
20	#14273658	—	4	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
22	#14273648	—	4	2	no	—
23	C₂²×C₈×C₅₉	—	2	1	no	—
24	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₉	—	2	1	no	—
26	C₂×C₁₆×C₅₉	—	4	2	no	—
27	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₂²⋊C₈×C₅₉	—	2	1	no	—
31	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	—	2	1	no	—
34	C₂×Dic₁₁₈	—	4	2	no	—
35	C₂×C₈×C₅₉	—	2	1	no	—
36	C₂×C₈×C₅₉	—	4	1	no	—
37	C₂×C₈×C₅₉	—	4	1	no	—
38	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
41	C₂²⋊C₄×C₅₉	—	2	1	no	—
42	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₅₉	—	4	2	no	—
44	C₄⋊Dic₅₉	—	4	2	no	—
45	C₂×C₅₉⋊C₈	—	4	2	no	—
46	C₁₆×C₅₉	—	2	1	no	—
47	C₁₆×C₅₉	—	4	1	no	—
48	C₁₆×C₅₉	SD₁₆	1	2	yes	—
49	C₂³×C₅₉	—	2	1	no	—
50	D₄×C₅₉	—	2	1	no	—
51	D₄×C₅₉	Q₃₂	1	2	yes	—
52	C₂×C₄×C₅₉	—	2	1	no	—
53	C₂×C₄×C₅₉	—	4	1	no	—
54	C₂×C₄×C₅₉	—	4	1	no	—
55	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
58	C₂×Dic₅₉	—	8	2	no	—
59	Dic₁₁₈	—	16	2	no	—
60	C₈×C₅₉	—	2	1	no	—
61	C₈×C₅₉	—	4	1	no	—
62	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₅₉	#137	1	1	yes	characteristic
64	C₅₉⋊C₈	—	8	2	no	—
65	#12452	—	59	1	no	maximal, Sylow
66	C₂²×C₅₉	—	2	2	no	—
67	C₂²×C₅₉	—	4	1	no	—
68	C₂²×C₅₉	—	4	1	no	—
69	C₂²×C₅₉	#466	1	1	yes	characteristic
70	C₄×C₅₉	—	2	1	no	—
71	C₄×C₅₉	—	4	1	no	—
72	C₄×C₅₉	#461	1	1	yes	characteristic
73	C₄×C₅₉	#473	1	1	yes	characteristic
74	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
75	C₄×C₅₉	#508	1	1	yes	characteristic
76	Dic₅₉	—	16	2	no	—
77	#2252	—	59	1	no	—
78	D₄×C₁₆	D₅₉	1	1	yes	characteristic
79	#1675	—	59	2	no	—
80	#1371	—	59	1	no	—
81	#1365	—	59	2	no	—
82	C₂×C₅₉	—	4	2	no	—
83	C₂×C₅₉	#1673	1	1	yes	characteristic
84	C₂×C₅₉	#2216	1	1	yes	characteristic
85	C₂×C₅₉	#2221	1	1	yes	characteristic
86	C₂²×C₁₆	—	2	1	no	—
87	#500	—	59	2	no	—
88	#476	—	118	2	no	—
89	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
90	#366	—	118	1	no	—
91	#366	—	118	1	no	—
92	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
93	C₂²⋊C₁₆	—	2	1	no	—
94	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
95	#335	—	59	2	no	—
96	#333	—	118	2	no	—
97	C₅₉	#12452	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	—	2	1	no	—
99	C₄⋊Q₈	—	118	2	no	—
100	C₄×D₄	D₂₃₆	1	1	yes	characteristic
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	4	2	no	—
103	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
105	#107	—	118	2	no	—
106	#107	—	118	2	no	—
107	#106	—	236	2	no	—
108	C₄⋊C₈	—	118	2	no	—
109	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
110	#102	—	236	2	no	—
111	C₂²⋊C₈	—	2	1	no	—
112	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
113	C₂×Q₈	—	236	2	no	—
114	C₂×D₄	D₄₇₂	1	1	yes	characteristic
115	C₂²×C₄	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	236	2	no	—
120	C₂×C₈	#11745685	1	1	yes	characteristic
121	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
122	C₄⋊C₄	—	236	2	no	—
123	C₄⋊C₄	—	236	2	no	—
124	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
125	C₂²⋊C₄	—	2	1	no	—
126	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	#11745670	1	2	yes	—
130	C₂³	—	2	1	no	—
131	Q₈	—	944	2	no	—
132	D₄	—	2	1	no	—
133	D₄	Dic₄₇₂	1	2	yes	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	472	2	no	—
138	C₂×C₄	#14273701	1	1	yes	characteristic
139	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
140	C₂×C₄	#14273749	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	4	1	no	—
143	C₈	—	472	2	no	—
144	C₈	C₂×#11745670	1	1	yes	characteristic
145	C₈	#14273757	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	#16837638	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	944	2	no	—
153	C₄	#16837607	1	1	yes	characteristic
154	C₄	#16837677	1	1	yes	characteristic
155	C₄	C₂×Dic₄₇₂	1	1	yes	characteristic
156	C₄	#16837834	1	1	yes	characteristic
157	C₂	—	4	2	no	—
158	C₂	#21423485	1	1	yes	characteristic
159	C₂	#21426187	1	1	yes	characteristic
160	C₂	#21426216	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426371

C₂

yes

maximal, characteristic, complemented

#21423495

C₂

yes

maximal

#21422168

C₂

yes

maximal, characteristic, complemented

#21422153

C₂

yes

maximal, complemented

D₄×C₁₆×C₅₉

C₂

yes

maximal, characteristic

#16837807

C₂²

yes

—

—

—

—

C₂²

yes

—

#16837150

—

C₂²×C₁₆×C₅₉

—

C₈×D₄×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

C₂²

yes

characteristic

C₅₉×C₂²⋊C₁₆

—

C₄×C₁₆×C₅₉

C₂²

yes

—

—

—

—

C₄⋊(C₅₉⋊C₈)

—

#14273648

—

C₂²×C₈×C₅₉

—

C₄×D₄×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₄⋊C₈×C₅₉

D₄

yes

characteristic

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×D₄×C₅₉

D₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

D₈

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

SD₁₆

yes

—

C₂³×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

Q₃₂

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×SD₁₆

yes

characteristic

C₈×C₅₉

#137

yes

characteristic

C₅₉⋊C₈

—

#12452

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#466

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×Q₃₂

yes

characteristic

C₄×C₅₉

#508

yes

characteristic

Dic₅₉

—

#2252

—

D₄×C₁₆

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂²×C₁₆

—

#500

—

#476

—

118

—

C₈×D₄

C₂×D₅₉

yes

characteristic

#366

—

118

—

#366

—

118

—

C₄⋊C₁₆

C₂×D₅₉

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

—

#333

—

118

—

C₅₉

#12452

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

118

—

100

C₄×D₄

D₂₃₆

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

105

#107

—

118

—

106

#107

—

118

—

107

#106

—

236

—

108

C₄⋊C₈

—

118

—

109

C₄⋊C₈

D₂₃₆

yes

characteristic

110

#102

—

236

—

111

C₂²⋊C₈

—

112

C₄×C₈

C₂²×D₅₉

yes

characteristic

113

C₂×Q₈

—

236

—

114

C₂×D₄

D₄₇₂

yes

characteristic

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

236

—

120

C₂×C₈

#11745685

yes

characteristic

121

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

122

C₄⋊C₄

—

236

—

123

C₄⋊C₄

—

236

—

124

C₄⋊C₄

D₄₇₂

yes

characteristic

125

C₂²⋊C₄

—

126

C₄²

C₂×D₂₃₆

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

#11745670

yes

—

130

C₂³

—

131

Q₈

—

944

—

132

D₄

—

133

D₄

Dic₄₇₂

yes

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

472

—

138

C₂×C₄

#14273701

yes

characteristic

139

C₂×C₄

C₂×D₄₇₂

yes

characteristic

140

C₂×C₄

#14273749

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

472

—

144

C₈

C₂×#11745670

yes

characteristic

145

C₈

#14273757

yes

characteristic

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#16837638

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

944

—

153

C₄

#16837607

yes

characteristic

154

C₄

#16837677

yes

characteristic

155

C₄

C₂×Dic₄₇₂

yes

characteristic

156

C₄

#16837834

yes

characteristic

157

C₂

—

158

C₂

#21423485

yes

characteristic

159

C₂

#21426187

yes

characteristic

160

C₂

#21426216

yes

characteristic

	subgroup	G/H
center	C₂²	#16837638
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	D₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	#466
prime radical	C₂×D₄×C₅₉	D₈
hypercenter	D₄×C₁₆	D₅₉
hypocenter	C₅₉	#12452
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆

subgroup

G/H

center

C₂²

#16837638

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

D₄×C₁₆×C₅₉

C₂

socle

C₂²×C₅₉

#466

prime radical

C₂×D₄×C₅₉

D₈

hypercenter

D₄×C₁₆

D₅₉

hypocenter

C₅₉

#12452

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆

N	H ≅ G/N	type
1. #21426371	C₂	semidirect
3. #21422168	C₂	semidirect
4. #21422153	C₂	semidirect
97. C₅₉	#12452	semidirect

H ≅ G/N

type

1. #21426371

C₂

semidirect

3. #21422168

C₂

semidirect

4. #21422153

C₂

semidirect

97. C₅₉

#12452

semidirect

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

subgroup	index
#21426371	2
#21423495	2
#21422168	2
#21422153	2
D₄×C₁₆×C₅₉	2
#12452	59

subgroup

index

D₄×C₁₆×C₅₉

Subgroups — #28503019

Conjugacy classes of subgroups

Complements

Maximal subgroups