Group #28505071 — subgroups

Subgroups — #28505071

Order 15104.

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

normal subgroups only

This group has 310 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×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#21424260	C₂	1	1	yes	maximal, characteristic
3	#21422187	C₂	1	4	yes	maximal
4	C₅₉×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14273661	C₂²	1	2	yes	—
7	C₂×#14273659	—	2	1	no	—
8	#16837268	—	2	1	no	—
9	C₄×C₅₉⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#107×C₅₉	C₂²	1	2	yes	—
11	C₂×#106×C₅₉	—	2	1	no	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#382	—	2	1	no	—
14	C₂×C₄⋊Dic₅₉	—	2	2	no	—
15	C₂×C₅₉⋊C₁₆	—	2	2	no	—
16	C₂×C₅₉⋊C₁₆	—	2	2	no	—
17	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
18	#14273661	—	2	4	no	—
19	#14273659	—	4	2	no	—
20	C₅₉⋊#95	—	4	2	no	—
21	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
24	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
25	#107×C₅₉	—	2	4	no	—
26	#106×C₅₉	—	4	2	no	—
27	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₅₉	Q₈	1	2	yes	—
29	#95×C₅₉	—	4	2	no	—
30	C₂²×C₄×C₅₉	—	4	2	no	—
31	C₂²×C₄×C₅₉	D₈	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₅₉	—	4	2	no	—
34	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
36	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
37	C₂×C₈×C₅₉	C₂²×C₄	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	C₄⋊C₄×C₅₉	—	4	2	no	—
42	C₄⋊C₄×C₅₉	—	4	2	no	—
43	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
44	C₄²×C₅₉	Q₁₆	1	2	yes	—
45	C₄⋊Dic₅₉	—	4	2	no	—
46	C₄⋊Dic₅₉	—	4	2	no	—
47	C₅₉⋊C₁₆	—	2	4	no	—
48	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₅₉	—	2	4	no	—
50	C₂×C₄×C₅₉	—	8	2	no	—
51	C₂×C₄×C₅₉	—	8	4	no	—
52	C₂×C₄×C₅₉	#107	1	4	yes	—
53	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
55	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₅₉	—	8	2	no	—
60	C₂×Dic₅₉	—	8	4	no	—
61	C₈×C₅₉	#106	1	4	yes	—
62	C₈×C₅₉	#117	1	2	yes	—
63	C₈×C₅₉	#125	1	2	yes	—
64	#12889	—	59	1	no	maximal, Sylow
65	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
66	C₂²×C₅₉	#486	1	1	yes	characteristic
67	C₂²×C₅₉	#486	1	1	yes	characteristic
68	C₂²×C₅₉	#508	1	4	yes	—
69	C₄×C₅₉	—	2	4	no	—
70	C₄×C₅₉	—	16	2	no	—
71	C₄×C₅₉	#382	1	1	yes	characteristic
72	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
73	C₄×C₅₉	#487	1	1	yes	characteristic
74	C₄×C₅₉	#487	1	1	yes	characteristic
75	Dic₅₉	—	16	2	no	—
76	C₂×C₄×C₁₆	—	59	1	no	—
77	#1846	—	59	1	no	—
78	#1846	D₅₉	1	1	yes	characteristic
79	#1380	—	59	4	no	—
80	C₂×C₅₉	#1834	1	1	yes	characteristic
81	C₂×C₅₉	#2158	1	2	yes	—
82	C₂×C₅₉	#2242	1	2	yes	—
83	C₂×C₅₉	#2242	1	2	yes	—
84	C₂²×C₁₆	—	59	2	no	—
85	C₂×#107	—	59	2	no	—
86	C₂×#107	C₂×D₅₉	1	2	yes	—
87	C₂×#106	—	2	1	no	—
88	C₂×#106	—	118	1	no	—
89	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
90	#382	—	2	1	no	—
91	#382	—	118	1	no	—
92	C₄×C₁₆	—	59	4	no	—
93	C₅₉	#12889	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	118	2	no	—
98	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
99	C₂×C₁₆	—	59	4	no	—
100	C₂×C₁₆	—	118	2	no	—
101	C₂×C₁₆	—	118	2	no	—
102	#107	—	2	4	no	—
103	#107	—	118	4	no	—
104	#106	—	4	2	no	—
105	#106	—	236	2	no	—
106	C₄×C₈	Dic₁₁₈	1	2	yes	—
107	C₄×C₈	C₄×D₅₉	1	2	yes	—
108	#95	—	4	2	no	—
109	#95	—	236	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	236	2	no	—
112	C₂²×C₄	#11745669	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
114	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
115	C₂×C₈	#11745670	1	2	yes	—
116	C₂×C₈	#11745671	1	2	yes	—
117	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
119	C₂×C₈	#11745687	1	2	yes	—
120	C₂×C₈	#11745689	1	2	yes	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	236	2	no	—
124	C₄⋊C₄	—	236	2	no	—
125	C₄²	Dic₅₉⋊C₄	1	2	yes	—
126	C₄²	#11745672	1	2	yes	—
127	C₁₆	—	118	4	no	—
128	C₂³	C₂×#11745669	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	—	8	4	no	—
132	C₂×C₄	—	472	2	no	—
133	C₂×C₄	—	472	4	no	—
134	C₂×C₄	#14273660	1	4	yes	—
135	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
136	C₂×C₄	#14273714	1	1	yes	characteristic
137	C₂×C₄	#14273716	1	1	yes	characteristic
138	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
139	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
140	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
141	C₈	#14273658	1	4	yes	—
142	C₈	#14273685	1	2	yes	—
143	C₈	#14273718	1	2	yes	—
144	C₂²	C₂×#14273660	1	1	yes	characteristic
145	C₂²	#16837748	1	1	yes	characteristic
146	C₂²	#16837750	1	1	yes	characteristic
147	C₂²	#16837832	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	944	2	no	—
151	C₄	#16837267	1	1	yes	characteristic
152	C₄	C₂×#14273658	1	1	yes	characteristic
153	C₄	#16837752	1	1	yes	characteristic
154	C₄	#16837754	1	1	yes	characteristic
155	C₂	#21424223	1	1	yes	characteristic
156	C₂	#21425878	1	2	yes	—
157	C₂	#21426327	1	2	yes	—
158	C₂	#21426330	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic

#21424260

C₂

yes

maximal, characteristic

#21422187

C₂

yes

maximal

C₅₉×#1846

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

—

C₂×#14273661

C₂²

yes

—

C₂×#14273659

—

#16837268

—

C₄×C₅₉⋊C₁₆

C₄

yes

complemented

C₂×#107×C₅₉

C₂²

yes

—

C₂×#106×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#382

—

C₂×C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

#14273661

—

#14273659

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

#107×C₅₉

—

#106×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

Q₁₆

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#107

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

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

—

C₈×C₅₉

#106

yes

—

C₈×C₅₉

#117

yes

—

C₈×C₅₉

#125

yes

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₅₉

yes

characteristic

#1380

—

C₂×C₅₉

#1834

yes

characteristic

C₂×C₅₉

#2158

yes

—

C₂×C₅₉

#2242

yes

—

C₂×C₅₉

#2242

yes

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₅₉

yes

—

C₂×#106

—

C₂×#106

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#382

—

#382

—

118

—

C₄×C₁₆

—

C₅₉

#12889

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

118

—

101

C₂×C₁₆

—

118

—

102

#107

—

103

#107

—

118

—

104

#106

—

105

#106

—

236

—

106

C₄×C₈

Dic₁₁₈

yes

—

107

C₄×C₈

C₄×D₅₉

yes

—

108

#95

—

109

#95

—

236

—

110

C₂²×C₄

—

111

C₂²×C₄

—

236

—

112

C₂²×C₄

#11745669

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

114

C₂×C₈

Dic₅₉⋊C₄

yes

—

115

C₂×C₈

#11745670

yes

—

116

C₂×C₈

#11745671

yes

—

117

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

118

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

119

C₂×C₈

#11745687

yes

—

120

C₂×C₈

#11745689

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

236

—

124

C₄⋊C₄

—

236

—

125

C₄²

Dic₅₉⋊C₄

yes

—

126

C₄²

#11745672

yes

—

127

C₁₆

—

118

—

128

C₂³

C₂×#11745669

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

472

—

133

C₂×C₄

—

472

—

134

C₂×C₄

#14273660

yes

—

135

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

136

C₂×C₄

#14273714

yes

characteristic

137

C₂×C₄

#14273716

yes

characteristic

138

C₂×C₄

C₂×#11745671

yes

characteristic

139

C₂×C₄

C₂×#11745670

yes

characteristic

140

C₂×C₄

C₂×#11745672

yes

characteristic

141

C₈

#14273658

yes

—

142

C₈

#14273685

yes

—

143

C₈

#14273718

yes

—

144

C₂²

C₂×#14273660

yes

characteristic

145

C₂²

#16837748

yes

characteristic

146

C₂²

#16837750

yes

characteristic

147

C₂²

#16837832

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

944

—

151

C₄

#16837267

yes

characteristic

152

C₄

C₂×#14273658

yes

characteristic

153

C₄

#16837752

yes

characteristic

154

C₄

#16837754

yes

characteristic

155

C₂

#21424223

yes

characteristic

156

C₂

#21425878

yes

—

157

C₂

#21426327

yes

—

158

C₂

#21426330

yes

—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1846	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1846	D₅₉
hypocenter	C₅₉	#12889
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#11745669

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1846

C₂

socle, prime radical

C₂³×C₅₉

C₂×D₈

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. C₄×C₅₉⋊C₁₆	C₄	semidirect
93. C₅₉	#12889	semidirect

H ≅ G/N

type

9. C₄×C₅₉⋊C₁₆

C₄

semidirect

93. C₅₉

#12889

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
#21424260	2
#21422187	2
C₅₉×#1846	2
#12889	59

subgroup

index

C₂×C₄×C₅₉⋊C₁₆

#21424260

#21422187

C₅₉×#1846

#12889

Subgroups — #28505071

Conjugacy classes of subgroups

Complements

Maximal subgroups