Group #28505049 — subgroups

Subgroups — #28505049

Order 15104.

342 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 143 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	#21425852	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#21422177	C₂	1	4	yes	maximal
4	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	2	yes	—
6	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₉	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
9	C₄²×Dic₅₉	C₄	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
11	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₉	—	2	2	no	—
16	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
17	#14273667	—	2	4	no	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	C₄×C₅₉⋊C₈	C₈	1	2	yes	—
20	C₄×C₅₉⋊C₈	C₈	1	2	yes	—
21	C₈×Dic₅₉	—	2	2	no	—
22	C₈×Dic₅₉	—	2	2	no	—
23	C₈×Dic₅₉	—	2	4	no	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₅₉	—	2	4	no	—
28	C₂×C₁₆×C₅₉	—	2	2	no	—
29	C₄×C₈×C₅₉	D₄	1	2	yes	—
30	C₄×C₈×C₅₉	Q₈	1	2	yes	—
31	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₅₉	—	2	2	no	—
34	C₂×C₈×C₅₉	—	2	2	no	—
35	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₅₉	#41	1	2	yes	—
39	C₂×C₈×C₅₉	#41	1	2	yes	—
40	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
41	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
43	C₄×Dic₅₉	—	2	8	no	—
44	C₄×Dic₅₉	—	2	2	no	—
45	C₄×Dic₅₉	M₁₆	1	4	yes	—
46	C₂×C₅₉⋊C₈	—	2	2	no	—
47	C₂×C₅₉⋊C₈	—	2	2	no	—
48	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
49	C₁₆×C₅₉	—	4	2	no	—
50	C₅₉⋊C₁₆	—	4	2	no	—
51	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	2	4	no	—
53	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#117	1	2	yes	—
57	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
59	C₂×Dic₅₉	—	2	8	no	—
60	C₂×Dic₅₉	—	2	4	no	—
61	C₈×C₅₉	—	2	2	no	—
62	C₈×C₅₉	#124	1	1	yes	characteristic
63	C₈×C₅₉	#124	1	1	yes	characteristic
64	C₈×C₅₉	#125	1	2	yes	—
65	C₅₉⋊C₈	—	2	4	no	—
66	#12883	—	59	1	no	maximal, Sylow
67	C₂²×C₅₉	—	2	2	no	—
68	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₅₉	#432	1	2	yes	—
70	C₄×C₅₉	—	2	2	no	—
71	C₄×C₅₉	#364	1	4	yes	—
72	C₄×C₅₉	#382	1	1	yes	characteristic
73	C₄×C₅₉	#423	1	1	yes	characteristic
74	C₄×C₅₉	#431	1	2	yes	—
75	Dic₅₉	—	2	8	no	—
76	#2150	—	59	1	no	—
77	#2150	D₅₉	1	1	yes	characteristic
78	C₄²×C₈	—	59	1	no	—
79	#1376	—	59	4	no	—
80	C₂×C₅₉	—	2	2	no	—
81	C₂×C₅₉	#1831	1	1	yes	characteristic
82	C₂×C₅₉	C₂×#364	1	2	yes	—
83	C₂×M₃₂	—	59	2	no	—
84	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
85	C₂×C₄×C₈	—	59	1	no	—
86	C₂×C₄×C₈	—	59	2	no	—
87	C₂×C₄×C₈	—	59	2	no	—
88	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
89	C₄³	—	59	1	no	—
90	C₄⋊C₁₆	—	2	2	no	—
91	C₄⋊C₁₆	—	118	2	no	—
92	C₅₉	#12883	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	—	59	2	no	—
94	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
96	C₂×C₄²	—	59	2	no	—
97	C₂×C₄²	—	118	2	no	—
98	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
99	M₃₂	—	2	4	no	—
100	M₃₂	—	118	4	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	118	2	no	—
103	C₄×C₈	—	59	2	no	—
104	C₄×C₈	—	59	2	no	—
105	C₄×C₈	—	118	4	no	—
106	C₄×C₈	—	118	2	no	—
107	C₄×C₈	—	118	2	no	—
108	C₄×C₈	Dic₁₁₈	1	2	yes	—
109	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
110	C₂²×C₄	—	118	2	no	—
111	C₂²×C₄	#11745659	1	2	yes	—
112	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	59	4	no	—
115	C₂×C₈	—	118	2	no	—
116	C₂×C₈	—	118	2	no	—
117	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
118	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
119	C₂×C₈	#11745687	1	2	yes	—
120	C₂×C₈	#11745689	1	2	yes	—
121	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
122	C₄²	—	59	4	no	—
123	C₄²	—	118	8	no	—
124	C₄²	—	118	2	no	—
125	C₄²	C₈×D₅₉	1	1	yes	characteristic
126	C₄²	C₈×D₅₉	1	1	yes	characteristic
127	C₄²	Dic₅₉⋊C₄	1	2	yes	—
128	C₁₆	—	4	2	no	—
129	C₁₆	—	236	2	no	—
130	C₂³	C₂×#11745659	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	118	8	no	—
133	C₂×C₄	—	118	4	no	—
134	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
135	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
136	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
137	C₂×C₄	#14273685	1	2	yes	—
138	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
139	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
140	C₈	—	2	2	no	—
141	C₈	—	118	4	no	—
142	C₈	#14273714	1	1	yes	characteristic
143	C₈	#14273716	1	1	yes	characteristic
144	C₈	#14273718	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
147	C₂²	#16837458	1	2	yes	—
148	C₄	—	2	2	no	—
149	C₄	—	118	8	no	—
150	C₄	#16837222	1	4	yes	—
151	C₄	#16837267	1	1	yes	characteristic
152	C₄	#16837425	1	1	yes	characteristic
153	C₄	#16837454	1	2	yes	—
154	C₂	—	2	2	no	—
155	C₂	#21424212	1	1	yes	characteristic
156	C₂	C₂×#16837222	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425852

C₂

yes

maximal, characteristic

C₄×C₈×Dic₅₉

C₂

yes

maximal, characteristic

#21422177

C₂

yes

maximal

C₅₉×#2150

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₅₉

C₄

yes

—

C₂×C₈×Dic₅₉

C₂²

yes

—

C₄²×Dic₅₉

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₁₆)

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₄×C₅₉⋊C₈

C₈

yes

—

C₄×C₅₉⋊C₈

C₈

yes

—

C₈×Dic₅₉

—

C₈×Dic₅₉

—

C₈×Dic₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

M₁₆

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

M₁₆

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₈×C₅₉

yes

—

C₅₉⋊C₈

—

#12883

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₉

#432

yes

—

C₄×C₅₉

—

C₄×C₅₉

#364

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

#2150

—

#2150

D₅₉

yes

characteristic

C₄²×C₈

—

#1376

—

C₂×C₅₉

—

C₂×C₅₉

#1831

yes

characteristic

C₂×C₅₉

C₂×#364

yes

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

C₅₉

#12883

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

118

—

C₂×C₄²

C₄×D₅₉

yes

characteristic

M₃₂

—

100

M₃₂

—

118

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

118

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

118

—

106

C₄×C₈

—

118

—

107

C₄×C₈

—

118

—

108

C₄×C₈

Dic₁₁₈

yes

—

109

C₄×C₈

C₂²⋊D₅₉

yes

—

110

C₂²×C₄

—

118

—

111

C₂²×C₄

#11745659

yes

—

112

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

118

—

116

C₂×C₈

—

118

—

117

C₂×C₈

Dic₅₉⋊C₄

yes

—

118

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

119

C₂×C₈

#11745687

yes

—

120

C₂×C₈

#11745689

yes

—

121

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

122

C₄²

—

123

C₄²

—

118

—

124

C₄²

—

118

—

125

C₄²

C₈×D₅₉

yes

characteristic

126

C₄²

C₈×D₅₉

yes

characteristic

127

C₄²

Dic₅₉⋊C₄

yes

—

128

C₁₆

—

129

C₁₆

—

236

—

130

C₂³

C₂×#11745659

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

118

—

133

C₂×C₄

—

118

—

134

C₂×C₄

Dic₅₉⋊C₈

yes

—

135

C₂×C₄

Dic₅₉⋊C₈

yes

—

136

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

137

C₂×C₄

#14273685

yes

—

138

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

139

C₂×C₄

M₁₆×D₅₉

yes

—

140

C₈

—

141

C₈

—

118

—

142

C₈

#14273714

yes

characteristic

143

C₈

#14273716

yes

characteristic

144

C₈

#14273718

yes

—

145

C₂²

—

146

C₂²

C₂×Dic₅₉⋊C₈

yes

characteristic

147

C₂²

#16837458

yes

—

148

C₄

—

149

C₄

—

118

—

150

C₄

#16837222

yes

—

151

C₄

#16837267

yes

characteristic

152

C₄

#16837425

yes

characteristic

153

C₄

#16837454

yes

—

154

C₂

—

155

C₂

#21424212

yes

characteristic

156

C₂

C₂×#16837222

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#12883
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#2150

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₈

prime radical

C₂³×C₅₉

C₂×M₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
92. C₅₉	#12883	semidirect

H ≅ G/N

type

92. C₅₉

#12883

semidirect

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

subgroup	index
#21425852	2
C₄×C₈×Dic₅₉	2
#21422177	2
C₅₉×#2150	2
#12883	59

subgroup

index

#21425852

C₄×C₈×Dic₅₉

#21422177

C₅₉×#2150

#12883

Subgroups — #28505049

Conjugacy classes of subgroups

Complements

Maximal subgroups