Group #28475547 — subgroups

Subgroups — #28475547

Order 15104.

266 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 103 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	#21425899	C₂	1	2	yes	maximal
2	C₄×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₅₉⋊C₁₆)	C₄	1	4	yes	complemented
4	C₂²⋊(C₅₉⋊C₁₆)	—	2	2	no	—
5	#16837178	—	2	2	no	—
6	C₂×#97×C₅₉	—	2	2	no	—
7	C₂×C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₄³×C₅₉	C₄	1	1	yes	characteristic
10	C₂×C₅₉⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₅₉	—	2	1	no	—
12	C₂²×C₈×C₅₉	D₄	1	2	yes	—
13	C₂×C₄²×C₅₉	—	2	1	no	—
14	C₂×C₄²×C₅₉	—	2	1	no	—
15	C₂×C₄²×C₅₉	C₈	1	2	yes	—
16	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
17	#97×C₅₉	—	2	8	no	—
18	C₄×C₈×C₅₉	—	2	1	no	—
19	C₄×C₈×C₅₉	C₈	1	2	yes	—
20	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	—	2	2	no	—
24	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₅₉	—	2	4	no	—
28	C₂×C₈×C₅₉	—	2	2	no	—
29	C₂×C₈×C₅₉	—	2	4	no	—
30	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
32	C₄²×C₅₉	—	2	4	no	—
33	C₄²×C₅₉	—	4	2	no	—
34	C₄²×C₅₉	—	4	2	no	—
35	C₄²×C₅₉	C₄²	1	1	yes	characteristic
36	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
39	C₅₉⋊C₁₆	—	4	4	no	—
40	C₂³×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	—	2	4	no	—
45	C₂×C₄×C₅₉	—	4	2	no	—
46	C₂×C₄×C₅₉	—	4	2	no	—
47	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₉	#97	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
53	C₈×C₅₉	—	2	4	no	—
54	C₈×C₅₉	—	2	2	no	—
55	C₈×C₅₉	#104	1	4	yes	—
56	#7149	—	59	1	no	maximal, Sylow
57	C₂²×C₅₉	—	2	1	no	—
58	C₂²×C₅₉	—	2	1	no	—
59	C₂²×C₅₉	#336	1	1	yes	characteristic
60	C₂²×C₅₉	#349	1	2	yes	—
61	C₄×C₅₉	—	2	1	no	—
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	—	2	1	no	—
64	C₄×C₅₉	—	2	1	no	—
65	C₄×C₅₉	—	4	4	no	—
66	C₄×C₅₉	#339	1	2	yes	—
67	C₄×C₅₉	#347	1	2	yes	—
68	#2163	—	59	2	no	—
69	C₄×#97	D₅₉	1	1	yes	characteristic
70	C₂×C₅₉	—	2	2	no	—
71	C₂×C₅₉	#1276	1	1	yes	characteristic
72	C₂×C₅₉	#1375	1	1	yes	characteristic
73	C₂×C₅₉	#1377	1	1	yes	characteristic
74	C₂×#97	—	2	2	no	—
75	C₂×C₄×C₈	Dic₅₉	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
77	C₄³	Dic₅₉	1	1	yes	characteristic
78	C₄⋊C₁₆	—	59	4	no	—
79	C₂²⋊C₁₆	—	118	2	no	—
80	#346	—	118	2	no	—
81	C₅₉	#7149	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₅₉⋊C₈	1	2	yes	—
87	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
88	C₂×C₁₆	—	118	4	no	—
89	#97	—	2	8	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₅₉⋊C₈	1	2	yes	—
92	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
93	C₄×C₈	C₄×D₅₉	1	2	yes	—
94	C₄×C₈	D₂₃₆	1	1	yes	characteristic
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
97	C₂²×C₄	#11745664	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
103	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
104	C₄²	—	2	4	no	—
105	C₄²	—	4	2	no	—
106	C₄²	—	4	2	no	—
107	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
108	C₄²	#11745664	1	1	yes	characteristic
109	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
110	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
111	C₁₆	—	236	4	no	—
112	C₂³	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
120	C₂×C₄	C₄×C₅₉⋊C₈	1	1	yes	characteristic
121	C₂×C₄	#14273638	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
123	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
124	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	—	2	2	no	—
127	C₈	#14273655	1	4	yes	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#16837157	1	1	yes	characteristic
131	C₂²	#16837184	1	2	yes	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	4	no	—
137	C₄	#16837162	1	2	yes	—
138	C₄	#16837180	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	#21421929	1	1	yes	characteristic
141	C₂	#21422176	1	1	yes	characteristic
142	C₂	#21422180	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425899

C₂

yes

maximal

C₄×#97×C₅₉

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₁₆)

C₄

yes

complemented

C₂²⋊(C₅₉⋊C₁₆)

—

#16837178

—

C₂×#97×C₅₉

—

C₂×C₄×C₈×C₅₉

C₄

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄³×C₅₉

C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₈

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₈

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₉

M₁₆

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₄²

yes

characteristic

C₄²×C₅₉

C₄⋊C₄

yes

characteristic

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄²×C₅₉

M₁₆

yes

characteristic

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#95

yes

characteristic

C₂×C₄×C₅₉

C₄×C₈

yes

characteristic

C₂×C₄×C₅₉

#97

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#104

yes

—

#7149

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#336

yes

characteristic

C₂²×C₅₉

#349

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#339

yes

—

C₄×C₅₉

#347

yes

—

#2163

—

C₄×#97

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×#97

—

C₂×C₄×C₈

Dic₅₉

yes

characteristic

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄³

Dic₅₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

118

—

#346

—

118

—

C₅₉

#7149

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₅₉⋊C₈

yes

—

C₂×C₄²

C₂×Dic₅₉

yes

characteristic

C₂×C₁₆

—

118

—

#97

—

C₄×C₈

—

C₄×C₈

C₅₉⋊C₈

yes

—

C₄×C₈

Dic₁₁₈

yes

characteristic

C₄×C₈

C₄×D₅₉

yes

—

C₄×C₈

D₂₃₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×C₅₉⋊C₈

yes

characteristic

C₂²×C₄

#11745664

yes

characteristic

C₂²×C₄

C₂²⋊Dic₅₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

Dic₅₉⋊C₄

yes

—

103

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂×C₅₉⋊C₈

yes

characteristic

108

C₄²

#11745664

yes

characteristic

109

C₄²

C₄×Dic₅₉

yes

characteristic

110

C₄²

C₄⋊Dic₅₉

yes

characteristic

111

C₁₆

—

236

—

112

C₂³

C₂²⋊(C₅₉⋊C₈)

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₅₉⋊#95

yes

characteristic

120

C₂×C₄

C₄×C₅₉⋊C₈

yes

characteristic

121

C₂×C₄

#14273638

yes

characteristic

122

C₂×C₄

C₂²⋊(C₅₉⋊C₈)

yes

characteristic

123

C₂×C₄

C₄⋊(C₅₉⋊C₈)

yes

characteristic

124

C₂×C₄

C₄⋊(C₅₉⋊C₈)

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

#14273655

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

#16837157

yes

characteristic

131

C₂²

#16837184

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#16837162

yes

—

138

C₄

#16837180

yes

—

139

C₂

—

140

C₂

#21421929

yes

characteristic

141

C₂

#21422176

yes

characteristic

142

C₂

#21422180

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₅₉⋊#95
commutator	C₂×C₄×C₅₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₄×#97×C₅₉	C₂
socle	C₂²×C₅₉	#336
prime radical	C₂³×C₅₉	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	C₄×#97	D₅₉
hypocenter	C₅₉	#7149
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#97

subgroup

G/H

center

C₂×C₄

C₅₉⋊#95

commutator

C₂×C₄×C₅₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₅₉

Fitting

C₄×#97×C₅₉

C₂

socle

C₂²×C₅₉

#336

prime radical

C₂³×C₅₉

C₂²⋊C₈

cosocle

C₂×C₄×C₈×C₅₉

C₂²

hypercenter

C₄×#97

D₅₉

hypocenter

C₅₉

#7149

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#97

N	H ≅ G/N	type
3. C₄⋊(C₅₉⋊C₁₆)	C₄	semidirect
81. C₅₉	#7149	semidirect

H ≅ G/N

type

3. C₄⋊(C₅₉⋊C₁₆)

C₄

semidirect

81. C₅₉

#7149

semidirect

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

subgroup	index
#21425899	2
C₄×#97×C₅₉	2
#7149	59

subgroup

index

#21425899

C₄×#97×C₅₉

#7149

Subgroups — #28475547

Conjugacy classes of subgroups

Complements

Maximal subgroups