Group #28475511 — subgroups

Subgroups — #28475511

Order 15104.

216 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 71 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	#21422180	C₂	1	2	yes	maximal
2	C₅₉×#1985	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273667	—	2	2	no	—
4	C₂×C₄×Q₈×C₅₉	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₈×C₅₉	C₄	1	1	yes	characteristic
6	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
7	C₅₉×#336	—	2	2	no	—
8	#14273667	—	2	4	no	—
9	#14273667	—	4	2	no	—
10	C₂×C₅₉⋊C₁₆	—	4	2	no	—
11	C₂²×Q₈×C₅₉	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₅₉	—	2	1	no	—
13	C₂²×C₈×C₅₉	D₄	1	2	yes	—
14	C₄×Q₈×C₅₉	—	4	1	no	—
15	C₄×Q₈×C₅₉	—	4	1	no	—
16	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₉	C₈	1	1	yes	characteristic
18	C₂×C₄²×C₅₉	—	2	1	no	—
19	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₈×C₅₉	—	4	1	no	—
21	#97×C₅₉	—	4	1	no	—
22	C₂×Q₈×C₅₉	—	4	2	no	—
23	C₂×Q₈×C₅₉	—	4	1	no	—
24	C₂²×C₄×C₅₉	—	2	1	no	—
25	C₂²×C₄×C₅₉	—	2	1	no	—
26	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
29	C₂×C₈×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	—	2	2	no	—
31	C₂×C₈×C₅₉	—	4	1	no	—
32	C₂×C₈×C₅₉	—	4	2	no	—
33	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
34	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
35	C₄⋊C₄×C₅₉	—	4	2	no	—
36	C₄⋊C₄×C₅₉	—	4	1	no	—
37	C₄²×C₅₉	—	2	1	no	—
38	C₄²×C₅₉	—	2	1	no	—
39	C₄²×C₅₉	—	4	1	no	—
40	C₄²×C₅₉	—	4	1	no	—
41	C₅₉⋊C₁₆	—	4	4	no	—
42	C₂³×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
43	Q₈×C₅₉	—	4	2	no	—
44	Q₈×C₅₉	—	4	2	no	—
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	—	2	1	no	—
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	—	4	1	no	—
49	C₂×C₄×C₅₉	—	4	1	no	—
50	C₂×C₄×C₅₉	—	4	1	no	—
51	C₂×C₄×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	—	4	2	no	—
53	C₂×C₄×C₅₉	—	4	1	no	—
54	C₂×C₄×C₅₉	#99	1	1	yes	characteristic
55	C₂×C₄×C₅₉	#100	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#103	1	2	yes	—
57	C₂×C₄×C₅₉	#104	1	2	yes	—
58	C₈×C₅₉	—	2	2	no	—
59	C₈×C₅₉	—	2	2	no	—
60	C₈×C₅₉	—	8	1	no	—
61	#7133	—	59	1	no	maximal, Sylow
62	C₂²×C₅₉	—	2	1	no	—
63	C₂²×C₅₉	—	2	1	no	—
64	C₂²×C₅₉	#323	1	1	yes	characteristic
65	C₂²×C₅₉	Q₈⋊C₈	1	2	yes	—
66	C₄×C₅₉	—	4	2	no	—
67	C₄×C₅₉	—	4	1	no	—
68	C₄×C₅₉	—	4	1	no	—
69	C₄×C₅₉	—	8	1	no	—
70	C₄×C₅₉	#328	1	2	yes	—
71	C₄×C₅₉	#329	1	1	yes	characteristic
72	C₄×C₅₉	#333	1	1	yes	characteristic
73	#1985	D₅₉	1	1	yes	characteristic
74	#1377	—	59	2	no	—
75	C₂×C₅₉	—	2	2	no	—
76	C₂×C₅₉	#1272	1	1	yes	characteristic
77	C₂×C₅₉	#1320	1	1	yes	characteristic
78	C₂×C₅₉	#1321	1	1	yes	characteristic
79	C₂×C₄×Q₈	Dic₅₉	1	1	yes	characteristic
80	C₂×M₃₂	—	118	2	no	—
81	C₂×C₄⋊C₈	Dic₅₉	1	1	yes	characteristic
82	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
83	#336	—	2	2	no	—
84	C₅₉	#7133	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	C₅₉⋊C₈	1	1	yes	characteristic
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
88	C₄×Q₈	—	4	1	no	—
89	C₄×Q₈	—	4	1	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	C₅₉⋊C₈	1	1	yes	characteristic
92	C₂×C₄²	—	2	1	no	—
93	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
94	M₃₂	—	118	4	no	—
95	M₃₂	—	236	2	no	—
96	C₂×C₁₆	—	236	2	no	—
97	C₄⋊C₈	—	4	1	no	—
98	#97	—	4	1	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
104	C₂²×C₄	#11745664	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	#11745671	1	2	yes	—
111	C₂×C₈	#11745672	1	2	yes	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₁₆	—	236	4	no	—
119	C₂³	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	#14273642	1	1	yes	characteristic
132	C₂×C₄	#14273644	1	1	yes	characteristic
133	C₂×C₄	#14273651	1	2	yes	—
134	C₂×C₄	#14273654	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	8	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	#16837126	1	1	yes	characteristic
141	C₂²	Q₈⋊(C₅₉⋊C₈)	1	2	yes	—
142	C₄	—	4	2	no	—
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	8	1	no	—
146	C₄	#16837139	1	2	yes	—
147	C₄	#16837142	1	1	yes	characteristic
148	C₄	#16837151	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#21421922	1	1	yes	characteristic
151	C₂	#21422032	1	1	yes	characteristic
152	C₂	#21422034	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21422180

C₂

yes

maximal

C₅₉×#1985

C₂

yes

maximal, characteristic

C₂×#14273667

—

C₂×C₄×Q₈×C₅₉

C₄

yes

characteristic

C₂×C₄⋊C₈×C₅₉

C₄

yes

characteristic

C₂×#97×C₅₉

C₂²

yes

characteristic

C₅₉×#336

—

#14273667

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₂²×Q₈×C₅₉

C₈

yes

characteristic

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₄×Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₈

yes

characteristic

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₉

—

#97×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

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

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂²⋊C₈

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

#104

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

#7133

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#323

yes

characteristic

C₂²×C₅₉

Q₈⋊C₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#328

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

characteristic

#1377

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₄×Q₈

Dic₅₉

yes

characteristic

C₂×M₃₂

—

118

—

C₂×C₄⋊C₈

Dic₅₉

yes

characteristic

C₂×#97

C₂×D₅₉

yes

characteristic

#336

—

C₅₉

#7133

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₅₉⋊C₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₅₉⋊C₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₅₉

yes

characteristic

M₃₂

—

118

—

M₃₂

—

236

—

C₂×C₁₆

—

236

—

C₄⋊C₈

—

#97

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂×C₅₉⋊C₈

yes

characteristic

104

C₂²×C₄

#11745664

yes

characteristic

105

C₂²×C₄

C₂²⋊Dic₅₉

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

#11745671

yes

—

111

C₂×C₈

#11745672

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₁₆

—

236

—

119

C₂³

C₂²⋊(C₅₉⋊C₈)

yes

characteristic

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#14273642

yes

characteristic

132

C₂×C₄

#14273644

yes

characteristic

133

C₂×C₄

#14273651

yes

—

134

C₂×C₄

#14273654

yes

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

#16837126

yes

characteristic

141

C₂²

Q₈⋊(C₅₉⋊C₈)

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#16837139

yes

—

147

C₄

#16837142

yes

characteristic

148

C₄

#16837151

yes

characteristic

149

C₂

—

150

C₂

#21421922

yes

characteristic

151

C₂

#21422032

yes

characteristic

152

C₂

#21422034

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14273642
commutator	C₂²×C₄×C₅₉	C₂×C₈
Frattini	C₂×#97	C₂×D₅₉
Fitting	C₅₉×#1985	C₂
socle	C₂²×C₅₉	#323
prime radical	C₂³×C₅₉	C₂²⋊C₈
cosocle	C₂×#97×C₅₉	C₂²
hypercenter	#1985	D₅₉
hypocenter	C₅₉	#7133
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1985

subgroup

G/H

center

C₂×C₄

#14273642

commutator

C₂²×C₄×C₅₉

C₂×C₈

Frattini

C₂×#97

C₂×D₅₉

Fitting

C₅₉×#1985

C₂

socle

C₂²×C₅₉

#323

prime radical

C₂³×C₅₉

C₂²⋊C₈

cosocle

C₂×#97×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1985

N	H ≅ G/N	type
84. C₅₉	#7133	semidirect

H ≅ G/N

type

84. C₅₉

#7133

semidirect

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

subgroup	index
#21422180	2
C₅₉×#1985	2
#7133	59

subgroup

index

#21422180

C₅₉×#1985

#7133

Subgroups — #28475511

Conjugacy classes of subgroups

Complements

Maximal subgroups