Group #28479136 — subgroups

Subgroups — #28479136

Order 15104.

330 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 139 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	#21423782	C₂	1	1	yes	maximal, characteristic
2	#21422388	C₂	1	1	yes	maximal, characteristic
3	#21421926	C₂	1	2	yes	maximal
4	#21421924	C₂	1	2	yes	maximal
5	C₅₉×#1452	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273638	C₂²	1	1	yes	characteristic
8	C₂×#14273637	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₉	C₄	1	2	yes	—
10	#16837248	C₂²	1	1	yes	characteristic
11	#16837124	C₄	1	4	yes	complemented
12	C₅₉×#431	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#322	C₄	1	4	yes	complemented
15	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
17	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
19	#14273638	—	2	2	no	—
20	#14273637	—	2	4	no	—
21	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
22	C₈×Dic₅₉	—	2	4	no	—
23	C₅₉⋊#95	—	2	2	no	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₅₉	—	2	2	no	—
28	C₂²⋊C₈×C₅₉	—	2	2	no	—
29	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	#41	1	2	yes	—
34	C₂²×Dic₅₉	—	2	2	no	—
35	C₂×C₈×C₅₉	—	2	2	no	—
36	C₂×C₈×C₅₉	—	2	2	no	—
37	C₂×C₈×C₅₉	—	2	4	no	—
38	C₂×C₈×C₅₉	C₄²	1	2	yes	—
39	C₂×C₈×C₅₉	C₄²	1	2	yes	—
40	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₅₉	#41	1	2	yes	—
43	C₄×Dic₅₉	—	2	2	no	—
44	C₄×Dic₅₉	M₁₆	1	4	yes	—
45	C₂×C₅₉⋊C₈	—	2	2	no	—
46	C₂×C₅₉⋊C₈	—	2	2	no	—
47	C₂×C₅₉⋊C₈	—	2	4	no	—
48	C₂×C₅₉⋊C₈	M₁₆	1	4	yes	—
49	C₂³×C₅₉	#117	1	1	yes	characteristic
50	C₂×C₄×C₅₉	—	4	2	no	—
51	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#117	1	2	yes	—
53	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
54	C₂×C₄×C₅₉	#117	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
57	C₂×Dic₅₉	—	2	4	no	—
58	C₂×Dic₅₉	—	4	4	no	—
59	C₈×C₅₉	—	2	2	no	—
60	C₈×C₅₉	—	2	2	no	—
61	C₈×C₅₉	—	4	4	no	—
62	C₅₉⋊C₈	—	2	4	no	—
63	C₅₉⋊C₈	—	4	4	no	—
64	#8139	—	59	1	no	maximal, Sylow
65	C₂²×C₅₉	—	2	2	no	—
66	C₂²×C₅₉	#376	1	1	yes	characteristic
67	C₂²×C₅₉	#432	1	2	yes	—
68	C₄×C₅₉	—	2	2	no	—
69	C₄×C₅₉	—	2	2	no	—
70	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
71	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
72	C₄×C₅₉	#432	1	2	yes	—
73	Dic₅₉	—	4	4	no	—
74	#1726	—	59	1	no	—
75	#1452	—	59	1	no	—
76	#1452	D₅₉	1	1	yes	characteristic
77	#1275	—	59	2	no	—
78	#1274	—	59	2	no	—
79	C₂×C₅₉	—	2	2	no	—
80	C₂×C₅₉	#1745	1	1	yes	characteristic
81	C₂×C₅₉	#1747	1	2	yes	—
82	#431	—	59	2	no	—
83	#431	C₂×D₅₉	1	2	yes	—
84	C₂×#97	—	59	2	no	—
85	C₂×#97	—	59	1	no	—
86	C₂×C₄×C₈	—	59	2	no	—
87	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
88	#376	—	59	1	no	—
89	#322	—	59	4	no	—
90	#322	Dic₅₉	1	4	yes	complemented
91	C₅₉	#8139	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	59	2	no	—
93	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
94	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
95	C₂×C₄²	—	59	2	no	—
96	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	118	2	no	—
99	C₂²⋊C₈	—	2	2	no	—
100	C₂²⋊C₈	—	118	2	no	—
101	#97	—	118	4	no	—
102	#97	—	118	2	no	—
103	C₄×C₈	—	59	4	no	—
104	C₄×C₈	—	118	4	no	—
105	C₄×C₈	C₄×D₅₉	1	2	yes	—
106	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
107	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
108	#95	—	118	2	no	—
109	C₂²×C₄	—	118	2	no	—
110	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
111	C₂²×C₄	#11745685	1	2	yes	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	59	4	no	—
116	C₂×C₈	—	118	2	no	—
117	C₂×C₈	—	118	2	no	—
118	C₂×C₈	—	118	4	no	—
119	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
120	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
121	C₄²	—	59	4	no	—
122	C₄²	—	118	2	no	—
123	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
124	C₄²	#11745685	1	2	yes	—
125	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
126	C₂³	#14273684	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	118	4	no	—
129	C₂×C₄	—	236	4	no	—
130	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
131	C₂×C₄	#14273684	1	2	yes	—
132	C₂×C₄	#14273684	1	1	yes	characteristic
133	C₂×C₄	#14273686	1	2	yes	—
134	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
135	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	—	4	4	no	—
139	C₈	—	118	4	no	—
140	C₈	—	236	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#16837248	1	1	yes	characteristic
143	C₂²	#16837457	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	236	4	no	—
147	C₄	M₁₆×Dic₅₉	1	1	yes	characteristic
148	C₄	M₁₆×Dic₅₉	1	1	yes	characteristic
149	C₄	#16837457	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#21423844	1	1	yes	characteristic
152	C₂	#21423851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21423782

C₂

yes

maximal, characteristic

#21422388

C₂

yes

maximal, characteristic

#21421926

C₂

yes

maximal

#21421924

C₂

yes

maximal

C₅₉×#1452

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

C₂×#14273638

C₂²

yes

characteristic

C₂×#14273637

C₄

yes

—

C₂×C₈×Dic₅₉

C₄

yes

—

#16837248

C₂²

yes

characteristic

#16837124

C₄

yes

complemented

C₅₉×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

C₄

yes

complemented

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

#14273637

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₈×Dic₅₉

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄²

yes

—

C₂×C₈×C₅₉

C₄²

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

M₁₆

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

M₁₆

yes

—

C₂³×C₅₉

#117

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#117

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#8139

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#376

yes

characteristic

C₂²×C₅₉

#432

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×M₁₆

yes

characteristic

C₄×C₅₉

C₄×M₁₆

yes

characteristic

C₄×C₅₉

#432

yes

—

Dic₅₉

—

#1726

—

#1452

—

#1452

D₅₉

yes

characteristic

#1275

—

#1274

—

C₂×C₅₉

—

C₂×C₅₉

#1745

yes

characteristic

C₂×C₅₉

#1747

yes

—

#431

—

#431

C₂×D₅₉

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

—

Dic₅₉

yes

complemented

C₅₉

#8139

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

characteristic

C₂²×C₈

C₄×D₅₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

118

—

101

#97

—

118

—

102

#97

—

118

—

103

C₄×C₈

—

104

C₄×C₈

—

118

—

105

C₄×C₈

C₄×D₅₉

yes

—

106

C₄×C₈

C₂×Dic₅₉

yes

—

107

C₄×C₈

C₂×Dic₅₉

yes

—

108

#95

—

118

—

109

C₂²×C₄

—

118

—

110

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

111

C₂²×C₄

#11745685

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

118

—

117

C₂×C₈

—

118

—

118

C₂×C₈

—

118

—

119

C₂×C₈

C₄×Dic₅₉

yes

—

120

C₂×C₈

C₄×Dic₅₉

yes

—

121

C₄²

—

122

C₄²

—

118

—

123

C₄²

C₂×C₄×D₅₉

yes

characteristic

124

C₄²

#11745685

yes

—

125

C₄²

C₂²×Dic₅₉

yes

characteristic

126

C₂³

#14273684

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

118

—

129

C₂×C₄

—

236

—

130

C₂×C₄

C₂×C₄×Dic₅₉

yes

characteristic

131

C₂×C₄

#14273684

yes

—

132

C₂×C₄

#14273684

yes

characteristic

133

C₂×C₄

#14273686

yes

—

134

C₂×C₄

M₁₆×D₅₉

yes

—

135

C₂×C₄

M₁₆×D₅₉

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

118

—

140

C₈

—

236

—

141

C₂²

—

142

C₂²

#16837248

yes

characteristic

143

C₂²

#16837457

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

236

—

147

C₄

M₁₆×Dic₅₉

yes

characteristic

148

C₄

M₁₆×Dic₅₉

yes

characteristic

149

C₄

#16837457

yes

—

150

C₂

—

151

C₂

#21423844

yes

characteristic

152

C₂

#21423851

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1452	C₂
socle	C₂²×C₅₉	#376
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1452	D₅₉
hypocenter	C₅₉	#8139
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #1452

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₅₉

commutator

C₂×C₄×C₅₉

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1452

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ #1452

N	H ≅ G/N	type
11. #16837124	C₄	semidirect
14. C₅₉×#322	C₄	semidirect
90. #322	Dic₅₉	semidirect
91. C₅₉	#8139	semidirect

H ≅ G/N

type

11. #16837124

C₄

semidirect

14. C₅₉×#322

C₄

semidirect

90. #322

Dic₅₉

semidirect

91. C₅₉

#8139

semidirect

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

subgroup	index
#21423782	2
#21422388	2
#21421926	2
#21421924	2
C₅₉×#1452	2
#8139	59

subgroup

index

C₅₉×#1452

Subgroups — #28479136

Conjugacy classes of subgroups

Complements

Maximal subgroups