Group #28488658 — subgroups

Subgroups — #28488658

Order 15104.

406 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 159 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	#21424268	C₂	1	1	yes	maximal, characteristic
2	#21422551	C₂	1	4	yes	maximal
3	C₂×#16837123	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273663	—	2	2	no	—
6	#16837426	C₂²	1	2	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
8	Q₈⋊(C₅₉⋊C₈)	—	2	8	no	—
9	#16837123	—	2	2	no	—
10	C₅₉×#533	C₂²	1	2	yes	—
11	C₅₉×#500	C₄	1	2	yes	—
12	C₅₉×#499	—	2	2	no	—
13	C₅₉×#498	C₄	1	2	yes	—
14	C₅₉×#427	—	2	2	no	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#11745664	—	2	2	no	—
17	C₂²×C₅₉⋊C₈	—	2	2	no	—
18	#14273663	—	4	4	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
20	C₄×C₅₉⋊C₈	—	2	2	no	—
21	C₄×C₅₉⋊C₈	D₄	1	4	yes	—
22	C₂²×C₈×C₅₉	D₄	1	2	yes	—
23	C₄⋊Q₈×C₅₉	C₂×C₄	1	4	yes	—
24	#125×C₅₉	—	2	2	no	—
25	C₂²⋊Q₈×C₅₉	—	2	4	no	—
26	C₄×Q₈×C₅₉	—	2	4	no	—
27	#117×C₅₉	—	2	2	no	—
28	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	#107×C₅₉	—	2	4	no	—
30	#106×C₅₉	—	2	4	no	—
31	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₉	D₄	1	2	yes	—
34	C₂×Q₈×C₅₉	—	2	4	no	—
35	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	#41	1	2	yes	—
37	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
40	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
41	C₄⋊C₄×C₅₉	—	2	4	no	—
42	C₄⋊C₄×C₅₉	—	2	4	no	—
43	C₄⋊C₄×C₅₉	—	4	4	no	—
44	C₂²⋊C₄×C₅₉	—	4	2	no	—
45	C₄²×C₅₉	—	4	2	no	—
46	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₂×D₄	1	2	yes	—
49	#11745664	—	4	4	no	—
50	C₂×C₅₉⋊C₈	—	2	4	no	—
51	C₂×C₅₉⋊C₈	—	4	4	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂³×C₅₉	#124	1	1	yes	characteristic
54	Q₈×C₅₉	—	4	8	no	—
55	C₂×C₄×C₅₉	—	2	4	no	—
56	C₂×C₄×C₅₉	—	4	4	no	—
57	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₅₉	#121	1	2	yes	—
60	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
63	C₈×C₅₉	#103	1	4	yes	—
64	C₈×C₅₉	#103	1	4	yes	—
65	C₅₉⋊C₈	—	4	4	no	—
66	C₅₉⋊C₈	—	8	2	no	—
67	#9861	—	59	1	no	maximal, Sylow
68	C₂²×C₅₉	—	2	2	no	—
69	C₂²×C₅₉	#390	1	1	yes	characteristic
70	C₂²×C₅₉	#487	1	2	yes	—
71	C₄×C₅₉	—	2	2	no	—
72	C₄×C₅₉	—	8	4	no	—
73	C₄×C₅₉	C₂×#103	1	2	yes	—
74	C₄×C₅₉	#444	1	4	yes	—
75	C₄×C₅₉	#492	1	1	yes	characteristic
76	C₄×C₅₉	#494	1	1	yes	characteristic
77	#3160	D₅₉	1	1	yes	characteristic
78	#1848	—	59	1	no	—
79	#1493	—	59	4	no	—
80	C₂×#322	—	59	1	no	—
81	C₂×C₅₉	—	2	2	no	—
82	C₂×C₅₉	#1956	1	1	yes	characteristic
83	C₂×C₅₉	#1975	1	2	yes	—
84	#533	C₂×D₅₉	1	2	yes	—
85	#500	Dic₅₉	1	2	yes	—
86	#499	—	2	2	no	—
87	#498	Dic₅₉	1	2	yes	—
88	C₂×#108	—	118	2	no	—
89	#427	—	2	2	no	—
90	#423	—	59	2	no	—
91	C₂×C₄×C₈	—	59	2	no	—
92	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
93	Q₈⋊C₈	—	118	8	no	—
94	#322	—	118	2	no	—
95	C₅₉	#9861	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₁₆	—	118	2	no	—
97	C₂²×C₈	—	118	2	no	—
98	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
99	C₄⋊Q₈	C₂×Dic₅₉	1	4	yes	—
100	#125	—	2	2	no	—
101	C₂²⋊Q₈	—	2	4	no	—
102	C₄×Q₈	—	2	4	no	—
103	#117	—	2	2	no	—
104	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
105	#108	—	236	4	no	—
106	#107	—	2	4	no	—
107	#106	—	2	4	no	—
108	C₄⋊C₈	—	118	4	no	—
109	C₄×C₈	—	59	4	no	—
110	C₄×C₈	—	118	2	no	—
111	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
114	C₂×Q₈	—	2	4	no	—
115	C₂²×C₄	#11745687	1	2	yes	—
116	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
117	M₁₆	—	236	4	no	—
118	C₂×C₈	—	118	4	no	—
119	C₂×C₈	—	236	2	no	—
120	C₂×C₈	—	236	4	no	—
121	C₂×C₈	#11745671	1	4	yes	—
122	C₂×C₈	#11745672	1	4	yes	—
123	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
124	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	4	2	no	—
130	C₄²	D₄×D₅₉	1	2	yes	—
131	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
132	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
133	C₂³	#14273714	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
138	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
139	C₂×C₄	#14273698	1	2	yes	—
140	C₂×C₄	#14273723	1	1	yes	characteristic
141	C₂×C₄	C₂×#11745671	1	2	yes	—
142	C₂×C₄	C₂×#11745672	1	2	yes	—
143	C₈	—	472	2	no	—
144	C₈	—	236	4	no	—
145	C₈	#14273651	1	4	yes	—
146	C₈	#14273651	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#16837305	1	1	yes	characteristic
149	C₂²	#16837752	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	8	4	no	—
152	C₄	C₂×#14273651	1	2	yes	—
153	C₄	#16837524	1	4	yes	—
154	C₄	#16837776	1	1	yes	characteristic
155	C₄	#16837781	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#21424899	1	1	yes	characteristic
158	C₂	#21424990	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424268

C₂

yes

maximal, characteristic

#21422551

C₂

yes

maximal

C₂×#16837123

C₂

yes

maximal, characteristic

C₅₉×#3160

C₂

yes

maximal, characteristic

C₂×#14273663

—

#16837426

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

Q₈⋊(C₅₉⋊C₈)

—

#16837123

—

C₅₉×#533

C₂²

yes

—

C₅₉×#500

C₄

yes

—

C₅₉×#499

—

C₅₉×#498

C₄

yes

—

C₅₉×#427

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

C₂²×C₅₉⋊C₈

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₄⋊Q₈×C₅₉

C₂×C₄

yes

—

#125×C₅₉

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#107×C₅₉

—

#106×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#124

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₈×C₅₉

#103

yes

—

C₈×C₅₉

#103

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9861

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#487

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#103

yes

—

C₄×C₅₉

#444

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

#1956

yes

characteristic

C₂×C₅₉

#1975

yes

—

#533

C₂×D₅₉

yes

—

#500

Dic₅₉

yes

—

#499

—

#498

Dic₅₉

yes

—

C₂×#108

—

118

—

#427

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

Q₈⋊C₈

—

118

—

#322

—

118

—

C₅₉

#9861

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

—

118

—

C₂²×C₈

C₂²⋊D₅₉

yes

—

C₄⋊Q₈

C₂×Dic₅₉

yes

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄²

C₂²×D₅₉

yes

characteristic

105

#108

—

236

—

106

#107

—

107

#106

—

108

C₄⋊C₈

—

118

—

109

C₄×C₈

—

110

C₄×C₈

—

118

—

111

C₄×C₈

C₂×Dic₅₉

yes

characteristic

112

C₄×C₈

C₂×Dic₅₉

yes

characteristic

113

C₄×C₈

C₂²⋊D₅₉

yes

—

114

C₂×Q₈

—

115

C₂²×C₄

#11745687

yes

—

116

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

117

M₁₆

—

236

—

118

C₂×C₈

—

118

—

119

C₂×C₈

—

236

—

120

C₂×C₈

—

236

—

121

C₂×C₈

#11745671

yes

—

122

C₂×C₈

#11745672

yes

—

123

C₂×C₈

C₂²⋊Dic₅₉

yes

—

124

C₂×C₈

C₂²⋊Dic₅₉

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

D₄×D₅₉

yes

—

131

C₄²

C₂²×Dic₅₉

yes

characteristic

132

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

133

C₂³

#14273714

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

138

C₂×C₄

D₄×Dic₅₉

yes

—

139

C₂×C₄

#14273698

yes

—

140

C₂×C₄

#14273723

yes

characteristic

141

C₂×C₄

C₂×#11745671

yes

—

142

C₂×C₄

C₂×#11745672

yes

—

143

C₈

—

472

—

144

C₈

—

236

—

145

C₈

#14273651

yes

—

146

C₈

#14273651

yes

—

147

C₂²

—

148

C₂²

#16837305

yes

characteristic

149

C₂²

#16837752

yes

—

150

C₄

—

151

C₄

—

152

C₄

C₂×#14273651

yes

—

153

C₄

#16837524

yes

—

154

C₄

#16837776

yes

characteristic

155

C₄

#16837781

yes

characteristic

156

C₂

—

157

C₂

#21424899

yes

characteristic

158

C₂

#21424990

yes

—

	subgroup	G/H
center	C₂×C₄	#14273723
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#3160	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂³×C₅₉	#124
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#3160	D₅₉
hypocenter	C₅₉	#9861
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#14273723

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#3160

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
95. C₅₉	#9861	semidirect

H ≅ G/N

type

95. C₅₉

#9861

semidirect

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

subgroup	index
#21424268	2
#21422551	2
C₂×#16837123	2
C₅₉×#3160	2
#9861	59

subgroup

index

C₅₉×#3160

Subgroups — #28488658

Conjugacy classes of subgroups

Complements

Maximal subgroups