Group #28481685 — subgroups

Subgroups — #28481685

Order 15104.

320 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 119 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	#21422859	C₂	1	2	yes	maximal
2	#21422734	C₂	1	2	yes	maximal
3	#21422007	C₂	1	2	yes	maximal
4	C₅₉×#2960	C₂	1	1	yes	maximal, characteristic
5	#16837459	C₂²	1	4	yes	—
6	#16837155	—	2	2	no	—
7	#16837153	—	2	2	no	—
8	#16837144	—	2	4	no	—
9	C₅₉×#533	C₂²	1	1	yes	characteristic
10	C₅₉×#446	—	2	2	no	—
11	C₈×Q₈×C₅₉	—	2	2	no	—
12	C₅₉×#424	C₄	1	2	yes	—
13	C₅₉×#423	C₂²	1	1	yes	characteristic
14	C₅₉×#405	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
17	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
18	#14273638	—	2	4	no	—
19	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₅₉	Q₈	1	2	yes	—
23	#125×C₅₉	D₄	1	2	yes	—
24	C₂²⋊Q₈×C₅₉	—	2	2	no	—
25	C₄×Q₈×C₅₉	—	2	2	no	—
26	#117×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₅₉	—	2	4	no	—
29	C₄⋊C₈×C₅₉	—	2	2	no	—
30	#97×C₅₉	—	2	2	no	—
31	C₄×C₈×C₅₉	—	2	2	no	—
32	C₄×C₈×C₅₉	D₄	1	2	yes	—
33	C₄×C₈×C₅₉	D₄	1	2	yes	—
34	C₂×Q₈×C₅₉	—	2	2	no	—
35	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	#41	1	2	yes	—
37	M₁₆×C₅₉	—	4	2	no	—
38	C₂×C₈×C₅₉	—	2	2	no	—
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	—	4	1	no	—
41	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₄⋊C₄×C₅₉	—	2	4	no	—
45	C₄⋊C₄×C₅₉	—	2	2	no	—
46	C₄⋊C₄×C₅₉	C₄⋊C₄	1	4	yes	—
47	C₂²⋊C₄×C₅₉	—	2	2	no	—
48	C₄²×C₅₉	—	2	2	no	—
49	C₄²×C₅₉	C₂×D₄	1	2	yes	—
50	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₅₉⋊C₈	—	4	4	no	—
53	C₂³×C₅₉	#117	1	1	yes	characteristic
54	Q₈×C₅₉	—	4	4	no	—
55	C₂×C₄×C₅₉	—	2	1	no	—
56	C₂×C₄×C₅₉	—	2	1	no	—
57	C₂×C₄×C₅₉	—	2	4	no	—
58	C₂×C₄×C₅₉	—	4	2	no	—
59	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#121	1	2	yes	—
62	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
63	C₈×C₅₉	—	2	1	no	—
64	C₈×C₅₉	—	2	2	no	—
65	C₈×C₅₉	—	4	2	no	—
66	C₈×C₅₉	#101	1	2	yes	—
67	C₅₉⋊C₈	—	8	4	no	—
68	#8631	—	59	1	no	maximal, Sylow
69	C₂²×C₅₉	—	2	1	no	—
70	C₂²×C₅₉	—	2	2	no	—
71	C₂²×C₅₉	#380	1	1	yes	characteristic
72	C₄×C₅₉	—	2	1	no	—
73	C₄×C₅₉	—	2	2	no	—
74	C₄×C₅₉	—	2	2	no	—
75	C₄×C₅₉	—	4	4	no	—
76	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
77	C₄×C₅₉	#413	1	1	yes	characteristic
78	#2960	D₅₉	1	1	yes	characteristic
79	#1568	—	59	2	no	—
80	#1537	—	59	2	no	—
81	#1307	—	59	2	no	—
82	C₂×C₅₉	—	4	1	no	—
83	C₂×C₅₉	#1801	1	1	yes	characteristic
84	C₂×C₅₉	#1814	1	2	yes	—
85	#533	C₂×D₅₉	1	1	yes	characteristic
86	#446	—	2	2	no	—
87	C₈×Q₈	—	2	2	no	—
88	#432	—	59	4	no	—
89	#424	Dic₅₉	1	2	yes	—
90	#423	C₂×D₅₉	1	1	yes	characteristic
91	#405	Dic₅₉	1	2	yes	—
92	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
93	#335	—	118	2	no	—
94	#334	—	118	2	no	—
95	#330	—	118	4	no	—
96	C₅₉	#8631	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
98	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
99	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
100	C₄⋊Q₈	Dic₁₁₈	1	2	yes	—
101	#125	D₂₃₆	1	2	yes	—
102	C₂²⋊Q₈	—	2	2	no	—
103	C₄×Q₈	—	2	2	no	—
104	#117	C₂×Dic₅₉	1	2	yes	—
105	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	118	4	no	—
109	C₂²⋊C₈	—	118	4	no	—
110	#97	—	2	2	no	—
111	#97	—	118	4	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
114	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
115	C₂×Q₈	—	2	2	no	—
116	C₂²×C₄	#11745685	1	2	yes	—
117	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
118	M₁₆	—	4	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	236	4	no	—
123	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
125	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	C₄⋊Dic₅₉	1	4	yes	—
129	C₂²⋊C₄	—	2	2	no	—
130	C₄²	—	2	2	no	—
131	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
132	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
133	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
134	C₂³	#14273686	1	1	yes	characteristic
135	Q₈	—	4	4	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
142	C₂×C₄	#14273701	1	2	yes	—
143	C₂×C₄	#14273706	1	2	yes	—
144	C₈	—	2	1	no	—
145	C₈	—	2	2	no	—
146	C₈	—	4	2	no	—
147	C₈	—	472	4	no	—
148	C₈	#14273646	1	2	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#16837262	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	4	no	—
156	C₄	C₂×#14273646	1	1	yes	characteristic
157	C₄	#16837386	1	1	yes	characteristic
158	C₂	—	4	1	no	—
159	C₂	#21424092	1	1	yes	characteristic
160	C₂	#21424150	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21422859

C₂

yes

maximal

#21422734

C₂

yes

maximal

#21422007

C₂

yes

maximal

C₅₉×#2960

C₂

yes

maximal, characteristic

#16837459

C₂²

yes

—

#16837155

—

#16837153

—

#16837144

—

C₅₉×#533

C₂²

yes

characteristic

C₅₉×#446

—

C₈×Q₈×C₅₉

—

C₅₉×#424

C₄

yes

—

C₅₉×#423

C₂²

yes

characteristic

C₅₉×#405

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₂×M₁₆×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₅₉

Q₈

yes

—

#125×C₅₉

D₄

yes

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₄⋊C₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#117

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#101

yes

—

C₅₉⋊C₈

—

#8631

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#380

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

#1801

yes

characteristic

C₂×C₅₉

#1814

yes

—

#533

C₂×D₅₉

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#424

Dic₅₉

yes

—

#423

C₂×D₅₉

yes

characteristic

#405

Dic₅₉

yes

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#335

—

118

—

#334

—

118

—

#330

—

118

—

C₅₉

#8631

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×Dic₅₉

yes

—

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

100

C₄⋊Q₈

Dic₁₁₈

yes

—

101

#125

D₂₃₆

yes

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

#117

C₂×Dic₅₉

yes

—

105

C₂×C₄²

C₂²×D₅₉

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

118

—

109

C₂²⋊C₈

—

118

—

110

#97

—

111

#97

—

118

—

112

C₄×C₈

—

113

C₄×C₈

C₂²⋊D₅₉

yes

—

114

C₄×C₈

C₂²⋊D₅₉

yes

—

115

C₂×Q₈

—

116

C₂²×C₄

#11745685

yes

—

117

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

236

—

123

C₂×C₈

C₂²⋊Dic₅₉

yes

characteristic

124

C₂×C₈

C₂²⋊Dic₅₉

yes

—

125

C₂×C₈

C₂²⋊Dic₅₉

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

C₄⋊Dic₅₉

yes

—

129

C₂²⋊C₄

—

130

C₄²

—

131

C₄²

C₂×Dic₁₁₈

yes

characteristic

132

C₄²

C₂×D₂₃₆

yes

characteristic

133

C₄²

C₂×C₂²⋊D₅₉

yes

—

134

C₂³

#14273686

yes

characteristic

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

141

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

142

C₂×C₄

#14273701

yes

—

143

C₂×C₄

#14273706

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

472

—

148

C₈

#14273646

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

#16837262

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₂×#14273646

yes

characteristic

157

C₄

#16837386

yes

characteristic

158

C₂

—

159

C₂

#21424092

yes

characteristic

160

C₂

#21424150

yes

—

	subgroup	G/H
center	C₂²	#16837262
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2960	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#2960	D₅₉
hypocenter	C₅₉	#8631
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

subgroup

G/H

center

C₂²

#16837262

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#2960

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

N	H ≅ G/N	type
96. C₅₉	#8631	semidirect

H ≅ G/N

type

96. C₅₉

#8631

semidirect

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

subgroup	index
#21422859	2
#21422734	2
#21422007	2
C₅₉×#2960	2
#8631	59

subgroup

index

C₅₉×#2960

Subgroups — #28481685

Conjugacy classes of subgroups

Complements

Maximal subgroups