Group #50313558 — subgroups

Subgroups — C₁₅₁×#8633 #50313558

Order 38656.

320 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 130 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	#2960×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#1568×C₁₅₁	C₂	1	2	yes	maximal
3	#1537×C₁₅₁	C₂	1	2	yes	maximal
4	#1307×C₁₅₁	C₂	1	2	yes	maximal
5	#533×C₁₅₁	C₂²	1	1	yes	characteristic
6	#446×C₁₅₁	—	2	2	no	—
7	C₈×Q₈×C₁₅₁	—	2	2	no	—
8	#432×C₁₅₁	C₂²	1	4	yes	—
9	#424×C₁₅₁	C₄	1	2	yes	—
10	#423×C₁₅₁	C₂²	1	1	yes	characteristic
11	#405×C₁₅₁	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
13	#335×C₁₅₁	—	2	2	no	—
14	#334×C₁₅₁	—	2	2	no	—
15	#330×C₁₅₁	—	2	4	no	—
16	C₂×M₁₆×C₁₅₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₅₁	Q₈	1	2	yes	—
20	#125×C₁₅₁	D₄	1	2	yes	—
21	C₂²⋊Q₈×C₁₅₁	—	2	2	no	—
22	C₄×Q₈×C₁₅₁	—	2	2	no	—
23	#117×C₁₅₁	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₁₅₁	—	2	4	no	—
26	C₄⋊C₈×C₁₅₁	—	2	2	no	—
27	C₄⋊C₈×C₁₅₁	—	2	4	no	—
28	C₂²⋊C₈×C₁₅₁	—	2	4	no	—
29	#97×C₁₅₁	—	2	2	no	—
30	#97×C₁₅₁	—	2	4	no	—
31	C₄×C₈×C₁₅₁	—	2	2	no	—
32	C₄×C₈×C₁₅₁	D₄	1	2	yes	—
33	C₄×C₈×C₁₅₁	D₄	1	2	yes	—
34	C₂×Q₈×C₁₅₁	—	2	2	no	—
35	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
37	M₁₆×C₁₅₁	—	4	2	no	—
38	C₂×C₈×C₁₅₁	—	2	2	no	—
39	C₂×C₈×C₁₅₁	—	2	2	no	—
40	C₂×C₈×C₁₅₁	—	4	1	no	—
41	C₂×C₈×C₁₅₁	—	4	4	no	—
42	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₁₅₁	—	2	4	no	—
46	C₄⋊C₄×C₁₅₁	—	2	2	no	—
47	C₄⋊C₄×C₁₅₁	C₄⋊C₄	1	4	yes	—
48	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
49	C₄²×C₁₅₁	—	2	2	no	—
50	C₄²×C₁₅₁	C₂×D₄	1	2	yes	—
51	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
53	C₂³×C₁₅₁	#117	1	1	yes	characteristic
54	Q₈×C₁₅₁	—	4	4	no	—
55	C₂×C₄×C₁₅₁	—	2	1	no	—
56	C₂×C₄×C₁₅₁	—	2	1	no	—
57	C₂×C₄×C₁₅₁	—	2	4	no	—
58	C₂×C₄×C₁₅₁	—	4	2	no	—
59	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₅₁	#121	1	2	yes	—
62	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
63	C₈×C₁₅₁	—	2	1	no	—
64	C₈×C₁₅₁	—	2	2	no	—
65	C₈×C₁₅₁	—	4	2	no	—
66	C₈×C₁₅₁	—	8	4	no	—
67	C₈×C₁₅₁	#101	1	2	yes	—
68	C₂²×C₁₅₁	—	2	1	no	—
69	C₂²×C₁₅₁	—	2	2	no	—
70	C₂²×C₁₅₁	#380	1	1	yes	characteristic
71	C₄×C₁₅₁	—	2	1	no	—
72	C₄×C₁₅₁	—	2	2	no	—
73	C₄×C₁₅₁	—	2	2	no	—
74	C₄×C₁₅₁	—	4	4	no	—
75	C₄×C₁₅₁	C₂×#101	1	1	yes	characteristic
76	C₄×C₁₅₁	#413	1	1	yes	characteristic
77	C₂×C₁₅₁	—	4	1	no	—
78	C₂×C₁₅₁	#1801	1	1	yes	characteristic
79	C₂×C₁₅₁	#1814	1	2	yes	—
80	#8633	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₁₅₁	#8633	1	1	yes	characteristic, direct product factor, Sylow
82	#2960	C₂×C₁₅₁	1	1	yes	characteristic
83	#1568	C₂×C₁₅₁	1	2	yes	—
84	#1537	C₂×C₁₅₁	1	2	yes	—
85	#1307	C₂×C₁₅₁	1	2	yes	—
86	#533	C₂²×C₁₅₁	1	1	yes	characteristic
87	#446	—	2	2	no	—
88	C₈×Q₈	—	2	2	no	—
89	#432	C₂²×C₁₅₁	1	4	yes	—
90	#424	C₄×C₁₅₁	1	2	yes	—
91	#423	C₂²×C₁₅₁	1	1	yes	characteristic
92	#405	C₄×C₁₅₁	1	2	yes	—
93	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
94	#335	—	2	2	no	—
95	#334	—	2	2	no	—
96	#330	—	2	4	no	—
97	C₂×M₁₆	C₂×C₄×C₁₅₁	1	2	yes	—
98	C₂²×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
99	C₂²×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
100	C₄⋊Q₈	Q₈×C₁₅₁	1	2	yes	—
101	#125	D₄×C₁₅₁	1	2	yes	—
102	C₂²⋊Q₈	—	2	2	no	—
103	C₄×Q₈	—	2	2	no	—
104	#117	C₂×C₄×C₁₅₁	1	2	yes	—
105	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	4	no	—
109	C₂²⋊C₈	—	2	4	no	—
110	#97	—	2	2	no	—
111	#97	—	2	4	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	D₄×C₁₅₁	1	2	yes	—
114	C₄×C₈	D₄×C₁₅₁	1	2	yes	—
115	C₂×Q₈	—	2	2	no	—
116	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
117	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
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	—	4	4	no	—
123	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	2	yes	—
125	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	C₄⋊C₄×C₁₅₁	1	4	yes	—
129	C₂²⋊C₄	—	2	2	no	—
130	C₄²	—	2	2	no	—
131	C₄²	C₂×D₄×C₁₅₁	1	2	yes	—
132	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
133	C₄²	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
134	C₂³	#117×C₁₅₁	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⋊C₄×C₁₅₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
142	C₂×C₄	#121×C₁₅₁	1	2	yes	—
143	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	2	yes	—
144	C₈	—	2	1	no	—
145	C₈	—	2	2	no	—
146	C₈	—	4	2	no	—
147	C₈	—	8	4	no	—
148	C₈	#101×C₁₅₁	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	2	no	—
151	C₂²	#380×C₁₅₁	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	4	no	—
156	C₄	C₂×#101×C₁₅₁	1	1	yes	characteristic
157	C₄	#413×C₁₅₁	1	1	yes	characteristic
158	C₂	—	4	1	no	—
159	C₂	#1801×C₁₅₁	1	1	yes	characteristic
160	C₂	#1814×C₁₅₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2960×C₁₅₁

C₂

yes

maximal, characteristic

#1568×C₁₅₁

C₂

yes

maximal

#1537×C₁₅₁

C₂

yes

maximal

#1307×C₁₅₁

C₂

yes

maximal

#533×C₁₅₁

C₂²

yes

characteristic

#446×C₁₅₁

—

C₈×Q₈×C₁₅₁

—

#432×C₁₅₁

C₂²

yes

—

#424×C₁₅₁

C₄

yes

—

#423×C₁₅₁

C₂²

yes

characteristic

#405×C₁₅₁

C₄

yes

—

C₂×C₄×C₈×C₁₅₁

C₂²

yes

characteristic

#335×C₁₅₁

—

#334×C₁₅₁

—

#330×C₁₅₁

—

C₂×M₁₆×C₁₅₁

C₂×C₄

yes

—

C₂²×C₈×C₁₅₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₅₁

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₁₅₁

Q₈

yes

—

#125×C₁₅₁

D₄

yes

—

C₂²⋊Q₈×C₁₅₁

—

C₄×Q₈×C₁₅₁

—

#117×C₁₅₁

C₂×C₄

yes

—

C₂×C₄²×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₈×C₁₅₁

—

C₄⋊C₈×C₁₅₁

—

C₄⋊C₈×C₁₅₁

—

C₂²⋊C₈×C₁₅₁

—

#97×C₁₅₁

—

#97×C₁₅₁

—

C₄×C₈×C₁₅₁

—

C₄×C₈×C₁₅₁

D₄

yes

—

C₄×C₈×C₁₅₁

D₄

yes

—

C₂×Q₈×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₅₁

#41

yes

—

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₈×C₁₅₁

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

C₄⋊C₄

yes

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

C₄²×C₁₅₁

C₂×D₄

yes

—

C₄²×C₁₅₁

C₂×D₄

yes

characteristic

C₄²×C₁₅₁

C₂×Q₈

yes

characteristic

C₂³×C₁₅₁

#117

yes

characteristic

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

#121

yes

—

C₂×C₄×C₁₅₁

C₂²⋊Q₈

yes

—

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

#101

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#380

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₂×C₁₅₁

—

C₂×C₁₅₁

#1801

yes

characteristic

C₂×C₁₅₁

#1814

yes

—

#8633

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#8633

yes

characteristic, direct product factor, Sylow

#2960

C₂×C₁₅₁

yes

characteristic

#1568

C₂×C₁₅₁

yes

—

#1537

C₂×C₁₅₁

yes

—

#1307

C₂×C₁₅₁

yes

—

#533

C₂²×C₁₅₁

yes

characteristic

#446

—

C₈×Q₈

—

#432

C₂²×C₁₅₁

yes

—

#424

C₄×C₁₅₁

yes

—

#423

C₂²×C₁₅₁

yes

characteristic

#405

C₄×C₁₅₁

yes

—

C₂×C₄×C₈

C₂²×C₁₅₁

yes

—

—

—

C₂×M₁₆

C₂×C₄×C₁₅₁

yes

—

C₂²×C₈

C₂×C₄×C₁₅₁

yes

characteristic

C₂²×C₈

C₂×C₄×C₁₅₁

yes

characteristic

100

C₄⋊Q₈

Q₈×C₁₅₁

yes

—

101

#125

D₄×C₁₅₁

yes

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

#117

C₂×C₄×C₁₅₁

yes

—

105

C₂×C₄²

C₂³×C₁₅₁

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₂²⋊C₈

—

110

#97

—

111

#97

—

112

C₄×C₈

—

113

C₄×C₈

D₄×C₁₅₁

yes

—

114

C₄×C₈

D₄×C₁₅₁

yes

—

115

C₂×Q₈

—

116

C₂²×C₄

C₂²×C₄×C₁₅₁

yes

characteristic

117

C₂²×C₄

#41×C₁₅₁

yes

—

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂²⋊C₄×C₁₅₁

yes

characteristic

124

C₂×C₈

C₂²⋊C₄×C₁₅₁

yes

—

125

C₂×C₈

C₂²⋊C₄×C₁₅₁

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

C₄⋊C₄×C₁₅₁

yes

—

129

C₂²⋊C₄

—

130

C₄²

—

131

C₄²

C₂×D₄×C₁₅₁

yes

—

132

C₄²

C₂×D₄×C₁₅₁

yes

characteristic

133

C₄²

C₂×Q₈×C₁₅₁

yes

characteristic

134

C₂³

#117×C₁₅₁

yes

characteristic

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂×C₂²⋊C₄×C₁₅₁

yes

characteristic

141

C₂×C₄

C₂×C₄⋊C₄×C₁₅₁

yes

characteristic

142

C₂×C₄

#121×C₁₅₁

yes

—

143

C₂×C₄

C₂²⋊Q₈×C₁₅₁

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

#101×C₁₅₁

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

#380×C₁₅₁

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₂×#101×C₁₅₁

yes

characteristic

157

C₄

#413×C₁₅₁

yes

characteristic

158

C₂

—

159

C₂

#1801×C₁₅₁

yes

characteristic

160

C₂

#1814×C₁₅₁

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₅₁	#380
commutator	C₂²×C₄	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₅₁
prime radical	C₂³×C₁₅₁	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂×C₄²×C₁₅₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₅₁

#380

commutator

C₂²×C₄

C₂²×C₄×C₁₅₁

Frattini, cosocle

C₂×C₄²

C₂³×C₁₅₁

prime radical

C₂³×C₁₅₁

#117

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₅₁ ◁ C₂×C₄²×C₁₅₁ ◁ G

N	H ≅ G/N	type
80. #8633	C₁₅₁	direct (N × H)
81. C₁₅₁	#8633	direct (N × H)

H ≅ G/N

type

80. #8633

C₁₅₁

direct (N × H)

81. C₁₅₁

#8633

direct (N × H)

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

subgroup	index
#2960×C₁₅₁	2
#1568×C₁₅₁	2
#1537×C₁₅₁	2
#1307×C₁₅₁	2
#8633	151

subgroup

index

#2960×C₁₅₁

#1568×C₁₅₁

#1537×C₁₅₁

#1307×C₁₅₁

#8633

151

Subgroups — C₁₅₁×#8633 #50313558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#8633 #50313558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#8633 #50313558