Group #50317454 — subgroups

Subgroups — C₁₅₁×#12529 #50317454

Order 38656.

318 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 102 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	#2248×C₁₅₁	C₂	1	2	yes	maximal, complemented
2	#2240×C₁₅₁	C₂	1	2	yes	maximal, complemented
3	#1712×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#1707×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1312×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₃₂×C₁₅₁	—	2	4	no	—
7	#500×C₁₅₁	C₂²	1	1	yes	characteristic
8	#498×C₁₅₁	—	2	1	no	—
9	#494×C₁₅₁	C₂²	1	1	yes	characteristic
10	#493×C₁₅₁	—	2	1	no	—
11	#493×C₁₅₁	—	2	1	no	—
12	#493×C₁₅₁	C₂²	1	2	yes	—
13	#487×C₁₅₁	—	2	1	no	—
14	#357×C₁₅₁	—	2	4	no	—
15	C₄×C₁₆×C₁₅₁	C₂²	1	2	yes	—
16	C₈²×C₁₅₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₅₁	—	2	2	no	—
18	C₂×D₈×C₁₅₁	—	2	2	no	—
19	C₂×D₈×C₁₅₁	—	2	2	no	—
20	C₂×D₈×C₁₅₁	—	4	2	no	—
21	C₂×D₈×C₁₅₁	—	4	2	no	—
22	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
23	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
24	#127×C₁₅₁	—	2	2	no	—
25	SD₃₂×C₁₅₁	—	4	8	no	—
26	C₂×C₁₆×C₁₅₁	D₄	1	4	yes	—
27	#107×C₁₅₁	—	2	2	no	—
28	#106×C₁₅₁	—	4	2	no	—
29	#103×C₁₅₁	—	4	2	no	—
30	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₅₁	—	4	1	no	—
34	C₂×Q₈×C₁₅₁	—	4	1	no	—
35	C₂×D₄×C₁₅₁	—	4	2	no	—
36	C₂×D₄×C₁₅₁	—	4	2	no	—
37	C₂×D₄×C₁₅₁	—	4	2	no	—
38	Q₁₆×C₁₅₁	—	4	4	no	—
39	D₈×C₁₅₁	—	4	4	no	—
40	D₈×C₁₅₁	—	4	4	no	—
41	D₈×C₁₅₁	—	8	4	no	—
42	D₈×C₁₅₁	—	8	4	no	—
43	C₂×C₈×C₁₅₁	D₈	1	2	yes	—
44	C₂×C₈×C₁₅₁	D₈	1	2	yes	—
45	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
46	C₄⋊C₄×C₁₅₁	—	4	1	no	—
47	C₄⋊C₄×C₁₅₁	—	4	1	no	—
48	C₄⋊C₄×C₁₅₁	—	4	1	no	—
49	C₄⋊C₄×C₁₅₁	—	4	1	no	—
50	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
51	C₁₆×C₁₅₁	—	2	4	no	—
52	C₂³×C₁₅₁	—	8	2	no	—
53	Q₈×C₁₅₁	—	8	2	no	—
54	Q₈×C₁₅₁	—	8	2	no	—
55	D₄×C₁₅₁	—	8	2	no	—
56	D₄×C₁₅₁	—	8	2	no	—
57	D₄×C₁₅₁	—	8	4	no	—
58	D₄×C₁₅₁	—	8	4	no	—
59	C₂×C₄×C₁₅₁	—	8	1	no	—
60	C₂×C₄×C₁₅₁	—	8	1	no	—
61	C₂×C₄×C₁₅₁	#127	1	1	yes	characteristic
62	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
64	C₈×C₁₅₁	—	2	2	no	—
65	C₈×C₁₅₁	SD₃₂	1	4	yes	complemented
66	C₈×C₁₅₁	AGL(1,ℤ/8)	1	4	yes	—
67	C₂²×C₁₅₁	—	16	2	no	—
68	C₂²×C₁₅₁	—	16	2	no	—
69	C₂²×C₁₅₁	—	16	2	no	—
70	C₂²×C₁₅₁	#493	1	1	yes	characteristic
71	C₄×C₁₅₁	—	16	1	no	—
72	C₄×C₁₅₁	—	16	1	no	—
73	C₄×C₁₅₁	#496	1	2	yes	—
74	C₄×C₁₅₁	C₂×SD₃₂	1	2	yes	—
75	C₄×C₁₅₁	#509	1	2	yes	—
76	C₂×C₁₅₁	—	32	2	no	—
77	C₂×C₁₅₁	#1721	1	1	yes	characteristic
78	C₂×C₁₅₁	#2248	1	1	yes	characteristic
79	C₂×C₁₅₁	#2250	1	1	yes	characteristic
80	#12529	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₁₅₁	#12529	1	1	yes	characteristic, direct product factor, Sylow
82	#2248	C₂×C₁₅₁	1	2	yes	complemented
83	#2240	C₂×C₁₅₁	1	2	yes	complemented
84	#1712	C₂×C₁₅₁	1	1	yes	characteristic
85	#1707	C₂×C₁₅₁	1	1	yes	characteristic, complemented
86	#1312	C₂×C₁₅₁	1	1	yes	characteristic, complemented
87	C₂×SD₃₂	—	2	4	no	—
88	#500	C₂²×C₁₅₁	1	1	yes	characteristic
89	#498	—	2	1	no	—
90	#494	C₂²×C₁₅₁	1	1	yes	characteristic
91	#493	—	2	1	no	—
92	#493	—	2	1	no	—
93	#493	C₂²×C₁₅₁	1	2	yes	—
94	#487	—	2	1	no	—
95	#357	—	2	4	no	—
96	C₄×C₁₆	C₂²×C₁₅₁	1	2	yes	—
97	C₈²	C₂²×C₁₅₁	1	1	yes	characteristic
98	C₂×Q₁₆	—	2	2	no	—
99	C₂×D₈	—	2	2	no	—
100	C₂×D₈	—	2	2	no	—
101	C₂×D₈	—	4	2	no	—
102	C₂×D₈	—	4	2	no	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	—	2	1	no	—
105	#127	—	2	2	no	—
106	SD₃₂	—	4	8	no	—
107	C₂×C₁₆	D₄×C₁₅₁	1	4	yes	—
108	#107	—	2	2	no	—
109	#106	—	4	2	no	—
110	#103	—	4	2	no	—
111	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
112	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
113	C₄×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
114	C₂×Q₈	—	4	1	no	—
115	C₂×Q₈	—	4	1	no	—
116	C₂×D₄	—	4	2	no	—
117	C₂×D₄	—	4	2	no	—
118	C₂×D₄	—	4	2	no	—
119	Q₁₆	—	4	4	no	—
120	D₈	—	4	4	no	—
121	D₈	—	4	4	no	—
122	D₈	—	8	4	no	—
123	D₈	—	8	4	no	—
124	C₂×C₈	D₈×C₁₅₁	1	2	yes	—
125	C₂×C₈	D₈×C₁₅₁	1	2	yes	—
126	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	4	1	no	—
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	4	1	no	—
131	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
132	C₁₆	—	2	4	no	—
133	C₂³	—	8	2	no	—
134	Q₈	—	8	2	no	—
135	Q₈	—	8	2	no	—
136	D₄	—	8	2	no	—
137	D₄	—	8	2	no	—
138	D₄	—	8	4	no	—
139	D₄	—	8	4	no	—
140	C₂×C₄	—	8	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	#127×C₁₅₁	1	1	yes	characteristic
143	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
144	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	SD₃₂×C₁₅₁	1	4	yes	complemented
147	C₈	AGL(1,ℤ/8)×C₁₅₁	1	4	yes	—
148	C₂²	—	16	2	no	—
149	C₂²	—	16	2	no	—
150	C₂²	—	16	2	no	—
151	C₂²	#493×C₁₅₁	1	1	yes	characteristic
152	C₄	—	16	1	no	—
153	C₄	—	16	1	no	—
154	C₄	#496×C₁₅₁	1	2	yes	—
155	C₄	C₂×SD₃₂×C₁₅₁	1	2	yes	—
156	C₄	#509×C₁₅₁	1	2	yes	—
157	C₂	—	32	2	no	—
158	C₂	#1721×C₁₅₁	1	1	yes	characteristic
159	C₂	#2248×C₁₅₁	1	1	yes	characteristic
160	C₂	#2250×C₁₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2248×C₁₅₁

C₂

yes

maximal, complemented

#2240×C₁₅₁

C₂

yes

maximal, complemented

#1712×C₁₅₁

C₂

yes

maximal, characteristic

#1707×C₁₅₁

C₂

yes

maximal, characteristic, complemented

#1312×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×SD₃₂×C₁₅₁

—

#500×C₁₅₁

C₂²

yes

characteristic

#498×C₁₅₁

—

#494×C₁₅₁

C₂²

yes

characteristic

#493×C₁₅₁

—

#493×C₁₅₁

—

#493×C₁₅₁

C₂²

yes

—

#487×C₁₅₁

—

#357×C₁₅₁

—

C₄×C₁₆×C₁₅₁

C₂²

yes

—

C₈²×C₁₅₁

C₂²

yes

characteristic

C₂×Q₁₆×C₁₅₁

—

C₂×D₈×C₁₅₁

—

C₂×D₈×C₁₅₁

—

C₂×D₈×C₁₅₁

—

C₂×D₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

#127×C₁₅₁

—

SD₃₂×C₁₅₁

—

C₂×C₁₆×C₁₅₁

D₄

yes

—

#107×C₁₅₁

—

#106×C₁₅₁

—

#103×C₁₅₁

—

C₄×C₈×C₁₅₁

D₄

yes

characteristic

C₄×C₈×C₁₅₁

D₄

yes

characteristic

C₄×C₈×C₁₅₁

C₂³

yes

characteristic

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

Q₁₆×C₁₅₁

—

D₈×C₁₅₁

—

D₈×C₁₅₁

—

D₈×C₁₅₁

—

D₈×C₁₅₁

—

C₂×C₈×C₁₅₁

D₈

yes

—

C₂×C₈×C₁₅₁

D₈

yes

—

C₂×C₈×C₁₅₁

C₂×D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂×D₄

yes

characteristic

C₁₆×C₁₅₁

—

C₂³×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

#127

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₈

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₈

yes

characteristic

C₈×C₁₅₁

—

C₈×C₁₅₁

SD₃₂

yes

complemented

C₈×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#493

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#496

yes

—

C₄×C₁₅₁

C₂×SD₃₂

yes

—

C₄×C₁₅₁

#509

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#12529

yes

characteristic, direct product factor, Sylow

#2248

C₂×C₁₅₁

yes

complemented

#2240

C₂×C₁₅₁

yes

complemented

#1712

C₂×C₁₅₁

yes

characteristic

#1707

C₂×C₁₅₁

yes

characteristic, complemented

#1312

C₂×C₁₅₁

yes

characteristic, complemented

C₂×SD₃₂

—

#500

C₂²×C₁₅₁

yes

characteristic

#498

—

#494

C₂²×C₁₅₁

yes

—

—

C₂²×C₁₅₁

yes

—

#487

—

#357

—

C₄×C₁₆

C₂²×C₁₅₁

yes

—

C₈²

C₂²×C₁₅₁

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

100

C₂×D₈

—

101

C₂×D₈

—

102

C₂×D₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

#127

—

106

SD₃₂

—

107

C₂×C₁₆

D₄×C₁₅₁

yes

—

108

#107

—

109

#106

—

110

#103

—

111

C₄×C₈

D₄×C₁₅₁

yes

characteristic

112

C₄×C₈

D₄×C₁₅₁

yes

characteristic

113

C₄×C₈

C₂³×C₁₅₁

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

Q₁₆

—

120

D₈

—

121

D₈

—

122

D₈

—

123

D₈

—

124

C₂×C₈

D₈×C₁₅₁

yes

—

125

C₂×C₈

D₈×C₁₅₁

yes

—

126

C₂×C₈

C₂×D₄×C₁₅₁

yes

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

C₂×D₄×C₁₅₁

yes

characteristic

132

C₁₆

—

133

C₂³

—

134

Q₈

—

135

Q₈

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#127×C₁₅₁

yes

characteristic

143

C₂×C₄

C₂×D₈×C₁₅₁

yes

characteristic

144

C₂×C₄

C₂×D₈×C₁₅₁

yes

characteristic

145

C₈

—

146

C₈

SD₃₂×C₁₅₁

yes

complemented

147

C₈

AGL(1,ℤ/8)×C₁₅₁

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#493×C₁₅₁

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

#496×C₁₅₁

yes

—

155

C₄

C₂×SD₃₂×C₁₅₁

yes

—

156

C₄

#509×C₁₅₁

yes

—

157

C₂

—

158

C₂

#1721×C₁₅₁

yes

characteristic

159

C₂

#2248×C₁₅₁

yes

characteristic

160

C₂

#2250×C₁₅₁

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₅₁	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₅₁
prime radical	#1712×C₁₅₁	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₄²×C₁₅₁ ◁ C₈²×C₁₅₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₅₁

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₅₁

prime radical

#1712×C₁₅₁

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₅₁ ◁ C₄²×C₁₅₁ ◁ C₈²×C₁₅₁ ◁ G

N	H ≅ G/N	type
1. #2248×C₁₅₁	C₂	semidirect
2. #2240×C₁₅₁	C₂	semidirect
4. #1707×C₁₅₁	C₂	semidirect
5. #1312×C₁₅₁	C₂	semidirect
65. C₈×C₁₅₁	SD₃₂	semidirect
80. #12529	C₁₅₁	direct (N × H)
81. C₁₅₁	#12529	direct (N × H)
82. #2248	C₂×C₁₅₁	semidirect
83. #2240	C₂×C₁₅₁	semidirect
85. #1707	C₂×C₁₅₁	semidirect
86. #1312	C₂×C₁₅₁	semidirect
146. C₈	SD₃₂×C₁₅₁	semidirect

H ≅ G/N

type

1. #2248×C₁₅₁

C₂

semidirect

2. #2240×C₁₅₁

C₂

semidirect

4. #1707×C₁₅₁

C₂

semidirect

5. #1312×C₁₅₁

C₂

semidirect

65. C₈×C₁₅₁

SD₃₂

semidirect

80. #12529

C₁₅₁

direct (N × H)

81. C₁₅₁

#12529

direct (N × H)

82. #2248

C₂×C₁₅₁

semidirect

83. #2240

C₂×C₁₅₁

semidirect

85. #1707

C₂×C₁₅₁

semidirect

86. #1312

C₂×C₁₅₁

semidirect

146. C₈

SD₃₂×C₁₅₁

semidirect

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

subgroup	index
#2248×C₁₅₁	2
#2240×C₁₅₁	2
#1712×C₁₅₁	2
#1707×C₁₅₁	2
#1312×C₁₅₁	2
#12529	151

subgroup

index

#2248×C₁₅₁

#2240×C₁₅₁

#1712×C₁₅₁

#1707×C₁₅₁

#1312×C₁₅₁

#12529

151

Subgroups — C₁₅₁×#12529 #50317454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#12529 #50317454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#12529 #50317454