Group #50317450 — subgroups

Subgroups — C₁₅₁×#12525 #50317450

Order 38656.

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

normal subgroups only

This group has 340 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	#2246×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, complemented
4	#1706×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1312×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×C₁₅₁	—	2	4	no	—
7	#500×C₁₅₁	C₂²	1	1	yes	characteristic
8	#493×C₁₅₁	—	2	1	no	—
9	#493×C₁₅₁	—	2	1	no	—
10	#493×C₁₅₁	C₂²	1	1	yes	characteristic
11	#493×C₁₅₁	C₂²	1	2	yes	—
12	#492×C₁₅₁	—	2	1	no	—
13	#486×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×SD₁₆×C₁₅₁	—	4	2	no	—
18	C₂×D₈×C₁₅₁	—	2	2	no	—
19	C₂×D₈×C₁₅₁	—	2	2	no	—
20	C₂×D₈×C₁₅₁	—	2	2	no	—
21	C₂×D₈×C₁₅₁	—	4	2	no	—
22	C₂×D₈×C₁₅₁	—	4	2	no	—
23	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
24	#127×C₁₅₁	—	2	1	no	—
25	#127×C₁₅₁	—	2	2	no	—
26	D₁₆×C₁₅₁	—	4	8	no	—
27	C₂×C₁₆×C₁₅₁	D₄	1	4	yes	—
28	#107×C₁₅₁	—	2	2	no	—
29	#102×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×D₄×C₁₅₁	—	4	1	no	—
35	C₂×D₄×C₁₅₁	—	4	2	no	—
36	C₂×D₄×C₁₅₁	—	4	1	no	—
37	C₂×D₄×C₁₅₁	—	4	2	no	—
38	C₂×D₄×C₁₅₁	—	4	2	no	—
39	C₂×D₄×C₁₅₁	—	4	1	no	—
40	SD₁₆×C₁₅₁	—	8	4	no	—
41	D₈×C₁₅₁	—	4	4	no	—
42	D₈×C₁₅₁	—	4	4	no	—
43	D₈×C₁₅₁	—	4	4	no	—
44	D₈×C₁₅₁	—	8	4	no	—
45	D₈×C₁₅₁	—	8	4	no	—
46	C₂×C₈×C₁₅₁	D₈	1	2	yes	—
47	C₂×C₈×C₁₅₁	D₈	1	2	yes	—
48	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
49	C₄⋊C₄×C₁₅₁	—	4	1	no	—
50	C₄⋊C₄×C₁₅₁	—	4	1	no	—
51	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
52	C₁₆×C₁₅₁	—	2	4	no	—
53	C₂³×C₁₅₁	—	8	1	no	—
54	C₂³×C₁₅₁	—	8	2	no	—
55	Q₈×C₁₅₁	—	8	2	no	—
56	D₄×C₁₅₁	—	8	2	no	—
57	D₄×C₁₅₁	—	8	2	no	—
58	D₄×C₁₅₁	—	8	2	no	—
59	D₄×C₁₅₁	—	8	4	no	—
60	D₄×C₁₅₁	—	8	2	no	—
61	D₄×C₁₅₁	—	8	4	no	—
62	D₄×C₁₅₁	—	8	2	no	—
63	C₂×C₄×C₁₅₁	—	8	1	no	—
64	C₂×C₄×C₁₅₁	#127	1	1	yes	characteristic
65	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
67	C₈×C₁₅₁	—	2	2	no	—
68	C₈×C₁₅₁	D₁₆	1	4	yes	complemented
69	C₈×C₁₅₁	AGL(1,ℤ/8)	1	4	yes	—
70	C₂²×C₁₅₁	—	16	1	no	—
71	C₂²×C₁₅₁	—	16	2	no	—
72	C₂²×C₁₅₁	—	16	2	no	—
73	C₂²×C₁₅₁	—	16	1	no	—
74	C₂²×C₁₅₁	—	16	2	no	—
75	C₂²×C₁₅₁	—	16	1	no	—
76	C₂²×C₁₅₁	#493	1	1	yes	characteristic
77	C₄×C₁₅₁	—	16	1	no	—
78	C₄×C₁₅₁	#496	1	2	yes	—
79	C₄×C₁₅₁	C₂×D₁₆	1	2	yes	—
80	C₄×C₁₅₁	#509	1	2	yes	—
81	C₂×C₁₅₁	—	32	2	no	—
82	C₂×C₁₅₁	—	32	1	no	—
83	C₂×C₁₅₁	#1721	1	1	yes	characteristic
84	C₂×C₁₅₁	#2246	1	1	yes	characteristic
85	C₂×C₁₅₁	#2250	1	1	yes	characteristic
86	#12525	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₁₅₁	#12525	1	1	yes	characteristic, direct product factor, Sylow
88	#2246	C₂×C₁₅₁	1	2	yes	complemented
89	#2240	C₂×C₁₅₁	1	2	yes	complemented
90	#1712	C₂×C₁₅₁	1	1	yes	characteristic, complemented
91	#1706	C₂×C₁₅₁	1	1	yes	characteristic, complemented
92	#1312	C₂×C₁₅₁	1	1	yes	characteristic, complemented
93	C₂×D₁₆	—	2	4	no	—
94	#500	C₂²×C₁₅₁	1	1	yes	characteristic
95	#493	—	2	1	no	—
96	#493	—	2	1	no	—
97	#493	C₂²×C₁₅₁	1	1	yes	characteristic
98	#493	C₂²×C₁₅₁	1	2	yes	—
99	#492	—	2	1	no	—
100	#486	—	2	1	no	—
101	#357	—	2	4	no	—
102	C₄×C₁₆	C₂²×C₁₅₁	1	2	yes	—
103	C₈²	C₂²×C₁₅₁	1	1	yes	characteristic
104	C₂×SD₁₆	—	4	2	no	—
105	C₂×D₈	—	2	2	no	—
106	C₂×D₈	—	2	2	no	—
107	C₂×D₈	—	2	2	no	—
108	C₂×D₈	—	4	2	no	—
109	C₂×D₈	—	4	2	no	—
110	C₄⋊Q₈	—	2	1	no	—
111	#127	—	2	1	no	—
112	#127	—	2	2	no	—
113	D₁₆	—	4	8	no	—
114	C₂×C₁₆	D₄×C₁₅₁	1	4	yes	—
115	#107	—	2	2	no	—
116	#102	—	4	2	no	—
117	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
118	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
119	C₄×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
120	C₂×Q₈	—	4	1	no	—
121	C₂×D₄	—	4	1	no	—
122	C₂×D₄	—	4	2	no	—
123	C₂×D₄	—	4	1	no	—
124	C₂×D₄	—	4	2	no	—
125	C₂×D₄	—	4	2	no	—
126	C₂×D₄	—	4	1	no	—
127	SD₁₆	—	8	4	no	—
128	D₈	—	4	4	no	—
129	D₈	—	4	4	no	—
130	D₈	—	4	4	no	—
131	D₈	—	8	4	no	—
132	D₈	—	8	4	no	—
133	C₂×C₈	D₈×C₁₅₁	1	2	yes	—
134	C₂×C₈	D₈×C₁₅₁	1	2	yes	—
135	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
139	C₁₆	—	2	4	no	—
140	C₂³	—	8	1	no	—
141	C₂³	—	8	2	no	—
142	Q₈	—	8	2	no	—
143	D₄	—	8	2	no	—
144	D₄	—	8	2	no	—
145	D₄	—	8	2	no	—
146	D₄	—	8	4	no	—
147	D₄	—	8	2	no	—
148	D₄	—	8	4	no	—
149	D₄	—	8	2	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	#127×C₁₅₁	1	1	yes	characteristic
152	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
153	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	D₁₆×C₁₅₁	1	4	yes	complemented
156	C₈	AGL(1,ℤ/8)×C₁₅₁	1	4	yes	—
157	C₂²	—	16	1	no	—
158	C₂²	—	16	2	no	—
159	C₂²	—	16	2	no	—
160	C₂²	—	16	1	no	—
161	C₂²	—	16	2	no	—
162	C₂²	—	16	1	no	—
163	C₂²	#493×C₁₅₁	1	1	yes	characteristic
164	C₄	—	16	1	no	—
165	C₄	#496×C₁₅₁	1	2	yes	—
166	C₄	C₂×D₁₆×C₁₅₁	1	2	yes	—
167	C₄	#509×C₁₅₁	1	2	yes	—
168	C₂	—	32	2	no	—
169	C₂	—	32	1	no	—
170	C₂	#1721×C₁₅₁	1	1	yes	characteristic
171	C₂	#2246×C₁₅₁	1	1	yes	characteristic
172	C₂	#2250×C₁₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2246×C₁₅₁

C₂

yes

maximal, complemented

#2240×C₁₅₁

C₂

yes

maximal, complemented

#1712×C₁₅₁

C₂

yes

maximal, characteristic, complemented

#1706×C₁₅₁

C₂

yes

maximal, characteristic, complemented

#1312×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₁₅₁

—

#500×C₁₅₁

C₂²

yes

characteristic

#493×C₁₅₁

—

#493×C₁₅₁

—

#493×C₁₅₁

C₂²

yes

characteristic

#493×C₁₅₁

C₂²

yes

—

#492×C₁₅₁

—

#486×C₁₅₁

—

#357×C₁₅₁

—

C₄×C₁₆×C₁₅₁

C₂²

yes

—

C₈²×C₁₅₁

C₂²

yes

characteristic

C₂×SD₁₆×C₁₅₁

—

C₂×D₈×C₁₅₁

—

C₂×D₈×C₁₅₁

—

C₂×D₈×C₁₅₁

—

C₂×D₈×C₁₅₁

—

C₂×D₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

#127×C₁₅₁

—

#127×C₁₅₁

—

D₁₆×C₁₅₁

—

C₂×C₁₆×C₁₅₁

D₄

yes

—

#107×C₁₅₁

—

#102×C₁₅₁

—

C₄×C₈×C₁₅₁

D₄

yes

characteristic

C₄×C₈×C₁₅₁

D₄

yes

characteristic

C₄×C₈×C₁₅₁

C₂³

yes

characteristic

C₂×Q₈×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

SD₁₆×C₁₅₁

—

D₈×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

—

C₄²×C₁₅₁

C₂×D₄

yes

characteristic

C₁₆×C₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×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

D₁₆

yes

complemented

C₈×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#493

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

#496

yes

—

C₄×C₁₅₁

C₂×D₁₆

yes

—

C₄×C₁₅₁

#509

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#12525

yes

characteristic, direct product factor, Sylow

#2246

C₂×C₁₅₁

yes

complemented

#2240

C₂×C₁₅₁

yes

complemented

#1712

C₂×C₁₅₁

yes

characteristic, complemented

#1706

C₂×C₁₅₁

yes

characteristic, complemented

#1312

C₂×C₁₅₁

yes

characteristic, complemented

C₂×D₁₆

—

#500

C₂²×C₁₅₁

yes

—

—

C₂²×C₁₅₁

yes

characteristic

#493

C₂²×C₁₅₁

yes

—

#492

—

100

#486

—

101

#357

—

102

C₄×C₁₆

C₂²×C₁₅₁

yes

—

103

C₈²

C₂²×C₁₅₁

yes

characteristic

104

C₂×SD₁₆

—

105

C₂×D₈

—

106

C₂×D₈

—

107

C₂×D₈

—

108

C₂×D₈

—

109

C₂×D₈

—

110

C₄⋊Q₈

—

111

#127

—

112

#127

—

113

D₁₆

—

114

C₂×C₁₆

D₄×C₁₅₁

yes

—

115

#107

—

116

#102

—

117

C₄×C₈

D₄×C₁₅₁

yes

characteristic

118

C₄×C₈

D₄×C₁₅₁

yes

characteristic

119

C₄×C₈

C₂³×C₁₅₁

yes

characteristic

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

SD₁₆

—

128

D₈

—

129

D₈

—

130

D₈

—

131

D₈

—

132

D₈

—

133

C₂×C₈

D₈×C₁₅₁

yes

—

134

C₂×C₈

D₈×C₁₅₁

yes

—

135

C₂×C₈

C₂×D₄×C₁₅₁

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

C₂×D₄×C₁₅₁

yes

characteristic

139

C₁₆

—

140

C₂³

—

141

C₂³

—

142

Q₈

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

—

148

D₄

—

149

D₄

—

150

C₂×C₄

—

151

C₂×C₄

#127×C₁₅₁

yes

characteristic

152

C₂×C₄

C₂×D₈×C₁₅₁

yes

characteristic

153

C₂×C₄

C₂×D₈×C₁₅₁

yes

characteristic

154

C₈

—

155

C₈

D₁₆×C₁₅₁

yes

complemented

156

C₈

AGL(1,ℤ/8)×C₁₅₁

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#493×C₁₅₁

yes

characteristic

164

C₄

—

165

C₄

#496×C₁₅₁

yes

—

166

C₄

C₂×D₁₆×C₁₅₁

yes

—

167

C₄

#509×C₁₅₁

yes

—

168

C₂

—

169

C₂

—

170

C₂

#1721×C₁₅₁

yes

characteristic

171

C₂

#2246×C₁₅₁

yes

characteristic

172

C₂

#2250×C₁₅₁

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₅₁	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₅₁
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

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
1. #2246×C₁₅₁	C₂	semidirect
2. #2240×C₁₅₁	C₂	semidirect
3. #1712×C₁₅₁	C₂	semidirect
4. #1706×C₁₅₁	C₂	semidirect
5. #1312×C₁₅₁	C₂	semidirect
68. C₈×C₁₅₁	D₁₆	semidirect
86. #12525	C₁₅₁	direct (N × H)
87. C₁₅₁	#12525	direct (N × H)
88. #2246	C₂×C₁₅₁	semidirect
89. #2240	C₂×C₁₅₁	semidirect
90. #1712	C₂×C₁₅₁	semidirect
91. #1706	C₂×C₁₅₁	semidirect
92. #1312	C₂×C₁₅₁	semidirect
155. C₈	D₁₆×C₁₅₁	semidirect

H ≅ G/N

type

1. #2246×C₁₅₁

C₂

semidirect

2. #2240×C₁₅₁

C₂

semidirect

3. #1712×C₁₅₁

C₂

semidirect

4. #1706×C₁₅₁

C₂

semidirect

5. #1312×C₁₅₁

C₂

semidirect

68. C₈×C₁₅₁

D₁₆

semidirect

86. #12525

C₁₅₁

direct (N × H)

87. C₁₅₁

#12525

direct (N × H)

88. #2246

C₂×C₁₅₁

semidirect

89. #2240

C₂×C₁₅₁

semidirect

90. #1712

C₂×C₁₅₁

semidirect

91. #1706

C₂×C₁₅₁

semidirect

92. #1312

C₂×C₁₅₁

semidirect

155. C₈

D₁₆×C₁₅₁

semidirect

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

subgroup	index
#2246×C₁₅₁	2
#2240×C₁₅₁	2
#1712×C₁₅₁	2
#1706×C₁₅₁	2
#1312×C₁₅₁	2
#12525	151

subgroup

index

#2246×C₁₅₁

#2240×C₁₅₁

#1712×C₁₅₁

#1706×C₁₅₁

#1312×C₁₅₁

#12525

151

Subgroups — C₁₅₁×#12525 #50317450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#12525 #50317450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#12525 #50317450