Group #50318467 — subgroups

Subgroups — C₁₅₁×#13542 #50318467

Order 38656.

252 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 106 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	#1945×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#1846×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#1386×C₁₅₁	C₂	1	2	yes	maximal
4	#1383×C₁₅₁	C₂	1	2	yes	maximal
5	#1376×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₁₅₁	C₂²	1	2	yes	—
7	C₂×#108×C₁₅₁	C₂²	1	1	yes	characteristic
8	#427×C₁₅₁	C₂²	1	1	yes	characteristic
9	C₂×#107×C₁₅₁	—	2	1	no	—
10	C₂×#106×C₁₅₁	C₂²	1	2	yes	—
11	#424×C₁₅₁	—	2	1	no	—
12	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
13	#382×C₁₅₁	—	2	1	no	—
14	#339×C₁₅₁	—	2	1	no	—
15	C₂×M₁₆×C₁₅₁	—	2	1	no	—
16	C₂²×C₈×C₁₅₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
18	#117×C₁₅₁	—	2	1	no	—
19	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
20	C₂×C₄²×C₁₅₁	D₄	1	1	yes	characteristic
21	M₃₂×C₁₅₁	—	2	2	no	—
22	M₃₂×C₁₅₁	—	2	2	no	—
23	C₂×C₁₆×C₁₅₁	—	2	2	no	—
24	#108×C₁₅₁	—	2	2	no	—
25	#107×C₁₅₁	—	4	2	no	—
26	#107×C₁₅₁	Q₈	1	2	yes	—
27	#106×C₁₅₁	—	2	1	no	—
28	#106×C₁₅₁	—	2	4	no	—
29	C₄⋊C₈×C₁₅₁	—	4	2	no	—
30	C₄×C₈×C₁₅₁	—	2	2	no	—
31	#95×C₁₅₁	—	4	2	no	—
32	C₂²×C₄×C₁₅₁	—	2	1	no	—
33	C₂²×C₄×C₁₅₁	—	4	2	no	—
34	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
35	M₁₆×C₁₅₁	—	4	1	no	—
36	M₁₆×C₁₅₁	—	4	1	no	—
37	C₂×C₈×C₁₅₁	—	2	1	no	—
38	C₂×C₈×C₁₅₁	—	2	1	no	—
39	C₂×C₈×C₁₅₁	—	4	1	no	—
40	C₂×C₈×C₁₅₁	SD₁₆	1	2	yes	—
41	C₂×C₈×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₁₅₁	#41	1	2	yes	—
43	C₂×C₈×C₁₅₁	#41	1	2	yes	—
44	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₁₅₁	—	2	2	no	—
46	C₄⋊C₄×C₁₅₁	—	4	2	no	—
47	C₄⋊C₄×C₁₅₁	—	4	2	no	—
48	C₂²⋊C₄×C₁₅₁	—	4	1	no	—
49	C₄²×C₁₅₁	—	2	1	no	—
50	C₄²×C₁₅₁	—	4	1	no	—
51	C₄²×C₁₅₁	Q₁₆	1	2	yes	—
52	C₁₆×C₁₅₁	—	4	2	no	—
53	C₂³×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×C₄×C₁₅₁	—	2	2	no	—
55	C₂×C₄×C₁₅₁	—	4	2	no	—
56	C₂×C₄×C₁₅₁	—	4	2	no	—
57	C₂×C₄×C₁₅₁	—	8	4	no	—
58	C₂×C₄×C₁₅₁	—	8	2	no	—
59	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₅₁	#123	1	1	yes	characteristic
61	C₂×C₄×C₁₅₁	#124	1	1	yes	characteristic
62	C₂×C₄×C₁₅₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₅₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₄×C₁₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₈×C₁₅₁	—	2	2	no	—
66	C₈×C₁₅₁	—	8	1	no	—
67	C₈×C₁₅₁	#125	1	2	yes	—
68	C₈×C₁₅₁	#126	1	2	yes	—
69	C₂²×C₁₅₁	—	2	2	no	—
70	C₂²×C₁₅₁	#476	1	1	yes	characteristic
71	C₂²×C₁₅₁	#484	1	1	yes	characteristic
72	C₂²×C₁₅₁	#490	1	1	yes	characteristic
73	C₄×C₁₅₁	—	4	2	no	—
74	C₄×C₁₅₁	—	8	2	no	—
75	C₄×C₁₅₁	—	16	2	no	—
76	C₄×C₁₅₁	#400	1	1	yes	characteristic
77	C₄×C₁₅₁	#477	1	1	yes	characteristic
78	C₄×C₁₅₁	#481	1	1	yes	characteristic
79	C₄×C₁₅₁	#487	1	1	yes	characteristic
80	C₂×C₁₅₁	—	2	1	no	—
81	C₂×C₁₅₁	—	2	1	no	—
82	C₂×C₁₅₁	#2100	1	1	yes	characteristic
83	C₂×C₁₅₁	#2231	1	2	yes	—
84	#13542	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₁₅₁	#13542	1	1	yes	characteristic, direct product factor, Sylow
86	#1945	C₂×C₁₅₁	1	1	yes	characteristic
87	#1846	C₂×C₁₅₁	1	1	yes	characteristic
88	#1386	C₂×C₁₅₁	1	2	yes	—
89	#1383	C₂×C₁₅₁	1	2	yes	—
90	#1376	C₂×C₁₅₁	1	1	yes	characteristic
91	C₂×M₃₂	C₂²×C₁₅₁	1	2	yes	—
92	C₂×#108	C₂²×C₁₅₁	1	1	yes	characteristic
93	#427	C₂²×C₁₅₁	1	1	yes	characteristic
94	C₂×#107	—	2	1	no	—
95	C₂×#106	C₂²×C₁₅₁	1	2	yes	—
96	#424	—	2	1	no	—
97	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
98	#382	—	2	1	no	—
99	#339	—	2	1	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂²×C₈	D₄×C₁₅₁	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
103	#117	—	2	1	no	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄²	D₄×C₁₅₁	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#108	—	2	2	no	—
110	#107	—	4	2	no	—
111	#107	Q₈×C₁₅₁	1	2	yes	—
112	#106	—	2	1	no	—
113	#106	—	2	4	no	—
114	C₄⋊C₈	—	4	2	no	—
115	C₄×C₈	—	2	2	no	—
116	#95	—	4	2	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
120	M₁₆	—	4	1	no	—
121	M₁₆	—	4	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	SD₁₆×C₁₅₁	1	2	yes	—
126	C₂×C₈	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
127	C₂×C₈	#41×C₁₅₁	1	2	yes	—
128	C₂×C₈	#41×C₁₅₁	1	2	yes	—
129	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₂²⋊C₄	—	4	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	4	1	no	—
136	C₄²	Q₁₆×C₁₅₁	1	2	yes	—
137	C₁₆	—	4	2	no	—
138	C₂³	AGL(1,ℤ/8)×C₁₅₁	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	4	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
145	C₂×C₄	#123×C₁₅₁	1	1	yes	characteristic
146	C₂×C₄	#124×C₁₅₁	1	1	yes	characteristic
147	C₂×C₄	C₂×SD₁₆×C₁₅₁	1	1	yes	characteristic
148	C₂×C₄	C₂×Q₁₆×C₁₅₁	1	1	yes	characteristic
149	C₂×C₄	AGL(1,ℤ/8)×C₁₅₁	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	8	1	no	—
152	C₈	#125×C₁₅₁	1	2	yes	—
153	C₈	#126×C₁₅₁	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	#476×C₁₅₁	1	1	yes	characteristic
156	C₂²	#484×C₁₅₁	1	1	yes	characteristic
157	C₂²	#490×C₁₅₁	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	8	2	no	—
160	C₄	—	16	2	no	—
161	C₄	#400×C₁₅₁	1	1	yes	characteristic
162	C₄	#477×C₁₅₁	1	1	yes	characteristic
163	C₄	#481×C₁₅₁	1	1	yes	characteristic
164	C₄	#487×C₁₅₁	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#2100×C₁₅₁	1	1	yes	characteristic
168	C₂	#2231×C₁₅₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#1945×C₁₅₁

C₂

yes

maximal, characteristic

#1846×C₁₅₁

C₂

yes

maximal, characteristic

#1386×C₁₅₁

C₂

yes

maximal

#1383×C₁₅₁

C₂

yes

maximal

#1376×C₁₅₁

C₂

yes

maximal, characteristic

C₂×M₃₂×C₁₅₁

C₂²

yes

—

C₂×#108×C₁₅₁

C₂²

yes

characteristic

#427×C₁₅₁

C₂²

yes

characteristic

C₂×#107×C₁₅₁

—

C₂×#106×C₁₅₁

C₂²

yes

—

#424×C₁₅₁

—

C₂×C₄×C₈×C₁₅₁

C₂²

yes

characteristic

#382×C₁₅₁

—

#339×C₁₅₁

—

C₂×M₁₆×C₁₅₁

—

C₂²×C₈×C₁₅₁

D₄

yes

characteristic

C₂²×C₈×C₁₅₁

C₂³

yes

characteristic

#117×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄²×C₁₅₁

D₄

yes

characteristic

M₃₂×C₁₅₁

—

M₃₂×C₁₅₁

—

C₂×C₁₆×C₁₅₁

—

#108×C₁₅₁

—

#107×C₁₅₁

—

#107×C₁₅₁

Q₈

yes

—

#106×C₁₅₁

—

#106×C₁₅₁

—

C₄⋊C₈×C₁₅₁

—

C₄×C₈×C₁₅₁

—

#95×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂×D₄

yes

characteristic

M₁₆×C₁₅₁

—

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

SD₁₆

yes

—

C₂×C₈×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₅₁

#41

yes

—

C₂×C₈×C₁₅₁

#41

yes

—

C₂×C₈×C₁₅₁

#41

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

C₄²×C₁₅₁

—

C₄²×C₁₅₁

Q₁₆

yes

—

C₁₆×C₁₅₁

—

C₂³×C₁₅₁

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂²⋊Q₈

yes

C₂×C₄×C₁₅₁

yes

C₂×C₄×C₁₅₁

yes

C₂×C₄×C₁₅₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₅₁

AGL(1,ℤ/8)

yes

characteristic

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

#125

yes

—

C₈×C₁₅₁

#126

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#2100

yes

characteristic

C₂×C₁₅₁

#2231

yes

—

#13542

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#13542

yes

characteristic, direct product factor, Sylow

#1945

C₂×C₁₅₁

yes

characteristic

#1846

C₂×C₁₅₁

yes

characteristic

#1386

C₂×C₁₅₁

yes

—

#1383

C₂×C₁₅₁

yes

—

#1376

C₂×C₁₅₁

yes

characteristic

C₂×M₃₂

C₂²×C₁₅₁

yes

—

C₂×#108

C₂²×C₁₅₁

yes

characteristic

#427

C₂²×C₁₅₁

yes

characteristic

C₂×#107

—

C₂×#106

C₂²×C₁₅₁

yes

—

#424

—

C₂×C₄×C₈

C₂²×C₁₅₁

yes

characteristic

#382

—

#339

—

100

C₂×M₁₆

—

101

C₂²×C₈

D₄×C₁₅₁

yes

characteristic

102

C₂²×C₈

C₂³×C₁₅₁

yes

characteristic

103

#117

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

D₄×C₁₅₁

yes

characteristic

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

#108

—

110

#107

—

111

#107

Q₈×C₁₅₁

yes

—

112

#106

—

113

#106

—

114

C₄⋊C₈

—

115

C₄×C₈

—

116

#95

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×D₄×C₁₅₁

yes

characteristic

120

M₁₆

—

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

SD₁₆×C₁₅₁

yes

—

126

C₂×C₈

C₂×Q₈×C₁₅₁

yes

characteristic

127

C₂×C₈

#41×C₁₅₁

yes

—

128

C₂×C₈

#41×C₁₅₁

yes

—

129

C₂×C₈

#41×C₁₅₁

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

Q₁₆×C₁₅₁

yes

—

137

C₁₆

—

138

C₂³

AGL(1,ℤ/8)×C₁₅₁

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂²⋊Q₈×C₁₅₁

yes

characteristic

145

C₂×C₄

#123×C₁₅₁

yes

characteristic

146

C₂×C₄

#124×C₁₅₁

yes

characteristic

147

C₂×C₄

C₂×SD₁₆×C₁₅₁

yes

characteristic

148

C₂×C₄

C₂×Q₁₆×C₁₅₁

yes

characteristic

149

C₂×C₄

AGL(1,ℤ/8)×C₁₅₁

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

#125×C₁₅₁

yes

—

153

C₈

#126×C₁₅₁

yes

—

154

C₂²

—

155

C₂²

#476×C₁₅₁

yes

characteristic

156

C₂²

#484×C₁₅₁

yes

characteristic

157

C₂²

#490×C₁₅₁

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#400×C₁₅₁

yes

characteristic

162

C₄

#477×C₁₅₁

yes

characteristic

163

C₄

#481×C₁₅₁

yes

characteristic

164

C₄

#487×C₁₅₁

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

#2100×C₁₅₁

yes

characteristic

168

C₂

#2231×C₁₅₁

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₅₁	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₅₁
prime radical	C₂³×C₁₅₁	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ C₂×C₄×C₈×C₁₅₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₅₁

#476

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₅₁

prime radical

C₂³×C₁₅₁

AGL(1,ℤ/8)

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ C₂×C₄×C₈×C₁₅₁ ◁ G

N	H ≅ G/N	type
84. #13542	C₁₅₁	direct (N × H)
85. C₁₅₁	#13542	direct (N × H)

H ≅ G/N

type

84. #13542

C₁₅₁

direct (N × H)

85. C₁₅₁

#13542

direct (N × H)

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

subgroup	index
#1945×C₁₅₁	2
#1846×C₁₅₁	2
#1386×C₁₅₁	2
#1383×C₁₅₁	2
#1376×C₁₅₁	2
#13542	151

subgroup

index

#1945×C₁₅₁

#1846×C₁₅₁

#1386×C₁₅₁

#1383×C₁₅₁

#1376×C₁₅₁

#13542

151

Subgroups — C₁₅₁×#13542 #50318467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#13542 #50318467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#13542 #50318467