Group #50318445 — subgroups

Subgroups — C₁₅₁×#13520 #50318445

Order 38656.

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

normal subgroups only

This group has 238 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	#2161×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2151×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#1953×C₁₅₁	C₂	1	2	yes	maximal
4	#1385×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#1381×C₁₅₁	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₅₁	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
8	#430×C₁₅₁	—	2	2	no	—
9	C₂×#108×C₁₅₁	C₂²	1	2	yes	—
10	C₂×#108×C₁₅₁	C₂²	1	2	yes	—
11	#405×C₁₅₁	C₂²	1	1	yes	characteristic
12	#368×C₁₅₁	—	2	2	no	—
13	C₄⋊C₁₆×C₁₅₁	—	2	2	no	—
14	#344×C₁₅₁	—	2	2	no	—
15	C₂×M₁₆×C₁₅₁	—	2	2	no	—
16	C₂×M₁₆×C₁₅₁	—	2	2	no	—
17	C₂×M₁₆×C₁₅₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
19	#117×C₁₅₁	D₄	1	1	yes	characteristic
20	M₃₂×C₁₅₁	—	2	2	no	—
21	C₂×C₁₆×C₁₅₁	—	2	1	no	—
22	C₂×C₁₆×C₁₅₁	—	2	2	no	—
23	C₂×C₁₆×C₁₅₁	D₄	1	2	yes	—
24	#108×C₁₅₁	—	2	2	no	—
25	#108×C₁₅₁	—	2	2	no	—
26	#108×C₁₅₁	—	2	2	no	—
27	#108×C₁₅₁	—	4	4	no	—
28	#108×C₁₅₁	Q₈	1	4	yes	—
29	#97×C₁₅₁	—	2	1	no	—
30	C₄×C₈×C₁₅₁	—	2	1	no	—
31	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₁₅₁	—	2	4	no	—
33	M₁₆×C₁₅₁	—	2	2	no	—
34	M₁₆×C₁₅₁	—	4	2	no	—
35	M₁₆×C₁₅₁	—	4	2	no	—
36	M₁₆×C₁₅₁	—	4	2	no	—
37	C₂×C₈×C₁₅₁	—	2	1	no	—
38	C₂×C₈×C₁₅₁	—	4	2	no	—
39	C₂×C₈×C₁₅₁	—	4	2	no	—
40	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₅₁	C₂×Q₈	1	2	yes	—
42	C₂×C₈×C₁₅₁	#41	1	2	yes	—
43	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₁₅₁	—	2	1	no	—
45	C₂²⋊C₄×C₁₅₁	Q₁₆	1	2	yes	—
46	C₄²×C₁₅₁	—	2	1	no	—
47	C₁₆×C₁₅₁	—	2	2	no	—
48	C₁₆×C₁₅₁	—	4	1	no	—
49	C₂³×C₁₅₁	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₁₅₁	—	2	2	no	—
51	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
52	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
53	C₂×C₄×C₁₅₁	#135	1	1	yes	characteristic
54	C₂×C₄×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₈×C₁₅₁	—	4	1	no	—
56	C₈×C₁₅₁	—	4	4	no	—
57	C₈×C₁₅₁	—	8	2	no	—
58	C₈×C₁₅₁	#123	1	1	yes	characteristic
59	C₈×C₁₅₁	#125	1	2	yes	—
60	C₈×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
61	C₂²×C₁₅₁	—	4	1	no	—
62	C₂²×C₁₅₁	#467	1	1	yes	characteristic
63	C₂²×C₁₅₁	#477	1	2	yes	—
64	C₄×C₁₅₁	—	4	2	no	—
65	C₄×C₁₅₁	#398	1	1	yes	characteristic
66	C₄×C₁₅₁	#469	1	1	yes	characteristic
67	C₄×C₁₅₁	#478	1	2	yes	—
68	C₂×C₁₅₁	—	2	1	no	—
69	C₂×C₁₅₁	—	2	2	no	—
70	C₂×C₁₅₁	#2087	1	1	yes	characteristic
71	#13520	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₅₁	#13520	1	1	yes	characteristic, direct product factor, Sylow
73	#2161	C₂×C₁₅₁	1	1	yes	characteristic
74	#2151	C₂×C₁₅₁	1	1	yes	characteristic
75	#1953	C₂×C₁₅₁	1	2	yes	—
76	#1385	C₂×C₁₅₁	1	1	yes	characteristic
77	#1381	C₂×C₁₅₁	1	2	yes	—
78	C₂×M₃₂	C₂²×C₁₅₁	1	1	yes	characteristic
79	C₂²×C₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
80	#430	—	2	2	no	—
81	C₂×#108	C₂²×C₁₅₁	1	2	yes	—
82	C₂×#108	C₂²×C₁₅₁	1	2	yes	—
83	#405	C₂²×C₁₅₁	1	1	yes	characteristic
84	#368	—	2	2	no	—
85	C₄⋊C₁₆	—	2	2	no	—
86	#344	—	2	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	2	2	no	—
89	C₂×M₁₆	D₄×C₁₅₁	1	1	yes	characteristic
90	C₂²×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
91	#117	D₄×C₁₅₁	1	1	yes	characteristic
92	M₃₂	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	D₄×C₁₅₁	1	2	yes	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	2	2	no	—
99	#108	—	4	4	no	—
100	#108	Q₈×C₁₅₁	1	4	yes	—
101	#97	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
104	M₁₆	—	2	4	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	4	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
113	C₂×C₈	C₂×Q₈×C₁₅₁	1	2	yes	—
114	C₂×C₈	#41×C₁₅₁	1	2	yes	—
115	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Q₁₆×C₁₅₁	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	4	1	no	—
121	C₂³	C₂×Q₁₆×C₁₅₁	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	2	yes	—
125	C₂×C₄	#135×C₁₅₁	1	1	yes	characteristic
126	C₂×C₄	AGL(1,ℤ/8)×C₁₅₁	1	2	yes	—
127	C₈	—	4	1	no	—
128	C₈	—	4	4	no	—
129	C₈	—	8	2	no	—
130	C₈	#123×C₁₅₁	1	1	yes	characteristic
131	C₈	#125×C₁₅₁	1	2	yes	—
132	C₈	C₄⋊Q₈×C₁₅₁	1	1	yes	characteristic
133	C₂²	—	4	1	no	—
134	C₂²	#467×C₁₅₁	1	1	yes	characteristic
135	C₂²	#477×C₁₅₁	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	#398×C₁₅₁	1	1	yes	characteristic
138	C₄	#469×C₁₅₁	1	1	yes	characteristic
139	C₄	#478×C₁₅₁	1	2	yes	—
140	C₂	—	2	1	no	—
141	C₂	—	2	2	no	—
142	C₂	#2087×C₁₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2161×C₁₅₁

C₂

yes

maximal, characteristic

#2151×C₁₅₁

C₂

yes

maximal, characteristic

#1953×C₁₅₁

C₂

yes

maximal

#1385×C₁₅₁

C₂

yes

maximal, characteristic

#1381×C₁₅₁

C₂

yes

maximal

C₂×M₃₂×C₁₅₁

C₂²

yes

characteristic

C₂²×C₁₆×C₁₅₁

C₂²

yes

characteristic

#430×C₁₅₁

—

C₂×#108×C₁₅₁

C₂²

yes

—

C₂×#108×C₁₅₁

C₂²

yes

—

#405×C₁₅₁

C₂²

yes

characteristic

#368×C₁₅₁

—

C₄⋊C₁₆×C₁₅₁

—

#344×C₁₅₁

—

C₂×M₁₆×C₁₅₁

—

C₂×M₁₆×C₁₅₁

—

C₂×M₁₆×C₁₅₁

D₄

yes

characteristic

C₂²×C₈×C₁₅₁

C₂³

yes

characteristic

#117×C₁₅₁

D₄

yes

characteristic

M₃₂×C₁₅₁

—

C₂×C₁₆×C₁₅₁

—

C₂×C₁₆×C₁₅₁

—

C₂×C₁₆×C₁₅₁

D₄

yes

—

#108×C₁₅₁

—

#108×C₁₅₁

—

#108×C₁₅₁

—

#108×C₁₅₁

—

#108×C₁₅₁

Q₈

yes

—

#97×C₁₅₁

—

C₄×C₈×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂×D₄

yes

characteristic

M₁₆×C₁₅₁

—

M₁₆×C₁₅₁

—

M₁₆×C₁₅₁

—

M₁₆×C₁₅₁

—

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₈×C₁₅₁

C₂×Q₈

yes

—

C₂×C₈×C₁₅₁

#41

yes

—

C₂×C₈×C₁₅₁

#41

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₂²⋊C₄×C₁₅₁

Q₁₆

yes

—

C₄²×C₁₅₁

—

C₁₆×C₁₅₁

—

C₁₆×C₁₅₁

—

C₂³×C₁₅₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

#121

yes

characteristic

C₂×C₄×C₁₅₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₅₁

#135

yes

characteristic

C₂×C₄×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

#123

yes

characteristic

C₈×C₁₅₁

#125

yes

—

C₈×C₁₅₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#467

yes

characteristic

C₂²×C₁₅₁

#477

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#2087

yes

characteristic

#13520

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#13520

yes

characteristic, direct product factor, Sylow

#2161

C₂×C₁₅₁

yes

characteristic

#2151

C₂×C₁₅₁

yes

characteristic

#1953

C₂×C₁₅₁

yes

—

#1385

C₂×C₁₅₁

yes

characteristic

#1381

C₂×C₁₅₁

yes

—

C₂×M₃₂

C₂²×C₁₅₁

yes

characteristic

C₂²×C₁₆

C₂²×C₁₅₁

yes

characteristic

#430

—

C₂×#108

C₂²×C₁₅₁

yes

—

C₂×#108

C₂²×C₁₅₁

yes

—

#405

C₂²×C₁₅₁

yes

characteristic

#368

—

C₄⋊C₁₆

—

#344

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄×C₁₅₁

yes

characteristic

C₂²×C₈

C₂³×C₁₅₁

yes

characteristic

#117

D₄×C₁₅₁

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₅₁

yes

—

—

—

—

—

100

#108

Q₈×C₁₅₁

yes

—

101

#97

—

102

C₄×C₈

—

103

C₂²×C₄

C₂×D₄×C₁₅₁

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂×D₄×C₁₅₁

yes

characteristic

113

C₂×C₈

C₂×Q₈×C₁₅₁

yes

—

114

C₂×C₈

#41×C₁₅₁

yes

—

115

C₂×C₈

#41×C₁₅₁

yes

characteristic

116

C₄⋊C₄

—

117

C₂²⋊C₄

Q₁₆×C₁₅₁

yes

—

118

C₄²

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

C₂×Q₁₆×C₁₅₁

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

#121×C₁₅₁

yes

characteristic

124

C₂×C₄

C₂²⋊Q₈×C₁₅₁

yes

—

125

C₂×C₄

#135×C₁₅₁

yes

characteristic

126

C₂×C₄

AGL(1,ℤ/8)×C₁₅₁

yes

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

#123×C₁₅₁

yes

characteristic

131

C₈

#125×C₁₅₁

yes

—

132

C₈

C₄⋊Q₈×C₁₅₁

yes

characteristic

133

C₂²

—

134

C₂²

#467×C₁₅₁

yes

characteristic

135

C₂²

#477×C₁₅₁

yes

—

136

C₄

—

137

C₄

#398×C₁₅₁

yes

characteristic

138

C₄

#469×C₁₅₁

yes

characteristic

139

C₄

#478×C₁₅₁

yes

—

140

C₂

—

141

C₂

—

142

C₂

#2087×C₁₅₁

yes

characteristic

	subgroup	G/H
center	C₄×C₁₅₁	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₅₁
socle	C₂×C₁₅₁	#2087
prime radical	C₂³×C₁₅₁	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ #405×C₁₅₁ ◁ G

subgroup

G/H

center

C₄×C₁₅₁

#469

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₅₁

socle

C₂×C₁₅₁

#2087

prime radical

C₂³×C₁₅₁

C₂×Q₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ #405×C₁₅₁ ◁ G

N	H ≅ G/N	type
71. #13520	C₁₅₁	direct (N × H)
72. C₁₅₁	#13520	direct (N × H)

H ≅ G/N

type

71. #13520

C₁₅₁

direct (N × H)

72. C₁₅₁

#13520

direct (N × H)

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

subgroup	index
#2161×C₁₅₁	2
#2151×C₁₅₁	2
#1953×C₁₅₁	2
#1385×C₁₅₁	2
#1381×C₁₅₁	2
#13520	151

subgroup

index

#2161×C₁₅₁

#2151×C₁₅₁

#1953×C₁₅₁

#1385×C₁₅₁

#1381×C₁₅₁

#13520

151

Subgroups — C₁₅₁×#13520 #50318445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#13520 #50318445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#13520 #50318445