Group #50317386 — subgroups

Subgroups — C₁₅₁×#12461 #50317386

Order 38656.

188 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 82 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	#2252×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2183×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#1676×C₁₅₁	C₂	1	2	yes	maximal
4	#1372×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#1365×C₁₅₁	C₂	1	2	yes	maximal
6	#500×C₁₅₁	C₂²	1	2	yes	—
7	#477×C₁₅₁	—	2	2	no	—
8	C₈×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
9	#366×C₁₅₁	—	2	1	no	—
10	#366×C₁₅₁	—	2	1	no	—
11	C₄⋊C₁₆×C₁₅₁	—	2	1	no	—
12	C₄⋊C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
13	#346×C₁₅₁	—	2	1	no	—
14	C₄×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
15	#335×C₁₅₁	C₂²	1	2	yes	—
16	#333×C₁₅₁	—	2	2	no	—
17	C₄⋊Q₈×C₁₅₁	—	2	2	no	—
18	C₄×Q₈×C₁₅₁	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₁₅₁	—	2	1	no	—
20	C₂×C₁₆×C₁₅₁	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₅₁	D₄	1	1	yes	characteristic
22	#107×C₁₅₁	—	2	2	no	—
23	#107×C₁₅₁	—	2	2	no	—
24	#107×C₁₅₁	—	4	2	no	—
25	C₄⋊C₈×C₁₅₁	—	2	1	no	—
26	C₄⋊C₈×C₁₅₁	—	2	2	no	—
27	C₄⋊C₈×C₁₅₁	D₄	1	1	yes	characteristic
28	#103×C₁₅₁	—	4	2	no	—
29	C₄×C₈×C₁₅₁	—	2	1	no	—
30	C₄×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₁₅₁	—	4	2	no	—
32	C₂×Q₈×C₁₅₁	D₈	1	1	yes	characteristic
33	C₂×C₈×C₁₅₁	—	2	1	no	—
34	C₂×C₈×C₁₅₁	—	4	2	no	—
35	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₅₁	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₁₅₁	—	2	1	no	—
38	C₄⋊C₄×C₁₅₁	—	4	2	no	—
39	C₄⋊C₄×C₁₅₁	—	4	2	no	—
40	C₄⋊C₄×C₁₅₁	D₈	1	1	yes	characteristic
41	C₄²×C₁₅₁	—	2	1	no	—
42	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₁₅₁	—	2	1	no	—
44	C₁₆×C₁₅₁	—	4	1	no	—
45	C₁₆×C₁₅₁	Q₁₆	1	2	yes	—
46	Q₈×C₁₅₁	—	2	2	no	—
47	Q₈×C₁₅₁	—	16	2	no	—
48	C₂×C₄×C₁₅₁	—	2	1	no	—
49	C₂×C₄×C₁₅₁	—	2	1	no	—
50	C₂×C₄×C₁₅₁	—	8	2	no	—
51	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
52	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₁₅₁	#135	1	1	yes	characteristic
54	C₈×C₁₅₁	—	2	1	no	—
55	C₈×C₁₅₁	—	4	1	no	—
56	C₈×C₁₅₁	—	8	2	no	—
57	C₈×C₁₅₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₁₅₁	#137	1	1	yes	characteristic
59	C₂²×C₁₅₁	#466	1	1	yes	characteristic
60	C₄×C₁₅₁	—	2	1	no	—
61	C₄×C₁₅₁	—	4	1	no	—
62	C₄×C₁₅₁	—	4	1	no	—
63	C₄×C₁₅₁	—	16	2	no	—
64	C₄×C₁₅₁	#462	1	1	yes	characteristic
65	C₄×C₁₅₁	#473	1	1	yes	characteristic
66	C₄×C₁₅₁	#510	1	2	yes	—
67	C₂×C₁₅₁	#1674	1	1	yes	characteristic
68	C₂×C₁₅₁	#2219	1	1	yes	characteristic
69	C₂×C₁₅₁	#2221	1	1	yes	characteristic
70	#12461	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₁₅₁	#12461	1	1	yes	characteristic, direct product factor, Sylow
72	#2252	C₂×C₁₅₁	1	1	yes	characteristic
73	#2183	C₂×C₁₅₁	1	1	yes	characteristic
74	#1676	C₂×C₁₅₁	1	2	yes	—
75	#1372	C₂×C₁₅₁	1	1	yes	characteristic
76	#1365	C₂×C₁₅₁	1	2	yes	—
77	#500	C₂²×C₁₅₁	1	2	yes	—
78	#477	—	2	2	no	—
79	C₈×Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
80	#366	—	2	1	no	—
81	#366	—	2	1	no	—
82	C₄⋊C₁₆	—	2	1	no	—
83	C₄⋊C₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
84	#346	—	2	1	no	—
85	C₄×C₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
86	#335	C₂²×C₁₅₁	1	2	yes	—
87	#333	—	2	2	no	—
88	C₄⋊Q₈	—	2	2	no	—
89	C₄×Q₈	D₄×C₁₅₁	1	1	yes	characteristic
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	D₄×C₁₅₁	1	1	yes	characteristic
92	C₂×C₁₆	D₄×C₁₅₁	1	1	yes	characteristic
93	#107	—	2	2	no	—
94	#107	—	2	2	no	—
95	#107	—	4	2	no	—
96	C₄⋊C₈	—	2	1	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	D₄×C₁₅₁	1	1	yes	characteristic
99	#103	—	4	2	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	D₈×C₁₅₁	1	1	yes	characteristic
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
107	C₂×C₈	#41×C₁₅₁	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	D₈×C₁₅₁	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	1	no	—
116	C₁₆	Q₁₆×C₁₅₁	1	2	yes	—
117	Q₈	—	2	2	no	—
118	Q₈	—	16	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	8	2	no	—
122	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
123	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
124	C₂×C₄	#135×C₁₅₁	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	8	2	no	—
128	C₈	C₂×Q₁₆×C₁₅₁	1	1	yes	characteristic
129	C₈	#137×C₁₅₁	1	1	yes	characteristic
130	C₂²	#466×C₁₅₁	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	16	2	no	—
135	C₄	#462×C₁₅₁	1	1	yes	characteristic
136	C₄	#473×C₁₅₁	1	1	yes	characteristic
137	C₄	#510×C₁₅₁	1	2	yes	—
138	C₂	#1674×C₁₅₁	1	1	yes	characteristic
139	C₂	#2219×C₁₅₁	1	1	yes	characteristic
140	C₂	#2221×C₁₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2252×C₁₅₁

C₂

yes

maximal, characteristic

#2183×C₁₅₁

C₂

yes

maximal, characteristic

#1676×C₁₅₁

C₂

yes

maximal

#1372×C₁₅₁

C₂

yes

maximal, characteristic

#1365×C₁₅₁

C₂

yes

maximal

#500×C₁₅₁

C₂²

yes

—

#477×C₁₅₁

—

C₈×Q₈×C₁₅₁

C₂²

yes

characteristic

#366×C₁₅₁

—

#366×C₁₅₁

—

C₄⋊C₁₆×C₁₅₁

—

C₄⋊C₁₆×C₁₅₁

C₂²

yes

characteristic

#346×C₁₅₁

—

C₄×C₁₆×C₁₅₁

C₂²

yes

characteristic

#335×C₁₅₁

C₂²

yes

—

#333×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₄×Q₈×C₁₅₁

D₄

yes

characteristic

C₂×C₁₆×C₁₅₁

—

C₂×C₁₆×C₁₅₁

D₄

yes

characteristic

C₂×C₁₆×C₁₅₁

D₄

yes

characteristic

#107×C₁₅₁

—

#107×C₁₅₁

—

#107×C₁₅₁

—

C₄⋊C₈×C₁₅₁

—

C₄⋊C₈×C₁₅₁

—

C₄⋊C₈×C₁₅₁

D₄

yes

characteristic

#103×C₁₅₁

—

C₄×C₈×C₁₅₁

—

C₄×C₈×C₁₅₁

C₂³

yes

characteristic

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

D₈

yes

characteristic

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₈×C₁₅₁

#41

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

D₈

yes

characteristic

C₄²×C₁₅₁

—

C₄²×C₁₅₁

C₂×D₄

yes

characteristic

C₁₆×C₁₅₁

—

C₁₆×C₁₅₁

—

C₁₆×C₁₅₁

Q₁₆

yes

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

#121

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₈

yes

characteristic

C₂×C₄×C₁₅₁

#135

yes

characteristic

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

C₂×Q₁₆

yes

C₈×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#12461

yes

characteristic, direct product factor, Sylow

#2252

C₂×C₁₅₁

yes

characteristic

#2183

C₂×C₁₅₁

yes

characteristic

#1676

C₂×C₁₅₁

yes

—

#1372

C₂×C₁₅₁

yes

characteristic

#1365

C₂×C₁₅₁

yes

—

#500

C₂²×C₁₅₁

yes

—

#477

—

C₈×Q₈

C₂²×C₁₅₁

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²×C₁₅₁

yes

characteristic

#346

—

C₄×C₁₆

C₂²×C₁₅₁

yes

characteristic

#335

C₂²×C₁₅₁

yes

—

#333

—

C₄⋊Q₈

—

C₄×Q₈

D₄×C₁₅₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₅₁

yes

characteristic

C₂×C₁₆

D₄×C₁₅₁

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄×C₁₅₁

yes

characteristic

#103

—

100

C₄×C₈

—

101

C₄×C₈

C₂³×C₁₅₁

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

D₈×C₁₅₁

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×D₄×C₁₅₁

yes

characteristic

107

C₂×C₈

#41×C₁₅₁

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

D₈×C₁₅₁

yes

characteristic

112

C₄²

—

113

C₄²

C₂×D₄×C₁₅₁

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

Q₁₆×C₁₅₁

yes

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#121×C₁₅₁

yes

characteristic

123

C₂×C₄

C₂×D₈×C₁₅₁

yes

characteristic

124

C₂×C₄

#135×C₁₅₁

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

C₂×Q₁₆×C₁₅₁

yes

characteristic

129

C₈

#137×C₁₅₁

yes

characteristic

130

C₂²

#466×C₁₅₁

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#462×C₁₅₁

yes

characteristic

136

C₄

#473×C₁₅₁

yes

characteristic

137

C₄

#510×C₁₅₁

yes

—

138

C₂

#1674×C₁₅₁

yes

characteristic

139

C₂

#2219×C₁₅₁

yes

characteristic

140

C₂

#2221×C₁₅₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₅₁	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₅₁
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₄²×C₁₅₁ ◁ C₈×Q₈×C₁₅₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₅₁

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₅₁

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₅₁ ◁ C₄²×C₁₅₁ ◁ C₈×Q₈×C₁₅₁ ◁ G

N	H ≅ G/N	type
70. #12461	C₁₅₁	direct (N × H)
71. C₁₅₁	#12461	direct (N × H)

H ≅ G/N

type

70. #12461

C₁₅₁

direct (N × H)

71. C₁₅₁

#12461

direct (N × H)

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

subgroup	index
#2252×C₁₅₁	2
#2183×C₁₅₁	2
#1676×C₁₅₁	2
#1372×C₁₅₁	2
#1365×C₁₅₁	2
#12461	151

subgroup

index

#2252×C₁₅₁

#2183×C₁₅₁

#1676×C₁₅₁

#1372×C₁₅₁

#1365×C₁₅₁

#12461

151

Subgroups — C₁₅₁×#12461 #50317386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#12461 #50317386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#12461 #50317386