Group #50317459 — subgroups

Subgroups — C₁₅₁×#12534 #50317459

Order 38656.

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

normal subgroups only

This group has 250 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	#2252×C₁₅₁	C₂	1	2	yes	maximal
2	#2241×C₁₅₁	C₂	1	2	yes	maximal
3	#1713×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#1707×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#1312×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	#500×C₁₅₁	—	2	1	no	—
7	#500×C₁₅₁	C₂²	1	1	yes	characteristic
8	#500×C₁₅₁	C₂²	1	2	yes	—
9	#498×C₁₅₁	—	2	1	no	—
10	#494×C₁₅₁	—	2	1	no	—
11	#494×C₁₅₁	C₂²	1	1	yes	characteristic
12	#487×C₁₅₁	—	2	1	no	—
13	#366×C₁₅₁	—	2	4	no	—
14	#358×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×Q₁₆×C₁₅₁	—	4	2	no	—
19	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
20	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
21	C₄⋊Q₈×C₁₅₁	—	2	2	no	—
22	C₂×C₁₆×C₁₅₁	D₄	1	4	yes	—
23	#107×C₁₅₁	—	2	2	no	—
24	#107×C₁₅₁	—	2	2	no	—
25	#107×C₁₅₁	—	2	2	no	—
26	#107×C₁₅₁	—	4	2	no	—
27	#106×C₁₅₁	—	4	2	no	—
28	#103×C₁₅₁	—	4	2	no	—
29	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₁₅₁	—	4	1	no	—
33	C₂×Q₈×C₁₅₁	—	4	1	no	—
34	C₂×Q₈×C₁₅₁	—	4	2	no	—
35	Q₁₆×C₁₅₁	—	4	4	no	—
36	Q₁₆×C₁₅₁	—	8	4	no	—
37	C₂×C₈×C₁₅₁	D₈	1	2	yes	—
38	C₂×C₈×C₁₅₁	D₈	1	2	yes	—
39	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
40	C₄⋊C₄×C₁₅₁	—	4	2	no	—
41	C₄⋊C₄×C₁₅₁	—	4	1	no	—
42	C₄⋊C₄×C₁₅₁	—	4	1	no	—
43	C₄⋊C₄×C₁₅₁	—	4	1	no	—
44	C₄⋊C₄×C₁₅₁	—	4	1	no	—
45	C₄⋊C₄×C₁₅₁	—	4	2	no	—
46	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
47	C₁₆×C₁₅₁	—	2	4	no	—
48	Q₈×C₁₅₁	—	8	2	no	—
49	Q₈×C₁₅₁	—	8	2	no	—
50	Q₈×C₁₅₁	—	8	4	no	—
51	C₂×C₄×C₁₅₁	—	8	1	no	—
52	C₂×C₄×C₁₅₁	—	8	1	no	—
53	C₂×C₄×C₁₅₁	—	8	2	no	—
54	C₂×C₄×C₁₅₁	#127	1	1	yes	characteristic
55	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
57	C₈×C₁₅₁	—	2	2	no	—
58	C₈×C₁₅₁	Q₃₂	1	4	yes	—
59	C₈×C₁₅₁	#137	1	4	yes	—
60	C₂²×C₁₅₁	#493	1	1	yes	characteristic
61	C₄×C₁₅₁	—	16	1	no	—
62	C₄×C₁₅₁	—	16	1	no	—
63	C₄×C₁₅₁	—	16	2	no	—
64	C₄×C₁₅₁	#497	1	2	yes	—
65	C₄×C₁₅₁	C₂×Q₃₂	1	2	yes	—
66	C₄×C₁₅₁	#509	1	2	yes	—
67	C₂×C₁₅₁	#1722	1	1	yes	characteristic
68	C₂×C₁₅₁	#2247	1	1	yes	characteristic
69	C₂×C₁₅₁	#2250	1	1	yes	characteristic
70	#12534	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₁₅₁	#12534	1	1	yes	characteristic, direct product factor, Sylow
72	#2252	C₂×C₁₅₁	1	2	yes	—
73	#2241	C₂×C₁₅₁	1	2	yes	—
74	#1713	C₂×C₁₅₁	1	1	yes	characteristic
75	#1707	C₂×C₁₅₁	1	1	yes	characteristic
76	#1312	C₂×C₁₅₁	1	1	yes	characteristic
77	#500	—	2	1	no	—
78	#500	C₂²×C₁₅₁	1	1	yes	characteristic
79	#500	C₂²×C₁₅₁	1	2	yes	—
80	#498	—	2	1	no	—
81	#494	—	2	1	no	—
82	#494	C₂²×C₁₅₁	1	1	yes	characteristic
83	#487	—	2	1	no	—
84	#366	—	2	4	no	—
85	#358	—	2	4	no	—
86	C₄×C₁₆	C₂²×C₁₅₁	1	2	yes	—
87	C₈²	C₂²×C₁₅₁	1	1	yes	characteristic
88	C₂×Q₁₆	—	2	2	no	—
89	C₂×Q₁₆	—	4	2	no	—
90	C₄⋊Q₈	—	2	1	no	—
91	C₄⋊Q₈	—	2	1	no	—
92	C₄⋊Q₈	—	2	2	no	—
93	C₂×C₁₆	D₄×C₁₅₁	1	4	yes	—
94	#107	—	2	2	no	—
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#107	—	4	2	no	—
98	#106	—	4	2	no	—
99	#103	—	4	2	no	—
100	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
101	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
102	C₄×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
103	C₂×Q₈	—	4	1	no	—
104	C₂×Q₈	—	4	1	no	—
105	C₂×Q₈	—	4	2	no	—
106	Q₁₆	—	4	4	no	—
107	Q₁₆	—	8	4	no	—
108	C₂×C₈	D₈×C₁₅₁	1	2	yes	—
109	C₂×C₈	D₈×C₁₅₁	1	2	yes	—
110	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
118	C₁₆	—	2	4	no	—
119	Q₈	—	8	2	no	—
120	Q₈	—	8	4	no	—
121	Q₈	—	8	2	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	#127×C₁₅₁	1	1	yes	characteristic
126	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
127	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	Q₃₂×C₁₅₁	1	4	yes	—
130	C₈	#137×C₁₅₁	1	4	yes	—
131	C₂²	#493×C₁₅₁	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	—	16	2	no	—
134	C₄	—	16	1	no	—
135	C₄	#497×C₁₅₁	1	2	yes	—
136	C₄	C₂×Q₃₂×C₁₅₁	1	2	yes	—
137	C₄	#509×C₁₅₁	1	2	yes	—
138	C₂	#1722×C₁₅₁	1	1	yes	characteristic
139	C₂	#2247×C₁₅₁	1	1	yes	characteristic
140	C₂	#2250×C₁₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2252×C₁₅₁

C₂

yes

maximal

#2241×C₁₅₁

C₂

yes

maximal

#1713×C₁₅₁

C₂

yes

maximal, characteristic

#1707×C₁₅₁

C₂

yes

maximal, characteristic

#1312×C₁₅₁

C₂

yes

maximal, characteristic

#500×C₁₅₁

—

#500×C₁₅₁

C₂²

yes

characteristic

#500×C₁₅₁

C₂²

yes

—

#498×C₁₅₁

—

#494×C₁₅₁

—

#494×C₁₅₁

C₂²

yes

characteristic

#487×C₁₅₁

—

#366×C₁₅₁

—

#358×C₁₅₁

—

C₄×C₁₆×C₁₅₁

C₂²

yes

—

C₈²×C₁₅₁

C₂²

yes

characteristic

C₂×Q₁₆×C₁₅₁

—

C₂×Q₁₆×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₂×C₁₆×C₁₅₁

D₄

yes

—

#107×C₁₅₁

—

#107×C₁₅₁

—

#107×C₁₅₁

—

#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×Q₈×C₁₅₁

—

Q₁₆×C₁₅₁

—

Q₁₆×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

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂×D₄

yes

characteristic

C₁₆×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

C₂×C₄×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

Q₃₂

yes

—

C₈×C₁₅₁

#137

yes

—

C₂²×C₁₅₁

#493

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#497

yes

—

C₄×C₁₅₁

C₂×Q₃₂

yes

—

C₄×C₁₅₁

#509

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#12534

yes

characteristic, direct product factor, Sylow

#2252

C₂×C₁₅₁

yes

—

#2241

C₂×C₁₅₁

yes

—

#1713

C₂×C₁₅₁

yes

characteristic

#1707

C₂×C₁₅₁

yes

characteristic

#1312

C₂×C₁₅₁

yes

characteristic

#500

—

#500

C₂²×C₁₅₁

yes

characteristic

#500

C₂²×C₁₅₁

yes

—

#498

—

#494

—

#494

C₂²×C₁₅₁

yes

—

—

—

C₄×C₁₆

C₂²×C₁₅₁

yes

—

C₈²

C₂²×C₁₅₁

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₄×C₁₅₁

yes

—

—

—

—

—

—

—

100

C₄×C₈

D₄×C₁₅₁

yes

characteristic

101

C₄×C₈

D₄×C₁₅₁

yes

characteristic

102

C₄×C₈

C₂³×C₁₅₁

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

Q₁₆

—

107

Q₁₆

—

108

C₂×C₈

D₈×C₁₅₁

yes

—

109

C₂×C₈

D₈×C₁₅₁

yes

—

110

C₂×C₈

C₂×D₄×C₁₅₁

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

C₂×D₄×C₁₅₁

yes

characteristic

118

C₁₆

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#127×C₁₅₁

yes

characteristic

126

C₂×C₄

C₂×D₈×C₁₅₁

yes

characteristic

127

C₂×C₄

C₂×D₈×C₁₅₁

yes

characteristic

128

C₈

—

129

C₈

Q₃₂×C₁₅₁

yes

—

130

C₈

#137×C₁₅₁

yes

—

131

C₂²

#493×C₁₅₁

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#497×C₁₅₁

yes

—

136

C₄

C₂×Q₃₂×C₁₅₁

yes

—

137

C₄

#509×C₁₅₁

yes

—

138

C₂

#1722×C₁₅₁

yes

characteristic

139

C₂

#2247×C₁₅₁

yes

characteristic

140

C₂

#2250×C₁₅₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	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, prime radical

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
70. #12534	C₁₅₁	direct (N × H)
71. C₁₅₁	#12534	direct (N × H)

H ≅ G/N

type

70. #12534

C₁₅₁

direct (N × H)

71. C₁₅₁

#12534

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
#2241×C₁₅₁	2
#1713×C₁₅₁	2
#1707×C₁₅₁	2
#1312×C₁₅₁	2
#12534	151

subgroup

index

#2252×C₁₅₁

#2241×C₁₅₁

#1713×C₁₅₁

#1707×C₁₅₁

#1312×C₁₅₁

#12534

151

Subgroups — C₁₅₁×#12534 #50317459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#12534 #50317459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#12534 #50317459