Group #50320136 — subgroups

Subgroups — C₁₅₁×#15211 #50320136

Order 38656.

1498 conjugacy classes of subgroups; only the 478 normal subgroups are shown.

normal subgroups only

This group has 1498 conjugacy classes of subgroups — too many to draw. Only its 478 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	C₂×#389×C₁₅₁	C₂	1	6	yes	maximal
2	#1444×C₁₅₁	C₂	1	4	yes	maximal
3	#1444×C₁₅₁	C₂	1	4	yes	maximal
4	C₄×#95×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊C₄×C₁₅₁	C₂²	1	6	yes	—
6	C₂²×C₄⋊C₄×C₁₅₁	C₂²	1	4	yes	—
7	C₂²×C₄⋊C₄×C₁₅₁	C₂²	1	6	yes	—
8	C₂²×C₄²×C₁₅₁	C₂²	1	3	yes	—
9	#389×C₁₅₁	—	2	24	no	—
10	C₂×#95×C₁₅₁	C₂²	1	12	yes	—
11	C₂×#95×C₁₅₁	C₂²	1	4	yes	—
12	C₂³×C₄×C₁₅₁	C₂³	1	4	yes	—
13	C₂³×C₄×C₁₅₁	C₂³	1	4	yes	—
14	C₂³×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
15	C₂³×C₄×C₁₅₁	C₂³	1	6	yes	—
16	C₂×C₄⋊C₄×C₁₅₁	—	2	12	no	—
17	C₂×C₄⋊C₄×C₁₅₁	—	2	24	no	—
18	C₂×C₄⋊C₄×C₁₅₁	—	2	12	no	—
19	C₂×C₄⋊C₄×C₁₅₁	—	2	24	no	—
20	C₂×C₄⋊C₄×C₁₅₁	—	4	12	no	—
21	C₂×C₄²×C₁₅₁	—	4	6	no	—
22	C₂×C₄²×C₁₅₁	D₄	1	12	yes	—
23	#95×C₁₅₁	—	2	24	no	—
24	#95×C₁₅₁	Q₈	1	16	yes	—
25	C₂⁴×C₁₅₁	C₂⁴	1	1	yes	characteristic
26	C₂²×C₄×C₁₅₁	—	2	12	no	—
27	C₂²×C₄×C₁₅₁	—	2	3	no	—
28	C₂²×C₄×C₁₅₁	—	2	12	no	—
29	C₂²×C₄×C₁₅₁	—	2	1	no	—
30	C₂²×C₄×C₁₅₁	—	2	12	no	—
31	C₂²×C₄×C₁₅₁	—	2	12	no	—
32	C₂²×C₄×C₁₅₁	—	2	12	no	—
33	C₂²×C₄×C₁₅₁	—	2	12	no	—
34	C₂²×C₄×C₁₅₁	—	2	4	no	—
35	C₂²×C₄×C₁₅₁	C₂×D₄	1	6	yes	—
36	C₂²×C₄×C₁₅₁	C₂×D₄	1	12	yes	—
37	C₂²×C₄×C₁₅₁	C₂×Q₈	1	24	yes	—
38	C₂²×C₄×C₁₅₁	#41	1	8	yes	—
39	C₄⋊C₄×C₁₅₁	—	2	48	no	—
40	C₄⋊C₄×C₁₅₁	—	4	24	no	—
41	C₄⋊C₄×C₁₅₁	—	4	16	no	—
42	C₄²×C₁₅₁	—	4	12	no	—
43	C₂³×C₁₅₁	C₂²×D₄	1	3	yes	—
44	C₂³×C₁₅₁	C₂²×Q₈	1	4	yes	—
45	C₂³×C₁₅₁	C₂×#41	1	4	yes	—
46	C₂³×C₁₅₁	H(5,𝔽₂)	1	3	yes	—
47	C₂³×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
48	C₂×C₄×C₁₅₁	—	2	12	no	—
49	C₂×C₄×C₁₅₁	—	2	12	no	—
50	C₂×C₄×C₁₅₁	—	2	6	no	—
51	C₂×C₄×C₁₅₁	—	2	6	no	—
52	C₂×C₄×C₁₅₁	—	2	48	no	—
53	C₂×C₄×C₁₅₁	—	2	24	no	—
54	C₂×C₄×C₁₅₁	—	4	12	no	—
55	C₂×C₄×C₁₅₁	—	4	4	no	—
56	C₂×C₄×C₁₅₁	—	4	4	no	—
57	C₂×C₄×C₁₅₁	—	4	12	no	—
58	C₂×C₄×C₁₅₁	—	4	12	no	—
59	C₂×C₄×C₁₅₁	#127	1	4	yes	—
60	C₂×C₄×C₁₅₁	C₄⋊Q₈	1	24	yes	—
61	C₂²×C₁₅₁	#525	1	6	yes	—
62	C₂²×C₁₅₁	C₂×#127	1	1	yes	characteristic
63	C₂²×C₁₅₁	C₂×C₄⋊Q₈	1	6	yes	—
64	C₂²×C₁₅₁	#534	1	3	yes	—
65	C₂²×C₁₅₁	#537	1	3	yes	—
66	C₂²×C₁₅₁	#554	1	12	yes	—
67	C₂²×C₁₅₁	#557	1	4	yes	—
68	C₄×C₁₅₁	—	2	24	no	—
69	C₄×C₁₅₁	—	2	4	no	—
70	C₄×C₁₅₁	—	4	8	no	—
71	C₄×C₁₅₁	—	8	4	no	—
72	C₂×C₁₅₁	#2397	1	1	yes	characteristic
73	C₂×C₁₅₁	#2583	1	6	yes	—
74	C₂×C₁₅₁	#2584	1	6	yes	—
75	C₂×C₁₅₁	#2656	1	1	yes	characteristic
76	C₂×C₁₅₁	#2868	1	1	yes	characteristic
77	#15211	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₁₅₁	#15211	1	1	yes	characteristic, direct product factor, Sylow
79	C₂×#389	C₂×C₁₅₁	1	6	yes	—
80	#1444	C₂×C₁₅₁	1	4	yes	—
81	#1444	C₂×C₁₅₁	1	4	yes	—
82	C₄×#95	C₂×C₁₅₁	1	1	yes	characteristic
83	C₂²×C₄⋊C₄	C₂²×C₁₅₁	1	6	yes	—
84	C₂²×C₄⋊C₄	C₂²×C₁₅₁	1	4	yes	—
85	C₂²×C₄⋊C₄	C₂²×C₁₅₁	1	6	yes	—
86	C₂²×C₄²	C₂²×C₁₅₁	1	3	yes	—
87	#389	—	2	24	no	—
88	C₂×#95	C₂²×C₁₅₁	1	12	yes	—
89	C₂×#95	C₂²×C₁₅₁	1	4	yes	—
90	C₂³×C₄	C₂³×C₁₅₁	1	4	yes	—
91	C₂³×C₄	C₂³×C₁₅₁	1	4	yes	—
92	C₂³×C₄	C₂³×C₁₅₁	1	6	yes	—
93	C₂³×C₄	C₂³×C₁₅₁	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	2	24	no	—
95	C₂×C₄⋊C₄	—	2	12	no	—
96	C₂×C₄⋊C₄	—	2	12	no	—
97	C₂×C₄⋊C₄	—	2	24	no	—
98	C₂×C₄⋊C₄	—	4	12	no	—
99	C₂×C₄²	—	4	6	no	—
100	C₂×C₄²	D₄×C₁₅₁	1	12	yes	—
101	#95	—	2	24	no	—
102	#95	Q₈×C₁₅₁	1	16	yes	—
103	C₂⁴	C₂⁴×C₁₅₁	1	1	yes	characteristic
104	C₂²×C₄	—	2	12	no	—
105	C₂²×C₄	—	2	3	no	—
106	C₂²×C₄	—	2	12	no	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	12	no	—
109	C₂²×C₄	—	2	12	no	—
110	C₂²×C₄	—	2	12	no	—
111	C₂²×C₄	—	2	12	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	C₂×D₄×C₁₅₁	1	6	yes	—
114	C₂²×C₄	C₂×D₄×C₁₅₁	1	12	yes	—
115	C₂²×C₄	C₂×Q₈×C₁₅₁	1	24	yes	—
116	C₂²×C₄	#41×C₁₅₁	1	8	yes	—
117	C₄⋊C₄	—	2	48	no	—
118	C₄⋊C₄	—	4	24	no	—
119	C₄⋊C₄	—	4	16	no	—
120	C₄²	—	4	12	no	—
121	C₂³	C₂²×D₄×C₁₅₁	1	3	yes	—
122	C₂³	C₂²×Q₈×C₁₅₁	1	4	yes	—
123	C₂³	C₂×#41×C₁₅₁	1	4	yes	—
124	C₂³	H(5,𝔽₂)×C₁₅₁	1	3	yes	—
125	C₂³	2₋¹⁺⁴×C₁₅₁	1	1	yes	characteristic
126	C₂×C₄	—	2	12	no	—
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	6	no	—
130	C₂×C₄	—	2	48	no	—
131	C₂×C₄	—	2	24	no	—
132	C₂×C₄	—	4	12	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	12	no	—
136	C₂×C₄	—	4	12	no	—
137	C₂×C₄	#127×C₁₅₁	1	4	yes	—
138	C₂×C₄	C₄⋊Q₈×C₁₅₁	1	24	yes	—
139	C₂²	#525×C₁₅₁	1	6	yes	—
140	C₂²	C₂×#127×C₁₅₁	1	1	yes	characteristic
141	C₂²	C₂×C₄⋊Q₈×C₁₅₁	1	6	yes	—
142	C₂²	#534×C₁₅₁	1	3	yes	—
143	C₂²	#537×C₁₅₁	1	3	yes	—
144	C₂²	#554×C₁₅₁	1	12	yes	—
145	C₂²	#557×C₁₅₁	1	4	yes	—
146	C₄	—	2	4	no	—
147	C₄	—	2	24	no	—
148	C₄	—	4	8	no	—
149	C₄	—	8	4	no	—
150	C₂	#2397×C₁₅₁	1	1	yes	characteristic
151	C₂	#2583×C₁₅₁	1	6	yes	—
152	C₂	#2584×C₁₅₁	1	6	yes	—
153	C₂	#2656×C₁₅₁	1	1	yes	characteristic
154	C₂	#2868×C₁₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#389×C₁₅₁

C₂

yes

maximal

#1444×C₁₅₁

C₂

yes

maximal

#1444×C₁₅₁

C₂

yes

maximal

C₄×#95×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂²×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂²×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂²×C₄²×C₁₅₁

C₂²

yes

—

#389×C₁₅₁

—

C₂×#95×C₁₅₁

C₂²

yes

—

C₂×#95×C₁₅₁

C₂²

yes

—

C₂³×C₄×C₁₅₁

C₂³

yes

—

C₂³×C₄×C₁₅₁

C₂³

yes

—

C₂³×C₄×C₁₅₁

C₂³

yes

characteristic

C₂³×C₄×C₁₅₁

C₂³

yes

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄²×C₁₅₁

—

C₂×C₄²×C₁₅₁

D₄

yes

—

#95×C₁₅₁

—

#95×C₁₅₁

Q₈

yes

—

C₂⁴×C₁₅₁

C₂⁴

yes

characteristic

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂×D₄

yes

—

C₂²×C₄×C₁₅₁

C₂×D₄

yes

—

C₂²×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂²×C₄×C₁₅₁

#41

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

C₂³×C₁₅₁

C₂²×D₄

yes

—

C₂³×C₁₅₁

C₂²×Q₈

yes

—

C₂³×C₁₅₁

C₂×#41

yes

—

C₂³×C₁₅₁

H(5,𝔽₂)

yes

—

C₂³×C₁₅₁

2₋¹⁺⁴

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

#127

yes

—

C₂×C₄×C₁₅₁

C₄⋊Q₈

yes

—

C₂²×C₁₅₁

#525

yes

—

C₂²×C₁₅₁

C₂×#127

yes

characteristic

C₂²×C₁₅₁

C₂×C₄⋊Q₈

yes

—

C₂²×C₁₅₁

#534

yes

—

C₂²×C₁₅₁

#537

yes

—

C₂²×C₁₅₁

#554

yes

—

C₂²×C₁₅₁

#557

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₂×C₁₅₁

#2397

yes

characteristic

C₂×C₁₅₁

#2583

yes

—

C₂×C₁₅₁

#2584

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#15211

yes

characteristic, direct product factor, Sylow

C₂×#389

C₂×C₁₅₁

yes

—

#1444

C₂×C₁₅₁

yes

—

#1444

C₂×C₁₅₁

yes

—

C₄×#95

C₂×C₁₅₁

yes

characteristic

C₂²×C₄⋊C₄

C₂²×C₁₅₁

yes

—

C₂²×C₄⋊C₄

C₂²×C₁₅₁

yes

—

C₂²×C₄⋊C₄

C₂²×C₁₅₁

yes

—

C₂²×C₄²

C₂²×C₁₅₁

yes

—

#389

—

C₂×#95

C₂²×C₁₅₁

yes

—

C₂×#95

C₂²×C₁₅₁

yes

—

C₂³×C₄

C₂³×C₁₅₁

yes

—

C₂³×C₄

C₂³×C₁₅₁

yes

—

C₂³×C₄

C₂³×C₁₅₁

yes

—

C₂³×C₄

C₂³×C₁₅₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

100

C₂×C₄²

D₄×C₁₅₁

yes

—

101

#95

—

102

#95

Q₈×C₁₅₁

yes

—

103

C₂⁴

C₂⁴×C₁₅₁

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×D₄×C₁₅₁

yes

—

114

C₂²×C₄

C₂×D₄×C₁₅₁

yes

—

115

C₂²×C₄

C₂×Q₈×C₁₅₁

yes

—

116

C₂²×C₄

#41×C₁₅₁

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

—

121

C₂³

C₂²×D₄×C₁₅₁

yes

—

122

C₂³

C₂²×Q₈×C₁₅₁

yes

—

123

C₂³

C₂×#41×C₁₅₁

yes

—

124

C₂³

H(5,𝔽₂)×C₁₅₁

yes

—

125

C₂³

2₋¹⁺⁴×C₁₅₁

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#127×C₁₅₁

yes

—

138

C₂×C₄

C₄⋊Q₈×C₁₅₁

yes

—

139

C₂²

#525×C₁₅₁

yes

—

140

C₂²

C₂×#127×C₁₅₁

yes

characteristic

141

C₂²

C₂×C₄⋊Q₈×C₁₅₁

yes

—

142

C₂²

#534×C₁₅₁

yes

—

143

C₂²

#537×C₁₅₁

yes

—

144

C₂²

#554×C₁₅₁

yes

—

145

C₂²

#557×C₁₅₁

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₂

#2397×C₁₅₁

yes

characteristic

151

C₂

#2583×C₁₅₁

yes

—

152

C₂

#2584×C₁₅₁

yes

—

153

C₂

#2656×C₁₅₁

yes

characteristic

154

C₂

#2868×C₁₅₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₅₁	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₅₁
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₅₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂⁴×C₁₅₁

C₂⁴

commutator, Frattini, cosocle

C₂⁴

C₂⁴×C₁₅₁

LCS

G ▷ C₂⁴ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₁₅₁ ◁ G

N	H ≅ G/N	type
77. #15211	C₁₅₁	direct (N × H)
78. C₁₅₁	#15211	direct (N × H)

H ≅ G/N

type

77. #15211

C₁₅₁

direct (N × H)

78. C₁₅₁

#15211

direct (N × H)

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

subgroup	index
C₂×#389×C₁₅₁	2
#1444×C₁₅₁	2
C₄×#95×C₁₅₁	2
#15211	151

subgroup

index

C₂×#389×C₁₅₁

#1444×C₁₅₁

C₄×#95×C₁₅₁

#15211

151

Subgroups — C₁₅₁×#15211 #50320136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#15211 #50320136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#15211 #50320136