Group #34096273 — subgroups

Subgroups — #1915×C₁₅₇ #34096273

Order 20096.

308 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 108 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	#568×C₁₅₇	C₂	1	1	yes	maximal, characteristic
2	#536×C₁₅₇	C₂	1	1	yes	maximal, characteristic
3	#413×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#101×C₁₅₇	C₂	1	2	yes	maximal, complemented
5	#344×C₁₅₇	C₂	1	2	yes	maximal, complemented
6	C₂×#41×C₁₅₇	C₂²	1	1	yes	characteristic
7	C₂²×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
8	#131×C₁₅₇	—	2	4	no	—
9	C₂×M₁₆×C₁₅₇	—	2	2	no	—
10	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
11	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₅₇	—	2	1	no	—
13	#124×C₁₅₇	—	2	1	no	—
14	#123×C₁₅₇	C₄	1	4	yes	—
15	C₂²⋊Q₈×C₁₅₇	—	2	2	no	—
16	#117×C₁₅₇	C₂²	1	1	yes	characteristic
17	#101×C₁₅₇	—	2	1	no	—
18	#101×C₁₅₇	—	2	4	no	—
19	#100×C₁₅₇	—	2	1	no	—
20	#41×C₁₅₇	—	2	2	no	—
21	#41×C₁₅₇	—	2	2	no	—
22	C₂×Q₈×C₁₅₇	—	2	2	no	—
23	C₂×Q₈×C₁₅₇	—	2	1	no	—
24	C₂×Q₈×C₁₅₇	—	2	1	no	—
25	C₂×Q₈×C₁₅₇	—	2	2	no	—
26	C₂×Q₈×C₁₅₇	D₄	1	1	yes	characteristic
27	C₂×D₄×C₁₅₇	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₁₅₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
31	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₅₇	—	2	4	no	—
33	M₁₆×C₁₅₇	—	2	2	no	—
34	M₁₆×C₁₅₇	—	2	2	no	—
35	M₁₆×C₁₅₇	—	2	2	no	—
36	M₁₆×C₁₅₇	D₄	1	4	yes	—
37	C₂×C₈×C₁₅₇	—	2	4	no	—
38	C₂×C₈×C₁₅₇	—	2	2	no	—
39	C₂×C₈×C₁₅₇	—	2	2	no	—
40	C₄⋊C₄×C₁₅₇	—	2	4	no	—
41	C₄⋊C₄×C₁₅₇	—	2	1	no	—
42	C₂²⋊C₄×C₁₅₇	—	2	4	no	—
43	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
44	C₄²×C₁₅₇	—	2	1	no	—
45	C₂³×C₁₅₇	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₁₅₇	—	2	2	no	—
48	Q₈×C₁₅₇	—	4	1	no	—
49	Q₈×C₁₅₇	—	4	2	no	—
50	Q₈×C₁₅₇	—	4	1	no	—
51	D₄×C₁₅₇	—	2	4	no	—
52	D₄×C₁₅₇	—	2	2	no	—
53	C₂×C₄×C₁₅₇	—	2	2	no	—
54	C₂×C₄×C₁₅₇	—	2	2	no	—
55	C₂×C₄×C₁₅₇	—	2	2	no	—
56	C₂×C₄×C₁₅₇	—	2	2	no	—
57	C₂×C₄×C₁₅₇	—	2	4	no	—
58	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₅₇	#41	1	2	yes	—
63	C₈×C₁₅₇	—	2	4	no	—
64	C₈×C₁₅₇	—	4	2	no	—
65	C₂²×C₁₅₇	—	2	2	no	—
66	C₂²×C₁₅₇	—	4	1	no	—
67	C₂²×C₁₅₇	—	4	2	no	—
68	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₅₇	C₄×D₄	1	2	yes	—
70	C₄×C₁₅₇	—	2	2	no	—
71	C₄×C₁₅₇	—	4	2	no	—
72	C₄×C₁₅₇	—	4	2	no	—
73	C₄×C₁₅₇	#120	1	1	yes	characteristic
74	C₄×C₁₅₇	#121	1	2	yes	—
75	C₄×C₁₅₇	#123	1	1	yes	characteristic
76	C₂×C₁₅₇	—	2	1	no	—
77	C₂×C₁₅₇	—	2	2	no	—
78	C₂×C₁₅₇	—	4	2	no	—
79	C₂×C₁₅₇	#386	1	1	yes	characteristic
80	C₁₅₇	#1915	1	1	yes	characteristic, direct product factor, Sylow
81	#1915	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	#568	C₂×C₁₅₇	1	1	yes	characteristic
83	#536	C₂×C₁₅₇	1	1	yes	characteristic
84	#413	C₂×C₁₅₇	1	1	yes	characteristic
85	C₂×#101	C₂×C₁₅₇	1	2	yes	complemented
86	#344	C₂×C₁₅₇	1	2	yes	complemented
87	C₂×#41	C₂²×C₁₅₇	1	1	yes	characteristic
88	C₂²×Q₈	C₂²×C₁₅₇	1	1	yes	characteristic
89	#131	—	2	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	C₂²×C₁₅₇	1	2	yes	—
92	C₂×M₁₆	C₂²×C₁₅₇	1	2	yes	—
93	C₄⋊Q₈	—	2	1	no	—
94	#124	—	2	1	no	—
95	#123	C₄×C₁₅₇	1	4	yes	—
96	C₂²⋊Q₈	—	2	2	no	—
97	#117	C₂²×C₁₅₇	1	1	yes	characteristic
98	#101	—	2	1	no	—
99	#101	—	2	4	no	—
100	#100	—	2	1	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	D₄×C₁₅₇	1	1	yes	characteristic
108	C₂×D₄	C₂×C₄×C₁₅₇	1	2	yes	—
109	C₂×D₄	D₄×C₁₅₇	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₄×C₁₅₇	1	2	yes	—
111	C₂²×C₄	D₄×C₁₅₇	1	2	yes	—
112	C₂²×C₄	C₂³×C₁₅₇	1	1	yes	characteristic
113	M₁₆	—	2	4	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	2	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	D₄×C₁₅₇	1	4	yes	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	C₂×C₄×C₁₅₇	1	2	yes	—
125	C₄²	—	2	1	no	—
126	C₂³	C₂²⋊C₄×C₁₅₇	1	2	yes	—
127	C₂³	C₂²×C₄×C₁₅₇	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂²⋊C₄×C₁₅₇	1	2	yes	—
140	C₂×C₄	C₂×D₄×C₁₅₇	1	2	yes	—
141	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
142	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
143	C₂×C₄	#41×C₁₅₇	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	4	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×C₂²⋊C₄×C₁₅₇	1	1	yes	characteristic
150	C₂²	C₄×D₄×C₁₅₇	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#120×C₁₅₇	1	1	yes	characteristic
155	C₄	#121×C₁₅₇	1	2	yes	—
156	C₄	#123×C₁₅₇	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#386×C₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#568×C₁₅₇

C₂

yes

maximal, characteristic

#536×C₁₅₇

C₂

yes

maximal, characteristic

#413×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#101×C₁₅₇

C₂

yes

maximal, complemented

#344×C₁₅₇

C₂

yes

maximal, complemented

C₂×#41×C₁₅₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₅₇

C₂²

yes

characteristic

#131×C₁₅₇

—

C₂×M₁₆×C₁₅₇

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

—

#124×C₁₅₇

—

#123×C₁₅₇

C₄

yes

—

C₂²⋊Q₈×C₁₅₇

—

#117×C₁₅₇

C₂²

yes

characteristic

#101×C₁₅₇

—

#101×C₁₅₇

—

#100×C₁₅₇

—

#41×C₁₅₇

—

#41×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

characteristic

C₂×D₄×C₁₅₇

C₂×C₄

yes

—

C₂×D₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

—

M₁₆×C₁₅₇

D₄

yes

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₂²⋊C₄×C₁₅₇

—

C₂²⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

—

C₂³×C₁₅₇

C₂²⋊C₄

yes

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#120

yes

characteristic

C₄×C₁₅₇

#121

yes

—

C₄×C₁₅₇

#123

yes

characteristic

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#386

yes

characteristic

C₁₅₇

#1915

yes

characteristic, direct product factor, Sylow

#1915

C₁₅₇

yes

maximal, characteristic, direct product factor, Sylow

#568

C₂×C₁₅₇

yes

characteristic

#536

C₂×C₁₅₇

yes

characteristic

#413

C₂×C₁₅₇

yes

characteristic

C₂×#101

C₂×C₁₅₇

yes

complemented

#344

C₂×C₁₅₇

yes

complemented

C₂×#41

C₂²×C₁₅₇

yes

characteristic

C₂²×Q₈

C₂²×C₁₅₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂²×C₁₅₇

yes

—

C₂×M₁₆

C₂²×C₁₅₇

yes

—

C₄⋊Q₈

—

#124

—

#123

C₄×C₁₅₇

yes

—

C₂²⋊Q₈

—

#117

C₂²×C₁₅₇

yes

characteristic

#101

—

#101

—

100

#100

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

D₄×C₁₅₇

yes

characteristic

108

C₂×D₄

C₂×C₄×C₁₅₇

yes

—

109

C₂×D₄

D₄×C₁₅₇

yes

characteristic

110

C₂²×C₄

C₂×C₄×C₁₅₇

yes

—

111

C₂²×C₄

D₄×C₁₅₇

yes

—

112

C₂²×C₄

C₂³×C₁₅₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

D₄×C₁₅₇

yes

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

C₂×C₄×C₁₅₇

yes

—

125

C₄²

—

126

C₂³

C₂²⋊C₄×C₁₅₇

yes

—

127

C₂³

C₂²×C₄×C₁₅₇

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊C₄×C₁₅₇

yes

—

140

C₂×C₄

C₂×D₄×C₁₅₇

yes

—

141

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

142

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

143

C₂×C₄

#41×C₁₅₇

yes

—

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊C₄×C₁₅₇

yes

characteristic

150

C₂²

C₄×D₄×C₁₅₇

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#120×C₁₅₇

yes

characteristic

155

C₄

#121×C₁₅₇

yes

—

156

C₄

#123×C₁₅₇

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#386×C₁₅₇

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₅₇	#386
commutator	C₂³	C₂²×C₄×C₁₅₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₅₇
prime radical	C₂×#41×C₁₅₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₅₇ ◁ C₂²×C₄×C₁₅₇ ◁ G

subgroup

G/H

center, socle

C₂×C₁₅₇

#386

commutator

C₂³

C₂²×C₄×C₁₅₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₅₇

prime radical

C₂×#41×C₁₅₇

C₂²

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₅₇ ◁ C₂²×C₄×C₁₅₇ ◁ G

N	H ≅ G/N	type
4. C₂×#101×C₁₅₇	C₂	semidirect
5. #344×C₁₅₇	C₂	semidirect
80. C₁₅₇	#1915	direct (N × H)
81. #1915	C₁₅₇	direct (N × H)
85. C₂×#101	C₂×C₁₅₇	semidirect
86. #344	C₂×C₁₅₇	semidirect

H ≅ G/N

type

4. C₂×#101×C₁₅₇

C₂

semidirect

5. #344×C₁₅₇

C₂

semidirect

80. C₁₅₇

#1915

direct (N × H)

81. #1915

C₁₅₇

direct (N × H)

85. C₂×#101

C₂×C₁₅₇

semidirect

86. #344

C₂×C₁₅₇

semidirect

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

subgroup	index
#568×C₁₅₇	2
#536×C₁₅₇	2
#413×C₁₅₇	2
C₂×#101×C₁₅₇	2
#344×C₁₅₇	2
#1915	157

subgroup

index

#568×C₁₅₇

#536×C₁₅₇

#413×C₁₅₇

C₂×#101×C₁₅₇

#344×C₁₅₇

#1915

157

Subgroups — #1915×C₁₅₇ #34096273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1915×C157 #34096273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1915×C₁₅₇ #34096273