Group #34096327 — subgroups

Subgroups — #1969×C₁₅₇ #34096327

Order 20096.

312 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 136 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	C₂²×Q₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#107×C₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#103×C₁₅₇	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	#391×C₁₅₇	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₁₅₇	C₂²	1	2	yes	—
7	C₂×Q₁₆×C₁₅₇	—	2	2	no	—
8	C₂×Q₁₆×C₁₅₇	—	2	2	no	—
9	C₂×Q₁₆×C₁₅₇	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
14	#107×C₁₅₇	—	2	2	no	—
15	#103×C₁₅₇	—	2	4	no	—
16	C₄×C₈×C₁₅₇	—	2	2	no	—
17	#95×C₁₅₇	—	2	4	no	—
18	C₂×Q₈×C₁₅₇	—	2	2	no	—
19	C₂×Q₈×C₁₅₇	—	2	2	no	—
20	C₂×Q₈×C₁₅₇	—	2	2	no	—
21	C₂×Q₈×C₁₅₇	—	2	2	no	—
22	C₂×Q₈×C₁₅₇	—	2	2	no	—
23	C₂×Q₈×C₁₅₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₅₇	—	2	2	no	—
25	C₂²×C₄×C₁₅₇	—	2	2	no	—
26	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
27	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
28	Q₁₆×C₁₅₇	—	2	8	no	—
29	C₂×C₈×C₁₅₇	—	2	1	no	—
30	C₂×C₈×C₁₅₇	—	2	1	no	—
31	C₂×C₈×C₁₅₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
33	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
34	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
35	C₄⋊C₄×C₁₅₇	—	2	2	no	—
36	C₄⋊C₄×C₁₅₇	—	2	2	no	—
37	C₄²×C₁₅₇	—	2	1	no	—
38	C₄²×C₁₅₇	—	2	1	no	—
39	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₅₇	—	2	4	no	—
41	Q₈×C₁₅₇	—	2	4	no	—
42	Q₈×C₁₅₇	—	4	2	no	—
43	Q₈×C₁₅₇	—	4	2	no	—
44	C₂×C₄×C₁₅₇	—	2	2	no	—
45	C₂×C₄×C₁₅₇	—	2	2	no	—
46	C₂×C₄×C₁₅₇	—	2	4	no	—
47	C₂×C₄×C₁₅₇	—	4	2	no	—
48	C₂×C₄×C₁₅₇	—	4	2	no	—
49	C₂×C₄×C₁₅₇	—	4	2	no	—
50	C₂×C₄×C₁₅₇	—	4	4	no	—
51	C₂×C₄×C₁₅₇	Q₁₆	1	4	yes	complemented
52	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
57	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
58	C₈×C₁₅₇	—	2	2	no	—
59	C₈×C₁₅₇	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₁₅₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₁₅₇	#121	1	1	yes	characteristic
62	C₂²×C₁₅₇	#127	1	1	yes	characteristic
63	C₂²×C₁₅₇	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₁₅₇	#135	1	2	yes	—
65	C₄×C₁₅₇	—	2	4	no	—
66	C₄×C₁₅₇	—	4	4	no	—
67	C₄×C₁₅₇	—	8	2	no	—
68	C₄×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₄×C₁₅₇	C₄×D₄	1	1	yes	characteristic
70	C₄×C₁₅₇	#121	1	1	yes	characteristic
71	C₄×C₁₅₇	#124	1	1	yes	characteristic
72	C₂×C₁₅₇	#390	1	1	yes	characteristic
73	C₂×C₁₅₇	C₄×Q₁₆	1	2	yes	—
74	C₂×C₁₅₇	#467	1	2	yes	—
75	C₂×C₁₅₇	#494	1	1	yes	characteristic
76	C₂×C₁₅₇	#495	1	1	yes	characteristic
77	C₁₅₇	#1969	1	1	yes	characteristic, direct product factor, Sylow
78	#1969	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×Q₁₆	C₂×C₁₅₇	1	1	yes	characteristic
80	C₂×#107	C₂×C₁₅₇	1	1	yes	characteristic
81	C₂×#103	C₂×C₁₅₇	1	2	yes	—
82	C₂×C₄×C₈	C₂×C₁₅₇	1	1	yes	characteristic
83	#391	C₂×C₁₅₇	1	2	yes	—
84	C₂²×Q₈	C₂²×C₁₅₇	1	2	yes	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×Q₁₆	—	2	2	no	—
87	C₂×Q₁₆	C₄×C₁₅₇	1	4	yes	complemented
88	C₂²×C₈	C₂²×C₁₅₇	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₁₅₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
91	C₂×C₄²	C₂²×C₁₅₇	1	1	yes	characteristic
92	#107	—	2	2	no	—
93	#103	—	2	4	no	—
94	C₄×C₈	—	2	2	no	—
95	#95	—	2	4	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×C₄×C₁₅₇	1	4	yes	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	D₄×C₁₅₇	1	2	yes	—
105	C₂²×C₄	C₂³×C₁₅₇	1	1	yes	characteristic
106	Q₁₆	—	2	8	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂×C₄×C₁₅₇	1	2	yes	—
110	C₂×C₈	D₄×C₁₅₇	1	2	yes	—
111	C₂×C₈	D₄×C₁₅₇	1	2	yes	—
112	C₂×C₈	D₄×C₁₅₇	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₂³	C₂×D₄×C₁₅₇	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	Q₁₆×C₁₅₇	1	4	yes	complemented
130	C₂×C₄	C₂²×C₄×C₁₅₇	1	1	yes	characteristic
131	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
132	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
133	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
134	C₂×C₄	#41×C₁₅₇	1	1	yes	characteristic
135	C₂×C₄	#41×C₁₅₇	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	C₂²⋊C₄×C₁₅₇	1	4	yes	—
138	C₂²	C₄×D₄×C₁₅₇	1	1	yes	characteristic
139	C₂²	#121×C₁₅₇	1	1	yes	characteristic
140	C₂²	#127×C₁₅₇	1	1	yes	characteristic
141	C₂²	C₂×Q₁₆×C₁₅₇	1	2	yes	—
142	C₂²	#135×C₁₅₇	1	2	yes	—
143	C₄	—	2	4	no	—
144	C₄	—	4	4	no	—
145	C₄	—	8	2	no	—
146	C₄	C₂×C₂²⋊C₄×C₁₅₇	1	1	yes	characteristic
147	C₄	C₄×D₄×C₁₅₇	1	1	yes	characteristic
148	C₄	#121×C₁₅₇	1	1	yes	characteristic
149	C₄	#124×C₁₅₇	1	1	yes	characteristic
150	C₂	#390×C₁₅₇	1	1	yes	characteristic
151	C₂	C₄×Q₁₆×C₁₅₇	1	2	yes	—
152	C₂	#467×C₁₅₇	1	2	yes	—
153	C₂	#494×C₁₅₇	1	1	yes	characteristic
154	C₂	#495×C₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₁₆×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#107×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#103×C₁₅₇

C₂

yes

maximal

C₂×C₄×C₈×C₁₅₇

C₂

yes

maximal, characteristic

#391×C₁₅₇

C₂

yes

maximal

C₂²×Q₈×C₁₅₇

C₂²

yes

—

C₂×Q₁₆×C₁₅₇

—

C₂×Q₁₆×C₁₅₇

—

C₂×Q₁₆×C₁₅₇

C₄

yes

complemented

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

#107×C₁₅₇

—

#103×C₁₅₇

—

C₄×C₈×C₁₅₇

—

#95×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

Q₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

Q₁₆

yes

complemented

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

C₂×C₄×C₁₅₇

yes

C₂×C₄×C₁₅₇

yes

C₈×C₁₅₇

—

C₈×C₁₅₇

C₂²⋊C₄

yes

—

C₂²×C₁₅₇

C₄×D₄

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

C₂×Q₁₆

yes

—

C₂²×C₁₅₇

#135

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₅₇

C₄×D₄

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

C₄×Q₁₆

yes

—

C₂×C₁₅₇

#467

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, direct product factor, Sylow

#1969

C₁₅₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂×C₁₅₇

yes

characteristic

C₂×#107

C₂×C₁₅₇

yes

characteristic

C₂×#103

C₂×C₁₅₇

yes

—

C₂×C₄×C₈

C₂×C₁₅₇

yes

characteristic

#391

C₂×C₁₅₇

yes

—

C₂²×Q₈

C₂²×C₁₅₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₄×C₁₅₇

yes

complemented

C₂²×C₈

C₂²×C₁₅₇

yes

characteristic

C₂²×C₈

C₂²×C₁₅₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₅₇

yes

—

C₂×C₄²

C₂²×C₁₅₇

yes

characteristic

#107

—

#103

—

C₄×C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×C₄×C₁₅₇

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

D₄×C₁₅₇

yes

—

105

C₂²×C₄

C₂³×C₁₅₇

yes

characteristic

106

Q₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×C₄×C₁₅₇

yes

—

110

C₂×C₈

D₄×C₁₅₇

yes

—

111

C₂×C₈

D₄×C₁₅₇

yes

—

112

C₂×C₈

D₄×C₁₅₇

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₂³

C₂×D₄×C₁₅₇

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

Q₁₆×C₁₅₇

yes

complemented

130

C₂×C₄

C₂²×C₄×C₁₅₇

yes

characteristic

131

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

132

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

133

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

134

C₂×C₄

#41×C₁₅₇

yes

characteristic

135

C₂×C₄

#41×C₁₅₇

yes

characteristic

136

C₈

—

137

C₈

C₂²⋊C₄×C₁₅₇

yes

—

138

C₂²

C₄×D₄×C₁₅₇

yes

characteristic

139

C₂²

#121×C₁₅₇

yes

characteristic

140

C₂²

#127×C₁₅₇

yes

characteristic

141

C₂²

C₂×Q₁₆×C₁₅₇

yes

—

142

C₂²

#135×C₁₅₇

yes

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

C₂×C₂²⋊C₄×C₁₅₇

yes

characteristic

147

C₄

C₄×D₄×C₁₅₇

yes

characteristic

148

C₄

#121×C₁₅₇

yes

characteristic

149

C₄

#124×C₁₅₇

yes

characteristic

150

C₂

#390×C₁₅₇

yes

characteristic

151

C₂

C₄×Q₁₆×C₁₅₇

yes

—

152

C₂

#467×C₁₅₇

yes

—

153

C₂

#494×C₁₅₇

yes

characteristic

154

C₂

#495×C₁₅₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₅₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₅₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₅₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₅₇ ◁ C₂×C₄²×C₁₅₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₅₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₅₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₅₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₅₇ ◁ C₂×C₄²×C₁₅₇ ◁ G

N	H ≅ G/N	type
9. C₂×Q₁₆×C₁₅₇	C₄	semidirect
51. C₂×C₄×C₁₅₇	Q₁₆	semidirect
77. C₁₅₇	#1969	direct (N × H)
78. #1969	C₁₅₇	direct (N × H)
87. C₂×Q₁₆	C₄×C₁₅₇	semidirect
129. C₂×C₄	Q₁₆×C₁₅₇	semidirect

H ≅ G/N

type

9. C₂×Q₁₆×C₁₅₇

C₄

semidirect

51. C₂×C₄×C₁₅₇

Q₁₆

semidirect

77. C₁₅₇

#1969

direct (N × H)

78. #1969

C₁₅₇

direct (N × H)

87. C₂×Q₁₆

C₄×C₁₅₇

semidirect

129. C₂×C₄

Q₁₆×C₁₅₇

semidirect

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

subgroup	index
C₂²×Q₁₆×C₁₅₇	2
C₂×#107×C₁₅₇	2
C₂×#103×C₁₅₇	2
C₂×C₄×C₈×C₁₅₇	2
#391×C₁₅₇	2
#1969	157

subgroup

index

C₂²×Q₁₆×C₁₅₇

C₂×#107×C₁₅₇

C₂×#103×C₁₅₇

C₂×C₄×C₈×C₁₅₇

#391×C₁₅₇

#1969

157

Subgroups — #1969×C₁₅₇ #34096327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1969×C157 #34096327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1969×C₁₅₇ #34096327