Group #34096414 — subgroups

Subgroups — #2056×C₁₅₇ #34096414

Order 20096.

288 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 104 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×C₄⋊C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#103×C₁₅₇	C₂	1	2	yes	maximal
4	#391×C₁₅₇	C₂	1	2	yes	maximal
5	#340×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₅₇	C₂²	1	2	yes	—
7	C₂×Q₁₆×C₁₅₇	—	2	4	no	—
8	C₂×Q₁₆×C₁₅₇	—	2	2	no	—
9	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₅₇	—	2	2	no	—
14	#103×C₁₅₇	—	2	4	no	—
15	#95×C₁₅₇	—	2	2	no	—
16	#95×C₁₅₇	—	2	2	no	—
17	C₂×Q₈×C₁₅₇	—	2	2	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	D₄	1	4	yes	—
23	C₂²×C₄×C₁₅₇	—	2	2	no	—
24	C₂²×C₄×C₁₅₇	—	2	2	no	—
25	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
28	Q₁₆×C₁₅₇	—	2	8	no	—
29	C₂×C₈×C₁₅₇	—	2	2	no	—
30	C₂×C₈×C₁₅₇	—	2	2	no	—
31	C₂×C₈×C₁₅₇	—	2	1	no	—
32	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₅₇	—	2	2	no	—
34	C₄⋊C₄×C₁₅₇	—	2	2	no	—
35	C₄²×C₁₅₇	—	2	1	no	—
36	C₄²×C₁₅₇	—	2	1	no	—
37	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₁₅₇	—	2	4	no	—
39	Q₈×C₁₅₇	—	2	4	no	—
40	Q₈×C₁₅₇	—	4	2	no	—
41	Q₈×C₁₅₇	—	4	2	no	—
42	C₂×C₄×C₁₅₇	—	2	1	no	—
43	C₂×C₄×C₁₅₇	—	2	1	no	—
44	C₂×C₄×C₁₅₇	—	2	1	no	—
45	C₂×C₄×C₁₅₇	—	2	4	no	—
46	C₂×C₄×C₁₅₇	—	2	2	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	2	no	—
51	C₂×C₄×C₁₅₇	—	4	2	no	—
52	C₂×C₄×C₁₅₇	Q₁₆	1	2	yes	complemented
53	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₅₇	#41	1	2	yes	—
57	C₈×C₁₅₇	—	2	2	no	—
58	C₈×C₁₅₇	—	4	1	no	—
59	C₂²×C₁₅₇	#121	1	2	yes	—
60	C₂²×C₁₅₇	#121	1	1	yes	characteristic
61	C₂²×C₁₅₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₅₇	#135	1	1	yes	characteristic
63	C₂²×C₁₅₇	#137	1	2	yes	—
64	C₄×C₁₅₇	—	2	2	no	—
65	C₄×C₁₅₇	—	4	4	no	—
66	C₄×C₁₅₇	—	4	1	no	—
67	C₄×C₁₅₇	—	8	2	no	—
68	C₄×C₁₅₇	#120	1	1	yes	characteristic
69	C₄×C₁₅₇	#123	1	2	yes	—
70	C₄×C₁₅₇	#124	1	1	yes	characteristic
71	C₂×C₁₅₇	#394	1	1	yes	characteristic
72	C₂×C₁₅₇	#462	1	2	yes	—
73	C₂×C₁₅₇	#464	1	2	yes	—
74	C₂×C₁₅₇	#467	1	1	yes	characteristic
75	C₂×C₁₅₇	#470	1	1	yes	characteristic
76	C₁₅₇	#2056	1	1	yes	characteristic, direct product factor, Sylow
77	#2056	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂²×Q₁₆	C₂×C₁₅₇	1	1	yes	characteristic
79	C₂×C₄⋊C₈	C₂×C₁₅₇	1	1	yes	characteristic
80	C₂×#103	C₂×C₁₅₇	1	2	yes	—
81	#391	C₂×C₁₅₇	1	2	yes	—
82	#340	C₂×C₁₅₇	1	1	yes	characteristic
83	C₂²×Q₈	C₂²×C₁₅₇	1	2	yes	—
84	C₂×Q₁₆	—	2	4	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂²×C₈	C₂²×C₁₅₇	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₁₅₇	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
89	C₂×C₄²	C₂²×C₁₅₇	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	2	4	no	—
92	#95	—	2	2	no	—
93	#95	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	D₄×C₁₅₇	1	4	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	D₄×C₁₅₇	1	1	yes	characteristic
103	C₂²×C₄	D₄×C₁₅₇	1	1	yes	characteristic
104	C₂²×C₄	C₂³×C₁₅₇	1	1	yes	characteristic
105	Q₁₆	—	2	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	D₄×C₁₅₇	1	2	yes	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₄×C₁₅₇	1	1	yes	characteristic
115	Q₈	—	2	4	no	—
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	Q₁₆×C₁₅₇	1	2	yes	complemented
130	C₂×C₄	C₂×D₄×C₁₅₇	1	2	yes	—
131	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
132	C₂×C₄	#41×C₁₅₇	1	1	yes	characteristic
133	C₂×C₄	#41×C₁₅₇	1	2	yes	—
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#121×C₁₅₇	1	2	yes	—
137	C₂²	#121×C₁₅₇	1	1	yes	characteristic
138	C₂²	C₂×Q₁₆×C₁₅₇	1	1	yes	characteristic
139	C₂²	#135×C₁₅₇	1	1	yes	characteristic
140	C₂²	#137×C₁₅₇	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	4	4	no	—
144	C₄	—	8	2	no	—
145	C₄	#120×C₁₅₇	1	1	yes	characteristic
146	C₄	#123×C₁₅₇	1	2	yes	—
147	C₄	#124×C₁₅₇	1	1	yes	characteristic
148	C₂	#394×C₁₅₇	1	1	yes	characteristic
149	C₂	#462×C₁₅₇	1	2	yes	—
150	C₂	#464×C₁₅₇	1	2	yes	—
151	C₂	#467×C₁₅₇	1	1	yes	characteristic
152	C₂	#470×C₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₁₆×C₁₅₇

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#103×C₁₅₇

C₂

yes

maximal

#391×C₁₅₇

C₂

yes

maximal

#340×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₅₇

C₂²

yes

—

C₂×Q₁₆×C₁₅₇

—

C₂×Q₁₆×C₁₅₇

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

#103×C₁₅₇

—

#95×C₁₅₇

—

#95×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

Q₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

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

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

Q₁₆

yes

complemented

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

C₂×C₄×C₁₅₇

yes

C₂×C₄×C₁₅₇

yes

—

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

#121

yes

—

C₂²×C₁₅₇

#121

yes

characteristic

C₂²×C₁₅₇

C₂×Q₁₆

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#120

yes

characteristic

C₄×C₁₅₇

#123

yes

—

C₄×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₂×C₁₅₇

#464

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, direct product factor, Sylow

#2056

C₁₅₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂×C₁₅₇

yes

characteristic

C₂×C₄⋊C₈

C₂×C₁₅₇

yes

characteristic

C₂×#103

C₂×C₁₅₇

yes

—

#391

C₂×C₁₅₇

yes

—

#340

C₂×C₁₅₇

yes

characteristic

C₂²×Q₈

C₂²×C₁₅₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂²×C₁₅₇

yes

characteristic

C₂²×C₈

C₂²×C₁₅₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₅₇

yes

—

C₂×C₄²

C₂²×C₁₅₇

yes

characteristic

C₄⋊C₈

—

#103

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₁₅₇

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

D₄×C₁₅₇

yes

characteristic

103

C₂²×C₄

D₄×C₁₅₇

yes

characteristic

104

C₂²×C₄

C₂³×C₁₅₇

yes

characteristic

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

D₄×C₁₅₇

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₄×C₁₅₇

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

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

C₂×D₄×C₁₅₇

yes

—

131

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

132

C₂×C₄

#41×C₁₅₇

yes

characteristic

133

C₂×C₄

#41×C₁₅₇

yes

—

134

C₈

—

135

C₈

—

136

C₂²

#121×C₁₅₇

yes

—

137

C₂²

#121×C₁₅₇

yes

characteristic

138

C₂²

C₂×Q₁₆×C₁₅₇

yes

characteristic

139

C₂²

#135×C₁₅₇

yes

characteristic

140

C₂²

#137×C₁₅₇

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#120×C₁₅₇

yes

characteristic

146

C₄

#123×C₁₅₇

yes

—

147

C₄

#124×C₁₅₇

yes

characteristic

148

C₂

#394×C₁₅₇

yes

characteristic

149

C₂

#462×C₁₅₇

yes

—

150

C₂

#464×C₁₅₇

yes

—

151

C₂

#467×C₁₅₇

yes

characteristic

152

C₂

#470×C₁₅₇

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³×C₁₅₇

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₅₇

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₅₇ ◁ C₂×C₄²×C₁₅₇ ◁ G

N	H ≅ G/N	type
52. C₂×C₄×C₁₅₇	Q₁₆	semidirect
76. C₁₅₇	#2056	direct (N × H)
77. #2056	C₁₅₇	direct (N × H)
129. C₂×C₄	Q₁₆×C₁₅₇	semidirect

H ≅ G/N

type

52. C₂×C₄×C₁₅₇

Q₁₆

semidirect

76. C₁₅₇

#2056

direct (N × H)

77. #2056

C₁₅₇

direct (N × H)

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×C₄⋊C₈×C₁₅₇	2
C₂×#103×C₁₅₇	2
#391×C₁₅₇	2
#340×C₁₅₇	2
#2056	157

subgroup

index

C₂²×Q₁₆×C₁₅₇

C₂×C₄⋊C₈×C₁₅₇

C₂×#103×C₁₅₇

#391×C₁₅₇

#340×C₁₅₇

#2056

157

Subgroups — #2056×C₁₅₇ #34096414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2056×C157 #34096414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2056×C₁₅₇ #34096414