Group #45698341 — subgroups

Subgroups — C₁₃₁×#12457 #45698341

Order 33536.

204 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 86 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×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1674×C₁₃₁	C₂	1	2	yes	maximal
4	#1371×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#1364×C₁₃₁	C₂	1	2	yes	maximal
6	#500×C₁₃₁	C₂²	1	2	yes	—
7	#462×C₁₃₁	—	2	2	no	—
8	C₈×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	#366×C₁₃₁	—	2	1	no	—
10	#366×C₁₃₁	—	2	1	no	—
11	C₄⋊C₁₆×C₁₃₁	—	2	1	no	—
12	C₄⋊C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₁₃₁	—	2	1	no	—
14	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
15	#335×C₁₃₁	C₂²	1	2	yes	—
16	#332×C₁₃₁	—	2	2	no	—
17	C₂×Q₁₆×C₁₃₁	—	4	2	no	—
18	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
19	C₄×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₃₁	—	2	1	no	—
21	C₂×C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
23	#107×C₁₃₁	—	2	2	no	—
24	#107×C₁₃₁	—	2	2	no	—
25	C₄⋊C₈×C₁₃₁	—	2	1	no	—
26	C₄⋊C₈×C₁₃₁	—	2	2	no	—
27	C₄⋊C₈×C₁₃₁	D₄	1	1	yes	characteristic
28	#103×C₁₃₁	—	4	2	no	—
29	C₄×C₈×C₁₃₁	—	2	1	no	—
30	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₁₃₁	—	4	2	no	—
32	C₂×Q₈×C₁₃₁	D₈	1	1	yes	characteristic
33	Q₁₆×C₁₃₁	—	8	4	no	—
34	C₂×C₈×C₁₃₁	—	2	1	no	—
35	C₂×C₈×C₁₃₁	—	4	2	no	—
36	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₁₃₁	—	2	1	no	—
39	C₄⋊C₄×C₁₃₁	—	4	2	no	—
40	C₄⋊C₄×C₁₃₁	—	4	2	no	—
41	C₄⋊C₄×C₁₃₁	D₈	1	1	yes	characteristic
42	C₄²×C₁₃₁	—	2	1	no	—
43	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₁₃₁	—	2	1	no	—
45	C₁₆×C₁₃₁	—	4	1	no	—
46	C₁₆×C₁₃₁	Q₁₆	1	2	yes	complemented
47	Q₈×C₁₃₁	—	2	1	no	—
48	Q₈×C₁₃₁	—	8	2	no	—
49	Q₈×C₁₃₁	—	8	2	no	—
50	Q₈×C₁₃₁	D₁₆	1	2	yes	—
51	C₂×C₄×C₁₃₁	—	2	1	no	—
52	C₂×C₄×C₁₃₁	—	2	1	no	—
53	C₂×C₄×C₁₃₁	—	8	2	no	—
54	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
55	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
57	C₈×C₁₃₁	—	2	1	no	—
58	C₈×C₁₃₁	—	4	1	no	—
59	C₈×C₁₃₁	—	8	2	no	—
60	C₈×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₈×C₁₃₁	#137	1	1	yes	characteristic
62	C₂²×C₁₃₁	#466	1	1	yes	characteristic
63	C₄×C₁₃₁	—	2	2	no	—
64	C₄×C₁₃₁	—	2	1	no	—
65	C₄×C₁₃₁	—	4	1	no	—
66	C₄×C₁₃₁	—	16	2	no	—
67	C₄×C₁₃₁	#462	1	1	yes	characteristic
68	C₄×C₁₃₁	#473	1	1	yes	characteristic
69	C₄×C₁₃₁	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₁₃₁	#508	1	1	yes	characteristic
71	C₂×C₁₃₁	#1674	1	1	yes	characteristic
72	C₂×C₁₃₁	#2215	1	1	yes	characteristic
73	C₂×C₁₃₁	#2222	1	1	yes	characteristic
74	#12457	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₃₁	#12457	1	1	yes	characteristic, direct product factor, Sylow
76	#2252	C₂×C₁₃₁	1	1	yes	characteristic
77	Q₈×C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
78	#1674	C₂×C₁₃₁	1	2	yes	—
79	#1371	C₂×C₁₃₁	1	1	yes	characteristic
80	#1364	C₂×C₁₃₁	1	2	yes	—
81	#500	C₂²×C₁₃₁	1	2	yes	—
82	#462	—	2	2	no	—
83	C₈×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
84	#366	—	2	1	no	—
85	#366	—	2	1	no	—
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
90	#335	C₂²×C₁₃₁	1	2	yes	—
91	#332	—	2	2	no	—
92	C₂×Q₁₆	—	4	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄×Q₈	D₄×C₁₃₁	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	D₄×C₁₃₁	1	1	yes	characteristic
97	C₂×C₁₆	D₄×C₁₃₁	1	1	yes	characteristic
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	D₄×C₁₃₁	1	1	yes	characteristic
103	#103	—	4	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	D₈×C₁₃₁	1	1	yes	characteristic
108	Q₁₆	—	8	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
112	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	D₈×C₁₃₁	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	Q₁₆×C₁₃₁	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	8	2	no	—
124	Q₈	—	8	2	no	—
125	Q₈	D₁₆×C₁₃₁	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	8	2	no	—
129	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
130	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
131	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	8	2	no	—
135	C₈	C₂×Q₁₆×C₁₃₁	1	1	yes	characteristic
136	C₈	#137×C₁₃₁	1	1	yes	characteristic
137	C₂²	#466×C₁₃₁	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	4	1	no	—
141	C₄	—	16	2	no	—
142	C₄	#462×C₁₃₁	1	1	yes	characteristic
143	C₄	#473×C₁₃₁	1	1	yes	characteristic
144	C₄	C₂×D₁₆×C₁₃₁	1	1	yes	characteristic
145	C₄	#508×C₁₃₁	1	1	yes	characteristic
146	C₂	#1674×C₁₃₁	1	1	yes	characteristic
147	C₂	#2215×C₁₃₁	1	1	yes	characteristic
148	C₂	#2222×C₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2252×C₁₃₁

C₂

yes

maximal, characteristic

Q₈×C₁₆×C₁₃₁

C₂

yes

maximal, characteristic

#1674×C₁₃₁

C₂

yes

maximal

#1371×C₁₃₁

C₂

yes

maximal, characteristic

#1364×C₁₃₁

C₂

yes

maximal

#500×C₁₃₁

C₂²

yes

—

#462×C₁₃₁

—

C₈×Q₈×C₁₃₁

C₂²

yes

characteristic

#366×C₁₃₁

—

#366×C₁₃₁

—

C₄⋊C₁₆×C₁₃₁

—

C₄⋊C₁₆×C₁₃₁

C₂²

yes

characteristic

C₄×C₁₆×C₁₃₁

—

C₄×C₁₆×C₁₃₁

C₂²

yes

characteristic

#335×C₁₃₁

C₂²

yes

—

#332×C₁₃₁

—

C₂×Q₁₆×C₁₃₁

—

C₄⋊Q₈×C₁₃₁

—

C₄×Q₈×C₁₃₁

D₄

yes

characteristic

C₂×C₁₆×C₁₃₁

—

C₂×C₁₆×C₁₃₁

D₄

yes

characteristic

C₂×C₁₆×C₁₃₁

D₄

yes

characteristic

#107×C₁₃₁

—

#107×C₁₃₁

—

C₄⋊C₈×C₁₃₁

—

C₄⋊C₈×C₁₃₁

—

C₄⋊C₈×C₁₃₁

D₄

yes

characteristic

#103×C₁₃₁

—

C₄×C₈×C₁₃₁

—

C₄×C₈×C₁₃₁

C₂³

yes

characteristic

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

D₈

yes

characteristic

Q₁₆×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃₁

#41

yes

characteristic

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

D₈

yes

characteristic

C₄²×C₁₃₁

—

C₄²×C₁₃₁

C₂×D₄

yes

characteristic

C₁₆×C₁₃₁

—

C₁₆×C₁₃₁

—

C₁₆×C₁₃₁

Q₁₆

yes

complemented

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

D₁₆

yes

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

#121

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃₁

#135

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

C₂×Q₁₆

yes

C₈×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

C₂×D₁₆

yes

C₄×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₁

#12457

yes

characteristic, direct product factor, Sylow

#2252

C₂×C₁₃₁

yes

characteristic

Q₈×C₁₆

C₂×C₁₃₁

yes

characteristic

#1674

C₂×C₁₃₁

yes

—

#1371

C₂×C₁₃₁

yes

characteristic

#1364

C₂×C₁₃₁

yes

—

#500

C₂²×C₁₃₁

yes

—

#462

—

C₈×Q₈

C₂²×C₁₃₁

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²×C₁₃₁

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂²×C₁₃₁

yes

characteristic

#335

C₂²×C₁₃₁

yes

—

#332

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

D₄×C₁₃₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₃₁

yes

characteristic

C₂×C₁₆

D₄×C₁₃₁

yes

characteristic

#107

—

#107

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

D₄×C₁₃₁

yes

characteristic

103

#103

—

104

C₄×C₈

—

105

C₄×C₈

C₂³×C₁₃₁

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

D₈×C₁₃₁

yes

characteristic

108

Q₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂×D₄×C₁₃₁

yes

characteristic

112

C₂×C₈

#41×C₁₃₁

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

D₈×C₁₃₁

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₄×C₁₃₁

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

Q₁₆×C₁₃₁

yes

complemented

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

D₁₆×C₁₃₁

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#121×C₁₃₁

yes

characteristic

130

C₂×C₄

C₂×D₈×C₁₃₁

yes

characteristic

131

C₂×C₄

#135×C₁₃₁

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

C₂×Q₁₆×C₁₃₁

yes

characteristic

136

C₈

#137×C₁₃₁

yes

characteristic

137

C₂²

#466×C₁₃₁

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#462×C₁₃₁

yes

characteristic

143

C₄

#473×C₁₃₁

yes

characteristic

144

C₄

C₂×D₁₆×C₁₃₁

yes

characteristic

145

C₄

#508×C₁₃₁

yes

characteristic

146

C₂

#1674×C₁₃₁

yes

characteristic

147

C₂

#2215×C₁₃₁

yes

characteristic

148

C₂

#2222×C₁₃₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₁	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃₁
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₄²×C₁₃₁ ◁ C₈×Q₈×C₁₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₃₁

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₃₁

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₁ ◁ C₄²×C₁₃₁ ◁ C₈×Q₈×C₁₃₁ ◁ G

N	H ≅ G/N	type
46. C₁₆×C₁₃₁	Q₁₆	semidirect
74. #12457	C₁₃₁	direct (N × H)
75. C₁₃₁	#12457	direct (N × H)
121. C₁₆	Q₁₆×C₁₃₁	semidirect

H ≅ G/N

type

46. C₁₆×C₁₃₁

Q₁₆

semidirect

74. #12457

C₁₃₁

direct (N × H)

75. C₁₃₁

#12457

direct (N × H)

121. C₁₆

Q₁₆×C₁₃₁

semidirect

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

subgroup	index
#2252×C₁₃₁	2
Q₈×C₁₆×C₁₃₁	2
#1674×C₁₃₁	2
#1371×C₁₃₁	2
#1364×C₁₃₁	2
#12457	131

subgroup

index

#2252×C₁₃₁

Q₈×C₁₆×C₁₃₁

#1674×C₁₃₁

#1371×C₁₃₁

#1364×C₁₃₁

#12457

131

Subgroups — C₁₃₁×#12457 #45698341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C131×#12457 #45698341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#12457 #45698341