Group #21468820 — subgroups

Subgroups — C₃₁×#12061 #21468820

Order 7936.

200 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₃₁×#2183	C₂	1	2	yes	maximal
2	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1367	C₂	1	1	yes	maximal, characteristic
4	C₃₁×Q₈⋊C₁₆	C₂	1	2	yes	maximal
5	C₃₁×#1312	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
7	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
8	C₃₁×C₄⋊C₁₆	—	2	2	no	—
9	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₃₁×#346	—	2	2	no	—
11	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#335	C₄	1	1	yes	characteristic
14	C₃₁×Q₈⋊C₈	C₄	1	2	yes	—
15	C₈²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₃₁	C₈	1	1	yes	characteristic
17	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₃₁	—	2	2	no	—
19	C₂×C₁₆×C₃₁	—	2	1	no	—
20	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
21	C₃₁×#107	C₈	1	1	yes	characteristic
22	C₃₁×C₄⋊C₈	—	2	2	no	—
23	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
24	C₃₁×#103	C₈	1	2	yes	—
25	C₄×C₈×C₃₁	—	2	2	no	—
26	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₃₁	C₂×C₈	1	2	yes	—
30	Q₁₆×C₃₁	—	2	2	no	—
31	C₂×C₈×C₃₁	—	2	2	no	—
32	C₂×C₈×C₃₁	—	2	1	no	—
33	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
39	C₄²×C₃₁	—	2	2	no	—
40	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₁₆×C₃₁	—	2	2	no	—
42	C₁₆×C₃₁	—	4	1	no	—
43	C₁₆×C₃₁	—	4	2	no	—
44	#12061	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	Q₈×C₃₁	—	2	2	no	—
46	Q₈×C₃₁	—	2	2	no	—
47	C₂×C₄×C₃₁	—	2	2	no	—
48	C₂×C₄×C₃₁	—	2	2	no	—
49	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₃₁	#131	1	1	yes	characteristic
52	C₈×C₃₁	—	2	1	no	—
53	C₈×C₃₁	—	2	1	no	—
54	C₈×C₃₁	—	4	1	no	—
55	C₈×C₃₁	—	4	2	no	—
56	C₈×C₃₁	M₃₂	1	2	yes	—
57	C₈×C₃₁	#137	1	2	yes	—
58	#2183	C₂×C₃₁	1	2	yes	—
59	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
60	#1367	C₂×C₃₁	1	1	yes	characteristic
61	Q₈⋊C₁₆	C₂×C₃₁	1	2	yes	—
62	#1312	C₂×C₃₁	1	1	yes	characteristic
63	C₂²×C₃₁	C₈×D₄	1	1	yes	characteristic
64	C₄×C₃₁	—	2	1	no	—
65	C₄×C₃₁	—	4	2	no	—
66	C₄×C₃₁	—	4	2	no	—
67	C₄×C₃₁	#441	1	1	yes	characteristic
68	C₄×C₃₁	#444	1	1	yes	characteristic
69	C₄×C₃₁	C₂×M₃₂	1	1	yes	characteristic
70	C₄×C₃₁	#504	1	1	yes	characteristic
71	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
72	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
73	C₄⋊C₁₆	—	2	2	no	—
74	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
75	#346	—	2	2	no	—
76	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
77	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
78	#335	C₄×C₃₁	1	1	yes	characteristic
79	Q₈⋊C₈	C₄×C₃₁	1	2	yes	—
80	C₈²	C₂²×C₃₁	1	1	yes	characteristic
81	C₂×C₃₁	#1579	1	1	yes	characteristic
82	C₂×C₃₁	#2169	1	1	yes	characteristic
83	C₂×C₃₁	#2171	1	1	yes	characteristic
84	C₂×Q₁₆	C₈×C₃₁	1	1	yes	characteristic
85	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
89	#107	C₈×C₃₁	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
92	#103	C₈×C₃₁	1	2	yes	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
95	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
96	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
97	C₃₁	#12061	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×Q₈	C₂×C₈×C₃₁	1	2	yes	—
99	Q₁₆	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
103	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
105	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	—	4	2	no	—
113	Q₈	—	2	2	no	—
114	Q₈	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
119	C₂×C₄	C₃₁×#131	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	4	2	no	—
124	C₈	C₃₁×M₃₂	1	2	yes	—
125	C₈	C₃₁×#137	1	2	yes	—
126	C₂²	C₈×D₄×C₃₁	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	C₃₁×#441	1	1	yes	characteristic
131	C₄	C₃₁×#444	1	1	yes	characteristic
132	C₄	C₂×C₃₁×M₃₂	1	1	yes	characteristic
133	C₄	C₃₁×#504	1	1	yes	characteristic
134	C₂	C₃₁×#1579	1	1	yes	characteristic
135	C₂	C₃₁×#2169	1	1	yes	characteristic
136	C₂	C₃₁×#2171	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2183

C₂

yes

maximal

C₈×Q₁₆×C₃₁

C₂

yes

maximal, characteristic

C₃₁×#1367

C₂

yes

maximal, characteristic

C₃₁×Q₈⋊C₁₆

C₂

yes

maximal

C₃₁×#1312

C₂

yes

maximal, characteristic

C₈×Q₈×C₃₁

C₂²

yes

—

C₄×Q₁₆×C₃₁

C₄

yes

characteristic

C₃₁×C₄⋊C₁₆

—

C₃₁×C₄⋊C₁₆

C₂²

yes

—

C₃₁×#346

—

C₄×C₁₆×C₃₁

C₂²

yes

characteristic

C₄×C₁₆×C₃₁

C₂²

yes

characteristic

C₃₁×#335

C₄

yes

characteristic

C₃₁×Q₈⋊C₈

C₄

yes

—

C₈²×C₃₁

C₂²

yes

characteristic

C₂×Q₁₆×C₃₁

C₈

yes

characteristic

C₄×Q₈×C₃₁

C₂×C₄

yes

—

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

D₄

yes

—

C₃₁×#107

C₈

yes

characteristic

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

C₂×C₄

yes

—

C₃₁×#103

C₈

yes

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

C₂×C₄

yes

characteristic

C₄×C₈×C₃₁

C₂×C₄

yes

characteristic

C₄×C₈×C₃₁

C₂³

yes

characteristic

C₂×Q₈×C₃₁

C₂×C₈

yes

—

Q₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₈

yes

characteristic

C₂×C₈×C₃₁

C₂×C₈

yes

characteristic

C₂×C₈×C₃₁

C₂×D₄

yes

characteristic

C₂×C₈×C₃₁

#41

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₈

yes

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂²×C₄

yes

characteristic

C₁₆×C₃₁

—

C₁₆×C₃₁

—

C₁₆×C₃₁

—

#12061

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₃₁

#131

yes

characteristic

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

M₃₂

yes

—

C₈×C₃₁

#137

yes

—

#2183

C₂×C₃₁

yes

—

C₈×Q₁₆

C₂×C₃₁

yes

characteristic

#1367

C₂×C₃₁

yes

characteristic

Q₈⋊C₁₆

C₂×C₃₁

yes

—

#1312

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

C₈×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

C₂×M₃₂

yes

characteristic

C₄×C₃₁

#504

yes

characteristic

C₈×Q₈

C₂²×C₃₁

yes

—

C₄×Q₁₆

C₄×C₃₁

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²×C₃₁

yes

—

#346

—

C₄×C₁₆

C₂²×C₃₁

yes

characteristic

C₄×C₁₆

C₂²×C₃₁

yes

characteristic

#335

C₄×C₃₁

yes

characteristic

Q₈⋊C₈

C₄×C₃₁

yes

—

C₈²

C₂²×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×Q₁₆

C₈×C₃₁

yes

characteristic

C₄×Q₈

C₂×C₄×C₃₁

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₃₁

yes

—

#107

C₈×C₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄×C₃₁

yes

—

#103

C₈×C₃₁

yes

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₃₁

yes

characteristic

C₄×C₈

C₂×C₄×C₃₁

yes

characteristic

C₄×C₈

C₂³×C₃₁

yes

characteristic

C₃₁

#12061

yes

characteristic, direct product factor, Sylow

C₂×Q₈

C₂×C₈×C₃₁

yes

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×C₈×C₃₁

yes

characteristic

103

C₂×C₈

C₂×C₈×C₃₁

yes

characteristic

104

C₂×C₈

C₂×D₄×C₃₁

yes

characteristic

105

C₂×C₈

#41×C₃₁

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×C₈×C₃₁

yes

—

108

C₄²

—

109

C₄²

C₂²×C₄×C₃₁

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₄×D₄×C₃₁

yes

characteristic

118

C₂×C₄

C₂²×C₈×C₃₁

yes

characteristic

119

C₂×C₄

C₃₁×#131

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₃₁×M₃₂

yes

—

125

C₈

C₃₁×#137

yes

—

126

C₂²

C₈×D₄×C₃₁

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

C₃₁×#441

yes

characteristic

131

C₄

C₃₁×#444

yes

characteristic

132

C₄

C₂×C₃₁×M₃₂

yes

characteristic

133

C₄

C₃₁×#504

yes

characteristic

134

C₂

C₃₁×#1579

yes

characteristic

135

C₂

C₃₁×#2169

yes

characteristic

136

C₂

C₃₁×#2171

yes

characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle, prime radical	C₂²×C₃₁	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₄×C₁₆×C₃₁ ◁ G

subgroup

G/H

center

C₂×C₈×C₃₁

C₂×D₄

commutator

C₂×C₄

C₂²×C₈×C₃₁

Frattini, cosocle

C₄×C₈

C₂³×C₃₁

socle, prime radical

C₂²×C₃₁

C₈×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₃₁ ◁ C₄×C₁₆×C₃₁ ◁ G

N	H ≅ G/N	type
44. #12061	C₃₁	direct (N × H)
97. C₃₁	#12061	direct (N × H)

H ≅ G/N

type

44. #12061

C₃₁

direct (N × H)

97. C₃₁

#12061

direct (N × H)

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

subgroup	index
C₃₁×#2183	2
C₈×Q₁₆×C₃₁	2
C₃₁×#1367	2
C₃₁×Q₈⋊C₁₆	2
C₃₁×#1312	2
#12061	31

subgroup

index

C₃₁×#2183

C₈×Q₁₆×C₃₁

C₃₁×#1367

C₃₁×Q₈⋊C₁₆

C₃₁×#1312

#12061

Subgroups — C₃₁×#12061 #21468820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#12061 #21468820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12061 #21468820