Group #21469639 — subgroups

Subgroups — C₃₁×#12880 #21469639

Order 7936.

322 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 182 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	C₂×C₃₁×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1374	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₃₁	C₂²	1	4	yes	—
5	C₂×C₃₁×#97	C₄	1	2	yes	—
6	C₂×C₃₁×#97	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#376	C₄	1	1	yes	characteristic
10	C₃₁×C₄⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₃₁	—	2	4	no	—
17	C₂×C₁₆×C₃₁	—	2	8	no	—
18	C₃₁×#97	—	2	4	no	—
19	C₃₁×#97	—	2	2	no	—
20	C₄×C₈×C₃₁	—	2	4	no	—
21	C₄×C₈×C₃₁	D₄	1	2	yes	—
22	C₄×C₈×C₃₁	Q₈	1	2	yes	—
23	C₃₁×#95	C₈	1	4	yes	—
24	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
25	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
26	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₃₁	—	2	2	no	—
28	C₂×C₈×C₃₁	—	2	2	no	—
29	C₂×C₈×C₃₁	—	2	2	no	—
30	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
32	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₃₁	#41	1	4	yes	—
35	C₄²×C₃₁	—	2	2	no	—
36	C₄²×C₃₁	—	2	2	no	—
37	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
38	C₄²×C₃₁	M₁₆	1	2	yes	—
39	C₁₆×C₃₁	—	4	4	no	—
40	#12880	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
42	C₂×C₄×C₃₁	—	2	4	no	—
43	C₂×C₄×C₃₁	—	2	4	no	—
44	C₂×C₄×C₃₁	—	2	8	no	—
45	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
46	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	#117	1	2	yes	—
48	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
49	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
50	C₈×C₃₁	—	2	2	no	—
51	C₈×C₃₁	—	2	4	no	—
52	C₈×C₃₁	#124	1	2	yes	—
53	C₈×C₃₁	#125	1	2	yes	—
54	C₂×C₄⋊C₁₆	C₂×C₃₁	1	2	yes	—
55	#1726	C₂×C₃₁	1	1	yes	characteristic
56	#1374	C₂×C₃₁	1	4	yes	—
57	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₂²×C₃₁	#431	1	2	yes	—
59	C₂²×C₃₁	#504	1	4	yes	—
60	C₄×C₃₁	—	2	4	no	—
61	C₄×C₃₁	—	4	4	no	—
62	C₄×C₃₁	#382	1	1	yes	characteristic
63	C₄×C₃₁	#423	1	1	yes	characteristic
64	C₄×C₃₁	#432	1	2	yes	—
65	C₂²×C₁₆	C₂²×C₃₁	1	4	yes	—
66	C₂×#97	C₄×C₃₁	1	2	yes	—
67	C₂×#97	C₄×C₃₁	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
69	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
70	#376	C₄×C₃₁	1	1	yes	characteristic
71	C₄⋊C₁₆	—	2	4	no	—
72	C₂×C₃₁	#1831	1	1	yes	characteristic
73	C₂×C₃₁	#2151	1	2	yes	—
74	C₂×C₃₁	#2164	1	4	yes	—
75	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
76	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
77	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
78	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
79	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
80	C₂×C₁₆	—	2	4	no	—
81	C₂×C₁₆	—	2	8	no	—
82	#97	—	2	4	no	—
83	#97	—	2	2	no	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	D₄×C₃₁	1	2	yes	—
86	C₄×C₈	Q₈×C₃₁	1	2	yes	—
87	#95	C₈×C₃₁	1	4	yes	—
88	C₃₁	#12880	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
90	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
91	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
96	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
97	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
98	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
99	C₂×C₈	#41×C₃₁	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
103	C₄²	M₁₆×C₃₁	1	2	yes	—
104	C₁₆	—	4	4	no	—
105	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	8	no	—
109	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
111	C₂×C₄	C₃₁×#117	1	2	yes	—
112	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
113	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	C₃₁×#124	1	2	yes	—
117	C₈	C₃₁×#125	1	2	yes	—
118	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
119	C₂²	C₃₁×#431	1	2	yes	—
120	C₂²	C₃₁×#504	1	4	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	C₃₁×#382	1	1	yes	characteristic
124	C₄	C₃₁×#423	1	1	yes	characteristic
125	C₄	C₃₁×#432	1	2	yes	—
126	C₂	C₃₁×#1831	1	1	yes	characteristic
127	C₂	C₃₁×#2151	1	2	yes	—
128	C₂	C₃₁×#2164	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×C₄⋊C₁₆

C₂

yes

maximal

C₃₁×#1726

C₂

yes

maximal, characteristic

C₃₁×#1374

C₂

yes

maximal

C₂²×C₁₆×C₃₁

C₂²

yes

—

C₂×C₃₁×#97

C₄

yes

—

C₂×C₃₁×#97

C₄

yes

characteristic

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#376

C₄

yes

characteristic

C₃₁×C₄⋊C₁₆

—

C₂²×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₃₁

C₂×C₄

yes

—

C₂²×C₈×C₃₁

C₂³

yes

characteristic

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

—

C₃₁×#97

—

C₃₁×#97

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

D₄

yes

—

C₄×C₈×C₃₁

Q₈

yes

—

C₃₁×#95

C₈

yes

—

C₂²×C₄×C₃₁

C₂×C₈

yes

—

C₂²×C₄×C₃₁

C₂×C₈

yes

—

C₂²×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₂×C₈×C₃₁

M₁₆

yes

—

C₂×C₈×C₃₁

C₂×D₄

yes

characteristic

C₂×C₈×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₃₁

#41

yes

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₄⋊C₄

yes

—

C₄²×C₃₁

M₁₆

yes

—

C₁₆×C₃₁

—

#12880

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₈

yes

—

C₂×C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

#117

yes

—

C₂×C₄×C₃₁

C₂×M₁₆

yes

—

C₂×C₄×C₃₁

C₂×M₁₆

yes

characteristic

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#124

yes

—

C₈×C₃₁

#125

yes

—

C₂×C₄⋊C₁₆

C₂×C₃₁

yes

—

#1726

C₂×C₃₁

yes

characteristic

#1374

C₂×C₃₁

yes

—

C₂²×C₃₁

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₃₁

#431

yes

—

C₂²×C₃₁

#504

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂²×C₁₆

C₂²×C₃₁

yes

—

C₂×#97

C₄×C₃₁

yes

—

C₂×#97

C₄×C₃₁

yes

characteristic

C₂×C₄×C₈

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

#376

C₄×C₃₁

yes

characteristic

C₄⋊C₁₆

—

C₂×C₃₁

#1831

yes

characteristic

C₂×C₃₁

#2151

yes

—

C₂×C₃₁

#2164

yes

—

C₂²×C₈

C₂×C₄×C₃₁

yes

characteristic

C₂²×C₈

C₂×C₄×C₃₁

yes

—

C₂²×C₈

C₂³×C₃₁

yes

characteristic

C₂×C₄²

C₂×C₄×C₃₁

yes

—

C₂×C₄²

C₂×C₄×C₃₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

D₄×C₃₁

yes

—

C₄×C₈

Q₈×C₃₁

yes

—

#95

C₈×C₃₁

yes

—

C₃₁

#12880

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₂×C₈×C₃₁

yes

—

C₂²×C₄

C₂×C₈×C₃₁

yes

—

C₂²×C₄

C₂²×C₄×C₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₂×C₈

M₁₆×C₃₁

yes

—

C₂×C₈

C₂×D₄×C₃₁

yes

characteristic

C₂×C₈

C₂×Q₈×C₃₁

yes

characteristic

C₂×C₈

#41×C₃₁

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₄⋊C₄×C₃₁

yes

—

103

C₄²

M₁₆×C₃₁

yes

—

104

C₁₆

—

105

C₂³

C₂²×C₈×C₃₁

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₃₁×C₄⋊C₈

yes

—

110

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

characteristic

111

C₂×C₄

C₃₁×#117

yes

—

112

C₂×C₄

C₂×M₁₆×C₃₁

yes

—

113

C₂×C₄

C₂×M₁₆×C₃₁

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

C₃₁×#124

yes

—

117

C₈

C₃₁×#125

yes

—

118

C₂²

C₂×C₃₁×C₄⋊C₈

yes

characteristic

119

C₂²

C₃₁×#431

yes

—

120

C₂²

C₃₁×#504

yes

—

121

C₄

—

122

C₄

—

123

C₄

C₃₁×#382

yes

characteristic

124

C₄

C₃₁×#423

yes

characteristic

125

C₄

C₃₁×#432

yes

—

126

C₂

C₃₁×#1831

yes

characteristic

127

C₂

C₃₁×#2151

yes

—

128

C₂

C₃₁×#2164

yes

—

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

subgroup

G/H

center

C₂²×C₈×C₃₁

C₂³

commutator

C₂³

C₂²×C₈×C₃₁

Frattini, cosocle

C₂²×C₈

C₂³×C₃₁

socle, prime radical

C₂³×C₃₁

C₂²×C₈

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
40. #12880	C₃₁	direct (N × H)
88. C₃₁	#12880	direct (N × H)

H ≅ G/N

type

40. #12880

C₃₁

direct (N × H)

88. C₃₁

#12880

direct (N × H)

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

subgroup	index
C₂×C₃₁×C₄⋊C₁₆	2
C₃₁×#1726	2
C₃₁×#1374	2
#12880	31

subgroup

index

C₂×C₃₁×C₄⋊C₁₆

C₃₁×#1726

C₃₁×#1374

#12880

Subgroups — C₃₁×#12880 #21469639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#12880 #21469639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12880 #21469639