Group #16841925 — subgroups

Subgroups — C₃₁×#1992 #16841925

Order 3968.

260 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 128 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#384	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#340	C₂	1	4	yes	maximal
5	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
6	C₃₁×#125	—	2	2	no	—
7	C₃₁×#125	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
9	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#97	—	2	2	no	—
14	C₂²×C₄×C₃₁	—	2	2	no	—
15	C₂²×C₄×C₃₁	—	2	2	no	—
16	C₂²×C₄×C₃₁	D₄	1	2	yes	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₃₁	—	2	2	no	—
19	C₂×C₈×C₃₁	—	2	2	no	—
20	C₂×C₈×C₃₁	Q₈	1	4	yes	—
21	C₄⋊C₄×C₃₁	—	2	8	no	—
22	C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₄⋊C₄×C₃₁	—	2	4	no	—
24	C₄⋊C₄×C₃₁	—	4	4	no	—
25	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
26	C₄²×C₃₁	C₂×C₄	1	2	yes	—
27	C₄²×C₃₁	D₄	1	2	yes	—
28	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁	—	2	2	no	—
30	C₂×C₄×C₃₁	—	2	2	no	—
31	C₂×C₄×C₃₁	—	2	4	no	—
32	C₂×C₄×C₃₁	—	4	2	no	—
33	C₂×C₄×C₃₁	—	4	2	no	—
34	C₂×C₄×C₃₁	—	4	2	no	—
35	C₂×C₄×C₃₁	—	4	4	no	—
36	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₃₁	#41	1	2	yes	—
41	C₈×C₃₁	—	2	4	no	—
42	#1992	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₃₁	#124	1	1	yes	characteristic
45	C₂²×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
46	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
47	C₂²×C₃₁	#137	1	2	yes	—
48	C₄×C₃₁	—	2	4	no	—
49	C₄×C₃₁	—	4	4	no	—
50	C₄×C₃₁	—	8	2	no	—
51	C₄×C₃₁	C₄×Q₈	1	2	yes	—
52	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
53	C₂×#125	C₂×C₃₁	1	1	yes	characteristic
54	C₂×#97	C₂×C₃₁	1	1	yes	characteristic
55	#384	C₂×C₃₁	1	1	yes	characteristic
56	#340	C₂×C₃₁	1	4	yes	—
57	C₂×C₃₁	#391	1	1	yes	characteristic
58	C₂×C₃₁	#417	1	2	yes	—
59	C₂×C₃₁	#490	1	1	yes	characteristic
60	C₂×C₃₁	#491	1	1	yes	characteristic
61	C₂×C₃₁	#501	1	2	yes	—
62	C₂²×C₈	C₂²×C₃₁	1	2	yes	—
63	#125	—	2	2	no	—
64	#125	C₄×C₃₁	1	4	yes	complemented
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
68	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
69	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
70	#97	—	2	2	no	—
71	C₃₁	#1992	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	D₄×C₃₁	1	2	yes	—
75	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Q₈×C₃₁	1	4	yes	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	4	4	no	—
83	C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
84	C₄²	C₂×C₄×C₃₁	1	2	yes	—
85	C₄²	D₄×C₃₁	1	2	yes	—
86	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
95	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
96	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
97	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
98	C₂×C₄	#41×C₃₁	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₂²	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
101	C₂²	C₃₁×#124	1	1	yes	characteristic
102	C₂²	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
103	C₂²	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
104	C₂²	C₃₁×#137	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	8	2	no	—
108	C₄	C₄×Q₈×C₃₁	1	2	yes	—
109	C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
110	C₂	C₃₁×#391	1	1	yes	characteristic
111	C₂	C₃₁×#417	1	2	yes	—
112	C₂	C₃₁×#490	1	1	yes	characteristic
113	C₂	C₃₁×#491	1	1	yes	characteristic
114	C₂	C₃₁×#501	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#125

C₂

yes

maximal, characteristic

C₂×C₃₁×#97

C₂

yes

maximal, characteristic

C₃₁×#384

C₂

yes

maximal, characteristic

C₃₁×#340

C₂

yes

maximal

C₂²×C₈×C₃₁

C₂²

yes

—

C₃₁×#125

—

C₃₁×#125

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₃₁×#97

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

Q₈

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₈×C₃₁

—

#1992

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

#124

yes

characteristic

C₂²×C₃₁

C₄⋊Q₈

yes

characteristic

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₂×#125

C₂×C₃₁

yes

characteristic

C₂×#97

C₂×C₃₁

yes

characteristic

#384

C₂×C₃₁

yes

characteristic

#340

C₂×C₃₁

yes

—

C₂×C₃₁

#391

yes

characteristic

C₂×C₃₁

#417

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂²×C₈

C₂²×C₃₁

yes

—

#125

—

#125

C₄×C₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₃₁

yes

—

C₂×C₄⋊C₄

C₂²×C₃₁

yes

—

C₂×C₄²

C₂²×C₃₁

yes

characteristic

#97

—

C₃₁

#1992

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₃₁

yes

—

C₂²×C₄

C₂³×C₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₈×C₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₃₁

yes

—

C₄²

C₂×C₄×C₃₁

yes

—

C₄²

D₄×C₃₁

yes

—

C₂³

C₂×D₄×C₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₃₁

yes

—

C₂×C₄

C₂²×C₄×C₃₁

yes

characteristic

C₂×C₄

C₂×D₄×C₃₁

yes

characteristic

C₂×C₄

C₂×Q₈×C₃₁

yes

—

C₂×C₄

#41×C₃₁

yes

—

C₈

—

100

C₂²

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

101

C₂²

C₃₁×#124

yes

characteristic

102

C₂²

C₃₁×C₄⋊Q₈

yes

characteristic

103

C₂²

C₃₁×AGL(1,ℤ/8)

yes

—

104

C₂²

C₃₁×#137

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

C₄×Q₈×C₃₁

yes

—

109

C₄

C₃₁×C₂²⋊Q₈

yes

—

110

C₂

C₃₁×#391

yes

characteristic

111

C₂

C₃₁×#417

yes

—

112

C₂

C₃₁×#490

yes

characteristic

113

C₂

C₃₁×#491

yes

characteristic

114

C₂

C₃₁×#501

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₁

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₃₁

Frattini, cosocle

C₂²×C₄

C₂³×C₃₁

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
7. C₃₁×#125	C₄	semidirect
42. #1992	C₃₁	direct (N × H)
64. #125	C₄×C₃₁	semidirect
71. C₃₁	#1992	direct (N × H)

H ≅ G/N

type

7. C₃₁×#125

C₄

semidirect

42. #1992

C₃₁

direct (N × H)

64. #125

C₄×C₃₁

semidirect

71. C₃₁

#1992

direct (N × H)

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

subgroup	index
C₂×C₃₁×#125	2
C₂×C₃₁×#97	2
C₃₁×#384	2
C₃₁×#340	2
#1992	31

subgroup

index

C₂×C₃₁×#125

C₂×C₃₁×#97

C₃₁×#384

C₃₁×#340

#1992

Subgroups — C₃₁×#1992 #16841925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#1992 #16841925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#1992 #16841925