Group #16846681 — subgroups

Subgroups — #16846681

Order 3968.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11758391	C₂	1	1	yes	maximal, characteristic
2	#14520919	C₂	1	1	yes	maximal, characteristic
3	#14520811	C₂	1	4	yes	maximal
4	C₂×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#125	—	2	2	no	—
6	C₃₁×#125	C₄	1	4	yes	complemented
7	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	—	2	2	no	—
11	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
12	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
13	#11758391	—	2	2	no	—
14	C₂²×C₄×C₃₁	—	2	2	no	—
15	C₂²×C₄×C₃₁	D₄	1	2	yes	—
16	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₁	—	2	2	no	—
18	C₄⋊C₄×C₃₁	—	2	8	no	—
19	C₄⋊C₄×C₃₁	—	2	2	no	—
20	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
21	C₄²×C₃₁	C₂×C₄	1	2	yes	—
22	C₄²×C₃₁	D₄	1	2	yes	—
23	C₄⋊Dic₃₁	—	2	4	no	—
24	C₄⋊Dic₃₁	—	4	4	no	—
25	C₂×C₃₁⋊C₈	—	2	2	no	—
26	C₂×C₃₁⋊C₈	—	2	2	no	—
27	C₂×C₃₁⋊C₈	Q₈	1	4	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	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₃₁	#41	1	2	yes	—
39	C₂×Dic₃₁	—	4	2	no	—
40	C₂×Dic₃₁	—	4	4	no	—
41	C₃₁⋊C₈	—	2	4	no	—
42	#1992	—	31	1	no	maximal, 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	C₄×Q₈	1	2	yes	—
51	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
52	Dic₃₁	—	8	2	no	—
53	C₂×#125	D₃₁	1	1	yes	characteristic
54	C₂×#97	—	31	1	no	—
55	#384	—	31	1	no	—
56	#340	—	31	4	no	—
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	—	31	2	no	—
63	#125	—	2	2	no	—
64	#125	Dic₃₁	1	4	yes	complemented
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄⋊C₄	—	31	2	no	—
67	C₂×C₄⋊C₄	—	62	2	no	—
68	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
69	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
70	#97	—	62	2	no	—
71	C₃₁	#1992	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	62	2	no	—
74	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
75	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
76	C₂×C₈	—	31	4	no	—
77	C₂×C₈	—	62	2	no	—
78	C₂×C₈	—	62	2	no	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	62	4	no	—
82	C₄⋊C₄	—	124	4	no	—
83	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
84	C₄²	C₂×Dic₃₁	1	2	yes	—
85	C₄²	C₂²⋊D₃₁	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₃₁	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	—	124	2	no	—
93	C₂×C₄	—	124	4	no	—
94	C₂×C₄	C₂²⋊Dic₃₁	1	4	yes	—
95	C₂×C₄	Q₈×D₃₁	1	2	yes	—
96	C₂×C₄	#92499	1	2	yes	—
97	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
99	C₈	—	62	4	no	—
100	C₂²	#11758521	1	1	yes	characteristic
101	C₂²	#11758524	1	1	yes	characteristic
102	C₂²	#11758526	1	2	yes	—
103	C₂²	#11758528	1	2	yes	—
104	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	248	2	no	—
108	C₄	Q₈×Dic₃₁	1	2	yes	—
109	C₄	#11758523	1	2	yes	—
110	C₂	#14520956	1	1	yes	characteristic
111	C₂	#14521050	1	2	yes	—
112	C₂	#14521415	1	1	yes	characteristic
113	C₂	#14521419	1	1	yes	characteristic
114	C₂	#14521459	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758391

C₂

yes

maximal, characteristic

#14520919

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal

C₂×C₃₁×#125

C₂

yes

maximal, characteristic

C₃₁×#125

—

C₃₁×#125

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758391

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

Q₈

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

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

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#1992

—

maximal, 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×Q₈

yes

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

Dic₃₁

—

C₂×#125

D₃₁

yes

characteristic

C₂×#97

—

#384

—

#340

—

C₂×C₃₁

#391

yes

characteristic

C₂×C₃₁

#417

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂²×C₈

—

#125

—

#125

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#97

—

C₃₁

#1992

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₄²

C₂×Dic₃₁

yes

—

C₄²

C₂²⋊D₃₁

yes

—

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

C₂²⋊Dic₃₁

yes

—

C₂×C₄

Q₈×D₃₁

yes

—

C₂×C₄

#92499

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

C₈

—

100

C₂²

#11758521

yes

characteristic

101

C₂²

#11758524

yes

characteristic

102

C₂²

#11758526

yes

—

103

C₂²

#11758528

yes

—

104

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

248

—

108

C₄

Q₈×Dic₃₁

yes

—

109

C₄

#11758523

yes

—

110

C₂

#14520956

yes

characteristic

111

C₂

#14521050

yes

—

112

C₂

#14521415

yes

characteristic

113

C₂

#14521419

yes

characteristic

114

C₂

#14521459

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#125	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#125	D₃₁
hypocenter	C₃₁	#1992
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#125

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
6. C₃₁×#125	C₄	semidirect
64. #125	Dic₃₁	semidirect
71. C₃₁	#1992	semidirect

H ≅ G/N

type

6. C₃₁×#125

C₄

semidirect

64. #125

Dic₃₁

semidirect

71. C₃₁

#1992

semidirect

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

subgroup	index
C₂×#11758391	2
#14520919	2
#14520811	2
C₂×C₃₁×#125	2
#1992	31

subgroup

index

C₂×C₃₁×#125

Subgroups — #16846681

Conjugacy classes of subgroups

Complements

Maximal subgroups