Group #16845845 — subgroups

Subgroups — #16845845

Order 3968.

224 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 109 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	#14520919	C₂	1	1	yes	maximal, characteristic
2	#14520915	C₂	1	1	yes	maximal, characteristic
3	#14520812	C₂	1	4	yes	maximal
4	C₂×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
6	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₃₁	—	2	2	no	—
8	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
10	C₃₁×#97	C₄	1	4	yes	complemented
11	C₃₁⋊#95	—	2	2	no	—
12	C₂²×C₄×C₃₁	D₄	1	2	yes	—
13	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
14	C₂²×Dic₃₁	—	2	2	no	—
15	C₂²×Dic₃₁	—	2	2	no	—
16	C₂×C₈×C₃₁	—	2	2	no	—
17	C₂×C₈×C₃₁	—	2	2	no	—
18	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
19	C₄²×C₃₁	C₂×C₄	1	2	yes	—
20	C₄²×C₃₁	Q₈	1	2	yes	—
21	C₄⋊Dic₃₁	—	2	4	no	—
22	C₄⋊Dic₃₁	—	2	4	no	—
23	C₄⋊Dic₃₁	—	4	4	no	—
24	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₃₁	—	2	2	no	—
26	C₂×C₄×C₃₁	—	2	2	no	—
27	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₃₁	#41	1	2	yes	—
31	C₂×C₄×C₃₁	#41	1	2	yes	—
32	C₂×Dic₃₁	—	4	2	no	—
33	C₂×Dic₃₁	—	4	2	no	—
34	C₂×Dic₃₁	—	4	4	no	—
35	C₂×Dic₃₁	—	4	2	no	—
36	C₂×Dic₃₁	—	4	2	no	—
37	C₈×C₃₁	—	2	4	no	—
38	#1836	—	31	1	no	maximal, Sylow
39	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₃₁	#124	1	1	yes	characteristic
41	C₂²×C₃₁	#124	1	1	yes	characteristic
42	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
43	C₂²×C₃₁	#137	1	2	yes	—
44	C₄×C₃₁	—	2	4	no	—
45	C₄×C₃₁	#117	1	2	yes	—
46	C₄×C₃₁	#125	1	2	yes	—
47	Dic₃₁	—	8	2	no	—
48	Dic₃₁	—	8	2	no	—
49	C₂×#97	D₃₁	1	1	yes	characteristic
50	#384	—	31	1	no	—
51	#382	—	31	1	no	—
52	#340	—	31	4	no	—
53	C₂×C₃₁	#382	1	1	yes	characteristic
54	C₂×C₃₁	#428	1	2	yes	—
55	C₂×C₃₁	#489	1	2	yes	—
56	C₂×C₃₁	#490	1	1	yes	characteristic
57	C₂×C₃₁	#491	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₃₁	1	2	yes	—
59	C₂×C₄⋊C₄	—	31	2	no	—
60	C₂×C₄⋊C₄	—	31	2	no	—
61	C₂×C₄⋊C₄	—	62	2	no	—
62	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
63	#97	Dic₃₁	1	4	yes	complemented
64	#95	—	62	2	no	—
65	C₃₁	#1836	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₄	—	62	2	no	—
67	C₂²×C₄	—	62	2	no	—
68	C₂²×C₄	D₁₂₄	1	2	yes	—
69	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×Dic₃₁	1	4	yes	—
73	C₄⋊C₄	—	62	4	no	—
74	C₄⋊C₄	—	62	4	no	—
75	C₄⋊C₄	—	124	4	no	—
76	C₄²	Dic₆₂	1	2	yes	—
77	C₄²	C₂×Dic₃₁	1	2	yes	—
78	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	124	2	no	—
82	C₂×C₄	—	124	2	no	—
83	C₂×C₄	—	124	4	no	—
84	C₂×C₄	—	124	2	no	—
85	C₂×C₄	—	124	2	no	—
86	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
87	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
88	C₂×C₄	#92495	1	2	yes	—
89	C₂×C₄	#92495	1	2	yes	—
90	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
91	C₈	—	2	4	no	—
92	C₂²	#11758452	1	1	yes	characteristic
93	C₂²	#11758452	1	1	yes	characteristic
94	C₂²	#11758485	1	2	yes	—
95	C₂²	#11758486	1	2	yes	—
96	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
97	C₄	—	2	4	no	—
98	C₄	—	248	2	no	—
99	C₄	—	248	2	no	—
100	C₄	#11758447	1	2	yes	—
101	C₄	#11758503	1	2	yes	—
102	C₂	#14520915	1	1	yes	characteristic
103	C₂	#14521091	1	2	yes	—
104	C₂	#14521409	1	2	yes	—
105	C₂	#14521414	1	1	yes	characteristic
106	C₂	#14521418	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14520919

C₂

yes

maximal, characteristic

#14520915

C₂

yes

maximal, characteristic

#14520812

C₂

yes

maximal

C₂×C₃₁×#97

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₃₁×#97

C₄

yes

complemented

C₃₁⋊#95

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

Q₈

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

—

#1836

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

#117

yes

—

C₄×C₃₁

#125

yes

—

Dic₃₁

—

Dic₃₁

—

C₂×#97

D₃₁

yes

—

—

—

C₂×C₃₁

#382

yes

characteristic

C₂×C₃₁

#428

yes

—

C₂×C₃₁

#489

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×C₈

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#97

Dic₃₁

yes

complemented

#95

—

C₃₁

#1836

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄²

Dic₆₂

yes

—

C₄²

C₂×Dic₃₁

yes

—

C₂³

C₂×D₁₂₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

C₄⋊Dic₃₁

yes

—

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

#92495

yes

—

C₂×C₄

#92495

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#11758486

yes

—

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

C₄

—

C₄

—

248

—

C₄

—

248

—

100

C₄

#11758447

yes

—

101

C₄

#11758503

yes

—

102

C₂

#14520915

yes

characteristic

103

C₂

#14521091

yes

—

104

C₂

#14521409

yes

—

105

C₂

#14521414

yes

characteristic

106

C₂

#14521418

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#97	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#97	D₃₁
hypocenter	C₃₁	#1836
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#97

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×#97

D₃₁

hypocenter

C₃₁

#1836

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
10. C₃₁×#97	C₄	semidirect
63. #97	Dic₃₁	semidirect
65. C₃₁	#1836	semidirect

H ≅ G/N

type

10. C₃₁×#97

C₄

semidirect

63. #97

Dic₃₁

semidirect

65. C₃₁

#1836

semidirect

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

subgroup	index
#14520919	2
#14520915	2
#14520812	2
C₂×C₃₁×#97	2
#1836	31

subgroup

index

C₂×C₃₁×#97

Subgroups — #16845845

Conjugacy classes of subgroups

Complements

Maximal subgroups