Group #16845861 — subgroups

Subgroups — #16845861

Order 3968.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic
2	#14520821	C₂	1	2	yes	maximal
3	#14520810	C₂	1	2	yes	maximal
4	#14520806	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#338	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	#11758503	C₂²	1	2	yes	—
9	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
10	C₂×#92474	C₂²	1	1	yes	characteristic
11	#11758403	C₄	1	2	yes	complemented
12	C₈×Dic₃₁	C₄	1	2	yes	complemented
13	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
14	C₂²×Dic₃₁	—	2	1	no	—
15	M₁₆×C₃₁	—	2	2	no	—
16	M₁₆×C₃₁	—	2	2	no	—
17	M₁₆×C₃₁	—	2	2	no	—
18	C₂×C₈×C₃₁	—	2	2	no	—
19	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²⋊Dic₃₁	—	2	2	no	—
21	C₄⋊Dic₃₁	—	2	2	no	—
22	C₄×Dic₃₁	—	2	1	no	—
23	C₄×Dic₃₁	—	2	2	no	—
24	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
26	#92474	—	2	2	no	—
27	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
28	C₂³×C₃₁	#41	1	1	yes	characteristic
29	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₃₁	#41	1	2	yes	—
31	C₂×C₄×C₃₁	#41	1	2	yes	—
32	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
33	C₂×Dic₃₁	—	2	1	no	—
34	C₂×Dic₃₁	—	2	1	no	—
35	C₂×Dic₃₁	—	4	2	no	—
36	C₂×Dic₃₁	—	4	2	no	—
37	C₈×C₃₁	—	2	2	no	—
38	C₈×C₃₁	—	4	2	no	—
39	C₃₁⋊C₈	—	2	2	no	—
40	#1841	—	31	1	no	maximal, Sylow
41	C₂²×C₃₁	—	4	1	no	—
42	C₂²×C₃₁	#117	1	1	yes	characteristic
43	C₂²×C₃₁	#126	1	2	yes	—
44	C₄×C₃₁	#117	1	1	yes	characteristic
45	C₄×C₃₁	#117	1	1	yes	characteristic
46	C₄×C₃₁	#126	1	2	yes	—
47	Dic₃₁	—	2	2	no	—
48	Dic₃₁	—	4	1	no	—
49	Dic₃₁	—	8	2	no	—
50	C₄×M₁₆	—	31	1	no	—
51	#344	—	31	2	no	—
52	#339	—	31	2	no	—
53	#338	D₃₁	1	1	yes	characteristic
54	H(3,ℤ/4)	—	31	1	no	—
55	C₂×C₃₁	—	2	1	no	—
56	C₂×C₃₁	—	2	2	no	—
57	C₂×C₃₁	#383	1	1	yes	characteristic
58	C₂×M₁₆	—	31	1	no	—
59	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
60	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
61	#117	—	31	2	no	—
62	C₂×C₄²	—	31	1	no	—
63	#97	—	31	2	no	—
64	C₄×C₈	—	31	2	no	—
65	C₃₁	#1841	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₄	—	62	1	no	—
67	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	62	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	31	2	no	—
74	C₂×C₈	C₄×D₃₁	1	2	yes	—
75	C₄⋊C₄	—	62	2	no	—
76	C₂²⋊C₄	—	62	2	no	—
77	C₄²	—	31	1	no	—
78	C₄²	—	31	1	no	—
79	C₄²	—	62	1	no	—
80	C₄²	—	62	2	no	—
81	C₂³	#92499	1	1	yes	characteristic
82	C₂×C₄	—	62	1	no	—
83	C₂×C₄	—	62	1	no	—
84	C₂×C₄	—	124	2	no	—
85	C₂×C₄	—	124	2	no	—
86	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
87	C₂×C₄	#92495	1	2	yes	—
88	C₂×C₄	#92497	1	2	yes	—
89	C₂×C₄	#92497	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	62	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#11758469	1	1	yes	characteristic
95	C₂²	#11758475	1	2	yes	—
96	C₄	—	62	2	no	—
97	C₄	—	124	1	no	—
98	C₄	—	248	2	no	—
99	C₄	#11758449	1	1	yes	characteristic
100	C₄	#11758454	1	1	yes	characteristic
101	C₄	#11758456	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#14520916	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₃₁

C₂

yes

maximal, characteristic

#14520821

C₂

yes

maximal

#14520810

C₂

yes

maximal

#14520806

C₂

yes

maximal, characteristic

C₃₁×#338

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×M₁₆×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

characteristic

#11758403

C₄

yes

complemented

C₈×Dic₃₁

C₄

yes

complemented

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

C₂×C₄

yes

characteristic

C₄×Dic₃₁

C₂×C₄

yes

characteristic

#92474

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

#41

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#1841

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

#117

yes

characteristic

C₂²×C₃₁

#126

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

C₄×M₁₆

—

#344

—

#339

—

#338

D₃₁

yes

characteristic

H(3,ℤ/4)

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#383

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂×M₁₆

C₂×D₃₁

yes

characteristic

#117

—

C₂×C₄²

—

#97

—

C₄×C₈

—

C₃₁

#1841

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

#92499

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

C₂×C₄

#92495

yes

—

C₂×C₄

#92497

yes

—

C₂×C₄

#92497

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

#11758469

yes

characteristic

C₂²

#11758475

yes

—

C₄

—

C₄

—

124

—

C₄

—

248

—

C₄

#11758449

yes

characteristic

100

C₄

#11758454

yes

characteristic

101

C₄

#11758456

yes

—

102

C₂

—

103

C₂

—

104

C₂

#14520916

yes

characteristic

	subgroup	G/H
center	C₄	#11758449
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#338	C₂
socle	C₂×C₃₁	#383
prime radical	C₂³×C₃₁	#41
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#338	D₃₁
hypocenter	C₃₁	#1841
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#11758449

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#338

C₂

C₂×C₃₁

C₂³×C₃₁

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
11. #11758403	C₄	semidirect
12. C₈×Dic₃₁	C₄	semidirect
65. C₃₁	#1841	semidirect

H ≅ G/N

type

11. #11758403

C₄

semidirect

12. C₈×Dic₃₁

C₄

semidirect

65. C₃₁

#1841

semidirect

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

subgroup	index
M₁₆×Dic₃₁	2
#14520821	2
#14520810	2
#14520806	2
C₃₁×#338	2
#1841	31

subgroup

index

M₁₆×Dic₃₁

C₃₁×#338

Subgroups — #16845861

Conjugacy classes of subgroups

Complements

Maximal subgroups