Group #16845839 — subgroups

Subgroups — #16845839

Order 3968.

248 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 141 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	2	yes	maximal
2	#14520812	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
5	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊Dic₃₁	—	2	2	no	—
7	C₂×C₄⋊Dic₃₁	—	2	2	no	—
8	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
9	C₄×C₈×C₃₁	C₄	1	4	yes	complemented
10	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
13	C₂²×Dic₃₁	—	2	4	no	—
14	C₂×C₈×C₃₁	—	2	4	no	—
15	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
16	C₄²×C₃₁	C₂×C₄	1	2	yes	—
17	C₄²×C₃₁	Q₈	1	2	yes	—
18	C₄⋊Dic₃₁	—	2	8	no	—
19	C₄⋊Dic₃₁	—	4	4	no	—
20	C₄⋊Dic₃₁	—	4	4	no	—
21	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
22	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₃₁	D₈	1	2	yes	—
25	C₂×C₄×C₃₁	SD₁₆	1	4	yes	—
26	C₂×C₄×C₃₁	Q₁₆	1	2	yes	—
27	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₄×C₃₁	#41	1	4	yes	—
30	C₂×Dic₃₁	—	4	4	no	—
31	C₂×Dic₃₁	—	4	4	no	—
32	C₂×Dic₃₁	—	4	4	no	—
33	C₈×C₃₁	—	2	4	no	—
34	#1834	—	31	1	no	maximal, Sylow
35	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₃₁	#124	1	2	yes	—
37	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
38	C₂²×C₃₁	C₂×SD₁₆	1	2	yes	—
39	C₂²×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
40	C₄×C₃₁	#106	1	4	yes	—
41	C₄×C₃₁	#107	1	4	yes	—
42	C₄×C₃₁	#117	1	2	yes	—
43	C₄×C₃₁	#125	1	2	yes	—
44	Dic₃₁	—	8	4	no	—
45	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic
46	#384	—	31	2	no	—
47	#340	—	31	4	no	—
48	C₂×C₃₁	#382	1	1	yes	characteristic
49	C₂×C₃₁	C₂×#106	1	1	yes	characteristic
50	C₂×C₃₁	C₂×#107	1	1	yes	characteristic
51	C₂×C₃₁	#486	1	2	yes	—
52	C₂×C₃₁	#487	1	2	yes	—
53	C₂²×C₈	C₂×D₃₁	1	2	yes	—
54	C₂×C₄⋊C₄	—	31	4	no	—
55	C₂×C₄⋊C₄	—	62	2	no	—
56	C₂×C₄⋊C₄	—	62	2	no	—
57	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
58	C₄×C₈	Dic₃₁	1	4	yes	complemented
59	C₃₁	#1834	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₄	—	62	4	no	—
61	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
62	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
63	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	C₂×Dic₃₁	1	4	yes	—
66	C₄⋊C₄	—	62	8	no	—
67	C₄⋊C₄	—	124	4	no	—
68	C₄⋊C₄	—	124	4	no	—
69	C₄²	Dic₆₂	1	2	yes	—
70	C₄²	C₂×Dic₃₁	1	2	yes	—
71	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
72	C₂×C₄	—	124	4	no	—
73	C₂×C₄	—	124	4	no	—
74	C₂×C₄	—	124	4	no	—
75	C₂×C₄	#92470	1	4	yes	—
76	C₂×C₄	D₂₄₈	1	2	yes	—
77	C₂×C₄	Dic₁₂₄	1	2	yes	—
78	C₂×C₄	C₄⋊Dic₃₁	1	2	yes	—
79	C₂×C₄	C₄⋊Dic₃₁	1	2	yes	—
80	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
81	C₂×C₄	#92495	1	4	yes	—
82	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
83	C₈	—	2	4	no	—
84	C₂²	#11758452	1	2	yes	—
85	C₂²	C₂×#92470	1	2	yes	—
86	C₂²	C₂×D₂₄₈	1	1	yes	characteristic
87	C₂²	C₂×Dic₁₂₄	1	1	yes	characteristic
88	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
89	C₄	—	248	4	no	—
90	C₄	#11758405	1	4	yes	—
91	C₄	#11758406	1	4	yes	—
92	C₄	#11758447	1	2	yes	—
93	C₄	#11758503	1	2	yes	—
94	C₂	#14520915	1	1	yes	characteristic
95	C₂	C₂×#11758405	1	1	yes	characteristic
96	C₂	C₂×#11758406	1	1	yes	characteristic
97	C₂	#14521396	1	2	yes	—
98	C₂	#14521400	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14520919

C₂

yes

maximal

#14520812

C₂

yes

maximal

C₂×C₄×C₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₄×C₈×C₃₁

C₄

yes

complemented

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

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

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

D₈

yes

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

—

#1834

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#124

yes

—

C₂²×C₃₁

C₂×D₈

yes

characteristic

C₂²×C₃₁

C₂×SD₁₆

yes

—

C₂²×C₃₁

C₂×Q₁₆

yes

characteristic

C₄×C₃₁

#106

yes

—

C₄×C₃₁

#107

yes

—

C₄×C₃₁

#117

yes

—

C₄×C₃₁

#125

yes

—

Dic₃₁

—

C₂×C₄×C₈

D₃₁

yes

characteristic

#384

—

#340

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#487

yes

—

C₂²×C₈

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄×C₈

Dic₃₁

yes

complemented

C₃₁

#1834

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄⋊C₄

—

124

—

C₄²

Dic₆₂

yes

—

C₄²

C₂×Dic₃₁

yes

—

C₂³

C₂×D₁₂₄

yes

characteristic

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

#92470

yes

—

C₂×C₄

D₂₄₈

yes

—

C₂×C₄

Dic₁₂₄

yes

—

C₂×C₄

C₄⋊Dic₃₁

yes

—

C₂×C₄

C₄⋊Dic₃₁

yes

—

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

#92495

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

C₈

—

C₂²

#11758452

yes

—

C₂²

C₂×#92470

yes

—

C₂²

C₂×D₂₄₈

yes

characteristic

C₂²

C₂×Dic₁₂₄

yes

characteristic

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

C₄

—

248

—

C₄

#11758405

yes

—

C₄

#11758406

yes

—

C₄

#11758447

yes

—

C₄

#11758503

yes

—

C₂

yes

C₂

yes

C₂

yes

C₂

yes

—

C₂

#14521400

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₄×C₈×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄×C₈	D₃₁
hypocenter	C₃₁	#1834
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₄×C₈×C₃₁

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄×C₈

D₃₁

hypocenter

C₃₁

#1834

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₄×C₈×C₃₁	C₄	semidirect
58. C₄×C₈	Dic₃₁	semidirect
59. C₃₁	#1834	semidirect

H ≅ G/N

type

9. C₄×C₈×C₃₁

C₄

semidirect

58. C₄×C₈

Dic₃₁

semidirect

59. C₃₁

#1834

semidirect

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

subgroup	index
#14520919	2
#14520812	2
C₂×C₄×C₈×C₃₁	2
#1834	31

subgroup

index

#14520919

#14520812

C₂×C₄×C₈×C₃₁

#1834

Subgroups — #16845839

Conjugacy classes of subgroups

Complements

Maximal subgroups