Group #24907613 — subgroups

Subgroups — #24907613

Order 10624.

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×Dic₈₃	C₂	1	1	yes	maximal, characteristic
2	#19127473	C₂	1	2	yes	maximal
3	#19127462	C₂	1	2	yes	maximal
4	#19127458	C₂	1	1	yes	maximal, characteristic
5	#338×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	1	yes	characteristic
7	#14581751	C₂²	1	2	yes	—
8	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
9	#14581702	C₄	1	2	yes	complemented
10	C₈×Dic₈₃	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₈₃	—	2	1	no	—
14	#13050574	—	2	2	no	—
15	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₈₃	—	2	2	no	—
17	C₂²⋊Dic₈₃	—	2	2	no	—
18	C₄×Dic₈₃	—	2	1	no	—
19	C₄×Dic₈₃	—	2	2	no	—
20	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₈₃	—	2	2	no	—
24	M₁₆×C₈₃	—	2	2	no	—
25	M₁₆×C₈₃	—	2	2	no	—
26	C₂×C₈×C₈₃	—	2	2	no	—
27	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
28	C₂³×C₈₃	#41	1	1	yes	characteristic
29	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈₃	#41	1	2	yes	—
31	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
32	C₂×C₄×C₈₃	#41	1	2	yes	—
33	C₂×Dic₈₃	—	2	1	no	—
34	C₂×Dic₈₃	—	2	1	no	—
35	C₂×Dic₈₃	—	4	2	no	—
36	C₂×Dic₈₃	—	4	2	no	—
37	C₈×C₈₃	—	2	2	no	—
38	C₈×C₈₃	—	4	2	no	—
39	C₈₃⋊C₈	—	2	2	no	—
40	C₂²×C₈₃	—	4	1	no	—
41	C₂²×C₈₃	#117	1	1	yes	characteristic
42	C₂²×C₈₃	#126	1	2	yes	—
43	C₄×C₈₃	#117	1	1	yes	characteristic
44	C₄×C₈₃	#117	1	1	yes	characteristic
45	C₄×C₈₃	#126	1	2	yes	—
46	Dic₈₃	—	2	2	no	—
47	Dic₈₃	—	4	1	no	—
48	Dic₈₃	—	8	2	no	—
49	C₂×C₈₃	—	2	1	no	—
50	C₂×C₈₃	—	2	2	no	—
51	C₂×C₈₃	#383	1	1	yes	characteristic
52	#1841	—	83	1	no	maximal, Sylow
53	C₈₃	#1841	1	1	yes	characteristic, complemented, Sylow
54	C₄×M₁₆	—	83	1	no	—
55	#344	—	83	2	no	—
56	#339	—	83	2	no	—
57	#338	D₈₃	1	1	yes	characteristic
58	H(3,ℤ/4)	—	83	1	no	—
59	C₂×M₁₆	—	83	1	no	—
60	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
61	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
62	#117	—	83	2	no	—
63	C₂×C₄²	—	83	1	no	—
64	#97	—	83	2	no	—
65	C₄×C₈	—	83	2	no	—
66	C₂²×C₄	—	166	1	no	—
67	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	166	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	83	2	no	—
74	C₂×C₈	C₄×D₈₃	1	2	yes	—
75	C₄⋊C₄	—	166	2	no	—
76	C₂²⋊C₄	—	166	2	no	—
77	C₄²	—	83	1	no	—
78	C₄²	—	83	1	no	—
79	C₄²	—	166	2	no	—
80	C₄²	—	166	1	no	—
81	C₂³	#13050591	1	1	yes	characteristic
82	C₂×C₄	—	332	2	no	—
83	C₂×C₄	—	332	2	no	—
84	C₂×C₄	—	166	1	no	—
85	C₂×C₄	—	166	1	no	—
86	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
87	C₂×C₄	#13050589	1	1	yes	characteristic
88	C₂×C₄	#13050589	1	2	yes	—
89	C₂×C₄	#13050590	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	166	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#14581750	1	1	yes	characteristic
95	C₂²	#14581787	1	2	yes	—
96	C₄	—	332	1	no	—
97	C₄	—	664	2	no	—
98	C₄	—	166	2	no	—
99	C₄	#14581748	1	1	yes	characteristic
100	C₄	#14581749	1	1	yes	characteristic
101	C₄	#14581785	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#19127568	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₈₃

C₂

yes

maximal, characteristic

#19127473

C₂

yes

maximal

#19127462

C₂

yes

maximal

#19127458

C₂

yes

maximal, characteristic

#338×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

characteristic

#14581751

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

#14581702

C₄

yes

complemented

C₈×Dic₈₃

C₄

yes

complemented

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂×C₄

yes

characteristic

C₄×Dic₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂³×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#117

yes

characteristic

C₂²×C₈₃

#126

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

#344

—

#339

—

#338

D₈₃

yes

characteristic

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

#117

—

C₂×C₄²

—

#97

—

C₄×C₈

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₈₃

yes

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

166

—

C₄²

—

C₄²

—

C₄²

—

166

—

C₄²

—

166

—

C₂³

#13050591

yes

characteristic

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

C₂×C₄×D₈₃

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#13050590

yes

—

C₈

—

C₈

—

C₈

—

166

—

C₂²

—

C₂²

#14581750

yes

characteristic

C₂²

#14581787

yes

—

C₄

—

332

—

C₄

—

664

—

C₄

—

166

—

C₄

#14581748

yes

characteristic

100

C₄

#14581749

yes

characteristic

101

C₄

#14581785

yes

—

102

C₂

—

103

C₂

—

104

C₂

#19127568

yes

characteristic

	subgroup	G/H
center	C₄	#14581749
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#338×C₈₃	C₂
socle	C₂×C₈₃	#383
prime radical	C₂³×C₈₃	#41
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#338	D₈₃
hypocenter	C₈₃	#1841
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#14581749

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#338×C₈₃

C₂

C₂×C₈₃

C₂³×C₈₃

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
9. #14581702	C₄	semidirect
10. C₈×Dic₈₃	C₄	semidirect
53. C₈₃	#1841	semidirect

H ≅ G/N

type

9. #14581702

C₄

semidirect

10. C₈×Dic₈₃

C₄

semidirect

53. C₈₃

#1841

semidirect

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

subgroup	index
M₁₆×Dic₈₃	2
#19127473	2
#19127462	2
#19127458	2
#338×C₈₃	2
#1841	83

subgroup

index

M₁₆×Dic₈₃

#338×C₈₃

Subgroups — #24907613

Conjugacy classes of subgroups

Complements

Maximal subgroups