Group #24908973 — subgroups

Subgroups — #24908973

Order 10624.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#19127744	C₂	1	2	yes	maximal
2	C₂×#14581724	C₂	1	1	yes	maximal, characteristic
3	#19127469	C₂	1	2	yes	maximal
4	#19127466	C₂	1	1	yes	maximal, characteristic
5	#424×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
8	#14581725	—	2	2	no	—
9	#14581724	—	2	2	no	—
10	#14581723	—	2	2	no	—
11	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	#117×C₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₃	—	2	2	no	—
15	C₂²×Dic₈₃	—	2	2	no	—
16	#13050574	—	2	2	no	—
17	#13050574	Q₈	1	4	yes	—
18	C₂×C₈₃⋊C₈	—	2	2	no	—
19	C₄⋊Dic₈₃	—	2	2	no	—
20	C₄⋊Dic₈₃	—	2	2	no	—
21	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₈₃	—	2	2	no	—
23	C₂×C₈×C₈₃	—	2	2	no	—
24	C₂×C₈×C₈₃	—	2	1	no	—
25	C₂×C₈×C₈₃	D₄	1	2	yes	—
26	C₄⋊C₄×C₈₃	D₄	1	2	yes	—
27	C₂²⋊C₄×C₈₃	—	2	1	no	—
28	C₄²×C₈₃	—	2	1	no	—
29	C₂³×C₈₃	#41	1	1	yes	characteristic
30	C₂×C₄×C₈₃	—	2	2	no	—
31	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₈₃	#41	1	2	yes	—
35	C₂×Dic₈₃	—	4	2	no	—
36	C₂×Dic₈₃	—	8	2	no	—
37	C₈×C₈₃	—	2	2	no	—
38	C₈×C₈₃	—	4	1	no	—
39	C₈₃⋊C₈	—	2	4	no	—
40	C₂²×C₈₃	—	2	2	no	—
41	C₂²×C₈₃	#121	1	1	yes	characteristic
42	C₂²×C₈₃	#125	1	2	yes	—
43	C₄×C₈₃	—	4	2	no	—
44	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₈₃	#123	1	1	yes	characteristic
46	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₈₃	—	8	2	no	—
48	C₂×C₈₃	—	2	2	no	—
49	C₂×C₈₃	#398	1	1	yes	characteristic
50	C₂×C₈₃	#471	1	2	yes	—
51	#2090	—	83	1	no	maximal, Sylow
52	C₈₃	#2090	1	1	yes	characteristic, complemented, Sylow
53	#428	—	83	2	no	—
54	C₂×#106	—	83	1	no	—
55	#424	D₈₃	1	1	yes	characteristic
56	#343	—	83	2	no	—
57	#341	—	83	1	no	—
58	C₂×M₁₆	—	83	2	no	—
59	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
61	#117	C₂×D₈₃	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	83	2	no	—
63	#107	—	166	2	no	—
64	#106	—	166	2	no	—
65	#106	—	166	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	166	2	no	—
68	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	83	4	no	—
71	M₁₆	—	166	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	166	2	no	—
75	C₂×C₈	C₂²⋊D₈₃	1	2	yes	—
76	C₄⋊C₄	—	166	2	no	—
77	C₄⋊C₄	—	166	2	no	—
78	C₄⋊C₄	D₃₃₂	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#13050590	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	332	2	no	—
84	C₂×C₄	—	664	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
86	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₈₃	1	2	yes	—
88	C₂×C₄	#13050589	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	166	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#14581766	1	1	yes	characteristic
94	C₂²	#14581782	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	664	2	no	—
97	C₄	#14581772	1	2	yes	—
98	C₄	#14581773	1	1	yes	characteristic
99	C₄	#14581790	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#19127631	1	1	yes	characteristic
102	C₂	#19127964	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19127744

C₂

yes

maximal

C₂×#14581724

C₂

yes

maximal, characteristic

#19127469

C₂

yes

maximal

#19127466

C₂

yes

maximal, characteristic

#424×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

#14581725

—

#14581724

—

#14581723

—

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²×Dic₈₃

—

#13050574

—

#13050574

Q₈

yes

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₄⋊C₄×C₈₃

D₄

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

#41

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#121

yes

characteristic

C₂²×C₈₃

#125

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

#123

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#398

yes

characteristic

C₂×C₈₃

#471

yes

—

#2090

—

maximal, Sylow

C₈₃

#2090

yes

characteristic, complemented, Sylow

#428

—

C₂×#106

—

#424

D₈₃

yes

characteristic

#343

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

C₂×D₈₃

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

166

—

#106

—

166

—

#106

—

166

—

C₄⋊C₈

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

C₂²⋊D₈₃

yes

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄⋊C₄

D₃₃₂

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#13050590

yes

characteristic

C₂×C₄

—

C₂×C₄

—

332

—

C₂×C₄

—

664

—

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

C₂×C₄

C₂×D₃₃₂

yes

characteristic

C₂×C₄

Q₈×D₈₃

yes

—

C₂×C₄

#13050589

yes

—

C₈

—

C₈

—

C₈

—

166

—

C₂²

—

C₂²

#14581766

yes

characteristic

C₂²

#14581782

yes

—

C₄

—

C₄

—

664

—

C₄

#14581772

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#19127631

yes

characteristic

102

C₂

#19127964

yes

—

	subgroup	G/H
center	C₂²	#14581766
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#424×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	C₂³×C₈₃	#41
hypercenter	#424	D₈₃
hypocenter	C₈₃	#2090
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#14581766

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#424×C₈₃

C₂

C₂²×C₈₃

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
52. C₈₃	#2090	semidirect

H ≅ G/N

type

52. C₈₃

#2090

semidirect

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

subgroup	index
#19127744	2
C₂×#14581724	2
#19127469	2
#19127466	2
#424×C₈₃	2
#2090	83

subgroup

index

#424×C₈₃

Subgroups — #24908973

Conjugacy classes of subgroups

Complements

Maximal subgroups