Group #24905564 — subgroups

Subgroups — #24905564

Order 10624.

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

normal subgroups only

This group has 132 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	C₂×#14581732	C₂	1	1	yes	maximal, characteristic
2	#19127739	C₂	1	1	yes	maximal, characteristic
3	#427×C₈₃	C₂	1	1	yes	maximal, characteristic
4	#14581751	—	2	1	no	—
5	#14581732	—	2	1	no	—
6	#14581732	—	2	1	no	—
7	C₂×C₈₃⋊C₁₆	C₄	1	2	yes	complemented
8	#14581726	—	2	1	no	—
9	#14581724	C₄	1	2	yes	complemented
10	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
11	#117×C₈₃	—	2	1	no	—
12	#107×C₈₃	—	2	1	no	—
13	#106×C₈₃	C₄	1	2	yes	complemented
14	C₄⋊Dic₈₃	—	2	2	no	—
15	C₂²⋊Dic₈₃	—	4	1	no	—
16	C₄×Dic₈₃	—	4	1	no	—
17	C₈₃⋊C₁₆	—	2	2	no	—
18	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₈₃	Q₈	1	1	yes	characteristic
25	C₄⋊C₄×C₈₃	—	2	2	no	—
26	C₂²⋊C₄×C₈₃	—	4	1	no	—
27	C₄²×C₈₃	—	4	1	no	—
28	C₂³×C₈₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₈₃	—	4	2	no	—
30	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₈₃	Q₁₆	1	1	yes	characteristic
36	C₂×Dic₈₃	—	4	2	no	—
37	C₈×C₈₃	C₄²	1	1	yes	characteristic
38	C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
40	C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₈₃	—	4	1	no	—
42	C₂²×C₈₃	#102	1	1	yes	characteristic
43	C₂²×C₈₃	#102	1	1	yes	characteristic
44	C₂²×C₈₃	#107	1	1	yes	characteristic
45	C₄×C₈₃	—	8	2	no	—
46	C₄×C₈₃	#95	1	1	yes	characteristic
47	C₄×C₈₃	#103	1	1	yes	characteristic
48	C₄×C₈₃	#103	1	1	yes	characteristic
49	C₄×C₈₃	#106	1	1	yes	characteristic
50	Dic₈₃	—	8	2	no	—
51	C₂×C₈₃	—	2	1	no	—
52	C₂×C₈₃	—	2	1	no	—
53	C₂×C₈₃	—	2	1	no	—
54	C₂×C₈₃	#340	1	1	yes	characteristic
55	#1384	—	83	1	no	maximal, Sylow
56	C₈₃	#1384	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	—	83	1	no	—
58	#427	—	83	1	no	—
59	#427	D₈₃	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
61	#117	—	2	1	no	—
62	#117	—	166	1	no	—
63	M₃₂	—	166	1	no	—
64	M₃₂	—	166	1	no	—
65	C₂×C₁₆	—	83	2	no	—
66	#107	—	2	1	no	—
67	#107	—	166	1	no	—
68	#106	—	83	2	no	—
69	#106	Dic₈₃	1	2	yes	complemented
70	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
71	C₂×C₈	Dic₁₆₆	1	1	yes	characteristic
72	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
73	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
74	C₂×C₈	D₃₃₂	1	1	yes	characteristic
75	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
76	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	166	2	no	—
79	C₂²⋊C₄	—	4	1	no	—
80	C₂²⋊C₄	—	332	1	no	—
81	C₄²	—	4	1	no	—
82	C₄²	—	332	1	no	—
83	C₁₆	—	166	2	no	—
84	C₂³	#13050575	1	1	yes	characteristic
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	332	2	no	—
87	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
89	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
90	C₂×C₄	#13050577	1	1	yes	characteristic
91	C₂×C₄	#13050578	1	1	yes	characteristic
92	C₂×C₄	#13050580	1	1	yes	characteristic
93	C₈	C₄×Dic₈₃	1	1	yes	characteristic
94	C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
95	C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
96	C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
97	C₂²	—	4	1	no	—
98	C₂²	#14581712	1	1	yes	characteristic
99	C₂²	#14581713	1	1	yes	characteristic
100	C₂²	#14581725	1	1	yes	characteristic
101	C₄	—	8	2	no	—
102	C₄	—	664	2	no	—
103	C₄	C₈₃⋊#95	1	1	yes	characteristic
104	C₄	#14581715	1	1	yes	characteristic
105	C₄	#14581716	1	1	yes	characteristic
106	C₄	#14581723	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	#19127463	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581732

C₂

yes

maximal, characteristic

#19127739

C₂

yes

maximal, characteristic

#427×C₈₃

C₂

yes

maximal, characteristic

#14581751

—

#14581732

—

#14581732

—

C₂×C₈₃⋊C₁₆

C₄

yes

complemented

#14581726

—

#14581724

C₄

yes

complemented

C₂²×C₈×C₈₃

C₂²

yes

characteristic

#117×C₈₃

—

#107×C₈₃

—

#106×C₈₃

C₄

yes

complemented

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂×C₈×C₈₃

Q₈

yes

characteristic

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

D₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

SD₁₆

yes

characteristic

C₂×C₄×C₈₃

SD₁₆

yes

characteristic

C₂×C₄×C₈₃

Q₁₆

yes

characteristic

C₂×Dic₈₃

—

C₈×C₈₃

C₄²

yes

characteristic

C₈×C₈₃

C₂²⋊C₄

yes

characteristic

C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#427

—

#427

D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

—

#117

—

166

—

M₃₂

—

166

—

M₃₂

—

166

—

C₂×C₁₆

—

#107

—

#107

—

166

—

#106

—

#106

Dic₈₃

yes

complemented

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

C₂×C₈

Dic₁₆₆

yes

characteristic

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

D₃₃₂

yes

characteristic

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

C₂²⋊C₄

—

332

—

C₄²

—

C₄²

—

332

—

C₁₆

—

166

—

C₂³

#13050575

yes

characteristic

C₂×C₄

—

C₂×C₄

—

332

—

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

C₂×C₄

Dic₈₃⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

C₄×Dic₈₃

yes

characteristic

C₈

(C₂×D₈₃)⋊C₄

yes

characteristic

C₈

Dic₈₃⋊C₄

yes

characteristic

C₈

C₄⋊Dic₈₃

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

100

C₂²

#14581725

yes

characteristic

101

C₄

—

102

C₄

—

664

—

103

C₄

C₈₃⋊#95

yes

characteristic

104

C₄

#14581715

yes

characteristic

105

C₄

#14581716

yes

characteristic

106

C₄

#14581723

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

110

C₂

#19127463

yes

characteristic

	subgroup	G/H
center	C₄	#14581723
commutator	C₈×C₈₃	C₄²
Frattini	C₂²×C₈	C₂×D₈₃
Fitting	#427×C₈₃	C₂
socle	C₂×C₈₃	#340
prime radical	C₂³×C₈₃	D₈
cosocle	C₂²×C₈×C₈₃	C₂²
hypercenter	#427	D₈₃
hypocenter	C₈₃	#1384
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₄

#14581723

commutator

C₈×C₈₃

C₄²

Frattini

C₂²×C₈

C₂×D₈₃

Fitting

#427×C₈₃

C₂

socle

C₂×C₈₃

#340

prime radical

C₂³×C₈₃

D₈

cosocle

C₂²×C₈×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
7. C₂×C₈₃⋊C₁₆	C₄	semidirect
9. #14581724	C₄	semidirect
13. #106×C₈₃	C₄	semidirect
56. C₈₃	#1384	semidirect
69. #106	Dic₈₃	semidirect

H ≅ G/N

type

7. C₂×C₈₃⋊C₁₆

C₄

semidirect

9. #14581724

C₄

semidirect

13. #106×C₈₃

C₄

semidirect

56. C₈₃

#1384

semidirect

69. #106

Dic₈₃

semidirect

Subgroups — #24905564

Conjugacy classes of subgroups

Complements