Group #24908970 — subgroups

Subgroups — #24908970

Order 10624.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#19127743	C₂	1	2	yes	maximal
2	C₂×#14581726	C₂	1	1	yes	maximal, characteristic
3	#19127727	C₂	1	1	yes	maximal, characteristic
4	#19127469	C₂	1	2	yes	maximal
5	#341×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	1	yes	characteristic
7	#14581751	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
9	#14581726	—	2	2	no	—
10	#14581726	—	2	2	no	—
11	#14581724	—	2	2	no	—
12	Dic₈₃⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₈₃	—	2	2	no	—
16	#13050574	—	2	2	no	—
17	C₂×C₈₃⋊C₈	—	2	1	no	—
18	C₄⋊Dic₈₃	—	2	2	no	—
19	C₄⋊Dic₈₃	—	2	2	no	—
20	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
21	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
22	C₂²⋊Dic₈₃	—	2	1	no	—
23	C₄×Dic₈₃	—	2	1	no	—
24	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
25	M₁₆×C₈₃	—	2	2	no	—
26	M₁₆×C₈₃	Q₈	1	4	yes	—
27	C₂×C₈×C₈₃	—	2	2	no	—
28	C₂×C₈×C₈₃	—	2	2	no	—
29	C₂×C₈×C₈₃	D₄	1	2	yes	—
30	C₂³×C₈₃	#41	1	1	yes	characteristic
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	—	2	1	no	—
36	C₂×Dic₈₃	—	2	1	no	—
37	C₂×Dic₈₃	—	4	2	no	—
38	C₂×Dic₈₃	—	8	2	no	—
39	C₈×C₈₃	—	2	4	no	—
40	C₈×C₈₃	—	2	2	no	—
41	C₈₃⋊C₈	—	4	1	no	—
42	C₂²×C₈₃	—	2	1	no	—
43	C₂²×C₈₃	—	2	1	no	—
44	C₂²×C₈₃	#121	1	1	yes	characteristic
45	C₂²×C₈₃	#125	1	2	yes	—
46	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₈₃	#123	1	1	yes	characteristic
48	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
49	Dic₈₃	—	4	1	no	—
50	Dic₈₃	—	4	1	no	—
51	Dic₈₃	—	8	2	no	—
52	C₂×C₈₃	—	2	2	no	—
53	C₂×C₈₃	#398	1	1	yes	characteristic
54	C₂×C₈₃	#472	1	1	yes	characteristic
55	C₂×C₈₃	#473	1	1	yes	characteristic
56	#2089	—	83	1	no	maximal, Sylow
57	C₈₃	#2089	1	1	yes	characteristic, complemented, Sylow
58	#428	—	83	2	no	—
59	C₂×#107	—	83	1	no	—
60	#424	—	83	1	no	—
61	#343	—	83	2	no	—
62	#341	D₈₃	1	1	yes	characteristic
63	C₂×M₁₆	—	83	1	no	—
64	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
65	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
66	#117	—	83	1	no	—
67	C₂×C₄⋊C₄	—	83	2	no	—
68	#107	—	166	2	no	—
69	#107	—	166	2	no	—
70	#106	—	166	2	no	—
71	C₄⋊C₈	—	166	2	no	—
72	C₂²×C₄	—	166	2	no	—
73	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	166	2	no	—
76	M₁₆	Dic₁₆₆	1	4	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	166	1	no	—
80	C₂×C₈	D₃₃₂	1	2	yes	—
81	C₄⋊C₄	—	83	1	no	—
82	C₄⋊C₄	—	83	1	no	—
83	C₄⋊C₄	—	166	2	no	—
84	C₄⋊C₄	—	166	2	no	—
85	C₂²⋊C₄	—	166	1	no	—
86	C₄²	—	166	1	no	—
87	C₂³	#13050591	1	1	yes	characteristic
88	C₂×C₄	—	332	2	no	—
89	C₂×C₄	—	664	2	no	—
90	C₂×C₄	—	166	1	no	—
91	C₂×C₄	—	166	1	no	—
92	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
93	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
95	C₂×C₄	#13050589	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	2	no	—
98	C₈	—	332	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#14581767	1	1	yes	characteristic
102	C₂²	#14581783	1	2	yes	—
103	C₄	—	332	1	no	—
104	C₄	—	332	1	no	—
105	C₄	—	664	2	no	—
106	C₄	#14581769	1	2	yes	—
107	C₄	#14581776	1	1	yes	characteristic
108	C₄	#14581792	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#19127633	1	1	yes	characteristic
111	C₂	#19127972	1	1	yes	characteristic
112	C₂	#19127977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127743

C₂

yes

maximal

C₂×#14581726

C₂

yes

maximal, characteristic

#19127727

C₂

yes

maximal, characteristic

#19127469

C₂

yes

maximal

#341×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

characteristic

#14581751

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

#14581726

—

#14581726

—

#14581724

—

Dic₈₃⋊C₈

—

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

characteristic

C₄⋊Dic₈₃

D₄

yes

characteristic

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

Q₈

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₂³×C₈₃

#41

yes

characteristic

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

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#121

yes

characteristic

C₂²×C₈₃

#125

yes

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

#123

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#107

—

166

—

#107

—

166

—

#106

—

166

—

C₄⋊C₈

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

166

—

M₁₆

Dic₁₆₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

D₃₃₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

166

—

C₄²

—

166

—

C₂³

#13050591

yes

characteristic

C₂×C₄

—

332

—

C₂×C₄

—

664

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

D₄×D₈₃

yes

characteristic

C₂×C₄

C₂×D₃₃₂

yes

characteristic

C₂×C₄

C₂×Dic₁₆₆

yes

—

C₂×C₄

#13050589

yes

—

C₈

—

C₈

—

C₈

—

332

—

C₂²

—

100

C₂²

—

101

C₂²

#14581767

yes

characteristic

102

C₂²

#14581783

yes

—

103

C₄

—

332

—

104

C₄

—

332

—

105

C₄

—

664

—

106

C₄

#14581769

yes

—

107

C₄

#14581776

yes

characteristic

108

C₄

#14581792

yes

characteristic

109

C₂

—

110

C₂

#19127633

yes

characteristic

111

C₂

#19127972

yes

characteristic

112

C₂

#19127977

yes

characteristic

	subgroup	G/H
center	C₂²	#14581767
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#341×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	C₂³×C₈₃	#41
hypercenter	#341	D₈₃
hypocenter	C₈₃	#2089
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#14581767

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#341×C₈₃

C₂

C₂²×C₈₃

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
57. C₈₃	#2089	semidirect

H ≅ G/N

type

57. C₈₃

#2089

semidirect

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

subgroup	index
#19127743	2
C₂×#14581726	2
#19127727	2
#19127469	2
#341×C₈₃	2
#2089	83

subgroup

index

#341×C₈₃

Subgroups — #24908970

Conjugacy classes of subgroups

Complements

Maximal subgroups