Group #24907149 — subgroups

Subgroups — #24907149

Order 10624.

328 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 205 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×#14581703	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581702	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₈₃	C₂	1	2	yes	maximal
4	#19127547	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
8	#14581703	C₄	1	4	yes	complemented
9	#14581702	C₄	1	8	yes	complemented
10	C₈×Dic₈₃	C₄	1	8	yes	complemented
11	C₈₃⋊#95	C₄	1	4	yes	—
12	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₈₃	C₄	1	4	yes	complemented
16	C₂²×Dic₈₃	C₂×C₄	1	4	yes	—
17	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×Dic₈₃	C₂×C₄	1	4	yes	—
21	C₄×Dic₈₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₈₃	C₂×C₄	1	4	yes	—
25	C₂×C₈×C₈₃	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
28	C₄²×C₈₃	C₂×C₄	1	2	yes	—
29	C₄²×C₈₃	C₂×C₄	1	2	yes	—
30	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	—	2	4	no	—
32	C₂×C₄×C₈₃	C₄²	1	4	yes	—
33	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
35	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₈₃	—	2	8	no	—
38	C₂×Dic₈₃	C₄²	1	8	yes	—
39	C₈×C₈₃	C₄²	1	4	yes	—
40	C₈×C₈₃	C₄²	1	4	yes	—
41	C₈₃⋊C₈	C₄²	1	8	yes	—
42	C₂²×C₈₃	C₂×C₄²	1	2	yes	—
43	C₂²×C₈₃	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₈₃	#131	1	4	yes	—
45	C₄×C₈₃	—	2	4	no	—
46	C₄×C₈₃	C₂×C₄²	1	2	yes	—
47	C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
49	Dic₈₃	—	2	8	no	—
50	C₂×C₈₃	C₄³	1	1	yes	characteristic
51	C₂×C₈₃	#405	1	4	yes	—
52	C₂×C₈₃	#405	1	2	yes	—
53	#1726	—	83	1	no	maximal, Sylow
54	C₈₃	#1726	1	1	yes	characteristic, complemented, Sylow
55	C₂×#97	—	83	2	no	—
56	C₂×#97	—	83	1	no	—
57	C₂×C₄×C₈	—	83	2	no	—
58	C₂×C₄×C₈	D₈₃	1	1	yes	characteristic
59	#376	—	83	1	no	—
60	C₂²×C₈	—	83	2	no	—
61	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
63	C₂×C₄²	—	83	2	no	—
64	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
65	#97	—	83	8	no	—
66	#97	—	83	4	no	—
67	C₄×C₈	—	83	8	no	—
68	C₄×C₈	Dic₈₃	1	4	yes	complemented
69	#95	—	83	4	no	—
70	C₂²×C₄	—	83	4	no	—
71	C₂²×C₄	C₄×D₈₃	1	2	yes	—
72	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
73	C₂×C₈	—	83	4	no	—
74	C₂×C₈	—	83	4	no	—
75	C₂×C₈	—	83	4	no	—
76	C₂×C₈	C₄×D₈₃	1	4	yes	—
77	C₂×C₈	C₄×D₈₃	1	4	yes	—
78	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
79	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
80	C₄²	—	83	4	no	—
81	C₄²	—	83	4	no	—
82	C₄²	C₄×D₈₃	1	2	yes	—
83	C₄²	C₂×Dic₈₃	1	2	yes	—
84	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	83	8	no	—
87	C₂×C₄	—	166	8	no	—
88	C₂×C₄	C₄×Dic₈₃	1	4	yes	—
89	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
91	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
92	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
93	C₈	—	83	8	no	—
94	C₈	C₄×Dic₈₃	1	4	yes	—
95	C₈	C₄×Dic₈₃	1	4	yes	—
96	C₂²	C₄²×D₈₃	1	2	yes	—
97	C₂²	C₂×C₄×Dic₈₃	1	1	yes	characteristic
98	C₂²	#14581798	1	4	yes	—
99	C₄	—	2	4	no	—
100	C₄	—	166	8	no	—
101	C₄	C₄²×D₈₃	1	2	yes	—
102	C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
103	C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
104	C₂	C₄²×Dic₈₃	1	1	yes	characteristic
105	C₂	#19127655	1	4	yes	—
106	C₂	#19127656	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581703

C₂

yes

maximal, characteristic

C₂×#14581702

C₂

yes

maximal

C₂×C₈×Dic₈₃

C₂

yes

maximal

#19127547

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

#14581703

C₄

yes

complemented

#14581702

C₄

yes

complemented

C₈×Dic₈₃

C₄

yes

complemented

C₈₃⋊#95

C₄

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₄

yes

complemented

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₈₃

C₂×C₄

yes

—

C₄×Dic₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄²

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

C₄²

yes

—

C₈×C₈₃

C₄²

yes

—

C₈×C₈₃

C₄²

yes

—

C₈₃⋊C₈

C₄²

yes

—

C₂²×C₈₃

C₂×C₄²

yes

—

C₂²×C₈₃

C₂×C₄²

yes

characteristic

C₂²×C₈₃

#131

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×C₄²

yes

—

C₄×C₈₃

C₂×C₄²

yes

characteristic

C₄×C₈₃

C₂×C₄²

yes

characteristic

Dic₈₃

—

C₂×C₈₃

C₄³

yes

characteristic

C₂×C₈₃

#405

yes

—

C₂×C₈₃

#405

yes

—

#1726

—

maximal, Sylow

C₈₃

#1726

yes

characteristic, complemented, Sylow

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₈₃

yes

characteristic

#376

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#97

—

#97

—

C₄×C₈

—

C₄×C₈

Dic₈₃

yes

complemented

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₈₃

yes

—

C₂×C₈

C₄×D₈₃

yes

—

C₂×C₈

C₂×Dic₈₃

yes

—

C₂×C₈

C₂×Dic₈₃

yes

—

C₄²

—

C₄²

—

C₄²

C₄×D₈₃

yes

—

C₄²

C₂×Dic₈₃

yes

—

C₂³

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

C₄×Dic₈₃

yes

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

C₂×C₄×D₈₃

yes

—

C₂×C₄

C₂×C₄×D₈₃

yes

—

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

C₈

—

C₈

C₄×Dic₈₃

yes

—

C₈

C₄×Dic₈₃

yes

—

C₂²

C₄²×D₈₃

yes

—

C₂²

C₂×C₄×Dic₈₃

yes

characteristic

C₂²

#14581798

yes

—

C₄

—

100

C₄

—

166

—

101

C₄

C₄²×D₈₃

yes

—

102

C₄

C₂×C₄×Dic₈₃

yes

characteristic

103

C₄

C₂×C₄×Dic₈₃

yes

characteristic

104

C₂

C₄²×Dic₈₃

yes

characteristic

105

C₂

#19127655

yes

—

106

C₂

#19127656

yes

—

	subgroup	G/H
center	C₂²×C₈	C₂×D₈₃
commutator	C₂×C₈₃	C₄³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂×C₄×C₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×C₄×C₈	D₈₃
hypocenter	C₈₃	#1726
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₈

C₂×D₈₃

commutator

C₂×C₈₃

C₄³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₂×C₄×C₈×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂²×C₄

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₂×C₄×C₈

D₈₃

C₈₃

G ▷ C₂×C₈₃ ▷ C₈₃

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
8. #14581703	C₄	semidirect
9. #14581702	C₄	semidirect
10. C₈×Dic₈₃	C₄	semidirect
15. C₄×C₈×C₈₃	C₄	semidirect
54. C₈₃	#1726	semidirect
68. C₄×C₈	Dic₈₃	semidirect

H ≅ G/N

type

8. #14581703

C₄

semidirect

9. #14581702

C₄

semidirect

10. C₈×Dic₈₃

C₄

semidirect

15. C₄×C₈×C₈₃

C₄

semidirect

54. C₈₃

#1726

semidirect

68. C₄×C₈

Dic₈₃

semidirect

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

subgroup	index
C₂×#14581703	2
C₂×#14581702	2
C₂×C₈×Dic₈₃	2
#19127547	2
C₂×C₄×C₈×C₈₃	2
#1726	83

subgroup

index

C₂×C₈×Dic₈₃

C₂×C₄×C₈×C₈₃

Subgroups — #24907149

Conjugacy classes of subgroups

Complements

Maximal subgroups