Group #24909271 — subgroups

Subgroups — #24909271

Order 10624.

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

normal subgroups only

This group has 138 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	#19127655	C₂	1	1	yes	maximal, characteristic
2	Dic₈₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	(C₂×D₈₃)⋊C₁₆	C₂	1	2	yes	maximal
4	C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13050573	C₄	1	1	yes	characteristic
7	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
8	#14581749	C₄	1	1	yes	characteristic
9	C₂×C₈₃⋊C₁₆	C₂²	1	2	yes	—
10	#14581703	C₄	1	1	yes	characteristic
11	#14581702	C₄	1	1	yes	characteristic
12	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₈₃	C₂²	1	2	yes	—
14	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
16	#13050573	—	2	2	no	—
17	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₈×D₈₃	—	2	2	no	—
20	Dic₈₃⋊C₄	C₈	1	2	yes	—
21	(C₂×D₈₃)⋊C₄	C₈	1	2	yes	—
22	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
23	C₈₃⋊C₁₆	—	2	2	no	—
24	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
26	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₈₃	—	2	2	no	—
28	C₂²×D₈₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₈₃	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₈₃	C₂×C₈	1	2	yes	—
32	C₂×Dic₈₃	C₂×C₈	1	1	yes	characteristic
33	C₄×D₈₃	—	2	1	no	—
34	C₄×D₈₃	—	2	1	no	—
35	C₈×C₈₃	M₁₆	1	2	yes	—
36	C₈×C₈₃	#41	1	2	yes	—
37	C₈₃⋊C₈	—	2	1	no	—
38	C₈₃⋊C₈	—	2	1	no	—
39	C₂²×C₈₃	C₂²×C₈	1	1	yes	characteristic
40	C₂×D₈₃	—	2	1	no	—
41	C₂×D₈₃	—	2	2	no	—
42	C₄×C₈₃	—	2	2	no	—
43	C₄×C₈₃	#117	1	1	yes	characteristic
44	C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
45	Dic₈₃	—	2	1	no	—
46	Dic₈₃	—	2	2	no	—
47	C₂×C₈₃	#431	1	1	yes	characteristic
48	C₂×C₈₃	#504	1	2	yes	—
49	D₈₃	—	4	1	no	—
50	#2164	—	83	1	no	maximal, Sylow
51	C₈₃	#2164	1	1	yes	characteristic, complemented, Sylow
52	#405	—	83	1	no	—
53	C₄⋊C₁₆	—	83	2	no	—
54	C₂²⋊C₁₆	—	83	2	no	—
55	#346	D₈₃	1	1	yes	characteristic, complemented
56	C₄×C₁₆	—	83	1	no	—
57	C₂×M₁₆	—	83	1	no	—
58	C₂²×C₈	—	83	1	no	—
59	#117	—	83	1	no	—
60	C₂×C₁₆	—	83	2	no	—
61	C₂×C₁₆	C₂×D₈₃	1	2	yes	—
62	#97	—	83	1	no	—
63	#97	—	83	1	no	—
64	C₄×C₈	—	83	1	no	—
65	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
66	C₂²×C₄	—	83	1	no	—
67	M₁₆	—	166	2	no	—
68	C₂×C₈	—	83	1	no	—
69	C₂×C₈	—	83	1	no	—
70	C₂×C₈	—	166	2	no	—
71	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
73	C₄⋊C₄	—	83	2	no	—
74	C₂²⋊C₄	—	83	2	no	—
75	C₄²	—	83	1	no	—
76	C₄²	C₄×D₈₃	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	166	2	no	—
79	C₂³	—	83	1	no	—
80	C₂×C₄	—	83	2	no	—
81	C₂×C₄	—	83	1	no	—
82	C₂×C₄	—	166	1	no	—
83	C₂×C₄	—	166	1	no	—
84	C₂×C₄	C₈×D₈₃	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
86	C₈	—	166	1	no	—
87	C₈	—	166	1	no	—
88	C₈	#13050573	1	2	yes	—
89	C₈	#13050590	1	2	yes	—
90	C₂²	—	166	2	no	—
91	C₂²	—	166	1	no	—
92	C₂²	C₂×C₈×D₈₃	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	166	2	no	—
95	C₄	—	166	1	no	—
96	C₄	#14581749	1	1	yes	characteristic
97	C₄	C₂×#13050573	1	1	yes	characteristic
98	C₂	—	332	1	no	—
99	C₂	#19127752	1	1	yes	characteristic
100	C₂	#19128119	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19127655

C₂

yes

maximal, characteristic

Dic₈₃⋊C₁₆

C₂

yes

maximal, complemented

(C₂×D₈₃)⋊C₁₆

C₂

yes

maximal

C₄×C₈₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#346×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050573

C₄

yes

characteristic

C₂×C₈×D₈₃

C₂²

yes

characteristic

#14581749

C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

C₂²

yes

—

#14581703

C₄

yes

characteristic

#14581702

C₄

yes

characteristic

C₈×Dic₈₃

C₂²

yes

characteristic

C₂×C₁₆×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄×D₈₃

C₂×C₄

yes

characteristic

#13050573

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₈×D₈₃

—

Dic₈₃⋊C₄

C₈

yes

—

(C₂×D₈₃)⋊C₄

C₈

yes

—

C₄×Dic₈₃

C₂×C₄

yes

characteristic

C₈₃⋊C₁₆

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₁₆×C₈₃

—

C₂²×D₈₃

C₂×C₈

yes

characteristic

C₂×C₄×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×Dic₈₃

C₂×C₈

yes

—

C₂×Dic₈₃

C₂×C₈

yes

characteristic

C₄×D₈₃

—

C₄×D₈₃

—

C₈×C₈₃

M₁₆

yes

—

C₈×C₈₃

#41

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂²×C₈

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#117

yes

characteristic

C₄×C₈₃

C₂×M₁₆

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#431

yes

characteristic

C₂×C₈₃

#504

yes

—

D₈₃

—

#2164

—

maximal, Sylow

C₈₃

#2164

yes

characteristic, complemented, Sylow

#405

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

D₈₃

yes

characteristic, complemented

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₃

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂²×C₄

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

C₂²×D₈₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₈₃

yes

characteristic

C₁₆

—

C₁₆

—

166

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

C₈×D₈₃

yes

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

C₈

—

166

—

C₈

—

166

—

C₈

#13050573

yes

—

C₈

#13050590

yes

—

C₂²

—

166

—

C₂²

—

166

—

C₂²

C₂×C₈×D₈₃

yes

characteristic

C₄

—

C₄

—

166

—

C₄

—

166

—

C₄

yes

C₄

yes

C₂

—

332

—

C₂

#19127752

yes

characteristic

100

C₂

#19128119

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₈₃
commutator, socle	C₂²×C₈₃	C₂²×C₈
Fitting	#346×C₈₃	C₂
prime radical	C₂²×D₈₃	C₂×C₈
cosocle	C₂×C₈×C₈₃	C₂³
hypercenter	#346	D₈₃
hypocenter	C₈₃	#2164
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₈₃

commutator, socle

C₂²×C₈₃

C₂²×C₈

Fitting

#346×C₈₃

C₂

prime radical

C₂²×D₈₃

C₂×C₈

cosocle

C₂×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
2. Dic₈₃⋊C₁₆	C₂	semidirect
4. C₄×C₈₃⋊C₁₆	C₂	semidirect
5. #346×C₈₃	C₂	semidirect
51. C₈₃	#2164	semidirect
55. #346	D₈₃	semidirect

H ≅ G/N

type

2. Dic₈₃⋊C₁₆

C₂

semidirect

4. C₄×C₈₃⋊C₁₆

C₂

semidirect

5. #346×C₈₃

C₂

semidirect

51. C₈₃

#2164

semidirect

55. #346

D₈₃

semidirect

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

subgroup	index
#19127655	2
Dic₈₃⋊C₁₆	2
(C₂×D₈₃)⋊C₁₆	2
C₄×C₈₃⋊C₁₆	2
#346×C₈₃	2
#2164	83

subgroup

index

#19127655

Dic₈₃⋊C₁₆

(C₂×D₈₃)⋊C₁₆

C₄×C₈₃⋊C₁₆

#346×C₈₃

#2164

Subgroups — #24909271

Conjugacy classes of subgroups

Complements

Maximal subgroups