Group #35313459 — subgroups

Subgroups — #35313459

Order 21248.

228 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 123 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	#24909266	C₂	1	2	yes	maximal
2	#24905384	C₂	1	4	yes	maximal, complemented
3	C₈₃×#1450	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₈₃⋊C₁₆)	C₄	1	4	yes	—
5	C₂²⋊(C₈₃⋊C₁₆)	C₄	1	4	yes	—
6	#19127477	C₂²	1	4	yes	—
7	#431×C₈₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
9	#322×C₈₃	C₂²	1	2	yes	—
10	C₈²×C₈₃	C₄	1	2	yes	—
11	C₂×C₈₃⋊C₁₆	C₂×C₄	1	8	yes	—
12	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
14	C₄⋊C₈×C₈₃	—	2	2	no	—
15	C₂²⋊C₈×C₈₃	—	2	2	no	—
16	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
17	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
20	C₈₃⋊C₁₆	—	2	8	no	—
21	C₂²×C₄×C₈₃	C₄²	1	1	yes	characteristic
22	C₂²×C₄×C₈₃	M₁₆	1	2	yes	—
23	C₂×C₈×C₈₃	—	2	4	no	—
24	C₂×C₈×C₈₃	—	2	4	no	—
25	C₂×C₈×C₈₃	C₄²	1	2	yes	—
26	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
27	C₄²×C₈₃	C₄²	1	1	yes	characteristic
28	C₄²×C₈₃	M₁₆	1	2	yes	—
29	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
30	C₂³×C₈₃	#97	1	1	yes	characteristic
31	C₂×C₄×C₈₃	—	2	2	no	—
32	C₂×C₄×C₈₃	—	4	2	no	—
33	C₂×C₄×C₈₃	#97	1	2	yes	—
34	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
37	C₈×C₈₃	—	2	2	no	—
38	C₈×C₈₃	—	4	4	no	—
39	C₈×C₈₃	#131	1	4	yes	—
40	C₂²×C₈₃	—	2	2	no	—
41	C₂²×C₈₃	—	2	1	no	—
42	C₂²×C₈₃	C₂×#97	1	1	yes	characteristic
43	C₄×C₈₃	—	2	2	no	—
44	C₄×C₈₃	—	2	2	no	—
45	C₄×C₈₃	—	2	1	no	—
46	C₄×C₈₃	#405	1	2	yes	—
47	#11416	—	83	1	no	maximal, Sylow
48	C₂×C₈₃	—	4	1	no	—
49	C₂×C₈₃	#1452	1	1	yes	characteristic
50	C₂×C₈₃	#2110	1	2	yes	—
51	#2163	—	83	2	no	—
52	#1450	D₈₃	1	1	yes	characteristic
53	#1313	—	83	4	no	—
54	C₈₃	#11416	1	1	yes	characteristic, complemented, Sylow
55	#431	Dic₈₃	1	2	yes	—
56	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
57	C₄⋊C₁₆	—	83	4	no	—
58	C₂²⋊C₁₆	—	83	4	no	—
59	#346	—	83	4	no	—
60	#322	C₂×D₈₃	1	2	yes	—
61	C₈²	Dic₈₃	1	2	yes	—
62	C₂²×C₈	C₄×D₈₃	1	2	yes	—
63	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
64	C₂×C₁₆	—	83	8	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₂²⋊C₈	—	2	2	no	—
67	C₄×C₈	C₄×D₈₃	1	2	yes	—
68	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
69	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
70	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
71	C₂²×C₄	C₄×Dic₈₃	1	1	yes	characteristic
72	C₂²×C₄	#13050574	1	2	yes	—
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
76	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
77	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
78	C₄²	#13050574	1	2	yes	—
79	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
80	C₁₆	—	166	8	no	—
81	C₂³	#14581703	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	#14581703	1	2	yes	—
85	C₂×C₄	#14581703	1	1	yes	characteristic
86	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
87	C₂×C₄	C₂×#13050574	1	2	yes	—
88	C₈	—	2	2	no	—
89	C₈	—	4	4	no	—
90	C₈	#14581798	1	4	yes	—
91	C₂²	—	2	2	no	—
92	C₂²	—	2	1	no	—
93	C₂²	C₂×#14581703	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	2	1	no	—
97	C₄	#19127656	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	#24905755	1	1	yes	characteristic
100	C₂	#24909068	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909266

C₂

yes

maximal

#24905384

C₂

yes

maximal, complemented

C₈₃×#1450

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₁₆)

C₄

yes

—

C₂²⋊(C₈₃⋊C₁₆)

C₄

yes

—

#19127477

C₂²

yes

—

#431×C₈₃

C₄

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#322×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₄

yes

—

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

—

C₂²×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

C₄²

yes

characteristic

C₂²×C₄×C₈₃

M₁₆

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄²

yes

—

C₂×C₈×C₈₃

C₂²×C₄

yes

—

C₄²×C₈₃

C₄²

yes

characteristic

C₄²×C₈₃

M₁₆

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₂³×C₈₃

#97

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#97

yes

—

C₂×C₄×C₈₃

#97

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₈₃

C₂×M₁₆

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#131

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×#97

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#405

yes

—

#11416

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1452

yes

characteristic

C₂×C₈₃

#2110

yes

—

#2163

—

#1450

D₈₃

yes

characteristic

#1313

—

C₈₃

#11416

yes

characteristic, complemented, Sylow

#431

Dic₈₃

yes

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

C₂×D₈₃

yes

—

C₈²

Dic₈₃

yes

—

C₂²×C₈

C₄×D₈₃

yes

—

C₂×C₄²

C₂×Dic₈₃

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₄×D₈₃

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₄×C₈

C₂×Dic₈₃

yes

characteristic

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂²×C₄

C₄×Dic₈₃

yes

characteristic

C₂²×C₄

#13050574

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×Dic₈₃

yes

—

C₂×C₈

C₂×C₄×D₈₃

yes

—

C₄²

C₄×Dic₈₃

yes

characteristic

C₄²

#13050574

yes

—

C₄²

C₂²×Dic₈₃

yes

characteristic

C₁₆

—

166

—

C₂³

#14581703

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14581703

yes

—

C₂×C₄

#14581703

yes

characteristic

C₂×C₄

C₂×C₄×Dic₈₃

yes

characteristic

C₂×C₄

C₂×#13050574

yes

—

C₈

—

C₈

—

C₈

#14581798

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#14581703

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#19127656

yes

—

C₂

—

C₂

#24905755

yes

characteristic

100

C₂

#24909068

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₈₃
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1450	C₂
socle	C₂²×C₈₃	C₂×#97
prime radical	C₂³×C₈₃	#97
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#1450	D₈₃
hypocenter	C₈₃	#11416
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ #1450

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₈₃

commutator

C₂×C₄×C₈₃

C₂×C₄²

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1450

C₂

socle

C₂²×C₈₃

C₂×#97

prime radical

C₂³×C₈₃

#97

cosocle

C₄×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₈₃

C₁ ◁ C₂×C₄ ◁ #1450

N	H ≅ G/N	type
2. #24905384	C₂	semidirect
54. C₈₃	#11416	semidirect

H ≅ G/N

type

2. #24905384

C₂

semidirect

54. C₈₃

#11416

semidirect

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

subgroup	index
#24909266	2
#24905384	2
C₈₃×#1450	2
#11416	83

subgroup

index

#24909266

#24905384

C₈₃×#1450

#11416

Subgroups — #35313459

Conjugacy classes of subgroups

Complements

Maximal subgroups