Group #35298120 — subgroups

Subgroups — #35298120

Order 21248.

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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	#24906226	C₂	1	2	yes	maximal
2	#24905325	C₂	1	4	yes	maximal
3	C₈₃×#1452	C₂	1	1	yes	maximal, characteristic
4	#19127758	C₂²	1	4	yes	—
5	#19127454	—	2	2	no	—
6	#19127452	—	2	2	no	—
7	#431×C₈₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
9	#322×C₈₃	C₄	1	4	yes	—
10	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
11	C₂²⋊(C₈₃⋊C₈)	—	2	4	no	—
12	#14581703	—	2	4	no	—
13	C₂²×C₈×C₈₃	Q₈	1	2	yes	—
14	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
15	C₄⋊C₈×C₈₃	—	2	2	no	—
16	C₂²⋊C₈×C₈₃	—	2	2	no	—
17	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₈₃	D₄	1	2	yes	—
20	C₂×C₈₃⋊C₈	—	4	4	no	—
21	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₈₃	#41	1	2	yes	—
23	C₂×C₈×C₈₃	—	2	4	no	—
24	C₂×C₈×C₈₃	—	4	2	no	—
25	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
26	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
27	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
28	C₄²×C₈₃	#41	1	2	yes	—
29	C₂³×C₈₃	#125	1	1	yes	characteristic
30	C₂×C₄×C₈₃	—	2	2	no	—
31	C₂×C₄×C₈₃	—	4	2	no	—
32	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	#117	1	2	yes	—
34	C₂×C₄×C₈₃	#124	1	2	yes	—
35	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
36	C₈×C₈₃	—	2	4	no	—
37	C₈×C₈₃	—	4	4	no	—
38	C₈₃⋊C₈	—	8	4	no	—
39	C₂²×C₈₃	—	2	2	no	—
40	C₂²×C₈₃	—	2	1	no	—
41	C₂²×C₈₃	#382	1	1	yes	characteristic
42	C₄×C₈₃	—	2	2	no	—
43	C₄×C₈₃	—	2	2	no	—
44	C₄×C₈₃	—	2	1	no	—
45	C₄×C₈₃	#430	1	2	yes	—
46	#8799	—	83	1	no	maximal, Sylow
47	C₂×C₈₃	—	4	1	no	—
48	C₂×C₈₃	#1833	1	2	yes	—
49	C₂×C₈₃	#1838	1	1	yes	characteristic
50	#1568	—	83	2	no	—
51	#1452	D₈₃	1	1	yes	characteristic
52	#1287	—	83	4	no	—
53	C₈₃	#8799	1	1	yes	characteristic, complemented, Sylow
54	#432	—	83	4	no	—
55	#431	C₂×D₈₃	1	2	yes	—
56	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
57	#335	—	166	2	no	—
58	#334	—	166	2	no	—
59	#322	Dic₈₃	1	4	yes	—
60	C₂²×C₈	Dic₁₆₆	1	2	yes	—
61	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	—	166	4	no	—
64	C₂²⋊C₈	—	2	2	no	—
65	C₂²⋊C₈	—	166	4	no	—
66	#97	—	166	4	no	—
67	C₄×C₈	D₃₃₂	1	2	yes	—
68	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
69	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
70	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
71	C₂²×C₄	#13050590	1	2	yes	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	4	2	no	—
74	C₂×C₈	—	332	4	no	—
75	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
76	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
77	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
78	C₄²	#13050590	1	2	yes	—
79	C₂³	#14581784	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
83	C₂×C₄	#14581751	1	2	yes	—
84	C₂×C₄	#14581780	1	2	yes	—
85	C₂×C₄	#14581784	1	1	yes	characteristic
86	C₈	—	2	4	no	—
87	C₈	—	4	4	no	—
88	C₈	—	664	4	no	—
89	C₂²	—	2	2	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#19127567	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	2	1	no	—
95	C₄	#19127750	1	2	yes	—
96	C₂	—	4	1	no	—
97	C₂	#24907587	1	2	yes	—
98	C₂	#24907604	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24906226

C₂

yes

maximal

#24905325

C₂

yes

maximal

C₈₃×#1452

C₂

yes

maximal, characteristic

#19127758

C₂²

yes

—

#19127454

—

#19127452

—

#431×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#322×C₈₃

C₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

—

C₂²×C₈×C₈₃

Q₈

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

#125

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#117

yes

—

C₂×C₄×C₈₃

#124

yes

—

C₂×C₄×C₈₃

#125

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#382

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#430

yes

—

#8799

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1833

yes

—

C₂×C₈₃

yes

—

D₈₃

yes

characteristic

#1287

—

C₈₃

#8799

yes

characteristic, complemented, Sylow

#432

—

#431

C₂×D₈₃

yes

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#335

—

166

—

#334

—

166

—

#322

Dic₈₃

yes

—

C₂²×C₈

Dic₁₆₆

yes

—

C₂×C₄²

C₂²×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₂²⋊C₈

—

C₂²⋊C₈

—

166

—

#97

—

166

—

C₄×C₈

D₃₃₂

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂×Dic₁₆₆

yes

characteristic

C₂²×C₄

#13050590

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₄²

C₂²×Dic₈₃

yes

characteristic

C₄²

C₂×D₃₃₂

yes

characteristic

C₄²

#13050590

yes

—

C₂³

#14581784

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

#14581751

yes

—

C₂×C₄

#14581780

yes

—

C₂×C₄

#14581784

yes

characteristic

C₈

—

C₈

—

C₈

—

664

—

C₂²

—

C₂²

—

C₂²

#19127567

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#19127750

yes

—

C₂

—

C₂

#24907587

yes

—

C₂

#24907604

yes

characteristic

	subgroup	G/H
center	C₂²	#19127567
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1452	C₂
socle	C₂²×C₈₃	#382
prime radical	C₂³×C₈₃	#125
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1452	D₈₃
hypocenter	C₈₃	#8799
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#19127567

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1452

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
53. C₈₃	#8799	semidirect

H ≅ G/N

type

53. C₈₃

#8799

semidirect

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

subgroup	index
#24906226	2
#24905325	2
C₈₃×#1452	2
#8799	83

subgroup

index

#24906226

#24905325

C₈₃×#1452

#8799

Subgroups — #35298120

Conjugacy classes of subgroups

Complements

Maximal subgroups