Group #35298117 — subgroups

Subgroups — #35298117

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	1	yes	characteristic
20	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
21	C₂×C₈₃⋊C₈	—	4	4	no	—
22	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
23	C₂²×C₄×C₈₃	#41	1	2	yes	—
24	C₂×C₈×C₈₃	—	2	4	no	—
25	C₂×C₈×C₈₃	—	4	2	no	—
26	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
28	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
30	C₄²×C₈₃	#41	1	2	yes	—
31	C₂³×C₈₃	#125	1	1	yes	characteristic
32	C₂×C₄×C₈₃	—	2	2	no	—
33	C₂×C₄×C₈₃	—	4	2	no	—
34	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	#117	1	2	yes	—
36	C₂×C₄×C₈₃	#124	1	2	yes	—
37	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
38	C₈×C₈₃	—	2	2	no	—
39	C₈×C₈₃	—	2	2	no	—
40	C₈×C₈₃	—	4	4	no	—
41	C₈₃⋊C₈	—	8	4	no	—
42	C₂²×C₈₃	—	2	2	no	—
43	C₂²×C₈₃	—	2	1	no	—
44	C₂²×C₈₃	#382	1	1	yes	characteristic
45	C₄×C₈₃	—	2	2	no	—
46	C₄×C₈₃	—	2	2	no	—
47	C₄×C₈₃	—	2	1	no	—
48	C₄×C₈₃	#430	1	2	yes	—
49	#8798	—	83	1	no	maximal, Sylow
50	C₂×C₈₃	—	4	1	no	—
51	C₂×C₈₃	#1833	1	2	yes	—
52	C₂×C₈₃	#1838	1	1	yes	characteristic
53	#1568	—	83	2	no	—
54	#1452	D₈₃	1	1	yes	characteristic
55	#1287	—	83	4	no	—
56	C₈₃	#8798	1	1	yes	characteristic, complemented, Sylow
57	#432	—	83	4	no	—
58	#431	C₂×D₈₃	1	2	yes	—
59	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
60	#335	—	166	2	no	—
61	#334	—	166	2	no	—
62	#322	Dic₈₃	1	4	yes	—
63	C₂²×C₈	Dic₁₆₆	1	2	yes	—
64	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	166	4	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	C₂²⋊C₈	—	166	4	no	—
69	#97	—	166	4	no	—
70	C₄×C₈	D₃₃₂	1	1	yes	characteristic
71	C₄×C₈	D₃₃₂	1	1	yes	characteristic
72	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
73	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
74	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
75	C₂²×C₄	#13050590	1	2	yes	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	332	4	no	—
79	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
80	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
81	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
82	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
83	C₄²	#13050590	1	2	yes	—
84	C₂³	#14581784	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
88	C₂×C₄	#14581751	1	2	yes	—
89	C₂×C₄	#14581780	1	2	yes	—
90	C₂×C₄	#14581784	1	1	yes	characteristic
91	C₈	—	2	2	no	—
92	C₈	—	2	2	no	—
93	C₈	—	4	4	no	—
94	C₈	—	664	4	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	#19127567	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	#19127750	1	2	yes	—
102	C₂	—	4	1	no	—
103	C₂	#24907587	1	2	yes	—
104	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

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

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

—

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

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#382

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#430

yes

—

#8798

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1833

yes

—

C₂×C₈₃

yes

—

D₈₃

yes

characteristic

#1287

—

C₈₃

#8798

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

characteristic

C₄×C₈

D₃₃₂

yes

characteristic

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

—

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

—

C₈

—

664

—

C₂²

—

C₂²

—

C₂²

#19127567

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

#19127750

yes

—

102

C₂

—

103

C₂

#24907587

yes

—

104

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	#8798
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
56. C₈₃	#8798	semidirect

H ≅ G/N

type

56. C₈₃

#8798

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
#8798	83

subgroup

index

#24906226

#24905325

C₈₃×#1452

#8798

Subgroups — #35298117

Conjugacy classes of subgroups

Complements

Maximal subgroups