Group #35298215 — subgroups

Subgroups — #35298215

Order 21248.

338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 187 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	#24907635	C₂	1	2	yes	maximal
2	#24906219	C₂	1	4	yes	maximal
3	C₂×#322×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581728	—	2	4	no	—
5	#19127725	C₂²	1	4	yes	—
6	#19127452	—	2	8	no	—
7	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
9	#322×C₈₃	C₄	1	4	yes	—
10	C₂×#13050574	—	2	4	no	—
11	#14581728	—	4	8	no	—
12	C₄⋊(C₈₃⋊C₈)	—	2	8	no	—
13	C₂²×C₈×C₈₃	D₄	1	2	yes	—
14	C₂²×C₈×C₈₃	Q₈	1	4	yes	—
15	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₈₃	D₄	1	4	yes	—
19	C₄×C₈×C₈₃	Q₈	1	2	yes	—
20	#13050574	—	4	8	no	—
21	C₂×C₈₃⋊C₈	—	4	4	no	—
22	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₃	C₂×Q₈	1	2	yes	—
24	C₂×C₈×C₈₃	—	4	4	no	—
25	C₂×C₈×C₈₃	C₄⋊C₄	1	8	yes	—
26	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
27	C₂×C₈×C₈₃	SD₁₆	1	8	yes	—
28	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₈₃	C₂×D₄	1	2	yes	—
30	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
31	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
32	C₂×C₄×C₈₃	—	2	4	no	—
33	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₈₃	C₂×SD₁₆	1	4	yes	—
38	C₈×C₈₃	—	2	8	no	—
39	C₈×C₈₃	#106	1	8	yes	—
40	C₈₃⋊C₈	—	8	4	no	—
41	C₂²×C₈₃	—	2	2	no	—
42	C₂²×C₈₃	#384	1	1	yes	characteristic
43	C₂²×C₈₃	#498	1	2	yes	—
44	C₄×C₈₃	—	2	2	no	—
45	C₄×C₈₃	C₂×#106	1	2	yes	—
46	C₄×C₈₃	#430	1	4	yes	—
47	C₄×C₈₃	#492	1	2	yes	—
48	#8825	—	83	1	no	maximal, Sylow
49	C₂×C₈₃	—	2	2	no	—
50	C₂×C₈₃	#1844	1	1	yes	characteristic
51	C₂×C₈₃	#1849	1	2	yes	—
52	#1848	—	83	2	no	—
53	#1566	—	83	4	no	—
54	C₂×#322	D₈₃	1	1	yes	characteristic
55	C₈₃	#8825	1	1	yes	characteristic, complemented, Sylow
56	C₂×#108	—	166	4	no	—
57	#423	—	83	4	no	—
58	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
60	#334	—	166	8	no	—
61	#322	Dic₈₃	1	4	yes	—
62	C₂×M₁₆	—	166	4	no	—
63	C₂²×C₈	Dic₁₆₆	1	4	yes	—
64	C₂²×C₈	D₃₃₂	1	2	yes	—
65	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
66	#108	—	332	8	no	—
67	C₄⋊C₈	—	166	8	no	—
68	C₄×C₈	Dic₁₆₆	1	2	yes	—
69	C₄×C₈	D₃₃₂	1	4	yes	—
70	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
71	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
72	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
73	C₂²×C₄	C₂×Dic₁₆₆	1	2	yes	—
74	M₁₆	—	332	8	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	332	4	no	—
77	C₂×C₈	C₄⋊Dic₈₃	1	8	yes	—
78	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
79	C₂×C₈	#13050579	1	8	yes	—
80	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
81	C₄²	C₂×D₃₃₂	1	2	yes	—
82	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
83	C₂³	#14581792	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
86	C₂×C₄	C₂×C₄⋊Dic₈₃	1	2	yes	—
87	C₂×C₄	#14581789	1	1	yes	characteristic
88	C₂×C₄	#14581792	1	2	yes	—
89	C₂×C₄	C₂×#13050579	1	4	yes	—
90	C₈	—	2	8	no	—
91	C₈	—	664	4	no	—
92	C₈	#14581724	1	8	yes	—
93	C₂²	—	2	2	no	—
94	C₂²	#19127571	1	1	yes	characteristic
95	C₂²	#19128099	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	C₂×#14581724	1	2	yes	—
98	C₄	#19127750	1	4	yes	—
99	C₄	#19128074	1	2	yes	—
100	C₂	—	2	2	no	—
101	C₂	#24907620	1	1	yes	characteristic
102	C₂	#24907638	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907635

C₂

yes

maximal

#24906219

C₂

yes

maximal

C₂×#322×C₈₃

C₂

yes

maximal, characteristic

C₂×#14581728

—

#19127725

C₂²

yes

—

#19127452

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#322×C₈₃

C₄

yes

—

C₂×#13050574

—

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂²×C₈×C₈₃

Q₈

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₂³×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₈×C₈₃

—

C₈×C₈₃

#106

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#384

yes

characteristic

C₂²×C₈₃

#498

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#106

yes

—

C₄×C₈₃

#430

yes

—

C₄×C₈₃

#492

yes

—

#8825

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1844

yes

characteristic

C₂×C₈₃

#1849

yes

—

#1848

—

#1566

—

C₂×#322

D₈₃

yes

characteristic

C₈₃

#8825

yes

characteristic, complemented, Sylow

C₂×#108

—

166

—

#423

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×C₄×C₈

C₂×D₈₃

yes

—

#334

—

166

—

#322

Dic₈₃

yes

—

C₂×M₁₆

—

166

—

C₂²×C₈

Dic₁₆₆

yes

—

C₂²×C₈

D₃₃₂

yes

—

C₂×C₄²

C₂²×D₈₃

yes

characteristic

#108

—

332

—

C₄⋊C₈

—

166

—

C₄×C₈

Dic₁₆₆

yes

—

C₄×C₈

D₃₃₂

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

C₂²×C₄

C₂×Dic₁₆₆

yes

—

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

#13050579

yes

—

C₄²

C₂²×Dic₈₃

yes

characteristic

C₄²

C₂×D₃₃₂

yes

—

C₄²

C₂×Dic₁₆₆

yes

characteristic

C₂³

#14581792

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

—

C₂×C₄

#14581789

yes

characteristic

C₂×C₄

#14581792

yes

—

C₂×C₄

C₂×#13050579

yes

—

C₈

—

C₈

—

664

—

C₈

#14581724

yes

—

C₂²

—

C₂²

#19127571

yes

characteristic

C₂²

#19128099

yes

—

C₄

—

C₄

C₂×#14581724

yes

—

C₄

#19127750

yes

—

C₄

#19128074

yes

—

100

C₂

—

101

C₂

#24907620

yes

characteristic

102

C₂

#24907638

yes

—

	subgroup	G/H
center	C₂×C₄	#14581789
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#322×C₈₃	C₂
socle	C₂²×C₈₃	#384
prime radical	C₂³×C₈₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#322	D₈₃
hypocenter	C₈₃	#8825
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14581789

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×#322×C₈₃

C₂

socle

C₂²×C₈₃

#384

prime radical

C₂³×C₈₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
55. C₈₃	#8825	semidirect

H ≅ G/N

type

55. C₈₃

#8825

semidirect

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

subgroup	index
#24907635	2
#24906219	2
C₂×#322×C₈₃	2
#8825	83

subgroup

index

#24907635

#24906219

C₂×#322×C₈₃

#8825

Subgroups — #35298215

Conjugacy classes of subgroups

Complements

Maximal subgroups