Group #35298178 — subgroups

Subgroups — #35298178

Order 21248.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#24907620	C₂	1	2	yes	maximal
2	#24907591	C₂	1	4	yes	maximal
3	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581724	—	2	4	no	—
5	#19127571	C₂²	1	4	yes	—
6	#19127464	—	2	8	no	—
7	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
9	C₈²×C₈₃	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₈₃	—	2	8	no	—
11	C₂×C₄⋊Dic₈₃	—	2	4	no	—
12	#14581724	—	4	8	no	—
13	C₂²×C₈×C₈₃	D₄	1	4	yes	—
14	C₂²×C₈×C₈₃	D₄	1	2	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	Q₈	1	4	yes	—
19	C₄×C₈×C₈₃	Q₈	1	2	yes	—
20	C₂²×Dic₈₃	—	4	4	no	—
21	C₄⋊Dic₈₃	—	4	8	no	—
22	C₄⋊Dic₈₃	—	4	16	no	—
23	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
25	C₂×C₈×C₈₃	—	2	8	no	—
26	C₂×C₈×C₈₃	C₄⋊C₄	1	8	yes	—
27	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
28	C₂×C₈×C₈₃	SD₁₆	1	8	yes	—
29	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
31	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
32	C₂³×C₈₃	#127	1	1	yes	characteristic
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	C₄⋊Q₈	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₈₃	C₂×SD₁₆	1	4	yes	—
38	C₂×C₄×C₈₃	#135	1	8	yes	—
39	C₂×Dic₈₃	—	8	4	no	—
40	C₂×Dic₈₃	—	8	8	no	—
41	C₈×C₈₃	—	2	8	no	—
42	C₈×C₈₃	#106	1	8	yes	—
43	C₂²×C₈₃	#384	1	1	yes	characteristic
44	C₂²×C₈₃	#492	1	2	yes	—
45	C₂²×C₈₃	#495	1	4	yes	—
46	C₄×C₈₃	C₂×#106	1	2	yes	—
47	C₄×C₈₃	#427	1	4	yes	—
48	C₄×C₈₃	#498	1	2	yes	—
49	C₄×C₈₃	#499	1	4	yes	—
50	Dic₈₃	—	16	4	no	—
51	#8815	—	83	1	no	maximal, Sylow
52	C₂×C₈₃	#1708	1	4	yes	—
53	C₂×C₈₃	#1844	1	1	yes	characteristic
54	C₂×C₈₃	#1846	1	2	yes	—
55	#1844	—	83	2	no	—
56	#1834	—	83	4	no	—
57	C₂×C₈²	D₈₃	1	1	yes	characteristic
58	C₈₃	#8815	1	1	yes	characteristic, complemented, Sylow
59	C₂×#106	—	166	4	no	—
60	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
61	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
62	#384	—	83	4	no	—
63	#340	—	166	8	no	—
64	C₈²	Dic₈₃	1	4	yes	complemented
65	C₂²×C₈	D₃₃₂	1	4	yes	—
66	C₂²×C₈	D₃₃₂	1	2	yes	—
67	C₂×C₄⋊C₄	—	166	8	no	—
68	C₂×C₄⋊C₄	—	166	4	no	—
69	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
70	#106	—	332	8	no	—
71	C₄×C₈	Dic₁₆₆	1	4	yes	—
72	C₄×C₈	Dic₁₆₆	1	2	yes	—
73	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
74	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
75	C₂²×C₄	—	332	4	no	—
76	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₃₃₂	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	C₄⋊Dic₈₃	1	8	yes	—
80	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
81	C₂×C₈	#13050579	1	8	yes	—
82	C₄⋊C₄	—	332	8	no	—
83	C₄⋊C₄	—	332	16	no	—
84	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
85	C₄²	C₂×Dic₁₆₆	1	2	yes	—
86	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
87	C₂³	#14581789	1	1	yes	characteristic
88	C₂×C₄	—	664	4	no	—
89	C₂×C₄	—	664	8	no	—
90	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊Dic₈₃	1	2	yes	—
92	C₂×C₄	#14581792	1	1	yes	characteristic
93	C₂×C₄	#14581792	1	2	yes	—
94	C₂×C₄	C₂×#13050579	1	4	yes	—
95	C₂×C₄	#14581814	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	#14581724	1	8	yes	—
98	C₂²	#19127571	1	1	yes	characteristic
99	C₂²	#19128074	1	2	yes	—
100	C₂²	#19128085	1	4	yes	—
101	C₄	—	1328	4	no	—
102	C₄	C₂×#14581724	1	2	yes	—
103	C₄	#19127739	1	4	yes	—
104	C₄	#19128099	1	2	yes	—
105	C₄	#19128103	1	4	yes	—
106	C₂	#24907064	1	4	yes	—
107	C₂	#24907620	1	1	yes	characteristic
108	C₂	#24907627	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907620

C₂

yes

maximal

#24907591

C₂

yes

maximal

C₂×C₈²×C₈₃

C₂

yes

maximal, characteristic

C₂×#14581724

—

#19127571

C₂²

yes

—

#19127464

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₈²×C₈₃

C₄

yes

complemented

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

#14581724

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

C₂²×Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

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×Q₈

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

—

C₂³×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

#135

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#106

yes

—

C₂²×C₈₃

#384

yes

characteristic

C₂²×C₈₃

#492

yes

—

C₂²×C₈₃

#495

yes

—

C₄×C₈₃

C₂×#106

yes

—

C₄×C₈₃

#427

yes

—

C₄×C₈₃

#498

yes

—

C₄×C₈₃

#499

yes

—

Dic₈₃

—

#8815

—

maximal, Sylow

C₂×C₈₃

#1708

yes

—

C₂×C₈₃

#1844

yes

characteristic

C₂×C₈₃

#1846

yes

—

#1844

—

#1834

—

C₂×C₈²

D₈₃

yes

characteristic

C₈₃

#8815

yes

characteristic, complemented, Sylow

C₂×#106

—

166

—

C₂×C₄×C₈

C₂×D₈₃

yes

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#384

—

#340

—

166

—

C₈²

Dic₈₃

yes

complemented

C₂²×C₈

D₃₃₂

yes

—

C₂²×C₈

D₃₃₂

yes

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄²

C₂²×D₈₃

yes

characteristic

#106

—

332

—

C₄×C₈

Dic₁₆₆

yes

—

C₄×C₈

Dic₁₆₆

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₂²×C₄

—

332

—

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

C₂²×C₄

C₂×D₃₃₂

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

#13050579

yes

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄²

C₂²×Dic₈₃

yes

characteristic

C₄²

C₂×Dic₁₆₆

yes

—

C₄²

C₂×Dic₁₆₆

yes

characteristic

C₂³

#14581789

yes

characteristic

C₂×C₄

—

664

—

C₂×C₄

—

664

—

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

—

C₂×C₄

#14581792

yes

characteristic

C₂×C₄

#14581792

yes

—

C₂×C₄

C₂×#13050579

yes

—

C₂×C₄

#14581814

yes

—

C₈

—

C₈

#14581724

yes

—

C₂²

#19127571

yes

characteristic

C₂²

#19128074

yes

—

100

C₂²

#19128085

yes

—

101

C₄

—

1328

—

102

C₄

C₂×#14581724

yes

—

103

C₄

#19127739

yes

—

104

C₄

#19128099

yes

—

105

C₄

#19128103

yes

—

106

C₂

#24907064

yes

—

107

C₂

#24907620

yes

characteristic

108

C₂

#24907627

yes

—

	subgroup	G/H
center	C₂³	#14581789
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×C₈²×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#127
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×C₈²	D₈₃
hypocenter	C₈₃	#8815
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14581789

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×C₈²×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

#127

cosocle

C₂×C₄²×C₈₃

C₂³

hypercenter

C₂×C₈²

D₈₃

hypocenter

C₈₃

#8815

LCS

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₈₃	C₄	semidirect
58. C₈₃	#8815	semidirect
64. C₈²	Dic₈₃	semidirect

H ≅ G/N

type

9. C₈²×C₈₃

C₄

semidirect

58. C₈₃

#8815

semidirect

64. C₈²

Dic₈₃

semidirect

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

subgroup	index
#24907620	2
#24907591	2
C₂×C₈²×C₈₃	2
#8815	83

subgroup

index

#24907620

#24907591

C₂×C₈²×C₈₃

#8815

Subgroups — #35298178

Conjugacy classes of subgroups

Complements

Maximal subgroups