Group #35298181 — subgroups

Subgroups — #35298181

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	#24907623	C₂	1	2	yes	maximal
2	#24907591	C₂	1	4	yes	maximal
3	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581726	—	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	#14581726	—	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	D₈	1	4	yes	—
29	C₂×C₈×C₈₃	Q₁₆	1	4	yes	—
30	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
33	C₂³×C₈₃	#127	1	1	yes	characteristic
34	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
40	C₂×C₄×C₈₃	#135	1	8	yes	—
41	C₂×Dic₈₃	—	8	4	no	—
42	C₂×Dic₈₃	—	8	8	no	—
43	C₈×C₈₃	—	2	8	no	—
44	C₈×C₈₃	#107	1	8	yes	—
45	C₂²×C₈₃	#384	1	1	yes	characteristic
46	C₂²×C₈₃	#493	1	1	yes	characteristic
47	C₂²×C₈₃	#494	1	1	yes	characteristic
48	C₂²×C₈₃	#495	1	4	yes	—
49	C₄×C₈₃	C₂×#107	1	2	yes	—
50	C₄×C₈₃	#427	1	4	yes	—
51	C₄×C₈₃	#499	1	4	yes	—
52	C₄×C₈₃	#500	1	2	yes	—
53	Dic₈₃	—	16	4	no	—
54	#8816	—	83	1	no	maximal, Sylow
55	C₂×C₈₃	#1709	1	2	yes	—
56	C₂×C₈₃	#1710	1	2	yes	—
57	C₂×C₈₃	#1845	1	1	yes	characteristic
58	C₂×C₈₃	#1846	1	2	yes	—
59	#1845	—	83	2	no	—
60	#1834	—	83	4	no	—
61	C₂×C₈²	D₈₃	1	1	yes	characteristic
62	C₈₃	#8816	1	1	yes	characteristic, complemented, Sylow
63	C₂×#107	—	166	4	no	—
64	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
65	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
66	#384	—	83	4	no	—
67	#340	—	166	8	no	—
68	C₈²	Dic₈₃	1	4	yes	complemented
69	C₂²×C₈	D₃₃₂	1	4	yes	—
70	C₂²×C₈	D₃₃₂	1	2	yes	—
71	C₂×C₄⋊C₄	—	166	8	no	—
72	C₂×C₄⋊C₄	—	166	4	no	—
73	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
74	#107	—	332	8	no	—
75	C₄×C₈	Dic₁₆₆	1	4	yes	—
76	C₄×C₈	Dic₁₆₆	1	2	yes	—
77	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
78	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
79	C₂²×C₄	—	332	4	no	—
80	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₃₃₂	1	2	yes	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	C₄⋊Dic₈₃	1	8	yes	—
84	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
85	C₂×C₈	D₆₆₄	1	4	yes	—
86	C₂×C₈	Dic₃₃₂	1	4	yes	—
87	C₄⋊C₄	—	332	8	no	—
88	C₄⋊C₄	—	332	16	no	—
89	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
90	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
91	C₄²	C₂×Dic₁₆₆	1	2	yes	—
92	C₂³	#14581789	1	1	yes	characteristic
93	C₂×C₄	—	664	4	no	—
94	C₂×C₄	—	664	8	no	—
95	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊Dic₈₃	1	2	yes	—
97	C₂×C₄	#14581792	1	2	yes	—
98	C₂×C₄	#14581792	1	1	yes	characteristic
99	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
100	C₂×C₄	C₂×Dic₃₃₂	1	2	yes	—
101	C₂×C₄	#14581814	1	8	yes	—
102	C₈	—	2	8	no	—
103	C₈	#14581726	1	8	yes	—
104	C₂²	#19127571	1	1	yes	characteristic
105	C₂²	#19128078	1	1	yes	characteristic
106	C₂²	#19128081	1	1	yes	characteristic
107	C₂²	#19128085	1	4	yes	—
108	C₄	—	1328	4	no	—
109	C₄	C₂×#14581726	1	2	yes	—
110	C₄	#19127739	1	4	yes	—
111	C₄	#19128103	1	4	yes	—
112	C₄	#19128106	1	2	yes	—
113	C₂	#24907069	1	2	yes	—
114	C₂	#24907073	1	2	yes	—
115	C₂	#24907623	1	1	yes	characteristic
116	C₂	#24907627	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907623

C₂

yes

maximal

#24907591

C₂

yes

maximal

C₂×C₈²×C₈₃

C₂

yes

maximal, characteristic

C₂×#14581726

—

#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

—

#14581726

—

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

D₈

yes

—

C₂×C₈×C₈₃

Q₁₆

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

—

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×C₄×C₈₃

C₂×Q₁₆

yes

—

C₂×C₄×C₈₃

#135

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#107

yes

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₄×C₈₃

C₂×#107

yes

—

C₄×C₈₃

#427

yes

—

C₄×C₈₃

#499

yes

—

C₄×C₈₃

#500

yes

—

Dic₈₃

—

#8816

—

maximal, Sylow

C₂×C₈₃

#1709

yes

—

C₂×C₈₃

#1710

yes

—

C₂×C₈₃

#1845

yes

characteristic

C₂×C₈₃

#1846

yes

—

#1845

—

#1834

—

C₂×C₈²

D₈₃

yes

characteristic

C₈₃

#8816

yes

characteristic, complemented, Sylow

C₂×#107

—

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

#107

—

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

D₆₆₄

yes

—

C₂×C₈

Dic₃₃₂

yes

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄²

C₂²×Dic₈₃

yes

characteristic

C₄²

C₂×Dic₁₆₆

yes

characteristic

C₄²

C₂×Dic₁₆₆

yes

—

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

—

C₂×C₄

#14581792

yes

characteristic

C₂×C₄

C₂×D₆₆₄

yes

—

100

C₂×C₄

C₂×Dic₃₃₂

yes

—

101

C₂×C₄

#14581814

yes

—

102

C₈

—

103

C₈

#14581726

yes

—

104

C₂²

#19127571

yes

characteristic

105

C₂²

#19128078

yes

characteristic

106

C₂²

#19128081

yes

characteristic

107

C₂²

#19128085

yes

—

108

C₄

—

1328

—

109

C₄

C₂×#14581726

yes

—

110

C₄

#19127739

yes

—

111

C₄

#19128103

yes

—

112

C₄

#19128106

yes

—

113

C₂

#24907069

yes

—

114

C₂

#24907073

yes

—

115

C₂

#24907623

yes

characteristic

116

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	#8816
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

#8816

LCS

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₈₃	C₄	semidirect
62. C₈₃	#8816	semidirect
68. C₈²	Dic₈₃	semidirect

H ≅ G/N

type

9. C₈²×C₈₃

C₄

semidirect

62. C₈₃

#8816

semidirect

68. C₈²

Dic₈₃

semidirect

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

subgroup	index
#24907623	2
#24907591	2
C₂×C₈²×C₈₃	2
#8816	83

subgroup

index

#24907623

#24907591

C₂×C₈²×C₈₃

#8816

Subgroups — #35298181

Conjugacy classes of subgroups

Complements

Maximal subgroups