Group #24906604 — subgroups

Subgroups — #24906604

Order 10624.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19128431	C₂	1	1	yes	maximal, characteristic
2	#19128080	C₂	1	1	yes	maximal, characteristic
3	#19127997	C₂	1	2	yes	maximal
4	#19127431	C₂	1	2	yes	maximal
5	#333×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050580	—	2	2	no	—
7	#14581792	C₂²	1	1	yes	characteristic
8	#14581791	—	2	2	no	—
9	#14581790	—	2	2	no	—
10	C₄×Dic₁₆₆	C₂²	1	2	yes	—
11	#14581757	—	2	2	no	—
12	#14581726	—	2	2	no	—
13	#14581717	—	2	2	no	—
14	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
17	C₂×Dic₁₆₆	—	2	1	no	—
18	C₂×Dic₁₆₆	—	2	2	no	—
19	C₂×Dic₁₆₆	D₄	1	2	yes	—
20	#13050580	—	4	4	no	—
21	C₂×C₈₃⋊C₈	—	2	1	no	—
22	C₄⋊Dic₈₃	—	2	2	no	—
23	C₄⋊Dic₈₃	D₄	1	2	yes	—
24	Dic₈₃⋊C₄	—	2	4	no	—
25	Dic₈₃⋊C₄	—	2	2	no	—
26	C₄×Dic₈₃	—	2	2	no	—
27	C₄×Dic₈₃	—	2	2	no	—
28	C₂×Q₈×C₈₃	D₄	1	2	yes	—
29	C₂×C₈×C₈₃	—	2	2	no	—
30	C₄⋊C₄×C₈₃	—	2	2	no	—
31	C₄²×C₈₃	C₂³	1	1	yes	characteristic
32	Q₈×C₈₃	—	4	2	no	—
33	C₂×C₄×C₈₃	—	2	2	no	—
34	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₈₃	—	2	2	no	—
37	C₂×Dic₈₃	—	2	2	no	—
38	C₂×Dic₈₃	—	4	1	no	—
39	Dic₁₆₆	—	2	2	no	—
40	Dic₁₆₆	—	4	2	no	—
41	Dic₁₆₆	—	8	2	no	—
42	Dic₁₆₆	Q₁₆	1	4	yes	complemented
43	C₈×C₈₃	—	4	2	no	—
44	C₈₃⋊C₈	—	2	2	no	—
45	C₂²×C₈₃	#120	1	1	yes	characteristic
46	C₄×C₈₃	—	2	1	no	—
47	C₄×C₈₃	—	4	2	no	—
48	C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
49	C₄×C₈₃	#137	1	2	yes	—
50	Dic₈₃	—	2	4	no	—
51	Dic₈₃	—	4	2	no	—
52	Dic₈₃	—	8	1	no	—
53	C₂×C₈₃	#451	1	2	yes	—
54	C₂×C₈₃	#453	1	1	yes	characteristic
55	#1633	—	83	1	no	maximal, Sylow
56	C₈₃	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	—	83	1	no	—
58	#494	—	83	1	no	—
59	#477	—	83	2	no	—
60	#333	D₈₃	1	1	yes	characteristic
61	Q₈⋊C₈	—	83	2	no	—
62	C₂×Q₁₆	—	166	2	no	—
63	C₄⋊Q₈	—	83	1	no	—
64	C₄⋊Q₈	—	166	2	no	—
65	C₄⋊Q₈	—	166	2	no	—
66	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
67	C₄×Q₈	—	83	2	no	—
68	C₄×Q₈	—	166	2	no	—
69	#107	—	166	2	no	—
70	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
71	#103	—	166	2	no	—
72	C₄×C₈	—	83	1	no	—
73	C₂×Q₈	—	83	2	no	—
74	C₂×Q₈	—	166	2	no	—
75	C₂×Q₈	—	166	1	no	—
76	C₂×Q₈	D₃₃₂	1	2	yes	—
77	Q₁₆	—	332	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	166	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	83	2	no	—
82	C₄⋊C₄	—	166	4	no	—
83	C₄⋊C₄	—	166	2	no	—
84	C₄⋊C₄	—	166	2	no	—
85	C₄²	—	166	2	no	—
86	C₄²	—	166	2	no	—
87	C₄²	C₂²×D₈₃	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	332	2	no	—
90	Q₈	—	83	4	no	—
91	Q₈	—	664	2	no	—
92	Q₈	—	166	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	332	1	no	—
95	C₂×C₄	—	166	2	no	—
96	C₂×C₄	—	166	2	no	—
97	C₂×C₄	D₄×D₈₃	1	2	yes	—
98	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
99	C₈	—	4	2	no	—
100	C₈	—	166	2	no	—
101	C₂²	#14581762	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	332	2	no	—
105	C₄	—	664	1	no	—
106	C₄	—	166	4	no	—
107	C₄	Q₁₆×D₈₃	1	2	yes	—
108	C₄	#14581821	1	2	yes	—
109	C₂	#19127858	1	2	yes	—
110	C₂	#19127871	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128431

C₂

yes

maximal, characteristic

#19128080

C₂

yes

maximal, characteristic

#19127997

C₂

yes

maximal

#19127431

C₂

yes

maximal

#333×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050580

—

#14581792

C₂²

yes

characteristic

#14581791

—

#14581790

—

C₄×Dic₁₆₆

C₂²

yes

—

#14581757

—

#14581726

—

#14581717

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

D₄

yes

—

#13050580

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

D₄

yes

—

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

Q₁₆

yes

complemented

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

#120

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×Q₁₆

yes

—

C₄×C₈₃

#137

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#451

yes

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

Q₈²

—

#494

—

#477

—

#333

D₈₃

yes

characteristic

Q₈⋊C₈

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

166

—

#107

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

—

#103

—

166

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂×Q₈

D₃₃₂

yes

—

Q₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

166

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄²

—

166

—

C₄²

—

166

—

C₄²

C₂²×D₈₃

yes

characteristic

Q₈

—

Q₈

—

332

—

Q₈

—

Q₈

—

664

—

Q₈

—

166

—

C₂×C₄

—

C₂×C₄

—

332

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

D₄×D₈₃

yes

—

C₂×C₄

C₂×D₃₃₂

yes

characteristic

C₈

—

100

C₈

—

166

—

101

C₂²

#14581762

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

332

—

105

C₄

—

664

—

106

C₄

—

166

—

107

C₄

Q₁₆×D₈₃

yes

—

108

C₄

#14581821

yes

—

109

C₂

#19127858

yes

—

110

C₂

#19127871

yes

characteristic

	subgroup	G/H
center	C₂²	#14581762
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#333×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	#120
hypercenter	#333	D₈₃
hypocenter	C₈₃	#1633
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#14581762

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#333×C₈₃

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
42. Dic₁₆₆	Q₁₆	semidirect
56. C₈₃	#1633	semidirect

H ≅ G/N

type

42. Dic₁₆₆

Q₁₆

semidirect

56. C₈₃

#1633

semidirect

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

subgroup	index
#19128431	2
#19128080	2
#19127997	2
#19127431	2
#333×C₈₃	2
#1633	83

subgroup

index

#333×C₈₃

Subgroups — #24906604

Conjugacy classes of subgroups

Complements

Maximal subgroups