Group #24915845 — subgroups

Subgroups — #24915845

Order 10624.

408 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 165 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	#19127607	C₂	1	3	yes	maximal
2	#19127599	C₂	1	3	yes	maximal
3	#19127585	C₂	1	6	yes	maximal
4	#19127569	C₂	1	2	yes	maximal
5	C₄³×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₈₃	C₂²	1	3	yes	—
8	C₂×Dic₈₃⋊C₄	C₂²	1	6	yes	—
9	C₈₃⋊#95	C₂²	1	6	yes	—
10	C₈₃⋊#95	C₂²	1	6	yes	—
11	C₈₃⋊#95	C₂²	1	6	yes	—
12	C₂×C₄²×C₈₃	C₂²	1	3	yes	—
13	C₂×C₄²×C₈₃	C₂²	1	3	yes	—
14	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₆₆	—	2	6	no	—
16	C₂²×Dic₈₃	C₂³	1	2	yes	—
17	C₂²×Dic₈₃	C₂³	1	6	yes	—
18	C₄⋊Dic₈₃	—	2	6	no	—
19	Dic₈₃⋊C₄	—	2	12	no	—
20	C₂²×C₄×C₈₃	C₂³	1	3	yes	—
21	C₂²×C₄×C₈₃	C₂³	1	3	yes	—
22	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
23	C₄²×C₈₃	—	2	6	no	—
24	C₄²×C₈₃	—	2	6	no	—
25	C₄²×C₈₃	Q₈	1	4	yes	complemented
26	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
27	C₂×C₄×C₈₃	—	2	3	no	—
28	C₂×C₄×C₈₃	—	2	3	no	—
29	C₂×C₄×C₈₃	—	2	3	no	—
30	C₂×C₄×C₈₃	—	2	3	no	—
31	C₂×C₄×C₈₃	C₂×Q₈	1	6	yes	—
32	C₂×C₄×C₈₃	#41	1	6	yes	—
33	C₂×C₄×C₈₃	#41	1	6	yes	—
34	C₂×Dic₈₃	—	2	12	no	—
35	C₂×Dic₈₃	—	2	6	no	—
36	C₂×Dic₈₃	—	2	6	no	—
37	Dic₁₆₆	—	4	4	no	—
38	C₂²×C₈₃	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₈₃	C₂×#41	1	3	yes	—
40	C₂²×C₈₃	C₂×#41	1	3	yes	—
41	C₄×C₈₃	—	2	6	no	—
42	C₄×C₈₃	—	2	6	no	—
43	C₄×C₈₃	#126	1	4	yes	—
44	Dic₈₃	—	4	2	no	—
45	Dic₈₃	—	4	6	no	—
46	C₂×C₈₃	C₂×#126	1	1	yes	characteristic
47	C₂×C₈₃	#529	1	3	yes	—
48	C₂×C₈₃	#533	1	3	yes	—
49	#2863	—	83	1	no	maximal, Sylow
50	C₈₃	#2863	1	1	yes	characteristic, complemented, Sylow
51	#391	—	83	3	no	—
52	#389	—	83	3	no	—
53	#387	—	83	6	no	—
54	#383	—	83	2	no	—
55	C₄³	D₈₃	1	1	yes	characteristic
56	C₂²×Q₈	—	83	1	no	—
57	C₂×C₄⋊C₄	—	83	6	no	—
58	C₂×C₄⋊C₄	—	83	3	no	—
59	C₂×C₄²	C₂×D₈₃	1	3	yes	—
60	C₂×C₄²	C₂×D₈₃	1	3	yes	—
61	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
62	#95	—	83	6	no	—
63	#95	—	83	6	no	—
64	#95	—	83	6	no	—
65	C₂×Q₈	—	166	6	no	—
66	C₂²×C₄	—	83	2	no	—
67	C₂²×C₄	—	83	6	no	—
68	C₂²×C₄	C₂²×D₈₃	1	3	yes	—
69	C₂²×C₄	C₂²×D₈₃	1	3	yes	—
70	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
71	C₄⋊C₄	—	166	12	no	—
72	C₄⋊C₄	—	166	6	no	—
73	C₄²	—	2	6	no	—
74	C₄²	—	2	6	no	—
75	C₄²	Dic₁₆₆	1	4	yes	complemented
76	C₂³	C₂³×D₈₃	1	1	yes	characteristic
77	Q₈	—	332	4	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	166	6	no	—
83	C₂×C₄	—	166	12	no	—
84	C₂×C₄	—	166	6	no	—
85	C₂×C₄	C₂×Dic₁₆₆	1	6	yes	—
86	C₂×C₄	#13050590	1	6	yes	—
87	C₂×C₄	#13050590	1	6	yes	—
88	C₂²	C₂²×Dic₁₆₆	1	1	yes	characteristic
89	C₂²	C₂×#13050590	1	3	yes	—
90	C₂²	C₂×#13050590	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	2	6	no	—
93	C₄	—	332	2	no	—
94	C₄	—	332	6	no	—
95	C₄	#14581786	1	4	yes	—
96	C₂	C₂×#14581786	1	1	yes	characteristic
97	C₂	#19128235	1	3	yes	—
98	C₂	#19128258	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19127607

C₂

yes

maximal

#19127599

C₂

yes

maximal

#19127585

C₂

yes

maximal

#19127569

C₂

yes

maximal

C₄³×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×Dic₈₃⋊C₄

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

C₂³

yes

—

C₂²×Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₂²×C₄×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

Q₈

yes

complemented

C₂³×C₈₃

C₂⁴

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

C₂²×C₈₃

C₂²×Q₈

yes

characteristic

C₂²×C₈₃

C₂×#41

yes

—

C₂²×C₈₃

C₂×#41

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#126

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

C₂×#126

yes

characteristic

C₂×C₈₃

#529

yes

—

C₂×C₈₃

#533

yes

—

#2863

—

maximal, Sylow

C₈₃

#2863

yes

characteristic, complemented, Sylow

—

—

—

—

C₄³

D₈₃

yes

characteristic

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

—

—

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄²

—

C₄²

—

C₄²

Dic₁₆₆

yes

complemented

C₂³

C₂³×D₈₃

yes

characteristic

Q₈

—

332

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

—

166

—

C₂×C₄

C₂×Dic₁₆₆

yes

—

C₂×C₄

#13050590

yes

—

C₂×C₄

#13050590

yes

—

C₂²

C₂²×Dic₁₆₆

yes

characteristic

C₂²

C₂×#13050590

yes

—

C₂²

C₂×#13050590

yes

—

C₄

—

C₄

—

C₄

—

332

—

C₄

—

332

—

C₄

#14581786

yes

—

C₂

C₂×#14581786

yes

characteristic

C₂

#19128235

yes

—

C₂

#19128258

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₃
commutator, socle, prime radical, cosocle	C₂³×C₈₃	C₂⁴
Fitting	C₄³×C₈₃	C₂
hypercenter	C₄³	D₈₃
hypocenter	C₈₃	#2863
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₃

commutator, socle, prime radical, cosocle

C₂³×C₈₃

C₂⁴

Fitting

C₄³×C₈₃

C₂

hypercenter

C₄³

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
25. C₄²×C₈₃	Q₈	semidirect
50. C₈₃	#2863	semidirect
75. C₄²	Dic₁₆₆	semidirect

H ≅ G/N

type

25. C₄²×C₈₃

Q₈

semidirect

50. C₈₃

#2863

semidirect

75. C₄²

Dic₁₆₆

semidirect

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

subgroup	index
#19127607	2
#19127599	2
#19127585	2
#19127569	2
C₄³×C₈₃	2
#2863	83

subgroup

index

C₄³×C₈₃

Subgroups — #24915845

Conjugacy classes of subgroups

Complements

Maximal subgroups