Group #24906372 — subgroups

Subgroups — #24906372

Order 10624.

148 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 71 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	#19127832	C₂	1	2	yes	maximal
2	#19127452	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal
4	#19127422	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
7	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	#14581703	—	2	2	no	—
9	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₈₃	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₈₃	C₂²	1	2	yes	—
13	#107×C₈₃	C₄	1	1	yes	characteristic
14	#103×C₈₃	C₄	1	2	yes	—
15	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×C₈₃⋊C₈	—	2	2	no	—
17	C₂×C₈₃⋊C₈	—	2	1	no	—
18	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
19	C₂×Q₈×C₈₃	C₂×C₄	1	2	yes	—
20	Q₁₆×C₈₃	—	2	2	no	—
21	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₈₃	—	2	2	no	—
24	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
25	C₄²×C₈₃	—	2	2	no	—
26	C₄²×C₈₃	C₂³	1	1	yes	characteristic
27	Q₈×C₈₃	—	2	2	no	—
28	Q₈×C₈₃	—	2	2	no	—
29	C₂×C₄×C₈₃	—	2	2	no	—
30	C₂×C₄×C₈₃	—	2	2	no	—
31	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
34	C₈×C₈₃	—	2	1	no	—
35	C₈×C₈₃	M₁₆	1	2	yes	—
36	C₈₃⋊C₈	—	2	2	no	—
37	C₈₃⋊C₈	—	4	1	no	—
38	C₈₃⋊C₈	—	4	2	no	—
39	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
40	C₄×C₈₃	—	2	1	no	—
41	C₄×C₈₃	—	4	2	no	—
42	C₄×C₈₃	—	4	2	no	—
43	C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₈₃	#131	1	1	yes	characteristic
45	C₄×C₈₃	#137	1	2	yes	—
46	C₂×C₈₃	#436	1	1	yes	characteristic
47	C₂×C₈₃	#441	1	1	yes	characteristic
48	C₂×C₈₃	#444	1	1	yes	characteristic
49	#1593	—	83	1	no	maximal, Sylow
50	C₈₃	#1593	1	1	yes	characteristic, complemented, Sylow
51	#446	—	83	2	no	—
52	C₄×Q₁₆	D₈₃	1	1	yes	characteristic
53	#334	—	83	1	no	—
54	Q₈⋊C₈	—	83	2	no	—
55	#322	—	83	1	no	—
56	C₂×Q₁₆	Dic₈₃	1	1	yes	characteristic
57	C₄×Q₈	C₂×D₈₃	1	2	yes	—
58	#107	Dic₈₃	1	1	yes	characteristic
59	C₄⋊C₈	—	83	2	no	—
60	C₄⋊C₈	—	166	2	no	—
61	#103	Dic₈₃	1	2	yes	—
62	#97	—	166	2	no	—
63	C₄×C₈	—	83	1	no	—
64	C₄×C₈	—	83	1	no	—
65	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
66	C₂×Q₈	C₂×Dic₈₃	1	2	yes	—
67	Q₁₆	—	2	2	no	—
68	C₂×C₈	—	83	2	no	—
69	C₂×C₈	—	166	2	no	—
70	C₂×C₈	—	166	1	no	—
71	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
72	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
75	C₄²	—	2	2	no	—
76	C₄²	C₂²×D₈₃	1	1	yes	characteristic
77	Q₈	—	2	2	no	—
78	Q₈	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
82	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
83	C₂×C₄	#13050589	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	332	1	no	—
86	C₈	—	332	2	no	—
87	C₈	—	166	2	no	—
88	C₈	#13050574	1	2	yes	—
89	C₂²	D₄×Dic₈₃	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	4	2	no	—
92	C₄	—	4	2	no	—
93	C₄	C₂×#13050574	1	1	yes	characteristic
94	C₄	#14581799	1	1	yes	characteristic
95	C₄	#14581824	1	2	yes	—
96	C₂	#19127779	1	1	yes	characteristic
97	C₂	#19127808	1	1	yes	characteristic
98	C₂	#19127823	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127832

C₂

yes

maximal

#19127452

C₂

yes

maximal, characteristic

Q₈⋊(C₈₃⋊C₈)

C₂

yes

maximal

#19127422

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₈₃

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

#14581703

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₈₃

C₄

yes

characteristic

C₄×Q₈×C₈₃

C₂²

yes

—

#107×C₈₃

C₄

yes

characteristic

#103×C₈₃

C₄

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₂×Q₈×C₈₃

C₂×C₄

yes

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

M₁₆

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×M₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#446

—

C₄×Q₁₆

D₈₃

yes

characteristic

#334

—

Q₈⋊C₈

—

#322

—

C₂×Q₁₆

Dic₈₃

yes

characteristic

C₄×Q₈

C₂×D₈₃

yes

—

#107

Dic₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

#103

Dic₈₃

yes

—

#97

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

C₂×Dic₈₃

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₈₃

yes

—

C₄²

—

C₄²

C₂²×D₈₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

C₂×C₄

D₄×D₈₃

yes

characteristic

C₂×C₄

#13050589

yes

characteristic

C₈

—

C₈

—

332

—

C₈

—

332

—

C₈

—

166

—

C₈

#13050574

yes

—

C₂²

D₄×Dic₈₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	C₄×Q₁₆×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₄×Q₁₆	D₈₃
hypocenter	C₈₃	#1593
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

C₄×Q₁₆×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₄×D₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₄×Q₁₆

D₈₃

hypocenter

C₈₃

#1593

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
50. C₈₃	#1593	semidirect

H ≅ G/N

type

50. C₈₃

#1593

semidirect

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

subgroup	index
#19127832	2
#19127452	2
Q₈⋊(C₈₃⋊C₈)	2
#19127422	2
C₄×Q₁₆×C₈₃	2
#1593	83

subgroup

index

Q₈⋊(C₈₃⋊C₈)

C₄×Q₁₆×C₈₃

Subgroups — #24906372

Conjugacy classes of subgroups

Complements

Maximal subgroups