Group #24906377 — subgroups

Subgroups — #24906377

Order 10624.

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

normal subgroups only

This group has 176 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	#19127779	C₂	1	2	yes	maximal, complemented
2	#19127452	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#19127422	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	—	2	2	no	—
7	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
9	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₈₃	C₄	1	1	yes	characteristic
12	C₄×D₄×C₈₃	C₂²	1	2	yes	—
13	#107×C₈₃	C₄	1	1	yes	characteristic
14	#102×C₈₃	C₄	1	2	yes	—
15	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#13050574	—	4	4	no	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₂×C₈₃⋊C₈	—	2	1	no	—
19	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
20	C₂×D₄×C₈₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₈₃	—	2	2	no	—
22	D₈×C₈₃	—	2	2	no	—
23	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
26	C₂²⋊C₄×C₈₃	—	2	2	no	—
27	C₄²×C₈₃	C₂³	1	1	yes	characteristic
28	C₂³×C₈₃	—	2	2	no	—
29	D₄×C₈₃	—	2	2	no	—
30	D₄×C₈₃	—	2	2	no	—
31	C₂×C₄×C₈₃	—	2	2	no	—
32	C₂×C₄×C₈₃	—	4	2	no	—
33	C₂×C₄×C₈₃	—	4	2	no	—
34	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
37	C₈×C₈₃	—	2	1	no	—
38	C₈×C₈₃	M₁₆	1	2	yes	complemented
39	C₈₃⋊C₈	—	2	2	no	—
40	C₈₃⋊C₈	—	4	1	no	—
41	C₈₃⋊C₈	—	4	2	no	—
42	C₂²×C₈₃	—	4	2	no	—
43	C₂²×C₈₃	—	4	2	no	—
44	C₂²×C₈₃	—	4	2	no	—
45	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
46	C₄×C₈₃	—	2	1	no	—
47	C₄×C₈₃	—	4	2	no	—
48	C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
49	C₄×C₈₃	#131	1	1	yes	characteristic
50	C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×C₈₃	—	8	2	no	—
52	C₂×C₈₃	#436	1	1	yes	characteristic
53	C₂×C₈₃	#442	1	1	yes	characteristic
54	C₂×C₈₃	#444	1	1	yes	characteristic
55	#1594	—	83	1	no	maximal, Sylow
56	C₈₃	#1594	1	1	yes	characteristic, complemented, Sylow
57	C₄×D₈	D₈₃	1	1	yes	characteristic
58	#436	—	83	2	no	—
59	#334	—	83	1	no	—
60	D₄⋊C₈	—	83	2	no	—
61	#322	—	83	1	no	—
62	C₂×D₈	Dic₈₃	1	1	yes	characteristic
63	C₂×M₁₆	—	166	2	no	—
64	C₄×D₄	C₂×D₈₃	1	2	yes	—
65	#107	Dic₈₃	1	1	yes	characteristic
66	C₄⋊C₈	—	83	2	no	—
67	#102	Dic₈₃	1	2	yes	—
68	C₂²⋊C₈	—	166	2	no	—
69	C₄×C₈	—	83	1	no	—
70	C₄×C₈	—	83	1	no	—
71	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
72	C₂×D₄	C₂×Dic₈₃	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	—	2	2	no	—
75	M₁₆	—	332	4	no	—
76	C₂×C₈	—	83	2	no	—
77	C₂×C₈	—	166	2	no	—
78	C₂×C₈	—	166	1	no	—
79	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
80	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
81	C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂²×D₈₃	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
91	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
92	C₂×C₄	#13050589	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	332	1	no	—
95	C₈	—	332	2	no	—
96	C₈	—	166	2	no	—
97	C₈	#13050574	1	2	yes	complemented
98	C₂²	—	4	2	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×Dic₈₃	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	C₂×#13050574	1	1	yes	characteristic
105	C₄	#14581799	1	1	yes	characteristic
106	C₄	#14581816	1	2	yes	—
107	C₂	—	8	2	no	—
108	C₂	#19127779	1	1	yes	characteristic
109	C₂	#19127813	1	1	yes	characteristic
110	C₂	#19127823	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127779

C₂

yes

maximal, complemented

#19127452

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, complemented

#19127422

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₈₃

C₄

yes

characteristic

C₄×D₄×C₈₃

C₂²

yes

—

#107×C₈₃

C₄

yes

characteristic

#102×C₈₃

C₄

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

—

D₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×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

complemented

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×M₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₄×D₈

D₈₃

yes

characteristic

#436

—

#334

—

D₄⋊C₈

—

#322

—

C₂×D₈

Dic₈₃

yes

characteristic

C₂×M₁₆

—

166

—

C₄×D₄

C₂×D₈₃

yes

—

#107

Dic₈₃

yes

characteristic

C₄⋊C₈

—

#102

Dic₈₃

yes

—

C₂²⋊C₈

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×D₄

C₂×Dic₈₃

yes

—

C₂²×C₄

—

D₈

—

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₄⋊C₄

C₂×Dic₈₃

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

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

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×Dic₈₃

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂×#13050574

yes

characteristic

105

C₄

#14581799

yes

characteristic

106

C₄

#14581816

yes

—

107

C₂

—

108

C₂

#19127779

yes

characteristic

109

C₂

#19127813

yes

characteristic

110

C₂

#19127823

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	C₄×D₈×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂×D₈×C₈₃	C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₄×D₈	D₈₃
hypocenter	C₈₃	#1594
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

C₄×D₈×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂×D₈×C₈₃

C₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₄×D₈

D₈₃

hypocenter

C₈₃

#1594

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #19127779	C₂	semidirect
2. #19127452	C₂	semidirect
3. D₄⋊(C₈₃⋊C₈)	C₂	semidirect
4. #19127422	C₂	semidirect
38. C₈×C₈₃	M₁₆	semidirect
56. C₈₃	#1594	semidirect
97. C₈	#13050574	semidirect

H ≅ G/N

type

1. #19127779

C₂

semidirect

2. #19127452

C₂

semidirect

3. D₄⋊(C₈₃⋊C₈)

C₂

semidirect

4. #19127422

C₂

semidirect

38. C₈×C₈₃

M₁₆

semidirect

56. C₈₃

#1594

semidirect

97. C₈

#13050574

semidirect

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

subgroup	index
#19127779	2
#19127452	2
D₄⋊(C₈₃⋊C₈)	2
#19127422	2
C₄×D₈×C₈₃	2
#1594	83

subgroup

index

D₄⋊(C₈₃⋊C₈)

C₄×D₈×C₈₃

Subgroups — #24906377

Conjugacy classes of subgroups

Complements

Maximal subgroups