Group #24906288 — subgroups

Subgroups — #24906288

Order 10624.

192 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×D₆₆₄	C₂	1	1	yes	maximal, characteristic
2	C₈×D₃₃₂	C₂	1	2	yes	maximal, complemented
3	#19127452	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127430	C₂	1	2	yes	maximal, complemented
5	#322×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₆₆₄	C₄	1	1	yes	characteristic
7	C₂×C₈×D₈₃	—	2	2	no	—
8	C₄×D₃₃₂	C₂²	1	2	yes	—
9	#14581726	C₄	1	1	yes	characteristic
10	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
11	#14581714	C₄	1	2	yes	—
12	(C₂×D₈₃)⋊C₈	—	2	2	no	—
13	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×D₃₃₂	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₈₃	—	2	2	no	—
18	D₆₆₄	C₈	1	4	yes	complemented
19	C₂×C₈₃⋊C₈	—	2	2	no	—
20	C₈×D₈₃	—	4	4	no	—
21	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
22	(C₂×D₈₃)⋊C₄	—	2	2	no	—
23	C₂×C₈×C₈₃	—	2	1	no	—
24	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₈₃	D₄	1	2	yes	—
27	C₄²×C₈₃	C₂³	1	1	yes	characteristic
28	C₂²×D₈₃	—	2	2	no	—
29	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
32	C₂×Dic₈₃	—	2	2	no	—
33	D₃₃₂	—	2	2	no	—
34	D₃₃₂	C₂×C₈	1	4	yes	complemented
35	C₄×D₈₃	—	4	2	no	—
36	C₄×D₈₃	—	4	2	no	—
37	C₈×C₈₃	—	2	1	no	—
38	C₈×C₈₃	—	2	2	no	—
39	C₈×C₈₃	—	4	1	no	—
40	C₈×C₈₃	C₂×C₈	1	2	yes	complemented
41	C₈₃⋊C₈	—	4	2	no	—
42	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
43	C₂×D₈₃	—	2	4	no	—
44	C₂×D₈₃	—	4	2	no	—
45	C₂×D₈₃	—	4	2	no	—
46	C₄×C₈₃	—	2	1	no	—
47	C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₈₃	#131	1	1	yes	characteristic
49	C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
50	Dic₈₃	—	4	2	no	—
51	C₂×C₈₃	C₈×D₄	1	1	yes	characteristic
52	C₂×C₈₃	#442	1	1	yes	characteristic
53	C₂×C₈₃	#444	1	1	yes	characteristic
54	D₈₃	—	4	4	no	—
55	#1580	—	83	1	no	maximal, Sylow
56	C₈₃	#1580	1	1	yes	characteristic, complemented, Sylow
57	C₄×D₈	—	83	1	no	—
58	C₈×D₄	—	83	2	no	—
59	#334	—	83	1	no	—
60	D₄⋊C₈	—	83	2	no	—
61	#322	D₈₃	1	1	yes	characteristic, complemented
62	C₂×D₈	—	83	1	no	—
63	C₂²×C₈	—	166	2	no	—
64	C₄×D₄	—	83	2	no	—
65	#107	—	83	1	no	—
66	C₄⋊C₈	—	83	2	no	—
67	#102	—	83	2	no	—
68	C₂²⋊C₈	—	166	2	no	—
69	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
70	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
71	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
72	C₂×D₄	—	83	2	no	—
73	C₂²×C₄	—	166	2	no	—
74	D₈	—	83	4	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	332	4	no	—
77	C₂×C₈	—	166	2	no	—
78	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
79	C₂×C₈	C₄×D₈₃	1	1	yes	characteristic
80	C₂×C₈	D₃₃₂	1	2	yes	—
81	C₄⋊C₄	—	83	2	no	—
82	C₂²⋊C₄	—	166	2	no	—
83	C₄²	C₂²×D₈₃	1	1	yes	characteristic
84	C₂³	—	166	2	no	—
85	D₄	—	83	4	no	—
86	D₄	—	166	2	no	—
87	C₂×C₄	—	332	2	no	—
88	C₂×C₄	—	332	2	no	—
89	C₂×C₄	—	166	2	no	—
90	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
91	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
92	C₂×C₄	#13050590	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	2	2	no	—
95	C₈	—	4	1	no	—
96	C₈	—	332	2	no	—
97	C₈	C₈×D₈₃	1	2	yes	complemented
98	C₂²	—	332	2	no	—
99	C₂²	—	332	2	no	—
100	C₂²	—	166	4	no	—
101	C₂²	C₄×D₃₃₂	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	332	2	no	—
104	C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
105	C₄	#14581798	1	1	yes	characteristic
106	C₄	#14581820	1	2	yes	—
107	C₂	—	332	4	no	—
108	C₂	C₈×D₃₃₂	1	1	yes	characteristic
109	C₂	#19127814	1	1	yes	characteristic
110	C₂	#19127824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₆₆₄

C₂

yes

maximal, characteristic

C₈×D₃₃₂

C₂

yes

maximal, complemented

#19127452

C₂

yes

maximal, characteristic, complemented

#19127430

C₂

yes

maximal, complemented

#322×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×D₆₆₄

C₄

yes

characteristic

C₂×C₈×D₈₃

—

C₄×D₃₃₂

C₂²

yes

—

#14581726

C₄

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

#14581714

C₄

yes

—

(C₂×D₈₃)⋊C₈

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×D₃₃₂

C₂×C₄

yes

—

C₂×C₄×D₈₃

—

D₆₆₄

C₈

yes

complemented

C₂×C₈₃⋊C₈

—

C₈×D₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

(C₂×D₈₃)⋊C₄

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

C₂×C₈

yes

complemented

C₄×D₈₃

—

C₄×D₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂×C₈

yes

complemented

C₈₃⋊C₈

—

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²×C₈

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

C₂×C₈₃

C₈×D₄

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

—

#1580

—

maximal, Sylow

C₈₃

#1580

yes

characteristic, complemented, Sylow

C₄×D₈

—

C₈×D₄

—

#334

—

D₄⋊C₈

—

#322

D₈₃

yes

characteristic, complemented

C₂×D₈

—

C₂²×C₈

—

166

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₄×C₈

C₂×D₈₃

yes

characteristic

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

166

—

D₈

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

—

166

—

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

C₄×D₈₃

yes

characteristic

C₂×C₈

D₃₃₂

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

166

—

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

—

166

—

D₄

—

D₄

—

166

—

C₂×C₄

—

332

—

C₂×C₄

—

332

—

C₂×C₄

—

166

—

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

C₂×C₄

C₂×D₃₃₂

yes

characteristic

C₂×C₄

#13050590

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

332

—

C₈

C₈×D₈₃

yes

complemented

C₂²

—

332

—

C₂²

—

332

—

100

C₂²

—

166

—

101

C₂²

C₄×D₃₃₂

yes

characteristic

102

C₄

—

103

C₄

—

332

—

104

C₄

C₂×C₈×D₈₃

yes

characteristic

105

C₄

#14581798

yes

characteristic

106

C₄

#14581820

yes

—

107

C₂

—

332

—

108

C₂

C₈×D₃₃₂

yes

characteristic

109

C₂

#19127814

yes

characteristic

110

C₂

#19127824

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₃₂
commutator	C₄×C₈₃	C₂²×C₈
Frattini	C₄²	C₂²×D₈₃
Fitting	#322×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂×D₆₆₄	C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	#322	D₈₃
hypocenter	C₈₃	#1580
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₃₃₂

commutator

C₄×C₈₃

C₂²×C₈

Frattini

C₄²

C₂²×D₈₃

Fitting

#322×C₈₃

C₂

socle

C₂²×C₈₃

C₄×D₄

prime radical

C₂×D₆₆₄

C₄

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
2. C₈×D₃₃₂	C₂	semidirect
3. #19127452	C₂	semidirect
4. #19127430	C₂	semidirect
5. #322×C₈₃	C₂	semidirect
18. D₆₆₄	C₈	semidirect
34. D₃₃₂	C₂×C₈	semidirect
40. C₈×C₈₃	C₂×C₈	semidirect
56. C₈₃	#1580	semidirect
61. #322	D₈₃	semidirect
97. C₈	C₈×D₈₃	semidirect

H ≅ G/N

type

2. C₈×D₃₃₂

C₂

semidirect

3. #19127452

C₂

semidirect

4. #19127430

C₂

semidirect

5. #322×C₈₃

C₂

semidirect

18. D₆₆₄

C₈

semidirect

34. D₃₃₂

C₂×C₈

semidirect

40. C₈×C₈₃

C₂×C₈

semidirect

56. C₈₃

#1580

semidirect

61. #322

D₈₃

semidirect

97. C₈

C₈×D₈₃

semidirect

Subgroups — #24906288

Conjugacy classes of subgroups

Complements