Group #24906207 — subgroups

Subgroups — C₂×#19127454 #24906207

Order 10624.

200 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×C₄⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal
2	#19127454	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈₃⋊C₈	—	2	2	no	—
5	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
6	C₄⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
7	C₂²×C₈×C₈₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₈₃	C₄	1	1	yes	characteristic
9	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₈₃	C₄	1	2	yes	—
11	C₄×C₈×C₈₃	C₂²	1	2	yes	—
12	C₂×C₈₃⋊C₈	—	2	4	no	—
13	C₂×C₈₃⋊C₈	—	4	4	no	—
14	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₈₃	—	2	2	no	—
18	C₂×C₈×C₈₃	C₈	1	4	yes	complemented
19	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
21	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₈₃	D₄	1	1	yes	characteristic
23	C₄²×C₈₃	Q₈	1	1	yes	characteristic
24	C₄²×C₈₃	C₂³	1	1	yes	characteristic
25	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₈₃	—	2	2	no	—
27	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₈₃	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₈₃	M₁₆	1	2	yes	—
32	C₂×C₄×C₈₃	D₈	1	2	yes	—
33	C₂×C₄×C₈₃	Q₁₆	1	2	yes	—
34	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
37	C₈×C₈₃	—	2	2	no	—
38	C₈×C₈₃	C₂×C₈	1	4	yes	complemented
39	C₈₃⋊C₈	—	4	4	no	—
40	C₂²×C₈₃	C₄⋊C₈	1	2	yes	—
41	C₂²×C₈₃	#107	1	2	yes	—
42	C₂²×C₈₃	#108	1	2	yes	—
43	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₈₃	—	2	1	no	—
45	C₄×C₈₃	—	2	1	no	—
46	C₄×C₈₃	C₄⋊C₈	1	2	yes	—
47	C₄×C₈₃	#107	1	2	yes	—
48	C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
51	C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₈₃	#335	1	4	yes	complemented
53	C₂×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₈₃	C₂×#107	1	1	yes	characteristic
55	C₂×C₈₃	C₂×#108	1	1	yes	characteristic
56	C₂×#335	—	83	1	no	maximal, Sylow
57	C₈₃	C₂×#335	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄⋊C₈	—	83	2	no	—
59	C₂×C₄×C₈	D₈₃	1	1	yes	characteristic
60	#335	—	83	4	no	—
61	C₂²×C₈	—	166	2	no	—
62	C₂²×C₈	Dic₈₃	1	1	yes	characteristic
63	C₂²×C₈	Dic₈₃	1	1	yes	characteristic
64	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
65	C₄⋊C₈	—	83	4	no	—
66	C₄⋊C₈	—	166	2	no	—
67	C₄×C₈	Dic₈₃	1	2	yes	—
68	C₄×C₈	C₂×D₈₃	1	2	yes	—
69	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
70	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
71	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	332	4	no	—
74	C₂×C₈	—	166	4	no	—
75	C₂×C₈	C₈₃⋊C₈	1	4	yes	complemented
76	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
77	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
78	C₄²	Dic₁₆₆	1	1	yes	characteristic
79	C₄²	D₃₃₂	1	1	yes	characteristic
80	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
81	C₄²	C₂²×D₈₃	1	1	yes	characteristic
82	C₂³	C₄⋊Dic₈₃	1	1	yes	characteristic
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
85	C₂×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
86	C₂×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
87	C₂×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
88	C₂×C₄	#13050574	1	2	yes	—
89	C₂×C₄	D₆₆₄	1	2	yes	—
90	C₂×C₄	Dic₃₃₂	1	2	yes	—
91	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
92	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
93	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
94	C₈	—	2	2	no	—
95	C₈	—	332	4	no	—
96	C₈	C₂×C₈₃⋊C₈	1	4	yes	complemented
97	C₂²	C₄⋊(C₈₃⋊C₈)	1	2	yes	—
98	C₂²	#14581726	1	2	yes	—
99	C₂²	#14581728	1	2	yes	—
100	C₂²	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	C₄⋊(C₈₃⋊C₈)	1	2	yes	—
104	C₄	#14581726	1	2	yes	—
105	C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
106	C₄	C₂×#13050574	1	1	yes	characteristic
107	C₄	C₂×D₆₆₄	1	1	yes	characteristic
108	C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
109	C₂	#19127454	1	4	yes	direct product factor
110	C₂	C₂×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
111	C₂	C₂×#14581726	1	1	yes	characteristic
112	C₂	C₂×#14581728	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₈₃⋊C₈)

C₂

yes

maximal

#19127454

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²×C₈×C₈₃

C₄

yes

characteristic

C₂²×C₈×C₈₃

C₄

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₄

yes

—

C₄×C₈×C₈₃

C₂²

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

Q₈

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₈

yes

complemented

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

Q₈

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

M₁₆

yes

—

C₂×C₄×C₈₃

D₈

yes

—

C₂×C₄×C₈₃

Q₁₆

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

C₂×C₈

yes

complemented

C₈₃⋊C₈

—

C₂²×C₈₃

C₄⋊C₈

yes

—

C₂²×C₈₃

#107

yes

—

C₂²×C₈₃

#108

yes

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₄⋊C₈

yes

—

C₄×C₈₃

#107

yes

—

C₄×C₈₃

C₂²×C₈

yes

characteristic

C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₄×C₈₃

C₂×D₈

yes

characteristic

C₄×C₈₃

C₂×Q₁₆

yes

characteristic

C₂×C₈₃

#335

yes

complemented

C₂×C₈₃

C₂×C₄⋊C₈

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₈₃

yes

characteristic

#335

—

C₂²×C₈

—

166

—

C₂²×C₈

Dic₈₃

yes

characteristic

C₂²×C₈

Dic₈₃

yes

characteristic

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄×C₈

Dic₈₃

yes

—

C₄×C₈

C₂×D₈₃

yes

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

—

166

—

C₂×C₈

C₈₃⋊C₈

yes

complemented

C₂×C₈

C₂×Dic₈₃

yes

—

C₂×C₈

C₂×Dic₈₃

yes

—

C₄²

Dic₁₆₆

yes

characteristic

C₄²

D₃₃₂

yes

characteristic

C₄²

C₂×Dic₈₃

yes

characteristic

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

C₂×C₈₃⋊C₈

yes

—

C₂×C₄

#13050574

yes

—

C₂×C₄

D₆₆₄

yes

—

C₂×C₄

Dic₃₃₂

yes

—

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

C₂×C₄

C₂×D₃₃₂

yes

characteristic

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

C₈

—

C₈

—

332

—

C₈

C₂×C₈₃⋊C₈

yes

complemented

C₂²

C₄⋊(C₈₃⋊C₈)

yes

—

C₂²

#14581726

yes

—

C₂²

#14581728

yes

—

100

C₂²

C₂×C₄⋊Dic₈₃

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

C₄⋊(C₈₃⋊C₈)

yes

—

104

C₄

#14581726

yes

—

105

C₄

C₂²×C₈₃⋊C₈

yes

characteristic

106

C₄

C₂×#13050574

yes

characteristic

107

C₄

C₂×D₆₆₄

yes

characteristic

108

C₄

C₂×Dic₃₃₂

yes

characteristic

109

C₂

#19127454

yes

direct product factor

110

C₂

C₂×C₄⋊(C₈₃⋊C₈)

yes

characteristic

111

C₂

C₂×#14581726

yes

characteristic

112

C₂

C₂×#14581728

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₃₃₂
commutator	C₄×C₈₃	C₂²×C₈
Frattini	C₄²	C₂²×D₈₃
Fitting	C₂×C₄×C₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₄⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₂×C₄×C₈	D₈₃
hypocenter	C₈₃	C₂×#335
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₃₃₂

commutator

C₄×C₈₃

C₂²×C₈

Frattini

C₄²

C₂²×D₈₃

Fitting

C₂×C₄×C₈×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₄⋊C₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₂×C₄×C₈

D₈₃

hypocenter

C₈₃

C₂×#335

LCS

G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #19127454	C₂	direct (N × H)
18. C₂×C₈×C₈₃	C₈	semidirect
38. C₈×C₈₃	C₂×C₈	semidirect
52. C₂×C₈₃	#335	semidirect
57. C₈₃	C₂×#335	semidirect
75. C₂×C₈	C₈₃⋊C₈	semidirect
96. C₈	C₂×C₈₃⋊C₈	semidirect
109. C₂	#19127454	direct (N × H)

H ≅ G/N

type

2. #19127454

C₂

direct (N × H)

18. C₂×C₈×C₈₃

C₈

semidirect

38. C₈×C₈₃

C₂×C₈

semidirect

52. C₂×C₈₃

#335

semidirect

57. C₈₃

C₂×#335

semidirect

75. C₂×C₈

C₈₃⋊C₈

semidirect

96. C₈

C₂×C₈₃⋊C₈

semidirect

109. C₂

#19127454

direct (N × H)

Subgroups — C₂×#19127454 #24906207

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19127454 #24906207

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19127454 #24906207