Group #27542256 — subgroups

Subgroups — C₂×C₄³×D₅₃ #27542256

Order 13568.

8094 conjugacy classes of subgroups; only the 4683 normal subgroups are shown.

normal subgroups only

This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 4683 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	C₂²×C₄²×D₅₃	C₂	1	7	yes	maximal
2	C₂×C₄²×Dic₅₃	C₂	1	7	yes	maximal, complemented
3	C₄³×D₅₃	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₄×D₅₃	C₂²	1	7	yes	—
6	C₂²×C₄×Dic₅₃	C₂²	1	21	yes	—
7	C₂×C₄²×D₅₃	C₄	1	112	yes	direct product factor
8	C₂×C₄²×D₅₃	C₂²	1	56	yes	—
9	C₄²×Dic₅₃	C₂²	1	56	yes	complemented
10	C₂²×C₄²×C₅₃	C₂²	1	7	yes	—
11	C₄³×C₅₃	C₂²	1	8	yes	complemented
12	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
13	C₂³×Dic₅₃	C₂³	1	7	yes	—
14	C₂²×C₄×D₅₃	C₂×C₄	1	168	yes	—
15	C₂²×C₄×D₅₃	C₂³	1	28	yes	—
16	C₂×C₄×Dic₅₃	C₂×C₄	1	168	yes	complemented
17	C₂×C₄×Dic₅₃	C₂³	1	84	yes	—
18	C₄²×D₅₃	C₂×C₄	1	448	yes	direct product factor
19	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	56	yes	complemented
21	C₂×C₄²×C₅₃	C₂³	1	28	yes	—
22	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
23	C₂³×D₅₃	C₂²×C₄	1	28	yes	—
24	C₂³×D₅₃	C₂⁴	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂²×C₄	1	84	yes	—
26	C₂²×C₄×C₅₃	C₂⁴	1	14	yes	—
27	C₂²×Dic₅₃	C₂²×C₄	1	84	yes	—
28	C₂²×Dic₅₃	C₂⁴	1	14	yes	—
29	C₂×C₄×D₅₃	C₄²	1	448	yes	direct product factor
30	C₂×C₄×D₅₃	C₂²×C₄	1	336	yes	—
31	C₄²×C₅₃	C₂²×C₄	1	112	yes	complemented
32	C₄×Dic₅₃	C₂²×C₄	1	336	yes	complemented
33	C₂³×C₅₃	C₂³×C₄	1	14	yes	—
34	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
35	C₂²×D₅₃	C₂×C₄²	1	112	yes	—
36	C₂²×D₅₃	C₂³×C₄	1	28	yes	—
37	C₂×C₄×C₅₃	C₂×C₄²	1	112	yes	complemented
38	C₂×C₄×C₅₃	C₂³×C₄	1	84	yes	—
39	C₂×Dic₅₃	C₂×C₄²	1	112	yes	complemented
40	C₂×Dic₅₃	C₂³×C₄	1	84	yes	—
41	C₄×D₅₃	C₂×C₄²	1	448	yes	direct product factor
42	C₂²×C₄³	—	53	1	no	maximal, Sylow
43	C₂²×C₅₃	C₂²×C₄²	1	28	yes	—
44	C₂²×C₅₃	C₂⁴×C₄	1	7	yes	—
45	C₂×D₅₃	C₄³	1	64	yes	direct product factor
46	C₂×D₅₃	C₂²×C₄²	1	56	yes	—
47	C₄×C₅₃	C₂²×C₄²	1	56	yes	complemented
48	Dic₅₃	C₂²×C₄²	1	56	yes	complemented
49	C₂³×C₄²	—	53	7	no	—
50	C₂×C₄³	—	53	7	no	—
51	C₂×C₄³	—	53	16	no	—
52	C₂×C₄³	D₅₃	1	1	yes	characteristic, direct product factor
53	C₂×C₅₃	C₂×C₄³	1	8	yes	complemented
54	C₂×C₅₃	C₂³×C₄²	1	7	yes	—
55	D₅₃	C₂×C₄³	1	16	yes	direct product factor
56	C₂⁴×C₄	—	53	7	no	—
57	C₂²×C₄²	—	53	21	no	—
58	C₂²×C₄²	—	53	112	no	—
59	C₂²×C₄²	—	53	56	no	—
60	C₂²×C₄²	C₂×D₅₃	1	7	yes	—
61	C₄³	—	53	56	no	—
62	C₄³	C₂×D₅₃	1	8	yes	direct product factor
63	C₅₃	C₂²×C₄³	1	1	yes	characteristic, complemented, Sylow
64	C₂⁵	—	53	1	no	—
65	C₂³×C₄	—	53	7	no	—
66	C₂³×C₄	—	53	168	no	—
67	C₂³×C₄	—	53	28	no	—
68	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
69	C₂×C₄²	—	53	168	no	—
70	C₂×C₄²	—	53	84	no	—
71	C₂×C₄²	—	53	448	no	—
72	C₂×C₄²	C₄×D₅₃	1	56	yes	direct product factor
73	C₂×C₄²	C₂²×D₅₃	1	28	yes	—
74	C₂⁴	—	53	28	no	—
75	C₂⁴	—	53	2	no	—
76	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
77	C₂²×C₄	—	53	84	no	—
78	C₂²×C₄	—	53	14	no	—
79	C₂²×C₄	—	53	448	no	—
80	C₂²×C₄	—	53	336	no	—
81	C₂²×C₄	C₂×C₄×D₅₃	1	84	yes	—
82	C₂²×C₄	C₂³×D₅₃	1	14	yes	—
83	C₄²	—	53	336	no	—
84	C₄²	C₂×C₄×D₅₃	1	112	yes	direct product factor
85	C₂³	—	53	112	no	—
86	C₂³	—	53	28	no	—
87	C₂³	C₂²×C₄×D₅₃	1	14	yes	—
88	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
89	C₂×C₄	—	53	112	no	—
90	C₂×C₄	—	53	84	no	—
91	C₂×C₄	—	53	448	no	—
92	C₂×C₄	C₄²×D₅₃	1	112	yes	direct product factor
93	C₂×C₄	C₂²×C₄×D₅₃	1	84	yes	—
94	C₂²	—	53	64	no	—
95	C₂²	—	53	56	no	—
96	C₂²	C₂×C₄²×D₅₃	1	28	yes	—
97	C₂²	C₂³×C₄×D₅₃	1	7	yes	—
98	C₄	—	53	56	no	—
99	C₄	C₂×C₄²×D₅₃	1	56	yes	direct product factor
100	C₂	—	53	16	no	—
101	C₂	C₄³×D₅₃	1	8	yes	direct product factor
102	C₂	C₂²×C₄²×D₅₃	1	7	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄²×D₅₃

C₂

yes

maximal

C₂×C₄²×Dic₅₃

C₂

yes

maximal, complemented

C₄³×D₅₃

C₂

yes

maximal, direct product factor

C₂×C₄³×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂³×C₄×D₅₃

C₂²

yes

—

C₂²×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄²×D₅₃

C₄

112

yes

direct product factor

C₂×C₄²×D₅₃

C₂²

yes

—

C₄²×Dic₅₃

C₂²

yes

complemented

C₂²×C₄²×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

complemented

C₂⁴×D₅₃

C₂³

yes

characteristic

C₂³×Dic₅₃

C₂³

yes

—

C₂²×C₄×D₅₃

C₂×C₄

168

yes

—

C₂²×C₄×D₅₃

C₂³

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

168

yes

complemented

C₂×C₄×Dic₅₃

C₂³

yes

—

C₄²×D₅₃

C₂×C₄

448

yes

direct product factor

C₂³×C₄×C₅₃

C₂³

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

complemented

C₂×C₄²×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

C₂⁴

yes

characteristic

C₂³×D₅₃

C₂²×C₄

yes

—

C₂³×D₅₃

C₂⁴

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

—

C₂²×Dic₅₃

C₂²×C₄

yes

—

C₂²×Dic₅₃

C₂⁴

yes

—

C₂×C₄×D₅₃

C₄²

448

yes

direct product factor

C₂×C₄×D₅₃

C₂²×C₄

336

yes

—

C₄²×C₅₃

C₂²×C₄

112

yes

complemented

C₄×Dic₅₃

C₂²×C₄

336

yes

complemented

C₂³×C₅₃

C₂³×C₄

yes

—

C₂³×C₅₃

C₂⁵

yes

characteristic

C₂²×D₅₃

C₂×C₄²

112

yes

—

C₂²×D₅₃

C₂³×C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄²

112

yes

complemented

C₂×C₄×C₅₃

C₂³×C₄

yes

—

C₂×Dic₅₃

C₂×C₄²

112

yes

complemented

C₂×Dic₅₃

C₂³×C₄

yes

—

C₄×D₅₃

C₂×C₄²

448

yes

direct product factor

C₂²×C₄³

—

maximal, Sylow

C₂²×C₅₃

C₂²×C₄²

yes

—

C₂²×C₅₃

C₂⁴×C₄

yes

—

C₂×D₅₃

C₄³

yes

direct product factor

C₂×D₅₃

C₂²×C₄²

yes

—

C₄×C₅₃

C₂²×C₄²

yes

complemented

Dic₅₃

C₂²×C₄²

yes

complemented

C₂³×C₄²

—

C₂×C₄³

—

C₂×C₄³

—

C₂×C₄³

D₅₃

yes

characteristic, direct product factor

C₂×C₅₃

C₂×C₄³

yes

complemented

C₂×C₅₃

C₂³×C₄²

yes

—

D₅₃

C₂×C₄³

yes

direct product factor

C₂⁴×C₄

—

C₂²×C₄²

—

C₂²×C₄²

—

112

—

C₂²×C₄²

—

C₂²×C₄²

C₂×D₅₃

yes

—

C₄³

—

C₄³

C₂×D₅₃

yes

direct product factor

C₅₃

C₂²×C₄³

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂³×C₄

—

C₂³×C₄

—

168

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₅₃

yes

—

C₂×C₄²

—

168

—

C₂×C₄²

—

C₂×C₄²

—

448

—

C₂×C₄²

C₄×D₅₃

yes

direct product factor

C₂×C₄²

C₂²×D₅₃

yes

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂³×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

448

—

C₂²×C₄

—

336

—

C₂²×C₄

C₂×C₄×D₅₃

yes

—

C₂²×C₄

C₂³×D₅₃

yes

—

C₄²

—

336

—

C₄²

C₂×C₄×D₅₃

112

yes

direct product factor

C₂³

—

112

—

C₂³

—

C₂³

C₂²×C₄×D₅₃

yes

—

C₂³

C₂⁴×D₅₃

yes

characteristic

C₂×C₄

—

112

—

C₂×C₄

—

C₂×C₄

—

448

—

C₂×C₄

C₄²×D₅₃

112

yes

direct product factor

C₂×C₄

C₂²×C₄×D₅₃

yes

—

C₂²

—

C₂²

—

C₂²

C₂×C₄²×D₅₃

yes

—

C₂²

C₂³×C₄×D₅₃

yes

—

C₄

—

C₄

C₂×C₄²×D₅₃

yes

direct product factor

100

C₂

—

101

C₂

C₄³×D₅₃

yes

direct product factor

102

C₂

C₂²×C₄²×D₅₃

yes

—

	subgroup	G/H
center, hypercenter	C₂×C₄³	D₅₃
commutator, hypocenter	C₅₃	C₂²×C₄³
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁴×D₅₃	C₂³
cosocle	C₂³×C₅₃	C₂⁵
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂×C₄³

subgroup

G/H

center, hypercenter

C₂×C₄³

D₅₃

commutator, hypocenter

C₅₃

C₂²×C₄³

Frattini

C₂³

C₂⁴×D₅₃

Fitting

C₂×C₄³×C₅₃

C₂

socle

C₂⁴×C₅₃

C₂⁴

prime radical

C₂⁴×D₅₃

C₂³

cosocle

C₂³×C₅₃

C₂⁵

LCS

G ▷ C₅₃

UCS

C₁ ◁ C₂×C₄³

N	H ≅ G/N	type
2. C₂×C₄²×Dic₅₃	C₂	semidirect
3. C₄³×D₅₃	C₂	direct (N × H)
4. C₂×C₄³×C₅₃	C₂	semidirect
7. C₂×C₄²×D₅₃	C₄	direct (N × H)
9. C₄²×Dic₅₃	C₂²	semidirect
11. C₄³×C₅₃	C₂²	semidirect
16. C₂×C₄×Dic₅₃	C₂×C₄	semidirect
18. C₄²×D₅₃	C₂×C₄	direct (N × H)
20. C₂×C₄²×C₅₃	C₂×C₄	semidirect
29. C₂×C₄×D₅₃	C₄²	direct (N × H)
31. C₄²×C₅₃	C₂²×C₄	semidirect
32. C₄×Dic₅₃	C₂²×C₄	semidirect
37. C₂×C₄×C₅₃	C₂×C₄²	semidirect
39. C₂×Dic₅₃	C₂×C₄²	semidirect
41. C₄×D₅₃	C₂×C₄²	direct (N × H)
45. C₂×D₅₃	C₄³	direct (N × H)
47. C₄×C₅₃	C₂²×C₄²	semidirect
48. Dic₅₃	C₂²×C₄²	semidirect
52. C₂×C₄³	D₅₃	direct (N × H)
53. C₂×C₅₃	C₂×C₄³	semidirect
55. D₅₃	C₂×C₄³	direct (N × H)
62. C₄³	C₂×D₅₃	direct (N × H)
63. C₅₃	C₂²×C₄³	semidirect
72. C₂×C₄²	C₄×D₅₃	direct (N × H)
84. C₄²	C₂×C₄×D₅₃	direct (N × H)
92. C₂×C₄	C₄²×D₅₃	direct (N × H)
99. C₄	C₂×C₄²×D₅₃	direct (N × H)
101. C₂	C₄³×D₅₃	direct (N × H)

H ≅ G/N

type

2. C₂×C₄²×Dic₅₃

C₂

semidirect

3. C₄³×D₅₃

C₂

direct (N × H)

4. C₂×C₄³×C₅₃

C₂

semidirect

7. C₂×C₄²×D₅₃

C₄

direct (N × H)

9. C₄²×Dic₅₃

C₂²

semidirect

11. C₄³×C₅₃

C₂²

semidirect

16. C₂×C₄×Dic₅₃

C₂×C₄

semidirect

18. C₄²×D₅₃

C₂×C₄

direct (N × H)

20. C₂×C₄²×C₅₃

C₂×C₄

semidirect

29. C₂×C₄×D₅₃

C₄²

direct (N × H)

31. C₄²×C₅₃

C₂²×C₄

semidirect

32. C₄×Dic₅₃

C₂²×C₄

semidirect

37. C₂×C₄×C₅₃

C₂×C₄²

semidirect

39. C₂×Dic₅₃

C₂×C₄²

semidirect

41. C₄×D₅₃

C₂×C₄²

direct (N × H)

45. C₂×D₅₃

C₄³

direct (N × H)

47. C₄×C₅₃

C₂²×C₄²

semidirect

48. Dic₅₃

C₂²×C₄²

semidirect

52. C₂×C₄³

D₅₃

direct (N × H)

53. C₂×C₅₃

C₂×C₄³

semidirect

55. D₅₃

C₂×C₄³

direct (N × H)

62. C₄³

C₂×D₅₃

direct (N × H)

63. C₅₃

C₂²×C₄³

semidirect

72. C₂×C₄²

C₄×D₅₃

direct (N × H)

84. C₄²

C₂×C₄×D₅₃

direct (N × H)

92. C₂×C₄

C₄²×D₅₃

direct (N × H)

99. C₄

C₂×C₄²×D₅₃

direct (N × H)

101. C₂

C₄³×D₅₃

direct (N × H)

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

subgroup	index
C₂²×C₄²×D₅₃	2
C₂×C₄²×Dic₅₃	2
C₄³×D₅₃	2
C₂×C₄³×C₅₃	2
C₂²×C₄³	53

subgroup

index

C₂²×C₄²×D₅₃

C₂×C₄²×Dic₅₃

C₄³×D₅₃

C₂×C₄³×C₅₃

C₂²×C₄³

Subgroups — C₂×C₄³×D₅₃ #27542256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C43×D53 #27542256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄³×D₅₃ #27542256