Group #20287989 — subgroups

Subgroups — C₂²×#13133215 #20287989

Order 6784.

1624 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 725 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	C₂³×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133215	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#11719919	C₂	1	4	yes	maximal
4	C₂²×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	—	2	8	no	—
7	C₂²×Dic₁₀₆	C₂²	1	12	yes	—
8	C₂³×Dic₅₃	C₂²	1	2	yes	—
9	C₂³×Dic₅₃	C₂²	1	2	yes	—
10	#13133215	C₂²	1	64	yes	direct product factor
11	C₂×#11719919	C₂²	1	48	yes	—
12	C₂×C₄×Dic₅₃	C₂²	1	12	yes	—
13	C₂²×Q₈×C₅₃	—	2	8	no	—
14	C₂²×Q₈×C₅₃	C₂²	1	12	yes	—
15	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
16	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×Q₈×C₅₃	—	2	48	no	—
19	C₂×Q₈×C₅₃	C₂³	1	16	yes	—
20	C₂²×C₄×C₅₃	—	2	8	no	—
21	C₂²×C₄×C₅₃	D₄	1	8	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	12	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
24	C₂²×Dic₅₃	—	2	8	no	—
25	C₂²×Dic₅₃	Q₈	1	16	yes	complemented
26	C₂²×Dic₅₃	C₂³	1	12	yes	—
27	C₂²×Dic₅₃	C₂³	1	12	yes	—
28	C₂×Dic₁₀₆	—	2	48	no	—
29	C₂×Dic₁₀₆	C₂³	1	16	yes	—
30	#11719919	C₂³	1	64	yes	—
31	C₄×Dic₅₃	C₂³	1	16	yes	—
32	C₂³×C₅₃	C₂×D₄	1	4	yes	—
33	C₂³×C₅₃	C₂×Q₈	1	8	yes	—
34	C₂³×C₅₃	C₂⁴	1	3	yes	—
35	Q₈×C₅₃	—	2	32	no	—
36	C₂×C₄×C₅₃	—	2	24	no	—
37	C₂×C₄×C₅₃	C₂×D₄	1	24	yes	—
38	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
39	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
40	C₂×Dic₅₃	—	2	24	no	—
41	C₂×Dic₅₃	C₂×Q₈	1	48	yes	complemented
42	C₂×Dic₅₃	C₂⁴	1	8	yes	—
43	C₂×Dic₅₃	C₂⁴	1	8	yes	—
44	Dic₁₀₆	—	2	32	no	—
45	C₂²×C₅₃	C₄⋊Q₈	1	16	yes	complemented
46	C₂²×C₅₃	C₂²×D₄	1	6	yes	—
47	C₂²×C₅₃	C₂²×Q₈	1	12	yes	—
48	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
49	C₄×C₅₃	—	2	8	no	—
50	C₄×C₅₃	C₂²×D₄	1	8	yes	—
51	Dic₅₃	—	2	8	no	—
52	Dic₅₃	C₂²×Q₈	1	16	yes	complemented
53	C₂²×C₄⋊Q₈	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	C₂×C₄⋊Q₈	1	12	yes	complemented
55	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
56	C₂×C₅₃	C₂³×Q₈	1	2	yes	—
57	C₂³×Q₈	—	53	1	no	—
58	C₂³×Q₈	D₅₃	1	1	yes	characteristic
59	C₂×C₄⋊Q₈	—	53	24	no	—
60	C₂²×C₄⋊C₄	—	53	4	no	—
61	C₂²×C₄²	—	53	1	no	—
62	C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
63	C₂²×Q₈	—	2	8	no	—
64	C₂²×Q₈	—	53	12	no	—
65	C₂²×Q₈	—	106	8	no	—
66	C₂²×Q₈	C₂×D₅₃	1	12	yes	—
67	C₂³×C₄	—	53	2	no	—
68	C₂³×C₄	—	53	2	no	—
69	C₂³×C₄	C₂×D₅₃	1	2	yes	—
70	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
71	C₄⋊Q₈	—	53	64	no	—
72	C₂×C₄⋊C₄	—	53	48	no	—
73	C₂×C₄²	—	53	12	no	—
74	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
75	C₂×Q₈	—	2	48	no	—
76	C₂×Q₈	—	53	16	no	—
77	C₂×Q₈	—	106	48	no	—
78	C₂×Q₈	C₂²×D₅₃	1	16	yes	—
79	C₂²×C₄	—	2	8	no	—
80	C₂²×C₄	—	53	16	no	—
81	C₂²×C₄	—	53	12	no	—
82	C₂²×C₄	—	53	12	no	—
83	C₂²×C₄	—	106	8	no	—
84	C₂²×C₄	C₂²⋊D₅₃	1	8	yes	—
85	C₂²×C₄	C₂²×D₅₃	1	12	yes	—
86	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
87	C₄⋊C₄	—	53	64	no	—
88	C₄²	—	53	16	no	—
89	C₂³	Q₈×D₅₃	1	8	yes	—
90	C₂³	C₂×C₂²⋊D₅₃	1	4	yes	—
91	C₂³	C₂³×D₅₃	1	3	yes	—
92	Q₈	—	2	32	no	—
93	Q₈	—	106	32	no	—
94	C₂×C₄	—	2	24	no	—
95	C₂×C₄	—	53	48	no	—
96	C₂×C₄	—	53	8	no	—
97	C₂×C₄	—	53	8	no	—
98	C₂×C₄	—	106	24	no	—
99	C₂×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
100	C₂×C₄	C₂³×D₅₃	1	8	yes	—
101	C₂×C₄	C₂³×D₅₃	1	4	yes	—
102	C₂²	#13133215	1	16	yes	direct product factor
103	C₂²	C₂²×C₂²⋊D₅₃	1	6	yes	—
104	C₂²	C₂×Q₈×D₅₃	1	12	yes	—
105	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
106	C₄	—	2	8	no	—
107	C₄	—	53	16	no	—
108	C₄	—	106	8	no	—
109	C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
110	C₂	C₂×#13133215	1	12	yes	direct product factor
111	C₂	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
112	C₂	C₂²×Q₈×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₀₆

C₂

yes

maximal, characteristic

C₂×#13133215

C₂

yes

maximal, direct product factor

C₂²×#11719919

C₂

yes

maximal

C₂²×C₄×Dic₅₃

C₂

yes

maximal, characteristic

C₂³×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

C₂²×Dic₁₀₆

C₂²

yes

—

C₂³×Dic₅₃

C₂²

yes

—

C₂³×Dic₅₃

C₂²

yes

—

#13133215

C₂²

yes

direct product factor

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

Q₈

yes

complemented

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂⁴

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×Q₈

yes

complemented

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

Dic₁₀₆

—

C₂²×C₅₃

C₄⋊Q₈

yes

complemented

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂²×D₄

yes

—

Dic₅₃

—

Dic₅₃

C₂²×Q₈

yes

complemented

C₂²×C₄⋊Q₈

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊Q₈

yes

complemented

C₂×C₅₃

C₂³×D₄

yes

characteristic

C₂×C₅₃

C₂³×Q₈

yes

—

C₂³×Q₈

—

C₂³×Q₈

D₅₃

yes

characteristic

C₂×C₄⋊Q₈

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

C₅₃

C₂²×C₄⋊Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

106

—

C₂²×Q₈

C₂×D₅₃

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

C₂²×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄²

—

C₂³

Q₈×D₅₃

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

—

C₂³

C₂³×D₅₃

yes

—

Q₈

—

Q₈

—

106

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

100

C₂×C₄

C₂³×D₅₃

yes

—

101

C₂×C₄

C₂³×D₅₃

yes

—

102

C₂²

#13133215

yes

direct product factor

103

C₂²

C₂²×C₂²⋊D₅₃

yes

—

104

C₂²

C₂×Q₈×D₅₃

yes

—

105

C₂²

C₂⁴×D₅₃

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

106

—

109

C₄

C₂²×C₂²⋊D₅₃

yes

—

110

C₂

C₂×#13133215

yes

direct product factor

111

C₂

C₂³×C₂²⋊D₅₃

yes

characteristic

112

C₂

C₂²×Q₈×D₅₃

yes

—

	subgroup	G/H
center	C₂⁴	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂³×Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂³×Q₈	D₅₃
hypocenter	C₅₃	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

subgroup

G/H

center

C₂⁴

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂⁵

Frattini

C₂²

C₂⁴×D₅₃

Fitting

C₂³×Q₈×C₅₃

C₂

socle

C₂⁴×C₅₃

C₂³

hypercenter

C₂³×Q₈

D₅₃

hypocenter

C₅₃

C₂²×C₄⋊Q₈

LCS

G ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂⁴ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. C₂×#13133215	C₂	direct (N × H)
10. #13133215	C₂²	direct (N × H)
25. C₂²×Dic₅₃	Q₈	semidirect
41. C₂×Dic₅₃	C₂×Q₈	semidirect
45. C₂²×C₅₃	C₄⋊Q₈	semidirect
52. Dic₅₃	C₂²×Q₈	semidirect
54. C₂×C₅₃	C₂×C₄⋊Q₈	semidirect
62. C₅₃	C₂²×C₄⋊Q₈	semidirect
102. C₂²	#13133215	direct (N × H)
110. C₂	C₂×#13133215	direct (N × H)

H ≅ G/N

type

2. C₂×#13133215

C₂

direct (N × H)

10. #13133215

C₂²

direct (N × H)

25. C₂²×Dic₅₃

Q₈

semidirect

41. C₂×Dic₅₃

C₂×Q₈

semidirect

45. C₂²×C₅₃

C₄⋊Q₈

semidirect

52. Dic₅₃

C₂²×Q₈

semidirect

54. C₂×C₅₃

C₂×C₄⋊Q₈

semidirect

62. C₅₃

C₂²×C₄⋊Q₈

semidirect

102. C₂²

#13133215

direct (N × H)

110. C₂

C₂×#13133215

direct (N × H)

Subgroups — C₂²×#13133215 #20287989

Conjugacy classes of subgroups

Complements

Subgroups — C22×#13133215 #20287989

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#13133215 #20287989