Group #20291349 — subgroups

Subgroups — C₂²×#13133284 #20291349

Order 6784.

640 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 338 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×#88324	C₂	1	2	yes	maximal
2	C₂×#13133284	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88324	—	2	8	no	—
5	C₂²×#88324	C₂²	1	12	yes	—
6	#13133284	C₂²	1	16	yes	direct product factor
7	C₂³×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₅₃	C₄	1	6	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
11	C₂³×C₄×C₅₃	C₄	1	1	yes	characteristic
12	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	—	2	4	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	6	yes	—
15	C₂²×Dic₅₃	C₈	1	8	yes	complemented
16	C₂²×Dic₅₃	C₂×C₄	1	3	yes	—
17	C₂²×Dic₅₃	C₂×C₄	1	6	yes	—
18	C₂²×Dic₅₃	D₄	1	4	yes	—
19	C₂²×Dic₅₃	Q₈	1	4	yes	—
20	C₂²×Dic₅₃	C₂³	1	3	yes	—
21	C₂×#88324	—	2	24	no	—
22	C₂×#88324	C₂³	1	8	yes	—
23	C₄×Dic₅₃	C₂×C₄	1	12	yes	—
24	C₄×Dic₅₃	C₂³	1	4	yes	—
25	C₂³×C₅₃	C₄⋊C₄	1	4	yes	—
26	C₂³×C₅₃	C₂×C₈	1	4	yes	—
27	C₂³×C₅₃	M₁₆	1	4	yes	—
28	C₂³×C₅₃	C₂²×C₄	1	3	yes	—
29	C₂×C₄×C₅₃	—	2	12	no	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
31	C₂×Dic₅₃	C₄⋊C₄	1	12	yes	—
32	C₂×Dic₅₃	C₂×C₈	1	24	yes	complemented
33	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
34	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
35	C₂×Dic₅₃	C₂×D₄	1	6	yes	—
36	C₂×Dic₅₃	C₂×Q₈	1	6	yes	—
37	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
38	#88324	—	2	8	no	—
39	C₂²×C₅₃	C₄⋊C₈	1	16	yes	complemented
40	C₂²×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
41	C₂²×C₅₃	C₂²×C₈	1	6	yes	—
42	C₂²×C₅₃	C₂×M₁₆	1	6	yes	—
43	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
44	C₄×C₅₃	—	2	4	no	—
45	Dic₅₃	C₂×C₄⋊C₄	1	6	yes	—
46	Dic₅₃	C₂²×C₈	1	8	yes	complemented
47	Dic₅₃	C₂²×D₄	1	1	yes	characteristic
48	Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
49	C₂²×C₄⋊C₈	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	C₂×C₄⋊C₈	1	12	yes	complemented
51	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
53	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
54	C₂³×C₈	—	53	2	no	—
55	C₂²×C₄²	—	53	1	no	—
56	C₂×C₄⋊C₈	—	53	12	no	—
57	C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
58	C₂³×C₄	—	53	1	no	—
59	C₂³×C₄	—	53	1	no	—
60	C₂³×C₄	#6157	1	1	yes	characteristic
61	C₂²×C₈	—	53	12	no	—
62	C₂²×C₈	—	106	8	no	—
63	C₂×C₄²	—	53	6	no	—
64	C₂×C₄²	—	53	6	no	—
65	C₄⋊C₈	—	53	16	no	—
66	C₂⁴	C₂×#6157	1	1	yes	characteristic
67	C₂²×C₄	—	2	4	no	—
68	C₂²×C₄	—	53	4	no	—
69	C₂²×C₄	—	53	4	no	—
70	C₂²×C₄	—	53	3	no	—
71	C₂²×C₄	—	53	6	no	—
72	C₂²×C₄	—	53	3	no	—
73	C₂²×C₄	—	53	8	no	—
74	C₂²×C₄	C₂×#6157	1	6	yes	—
75	C₂×C₈	—	53	8	no	—
76	C₂×C₈	—	106	24	no	—
77	C₄²	—	53	4	no	—
78	C₄²	—	53	12	no	—
79	C₂³	D₅₃⋊C₈	1	4	yes	—
80	C₂³	C₄⋊#6157	1	4	yes	—
81	C₂³	#11719940	1	4	yes	—
82	C₂³	C₂²×#6157	1	3	yes	—
83	C₂×C₄	—	2	12	no	—
84	C₂×C₄	—	53	12	no	—
85	C₂×C₄	—	53	6	no	—
86	C₂×C₄	—	53	6	no	—
87	C₂×C₄	—	53	24	no	—
88	C₂×C₄	—	53	3	no	—
89	C₂×C₄	—	53	4	no	—
90	C₂×C₄	—	53	1	no	—
91	C₂×C₄	C₂²×#6157	1	4	yes	—
92	C₈	—	106	8	no	—
93	C₂²	#13133284	1	16	yes	direct product factor
94	C₂²	C₂×C₄⋊#6157	1	6	yes	—
95	C₂²	C₂×D₅₃⋊C₈	1	6	yes	—
96	C₂²	C₂×#11719940	1	6	yes	—
97	C₂²	C₂³×#6157	1	1	yes	characteristic
98	C₄	—	2	4	no	—
99	C₄	—	53	1	no	—
100	C₄	—	53	6	no	—
101	C₄	—	53	1	no	—
102	C₄	—	53	8	no	—
103	C₂	C₂×#13133284	1	12	yes	direct product factor
104	C₂	C₂²×C₄⋊#6157	1	1	yes	characteristic
105	C₂	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
106	C₂	C₂²×#11719940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#88324

C₂

yes

maximal

C₂×#13133284

C₂

yes

maximal, direct product factor

C₂²×C₄×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×#88324

—

C₂²×#88324

C₂²

yes

—

#13133284

C₂²

yes

direct product factor

C₂³×Dic₅₃

C₄

yes

characteristic

C₂³×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₄

yes

characteristic

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₈

yes

complemented

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

D₄

yes

—

C₂²×Dic₅₃

Q₈

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×#88324

—

C₂×#88324

C₂³

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×C₈

yes

—

C₂³×C₅₃

M₁₆

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂×C₈

yes

complemented

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

C₂⁴

yes

characteristic

#88324

—

C₂²×C₅₃

C₄⋊C₈

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂²×C₈

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

Dic₅₃

C₂×C₄⋊C₄

yes

—

Dic₅₃

C₂²×C₈

yes

complemented

Dic₅₃

C₂²×D₄

yes

characteristic

Dic₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₄⋊C₈

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊C₈

yes

complemented

C₂×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₅₃

C₂³×C₈

yes

characteristic

C₂×C₅₃

C₂²×M₁₆

yes

characteristic

C₂³×C₈

—

C₂²×C₄²

—

C₂×C₄⋊C₈

—

C₅₃

C₂²×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

#6157

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂⁴

C₂×#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6157

yes

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₂³

D₅₃⋊C₈

yes

—

C₂³

C₄⋊#6157

yes

—

C₂³

#11719940

yes

—

C₂³

C₂²×#6157

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×#6157

yes

—

C₈

—

106

—

C₂²

#13133284

yes

direct product factor

C₂²

C₂×C₄⋊#6157

yes

—

C₂²

C₂×D₅₃⋊C₈

yes

—

C₂²

C₂×#11719940

yes

—

C₂²

C₂³×#6157

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₂

C₂×#13133284

yes

direct product factor

104

C₂

C₂²×C₄⋊#6157

yes

characteristic

105

C₂

C₂²×D₅₃⋊C₈

yes

characteristic

106

C₂

C₂²×#11719940

yes

characteristic

	subgroup	G/H
center	C₂⁴	C₂×#6157
commutator	C₂×C₅₃	C₂³×C₈
Frattini	C₂²	C₂³×#6157
Fitting	C₂³×C₄×C₅₃	C₄
socle	C₂⁴×C₅₃	C₂×C₄
hypercenter	C₂³×C₄	#6157
hypocenter	C₅₃	C₂²×C₄⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

subgroup

G/H

center

C₂⁴

C₂×#6157

commutator

C₂×C₅₃

C₂³×C₈

Frattini

C₂²

C₂³×#6157

Fitting

C₂³×C₄×C₅₃

C₄

socle

C₂⁴×C₅₃

C₂×C₄

hypercenter

C₂³×C₄

#6157

hypocenter

C₅₃

C₂²×C₄⋊C₈

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂⁴ ◁ C₂³×C₄

N	H ≅ G/N	type
2. C₂×#13133284	C₂	direct (N × H)
6. #13133284	C₂²	direct (N × H)
15. C₂²×Dic₅₃	C₈	semidirect
32. C₂×Dic₅₃	C₂×C₈	semidirect
39. C₂²×C₅₃	C₄⋊C₈	semidirect
46. Dic₅₃	C₂²×C₈	semidirect
50. C₂×C₅₃	C₂×C₄⋊C₈	semidirect
57. C₅₃	C₂²×C₄⋊C₈	semidirect
93. C₂²	#13133284	direct (N × H)
103. C₂	C₂×#13133284	direct (N × H)

H ≅ G/N

type

2. C₂×#13133284

C₂

direct (N × H)

6. #13133284

C₂²

direct (N × H)

15. C₂²×Dic₅₃

C₈

semidirect

32. C₂×Dic₅₃

C₂×C₈

semidirect

39. C₂²×C₅₃

C₄⋊C₈

semidirect

46. Dic₅₃

C₂²×C₈

semidirect

50. C₂×C₅₃

C₂×C₄⋊C₈

semidirect

57. C₅₃

C₂²×C₄⋊C₈

semidirect

93. C₂²

#13133284

direct (N × H)

103. C₂

C₂×#13133284

direct (N × H)

Subgroups — C₂²×#13133284 #20291349

Conjugacy classes of subgroups

Complements

Subgroups — C22×#13133284 #20291349

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#13133284 #20291349