Group #20277065 — subgroups

Subgroups — C₄²×Dic₁₀₆ #20277065

Order 6784.

684 conjugacy classes of subgroups; only the 381 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 381 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×Dic₁₀₆	C₂	1	3	yes	maximal
2	C₄×#11719919	C₂	1	6	yes	maximal
3	C₄×C₄⋊Dic₅₃	C₂	1	3	yes	maximal
4	C₄²×Dic₅₃	C₂	1	2	yes	maximal
5	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₀₆	C₄	1	24	yes	direct product factor
8	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
9	C₂×#11719919	C₂²	1	6	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
12	#13133124	C₂²	1	6	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
19	C₂²×Dic₅₃	C₂³	1	6	yes	—
20	C₂²×Dic₅₃	C₂³	1	2	yes	—
21	C₂×Dic₁₀₆	C₂×C₄	1	12	yes	—
22	C₄²×C₅₃	—	2	6	no	—
23	C₄²×C₅₃	C₂×C₄	1	12	yes	—
24	C₄²×C₅₃	Q₈	1	4	yes	complemented
25	C₄⋊Dic₅₃	C₂×C₄	1	12	yes	—
26	#11719919	C₂×C₄	1	24	yes	—
27	C₄×Dic₅₃	—	2	12	no	—
28	C₄×Dic₅₃	C₂×C₄	1	24	yes	—
29	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	6	no	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
32	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
34	C₂×C₄×C₅₃	C₂×Q₈	1	6	yes	—
35	C₂×C₄×C₅₃	#41	1	6	yes	—
36	C₂×Dic₅₃	—	2	12	no	—
37	C₂×Dic₅₃	C₂²×C₄	1	12	yes	—
38	C₂×Dic₅₃	C₂²×C₄	1	12	yes	—
39	Dic₁₀₆	C₄²	1	16	yes	direct product factor
40	C₂²×C₅₃	C₂³×C₄	1	3	yes	—
41	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
42	C₂²×C₅₃	C₂×#41	1	3	yes	—
43	C₄×C₅₃	—	2	6	no	—
44	C₄×C₅₃	C₂×C₄²	1	4	yes	—
45	C₄×C₅₃	C₄×Q₈	1	12	yes	complemented
46	Dic₅₃	—	2	12	no	—
47	Dic₅₃	C₂×C₄²	1	8	yes	—
48	C₄²×Q₈	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
50	C₂×C₅₃	C₂×C₄×Q₈	1	3	yes	—
51	C₂×C₅₃	C₄×#41	1	3	yes	—
52	C₂×C₄×Q₈	—	53	3	no	—
53	C₄×C₄⋊C₄	—	53	3	no	—
54	C₄×C₄⋊C₄	—	53	6	no	—
55	C₄³	—	53	2	no	—
56	C₄³	D₅₃	1	1	yes	characteristic
57	C₅₃	C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
58	C₂²×Q₈	—	53	1	no	—
59	C₄×Q₈	—	53	24	no	—
60	C₂×C₄⋊C₄	—	53	6	no	—
61	C₂×C₄⋊C₄	—	53	3	no	—
62	C₂×C₄²	—	53	6	no	—
63	C₂×C₄²	—	53	6	no	—
64	C₂×C₄²	C₂×D₅₃	1	3	yes	—
65	C₂×C₄²	C₂×D₅₃	1	3	yes	—
66	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
67	#95	—	53	6	no	—
68	C₂×Q₈	—	53	12	no	—
69	C₂²×C₄	—	53	2	no	—
70	C₂²×C₄	—	53	6	no	—
71	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
72	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
73	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
74	C₄⋊C₄	—	53	24	no	—
75	C₄⋊C₄	—	53	12	no	—
76	C₄²	—	2	6	no	—
77	C₄²	—	53	24	no	—
78	C₄²	—	106	12	no	—
79	C₄²	Dic₁₀₆	1	4	yes	direct product factor
80	C₄²	C₄×D₅₃	1	12	yes	—
81	C₂³	C₂³×D₅₃	1	1	yes	characteristic
82	Q₈	—	53	16	no	—
83	C₂×C₄	—	2	6	no	—
84	C₂×C₄	—	53	12	no	—
85	C₂×C₄	—	53	12	no	—
86	C₂×C₄	—	106	12	no	—
87	C₂×C₄	C₂×Dic₁₀₆	1	6	yes	—
88	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
89	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
90	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
91	C₂×C₄	#11719945	1	6	yes	—
92	C₂²	C₂²×C₄×D₅₃	1	3	yes	—
93	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
94	C₂²	C₂×#11719945	1	3	yes	—
95	C₄	—	2	6	no	—
96	C₄	—	53	8	no	—
97	C₄	—	106	12	no	—
98	C₄	C₄²×D₅₃	1	4	yes	—
99	C₄	C₄×Dic₁₀₆	1	12	yes	direct product factor
100	C₂	C₂×C₄²×D₅₃	1	1	yes	characteristic
101	C₂	C₂×C₄×Dic₁₀₆	1	3	yes	—
102	C₂	C₄×#11719945	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal

C₄×#11719919

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal

C₄²×Dic₅₃

C₂

yes

maximal

C₄³×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₄

yes

direct product factor

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

Q₈

yes

complemented

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

Dic₁₀₆

C₄²

yes

direct product factor

C₂²×C₅₃

C₂³×C₄

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

C₄×C₅₃

C₄×Q₈

yes

complemented

Dic₅₃

—

Dic₅₃

C₂×C₄²

yes

—

C₄²×Q₈

—

maximal, Sylow

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

C₂×C₄×Q₈

yes

—

C₂×C₅₃

C₄×#41

yes

—

C₂×C₄×Q₈

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

—

C₄³

D₅₃

yes

characteristic

C₅₃

C₄²×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₄²

Dic₁₀₆

yes

direct product factor

C₄²

C₄×D₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

#11719945

yes

—

C₂²

C₂²×C₄×D₅₃

yes

—

C₂²

C₂²×Dic₁₀₆

yes

characteristic

C₂²

C₂×#11719945

yes

—

C₄

—

C₄

—

C₄

—

106

—

C₄

C₄²×D₅₃

yes

—

C₄

C₄×Dic₁₀₆

yes

direct product factor

100

C₂

C₂×C₄²×D₅₃

yes

characteristic

101

C₂

C₂×C₄×Dic₁₀₆

yes

—

102

C₂

C₄×#11719945

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₅₃
Fitting	C₄³×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	C₄²×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄²

C₂×D₅₃

commutator

C₂×C₅₃

C₂²×C₄²

Frattini

C₂³

C₂³×D₅₃

Fitting

C₄³×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₄³

D₅₃

hypocenter

C₅₃

C₄²×Q₈

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
7. C₄×Dic₁₀₆	C₄	direct (N × H)
24. C₄²×C₅₃	Q₈	semidirect
39. Dic₁₀₆	C₄²	direct (N × H)
45. C₄×C₅₃	C₄×Q₈	semidirect
57. C₅₃	C₄²×Q₈	semidirect
79. C₄²	Dic₁₀₆	direct (N × H)
99. C₄	C₄×Dic₁₀₆	direct (N × H)

H ≅ G/N

type

7. C₄×Dic₁₀₆

C₄

direct (N × H)

24. C₄²×C₅₃

Q₈

semidirect

39. Dic₁₀₆

C₄²

direct (N × H)

45. C₄×C₅₃

C₄×Q₈

semidirect

57. C₅₃

C₄²×Q₈

semidirect

79. C₄²

Dic₁₀₆

direct (N × H)

99. C₄

C₄×Dic₁₀₆

direct (N × H)

Subgroups — C₄²×Dic₁₀₆ #20277065

Conjugacy classes of subgroups

Complements

Subgroups — C42×Dic106 #20277065

Conjugacy classes of subgroups

Complements

Subgroups — C₄²×Dic₁₀₆ #20277065