Group #20277066 — subgroups

Subgroups — C₄×Q₈×Dic₅₃ #20277066

Order 6784.

684 conjugacy classes of subgroups; only the 425 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 425 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×Q₈×Dic₅₃	C₂	1	2	yes	maximal
2	C₄⋊C₄×Dic₅₃	C₂	1	6	yes	maximal
3	C₄×C₄⋊Dic₅₃	C₂	1	3	yes	maximal
4	C₄²×Dic₅₃	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₅₃	C₄	1	16	yes	direct product factor
7	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
8	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	#13133124	C₂²	1	6	yes	—
13	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₅₃	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
17	C₂×Q₈×C₅₃	C₂×C₄	1	8	yes	—
18	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₃	C₂³	1	6	yes	—
23	C₂²×Dic₅₃	C₂³	1	2	yes	—
24	C₄⋊C₄×C₅₃	C₂×C₄	1	12	yes	—
25	C₄²×C₅₃	C₂×C₄	1	12	yes	—
26	C₄⋊Dic₅₃	C₂×C₄	1	24	yes	—
27	C₄×Dic₅₃	—	2	6	no	—
28	C₄×Dic₅₃	—	2	12	no	—
29	C₄×Dic₅₃	C₂×C₄	1	24	yes	—
30	C₄×Dic₅₃	Q₈	1	4	yes	direct product factor
31	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
32	Q₈×C₅₃	C₄²	1	16	yes	complemented
33	C₂×C₄×C₅₃	—	2	6	no	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	12	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₅₃	#41	1	2	yes	—
40	C₂×Dic₅₃	—	2	6	no	—
41	C₂×Dic₅₃	—	2	6	no	—
42	C₂×Dic₅₃	C₂²×C₄	1	12	yes	—
43	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
44	C₂×Dic₅₃	C₂×Q₈	1	4	yes	—
45	C₂×Dic₅₃	#41	1	4	yes	—
46	C₂²×C₅₃	C₂³×C₄	1	2	yes	—
47	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
48	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
49	C₂²×C₅₃	C₂×#41	1	2	yes	—
50	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
51	C₄×C₅₃	—	2	6	no	—
52	C₄×C₅₃	C₂×C₄²	1	12	yes	—
53	C₄×C₅₃	C₄×Q₈	1	4	yes	complemented
54	Dic₅₃	—	2	12	no	—
55	Dic₅₃	C₄×Q₈	1	8	yes	direct product factor
56	C₄²×Q₈	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
58	C₂×C₅₃	C₂×C₄×Q₈	1	2	yes	—
59	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
60	C₂×C₅₃	C₄×#41	1	2	yes	—
61	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
62	C₂×C₄×Q₈	—	53	2	no	—
63	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
64	C₄×C₄⋊C₄	—	53	6	no	—
65	C₄×C₄⋊C₄	—	53	3	no	—
66	C₄³	—	53	3	no	—
67	C₅₃	C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
68	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
69	C₄×Q₈	—	53	16	no	—
70	C₄×Q₈	Dic₅₃	1	8	yes	direct product factor
71	C₂×C₄⋊C₄	—	53	6	no	—
72	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
73	C₂×C₄²	—	53	6	no	—
74	C₂×C₄²	—	53	1	no	—
75	C₂×C₄²	—	53	6	no	—
76	C₂×C₄²	—	53	3	no	—
77	C₂×C₄²	C₂×D₅₃	1	3	yes	—
78	#95	—	53	6	no	—
79	C₂×Q₈	C₄×D₅₃	1	8	yes	—
80	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
81	C₂²×C₄	—	53	2	no	—
82	C₂²×C₄	—	53	6	no	—
83	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
84	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
85	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
86	C₄⋊C₄	—	53	24	no	—
87	C₄⋊C₄	C₂×Dic₅₃	1	12	yes	—
88	C₄²	—	53	24	no	—
89	C₄²	—	53	4	no	—
90	C₄²	—	106	12	no	—
91	C₄²	—	106	6	no	—
92	C₄²	C₂×Dic₅₃	1	12	yes	—
93	C₂³	C₂³×D₅₃	1	1	yes	characteristic
94	Q₈	C₄×Dic₅₃	1	16	yes	direct product factor
95	C₂×C₄	—	2	6	no	—
96	C₂×C₄	—	53	4	no	—
97	C₂×C₄	—	53	4	no	—
98	C₂×C₄	—	53	4	no	—
99	C₂×C₄	—	53	12	no	—
100	C₂×C₄	—	106	6	no	—
101	C₂×C₄	—	106	6	no	—
102	C₂×C₄	C₂×C₄×D₅₃	1	12	yes	—
103	C₂×C₄	Q₈×D₅₃	1	2	yes	—
104	C₂×C₄	#11719949	1	2	yes	—
105	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
106	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
107	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
108	C₂²	C₂²×C₄×D₅₃	1	2	yes	—
109	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
110	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
111	C₂²	#41×D₅₃	1	2	yes	—
112	C₂²	C₂×#11719949	1	1	yes	characteristic
113	C₄	—	2	6	no	—
114	C₄	—	53	8	no	—
115	C₄	—	106	12	no	—
116	C₄	C₂×C₄×Dic₅₃	1	12	yes	—
117	C₄	Q₈×Dic₅₃	1	4	yes	direct product factor
118	C₂	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
119	C₂	C₄×Q₈×D₅₃	1	2	yes	—
120	C₂	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
121	C₂	C₄×#11719949	1	2	yes	—
122	C₂	#41×Dic₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₅₃

C₂

yes

maximal

C₄⋊C₄×Dic₅₃

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal

C₄²×Dic₅₃

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

Q₈×Dic₅₃

C₄

yes

direct product factor

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

Q₈

yes

direct product factor

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

C₄²

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

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

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

#41

yes

—

C₂²×C₅₃

C₂³×C₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

C₄×C₅₃

C₄×Q₈

yes

complemented

Dic₅₃

—

Dic₅₃

C₄×Q₈

yes

direct product factor

C₄²×Q₈

—

maximal, Sylow

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

C₂×C₄×Q₈

yes

—

C₂×C₅₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₅₃

C₄×#41

yes

—

C₂×C₅₃

C₄×#41

yes

characteristic

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₅₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

—

C₅₃

C₄²×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

Dic₅₃

yes

direct product factor

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

#95

—

C₂×Q₈

C₄×D₅₃

yes

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

C₂×Dic₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

C₄×Dic₅₃

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

106

—

101

C₂×C₄

—

106

—

102

C₂×C₄

C₂×C₄×D₅₃

yes

—

103

C₂×C₄

Q₈×D₅₃

yes

—

104

C₂×C₄

#11719949

yes

—

105

C₂×C₄

C₂²×Dic₅₃

yes

—

106

C₂×C₄

C₂²×Dic₅₃

yes

—

107

C₂×C₄

C₂²×Dic₅₃

yes

—

108

C₂²

C₂²×C₄×D₅₃

yes

—

109

C₂²

C₂³×Dic₅₃

yes

characteristic

110

C₂²

C₂×Q₈×D₅₃

yes

characteristic

111

C₂²

#41×D₅₃

yes

—

112

C₂²

C₂×#11719949

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

106

—

116

C₄

C₂×C₄×Dic₅₃

yes

—

117

C₄

Q₈×Dic₅₃

yes

direct product factor

118

C₂

C₂²×C₄×Dic₅₃

yes

characteristic

119

C₂

C₄×Q₈×D₅₃

yes

—

120

C₂

C₂×Q₈×Dic₅₃

yes

characteristic

121

C₂

C₄×#11719949

yes

—

122

C₂

#41×Dic₅₃

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₄²×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₅₃

commutator

C₂×C₅₃

C₂²×C₄²

Frattini

C₂³

C₂³×D₅₃

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂×C₄×Q₈

D₅₃

hypocenter

C₅₃

C₄²×Q₈

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
6. Q₈×Dic₅₃	C₄	direct (N × H)
14. C₄×Q₈×C₅₃	C₄	semidirect
30. C₄×Dic₅₃	Q₈	direct (N × H)
32. Q₈×C₅₃	C₄²	semidirect
53. C₄×C₅₃	C₄×Q₈	semidirect
55. Dic₅₃	C₄×Q₈	direct (N × H)
67. C₅₃	C₄²×Q₈	semidirect
70. C₄×Q₈	Dic₅₃	direct (N × H)
94. Q₈	C₄×Dic₅₃	direct (N × H)
117. C₄	Q₈×Dic₅₃	direct (N × H)

H ≅ G/N

type

6. Q₈×Dic₅₃

C₄

direct (N × H)

14. C₄×Q₈×C₅₃

C₄

semidirect

30. C₄×Dic₅₃

Q₈

direct (N × H)

32. Q₈×C₅₃

C₄²

semidirect

53. C₄×C₅₃

C₄×Q₈

semidirect

55. Dic₅₃

C₄×Q₈

direct (N × H)

67. C₅₃

C₄²×Q₈

semidirect

70. C₄×Q₈

Dic₅₃

direct (N × H)

94. Q₈

C₄×Dic₅₃

direct (N × H)

117. C₄

Q₈×Dic₅₃

direct (N × H)

Subgroups — C₄×Q₈×Dic₅₃ #20277066

Conjugacy classes of subgroups

Complements

Subgroups — C4×Q8×Dic53 #20277066

Conjugacy classes of subgroups

Complements

Subgroups — C₄×Q₈×Dic₅₃ #20277066