Group #20274608 — subgroups

Subgroups — C₄×#13133151 #20274608

Order 6784.

280 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133151	C₂	1	2	yes	maximal
2	C₄×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
3	#16810386	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
6	C₂×C₄×Dic₅₃	—	2	2	no	—
7	#13133151	C₄	1	8	yes	direct product factor
8	#13133124	—	2	2	no	—
9	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₅₃	C₄	1	4	yes	complemented
13	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₃	—	2	4	no	—
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₄²×C₅₃	Q₈	1	2	yes	—
23	C₄⋊Dic₅₃	—	2	4	no	—
24	C₄⋊Dic₅₃	C₂×C₄	1	8	yes	—
25	C₄×Dic₅₃	—	4	4	no	—
26	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	2	2	no	—
28	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₅₃	D₈	1	2	yes	—
31	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
32	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	#41	1	2	yes	—
36	C₂×Dic₅₃	—	2	8	no	—
37	C₂×Dic₅₃	—	4	4	no	—
38	C₂×Dic₅₃	—	4	4	no	—
39	C₈×C₅₃	—	2	2	no	—
40	C₈×C₅₃	C₄²	1	4	yes	complemented
41	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₅₃	C₄×D₄	1	2	yes	—
43	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
44	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
45	C₂²×C₅₃	#135	1	2	yes	—
46	C₄×C₅₃	—	2	2	no	—
47	C₄×C₅₃	#107	1	4	yes	complemented
48	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
49	C₄×C₅₃	#117	1	1	yes	characteristic
50	C₄×C₅₃	C₄×Q₈	1	2	yes	—
51	Dic₅₃	—	4	8	no	—
52	C₄×#107	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
55	C₂×C₅₃	#427	1	1	yes	characteristic
56	C₂×C₅₃	C₄×D₈	1	2	yes	—
57	C₂×C₅₃	C₄×Q₁₆	1	2	yes	—
58	C₂×#107	—	53	2	no	—
59	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
60	C₄×C₄⋊C₄	—	53	2	no	—
61	#340	—	53	2	no	—
62	C₅₃	C₄×#107	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	—	53	4	no	—
66	C₂×C₄²	—	106	2	no	—
67	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
68	#107	—	53	8	no	—
69	C₄×C₈	Dic₅₃	1	4	yes	complemented
70	#95	—	106	2	no	—
71	C₂²×C₄	—	106	4	no	—
72	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
73	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
74	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	C₄×D₅₃	1	4	yes	—
77	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
78	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
79	C₄⋊C₄	—	53	8	no	—
80	C₄⋊C₄	—	106	4	no	—
81	C₄²	—	212	4	no	—
82	C₄²	Dic₁₀₆	1	2	yes	—
83	C₄²	C₂×Dic₅₃	1	2	yes	—
84	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	106	8	no	—
87	C₂×C₄	—	212	4	no	—
88	C₂×C₄	—	212	4	no	—
89	C₂×C₄	D₄₂₄	1	2	yes	—
90	C₂×C₄	Dic₂₁₂	1	2	yes	—
91	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
92	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
93	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
94	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
95	C₂×C₄	#11719945	1	2	yes	—
96	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	C₄×Dic₅₃	1	4	yes	complemented
99	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
100	C₂²	C₄×D₂₁₂	1	2	yes	—
101	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
102	C₂²	C₂×Dic₂₁₂	1	1	yes	characteristic
103	C₂²	#13133239	1	2	yes	—
104	C₄	—	2	2	no	—
105	C₄	—	212	8	no	—
106	C₄	#13133151	1	4	yes	direct product factor
107	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
108	C₄	#13133176	1	1	yes	characteristic
109	C₄	C₄×Dic₁₀₆	1	2	yes	—
110	C₂	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
111	C₂	C₂×#13133151	1	1	yes	characteristic
112	C₂	#16810661	1	1	yes	characteristic
113	C₂	C₄×D₄₂₄	1	2	yes	—
114	C₂	C₄×Dic₂₁₂	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal

#16810386

C₂

yes

maximal

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

—

#13133151

C₄

yes

direct product factor

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄²

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

characteristic

C₂²×C₅₃

#135

yes

—

C₄×C₅₃

—

C₄×C₅₃

#107

yes

complemented

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

—

Dic₅₃

—

C₄×#107

—

maximal, Sylow

C₂×C₅₃

C₄×C₄⋊C₄

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

C₄×D₈

yes

—

C₂×C₅₃

C₄×Q₁₆

yes

—

C₂×#107

—

C₂×C₄×C₈

D₅₃

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₅₃

C₄×#107

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#107

—

C₄×C₈

Dic₅₃

yes

complemented

#95

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

—

212

—

C₄²

Dic₁₀₆

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₂³

C₂×D₂₁₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

D₄₂₄

yes

—

C₂×C₄

Dic₂₁₂

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

#11719945

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

C₄×Dic₅₃

yes

complemented

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

100

C₂²

C₄×D₂₁₂

yes

—

101

C₂²

C₂×D₄₂₄

yes

characteristic

102

C₂²

C₂×Dic₂₁₂

yes

characteristic

103

C₂²

#13133239

yes

—

104

C₄

—

105

C₄

—

212

—

106

C₄

#13133151

yes

direct product factor

107

C₄

C₂×C₄×Dic₅₃

yes

characteristic

108

C₄

#13133176

yes

characteristic

109

C₄

C₄×Dic₁₀₆

yes

—

110

C₂

C₄×C₄⋊Dic₅₃

yes

characteristic

111

C₂

C₂×#13133151

yes

characteristic

112

C₂

#16810661

yes

characteristic

113

C₂

C₄×D₄₂₄

yes

—

114

C₂

C₄×Dic₂₁₂

yes

—

	subgroup	G/H
center	C₂²×C₄	D₂₁₂
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	C₄×#107
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₂₁₂

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

C₄×#107

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #13133151	C₄	direct (N × H)
12. C₄×C₈×C₅₃	C₄	semidirect
40. C₈×C₅₃	C₄²	semidirect
47. C₄×C₅₃	#107	semidirect
62. C₅₃	C₄×#107	semidirect
69. C₄×C₈	Dic₅₃	semidirect
98. C₈	C₄×Dic₅₃	semidirect
106. C₄	#13133151	direct (N × H)

H ≅ G/N

type

7. #13133151

C₄

direct (N × H)

12. C₄×C₈×C₅₃

C₄

semidirect

40. C₈×C₅₃

C₄²

semidirect

47. C₄×C₅₃

#107

semidirect

62. C₅₃

C₄×#107

semidirect

69. C₄×C₈

Dic₅₃

semidirect

98. C₈

C₄×Dic₅₃

semidirect

106. C₄

#13133151

direct (N × H)

Subgroups — C₄×#13133151 #20274608

Conjugacy classes of subgroups

Complements

Subgroups — C4×#13133151 #20274608

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13133151 #20274608