Group #16811343 — subgroups

Subgroups — #16811343

Order 3392.

212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13133217	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133210	C₂	1	4	yes	maximal, complemented
3	#13133204	C₂	1	2	yes	maximal
4	#13133196	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133182	C₂	1	2	yes	maximal, complemented
7	#13133176	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	C₂²	1	2	yes	—
11	C₂×Dic₁₀₆	C₂²	1	2	yes	—
12	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²⋊Dic₅₃	C₂²	1	4	yes	—
17	C₂²⋊Dic₅₃	C₂²	1	4	yes	—
18	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₅₃	C₂²	1	2	yes	—
20	C₄⋊Dic₅₃	C₂²	1	2	yes	—
21	#11719919	C₂²	1	4	yes	—
22	#11719919	C₂²	1	2	yes	—
23	C₄×Dic₅₃	C₂²	1	4	yes	—
24	C₄×Dic₅₃	C₂²	1	2	yes	—
25	C₂³×C₅₃	C₂³	1	2	yes	—
26	D₄×C₅₃	—	2	2	no	—
27	C₂×C₄×C₅₃	—	2	4	no	—
28	C₂×C₄×C₅₃	C₂³	1	2	yes	—
29	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₅₃	C₂³	1	2	yes	—
33	C₂×Dic₅₃	C₂³	1	4	yes	—
34	C₂×Dic₅₃	C₂³	1	2	yes	—
35	Dic₁₀₆	—	2	4	no	—
36	Dic₁₀₆	—	2	2	no	—
37	C₂²×C₅₃	—	2	2	no	—
38	C₂²×C₅₃	—	2	2	no	—
39	C₂²×C₅₃	—	2	2	no	—
40	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₄×C₅₃	—	2	2	no	—
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	#41	1	2	yes	—
44	C₄×C₅₃	#41	1	2	yes	—
45	Dic₅₃	—	2	2	no	—
46	Dic₅₃	—	2	4	no	—
47	Dic₅₃	—	2	2	no	—
48	C₂×C₅₃	—	4	2	no	—
49	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
50	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
51	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
52	#556	—	53	1	no	maximal, Sylow
53	C₅₃	#556	1	1	yes	characteristic, complemented, Sylow
54	C₄⋊Q₈	—	53	1	no	—
55	#126	—	53	4	no	—
56	#124	—	53	2	no	—
57	C₂²⋊Q₈	—	53	2	no	—
58	C₄×Q₈	—	53	2	no	—
59	C₄×Q₈	—	53	1	no	—
60	C₄×D₄	D₅₃	1	1	yes	characteristic
61	#117	—	53	2	no	—
62	C₂×Q₈	—	53	1	no	—
63	C₂×Q₈	—	53	2	no	—
64	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₅₃	1	2	yes	—
66	C₄⋊C₄	—	53	1	no	—
67	C₄⋊C₄	—	53	2	no	—
68	C₄⋊C₄	—	53	4	no	—
69	C₄⋊C₄	—	53	2	no	—
70	C₄⋊C₄	—	53	2	no	—
71	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
72	C₂²⋊C₄	—	53	4	no	—
73	C₂²⋊C₄	—	53	4	no	—
74	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
75	C₄²	—	53	4	no	—
76	C₄²	—	53	2	no	—
77	C₄²	C₂×D₅₃	1	1	yes	characteristic
78	C₂³	C₂²×D₅₃	1	2	yes	—
79	Q₈	—	106	4	no	—
80	Q₈	—	106	2	no	—
81	D₄	—	2	2	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	53	2	no	—
84	C₂×C₄	—	53	2	no	—
85	C₂×C₄	—	53	4	no	—
86	C₂×C₄	C₂²×D₅₃	1	2	yes	—
87	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	—	2	2	no	—
92	C₂²	—	2	2	no	—
93	C₂²	C₂³×D₅₃	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	106	2	no	—
97	C₄	—	106	4	no	—
98	C₄	—	106	2	no	—
99	C₄	#11719945	1	2	yes	—
100	C₄	#11719947	1	2	yes	—
101	C₂	—	4	2	no	—
102	C₂	C₂×#11719947	1	1	yes	characteristic
103	C₂	C₂×#11719945	1	1	yes	characteristic
104	C₂	#13133264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133217

C₂

yes

maximal, characteristic, complemented

#13133210

C₂

yes

maximal, complemented

#13133204

C₂

yes

maximal

#13133196

C₂

yes

maximal, complemented

C₄×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

#13133182

C₂

yes

maximal, complemented

#13133176

C₂

yes

maximal, complemented

C₄×D₄×C₅₃

C₂

yes

maximal, characteristic

C₂×D₄×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂²⋊C₄×C₅₃

C₂²

yes

—

C₄²×C₅₃

C₂²

yes

characteristic

C₂²⋊Dic₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

characteristic

C₄⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₂³×C₅₃

C₂³

yes

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#41

yes

—

C₄×C₅₃

#41

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#41

yes

characteristic

C₂×C₅₃

C₂×#41

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

—

C₅₃

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#126

—

#124

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

D₅₃

yes

characteristic

#117

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₅₃

yes

characteristic

C₂²×C₄

C₂×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅₃

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₅₃

yes

characteristic

C₂³

C₂²×D₅₃

yes

—

Q₈

—

106

—

Q₈

—

106

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₅₃

yes

—

C₂×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₄

C₂²×D₅₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×D₅₃

yes

characteristic

C₄

—

C₄

—

C₄

—

106

—

C₄

—

106

—

C₄

—

106

—

C₄

#11719945

yes

—

100

C₄

#11719947

yes

—

101

C₂

—

102

C₂

C₂×#11719947

yes

characteristic

103

C₂

C₂×#11719945

yes

characteristic

104

C₂

#13133264

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₄×D₄×C₅₃	C₂
prime radical	C₂×D₄×C₅₃	C₂²
hypercenter	C₄×D₄	D₅₃
hypocenter	C₅₃	#556
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting

C₄×D₄×C₅₃

C₂

prime radical

C₂×D₄×C₅₃

C₂²

hypercenter

C₄×D₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #13133217	C₂	semidirect
2. #13133210	C₂	semidirect
4. #13133196	C₂	semidirect
5. C₄×Dic₁₀₆	C₂	semidirect
6. #13133182	C₂	semidirect
7. #13133176	C₂	semidirect
53. C₅₃	#556	semidirect

H ≅ G/N

type

1. #13133217

C₂

semidirect

2. #13133210

C₂

semidirect

4. #13133196

C₂

semidirect

5. C₄×Dic₁₀₆

C₂

semidirect

6. #13133182

C₂

semidirect

7. #13133176

C₂

semidirect

53. C₅₃

#556

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13133217	2
#13133210	2
#13133204	2
#13133196	2
C₄×Dic₁₀₆	2
#13133182	2
#13133176	2
C₄×D₄×C₅₃	2
#556	53

subgroup

index

C₄×Dic₁₀₆

C₄×D₄×C₅₃

Subgroups — #16811343

Conjugacy classes of subgroups

Complements

Maximal subgroups