Group #16811251 — subgroups

Subgroups — #16811251

Order 3392.

206 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 111 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	2	yes	maximal, complemented
2	#13133216	C₂	1	2	yes	maximal, complemented
3	#13133209	C₂	1	2	yes	maximal, complemented
4	#13133208	C₂	1	2	yes	maximal, complemented
5	#13133194	C₂	1	4	yes	maximal
6	#13133176	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
8	#117×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₅₃	C₂²	1	2	yes	—
11	C₂×Dic₁₀₆	C₂²	1	4	yes	—
12	C₄⋊C₄×C₅₃	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	2	yes	—
16	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
17	C₄⋊Dic₅₃	C₂²	1	2	yes	—
18	C₄⋊Dic₅₃	C₂²	1	2	yes	—
19	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₅₃	C₂²	1	4	yes	—
22	#11719919	C₂²	1	4	yes	—
23	#11719919	C₂²	1	4	yes	—
24	C₄×Dic₅₃	C₂²	1	2	yes	—
25	C₂³×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₅₃	Q₈	1	4	yes	complemented
27	C₂×C₄×C₅₃	C₂³	1	2	yes	—
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×Dic₅₃	—	4	2	no	—
32	C₂×Dic₅₃	C₂³	1	2	yes	—
33	C₂×Dic₅₃	C₂³	1	4	yes	—
34	C₂×Dic₅₃	C₂³	1	2	yes	—
35	Dic₁₀₆	—	4	4	no	—
36	C₂²×C₅₃	—	2	1	no	—
37	C₂²×C₅₃	—	2	1	no	—
38	C₂²×C₅₃	C₂×Q₈	1	2	yes	—
39	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	—	2	2	no	—
42	C₄×C₅₃	C₂×Q₈	1	2	yes	—
43	C₄×C₅₃	C₂×Q₈	1	2	yes	—
44	Dic₅₃	—	2	2	no	—
45	Dic₅₃	—	2	4	no	—
46	Dic₅₃	—	2	2	no	—
47	C₂×C₅₃	—	2	2	no	—
48	C₂×C₅₃	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
50	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
51	#544	—	53	1	no	maximal, Sylow
52	C₅₃	#544	1	1	yes	characteristic, complemented, Sylow
53	C₄⋊Q₈	—	53	2	no	—
54	C₄⋊Q₈	—	53	2	no	—
55	#125	—	53	2	no	—
56	#125	—	53	2	no	—
57	C₂²⋊Q₈	—	53	4	no	—
58	#117	—	53	1	no	—
59	#117	D₅₃	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	53	1	no	—
61	C₂×Q₈	—	53	4	no	—
62	C₂²×C₄	—	53	2	no	—
63	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
64	C₄⋊C₄	—	53	2	no	—
65	C₄⋊C₄	—	53	2	no	—
66	C₄⋊C₄	—	53	1	no	—
67	C₄⋊C₄	—	53	1	no	—
68	C₄⋊C₄	—	53	4	no	—
69	C₄⋊C₄	—	53	4	no	—
70	C₄⋊C₄	—	53	4	no	—
71	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
72	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
73	C₂²⋊C₄	—	53	2	no	—
74	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
75	C₄²	—	53	2	no	—
76	C₄²	C₂×D₅₃	1	2	yes	—
77	C₂³	C₂²×D₅₃	1	1	yes	characteristic
78	Q₈	—	212	4	no	—
79	C₂×C₄	—	53	2	no	—
80	C₂×C₄	—	53	2	no	—
81	C₂×C₄	—	53	4	no	—
82	C₂×C₄	—	212	2	no	—
83	C₂×C₄	Dic₁₀₆	1	4	yes	complemented
84	C₂×C₄	C₂²×D₅₃	1	2	yes	—
85	C₂×C₄	C₂²×D₅₃	1	2	yes	—
86	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
87	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	C₂×Dic₁₀₆	1	2	yes	—
91	C₂²	C₂³×D₅₃	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	106	4	no	—
95	C₄	—	106	2	no	—
96	C₄	—	106	2	no	—
97	C₄	C₂×Dic₁₀₆	1	2	yes	—
98	C₄	C₂×Dic₁₀₆	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	C₂²×Dic₁₀₆	1	1	yes	characteristic
101	C₂	#13133263	1	1	yes	characteristic
102	C₂	#13133264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133217

C₂

yes

maximal, complemented

#13133216

C₂

yes

maximal, complemented

#13133209

C₂

yes

maximal, complemented

#13133208

C₂

yes

maximal, complemented

#13133194

C₂

yes

maximal

#13133176

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

#117×C₅₃

C₂

yes

maximal, characteristic

C₂²×C₄×C₅₃

C₂²

yes

characteristic

C₂²×Dic₅₃

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

—

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂²⋊C₄×C₅₃

C₂²

yes

—

C₄²×C₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

characteristic

C₄⋊Dic₅₃

C₂²

yes

characteristic

C₄⋊Dic₅₃

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₂³×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

Q₈

yes

complemented

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×Q₈

yes

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₈

yes

—

C₄×C₅₃

C₂×Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×Q₈

yes

characteristic

C₂×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

—

C₅₃

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅₃

yes

—

C₄²

—

C₄²

C₂×D₅₃

yes

—

C₂³

C₂²×D₅₃

yes

characteristic

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

Dic₁₀₆

yes

complemented

C₂×C₄

C₂²×D₅₃

yes

—

C₂×C₄

C₂²×D₅₃

yes

—

C₂×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₄

C₂²×D₅₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₁₀₆

yes

—

C₂²

C₂³×D₅₃

yes

characteristic

C₄

—

C₄

—

C₄

—

106

—

C₄

—

106

—

C₄

—

106

—

C₄

C₂×Dic₁₀₆

yes

—

C₄

C₂×Dic₁₀₆

yes

—

C₂

—

100

C₂

C₂²×Dic₁₀₆

yes

characteristic

101

C₂

#13133263

yes

characteristic

102

C₂

#13133264

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	#117×C₅₃	C₂
prime radical	C₂³×C₅₃	C₂³
hypercenter	#117	D₅₃
hypocenter	C₅₃	#544
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting

#117×C₅₃

C₂

prime radical

C₂³×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #13133217	C₂	semidirect
2. #13133216	C₂	semidirect
3. #13133209	C₂	semidirect
4. #13133208	C₂	semidirect
26. C₂×C₄×C₅₃	Q₈	semidirect
52. C₅₃	#544	semidirect
83. C₂×C₄	Dic₁₀₆	semidirect

H ≅ G/N

type

1. #13133217

C₂

semidirect

2. #13133216

C₂

semidirect

3. #13133209

C₂

semidirect

4. #13133208

C₂

semidirect

26. C₂×C₄×C₅₃

Q₈

semidirect

52. C₅₃

#544

semidirect

83. C₂×C₄

Dic₁₀₆

semidirect

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

subgroup	index
#13133217	2
#13133216	2
#13133209	2
#13133208	2
#13133194	2
#13133176	2
C₂×C₄⋊Dic₅₃	2
#117×C₅₃	2
#544	53

subgroup

index

C₂×C₄⋊Dic₅₃

#117×C₅₃

#544

Subgroups — #16811251

Conjugacy classes of subgroups

Complements

Maximal subgroups