Group #20290926 — subgroups

Subgroups — #20290926

Order 6784.

184 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 114 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×#13133291	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11719910	C₂	1	1	yes	maximal, characteristic
3	#16811512	C₂	1	2	yes	maximal, complemented
4	#16811509	C₂	1	2	yes	maximal, complemented
5	#16810581	C₂	1	1	yes	maximal, characteristic
6	#13133291	C₄	1	4	yes	complemented
7	C₂×#11719910	C₄	1	4	yes	complemented
8	C₂×#11719910	C₂²	1	1	yes	characteristic
9	C₂×#11719910	C₂²	1	1	yes	characteristic
10	C₂×#11719910	C₂²	1	2	yes	—
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	#13133176	C₄	1	1	yes	characteristic
13	#13133127	C₄	1	2	yes	—
14	C₈×Dic₅₃	C₂²	1	2	yes	—
15	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	M₁₆×C₅₃	C₈	1	4	yes	—
18	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²⋊Dic₅₃	C₈	1	2	yes	—
20	C₄⋊Dic₅₃	C₈	1	2	yes	—
21	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
22	C₂×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	C₂×C₄	1	2	yes	—
24	C₂×#88322	C₂×C₄	1	2	yes	—
25	C₂×#88322	C₂³	1	1	yes	characteristic
26	#11719910	C₂×C₄	1	2	yes	—
27	#11719910	C₂×C₄	1	2	yes	—
28	#11719910	C₂×C₄	1	4	yes	—
29	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₄²	1	2	yes	—
31	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
35	C₈×C₅₃	C₂×C₈	1	4	yes	—
36	#88322	C₄²	1	2	yes	—
37	#88322	C₂²×C₄	1	1	yes	characteristic
38	#88322	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₅₃	—	2	2	no	—
40	C₂²×C₅₃	C₄×C₈	1	2	yes	—
41	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₅₃	C₄×C₈	1	2	yes	—
43	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
45	Dic₅₃	—	2	4	no	—
46	#2108	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	—	2	2	no	—
48	C₂×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
49	C₂×C₅₃	#504	1	2	yes	—
50	C₂×M₃₂	—	53	1	no	—
51	C₂²×C₁₆	—	53	1	no	—
52	#405	—	53	1	no	—
53	#346	—	53	2	no	—
54	C₄×C₁₆	—	53	2	no	—
55	C₅₃	#2108	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₁₆	#6157	1	1	yes	characteristic
57	C₂²×C₈	—	53	1	no	—
58	#117	—	53	1	no	—
59	M₃₂	—	53	4	no	—
60	C₂×C₁₆	—	53	1	no	—
61	C₂×C₁₆	—	53	2	no	—
62	C₂×C₁₆	—	53	1	no	—
63	C₂×C₁₆	—	53	4	no	—
64	#97	—	53	2	no	—
65	C₄×C₈	—	53	2	no	—
66	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
67	M₁₆	#88324	1	4	yes	—
68	C₂×C₈	—	53	2	no	—
69	C₂×C₈	—	53	2	no	—
70	C₂×C₈	—	53	1	no	—
71	C₂×C₈	—	53	1	no	—
72	C₂×C₈	C₂×#6157	1	2	yes	—
73	C₄⋊C₄	—	53	2	no	—
74	C₂²⋊C₄	—	53	2	no	—
75	C₄²	—	53	2	no	—
76	C₁₆	—	53	2	no	—
77	C₁₆	—	53	2	no	—
78	C₁₆	—	53	4	no	—
79	C₂³	D₅₃⋊C₈	1	1	yes	characteristic
80	C₂×C₄	—	53	4	no	—
81	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
82	C₂×C₄	C₄×#6157	1	2	yes	—
83	C₂×C₄	C₂×#88324	1	2	yes	—
84	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
85	C₈	—	53	2	no	—
86	C₈	—	53	1	no	—
87	C₈	—	53	1	no	—
88	C₈	C₂×#88324	1	4	yes	—
89	C₂²	—	2	2	no	—
90	C₂²	C₄×#88324	1	2	yes	—
91	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
92	C₄	—	106	4	no	—
93	C₄	C₄×#88324	1	2	yes	—
94	C₄	C₂×C₄×#6157	1	1	yes	characteristic
95	C₄	C₂²×#88324	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₄×#88324	1	1	yes	characteristic
98	C₂	#16811682	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133291

C₂

yes

maximal, characteristic

C₂²×#11719910

C₂

yes

maximal, characteristic

#16811512

C₂

yes

maximal, complemented

#16811509

C₂

yes

maximal, complemented

#16810581

C₂

yes

maximal, characteristic

#13133291

C₄

yes

complemented

C₂×#11719910

C₄

yes

complemented

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

#13133176

C₄

yes

characteristic

#13133127

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

M₁₆×C₅₃

C₈

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

C₈

yes

—

C₄⋊Dic₅₃

C₈

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂³

yes

characteristic

#11719910

C₂×C₄

yes

—

#11719910

C₂×C₄

yes

—

#11719910

C₂×C₄

yes

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

—

C₈×C₅₃

C₂×C₈

yes

—

#88322

C₄²

yes

—

#88322

C₂²×C₄

yes

characteristic

#88322

C₂²×C₄

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

C₄×C₈

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₄×C₈

yes

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

Dic₅₃

—

#2108

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂×C₅₃

#504

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#346

—

C₄×C₁₆

—

C₅₃

#2108

yes

characteristic, complemented, Sylow

C₂×M₁₆

#6157

yes

characteristic

C₂²×C₈

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₂²×C₄

C₂×#6157

yes

characteristic

M₁₆

#88324

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#6157

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

D₅₃⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

D₅₃⋊C₈

yes

characteristic

C₂×C₄

C₄×#6157

yes

—

C₂×C₄

C₂×#88324

yes

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

C₂×#88324

yes

—

C₂²

—

C₂²

C₄×#88324

yes

—

C₂²

C₂×D₅₃⋊C₈

yes

characteristic

C₄

—

106

—

C₄

C₄×#88324

yes

—

C₄

C₂×C₄×#6157

yes

characteristic

C₄

C₂²×#88324

yes

characteristic

C₂

—

C₂

C₂×C₄×#88324

yes

characteristic

C₂

#16811682

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂×C₄×C₈
Fitting	C₂×M₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₈
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	#2108
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator

C₂×C₅₃

C₂×C₄×C₈

Fitting

C₂×M₁₆×C₅₃

C₄

socle

C₂²×C₅₃

C₂²×C₈

C₂×M₁₆

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
3. #16811512	C₂	semidirect
4. #16811509	C₂	semidirect
6. #13133291	C₄	semidirect
7. C₂×#11719910	C₄	semidirect
55. C₅₃	#2108	semidirect

H ≅ G/N

type

3. #16811512

C₂

semidirect

4. #16811509

C₂

semidirect

6. #13133291

C₄

semidirect

7. C₂×#11719910

C₄

semidirect

55. C₅₃

#2108

semidirect

Subgroups — #20290926

Conjugacy classes of subgroups

Complements