Group #20276314 — subgroups

Subgroups — #20276314

Order 6784.

184 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂²×#11719908	C₂	1	1	yes	maximal, characteristic
2	#16810578	C₂	1	1	yes	maximal, characteristic
3	#16810398	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₅₃	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₄	1	1	yes	characteristic
7	#13133176	C₄	1	1	yes	characteristic
8	C₂×#11719908	C₄	1	4	yes	complemented
9	C₂×#11719908	C₂²	1	1	yes	characteristic
10	C₂×#11719908	C₂²	1	1	yes	characteristic
11	#13133127	C₄	1	2	yes	—
12	C₈×Dic₅₃	C₂²	1	2	yes	—
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	M₃₂×C₅₃	C₄	1	4	yes	complemented
15	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²⋊Dic₅₃	C₈	1	2	yes	—
22	C₄⋊Dic₅₃	C₈	1	2	yes	—
23	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
24	#11719917	C₈	1	4	yes	—
25	C₂×#88322	C₂×C₄	1	2	yes	—
26	C₁₆×C₅₃	C₂×C₄	1	4	yes	—
27	#11719908	C₂×C₄	1	2	yes	—
28	#11719908	C₂×C₄	1	2	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	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
35	C₈×C₅₃	C₄²	1	2	yes	—
36	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	#88322	C₂×C₈	1	4	yes	—
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	D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
57	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
58	#117	—	53	1	no	—
59	M₃₂	Dic₅₃	1	4	yes	complemented
60	C₂×C₁₆	—	53	4	no	—
61	C₂×C₁₆	—	53	1	no	—
62	C₂×C₁₆	—	53	1	no	—
63	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
64	#97	—	53	2	no	—
65	C₄×C₈	—	53	2	no	—
66	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
67	M₁₆	—	53	4	no	—
68	C₂×C₈	—	53	2	no	—
69	C₂×C₈	C₄×D₅₃	1	2	yes	—
70	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
71	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
72	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
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	C₂×Dic₅₃	1	4	yes	—
79	C₂³	C₈×D₅₃	1	1	yes	characteristic
80	C₂×C₄	—	53	4	no	—
81	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
82	C₂×C₄	C₈×D₅₃	1	2	yes	—
83	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
84	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
85	C₈	—	53	4	no	—
86	C₈	C₄×Dic₅₃	1	2	yes	—
87	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
88	C₈	C₂²×Dic₅₃	1	1	yes	characteristic
89	C₂²	—	2	2	no	—
90	C₂²	C₈×Dic₅₃	1	2	yes	—
91	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
92	C₄	—	106	4	no	—
93	C₄	C₈×Dic₅₃	1	2	yes	—
94	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
95	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₈×Dic₅₃	1	1	yes	characteristic
98	C₂	#16811041	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719908

C₂

yes

maximal, characteristic

#16810578

C₂

yes

maximal, characteristic

#16810398

C₂

yes

maximal, complemented

C₁₆×Dic₅₃

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

characteristic

#13133176

C₄

yes

characteristic

C₂×#11719908

C₄

yes

complemented

C₂×#11719908

C₂²

yes

characteristic

C₂×#11719908

C₂²

yes

characteristic

#13133127

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₄

yes

complemented

C₂×C₁₆×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

C₈

yes

—

C₄⋊Dic₅₃

C₈

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

#11719917

C₈

yes

—

C₂×#88322

C₂×C₄

yes

—

C₁₆×C₅₃

C₂×C₄

yes

—

#11719908

C₂×C₄

yes

—

#11719908

C₂×C₄

yes

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

—

C₈×C₅₃

C₄²

yes

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

#88322

C₂×C₈

yes

—

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

D₅₃

yes

characteristic

C₂²×C₁₆

—

#405

—

#346

—

C₄×C₁₆

—

C₅₃

#2108

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

M₃₂

Dic₅₃

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

—

#97

—

C₄×C₈

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

C₂×Dic₅₃

yes

—

C₂³

C₈×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₈×D₅₃

yes

characteristic

C₂×C₄

C₈×D₅₃

yes

—

C₂×C₄

C₄×Dic₅₃

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₈

—

C₈

C₄×Dic₅₃

yes

—

C₈

C₂×C₄×D₅₃

yes

characteristic

C₈

C₂²×Dic₅₃

yes

characteristic

C₂²

—

C₂²

C₈×Dic₅₃

yes

—

C₂²

C₂×C₈×D₅₃

yes

characteristic

C₄

—

106

—

C₄

C₈×Dic₅₃

yes

—

C₄

C₂×C₄×Dic₅₃

yes

characteristic

C₄

C₂×C₈×D₅₃

yes

characteristic

C₂

—

C₂

C₂×C₈×Dic₅₃

yes

characteristic

C₂

#16811041

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₈
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₈
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#2108
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₃

commutator

C₂×C₅₃

C₂×C₄×C₈

Fitting

C₂×M₃₂×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×C₈

hypercenter

C₂×M₃₂

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16810398	C₂	semidirect
4. C₁₆×Dic₅₃	C₂	semidirect
8. C₂×#11719908	C₄	semidirect
14. M₃₂×C₅₃	C₄	semidirect
55. C₅₃	#2108	semidirect
59. M₃₂	Dic₅₃	semidirect

H ≅ G/N

type

3. #16810398

C₂

semidirect

4. C₁₆×Dic₅₃

C₂

semidirect

8. C₂×#11719908

C₄

semidirect

14. M₃₂×C₅₃

C₄

semidirect

55. C₅₃

#2108

semidirect

59. M₃₂

Dic₅₃

semidirect

Subgroups — #20276314

Conjugacy classes of subgroups

Complements