Group #20276547 — subgroups

Subgroups — #20276547

Order 6784.

162 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 87 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×#13133157	C₂	1	2	yes	maximal, complemented
2	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#11719908	C₂	1	2	yes	maximal, complemented
4	C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#13133157	—	2	4	no	—
7	C₂×#11719908	C₂²	1	2	yes	—
8	C₂×#11719908	C₂²	1	1	yes	characteristic
9	C₂×#11719908	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
11	C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
12	C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₅₃	C₈	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
23	C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
24	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	#11719908	—	2	2	no	—
26	#11719908	—	2	1	no	—
27	#11719908	D₄	1	2	yes	complemented
28	C₂³×C₅₃	C₂×C₈	1	2	yes	—
29	D₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₅₃	—	2	2	no	—
37	C₈×C₅₃	—	2	1	no	—
38	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₅₃	#41	1	1	yes	characteristic
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	—	2	2	no	—
42	C₂²×C₅₃	—	2	2	no	—
43	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	M₃₂	1	2	yes	complemented
47	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
48	C₄×C₅₃	#131	1	1	yes	characteristic
49	#2169	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	4	2	no	—
51	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
52	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
53	C₂×C₅₃	#504	1	1	yes	characteristic
54	C₂×M₃₂	—	53	2	no	—
55	C₈×D₄	D₅₃	1	1	yes	characteristic
56	C₄⋊C₁₆	—	53	1	no	—
57	C₂²⋊C₁₆	—	53	2	no	—
58	C₄×C₁₆	—	53	1	no	—
59	C₅₃	#2169	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	C₂×D₅₃	1	2	yes	—
61	C₄×D₄	Dic₅₃	1	1	yes	characteristic
62	M₃₂	—	106	4	no	—
63	C₂×C₁₆	—	53	1	no	—
64	C₂×C₁₆	—	53	1	no	—
65	C₂×C₁₆	—	53	2	no	—
66	C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
67	C₂²⋊C₈	Dic₅₃	1	2	yes	—
68	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
69	C₂×D₄	#88322	1	1	yes	characteristic
70	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
74	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
75	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
76	C₄⋊C₄	#88322	1	1	yes	characteristic
77	C₂²⋊C₄	#88322	1	2	yes	—
78	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
79	C₁₆	—	53	2	no	—
80	C₁₆	—	106	2	no	—
81	C₁₆	—	106	1	no	—
82	C₂³	C₂×#88322	1	2	yes	—
83	D₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂×#88322	1	1	yes	characteristic
87	C₂×C₄	C₂×#88322	1	2	yes	—
88	C₂×C₄	C₂×#88322	1	1	yes	characteristic
89	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	2	1	no	—
92	C₈	D₄×D₅₃	1	1	yes	characteristic
93	C₈	#11719947	1	1	yes	characteristic
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₂²×#88322	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	2	no	—
100	C₄	#13133157	1	2	yes	complemented
101	C₄	D₄×Dic₅₃	1	1	yes	characteristic
102	C₄	#13133224	1	1	yes	characteristic
103	C₂	—	4	2	no	—
104	C₂	D₄×#88322	1	1	yes	characteristic
105	C₂	C₂×#13133157	1	1	yes	characteristic
106	C₂	#16811040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133157

C₂

yes

maximal, complemented

C₄⋊#11719908

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11719908

C₂

yes

maximal, complemented

C₄×#11719908

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₅₃

C₂

yes

maximal, characteristic

#13133157

—

C₂×#11719908

C₂²

yes

—

C₂×#11719908

C₂²

yes

characteristic

C₂×#11719908

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

—

C₄×D₄×C₅₃

C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂²⋊C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₈

yes

characteristic

C₂²⋊C₄×C₅₃

C₈

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

—

D₄

yes

complemented

C₂³×C₅₃

C₂×C₈

yes

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

M₃₂

yes

complemented

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₈×D₄

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂×C₅₃

#504

yes

characteristic

C₂×M₃₂

—

C₈×D₄

D₅₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₅₃

#2169

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₃

yes

—

C₄×D₄

Dic₅₃

yes

characteristic

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

Dic₅₃

yes

characteristic

C₂²⋊C₈

Dic₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

#88322

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂²×D₅₃

yes

C₄⋊C₄

yes

C₂²⋊C₄

yes

—

C₄²

C₂×Dic₅₃

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₁₆

—

106

—

C₂³

C₂×#88322

yes

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#88322

yes

characteristic

C₂×C₄

C₂×#88322

yes

—

C₂×C₄

C₂×#88322

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

—

C₈

D₄×D₅₃

yes

characteristic

C₈

#11719947

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×#88322

yes

characteristic

C₄

—

C₄

—

100

C₄

#13133157

yes

complemented

101

C₄

D₄×Dic₅₃

yes

characteristic

102

C₄

#13133224

yes

characteristic

103

C₂

—

104

C₂

D₄×#88322

yes

characteristic

105

C₂

C₂×#13133157

yes

characteristic

106

C₂

#16811040

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₈×D₄×C₅₃	C₂
hypercenter	C₈×D₄	D₅₃
hypocenter	C₅₃	#2169
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

C₈×D₄×C₅₃

C₂

hypercenter

C₈×D₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₈×D₄

N	H ≅ G/N	type
1. C₂×#13133157	C₂	semidirect
2. C₄⋊#11719908	C₂	semidirect
3. C₂²⋊#11719908	C₂	semidirect
4. C₄×#11719908	C₂	semidirect
27. #11719908	D₄	semidirect
46. C₄×C₅₃	M₃₂	semidirect
59. C₅₃	#2169	semidirect
100. C₄	#13133157	semidirect

H ≅ G/N

type

1. C₂×#13133157

C₂

semidirect

2. C₄⋊#11719908

C₂

semidirect

3. C₂²⋊#11719908

C₂

semidirect

4. C₄×#11719908

C₂

semidirect

27. #11719908

D₄

semidirect

46. C₄×C₅₃

M₃₂

semidirect

59. C₅₃

#2169

semidirect

100. C₄

#13133157

semidirect

Subgroups — #20276547

Conjugacy classes of subgroups

Complements