Group #20272991 — subgroups

Subgroups — C₂×#16810345 #20272991

Order 6784.

248 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 173 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×C₄×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#16810345	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₄	1	2	yes	—
6	C₂²×#88322	C₄	1	2	yes	—
7	C₄×#88322	C₄	1	2	yes	—
8	C₄×#88322	C₄	1	2	yes	—
9	C₄×#88322	C₂²	1	2	yes	—
10	C₄×#88322	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₄	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₄	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	—	2	4	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×#88322	—	2	4	no	—
25	C₂×#88322	C₈	1	8	yes	complemented
26	C₂×#88322	C₂×C₄	1	4	yes	—
27	C₂×#88322	C₂×C₄	1	4	yes	—
28	C₂³×C₅₃	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
33	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
34	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₅₃	—	2	4	no	—
39	#88322	—	2	4	no	—
40	#88322	C₂×C₈	1	8	yes	complemented
41	C₂²×C₅₃	C₄×C₈	1	2	yes	—
42	C₂²×C₅₃	#97	1	2	yes	—
43	C₂²×C₅₃	#97	1	2	yes	—
44	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
45	C₄×C₅₃	C₄×C₈	1	2	yes	—
46	C₄×C₅₃	#97	1	2	yes	—
47	C₄×C₅₃	#97	1	2	yes	—
48	C₄×C₅₃	C₂²×C₈	1	2	yes	—
49	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
50	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
51	C₂×#322	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	#322	1	4	yes	complemented
53	C₂×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
54	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
55	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
56	C₂×C₄×C₈	—	53	1	no	—
57	C₂×C₄×C₈	—	53	1	no	—
58	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
59	#322	—	53	4	no	—
60	C₅₃	C₂×#322	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₈	—	53	2	no	—
62	C₂²×C₈	—	53	2	no	—
63	C₂²×C₈	Dic₅₃	1	2	yes	—
64	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
65	C₄×C₈	—	53	2	no	—
66	C₄×C₈	—	53	2	no	—
67	C₄×C₈	—	53	2	no	—
68	C₄×C₈	—	53	2	no	—
69	C₄×C₈	Dic₅₃	1	2	yes	—
70	C₄×C₈	C₂×D₅₃	1	2	yes	—
71	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
73	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	53	4	no	—
76	C₂×C₈	—	53	8	no	—
77	C₂×C₈	—	53	4	no	—
78	C₂×C₈	—	106	4	no	—
79	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
80	C₄²	C₄×D₅₃	1	1	yes	characteristic
81	C₄²	C₄×D₅₃	1	1	yes	characteristic
82	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
83	C₄²	C₂²×D₅₃	1	1	yes	characteristic
84	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
85	C₂×C₄	C₈×D₅₃	1	4	yes	—
86	C₂×C₄	#11719912	1	4	yes	—
87	C₂×C₄	#11719917	1	4	yes	—
88	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
89	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
90	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
92	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
93	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
94	C₈	—	2	4	no	—
95	C₈	—	53	8	no	—
96	C₈	—	106	4	no	—
97	C₂²	C₈×Dic₅₃	1	2	yes	—
98	C₂²	#13133127	1	2	yes	—
99	C₂²	#13133128	1	2	yes	—
100	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
101	C₄	C₈×Dic₅₃	1	2	yes	—
102	C₄	#13133127	1	2	yes	—
103	C₄	#13133128	1	2	yes	—
104	C₄	C₂×C₈×D₅₃	1	2	yes	—
105	C₄	C₂×#11719912	1	2	yes	—
106	C₄	C₂×#11719917	1	2	yes	—
107	C₂	#16810345	1	4	yes	direct product factor
108	C₂	C₂×C₈×Dic₅₃	1	1	yes	characteristic
109	C₂	C₂×#13133127	1	1	yes	characteristic
110	C₂	C₂×#13133128	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂

yes

maximal, characteristic

#16810345

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₄

yes

—

C₂²×#88322

C₄

yes

—

C₄×#88322

C₄

yes

—

C₄×#88322

C₄

yes

—

C₄×#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂²×C₈×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂×#88322

—

C₂×#88322

C₈

yes

complemented

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

—

#88322

—

#88322

C₂×C₈

yes

complemented

C₂²×C₅₃

C₄×C₈

yes

—

C₂²×C₅₃

#97

yes

—

C₂²×C₅₃

#97

yes

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

C₄×C₈

yes

—

C₄×C₅₃

#97

yes

—

C₄×C₅₃

#97

yes

—

C₄×C₅₃

C₂²×C₈

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂×C₅₃

C₂×#97

yes

characteristic

C₂×C₅₃

C₂×#97

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₅₃

yes

characteristic

#322

—

C₅₃

C₂×#322

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₈×D₅₃

yes

—

C₂×C₄

#11719912

yes

—

C₂×C₄

#11719917

yes

—

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

C₂²

C₈×Dic₅₃

yes

—

C₂²

#13133127

yes

—

C₂²

#13133128

yes

—

100

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

101

C₄

C₈×Dic₅₃

yes

—

102

C₄

#13133127

yes

—

103

C₄

#13133128

yes

—

104

C₄

C₂×C₈×D₅₃

yes

—

105

C₄

C₂×#11719912

yes

—

106

C₄

C₂×#11719917

yes

—

107

C₂

#16810345

yes

direct product factor

108

C₂

C₂×C₈×Dic₅₃

yes

characteristic

109

C₂

C₂×#13133127

yes

characteristic

110

C₂

C₂×#13133128

yes

characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₄²
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#322
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄²

C₂×D₅₃

commutator

C₂×C₅₃

C₂×C₄×C₈

Frattini

C₄²

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₄²

hypercenter

C₂×C₄×C₈

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #16810345	C₂	direct (N × H)
25. C₂×#88322	C₈	semidirect
40. #88322	C₂×C₈	semidirect
52. C₂×C₅₃	#322	semidirect
60. C₅₃	C₂×#322	semidirect
107. C₂	#16810345	direct (N × H)

H ≅ G/N

type

3. #16810345

C₂

direct (N × H)

25. C₂×#88322

C₈

semidirect

40. #88322

C₂×C₈

semidirect

52. C₂×C₅₃

#322

semidirect

60. C₅₃

C₂×#322

semidirect

107. C₂

#16810345

direct (N × H)

Subgroups — C₂×#16810345 #20272991

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810345 #20272991

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810345 #20272991