Group #20277924 — subgroups

Subgroups — C₂×#16810542 #20277924

Order 6784.

908 conjugacy classes of subgroups; only the 285 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 285 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×#11719919	C₂	1	3	yes	maximal
2	#16810542	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133124	C₂	1	3	yes	maximal
4	C₂³×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic
6	C₂³×Dic₅₃	C₂²	1	3	yes	—
7	C₂×#11719919	—	2	12	no	—
8	C₂×#11719919	C₂²	1	12	yes	—
9	#13133124	C₂²	1	12	yes	—
10	C₂²×Q₈×C₅₃	—	2	12	no	—
11	C₂²×Q₈×C₅₃	C₂²	1	4	yes	—
12	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
13	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
14	C₂×Q₈×C₅₃	—	2	32	no	—
15	C₂×Q₈×C₅₃	—	2	24	no	—
16	C₂²×C₄×C₅₃	—	2	12	no	—
17	C₂²×C₄×C₅₃	D₄	1	12	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
19	C₂²×Dic₅₃	—	2	6	no	—
20	C₂²×Dic₅₃	—	2	6	no	—
21	C₂²×Dic₅₃	—	2	6	no	—
22	C₂²×Dic₅₃	Q₈	1	12	yes	complemented
23	C₂²×Dic₅₃	C₂³	1	2	yes	—
24	C₂²×Dic₅₃	C₂³	1	6	yes	—
25	#11719919	—	2	24	no	—
26	C₂³×C₅₃	C₂×D₄	1	6	yes	—
27	C₂³×C₅₃	C₂×Q₈	1	6	yes	—
28	C₂³×C₅₃	#41	1	2	yes	—
29	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
30	Q₈×C₅₃	—	2	32	no	—
31	C₂×C₄×C₅₃	—	2	24	no	—
32	C₂×C₄×C₅₃	—	2	12	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
34	C₂×Dic₅₃	—	2	24	no	—
35	C₂×Dic₅₃	—	2	6	no	—
36	C₂×Dic₅₃	—	2	6	no	—
37	C₂×Dic₅₃	—	2	6	no	—
38	C₂×Dic₅₃	—	4	4	no	—
39	C₂×Dic₅₃	C₂×Q₈	1	12	yes	complemented
40	C₂²×C₅₃	#120	1	4	yes	—
41	C₂²×C₅₃	C₂²⋊Q₈	1	12	yes	—
42	C₂²×C₅₃	C₄⋊Q₈	1	12	yes	—
43	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
44	C₂²×C₅₃	C₂²×Q₈	1	3	yes	—
45	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
46	C₄×C₅₃	—	2	12	no	—
47	Dic₅₃	—	2	12	no	—
48	Dic₅₃	—	4	2	no	—
49	C₂×#395	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	#395	1	8	yes	complemented
51	C₂×C₅₃	C₂×#120	1	1	yes	characteristic
52	C₂×C₅₃	C₂×C₂²⋊Q₈	1	3	yes	—
53	C₂×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
54	C₂³×Q₈	D₅₃	1	1	yes	characteristic
55	C₂²×C₄⋊C₄	—	53	3	no	—
56	#395	—	53	8	no	—
57	C₂×#95	—	53	3	no	—
58	C₅₃	C₂×#395	1	1	yes	characteristic, complemented, Sylow
59	C₂²×Q₈	—	2	12	no	—
60	C₂²×Q₈	C₂×D₅₃	1	4	yes	—
61	C₂³×C₄	—	53	3	no	—
62	C₂³×C₄	—	53	1	no	—
63	C₂³×C₄	C₂×D₅₃	1	3	yes	—
64	C₂×C₄⋊C₄	—	53	12	no	—
65	C₂×C₄⋊C₄	—	106	12	no	—
66	#95	—	53	12	no	—
67	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
68	C₂×Q₈	—	2	32	no	—
69	C₂×Q₈	—	2	24	no	—
70	C₂²×C₄	—	2	12	no	—
71	C₂²×C₄	—	53	12	no	—
72	C₂²×C₄	—	53	6	no	—
73	C₂²×C₄	—	53	2	no	—
74	C₂²×C₄	—	106	6	no	—
75	C₂²×C₄	—	106	6	no	—
76	C₂²×C₄	—	106	6	no	—
77	C₂²×C₄	C₂²⋊D₅₃	1	12	yes	—
78	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
79	C₄⋊C₄	—	106	24	no	—
80	C₂³	Q₈×D₅₃	1	6	yes	—
81	C₂³	#11719949	1	2	yes	—
82	C₂³	C₂×C₂²⋊D₅₃	1	6	yes	—
83	C₂³	C₂³×D₅₃	1	1	yes	characteristic
84	Q₈	—	2	32	no	—
85	C₂×C₄	—	2	24	no	—
86	C₂×C₄	—	2	12	no	—
87	C₂×C₄	—	53	12	no	—
88	C₂×C₄	—	106	24	no	—
89	C₂×C₄	—	106	6	no	—
90	C₂×C₄	—	106	6	no	—
91	C₂×C₄	—	106	6	no	—
92	C₂×C₄	—	212	4	no	—
93	C₂×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
94	C₂²	#13133186	1	4	yes	—
95	C₂²	#13133193	1	12	yes	—
96	C₂²	#13133215	1	12	yes	—
97	C₂²	C₂²×C₂²⋊D₅₃	1	3	yes	—
98	C₂²	C₂×Q₈×D₅₃	1	3	yes	—
99	C₂²	C₂×#11719949	1	1	yes	characteristic
100	C₄	—	2	12	no	—
101	C₄	—	106	12	no	—
102	C₄	—	212	2	no	—
103	C₂	#16810542	1	8	yes	direct product factor
104	C₂	C₂×#13133186	1	1	yes	characteristic
105	C₂	C₂×#13133193	1	3	yes	—
106	C₂	C₂×#13133215	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719919

C₂

yes

maximal

#16810542

C₂

yes

maximal, direct product factor

C₂×#13133124

C₂

yes

maximal

C₂³×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₂²

yes

characteristic

C₂³×Dic₅₃

C₂²

yes

—

C₂×#11719919

—

C₂×#11719919

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

Q₈

yes

complemented

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

#11719919

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

#41

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×Q₈

yes

complemented

C₂²×C₅₃

#120

yes

—

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

C₄⋊Q₈

yes

—

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₂²⋊Q₈

yes

—

C₂×C₅₃

C₂×C₄⋊Q₈

yes

—

C₂³×Q₈

D₅₃

yes

characteristic

C₂²×C₄⋊C₄

—

#395

—

C₂×#95

—

C₅₃

C₂×#395

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

#95

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₂³

Q₈×D₅₃

yes

—

C₂³

#11719949

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂²

#13133186

yes

—

C₂²

#13133193

yes

—

C₂²

#13133215

yes

—

C₂²

C₂²×C₂²⋊D₅₃

yes

—

C₂²

C₂×Q₈×D₅₃

yes

—

C₂²

C₂×#11719949

yes

characteristic

100

C₄

—

101

C₄

—

106

—

102

C₄

—

212

—

103

C₂

#16810542

yes

direct product factor

104

C₂

C₂×#13133186

yes

characteristic

105

C₂

C₂×#13133193

yes

—

106

C₂

C₂×#13133215

yes

—

	subgroup	G/H
center	C₂⁴	C₂²×D₅₃
commutator	C₂³×C₅₃	C₂⁴
Frattini	C₂³	C₂³×D₅₃
Fitting	C₂³×Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂³×Q₈	D₅₃
hypocenter	C₅₃	C₂×#395
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

subgroup

G/H

center

C₂⁴

C₂²×D₅₃

commutator

C₂³×C₅₃

C₂⁴

Frattini

C₂³

C₂³×D₅₃

Fitting

C₂³×Q₈×C₅₃

C₂

socle

C₂⁴×C₅₃

C₂³

hypercenter

C₂³×Q₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂⁴ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #16810542	C₂	direct (N × H)
22. C₂²×Dic₅₃	Q₈	semidirect
39. C₂×Dic₅₃	C₂×Q₈	semidirect
50. C₂×C₅₃	#395	semidirect
58. C₅₃	C₂×#395	semidirect
103. C₂	#16810542	direct (N × H)

H ≅ G/N

type

2. #16810542

C₂

direct (N × H)

22. C₂²×Dic₅₃

Q₈

semidirect

39. C₂×Dic₅₃

C₂×Q₈

semidirect

50. C₂×C₅₃

#395

semidirect

58. C₅₃

C₂×#395

semidirect

103. C₂

#16810542

direct (N × H)

Subgroups — C₂×#16810542 #20277924

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810542 #20277924

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810542 #20277924