Group #20277102 — subgroups

Subgroups — C₂×#16810489 #20277102

Order 6784.

712 conjugacy classes of subgroups; only the 389 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 389 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×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
2	#16810489	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133124	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₅₃	C₂²	1	4	yes	—
6	C₂×C₄⋊Dic₅₃	—	2	8	no	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	8	yes	—
8	#13133124	C₂²	1	16	yes	—
9	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
10	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₅₃	C₄	1	8	yes	complemented
12	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
13	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	—	2	8	no	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₅₃	D₄	1	4	yes	—
18	C₂²×C₄×C₅₃	Q₈	1	4	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
21	C₂²×Dic₅₃	—	2	8	no	—
22	C₂²×Dic₅₃	—	2	16	no	—
23	C₂²×Dic₅₃	C₂³	1	8	yes	—
24	C₄²×C₅₃	C₂×C₄	1	16	yes	complemented
25	C₄⋊Dic₅₃	—	2	16	no	—
26	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
27	C₂³×C₅₃	C₂×D₄	1	2	yes	—
28	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
29	C₂³×C₅₃	#41	1	8	yes	—
30	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	16	no	—
32	C₂×C₄×C₅₃	—	2	8	no	—
33	C₂×C₄×C₅₃	C₄⋊C₄	1	16	yes	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
36	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
38	C₂×Dic₅₃	—	2	8	no	—
39	C₂×Dic₅₃	—	2	16	no	—
40	C₂×Dic₅₃	—	4	16	no	—
41	C₂²×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
42	C₂²×C₅₃	#117	1	8	yes	—
43	C₂²×C₅₃	#124	1	8	yes	—
44	C₂²×C₅₃	#125	1	8	yes	—
45	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
46	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
47	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
48	C₂²×C₅₃	C₂×#41	1	4	yes	—
49	C₄×C₅₃	—	2	8	no	—
50	C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
51	Dic₅₃	—	4	8	no	—
52	C₂×#382	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	#382	1	8	yes	complemented
54	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₅₃	C₂×#117	1	2	yes	—
56	C₂×C₅₃	C₂×#124	1	2	yes	—
57	C₂×C₅₃	C₂×#125	1	2	yes	—
58	C₂²×C₄⋊C₄	—	53	2	no	—
59	C₂²×C₄²	D₅₃	1	1	yes	characteristic
60	#382	—	53	8	no	—
61	C₂×#95	—	53	4	no	—
62	C₅₃	C₂×#382	1	1	yes	characteristic, complemented, Sylow
63	C₂³×C₄	—	53	4	no	—
64	C₂³×C₄	C₂×D₅₃	1	2	yes	—
65	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
66	C₂×C₄⋊C₄	—	53	8	no	—
67	C₂×C₄⋊C₄	—	106	8	no	—
68	C₂×C₄²	Dic₅₃	1	8	yes	complemented
69	C₂×C₄²	C₂×D₅₃	1	4	yes	—
70	#95	—	53	16	no	—
71	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
72	C₂²×C₄	—	2	8	no	—
73	C₂²×C₄	—	53	8	no	—
74	C₂²×C₄	—	106	8	no	—
75	C₂²×C₄	—	106	16	no	—
76	C₂²×C₄	Dic₁₀₆	1	4	yes	—
77	C₂²×C₄	D₂₁₂	1	4	yes	—
78	C₂²×C₄	C₂×Dic₅₃	1	8	yes	—
79	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
80	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
81	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
82	C₄⋊C₄	—	106	16	no	—
83	C₄²	C₂×Dic₅₃	1	16	yes	complemented
84	C₂³	C₂×Dic₁₀₆	1	2	yes	—
85	C₂³	C₂×D₂₁₂	1	2	yes	—
86	C₂³	#11719945	1	8	yes	—
87	C₂³	C₂²×Dic₅₃	1	2	yes	—
88	C₂³	C₂³×D₅₃	1	1	yes	characteristic
89	C₂×C₄	—	2	16	no	—
90	C₂×C₄	—	2	8	no	—
91	C₂×C₄	—	106	8	no	—
92	C₂×C₄	—	106	16	no	—
93	C₂×C₄	—	212	16	no	—
94	C₂×C₄	C₄⋊Dic₅₃	1	16	yes	—
95	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
96	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
97	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
98	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
99	C₂²	C₂×C₄⋊Dic₅₃	1	4	yes	—
100	C₂²	#13133176	1	8	yes	—
101	C₂²	#13133205	1	8	yes	—
102	C₂²	#13133209	1	8	yes	—
103	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
104	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
105	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
106	C₂²	C₂×#11719945	1	4	yes	—
107	C₄	—	2	8	no	—
108	C₄	—	212	8	no	—
109	C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
110	C₂	#16810489	1	8	yes	direct product factor
111	C₂	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
112	C₂	C₂×#13133176	1	2	yes	—
113	C₂	C₂×#13133205	1	2	yes	—
114	C₂	C₂×#13133209	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₅₃

C₂

yes

maximal

#16810489

C₂

yes

maximal, direct product factor

C₂×#13133124

C₂

yes

maximal

C₂²×C₄²×C₅₃

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₄

yes

complemented

C₂×C₄²×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

complemented

C₄⋊Dic₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

#41

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

#125

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₅₃

C₂×#117

yes

—

C₂×C₅₃

C₂×#124

yes

—

C₂×C₅₃

C₂×#125

yes

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

D₅₃

yes

characteristic

#382

—

C₂×#95

—

C₅₃

C₂×#382

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

Dic₅₃

yes

complemented

C₂×C₄²

C₂×D₅₃

yes

—

#95

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

Dic₁₀₆

yes

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₄²

C₂×Dic₅₃

yes

complemented

C₂³

C₂×Dic₁₀₆

yes

—

C₂³

C₂×D₂₁₂

yes

—

C₂³

#11719945

yes

—

C₂³

C₂²×Dic₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

—

C₂²

C₂×C₄⋊Dic₅₃

yes

—

100

C₂²

#13133176

yes

—

101

C₂²

#13133205

yes

—

102

C₂²

#13133209

yes

—

103

C₂²

C₂³×Dic₅₃

yes

characteristic

104

C₂²

C₂²×D₂₁₂

yes

characteristic

105

C₂²

C₂²×Dic₁₀₆

yes

characteristic

106

C₂²

C₂×#11719945

yes

—

107

C₄

—

108

C₄

—

212

—

109

C₄

C₂×C₄⋊Dic₅₃

yes

—

110

C₂

#16810489

yes

direct product factor

111

C₂

C₂²×C₄⋊Dic₅₃

yes

characteristic

112

C₂

C₂×#13133176

yes

—

113

C₂

C₂×#13133205

yes

—

114

C₂

C₂×#13133209

yes

—

	subgroup	G/H
center	C₂⁴	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Frattini	C₂³	C₂³×D₅₃
Fitting	C₂²×C₄²×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂²×C₄²	D₅₃
hypocenter	C₅₃	C₂×#382
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

subgroup

G/H

center

C₂⁴

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Frattini

C₂³

C₂³×D₅₃

Fitting

C₂²×C₄²×C₅₃

C₂

socle

C₂⁴×C₅₃

C₂³

hypercenter

C₂²×C₄²

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂⁴ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. #16810489	C₂	direct (N × H)
11. C₂×C₄²×C₅₃	C₄	semidirect
24. C₄²×C₅₃	C₂×C₄	semidirect
53. C₂×C₅₃	#382	semidirect
62. C₅₃	C₂×#382	semidirect
68. C₂×C₄²	Dic₅₃	semidirect
83. C₄²	C₂×Dic₅₃	semidirect
110. C₂	#16810489	direct (N × H)

H ≅ G/N

type

2. #16810489

C₂

direct (N × H)

11. C₂×C₄²×C₅₃

C₄

semidirect

24. C₄²×C₅₃

C₂×C₄

semidirect

53. C₂×C₅₃

#382

semidirect

62. C₅₃

C₂×#382

semidirect

68. C₂×C₄²

Dic₅₃

semidirect

83. C₄²

C₂×Dic₅₃

semidirect

110. C₂

#16810489

direct (N × H)

Subgroups — C₂×#16810489 #20277102

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810489 #20277102

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810489 #20277102