Group #20277964 — subgroups

Subgroups — C₂×#16810560 #20277964

Order 6784.

908 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 269 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	3	yes	maximal, complemented
2	C₂²×#11719921	C₂	1	3	yes	maximal
3	#16810560	C₂	1	8	yes	maximal, direct product factor
4	C₂×#95×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₅₃	C₂²	1	1	yes	characteristic
6	C₂³×Dic₅₃	C₂²	1	3	yes	—
7	C₂×C₄⋊Dic₅₃	—	2	12	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	12	yes	complemented
9	C₂×#11719921	—	2	12	no	—
10	C₂×#11719921	C₂²	1	12	yes	—
11	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
12	#95×C₅₃	C₂²	1	4	yes	complemented
13	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
14	C₂³×D₅₃	—	2	6	no	—
15	C₂³×D₅₃	—	2	6	no	—
16	C₂³×D₅₃	Q₈	1	4	yes	—
17	C₂³×D₅₃	C₂³	1	2	yes	—
18	C₂²×C₄×C₅₃	—	2	12	no	—
19	C₂²×C₄×C₅₃	D₄	1	12	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
21	C₂²×Dic₅₃	—	2	6	no	—
22	C₂²×Dic₅₃	—	2	12	no	—
23	C₂²×Dic₅₃	C₂³	1	6	yes	—
24	#11719921	—	2	24	no	—
25	C₄⋊Dic₅₃	—	2	24	no	—
26	C₂³×C₅₃	C₂×D₄	1	6	yes	—
27	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
28	C₂³×C₅₃	#41	1	6	yes	—
29	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
30	C₂²×D₅₃	—	2	24	no	—
31	C₂²×D₅₃	—	2	6	no	—
32	C₂²×D₅₃	—	2	6	no	—
33	C₂²×D₅₃	—	4	12	no	—
34	C₂²×D₅₃	—	4	4	no	—
35	C₂²×D₅₃	C₂×Q₈	1	4	yes	—
36	C₂×C₄×C₅₃	—	2	24	no	—
37	C₂×C₄×C₅₃	—	2	12	no	—
38	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
39	C₂×Dic₅₃	—	2	12	no	—
40	C₂×Dic₅₃	—	2	6	no	—
41	C₂×Dic₅₃	—	4	12	no	—
42	C₂²×C₅₃	C₂²⋊Q₈	1	12	yes	—
43	C₂²×C₅₃	#123	1	12	yes	—
44	C₂²×C₅₃	#127	1	4	yes	—
45	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
46	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
47	C₂²×C₅₃	C₂×#41	1	3	yes	—
48	C₂×D₅₃	—	2	12	no	—
49	C₂×D₅₃	—	4	6	no	—
50	C₂×D₅₃	—	4	2	no	—
51	C₂×D₅₃	—	4	16	no	—
52	C₄×C₅₃	—	2	12	no	—
53	Dic₅₃	—	4	6	no	—
54	C₂×#399	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	#399	1	8	yes	complemented
56	C₂×C₅₃	C₂×C₂²⋊Q₈	1	3	yes	—
57	C₂×C₅₃	C₂×#123	1	3	yes	—
58	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
59	D₅₃	—	4	4	no	—
60	C₂²×C₄⋊C₄	—	53	3	no	—
61	C₂²×C₂²⋊C₄	—	53	3	no	—
62	#399	—	53	8	no	—
63	C₂×#95	D₅₃	1	1	yes	characteristic, complemented
64	C₅₃	C₂×#399	1	1	yes	characteristic, complemented, Sylow
65	C₂⁵	—	53	1	no	—
66	C₂³×C₄	—	53	3	no	—
67	C₂³×C₄	C₂×D₅₃	1	3	yes	—
68	C₂×C₄⋊C₄	—	53	12	no	—
69	C₂×C₄⋊C₄	—	106	12	no	—
70	C₂×C₂²⋊C₄	—	53	12	no	—
71	C₂×C₂²⋊C₄	—	106	12	no	—
72	#95	C₂×D₅₃	1	4	yes	complemented
73	C₂⁴	—	53	4	no	—
74	C₂⁴	—	53	2	no	—
75	C₂⁴	—	106	6	no	—
76	C₂⁴	—	106	6	no	—
77	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
78	C₂²×C₄	—	2	12	no	—
79	C₂²×C₄	—	53	6	no	—
80	C₂²×C₄	—	106	6	no	—
81	C₂²×C₄	—	106	12	no	—
82	C₂²×C₄	D₂₁₂	1	12	yes	—
83	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
84	C₄⋊C₄	—	106	24	no	—
85	C₂²⋊C₄	—	106	24	no	—
86	C₂³	—	53	4	no	—
87	C₂³	—	106	24	no	—
88	C₂³	—	106	6	no	—
89	C₂³	—	106	6	no	—
90	C₂³	—	212	12	no	—
91	C₂³	—	212	4	no	—
92	C₂³	C₂×D₂₁₂	1	6	yes	—
93	C₂³	#11719947	1	6	yes	—
94	C₂³	Q₈×D₅₃	1	2	yes	—
95	C₂³	C₂³×D₅₃	1	1	yes	characteristic
96	C₂×C₄	—	2	24	no	—
97	C₂×C₄	—	2	12	no	—
98	C₂×C₄	—	106	6	no	—
99	C₂×C₄	—	106	12	no	—
100	C₂×C₄	—	212	12	no	—
101	C₂×C₄	C₂×D₂₁₂	1	12	yes	—
102	C₂²	—	106	12	no	—
103	C₂²	—	212	16	no	—
104	C₂²	—	212	6	no	—
105	C₂²	—	212	2	no	—
106	C₂²	#13133197	1	12	yes	—
107	C₂²	#13133201	1	12	yes	—
108	C₂²	#13133214	1	4	yes	—
109	C₂²	C₂²×D₂₁₂	1	3	yes	—
110	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
111	C₂²	C₂×#11719947	1	3	yes	—
112	C₄	—	2	12	no	—
113	C₄	—	212	6	no	—
114	C₂	—	212	4	no	—
115	C₂	#16810560	1	8	yes	direct product factor
116	C₂	C₂×#13133197	1	3	yes	—
117	C₂	C₂×#13133201	1	3	yes	—
118	C₂	C₂×#13133214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₅₃

C₂

yes

maximal, complemented

C₂²×#11719921

C₂

yes

maximal

#16810560

C₂

yes

maximal, direct product factor

C₂×#95×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₅₃

C₂²

yes

characteristic

C₂³×Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

complemented

C₂×#11719921

—

C₂×#11719921

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

#95×C₅₃

C₂²

yes

complemented

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂³×D₅₃

—

C₂³×D₅₃

—

C₂³×D₅₃

Q₈

yes

—

C₂³×D₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

#41

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

#123

yes

—

C₂²×C₅₃

#127

yes

—

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₂²⋊Q₈

yes

—

C₂×C₅₃

C₂×#123

yes

—

C₂×C₅₃

C₂×#127

yes

characteristic

D₅₃

—

C₂²×C₄⋊C₄

—

C₂²×C₂²⋊C₄

—

#399

—

C₂×#95

D₅₃

yes

characteristic, complemented

C₅₃

C₂×#399

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

106

—

#95

C₂×D₅₃

yes

complemented

C₂⁴

—

C₂⁴

—

C₂⁴

—

106

—

C₂⁴

—

106

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₂³

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

212

—

C₂³

—

212

—

C₂³

C₂×D₂₁₂

yes

—

C₂³

#11719947

yes

—

C₂³

Q₈×D₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

100

C₂×C₄

—

212

—

101

C₂×C₄

C₂×D₂₁₂

yes

—

102

C₂²

—

106

—

103

C₂²

—

212

—

104

C₂²

—

212

—

105

C₂²

—

212

—

106

C₂²

#13133197

yes

—

107

C₂²

#13133201

yes

—

108

C₂²

#13133214

yes

—

109

C₂²

C₂²×D₂₁₂

yes

—

110

C₂²

C₂×Q₈×D₅₃

yes

characteristic

111

C₂²

C₂×#11719947

yes

—

112

C₄

—

113

C₄

—

212

—

114

C₂

—

212

—

115

C₂

#16810560

yes

direct product factor

116

C₂

C₂×#13133197

yes

—

117

C₂

C₂×#13133201

yes

—

118

C₂

C₂×#13133214

yes

characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₅₃
commutator	C₂³×C₅₃	C₂⁴
Frattini	C₂³	C₂³×D₅₃
Fitting	C₂×#95×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂×#95	D₅₃
hypocenter	C₅₃	C₂×#399
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

subgroup

G/H

center

C₂⁴

C₂²×D₅₃

commutator

C₂³×C₅₃

C₂⁴

Frattini

C₂³

C₂³×D₅₃

Fitting

C₂×#95×C₅₃

C₂

socle

C₂⁴×C₅₃

C₂³

hypercenter

C₂×#95

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂⁴ ◁ C₂×#95

N	H ≅ G/N	type
1. C₂²×C₄⋊Dic₅₃	C₂	semidirect
3. #16810560	C₂	direct (N × H)
4. C₂×#95×C₅₃	C₂	semidirect
8. C₂×C₄⋊Dic₅₃	C₂²	semidirect
12. #95×C₅₃	C₂²	semidirect
55. C₂×C₅₃	#399	semidirect
63. C₂×#95	D₅₃	semidirect
64. C₅₃	C₂×#399	semidirect
72. #95	C₂×D₅₃	semidirect
115. C₂	#16810560	direct (N × H)

H ≅ G/N

type

1. C₂²×C₄⋊Dic₅₃

C₂

semidirect

3. #16810560

C₂

direct (N × H)

4. C₂×#95×C₅₃

C₂

semidirect

8. C₂×C₄⋊Dic₅₃

C₂²

semidirect

12. #95×C₅₃

C₂²

semidirect

55. C₂×C₅₃

#399

semidirect

63. C₂×#95

D₅₃

semidirect

64. C₅₃

C₂×#399

semidirect

72. #95

C₂×D₅₃

semidirect

115. C₂

#16810560

direct (N × H)

Subgroups — C₂×#16810560 #20277964

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810560 #20277964

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810560 #20277964