Group #20277092 — subgroups

Subgroups — C₂×#16810486 #20277092

Order 6784.

1144 conjugacy classes of subgroups; only the 389 normal subgroups are shown.

normal subgroups only

This group has 1144 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×C₂²⋊Dic₅₃	C₂	1	2	yes	maximal
2	#16810486	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133124	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₅₃	C₂²	1	4	yes	—
6	C₂×C₂²⋊Dic₅₃	—	2	8	no	—
7	C₂×C₂²⋊Dic₅₃	C₂²	1	8	yes	—
8	#13133124	C₂²	1	16	yes	complemented
9	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₅₃	—	2	8	no	—
11	C₂³×C₄×C₅₃	C₄	1	8	yes	complemented
12	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₅₃	C₂²	1	4	yes	—
14	C₂⁴×C₅₃	—	2	8	no	—
15	C₂⁴×C₅₃	C₂×C₄	1	4	yes	—
16	C₂⁴×C₅₃	D₄	1	8	yes	—
17	C₂⁴×C₅₃	C₂³	1	2	yes	—
18	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	—	2	32	no	—
20	C₂²×C₄×C₅₃	—	2	8	no	—
21	C₂²×C₄×C₅₃	—	2	16	no	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	16	yes	complemented
24	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
25	C₂²×Dic₅₃	—	2	8	no	—
26	C₂²×Dic₅₃	—	2	16	no	—
27	C₂²×Dic₅₃	C₂³	1	8	yes	—
28	C₂²⋊Dic₅₃	—	2	16	no	—
29	C₂³×C₅₃	—	2	16	no	—
30	C₂³×C₅₃	—	2	32	no	—
31	C₂³×C₅₃	—	2	8	no	—
32	C₂³×C₅₃	C₂²⋊C₄	1	16	yes	—
33	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
34	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
35	C₂³×C₅₃	C₂×D₄	1	8	yes	—
36	C₂³×C₅₃	C₂×D₄	1	4	yes	—
37	C₂³×C₅₃	#41	1	8	yes	—
38	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	16	no	—
40	C₂×C₄×C₅₃	—	2	32	no	—
41	C₂×C₄×C₅₃	—	2	8	no	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
43	C₂×Dic₅₃	—	2	8	no	—
44	C₂×Dic₅₃	—	2	16	no	—
45	C₂×Dic₅₃	—	4	16	no	—
46	C₂²×C₅₃	—	2	8	no	—
47	C₂²×C₅₃	—	2	16	no	—
48	C₂²×C₅₃	—	2	32	no	—
49	C₂²×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
50	C₂²×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
51	C₂²×C₅₃	#117	1	8	yes	—
52	C₂²×C₅₃	#123	1	16	yes	—
53	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
55	C₂²×C₅₃	C₂×#41	1	4	yes	—
56	C₄×C₅₃	—	2	8	no	—
57	Dic₅₃	—	4	8	no	—
58	C₂×#381	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	8	no	—
60	C₂×C₅₃	#381	1	8	yes	complemented
61	C₂×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₅₃	C₂×#117	1	2	yes	—
63	C₂×C₅₃	C₂×#123	1	4	yes	—
64	C₂⁴×C₄	D₅₃	1	1	yes	characteristic
65	C₂²×C₂²⋊C₄	—	53	2	no	—
66	#381	—	53	8	no	—
67	C₂×#95	—	53	4	no	—
68	C₅₃	C₂×#381	1	1	yes	characteristic, complemented, Sylow
69	C₂⁵	C₂×D₅₃	1	1	yes	characteristic
70	C₂³×C₄	—	2	8	no	—
71	C₂³×C₄	—	53	4	no	—
72	C₂³×C₄	Dic₅₃	1	8	yes	complemented
73	C₂³×C₄	C₂×D₅₃	1	2	yes	—
74	C₂³×C₄	C₂×D₅₃	1	4	yes	—
75	C₂×C₂²⋊C₄	—	53	8	no	—
76	C₂×C₂²⋊C₄	—	106	8	no	—
77	#95	—	53	16	no	—
78	C₂⁴	—	2	8	no	—
79	C₂⁴	C₂×Dic₅₃	1	4	yes	—
80	C₂⁴	C₂²⋊D₅₃	1	8	yes	—
81	C₂⁴	C₂²×D₅₃	1	2	yes	—
82	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
83	C₂²×C₄	—	2	8	no	—
84	C₂²×C₄	—	2	16	no	—
85	C₂²×C₄	—	2	32	no	—
86	C₂²×C₄	—	53	8	no	—
87	C₂²×C₄	—	106	16	no	—
88	C₂²×C₄	—	106	8	no	—
89	C₂²×C₄	C₂×Dic₅₃	1	8	yes	—
90	C₂²×C₄	C₂×Dic₅₃	1	16	yes	complemented
91	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
92	C₂²⋊C₄	—	106	16	no	—
93	C₂³	—	2	16	no	—
94	C₂³	—	2	32	no	—
95	C₂³	—	2	8	no	—
96	C₂³	C₂²⋊Dic₅₃	1	16	yes	—
97	C₂³	#11719945	1	8	yes	—
98	C₂³	C₂²×Dic₅₃	1	4	yes	—
99	C₂³	C₂²×Dic₅₃	1	2	yes	—
100	C₂³	C₂×C₂²⋊D₅₃	1	8	yes	—
101	C₂³	C₂×C₂²⋊D₅₃	1	4	yes	—
102	C₂³	C₂³×D₅₃	1	1	yes	characteristic
103	C₂×C₄	—	2	16	no	—
104	C₂×C₄	—	2	32	no	—
105	C₂×C₄	—	2	8	no	—
106	C₂×C₄	—	106	8	no	—
107	C₂×C₄	—	106	16	no	—
108	C₂×C₄	—	212	16	no	—
109	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
110	C₂²	—	2	8	no	—
111	C₂²	—	2	16	no	—
112	C₂²	—	2	32	no	—
113	C₂²	C₂×C₂²⋊Dic₅₃	1	8	yes	—
114	C₂²	C₂×C₂²⋊Dic₅₃	1	4	yes	—
115	C₂²	#13133176	1	8	yes	—
116	C₂²	#13133200	1	16	yes	—
117	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
118	C₂²	C₂²×C₂²⋊D₅₃	1	2	yes	—
119	C₂²	C₂×#11719945	1	4	yes	—
120	C₄	—	2	8	no	—
121	C₄	—	212	8	no	—
122	C₂	—	2	8	no	—
123	C₂	#16810486	1	8	yes	direct product factor
124	C₂	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
125	C₂	C₂×#13133176	1	2	yes	—
126	C₂	C₂×#13133200	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊Dic₅₃

C₂

yes

maximal

#16810486

C₂

yes

maximal, direct product factor

C₂×#13133124

C₂

yes

maximal, complemented

C₂⁴×C₄×C₅₃

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

complemented

C₂⁵×C₅₃

C₂²

yes

characteristic

C₂³×C₄×C₅₃

—

C₂³×C₄×C₅₃

C₄

yes

complemented

C₂³×C₄×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

C₂×C₄

yes

—

C₂⁴×C₅₃

D₄

yes

—

C₂⁴×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

#41

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

#123

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂×#41

yes

—

C₄×C₅₃

—

Dic₅₃

—

C₂×#381

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#381

yes

complemented

C₂×C₅₃

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₅₃

C₂×#117

yes

—

C₂×C₅₃

C₂×#123

yes

—

C₂⁴×C₄

D₅₃

yes

characteristic

C₂²×C₂²⋊C₄

—

#381

—

C₂×#95

—

C₅₃

C₂×#381

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₅₃

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

Dic₅₃

yes

complemented

C₂³×C₄

C₂×D₅₃

yes

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

106

—

#95

—

C₂⁴

—

C₂⁴

C₂×Dic₅₃

yes

—

C₂⁴

C₂²⋊D₅₃

yes

—

C₂⁴

C₂²×D₅₃

yes

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

complemented

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²⋊C₄

—

106

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₅₃

yes

—

C₂³

#11719945

yes

—

C₂³

C₂²×Dic₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

—

100

C₂³

C₂×C₂²⋊D₅₃

yes

—

101

C₂³

C₂×C₂²⋊D₅₃

yes

—

102

C₂³

C₂³×D₅₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

212

—

109

C₂×C₄

C₂²×Dic₅₃

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₂²⋊Dic₅₃

yes

—

114

C₂²

C₂×C₂²⋊Dic₅₃

yes

—

115

C₂²

#13133176

yes

—

116

C₂²

#13133200

yes

—

117

C₂²

C₂³×Dic₅₃

yes

characteristic

118

C₂²

C₂²×C₂²⋊D₅₃

yes

—

119

C₂²

C₂×#11719945

yes

—

120

C₄

—

121

C₄

—

212

—

122

C₂

—

123

C₂

#16810486

yes

direct product factor

124

C₂

C₂²×C₂²⋊Dic₅₃

yes

characteristic

125

C₂

C₂×#13133176

yes

—

126

C₂

C₂×#13133200

yes

—

	subgroup	G/H
center	C₂⁴	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Frattini	C₂³	C₂³×D₅₃
Fitting	C₂⁴×C₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂⁴×C₄	D₅₃
hypocenter	C₅₃	C₂×#381
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄

subgroup

G/H

center

C₂⁴

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Frattini

C₂³

C₂³×D₅₃

Fitting

C₂⁴×C₄×C₅₃

C₂

socle

C₂⁴×C₅₃

C₂³

hypercenter

C₂⁴×C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂⁴ ◁ C₂⁴×C₄

N	H ≅ G/N	type
2. #16810486	C₂	direct (N × H)
3. C₂×#13133124	C₂	semidirect
8. #13133124	C₂²	semidirect
11. C₂³×C₄×C₅₃	C₄	semidirect
23. C₂²×C₄×C₅₃	C₂×C₄	semidirect
60. C₂×C₅₃	#381	semidirect
68. C₅₃	C₂×#381	semidirect
72. C₂³×C₄	Dic₅₃	semidirect
90. C₂²×C₄	C₂×Dic₅₃	semidirect
123. C₂	#16810486	direct (N × H)

H ≅ G/N

type

2. #16810486

C₂

direct (N × H)

3. C₂×#13133124

C₂

semidirect

8. #13133124

C₂²

semidirect

11. C₂³×C₄×C₅₃

C₄

semidirect

23. C₂²×C₄×C₅₃

C₂×C₄

semidirect

60. C₂×C₅₃

#381

semidirect

68. C₅₃

C₂×#381

semidirect

72. C₂³×C₄

Dic₅₃

semidirect

90. C₂²×C₄

C₂×Dic₅₃

semidirect

123. C₂

#16810486

direct (N × H)

Subgroups — C₂×#16810486 #20277092

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810486 #20277092

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810486 #20277092