Group #20277914 — subgroups

Subgroups — C₂×#16810534 #20277914

Order 6784.

1036 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 1036 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×#11719921	C₂	1	3	yes	maximal
2	#16810534	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133124	C₂	1	3	yes	maximal, complemented
4	C₂³×Q₈×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×#11719921	—	2	12	no	—
8	C₂×#11719921	C₂²	1	12	yes	—
9	#13133124	C₂²	1	12	yes	complemented
10	C₂²×Q₈×C₅₃	—	2	12	no	—
11	C₂²×Q₈×C₅₃	C₂²	1	4	yes	complemented
12	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
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	complemented
17	C₂³×D₅₃	C₂³	1	2	yes	—
18	C₂×Q₈×C₅₃	—	2	32	no	—
19	C₂×Q₈×C₅₃	—	2	24	no	—
20	C₂²×C₄×C₅₃	—	2	12	no	—
21	C₂²×C₄×C₅₃	D₄	1	12	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
23	C₂²×Dic₅₃	—	2	6	no	—
24	C₂²×Dic₅₃	—	2	12	no	—
25	C₂²×Dic₅₃	C₂³	1	6	yes	—
26	#11719921	—	2	24	no	—
27	C₂³×C₅₃	C₂×D₄	1	6	yes	—
28	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
29	C₂³×C₅₃	#41	1	6	yes	—
30	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
31	C₂²×D₅₃	—	2	24	no	—
32	C₂²×D₅₃	—	2	6	no	—
33	C₂²×D₅₃	—	2	6	no	—
34	C₂²×D₅₃	—	4	4	no	—
35	C₂²×D₅₃	—	4	12	no	—
36	C₂²×D₅₃	C₂×Q₈	1	4	yes	complemented
37	Q₈×C₅₃	—	2	32	no	—
38	C₂×C₄×C₅₃	—	2	24	no	—
39	C₂×C₄×C₅₃	—	2	12	no	—
40	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
41	C₂×Dic₅₃	—	2	6	no	—
42	C₂×Dic₅₃	—	2	12	no	—
43	C₂×Dic₅₃	—	4	12	no	—
44	C₂²×C₅₃	#120	1	4	yes	—
45	C₂²×C₅₃	C₂²⋊Q₈	1	12	yes	—
46	C₂²×C₅₃	#124	1	12	yes	—
47	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
48	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
49	C₂²×C₅₃	C₂×#41	1	3	yes	—
50	C₂×D₅₃	—	2	12	no	—
51	C₂×D₅₃	—	4	2	no	—
52	C₂×D₅₃	—	4	6	no	—
53	C₂×D₅₃	—	4	16	no	—
54	C₄×C₅₃	—	2	12	no	—
55	Dic₅₃	—	4	6	no	—
56	C₂×#393	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	#393	1	8	yes	complemented
58	C₂×C₅₃	C₂×#120	1	1	yes	characteristic
59	C₂×C₅₃	C₂×C₂²⋊Q₈	1	3	yes	—
60	C₂×C₅₃	C₂×#124	1	3	yes	—
61	D₅₃	—	4	4	no	—
62	C₂³×Q₈	D₅₃	1	1	yes	characteristic, complemented
63	C₂²×C₂²⋊C₄	—	53	3	no	—
64	#393	—	53	8	no	—
65	C₂×#95	—	53	3	no	—
66	C₅₃	C₂×#393	1	1	yes	characteristic, complemented, Sylow
67	C₂⁵	—	53	1	no	—
68	C₂²×Q₈	—	2	12	no	—
69	C₂²×Q₈	C₂×D₅₃	1	4	yes	complemented
70	C₂³×C₄	—	53	3	no	—
71	C₂³×C₄	C₂×D₅₃	1	3	yes	—
72	C₂×C₂²⋊C₄	—	53	12	no	—
73	C₂×C₂²⋊C₄	—	106	12	no	—
74	#95	—	53	12	no	—
75	C₂⁴	—	53	4	no	—
76	C₂⁴	—	53	2	no	—
77	C₂⁴	—	106	6	no	—
78	C₂⁴	—	106	6	no	—
79	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
80	C₂×Q₈	—	2	32	no	—
81	C₂×Q₈	—	2	24	no	—
82	C₂²×C₄	—	2	12	no	—
83	C₂²×C₄	—	53	6	no	—
84	C₂²×C₄	—	106	12	no	—
85	C₂²×C₄	—	106	6	no	—
86	C₂²×C₄	C₂²⋊D₅₃	1	12	yes	—
87	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
88	C₂²⋊C₄	—	106	24	no	—
89	C₂³	—	53	4	no	—
90	C₂³	—	106	24	no	—
91	C₂³	—	106	6	no	—
92	C₂³	—	106	6	no	—
93	C₂³	—	212	4	no	—
94	C₂³	—	212	12	no	—
95	C₂³	Q₈×D₅₃	1	2	yes	—
96	C₂³	#11719949	1	6	yes	—
97	C₂³	C₂×C₂²⋊D₅₃	1	6	yes	—
98	C₂³	C₂³×D₅₃	1	1	yes	characteristic
99	Q₈	—	2	32	no	—
100	C₂×C₄	—	2	24	no	—
101	C₂×C₄	—	2	12	no	—
102	C₂×C₄	—	106	6	no	—
103	C₂×C₄	—	106	12	no	—
104	C₂×C₄	—	212	12	no	—
105	C₂×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
106	C₂²	—	106	12	no	—
107	C₂²	—	212	16	no	—
108	C₂²	—	212	2	no	—
109	C₂²	—	212	6	no	—
110	C₂²	#13133186	1	4	yes	—
111	C₂²	#13133193	1	12	yes	—
112	C₂²	#13133206	1	12	yes	—
113	C₂²	C₂²×C₂²⋊D₅₃	1	3	yes	—
114	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
115	C₂²	C₂×#11719949	1	3	yes	—
116	C₄	—	2	12	no	—
117	C₄	—	212	6	no	—
118	C₂	—	212	4	no	—
119	C₂	#16810534	1	8	yes	direct product factor
120	C₂	C₂×#13133186	1	1	yes	characteristic
121	C₂	C₂×#13133193	1	3	yes	—
122	C₂	C₂×#13133206	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719921

C₂

yes

maximal

#16810534

C₂

yes

maximal, direct product factor

C₂×#13133124

C₂

yes

maximal, complemented

C₂³×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₅₃

C₂²

yes

characteristic

C₂³×Dic₅₃

C₂²

yes

—

C₂×#11719921

—

C₂×#11719921

C₂²

yes

—

#13133124

C₂²

yes

complemented

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

C₂²

yes

complemented

C₂³×C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂³×D₅₃

—

C₂³×D₅₃

—

C₂³×D₅₃

Q₈

yes

complemented

C₂³×D₅₃

C₂³

yes

—

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

C₂³

yes

—

#11719921

—

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

complemented

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

#120

yes

—

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

#124

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

yes

C₂×C₅₃

C₂×C₂²⋊Q₈

yes

—

C₂×C₅₃

C₂×#124

yes

—

D₅₃

—

C₂³×Q₈

D₅₃

yes

characteristic, complemented

C₂²×C₂²⋊C₄

—

#393

—

C₂×#95

—

C₅₃

C₂×#393

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

complemented

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

106

—

#95

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

106

—

C₂⁴

—

106

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²⋊C₄

—

106

—

C₂³

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

212

—

C₂³

—

212

—

C₂³

Q₈×D₅₃

yes

—

C₂³

#11719949

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

106

—

103

C₂×C₄

—

106

—

104

C₂×C₄

—

212

—

105

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

106

C₂²

—

106

—

107

C₂²

—

212

—

108

C₂²

—

212

—

109

C₂²

—

212

—

110

C₂²

#13133186

yes

—

111

C₂²

#13133193

yes

—

112

C₂²

#13133206

yes

—

113

C₂²

C₂²×C₂²⋊D₅₃

yes

—

114

C₂²

C₂×Q₈×D₅₃

yes

characteristic

115

C₂²

C₂×#11719949

yes

—

116

C₄

—

117

C₄

—

212

—

118

C₂

—

212

—

119

C₂

#16810534

yes

direct product factor

120

C₂

C₂×#13133186

yes

characteristic

121

C₂

C₂×#13133193

yes

—

122

C₂

C₂×#13133206

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×#393
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. #16810534	C₂	direct (N × H)
3. C₂×#13133124	C₂	semidirect
4. C₂³×Q₈×C₅₃	C₂	semidirect
9. #13133124	C₂²	semidirect
11. C₂²×Q₈×C₅₃	C₂²	semidirect
16. C₂³×D₅₃	Q₈	semidirect
36. C₂²×D₅₃	C₂×Q₈	semidirect
57. C₂×C₅₃	#393	semidirect
62. C₂³×Q₈	D₅₃	semidirect
66. C₅₃	C₂×#393	semidirect
69. C₂²×Q₈	C₂×D₅₃	semidirect
119. C₂	#16810534	direct (N × H)

H ≅ G/N

type

2. #16810534

C₂

direct (N × H)

3. C₂×#13133124

C₂

semidirect

4. C₂³×Q₈×C₅₃

C₂

semidirect

9. #13133124

C₂²

semidirect

11. C₂²×Q₈×C₅₃

C₂²

semidirect

16. C₂³×D₅₃

Q₈

semidirect

36. C₂²×D₅₃

C₂×Q₈

semidirect

57. C₂×C₅₃

#393

semidirect

62. C₂³×Q₈

D₅₃

semidirect

66. C₅₃

C₂×#393

semidirect

69. C₂²×Q₈

C₂×D₅₃

semidirect

119. C₂

#16810534

direct (N × H)

Subgroups — C₂×#16810534 #20277914

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810534 #20277914

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810534 #20277914