Group #20288288 — subgroups

Subgroups — C₂×#16811352 #20288288

Order 6784.

1048 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 581 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	#16811352	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133217	C₂	1	3	yes	maximal
3	C₂×#13133208	C₂	1	6	yes	maximal
4	C₂×Q₈×Dic₅₃	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₁₀₆	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₆	C₂²	1	3	yes	—
8	#13133217	C₂²	1	24	yes	—
9	#13133208	C₂²	1	48	yes	—
10	Q₈×Dic₅₃	C₂²	1	16	yes	—
11	C₄×Dic₁₀₆	C₂²	1	24	yes	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
14	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
15	C₂×#11719919	C₂²	1	6	yes	—
16	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
17	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₅₃	C₂²	1	8	yes	—
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
21	C₂×Q₈×C₅₃	Q₈	1	8	yes	complemented
22	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₃	C₂³	1	2	yes	—
27	C₂²×Dic₅₃	C₂³	1	6	yes	—
28	C₂×Dic₁₀₆	—	2	12	no	—
29	C₂×Dic₁₀₆	C₂³	1	12	yes	—
30	C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
31	C₄²×C₅₃	C₂³	1	12	yes	—
32	C₄⋊Dic₅₃	C₂³	1	4	yes	—
33	C₄⋊Dic₅₃	C₂³	1	24	yes	—
34	C₄⋊Dic₅₃	C₂³	1	24	yes	—
35	#11719919	C₂³	1	24	yes	—
36	C₄×Dic₅₃	C₂³	1	24	yes	—
37	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
38	Q₈×C₅₃	C₂×Q₈	1	16	yes	complemented
39	C₂×C₄×C₅₃	—	2	6	no	—
40	C₂×C₄×C₅₃	C₂×Q₈	1	12	yes	—
41	C₂×C₄×C₅₃	#41	1	4	yes	—
42	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
43	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
44	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
45	C₂×Dic₅₃	—	2	12	no	—
46	C₂×Dic₅₃	—	2	4	no	—
47	C₂×Dic₅₃	C₂⁴	1	12	yes	—
48	C₂×Dic₅₃	C₂⁴	1	4	yes	—
49	Dic₁₀₆	—	2	24	no	—
50	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
51	C₂²×C₅₃	C₂×#41	1	2	yes	—
52	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
53	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
54	C₄×C₅₃	—	2	6	no	—
55	C₄×C₅₃	C₂²×Q₈	1	12	yes	—
56	C₄×C₅₃	C₂×#41	1	4	yes	—
57	Dic₅₃	—	2	12	no	—
58	Dic₅₃	—	2	4	no	—
59	C₂×#557	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	#557	1	4	yes	complemented
61	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
62	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
63	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
64	#557	—	53	16	no	—
65	C₂×C₄⋊Q₈	—	53	3	no	—
66	C₂×#125	—	53	6	no	—
67	C₂×C₄×Q₈	—	53	2	no	—
68	C₂×C₄×Q₈	—	53	3	no	—
69	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
70	C₅₃	C₂×#557	1	1	yes	characteristic, complemented, Sylow
71	C₂²×Q₈	—	53	3	no	—
72	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
73	C₄⋊Q₈	—	53	24	no	—
74	#125	—	53	48	no	—
75	C₄×Q₈	—	53	16	no	—
76	C₄×Q₈	—	53	24	no	—
77	C₄×Q₈	C₂×D₅₃	1	8	yes	—
78	C₂×C₄⋊C₄	—	53	1	no	—
79	C₂×C₄⋊C₄	—	53	6	no	—
80	C₂×C₄⋊C₄	—	53	6	no	—
81	C₂×C₄⋊C₄	—	53	6	no	—
82	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
83	C₂×C₄²	—	53	6	no	—
84	C₂×C₄²	C₂×D₅₃	1	3	yes	—
85	C₂×Q₈	—	53	12	no	—
86	C₂×Q₈	—	106	12	no	—
87	C₂×Q₈	Dic₁₀₆	1	8	yes	complemented
88	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
89	C₂²×C₄	—	53	2	no	—
90	C₂²×C₄	—	53	6	no	—
91	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
92	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
93	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
94	C₄⋊C₄	—	53	4	no	—
95	C₄⋊C₄	—	53	24	no	—
96	C₄⋊C₄	—	53	24	no	—
97	C₄⋊C₄	—	53	24	no	—
98	C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
99	C₄²	—	53	24	no	—
100	C₄²	C₂²×D₅₃	1	12	yes	—
101	C₂³	C₂³×D₅₃	1	1	yes	characteristic
102	Q₈	—	106	24	no	—
103	Q₈	C₂×Dic₁₀₆	1	16	yes	complemented
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	53	4	no	—
106	C₂×C₄	—	53	12	no	—
107	C₂×C₄	—	106	12	no	—
108	C₂×C₄	—	106	4	no	—
109	C₂×C₄	C₂×Dic₁₀₆	1	12	yes	—
110	C₂×C₄	#11719949	1	4	yes	—
111	C₂×C₄	C₂³×D₅₃	1	6	yes	—
112	C₂×C₄	C₂³×D₅₃	1	6	yes	—
113	C₂×C₄	C₂³×D₅₃	1	2	yes	—
114	C₂²	C₂²×Dic₁₀₆	1	2	yes	—
115	C₂²	C₂×#11719949	1	2	yes	—
116	C₂²	#13133264	1	2	yes	—
117	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
118	C₄	—	2	6	no	—
119	C₄	—	106	12	no	—
120	C₄	—	106	4	no	—
121	C₄	C₂²×Dic₁₀₆	1	12	yes	—
122	C₄	C₂×#11719949	1	4	yes	—
123	C₂	#16811352	1	4	yes	direct product factor
124	C₂	C₂³×Dic₁₀₆	1	1	yes	characteristic
125	C₂	C₂²×#11719949	1	1	yes	characteristic
126	C₂	C₂×#13133264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811352

C₂

yes

maximal, direct product factor

C₂×#13133217

C₂

yes

maximal

C₂×#13133208

C₂

yes

maximal

C₂×Q₈×Dic₅₃

C₂

yes

maximal

C₂×C₄×Dic₁₀₆

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

#13133217

C₂²

yes

—

#13133208

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

Q₈

yes

complemented

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

C₂×Q₈

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

Dic₁₀₆

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂²×Q₈

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂³×Q₈

yes

characteristic

C₂×C₅₃

C₂²×#41

yes

characteristic

C₂×C₅₃

C₂×2₋¹⁺⁴

yes

characteristic

#557

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₅₃

yes

characteristic

C₅₃

C₂×#557

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

Dic₁₀₆

yes

complemented

C₂×Q₈

C₂²×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₄²

—

100

C₄²

C₂²×D₅₃

yes

—

101

C₂³

C₂³×D₅₃

yes

characteristic

102

Q₈

—

106

—

103

Q₈

C₂×Dic₁₀₆

yes

complemented

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

C₂×Dic₁₀₆

yes

—

110

C₂×C₄

#11719949

yes

—

111

C₂×C₄

C₂³×D₅₃

yes

—

112

C₂×C₄

C₂³×D₅₃

yes

—

113

C₂×C₄

C₂³×D₅₃

yes

—

114

C₂²

C₂²×Dic₁₀₆

yes

—

115

C₂²

C₂×#11719949

yes

—

116

C₂²

#13133264

yes

—

117

C₂²

C₂⁴×D₅₃

yes

characteristic

118

C₄

—

119

C₄

—

106

—

120

C₄

—

106

—

121

C₄

C₂²×Dic₁₀₆

yes

—

122

C₄

C₂×#11719949

yes

—

123

C₂

#16811352

yes

direct product factor

124

C₂

C₂³×Dic₁₀₆

yes

characteristic

125

C₂

C₂²×#11719949

yes

characteristic

126

C₂

C₂×#13133264

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂×#557
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂⁵

Frattini

C₂²

C₂⁴×D₅₃

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂×C₄×Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #16811352	C₂	direct (N × H)
21. C₂×Q₈×C₅₃	Q₈	semidirect
38. Q₈×C₅₃	C₂×Q₈	semidirect
60. C₂×C₅₃	#557	semidirect
70. C₅₃	C₂×#557	semidirect
87. C₂×Q₈	Dic₁₀₆	semidirect
103. Q₈	C₂×Dic₁₀₆	semidirect
123. C₂	#16811352	direct (N × H)

H ≅ G/N

type

1. #16811352

C₂

direct (N × H)

21. C₂×Q₈×C₅₃

Q₈

semidirect

38. Q₈×C₅₃

C₂×Q₈

semidirect

60. C₂×C₅₃

#557

semidirect

70. C₅₃

C₂×#557

semidirect

87. C₂×Q₈

Dic₁₀₆

semidirect

103. Q₈

C₂×Dic₁₀₆

semidirect

123. C₂

#16811352

direct (N × H)

Subgroups — C₂×#16811352 #20288288

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811352 #20288288

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811352 #20288288