Group #20274409 — subgroups

Subgroups — #20274409

Order 6784.

480 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 221 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×#11719917	C₂	1	2	yes	maximal
2	#16810378	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	4	no	—
5	C₂×#11719917	—	2	4	no	—
6	C₂×#11719917	C₄	1	8	yes	complemented
7	C₂²×#88322	C₂²	1	4	yes	—
8	C₂³×C₄×C₅₃	C₄	1	2	yes	—
9	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₅₃	—	2	4	no	—
11	C₂×C₄²×C₅₃	C₄	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
13	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	—	2	4	no	—
15	C₂²×C₄×C₅₃	—	2	8	no	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	D₄	1	4	yes	—
21	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₄²×C₅₃	—	2	4	no	—
24	C₄²×C₅₃	—	2	4	no	—
25	#11719917	—	2	16	no	—
26	C₂×#88322	—	2	4	no	—
27	C₂×#88322	—	2	4	no	—
28	C₂×#88322	C₂×C₄	1	8	yes	complemented
29	C₂³×C₅₃	—	2	4	no	—
30	C₂³×C₅₃	C₄²	1	2	yes	—
31	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
32	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
33	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	16	no	—
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	—	2	4	no	—
37	C₂×C₄×C₅₃	C₄²	1	2	yes	—
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
39	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
41	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	M₁₆	1	8	yes	complemented
43	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	#88322	—	2	8	no	—
48	C₂²×C₅₃	—	2	4	no	—
49	C₂²×C₅₃	—	2	4	no	—
50	C₂²×C₅₃	—	2	4	no	—
51	C₂²×C₅₃	#95	1	4	yes	—
52	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
53	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₅₃	C₂×M₁₆	1	4	yes	—
56	C₄×C₅₃	—	2	8	no	—
57	C₄×C₅₃	#95	1	4	yes	—
58	C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
59	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
60	#1728	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	4	no	—
62	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
63	C₂×C₅₃	C₄×M₁₆	1	2	yes	—
64	C₂×C₅₃	#407	1	2	yes	—
65	C₂×C₅₃	#423	1	2	yes	—
66	C₂²×M₁₆	—	53	2	no	—
67	C₂²×C₄²	D₅₃	1	1	yes	characteristic
68	#336	—	53	4	no	—
69	C₅₃	#1728	1	1	yes	characteristic, complemented, Sylow
70	C₂³×C₄	Dic₅₃	1	2	yes	—
71	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
72	C₂×M₁₆	—	53	8	no	—
73	C₂×M₁₆	—	106	4	no	—
74	C₂×M₁₆	—	106	4	no	—
75	C₂²×C₈	—	53	4	no	—
76	C₂×C₄²	—	2	4	no	—
77	C₂×C₄²	Dic₅₃	1	2	yes	—
78	C₂×C₄²	C₂×D₅₃	1	2	yes	—
79	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
80	C₂²×C₄	—	2	4	no	—
81	C₂²×C₄	—	2	8	no	—
82	C₂²×C₄	Dic₁₀₆	1	2	yes	—
83	C₂²×C₄	C₄×D₅₃	1	4	yes	—
84	C₂²×C₄	D₂₁₂	1	2	yes	—
85	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
86	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₅₃	1	4	yes	—
88	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
89	M₁₆	—	106	16	no	—
90	C₂×C₈	—	53	8	no	—
91	C₂×C₈	—	106	4	no	—
92	C₂×C₈	—	106	4	no	—
93	C₄²	—	2	4	no	—
94	C₄²	—	2	4	no	—
95	C₂³	—	2	4	no	—
96	C₂³	C₄×Dic₅₃	1	2	yes	—
97	C₂³	C₄⋊Dic₅₃	1	2	yes	—
98	C₂³	C₂²⋊Dic₅₃	1	2	yes	—
99	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
100	C₂×C₄	—	2	16	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	#11719917	1	8	yes	complemented
104	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
105	C₂×C₄	#11719919	1	8	yes	—
106	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
107	C₂×C₄	#11719921	1	8	yes	—
108	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
109	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
111	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
113	C₈	—	106	8	no	—
114	C₂²	—	2	4	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	4	no	—
117	C₂²	#13133124	1	4	yes	—
118	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
119	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
120	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
121	C₂²	C₂×#11719917	1	4	yes	—
122	C₄	—	2	8	no	—
123	C₄	#13133124	1	4	yes	—
124	C₄	C₂×#11719921	1	2	yes	—
125	C₄	C₂×#11719919	1	2	yes	—
126	C₂	—	2	4	no	—
127	C₂	C₂×#13133124	1	1	yes	characteristic
128	C₂	C₄×#11719917	1	2	yes	—
129	C₂	#16810587	1	2	yes	—
130	C₂	#16810647	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719917

C₂

yes

maximal

#16810378

C₂

yes

maximal, complemented

C₂²×C₄²×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂×#11719917

—

C₂×#11719917

C₄

yes

complemented

C₂²×#88322

C₂²

yes

—

C₂³×C₄×C₅₃

C₄

yes

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

—

—

—

—

C₂×C₄

yes

complemented

C₂³×C₅₃

—

C₂³×C₅₃

C₄²

yes

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

M₁₆

yes

complemented

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

#95

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

#1728

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#95

yes

characteristic

C₂×C₅₃

C₄×M₁₆

yes

—

C₂×C₅₃

#407

yes

—

C₂×C₅₃

#423

yes

—

C₂²×M₁₆

—

C₂²×C₄²

D₅₃

yes

characteristic

#336

—

C₅₃

#1728

yes

characteristic, complemented, Sylow

C₂³×C₄

Dic₅₃

yes

—

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂⁴

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₀₆

yes

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₂³

—

C₂³

C₄×Dic₅₃

yes

—

C₂³

C₄⋊Dic₅₃

yes

—

C₂³

C₂²⋊Dic₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#11719917

yes

complemented

104

C₂×C₄

C₄×Dic₅₃

yes

—

105

C₂×C₄

#11719919

yes

—

106

C₂×C₄

C₄⋊Dic₅₃

yes

—

107

C₂×C₄

#11719921

yes

—

108

C₂×C₄

C₂²⋊Dic₅₃

yes

—

109

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

110

C₂×C₄

C₂×C₄×D₅₃

yes

—

111

C₂×C₄

C₂×D₂₁₂

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

113

C₈

—

106

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#13133124

yes

—

118

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

119

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

120

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

121

C₂²

C₂×#11719917

yes

—

122

C₄

—

123

C₄

#13133124

yes

—

124

C₄

C₂×#11719921

yes

—

125

C₄

C₂×#11719919

yes

—

126

C₂

—

127

C₂

C₂×#13133124

yes

characteristic

128

C₂

C₄×#11719917

yes

—

129

C₂

#16810587

yes

—

130

C₂

#16810647

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Fitting	C₂²×C₄²×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂²×C₄²	D₅₃
hypocenter	C₅₃	#1728
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂×C₄²

Fitting

C₂²×C₄²×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂²×C₄²

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. #16810378	C₂	semidirect
6. C₂×#11719917	C₄	semidirect
28. C₂×#88322	C₂×C₄	semidirect
42. C₂×C₄×C₅₃	M₁₆	semidirect
69. C₅₃	#1728	semidirect
103. C₂×C₄	#11719917	semidirect

H ≅ G/N

type

2. #16810378

C₂

semidirect

6. C₂×#11719917

C₄

semidirect

28. C₂×#88322

C₂×C₄

semidirect

42. C₂×C₄×C₅₃

M₁₆

semidirect

69. C₅₃

#1728

semidirect

103. C₂×C₄

#11719917

semidirect

Subgroups — #20274409

Conjugacy classes of subgroups

Complements