Group #20273468 — subgroups

Subgroups — #20273468

Order 6784.

180 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#16810375	C₂	1	2	yes	maximal, complemented
4	#16810373	C₂	1	2	yes	maximal, complemented
5	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₄	1	1	yes	characteristic
7	C₂×#11719917	C₄	1	1	yes	characteristic
8	C₂²×#88322	—	2	1	no	—
9	C₄⋊#88322	—	2	1	no	—
10	C₄⋊#88322	C₂²	1	2	yes	—
11	#13133128	C₄	1	2	yes	—
12	C₄×#88322	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²⋊C₈×C₅₃	—	2	1	no	—
17	C₄×C₈×C₅₃	—	2	1	no	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₅₃	—	2	1	no	—
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₄²×C₅₃	Q₈	1	1	yes	characteristic
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	#11719917	—	2	2	no	—
28	#11719917	C₈	1	4	yes	complemented
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	—	4	2	no	—
31	C₂×#88322	C₂×C₄	1	2	yes	—
32	C₂×#88322	C₂×C₄	1	2	yes	—
33	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
41	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₅₃	—	4	1	no	—
46	C₈×C₅₃	—	4	1	no	—
47	#88322	—	2	2	no	—
48	#88322	—	4	2	no	—
49	#88322	C₂×C₈	1	4	yes	complemented
50	C₂²×C₅₃	—	2	1	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
53	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
58	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₅₃	#137	1	1	yes	characteristic
62	#1565	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₅₃	#428	1	1	yes	characteristic
66	C₂×C₅₃	#430	1	1	yes	characteristic
67	#431	D₅₃	1	1	yes	characteristic
68	C₂×C₄⋊C₈	—	53	1	no	—
69	C₄×M₁₆	—	53	1	no	—
70	#335	—	53	2	no	—
71	#334	—	53	2	no	—
72	C₅₃	#1565	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	—	53	1	no	—
74	C₂×M₁₆	—	53	1	no	—
75	C₂²×C₈	—	106	1	no	—
76	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
77	C₄⋊C₈	—	53	2	no	—
78	C₄⋊C₈	—	106	1	no	—
79	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
80	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
81	C₂²⋊C₈	—	2	1	no	—
82	#97	—	53	2	no	—
83	C₄×C₈	—	2	1	no	—
84	C₄×C₈	—	53	2	no	—
85	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
88	M₁₆	—	53	4	no	—
89	M₁₆	—	106	2	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	53	2	no	—
93	C₂×C₈	—	53	2	no	—
94	C₂×C₈	—	106	2	no	—
95	C₂×C₈	—	212	2	no	—
96	C₄²	Dic₁₀₆	1	1	yes	characteristic
97	C₄²	C₄×D₅₃	1	1	yes	characteristic
98	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
99	C₄²	C₂²×D₅₃	1	1	yes	characteristic
100	C₂³	#11719919	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	C₈×D₅₃	1	2	yes	—
105	C₂×C₄	#11719912	1	2	yes	—
106	C₂×C₄	#11719919	1	1	yes	characteristic
107	C₂×C₄	#11719919	1	1	yes	characteristic
108	C₂×C₄	#11719919	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
112	C₈	—	4	1	no	—
113	C₈	—	4	1	no	—
114	C₈	—	53	4	no	—
115	C₈	—	106	2	no	—
116	C₈	—	212	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#13133146	1	2	yes	—
120	C₂²	C₂×#11719919	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	#13133146	1	2	yes	—
125	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
126	C₄	C₂×#11719912	1	1	yes	characteristic
127	C₄	#13133242	1	1	yes	characteristic
128	C₄	#13133247	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×#13133146	1	1	yes	characteristic
131	C₂	#16810666	1	1	yes	characteristic
132	C₂	#16810671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₄×#11719917

C₂

yes

maximal, characteristic

#16810375

C₂

yes

maximal, complemented

#16810373

C₂

yes

maximal, complemented

C₅₃×#431

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

characteristic

C₂×#11719917

C₄

yes

characteristic

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

#13133128

C₄

yes

—

C₄×#88322

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719917

—

#11719917

C₈

yes

—

—

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

C₂×C₈

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₈

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#335

—

#334

—

C₅₃

#1565

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

characteristic

100

C₂³

#11719919

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₈×D₅₃

yes

—

105

C₂×C₄

#11719912

yes

—

106

C₂×C₄

#11719919

yes

characteristic

107

C₂×C₄

#11719919

yes

characteristic

108

C₂×C₄

#11719919

yes

characteristic

109

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

110

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

106

—

116

C₈

—

212

—

117

C₂²

—

118

C₂²

—

119

C₂²

#13133146

yes

—

120

C₂²

C₂×#11719919

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#13133146

yes

—

125

C₄

C₂×C₈×D₅₃

yes

characteristic

126

C₄

C₂×#11719912

yes

characteristic

127

C₄

#13133242

yes

characteristic

128

C₄

#13133247

yes

characteristic

129

C₂

—

130

C₂

C₂×#13133146

yes

characteristic

131

C₂

#16810666

yes

characteristic

132

C₂

#16810671

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1565
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16810375	C₂	semidirect
4. #16810373	C₂	semidirect
28. #11719917	C₈	semidirect
49. #88322	C₂×C₈	semidirect
72. C₅₃	#1565	semidirect

H ≅ G/N

type

3. #16810375

C₂

semidirect

4. #16810373

C₂

semidirect

28. #11719917

C₈

semidirect

49. #88322

C₂×C₈

semidirect

72. C₅₃

#1565

semidirect

Subgroups — #20273468

Conjugacy classes of subgroups

Complements