Group #20273480 — subgroups

Subgroups — #20273480

Order 6784.

166 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16810680	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×#423	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₄	1	2	yes	—
7	C₄⋊#88322	C₂²	1	1	yes	characteristic
8	C₄⋊#88322	C₂²	1	1	yes	characteristic
9	C₂²⋊#88322	—	2	1	no	—
10	#13133128	—	2	1	no	—
11	#13133128	C₄	1	1	yes	characteristic
12	#13133128	C₄	1	1	yes	characteristic
13	C₄×#88322	C₂²	1	2	yes	—
14	C₂×M₁₆×C₅₃	—	2	1	no	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	—	2	1	no	—
17	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
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	M₁₆×C₅₃	—	4	2	no	—
22	C₂×C₈×C₅₃	—	2	2	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	4	no	—
28	C₂×#88322	—	2	1	no	—
29	C₂×#88322	—	2	1	no	—
30	C₂×#88322	C₂×C₄	1	2	yes	—
31	C₂×#88322	C₂×C₄	1	2	yes	—
32	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₅₃	—	4	2	no	—
44	#88322	—	2	2	no	—
45	#88322	—	4	1	no	—
46	#88322	—	4	1	no	—
47	#88322	M₁₆	1	4	yes	complemented
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	—	2	1	no	—
50	C₂²×C₅₃	—	2	1	no	—
51	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
57	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
58	C₄×C₅₃	#137	1	1	yes	characteristic
59	#1569	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	—	4	1	no	—
61	C₂×C₅₃	#423	1	1	yes	characteristic
62	C₂×C₅₃	#428	1	1	yes	characteristic
63	C₂×C₅₃	#430	1	1	yes	characteristic
64	#432	—	53	1	no	—
65	#423	D₅₃	1	1	yes	characteristic
66	C₄×M₁₆	—	53	1	no	—
67	#335	—	53	2	no	—
68	#334	—	53	2	no	—
69	C₅₃	#1569	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₁₆	—	2	1	no	—
71	C₂×M₁₆	—	53	2	no	—
72	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
73	C₄⋊C₈	—	2	1	no	—
74	C₄⋊C₈	—	53	1	no	—
75	C₄⋊C₈	—	53	1	no	—
76	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
77	C₂²⋊C₈	—	106	1	no	—
78	#97	—	53	1	no	—
79	#97	—	53	1	no	—
80	#97	—	106	1	no	—
81	C₄×C₈	—	53	2	no	—
82	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
85	M₁₆	—	4	2	no	—
86	M₁₆	—	106	4	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	53	2	no	—
89	C₂×C₈	—	53	2	no	—
90	C₂×C₈	—	106	1	no	—
91	C₂×C₈	—	106	1	no	—
92	C₄²	Dic₁₀₆	1	1	yes	characteristic
93	C₄²	C₄×D₅₃	1	1	yes	characteristic
94	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
95	C₄²	C₂²×D₅₃	1	1	yes	characteristic
96	C₂³	#11719919	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	#11719919	1	1	yes	characteristic
102	C₂×C₄	#11719919	1	1	yes	characteristic
103	C₂×C₄	#11719919	1	1	yes	characteristic
104	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	53	4	no	—
109	C₈	—	106	2	no	—
110	C₈	—	212	1	no	—
111	C₈	—	212	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	C₂×#11719919	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	M₁₆×D₅₃	1	2	yes	—
121	C₄	#13133243	1	1	yes	characteristic
122	C₄	#13133248	1	1	yes	characteristic
123	C₂	—	4	1	no	—
124	C₂	#16810646	1	1	yes	characteristic
125	C₂	#16810663	1	1	yes	characteristic
126	C₂	#16810671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic

C₄×#11719917

C₂

yes

maximal, characteristic

#16810375

C₂

yes

maximal, complemented

#16810373

C₂

yes

maximal, complemented

C₅₃×#423

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

C₄

yes

characteristic

#13133128

C₄

yes

characteristic

C₄×#88322

C₂²

yes

—

C₂×M₁₆×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₄²×C₅₃

C₂³

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

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

—

—

—

—

M₁₆

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₄×M₁₆

—

#335

—

#334

—

C₅₃

#1569

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₂²⋊C₈

—

106

—

#97

—

#97

—

#97

—

106

—

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

—

106

—

C₂×C₈

—

106

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

#11719919

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#11719919

yes

characteristic

102

C₂×C₄

#11719919

yes

characteristic

103

C₂×C₄

#11719919

yes

characteristic

104

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

105

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

106

—

110

C₈

—

212

—

111

C₈

—

212

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×#11719919

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

M₁₆×D₅₃

yes

—

121

C₄

#13133243

yes

characteristic

122

C₄

#13133248

yes

characteristic

123

C₂

—

124

C₂

#16810646

yes

characteristic

125

C₂

#16810663

yes

characteristic

126

C₂

#16810671

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1569
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#423

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #16810375	C₂	semidirect
4. #16810373	C₂	semidirect
47. #88322	M₁₆	semidirect
69. C₅₃	#1569	semidirect

H ≅ G/N

type

3. #16810375

C₂

semidirect

4. #16810373

C₂

semidirect

47. #88322

M₁₆

semidirect

69. C₅₃

#1569

semidirect

Subgroups — #20273480

Conjugacy classes of subgroups

Complements