Group #20273439 — subgroups

Subgroups — #20273439

Order 6784.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16810679	C₂	1	2	yes	maximal
3	#16810372	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810371	C₂	1	2	yes	maximal, complemented
5	C₅₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₅₃	C₄	1	2	yes	—
7	#13133217	C₂²	1	1	yes	characteristic
8	#13133216	—	2	2	no	—
9	#13133215	C₄	1	2	yes	—
10	C₄⋊C₄×D₅₃	—	2	2	no	—
11	C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	2	yes	—
13	#13133129	—	2	2	no	—
14	#13133127	—	2	2	no	—
15	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
18	Q₈×D₅₃	—	4	4	no	—
19	C₂×C₄×D₅₃	—	2	2	no	—
20	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₅₃	D₄	1	2	yes	—
22	C₂×Dic₁₀₆	—	2	2	no	—
23	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₅₃	—	2	2	no	—
28	#11719919	—	2	4	no	—
29	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₅₃	D₄	1	2	yes	—
31	C₂×#88322	—	2	2	no	—
32	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	Q₈×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	C₂×Dic₅₃	—	2	2	no	—
38	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
40	C₄×D₅₃	—	2	2	no	—
41	C₄×D₅₃	—	2	2	no	—
42	C₄×D₅₃	—	2	2	no	—
43	C₄×D₅₃	—	4	4	no	—
44	Dic₁₀₆	—	4	4	no	—
45	Dic₁₀₆	—	4	2	no	—
46	C₈×C₅₃	—	4	2	no	—
47	#88322	—	4	2	no	—
48	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₅₃	—	2	2	no	—
50	C₂×D₅₃	#100	1	2	yes	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	4	2	no	—
53	C₄×C₅₃	#137	1	2	yes	—
54	C₄×C₅₃	#137	1	2	yes	—
55	Dic₅₃	—	2	1	no	—
56	Dic₅₃	—	2	2	no	—
57	Dic₅₃	—	4	2	no	—
58	#1558	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
60	C₂×C₅₃	#419	1	2	yes	—
61	D₅₃	—	2	2	no	—
62	C₂×C₄⋊Q₈	—	53	1	no	—
63	#432	—	53	2	no	—
64	#333	—	53	1	no	—
65	#333	—	53	2	no	—
66	#333	D₅₃	1	1	yes	characteristic, complemented
67	C₅₃	#1558	1	1	yes	characteristic, complemented, Sylow
68	C₂²×Q₈	—	53	2	no	—
69	C₄⋊Q₈	—	53	2	no	—
70	C₄⋊Q₈	—	53	1	no	—
71	C₄⋊Q₈	—	106	2	no	—
72	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	106	2	no	—
74	C₂×C₄²	—	53	1	no	—
75	C₄⋊C₈	—	53	2	no	—
76	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
77	C₂²⋊C₈	—	106	2	no	—
78	#97	—	106	2	no	—
79	C₂×Q₈	—	53	2	no	—
80	C₂×Q₈	—	106	2	no	—
81	C₂×Q₈	—	212	4	no	—
82	C₂×Q₈	C₄×D₅₃	1	2	yes	—
83	C₂²×C₄	—	53	1	no	—
84	C₂²×C₄	—	53	2	no	—
85	C₂²×C₄	—	106	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	106	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	106	4	no	—
90	C₄⋊C₄	—	106	2	no	—
91	C₄²	—	53	1	no	—
92	C₄²	—	53	2	no	—
93	C₄²	C₂²×D₅₃	1	1	yes	characteristic
94	C₂³	—	53	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	212	2	no	—
97	Q₈	—	212	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	53	2	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	106	2	no	—
102	C₂×C₄	—	106	2	no	—
103	C₂×C₄	—	106	2	no	—
104	C₂×C₄	—	106	2	no	—
105	C₂×C₄	—	212	4	no	—
106	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
107	C₂×C₄	D₄×D₅₃	1	2	yes	—
108	C₈	—	4	2	no	—
109	C₈	—	212	2	no	—
110	C₂²	—	53	2	no	—
111	C₂²	—	106	2	no	—
112	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	106	1	no	—
116	C₄	—	106	2	no	—
117	C₄	—	212	2	no	—
118	C₄	#13133246	1	2	yes	—
119	C₄	#13133249	1	2	yes	—
120	C₂	—	106	2	no	—
121	C₂	#100×D₅₃	1	1	yes	characteristic
122	C₂	#16810630	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic

#16810679

C₂

yes

maximal

#16810372

C₂

yes

maximal, characteristic, complemented

#16810371

C₂

yes

maximal, complemented

C₅₃×#333

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₃

C₄

yes

—

#13133217

C₂²

yes

characteristic

#13133216

—

#13133215

C₄

yes

—

C₄⋊C₄×D₅₃

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

#13133129

—

#13133127

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

C₂×#88322

—

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

#100

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#137

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

C₂×C₄⋊Q₈

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1558

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂×Q₈

C₄×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

C₄²

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

106

—

102

C₂×C₄

—

106

—

103

C₂×C₄

—

106

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

212

—

106

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

107

C₂×C₄

D₄×D₅₃

yes

—

108

C₈

—

109

C₈

—

212

—

110

C₂²

—

111

C₂²

—

106

—

112

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

106

—

116

C₄

—

106

—

117

C₄

—

212

—

118

C₄

#13133246

yes

—

119

C₄

#13133249

yes

—

120

C₂

—

106

—

121

C₂

#100×D₅₃

yes

characteristic

122

C₂

#16810630

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#333	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#333	D₅₃
hypocenter	C₅₃	#1558
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#333

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
3. #16810372	C₂	semidirect
4. #16810371	C₂	semidirect
5. C₅₃×#333	C₂	semidirect
66. #333	D₅₃	semidirect
67. C₅₃	#1558	semidirect

H ≅ G/N

type

3. #16810372

C₂

semidirect

4. #16810371

C₂

semidirect

5. C₅₃×#333

C₂

semidirect

66. #333

D₅₃

semidirect

67. C₅₃

#1558

semidirect

Subgroups — #20273439

Conjugacy classes of subgroups

Complements