Group #20289922 — subgroups

Subgroups — #20289922

Order 6784.

208 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 98 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	#16811639	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#88324	C₂	1	1	yes	maximal, characteristic
3	#88324⋊C₈	C₂	1	2	yes	maximal, complemented
4	#88324⋊C₈	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88324	C₄	1	2	yes	—
7	C₄⋊#88324	C₄	1	2	yes	—
8	C₂²⋊#88324	C₄	1	2	yes	—
9	C₄×#88324	C₄	1	2	yes	—
10	C₄×#88324	C₂²	1	1	yes	characteristic
11	C₄×#88324	C₂²	1	1	yes	characteristic
12	C₄×#88324	C₂²	1	2	yes	—
13	C₂×#11719917	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	#13133127	C₄	1	2	yes	—
16	C₈×Dic₅₃	C₂²	1	2	yes	—
17	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#88324	—	2	4	no	—
22	C₂×#88324	C₂×C₄	1	2	yes	—
23	C₂×#88324	C₂×C₄	1	4	yes	—
24	C₂×#88324	C₂×C₄	1	2	yes	—
25	M₁₆×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
31	#11719917	—	2	2	no	—
32	C₂×#88322	C₂×C₄	1	2	yes	—
33	C₂³×C₅₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
36	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₅₃	—	2	2	no	—
39	C₂×Dic₅₃	—	2	2	no	—
40	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
41	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
42	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
44	#88324	—	2	2	no	—
45	#88324	—	2	4	no	—
46	#88324	M₁₆	1	4	yes	complemented
47	C₈×C₅₃	—	2	2	no	—
48	#88322	—	2	2	no	—
49	C₂²×C₅₃	—	2	1	no	—
50	C₂²×C₅₃	—	2	1	no	—
51	C₂²×C₅₃	#97	1	2	yes	—
52	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
53	C₄×C₅₃	#97	1	2	yes	—
54	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
56	Dic₅₃	—	2	1	no	—
57	Dic₅₃	—	2	1	no	—
58	Dic₅₃	C₂×M₁₆	1	2	yes	—
59	Dic₅₃	#131	1	2	yes	—
60	#1451	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	2	no	—
62	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
63	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
64	C₂×C₅₃	#405	1	1	yes	characteristic
65	#431	—	53	1	no	—
66	C₄×M₁₆	—	53	1	no	—
67	C₂×C₄×C₈	—	53	1	no	—
68	#322	—	53	2	no	—
69	#322	—	53	2	no	—
70	C₅₃	#1451	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	53	1	no	—
72	C₂×M₁₆	#6157	1	1	yes	characteristic
73	C₂²×C₈	—	53	2	no	—
74	C₂×C₄²	—	53	1	no	—
75	C₄⋊C₈	—	53	2	no	—
76	C₂²⋊C₈	—	53	2	no	—
77	#97	—	53	2	no	—
78	C₄×C₈	—	53	1	no	—
79	C₄×C₈	—	53	1	no	—
80	C₄×C₈	—	53	2	no	—
81	C₄×C₈	—	53	2	no	—
82	C₄×C₈	—	53	2	no	—
83	C₂²×C₄	—	53	1	no	—
84	C₂²×C₄	—	53	1	no	—
85	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	106	2	no	—
88	C₂×C₈	—	53	2	no	—
89	C₂×C₈	—	53	2	no	—
90	C₂×C₈	—	53	4	no	—
91	C₂×C₈	—	53	2	no	—
92	C₂×C₈	—	106	4	no	—
93	C₂×C₈	C₂×#6157	1	2	yes	—
94	C₄²	—	53	1	no	—
95	C₄²	—	53	1	no	—
96	C₄²	—	53	1	no	—
97	C₄²	—	53	1	no	—
98	C₂³	C₄×#6157	1	1	yes	characteristic
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	53	1	no	—
103	C₂×C₄	—	106	2	no	—
104	C₂×C₄	—	106	2	no	—
105	C₂×C₄	#11719936	1	2	yes	—
106	C₂×C₄	C₄×#6157	1	1	yes	characteristic
107	C₂×C₄	#11719940	1	2	yes	—
108	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	53	4	no	—
111	C₈	—	106	4	no	—
112	C₈	—	106	2	no	—
113	C₈	—	106	2	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#13133272	1	2	yes	—
117	C₂²	C₂×C₄×#6157	1	1	yes	characteristic
118	C₄	—	53	2	no	—
119	C₄	—	53	2	no	—
120	C₄	—	106	1	no	—
121	C₄	—	106	1	no	—
122	C₄	#13133272	1	2	yes	—
123	C₄	C₂×#11719940	1	1	yes	characteristic
124	C₄	C₂×#11719936	1	1	yes	characteristic
125	C₂	—	2	2	no	—
126	C₂	C₂×#13133272	1	1	yes	characteristic
127	C₂	M₁₆×#6157	1	1	yes	characteristic
128	C₂	#16811589	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811639

C₂

yes

maximal, characteristic

C₂×C₄×#88324

C₂

yes

maximal, characteristic

#88324⋊C₈

C₂

yes

maximal, complemented

#88324⋊C₈

C₂

yes

maximal, complemented

M₁₆×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×#88324

C₄

yes

—

C₄⋊#88324

C₄

yes

—

C₂²⋊#88324

C₄

yes

—

C₄×#88324

C₄

yes

—

C₄×#88324

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

—

C₂×#11719917

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133127

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88324

—

C₂×#88324

C₂×C₄

yes

—

C₂×#88324

C₂×C₄

yes

—

C₂×#88324

C₂×C₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

#11719917

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

—

M₁₆

yes

complemented

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#97

yes

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

#97

yes

—

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×M₁₆

yes

—

Dic₅₃

#131

yes

—

#1451

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#97

yes

characteristic

C₂×C₅₃

C₄×M₁₆

yes

C₂×C₅₃

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

#322

—

C₅₃

#1451

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

#6157

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6157

yes

characteristic

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₄×#6157

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

106

—

104

C₂×C₄

—

106

—

105

C₂×C₄

#11719936

yes

—

106

C₂×C₄

C₄×#6157

yes

characteristic

107

C₂×C₄

#11719940

yes

—

108

C₂×C₄

C₂²×#6157

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

106

—

112

C₈

—

106

—

113

C₈

—

106

—

114

C₂²

—

115

C₂²

—

116

C₂²

#13133272

yes

—

117

C₂²

C₂×C₄×#6157

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

106

—

121

C₄

—

106

—

122

C₄

#13133272

yes

—

123

C₄

C₂×#11719940

yes

characteristic

124

C₄

C₂×#11719936

yes

characteristic

125

C₂

—

126

C₂

C₂×#13133272

yes

characteristic

127

C₂

M₁₆×#6157

yes

characteristic

128

C₂

#16811589

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₂×M₁₆×C₅₃	C₄
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	#1451
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂×C₄²

Fitting

C₂×M₁₆×C₅₃

C₄

C₂×M₁₆

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
3. #88324⋊C₈	C₂	semidirect
4. #88324⋊C₈	C₂	semidirect
46. #88324	M₁₆	semidirect
70. C₅₃	#1451	semidirect

H ≅ G/N

type

3. #88324⋊C₈

C₂

semidirect

4. #88324⋊C₈

C₂

semidirect

46. #88324

M₁₆

semidirect

70. C₅₃

#1451

semidirect

Subgroups — #20289922

Conjugacy classes of subgroups

Complements