Group #20290848 — subgroups

Subgroups — #20290848

Order 6784.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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	#16811627	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133282	C₂	1	4	yes	maximal, complemented
3	C₄²×#6157	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811174	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719936	C₂²	1	2	yes	—
6	C₂×C₄×#6157	—	2	2	no	—
7	C₂×C₄×#6157	C₂²	1	2	yes	—
8	C₄⋊#88324	—	2	2	no	—
9	#13133282	—	2	8	no	—
10	C₂×#11719949	C₂²	1	2	yes	—
11	#13133214	C₄	1	1	yes	characteristic, complemented
12	#13133206	C₄	1	2	yes	complemented
13	#13133192	—	2	2	no	—
14	#13133179	—	2	2	no	—
15	C₄²×D₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×#6157	—	2	2	no	—
18	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
19	#11719949	—	2	4	no	—
20	#11719949	—	2	4	no	—
21	C₂×D₂₁₂	—	2	2	no	—
22	C₂×D₂₁₂	—	2	2	no	—
23	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
24	C₂×C₄×D₅₃	—	2	2	no	—
25	C₂×C₄×D₅₃	D₄	1	2	yes	—
26	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
27	C₂×#88324	—	2	2	no	—
28	C₄×#6157	—	2	8	no	—
29	C₄×#6157	—	2	2	no	—
30	C₄×#6157	D₄	1	4	yes	complemented
31	#11719936	—	2	4	no	—
32	C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	#11719921	—	2	4	no	—
35	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₅₃	D₄	1	2	yes	—
37	C₂²×D₅₃	—	2	2	no	—
38	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×#6157	—	2	8	no	—
40	C₂×#6157	—	2	4	no	—
41	Q₈×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
47	D₂₁₂	—	2	4	no	—
48	D₂₁₂	—	4	4	no	—
49	D₂₁₂	—	4	4	no	—
50	C₄×D₅₃	—	2	4	no	—
51	C₄×D₅₃	—	4	2	no	—
52	C₄×D₅₃	—	4	2	no	—
53	C₄×D₅₃	C₂×D₄	1	2	yes	—
54	C₄×D₅₃	#41	1	2	yes	—
55	#88324	—	4	2	no	—
56	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×D₅₃	—	2	2	no	—
58	C₂×D₅₃	—	4	2	no	—
59	C₂×D₅₃	—	4	4	no	—
60	C₂×D₅₃	#121	1	2	yes	—
61	#6157	—	2	8	no	—
62	C₄×C₅₃	—	4	2	no	—
63	C₄×C₅₃	#104	1	4	yes	complemented
64	C₄×C₅₃	C₄×D₄	1	2	yes	—
65	Dic₅₃	—	2	2	no	—
66	Dic₅₃	#124	1	1	yes	characteristic
67	Dic₅₃	#127	1	1	yes	characteristic
68	#1962	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	#390	1	1	yes	characteristic
70	C₂×C₅₃	C₂×#104	1	2	yes	—
71	D₅₃	—	2	2	no	—
72	D₅₃	—	8	2	no	—
73	#532	—	53	1	no	—
74	#423	—	53	1	no	—
75	C₂×#104	—	53	4	no	—
76	C₄³	—	53	1	no	—
77	C₅₃	#1962	1	1	yes	characteristic, complemented, Sylow
78	C₂×#41	—	53	2	no	—
79	C₂×M₁₆	—	53	2	no	—
80	C₄⋊Q₈	#6157	1	1	yes	characteristic, complemented
81	#127	—	53	1	no	—
82	#124	—	53	2	no	—
83	#121	—	106	2	no	—
84	C₄×D₄	—	106	2	no	—
85	C₂×C₄²	—	53	1	no	—
86	C₂×C₄²	—	53	2	no	—
87	C₂×C₄²	—	106	2	no	—
88	C₄⋊C₈	—	106	2	no	—
89	#104	—	106	8	no	—
90	#41	—	106	4	no	—
91	#41	—	106	4	no	—
92	C₂×Q₈	C₂×#6157	1	2	yes	—
93	C₂×D₄	—	53	2	no	—
94	C₂×D₄	—	106	2	no	—
95	C₂×D₄	—	106	2	no	—
96	C₂²×C₄	—	53	1	no	—
97	C₂²×C₄	—	53	2	no	—
98	C₂²×C₄	—	106	2	no	—
99	C₂²×C₄	—	106	2	no	—
100	M₁₆	—	106	4	no	—
101	C₂×C₈	—	106	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	106	4	no	—
104	C₄²	—	53	2	no	—
105	C₄²	—	53	1	no	—
106	C₄²	—	53	4	no	—
107	C₄²	—	106	2	no	—
108	C₄²	—	106	8	no	—
109	C₄²	C₂×#6157	1	1	yes	characteristic
110	C₂³	—	53	1	no	—
111	C₂³	—	106	2	no	—
112	Q₈	—	2	4	no	—
113	D₄	—	106	4	no	—
114	D₄	—	212	4	no	—
115	D₄	—	212	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	53	2	no	—
119	C₂×C₄	—	53	2	no	—
120	C₂×C₄	—	53	2	no	—
121	C₂×C₄	—	106	4	no	—
122	C₂×C₄	—	106	8	no	—
123	C₂×C₄	—	106	4	no	—
124	C₂×C₄	—	212	2	no	—
125	C₂×C₄	—	212	2	no	—
126	C₂×C₄	C₂²⋊#6157	1	2	yes	—
127	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
128	C₈	—	212	2	no	—
129	C₂²	—	53	2	no	—
130	C₂²	—	106	2	no	—
131	C₂²	—	212	2	no	—
132	C₂²	—	212	4	no	—
133	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
134	C₄	—	4	2	no	—
135	C₄	—	53	1	no	—
136	C₄	—	53	1	no	—
137	C₄	—	106	8	no	—
138	C₄	—	106	2	no	—
139	C₄	#13133282	1	4	yes	complemented
140	C₄	D₄×#6157	1	2	yes	—
141	C₂	—	106	2	no	—
142	C₂	—	424	2	no	—
143	C₂	#16811575	1	1	yes	characteristic
144	C₂	C₂×#13133282	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811627

C₂

yes

maximal, characteristic, complemented

C₂×#13133282

C₂

yes

maximal, complemented

C₄²×#6157

C₂

yes

maximal, characteristic, complemented

#16811174

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

C₂×C₄×#6157

—

C₂×C₄×#6157

C₂²

yes

—

C₄⋊#88324

—

#13133282

—

C₂×#11719949

C₂²

yes

—

#13133214

C₄

yes

characteristic, complemented

#13133206

C₄

yes

complemented

#13133192

—

#13133179

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

#11719949

—

#11719949

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

—

—

D₄

yes

complemented

#11719936

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

C₂×#6157

—

C₂×#6157

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

—

C₄×D₅₃

#41

yes

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#121

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

#104

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

D₅₃

—

#532

—

#423

—

C₂×#104

—

C₄³

—

C₅₃

#1962

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

#6157

yes

characteristic, complemented

#127

—

#124

—

#121

—

106

—

C₄×D₄

—

106

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

106

—

C₄⋊C₈

—

106

—

#104

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

C₂×#6157

yes

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

100

M₁₆

—

106

—

101

C₂×C₈

—

106

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

106

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

106

—

108

C₄²

—

106

—

109

C₄²

C₂×#6157

yes

characteristic

110

C₂³

—

111

C₂³

—

106

—

112

Q₈

—

113

D₄

—

106

—

114

D₄

—

212

—

115

D₄

—

212

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

212

—

125

C₂×C₄

—

212

—

126

C₂×C₄

C₂²⋊#6157

yes

—

127

C₂×C₄

C₂²×#6157

yes

characteristic

128

C₈

—

212

—

129

C₂²

—

130

C₂²

—

106

—

131

C₂²

—

212

—

132

C₂²

—

212

—

133

C₂²

C₂×C₂²⋊#6157

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

106

—

138

C₄

—

106

—

139

C₄

#13133282

yes

complemented

140

C₄

D₄×#6157

yes

—

141

C₂

—

106

—

142

C₂

—

424

—

143

C₂

#16811575

yes

characteristic

144

C₂

C₂×#13133282

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄⋊Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#1962
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄⋊Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #16811627	C₂	semidirect
2. C₂×#13133282	C₂	semidirect
3. C₄²×#6157	C₂	semidirect
11. #13133214	C₄	semidirect
12. #13133206	C₄	semidirect
16. C₄⋊Q₈×C₅₃	C₄	semidirect
30. C₄×#6157	D₄	semidirect
63. C₄×C₅₃	#104	semidirect
77. C₅₃	#1962	semidirect
80. C₄⋊Q₈	#6157	semidirect
139. C₄	#13133282	semidirect

H ≅ G/N

type

1. #16811627

C₂

semidirect

2. C₂×#13133282

C₂

semidirect

3. C₄²×#6157

C₂

semidirect

11. #13133214

C₄

semidirect

12. #13133206

C₄

semidirect

16. C₄⋊Q₈×C₅₃

C₄

semidirect

30. C₄×#6157

D₄

semidirect

63. C₄×C₅₃

#104

semidirect

77. C₅₃

#1962

semidirect

80. C₄⋊Q₈

#6157

semidirect

139. C₄

#13133282

semidirect

Subgroups — #20290848

Conjugacy classes of subgroups

Complements