Group #20287382 — subgroups

Subgroups — #20287382

Order 6784.

392 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810969	C₂	1	3	yes	maximal, complemented
3	#16810913	C₂	1	3	yes	maximal, complemented
4	#16810820	C₂	1	3	yes	maximal, complemented
5	Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11719923	C₂	1	3	yes	maximal, complemented
7	C₅₃×#550	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719923	C₂²	1	3	yes	—
9	#13133217	C₂²	1	3	yes	—
10	#13133216	—	2	3	no	—
11	Q₈×Dic₅₃	—	2	1	no	—
12	C₄×Dic₁₀₆	C₂²	1	3	yes	—
13	#13133148	C₂²	1	3	yes	—
14	C₄⋊#88322	C₂²	1	3	yes	—
15	#13133140	C₂²	1	1	yes	characteristic
16	#13133138	C₂²	1	6	yes	—
17	#13133138	C₂²	1	3	yes	—
18	C₄×#88322	C₂²	1	3	yes	—
19	C₂×#41×C₅₃	—	2	1	no	—
20	#127×C₅₃	C₂²	1	3	yes	—
21	#121×C₅₃	—	2	3	no	—
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₃	C₂²	1	3	yes	—
24	#41×C₅₃	—	4	4	no	—
25	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂×D₄×C₅₃	—	2	3	no	—
27	C₂×D₄×C₅₃	—	2	3	no	—
28	C₂×D₄×C₅₃	C₂³	1	3	yes	—
29	C₂²×C₄×C₅₃	—	2	3	no	—
30	C₂×Dic₁₀₆	—	2	3	no	—
31	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₅₃	D₄	1	3	yes	—
33	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²⋊C₄×C₅₃	—	2	3	no	—
35	C₄²×C₅₃	C₂³	1	3	yes	—
36	#11719923	—	2	6	no	—
37	C₄⋊Dic₅₃	—	2	3	no	—
38	C₄⋊Dic₅₃	C₂³	1	3	yes	—
39	#11719919	—	2	3	no	—
40	C₄×Dic₅₃	—	2	3	no	—
41	C₂×#88322	C₂³	1	1	yes	characteristic
42	C₂×#88322	C₂³	1	3	yes	—
43	C₂³×C₅₃	—	2	3	no	—
44	Q₈×C₅₃	SD₁₆	1	4	yes	complemented
45	D₄×C₅₃	—	2	3	no	—
46	D₄×C₅₃	—	2	3	no	—
47	D₄×C₅₃	—	4	6	no	—
48	D₄×C₅₃	—	8	3	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	4	6	no	—
51	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
53	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	1	no	—
55	C₂×Dic₅₃	—	2	3	no	—
56	Dic₁₀₆	—	2	1	no	—
57	Dic₁₀₆	—	8	3	no	—
58	Dic₁₀₆	#41	1	2	yes	complemented
59	#88322	—	2	3	no	—
60	#88322	—	2	1	no	—
61	C₂²×C₅₃	—	4	3	no	—
62	C₂²×C₅₃	—	4	3	no	—
63	C₂²×C₅₃	—	4	3	no	—
64	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₅₃	—	2	3	no	—
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
68	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
69	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₅₃	—	2	2	no	—
71	Dic₅₃	—	4	3	no	—
72	#3362	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	—	8	3	no	—
74	C₂×C₅₃	#548	1	1	yes	characteristic
75	C₂×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
76	C₂×C₅₃	#576	1	1	yes	characteristic
77	Q₈²	—	53	1	no	—
78	#550	D₅₃	1	1	yes	characteristic
79	#486	—	53	3	no	—
80	#476	—	53	3	no	—
81	#460	—	53	3	no	—
82	C₈×Q₈	—	53	1	no	—
83	C₄×SD₁₆	—	53	3	no	—
84	C₅₃	#3362	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	2	1	no	—
86	C₂×SD₁₆	—	53	3	no	—
87	C₄⋊Q₈	—	53	3	no	—
88	C₄⋊Q₈	—	106	3	no	—
89	#127	C₂×D₅₃	1	3	yes	—
90	#121	—	2	3	no	—
91	C₄×Q₈	—	53	3	no	—
92	C₄×Q₈	—	106	1	no	—
93	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
94	C₄×D₄	C₂×D₅₃	1	3	yes	—
95	#106	—	53	3	no	—
96	C₄⋊C₈	—	53	3	no	—
97	#103	—	53	1	no	—
98	#102	—	53	3	no	—
99	#102	—	53	6	no	—
100	C₄×C₈	—	53	3	no	—
101	#41	—	4	4	no	—
102	C₂×Q₈	—	53	1	no	—
103	C₂×Q₈	—	106	3	no	—
104	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	C₂²×D₅₃	1	3	yes	—
108	C₂²×C₄	—	2	3	no	—
109	SD₁₆	—	106	6	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	3	no	—
112	C₄⋊C₄	—	53	3	no	—
113	C₄⋊C₄	—	106	3	no	—
114	C₄⋊C₄	—	106	3	no	—
115	C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
116	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	3	no	—
118	C₄²	—	106	3	no	—
119	C₄²	C₂²×D₅₃	1	3	yes	—
120	C₂³	—	2	3	no	—
121	Q₈	—	53	2	no	—
122	Q₈	—	106	1	no	—
123	Q₈	—	424	3	no	—
124	Q₈	#11719923	1	4	yes	complemented
125	D₄	—	2	3	no	—
126	D₄	—	2	3	no	—
127	D₄	—	4	6	no	—
128	D₄	—	8	3	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	—	106	1	no	—
132	C₂×C₄	—	106	3	no	—
133	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
134	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
135	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
136	C₈	—	106	3	no	—
137	C₈	—	106	1	no	—
138	C₂²	—	4	3	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	3	no	—
141	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	106	2	no	—
145	C₄	—	212	3	no	—
146	C₄	C₂×#11719923	1	6	yes	—
147	C₄	#41×D₅₃	1	1	yes	characteristic
148	C₄	#13133264	1	1	yes	characteristic
149	C₂	—	8	3	no	—
150	C₂	#16811280	1	1	yes	characteristic
151	C₂	C₂²×#11719923	1	1	yes	characteristic
152	C₂	#16811426	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

#16810969

C₂

yes

maximal, complemented

#16810913

C₂

yes

maximal, complemented

#16810820

C₂

yes

maximal, complemented

Q₈×#88322

C₂

yes

maximal, characteristic, complemented

C₄×#11719923

C₂

yes

maximal, complemented

C₅₃×#550

C₂

yes

maximal, characteristic

C₂×#11719923

C₂²

yes

—

#13133217

C₂²

yes

—

#13133216

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133148

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133140

C₂²

yes

characteristic

#13133138

C₂²

yes

—

#13133138

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂×#41×C₅₃

—

#127×C₅₃

C₂²

yes

—

#121×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

Q₈×C₅₃

SD₁₆

yes

complemented

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#3362

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#548

yes

characteristic

C₂×C₅₃

C₂²×SD₁₆

yes

characteristic

C₂×C₅₃

#576

yes

characteristic

Q₈²

—

#550

D₅₃

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₅₃

#3362

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

#127

C₂×D₅₃

yes

—

#121

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

C₂×D₅₃

yes

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

#102

—

100

C₄×C₈

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

106

—

104

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₅₃

yes

—

108

C₂²×C₄

—

109

SD₁₆

—

106

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

C₂²⋊D₅₃

yes

—

116

C₄⋊C₄

C₂²×D₅₃

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

—

106

—

119

C₄²

C₂²×D₅₃

yes

—

120

C₂³

—

121

Q₈

—

122

Q₈

—

106

—

123

Q₈

—

424

—

124

Q₈

#11719923

yes

complemented

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

134

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

135

C₂×C₄

C₂³×D₅₃

yes

characteristic

136

C₈

—

106

—

137

C₈

—

106

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

106

—

145

C₄

—

212

—

146

C₄

C₂×#11719923

yes

—

147

C₄

#41×D₅₃

yes

characteristic

148

C₄

#13133264

yes

characteristic

149

C₂

—

150

C₂

#16811280

yes

characteristic

151

C₂

C₂²×#11719923

yes

characteristic

152

C₂

#16811426

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#550	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#550	D₅₃
hypocenter	C₅₃	#3362
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#550

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×Dic₁₀₆	C₂	semidirect
2. #16810969	C₂	semidirect
3. #16810913	C₂	semidirect
4. #16810820	C₂	semidirect
5. Q₈×#88322	C₂	semidirect
6. C₄×#11719923	C₂	semidirect
44. Q₈×C₅₃	SD₁₆	semidirect
58. Dic₁₀₆	#41	semidirect
84. C₅₃	#3362	semidirect
124. Q₈	#11719923	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₀₆

C₂

semidirect

2. #16810969

C₂

semidirect

3. #16810913

C₂

semidirect

4. #16810820

C₂

semidirect

5. Q₈×#88322

C₂

semidirect

6. C₄×#11719923

C₂

semidirect

44. Q₈×C₅₃

SD₁₆

semidirect

58. Dic₁₀₆

#41

semidirect

84. C₅₃

#3362

semidirect

124. Q₈

#11719923

semidirect

Subgroups — #20287382

Conjugacy classes of subgroups

Complements