Group #20273162 — subgroups

Subgroups — #20273162

Order 6784.

252 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 93 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	Q₈⋊#88322	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	1	no	—
6	C₄⋊#88322	C₂²	1	2	yes	—
7	C₂²⋊#88322	—	2	1	no	—
8	#13133128	—	2	1	no	—
9	#13133128	—	2	1	no	—
10	C₄×#88322	C₂²	1	2	yes	—
11	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₅₃	—	2	1	no	—
13	C₄×Q₈×C₅₃	—	2	1	no	—
14	C₄×Q₈×C₅₃	C₄	1	2	yes	—
15	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	—	2	1	no	—
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	—	2	4	no	—
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	—	2	1	no	—
24	C₂²×C₄×C₅₃	—	2	1	no	—
25	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₄²×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#11719917	—	2	4	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₅₃	—	2	4	no	—
43	Q₈×C₅₃	—	2	2	no	—
44	Q₈×C₅₃	M₁₆	1	4	yes	complemented
45	C₂×C₄×C₅₃	—	2	4	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	4	2	no	—
52	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	#88322	—	2	4	no	—
59	#88322	—	4	2	no	—
60	C₂²×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	#104	1	2	yes	—
63	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₅₃	—	2	4	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
71	C₄×C₅₃	#137	1	2	yes	—
72	#1491	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	—	2	2	no	—
74	C₂×C₅₃	#407	1	1	yes	characteristic
75	C₂×C₅₃	#419	1	1	yes	characteristic
76	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
77	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
78	#432	—	53	1	no	—
79	C₄×M₁₆	—	53	1	no	—
80	Q₈⋊C₈	—	53	4	no	—
81	C₅₃	#1491	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic
83	C₂×M₁₆	—	106	1	no	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	—	2	1	no	—
86	C₄×Q₈	Dic₅₃	1	2	yes	—
87	C₄×Q₈	C₂×D₅₃	1	2	yes	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
90	C₂×C₄²	—	2	1	no	—
91	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
92	C₄⋊C₈	—	53	2	no	—
93	C₂²⋊C₈	—	106	1	no	—
94	#97	—	106	1	no	—
95	#97	—	106	1	no	—
96	C₄×C₈	—	53	2	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
105	M₁₆	—	106	4	no	—
106	C₂×C₈	—	106	2	no	—
107	C₂×C₈	—	106	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
112	C₄²	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
115	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₄²	C₂²×D₅₃	1	1	yes	characteristic
118	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	2	2	no	—
121	Q₈	#11719917	1	4	yes	complemented
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
135	C₈	—	106	4	no	—
136	C₈	—	212	2	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	#13133144	1	2	yes	—
140	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	4	2	no	—
147	C₄	C₂×#11719917	1	2	yes	—
148	C₄	#13133248	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#16810587	1	1	yes	characteristic
151	C₂	#16810632	1	1	yes	characteristic
152	C₂	C₂×#13133144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic

C₄×#11719917

C₂

yes

maximal, characteristic

Q₈⋊#88322

C₂

yes

maximal, complemented

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₄⋊#88322

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂²×Q₈×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₄

yes

—

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

M₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#104

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

#137

yes

—

#1491

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

D₅₃

yes

characteristic

#432

—

C₄×M₁₆

—

Q₈⋊C₈

—

C₅₃

#1491

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

characteristic

C₂×M₁₆

—

106

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

Dic₅₃

yes

—

C₄×Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

#97

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

103

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

104

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

105

M₁₆

—

106

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂×Dic₅₃

yes

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×Dic₅₃

yes

characteristic

115

C₄²

C₂²⋊D₅₃

yes

characteristic

116

C₄²

C₂²⋊D₅₃

yes

characteristic

117

C₄²

C₂²×D₅₃

yes

characteristic

118

C₂³

C₂²⋊Dic₅₃

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

#11719917

yes

complemented

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

130

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

131

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

132

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

135

C₈

—

106

—

136

C₈

—

212

—

137

C₂²

—

138

C₂²

—

139

C₂²

#13133144

yes

—

140

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂×#11719917

yes

—

148

C₄

#13133248

yes

—

149

C₂

—

150

C₂

#16810587

yes

characteristic

151

C₂

#16810632

yes

characteristic

152

C₂

C₂×#13133144

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#1491
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×Q₈

D₅₃

hypocenter

C₅₃

#1491

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
3. Q₈⋊#88322	C₂	semidirect
44. Q₈×C₅₃	M₁₆	semidirect
81. C₅₃	#1491	semidirect
121. Q₈	#11719917	semidirect

H ≅ G/N

type

3. Q₈⋊#88322

C₂

semidirect

44. Q₈×C₅₃

M₁₆

semidirect

81. C₅₃

#1491

semidirect

121. Q₈

#11719917

semidirect

Subgroups — #20273162

Conjugacy classes of subgroups

Complements