Group #20275705 — subgroups

Subgroups — #20275705

Order 6784.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	#16810647	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#16810392	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	2	yes	—
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₄⋊#88322	—	2	1	no	—
8	C₄⋊#88322	—	2	1	no	—
9	#13133128	—	2	1	no	—
10	C₄×#88322	—	2	1	no	—
11	C₂²×Q₈×C₅₃	C₄	1	2	yes	—
12	C₄⋊Q₈×C₅₃	—	2	1	no	—
13	C₄⋊Q₈×C₅₃	—	2	1	no	—
14	C₄⋊Q₈×C₅₃	—	2	1	no	—
15	C₄⋊Q₈×C₅₃	—	2	1	no	—
16	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	—	2	2	no	—
20	C₂×Q₈×C₅₃	—	2	2	no	—
21	C₂×Q₈×C₅₃	—	4	4	no	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₄²×C₅₃	D₄	1	1	yes	characteristic
34	#11719917	—	2	2	no	—
35	#11719917	—	2	4	no	—
36	#11719917	Q₈	1	4	yes	complemented
37	C₂×#88322	—	2	2	no	—
38	C₂×#88322	—	2	2	no	—
39	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₅₃	—	4	4	no	—
41	Q₈×C₅₃	—	4	4	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	—	4	4	no	—
46	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₅₃	#41	1	2	yes	—
54	#88322	—	2	4	no	—
55	#88322	—	4	2	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	4	4	no	—
63	C₄×C₅₃	#100	1	2	yes	—
64	C₄×C₅₃	#101	1	2	yes	—
65	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₅₃	#124	1	1	yes	characteristic
67	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
68	#1997	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	2	no	—
70	C₂×C₅₃	#391	1	1	yes	characteristic
71	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
72	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
73	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
74	#423	—	53	1	no	—
75	C₄×M₁₆	—	53	1	no	—
76	#343	—	53	4	no	—
77	C₅₃	#1997	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₈	Dic₅₃	1	2	yes	—
79	C₂×M₁₆	—	53	2	no	—
80	C₂×M₁₆	—	53	2	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	—	2	1	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
87	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
88	C₄⋊C₈	—	106	1	no	—
89	C₄⋊C₈	—	106	1	no	—
90	#97	—	106	1	no	—
91	C₄×C₈	—	106	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	4	no	—
95	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	M₁₆	—	53	4	no	—
100	M₁₆	—	106	2	no	—
101	M₁₆	—	106	4	no	—
102	C₂×C₈	—	106	2	no	—
103	C₂×C₈	—	106	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
109	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
111	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
112	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	Q₈	—	4	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
123	C₂×C₄	Q₈×D₅₃	1	2	yes	—
124	C₂×C₄	#11719949	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
127	C₈	—	106	4	no	—
128	C₈	—	212	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
132	C₂²	Q₈×Dic₅₃	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	4	no	—
136	C₄	#13133134	1	2	yes	—
137	C₄	#13133136	1	2	yes	—
138	C₄	#13133193	1	2	yes	—
139	C₄	#13133206	1	1	yes	characteristic
140	C₄	#13133215	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#16810530	1	1	yes	characteristic
143	C₂	C₂×#13133134	1	1	yes	characteristic
144	C₂	C₂×#13133136	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal, characteristic

C₄×#11719917

C₂

yes

maximal, characteristic

#16810392

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₅₃

C₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

—

Q₈

yes

complemented

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×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

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#100

yes

—

C₄×C₅₃

#101

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#124

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

#1997

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

#423

—

C₄×M₁₆

—

#343

—

C₅₃

#1997

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

100

M₁₆

—

106

—

101

M₁₆

—

106

—

102

C₂×C₈

—

106

—

103

C₂×C₈

—

106

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₂²⋊D₅₃

yes

characteristic

109

C₄²

C₂²⋊D₅₃

yes

characteristic

110

C₄²

C₂²⋊D₅₃

yes

characteristic

111

C₄²

C₂²⋊D₅₃

yes

characteristic

112

C₂³

C₂²×Dic₅₃

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

120

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

121

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

123

C₂×C₄

Q₈×D₅₃

yes

—

124

C₂×C₄

#11719949

yes

—

125

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

127

C₈

—

106

—

128

C₈

—

212

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

132

C₂²

Q₈×Dic₅₃

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#13133134

yes

—

137

C₄

#13133136

yes

—

138

C₄

#13133193

yes

—

139

C₄

#13133206

yes

characteristic

140

C₄

#13133215

yes

characteristic

141

C₂

—

142

C₂

#16810530

yes

characteristic

143

C₂

C₂×#13133134

yes

characteristic

144

C₂

C₂×#13133136

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#1997
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#1997

LCS

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
36. #11719917	Q₈	semidirect
77. C₅₃	#1997	semidirect

H ≅ G/N

type

36. #11719917

Q₈

semidirect

77. C₅₃

#1997

semidirect

Subgroups — #20275705

Conjugacy classes of subgroups

Complements