Group #20274493 — subgroups

Subgroups — #20274493

Order 6784.

360 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂³×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133128	C₂	1	2	yes	maximal, complemented
4	#16810378	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	—	2	4	no	—
7	C₂²×#88322	C₄	1	4	yes	complemented
8	C₂²×#88322	C₂²	1	2	yes	—
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂²⋊#88322	C₄	1	4	yes	complemented
12	#13133128	—	2	4	no	—
13	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
16	#95×C₅₃	C₄	1	2	yes	—
17	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²⋊C₄×C₅₃	—	2	4	no	—
26	C₄²×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	—	2	2	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	—	2	8	no	—
32	C₂×#88322	C₂×C₄	1	4	yes	—
33	C₂×#88322	C₂×C₄	1	2	yes	—
34	C₂×#88322	C₂×C₄	1	2	yes	—
35	C₂×#88322	D₄	1	4	yes	—
36	C₂³×C₅₃	—	2	2	no	—
37	C₂³×C₅₃	—	2	2	no	—
38	C₂³×C₅₃	C₄²	1	2	yes	—
39	C₂³×C₅₃	M₁₆	1	4	yes	—
40	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	2	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	—	2	4	no	—
46	C₂×C₄×C₅₃	—	2	4	no	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	C₄²	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₃	#41	1	2	yes	—
53	#88322	—	2	2	no	—
54	#88322	—	2	4	no	—
55	#88322	C₂²⋊C₄	1	4	yes	complemented
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	—	2	2	no	—
58	C₂²×C₅₃	—	2	8	no	—
59	C₂²×C₅₃	#97	1	4	yes	complemented
60	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
61	C₂²×C₅₃	C₄×D₄	1	2	yes	—
62	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
63	C₂²×C₅₃	#131	1	2	yes	—
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	—	4	4	no	—
66	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₅₃	#117	1	1	yes	characteristic
68	C₄×C₅₃	C₄×D₄	1	2	yes	—
69	#1752	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	4	no	—
71	C₂×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
73	C₂×C₅₃	#405	1	1	yes	characteristic
74	C₂×C₅₃	#435	1	4	yes	—
75	C₂³×C₈	—	53	1	no	—
76	C₂×C₂²⋊C₈	—	53	1	no	—
77	C₂×#97	—	53	2	no	—
78	C₄×C₂²⋊C₄	D₅₃	1	1	yes	characteristic
79	#336	—	53	2	no	—
80	C₅₃	#1752	1	1	yes	characteristic, complemented, Sylow
81	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
82	C₂²×C₈	—	53	1	no	—
83	C₂²×C₈	—	53	4	no	—
84	C₂²×C₈	—	53	1	no	—
85	C₂²×C₈	—	53	2	no	—
86	C₂²×C₈	—	106	4	no	—
87	C₂×C₂²⋊C₄	Dic₅₃	1	2	yes	—
88	C₂×C₄²	C₂×D₅₃	1	2	yes	—
89	C₂²⋊C₈	—	53	4	no	—
90	#97	—	106	4	no	—
91	#95	Dic₅₃	1	2	yes	—
92	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₄×D₅₃	1	2	yes	—
96	C₂²×C₄	C₄×D₅₃	1	2	yes	—
97	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
98	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	C₂×C₈	—	53	2	no	—
101	C₂×C₈	—	53	4	no	—
102	C₂×C₈	—	53	4	no	—
103	C₂×C₈	—	53	2	no	—
104	C₂×C₈	—	106	8	no	—
105	C₂×C₈	—	106	2	no	—
106	C₂×C₈	—	106	2	no	—
107	C₂×C₈	—	106	2	no	—
108	C₂²⋊C₄	—	2	4	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	#11719917	1	4	yes	—
114	C₂³	C₄×Dic₅₃	1	2	yes	—
115	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
126	C₂×C₄	D₄×D₅₃	1	2	yes	—
127	C₂×C₄	#11719947	1	2	yes	—
128	C₈	—	53	4	no	—
129	C₈	—	106	2	no	—
130	C₈	—	106	4	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	8	no	—
133	C₂²	—	2	2	no	—
134	C₂²	#13133128	1	4	yes	complemented
135	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
136	C₂²	D₄×Dic₅₃	1	2	yes	—
137	C₂²	C₂×#11719917	1	2	yes	—
138	C₂²	#13133224	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	4	4	no	—
141	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
142	C₄	#13133173	1	1	yes	characteristic
143	C₄	#13133177	1	2	yes	—
144	C₂	—	2	4	no	—
145	C₂	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
146	C₂	C₂×#13133128	1	1	yes	characteristic
147	C₂	#16810579	1	1	yes	characteristic
148	C₂	#16810695	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#88322

C₂

yes

maximal, characteristic

C₂×C₂²⋊#88322

C₂

yes

maximal, characteristic

C₂×#13133128

C₂

yes

maximal, complemented

#16810378

C₂

yes

maximal, complemented

C₄×C₂²⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₂²×#88322

C₄

yes

complemented

C₂²×#88322

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂²⋊#88322

C₄

yes

complemented

#13133128

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

#95×C₅₃

C₄

yes

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

—

—

—

—

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄²

yes

—

C₂³×C₅₃

M₁₆

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

#88322

—

#88322

—

#88322

C₂²⋊C₄

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#97

yes

complemented

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

—

#1752

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₅₃

C₂×#97

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂³×C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

—

C₄×C₂²⋊C₄

D₅₃

yes

characteristic

#336

—

C₅₃

#1752

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₂²⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂²⋊C₈

—

#97

—

106

—

#95

Dic₅₃

yes

—

C₂⁴

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

106

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₂³

—

112

C₂³

—

113

C₂³

#11719917

yes

—

114

C₂³

C₄×Dic₅₃

yes

—

115

C₂³

C₂²×Dic₅₃

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₄×Dic₅₃

yes

—

124

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

125

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

126

C₂×C₄

D₄×D₅₃

yes

—

127

C₂×C₄

#11719947

yes

—

128

C₈

—

129

C₈

—

106

—

130

C₈

—

106

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

#13133128

yes

complemented

135

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

136

C₂²

D₄×Dic₅₃

yes

—

137

C₂²

C₂×#11719917

yes

—

138

C₂²

#13133224

yes

—

139

C₄

—

140

C₄

—

141

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

142

C₄

#13133173

yes

characteristic

143

C₄

#13133177

yes

—

144

C₂

—

145

C₂

C₂²⋊C₄×Dic₅₃

yes

characteristic

146

C₂

C₂×#13133128

yes

characteristic

147

C₂

#16810579

yes

characteristic

148

C₂

#16810695

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Fitting	C₄×C₂²⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₄×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	#1752
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂×C₄²

Fitting

C₄×C₂²⋊C₄×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₄×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
3. C₂×#13133128	C₂	semidirect
4. #16810378	C₂	semidirect
7. C₂²×#88322	C₄	semidirect
11. C₂²⋊#88322	C₄	semidirect
55. #88322	C₂²⋊C₄	semidirect
59. C₂²×C₅₃	#97	semidirect
80. C₅₃	#1752	semidirect
134. C₂²	#13133128	semidirect

H ≅ G/N

type

3. C₂×#13133128

C₂

semidirect

4. #16810378

C₂

semidirect

7. C₂²×#88322

C₄

semidirect

11. C₂²⋊#88322

C₄

semidirect

55. #88322

C₂²⋊C₄

semidirect

59. C₂²×C₅₃

#97

semidirect

80. C₅₃

#1752

semidirect

134. C₂²

#13133128

semidirect

Subgroups — #20274493

Conjugacy classes of subgroups

Complements