Group #28309906 — subgroups

Subgroups — #28309906

Order 13568.

216 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 84 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	#20290275	C₂	1	2	yes	maximal
2	#20289741	C₂	1	2	yes	maximal, complemented
3	#20289740	C₂	1	1	yes	maximal, characteristic, complemented
4	#20289739	C₂	1	1	yes	maximal, characteristic, complemented
5	M₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811644	—	2	2	no	—
7	#16811487	—	2	2	no	—
8	#16811484	C₂²	1	2	yes	—
9	#16811484	C₂²	1	2	yes	—
10	C₂×#13133157	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810398	C₂²	1	1	yes	characteristic
13	#16810398	C₂²	1	1	yes	characteristic
14	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
15	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
16	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
17	C₄⋊#88324	—	2	2	no	—
18	C₄⋊#88324	—	2	2	no	—
19	C₂²⋊#88324	—	4	2	no	—
20	#13133272	—	4	2	no	—
21	C₂²×#88322	Q₈	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
23	#13133157	—	2	2	no	—
24	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
25	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₅₃	D₄	1	1	yes	characteristic
28	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
29	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	M₃₂×C₅₃	—	2	2	no	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	SD₁₆	1	2	yes	—
36	C₂×#88324	—	4	2	no	—
37	C₂×#88324	—	4	2	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	SD₁₆	1	2	yes	—
44	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
47	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₅₃	—	2	1	no	—
49	C₁₆×C₅₃	M₁₆	1	2	yes	—
50	#11719908	—	2	1	no	—
51	#11719908	M₁₆	1	2	yes	—
52	C₂³×C₅₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×Dic₅₃	—	4	2	no	—
58	C₂×Dic₅₃	#106	1	2	yes	—
59	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
60	#88324	—	8	2	no	—
61	#88324	—	8	2	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
65	#88322	—	2	1	no	—
66	#88322	#108	1	2	yes	—
67	#12021	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	—	2	2	no	—
70	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	#423	1	1	yes	characteristic
73	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
74	Dic₅₃	—	2	2	no	—
75	Dic₅₃	—	2	2	no	—
76	C₄×M₃₂	—	53	1	no	—
77	#1568	—	53	2	no	—
78	#1368	—	53	1	no	—
79	#1368	—	53	1	no	—
80	#1368	—	53	2	no	—
81	C₂×C₅₃	—	4	1	no	—
82	C₂×C₅₃	#1566	1	1	yes	characteristic
83	C₂×C₅₃	#2155	1	2	yes	—
84	C₂×M₃₂	—	53	1	no	—
85	C₂×M₃₂	#6157	1	1	yes	characteristic
86	#432	—	106	2	no	—
87	C₂×C₄×C₈	—	53	1	no	—
88	#346	—	53	1	no	—
89	#346	—	53	1	no	—
90	C₄×C₁₆	—	53	1	no	—
91	C₄×C₁₆	—	53	1	no	—
92	#335	—	106	2	no	—
93	#334	—	53	2	no	—
94	#334	—	53	2	no	—
95	C₅₃	#12021	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	53	1	no	—
97	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
98	C₂×C₄²	—	53	1	no	—
99	M₃₂	—	2	2	no	—
100	M₃₂	—	106	2	no	—
101	C₂×C₁₆	—	53	1	no	—
102	C₂×C₁₆	—	53	1	no	—
103	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
104	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
105	C₄⋊C₈	—	106	2	no	—
106	C₄⋊C₈	—	106	2	no	—
107	C₂²⋊C₈	—	212	2	no	—
108	#97	—	212	2	no	—
109	C₄×C₈	—	53	1	no	—
110	C₄×C₈	—	53	1	no	—
111	C₄×C₈	—	53	1	no	—
112	C₄×C₈	—	53	1	no	—
113	C₂²×C₄	—	53	2	no	—
114	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	53	1	no	—
118	C₂×C₈	—	53	1	no	—
119	C₂×C₈	—	106	2	no	—
120	C₂×C₈	—	212	2	no	—
121	C₂×C₈	—	212	2	no	—
122	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
123	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
124	C₄²	—	53	2	no	—
125	C₄²	—	53	1	no	—
126	C₄²	—	53	1	no	—
127	C₁₆	—	2	1	no	—
128	C₁₆	—	53	2	no	—
129	C₁₆	—	106	1	no	—
130	C₁₆	#11719940	1	2	yes	—
131	C₂³	#13133285	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	53	2	no	—
135	C₂×C₄	—	53	2	no	—
136	C₂×C₄	—	212	2	no	—
137	C₂×C₄	#13133285	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	53	2	no	—
141	C₈	—	106	1	no	—
142	C₈	—	424	2	no	—
143	C₈	—	424	2	no	—
144	C₈	C₂×#11719940	1	1	yes	characteristic
145	C₈	D₅₃⋊M₁₆	1	1	yes	characteristic
146	C₂²	—	2	1	no	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×#13133285	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	106	2	no	—
151	C₄	—	106	2	no	—
152	C₄	#16811626	1	1	yes	characteristic
153	C₄	D₅₃⋊#108	1	1	yes	characteristic
154	C₂	—	4	1	no	—
155	C₂	#20290269	1	1	yes	characteristic
156	C₂	#20291063	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290275

C₂

yes

maximal

#20289741

C₂

yes

maximal, complemented

#20289740

C₂

yes

maximal, characteristic, complemented

#20289739

C₂

yes

maximal, characteristic, complemented

M₃₂×Dic₅₃

C₂

yes

maximal, characteristic

#16811644

—

#16811487

—

#16811484

C₂²

yes

—

#16811484

C₂²

yes

—

C₂×#13133157

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810398

C₂²

yes

characteristic

#16810398

C₂²

yes

characteristic

C₁₆×Dic₅₃

C₄

yes

characteristic

C₁₆×Dic₅₃

C₄

yes

characteristic

C₂×M₃₂×C₅₃

C₄

yes

—

—

—

—

C₂²×#88322

Q₈

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

characteristic

#13133157

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₈×Dic₅₃

C₂³

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₅₃

SD₁₆

yes

—

C₂×#88324

—

C₂×#88324

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

—

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×Q₈

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

M₁₆

yes

—

#11719908

—

#11719908

M₁₆

yes

—

C₂³×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#106

yes

—

C₂×Dic₅₃

C₂×SD₁₆

yes

—

#88324

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

C₂×M₁₆

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

—

yes

—

#12021

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#106

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

C₄×M₃₂

—

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1566

yes

characteristic

C₂×C₅₃

#2155

yes

—

C₂×M₃₂

—

C₂×M₃₂

#6157

yes

characteristic

#432

—

106

—

C₂×C₄×C₈

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

106

—

#334

—

#334

—

C₅₃

#12021

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

characteristic

C₂×C₄²

—

M₃₂

—

100

M₃₂

—

106

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂×#6157

yes

characteristic

104

C₂×C₁₆

C₂×#6157

yes

characteristic

105

C₄⋊C₈

—

106

—

106

C₄⋊C₈

—

106

—

107

C₂²⋊C₈

—

212

—

108

#97

—

212

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₂²×C₄

—

114

C₂²×C₄

C₄⋊#6157

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

106

—

120

C₂×C₈

—

212

—

121

C₂×C₈

—

212

—

122

C₂×C₈

C₄⋊#6157

yes

characteristic

123

C₂×C₈

C₂²×#6157

yes

characteristic

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

106

—

130

C₁₆

#11719940

yes

—

131

C₂³

#13133285

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

212

—

137

C₂×C₄

#13133285

yes

characteristic

138

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

106

—

142

C₈

—

424

—

143

C₈

—

424

—

144

C₈

C₂×#11719940

yes

characteristic

145

C₈

D₅₃⋊M₁₆

yes

characteristic

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×#13133285

yes

characteristic

149

C₄

—

150

C₄

—

106

—

151

C₄

—

106

—

152

C₄

#16811626

yes

characteristic

153

C₄

D₅₃⋊#108

yes

characteristic

154

C₂

—

155

C₂

#20290269

yes

characteristic

156

C₂

#20291063

yes

—

	subgroup	G/H
center	C₂²	C₂×#13133285
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂×M₃₂×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	#106
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	#12021
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×#13133285

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₂×M₃₂×C₅₃

C₄

C₂²×C₅₃

C₂³×C₅₃

C₈×Dic₅₃

C₂³

C₂×M₃₂

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. #20289741	C₂	semidirect
3. #20289740	C₂	semidirect
4. #20289739	C₂	semidirect
95. C₅₃	#12021	semidirect

H ≅ G/N

type

2. #20289741

C₂

semidirect

3. #20289740

C₂

semidirect

4. #20289739

C₂

semidirect

95. C₅₃

#12021

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20290275	2
#20289741	2
#20289740	2
#20289739	2
M₃₂×Dic₅₃	2
#12021	53

subgroup

index

M₃₂×Dic₅₃

Subgroups — #28309906

Conjugacy classes of subgroups

Complements

Maximal subgroups