Finite Group Databasebeta

← #20276722

Subgroups — #20276722

Order 6784.

260 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810812C211yesmaximal, characteristic, complemented
2#16810809C211yesmaximal, characteristic, complemented
3#16810790C211yesmaximal, characteristic, complemented
4#16810416C211yesmaximal, characteristic, complemented
5#16810413C211yesmaximal, characteristic, complemented
6#16810410C211yesmaximal, characteristic, complemented
7C53×#560C211yesmaximal, characteristic, complemented
8#13133262C2211yescharacteristic
9#13133243—21no—
10#13133200—21no—
11#13133186—21no—
12#13133145—21no—
13#13133134C2211yescharacteristic
14#13133132C2211yescharacteristic
15#13133132C2211yescharacteristic
16#13133131—21no—
17#13133131—21no—
18#127×C53C2211yescharacteristic
19#126×C53—21no—
20#123×C53—21no—
21#123×C53C2211yescharacteristic
22#121×C53—21no—
23#121×C53—21no—
24#121×C53—21no—
25#120×C53—21no—
26#120×C53C2211yescharacteristic
27C24×C53D411yescharacteristic
28C2×D4×C53—21no—
29C2×D4×C53—21no—
30C2×D4×C53—21no—
31C2×D4×C53—21no—
32C2×D4×C53—21no—
33C2×D4×C53D411yescharacteristic, complemented
34C2×D4×C53C2311yescharacteristic
35C22×C4×C53—21no—
36C22×C4×C53D411yescharacteristic, complemented
37C2×C22⋊D53—21no—
38C2×C22⋊D53—21no—
39C2×C22⋊D53—41no—
40#11719947—41no—
41D4×D53—41no—
42#11719945—21no—
43C4⋊C4×C53—21no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C22⋊C4×C53—21no—
50C22⋊C4×C53—21no—
51C22⋊C4×C53D411yescharacteristic
52C22⋊C4×C53D411yescharacteristic, complemented
53C42×C53D411yescharacteristic, complemented
54C22⋊Dic53—21no—
55C22⋊Dic53—21no—
56C22⋊Dic53—41no—
57#11719923—41no—
58#11719922—41no—
59#11719921—41no—
60#11719919—41no—
61#11719917—21no—
62C23×C53—21no—
63C23×C53—21no—
64C23×C53—41no—
65C23×C53—81no—
66C23×C53C2×D411yescharacteristic
67C23×C53C2×D411yescharacteristic
68C22×D53—21no—
69C22×D53—21no—
70D4×C53—41no—
71D4×C53—41no—
72D4×C53—81no—
73D4×C53—81no—
74D4×C53—81no—
75D4×C53—81no—
76D4×C53—81no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—41no—
82C2×C4×C53—81no—
83C2×C4×C53C2×D411yescharacteristic
84C22⋊D53—41no—
85C22⋊D53—41no—
86C22⋊D53—41no—
87C22⋊D53—82no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—41no—
91C2×Dic53—41no—
92D212—21no—
93C4×D53—41no—
94Dic106—21no—
95#88322—41no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—42no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—81no—
108C22×C53#12011yescharacteristic
109C2×D53—41no—
110C2×D53—81no—
111C2×D53—81no—
112C4×C53—21no—
113C4×C53—41no—
114C4×C53—41no—
115C4×C53—41no—
116C4×C53—41no—
117Dic53—41no—
118Dic53—81no—
119Dic53—81no—
120#2200—531nomaximal, Sylow
121C2×C53—21no—
122C2×C53—41no—
123C2×C53—41no—
124C2×C53—81no—
125C2×C53—81no—
126C2×C53B(4,𝔽2)11yescharacteristic
127D53—81no—
128#560D5311yescharacteristic, complemented
129#458—531no—
130B(4,𝔽2)—531no—
131#453—531no—
132#353—531no—
133#352—531no—
134AΣL(1,𝔽16)—531no—
135C53#220011yescharacteristic, complemented, Sylow
136H(5,𝔽2)—531no—
137AGL(1,ℤ/8)—1061no—
138#127C2×D5311yescharacteristic
139#126—21no—
140#123—21no—
141#123—1061no—
142#123C2×D5311yescharacteristic
143#121—21no—
144#121—21no—
145#121—21no—
146#120—21no—
147#120—1061no—
148#120C2×D5311yescharacteristic
149#104—1061no—
150#100—531no—
151#99—531no—
152#99—531no—
153#99—1061no—
154#99—1061no—
155C24C22⋊D5311yescharacteristic
156#41—1061no—
157#41—2121no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—1061no—
164C2×D4—1061no—
165C2×D4—2121no—
166C2×D4—2121no—
167C2×D4C22⋊D5311yescharacteristic, complemented
168C2×D4C22×D5311yescharacteristic
169C22×C4—21no—
170C22×C4C22⋊D5311yescharacteristic, complemented
171SD16—2121no—
172D8—2121no—
173M16—1061no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—2121no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—1061no—
184C22⋊C4—1061no—
185C22⋊C4—2121no—
186C22⋊C4—2121no—
187C22⋊C4C22⋊D5311yescharacteristic
188C22⋊C4C22⋊D5311yescharacteristic, complemented
189C42C22⋊D5311yescharacteristic, complemented
190C23—21no—
191C23—21no—
192C23—41no—
193C23—81no—
194C23—1061no—
195C23—1061no—
196C23C2×C22⋊D5311yescharacteristic
197C23C2×C22⋊D5311yescharacteristic
198Q8—1061no—
199D4—41no—
200D4—41no—
201D4—81no—
202D4—81no—
203D4—81no—
204D4—81no—
205D4—81no—
206D4—1061no—
207D4—4242no—
208D4—2121no—
209D4—2121no—
210D4—2121no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—81no—
217C2×C4—1061no—
218C2×C4—1061no—
219C2×C4—2121no—
220C2×C4—2121no—
221C2×C4—2121no—
222C2×C4C2×C22⋊D5311yescharacteristic
223C8—2121no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—81no—
236C22—4241no—
237C22—4241no—
238C22—2121no—
239C22#1313318611yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—4241no—
246C4—4241no—
247C4—2121no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—4241no—
254C2#1681081011yescharacteristic
subgroupG/H
centerC2#16810810
commutatorC2×D4×C53C23
FrattiniC2×D4C22×D53
FittingC53×#560C2
socleC2×C53B(4,𝔽2)
hypercenter#560D53
hypocenterC53#2200
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #560

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #16810812C2semidirect
2. #16810809C2semidirect
3. #16810790C2semidirect
4. #16810416C2semidirect
5. #16810413C2semidirect
6. #16810410C2semidirect
7. C53×#560C2semidirect
33. C2×D4×C53D4semidirect
36. C22×C4×C53D4semidirect
52. C22⋊C4×C53D4semidirect
53. C42×C53D4semidirect
128. #560D53semidirect
135. C53#2200semidirect
167. C2×D4C22⋊D53semidirect
170. C22×C4C22⋊D53semidirect
188. C22⋊C4C22⋊D53semidirect
189. C42C22⋊D53semidirect