Finite Group Databasebeta

← #27287181

Subgroups — #27287181

Order 13568.

454 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 97 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#20283093C211yesmaximal, characteristic, complemented
2#20275543C211yesmaximal, characteristic
3#20273276C212yesmaximal
4#20272574C212yesmaximal, complemented
5C53×#1276C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7C2×#13133217—21no—
8#16810647C2211yescharacteristic
9C2×#13133145—21no—
10C2×#13133145—21no—
11#16810529—24no—
12#16810489—21no—
13#16810489C2212yes—
14#16810358—24no—
15C53×#431C2212yes—
16C43×C53C2211yescharacteristic
17C2×#11719945—21no—
18C22×Dic106—22no—
19C2×#11719917—21no—
20#13133217—42no—
21#13133217—42no—
22#13133217Q811yescharacteristic, complemented
23#13133214Q811yescharacteristic, complemented
24#13133205—21no—
25#13133191—22no—
26C4×D212—22no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—22no—
29C4⋊#88322—22no—
30#13133145—42no—
31#13133145—42no—
32#13133124—24no—
33#13133124—22no—
34#13133124D414yes—
35C2×C42×C53—22no—
36C2×C42×C53D411yescharacteristic
37C2×C42×C53D411yescharacteristic
38C2×C42×C53C2311yescharacteristic
39C4⋊C8×C53—22no—
40C22⋊C8×C53—22no—
41C4×C8×C53—22no—
42C22×C4×C53—21no—
43C22×C4×C53—21no—
44C22×C4×C53C2×D412yes—
45C22×C4×C53C2×D411yescharacteristic
46C2×C22⋊D53—41no—
47C22×Dic53—24no—
48#11719945—41no—
49#11719945—41no—
50#11719945—41no—
51#11719945—41no—
52C2×D212—21no—
53C2×D212—22no—
54C2×C4×D53—41no—
55C2×Dic106—21no—
56C2×Dic106—44no—
57C2×Dic106—48no—
58C2×C8×C53—42no—
59C42×C53—21no—
60C42×C53—21no—
61C42×C53—44no—
62C42×C53D812yes—
63C42×C53SD1612yes—
64C42×C53C2×Q811yescharacteristic
65C42×C53#4112yes—
66C42×C53#4111yescharacteristic
67#11719921—42no—
68C4⋊Dic53—22no—
69C4⋊Dic53—44no—
70C4⋊Dic53—42no—
71C4⋊Dic53—42no—
72#11719917—41no—
73#11719917—41no—
74C2×#88322—41no—
75C23×C53#12011yescharacteristic
76C22×D53—41no—
77C2×C4×C53—24no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53C22⋊Q812yes—
83C2×C4×C53C22⋊Q811yescharacteristic
84C2×C4×C53#12412yes—
85C2×C4×C53#12411yescharacteristic
86C2×C4×C53C2×D811yescharacteristic
87C2×C4×C53C2×SD1611yescharacteristic
88C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C53#13711yescharacteristic
90C22⋊D53—81no—
91C22⋊D53—81no—
92C2×Dic53—48no—
93C2×Dic53—41no—
94C2×Dic53—44no—
95D212—41no—
96D212—41no—
97D212—82no—
98C4×D53—81no—
99C4×D53—81no—
100Dic106—41no—
101Dic106—41no—
102Dic106—88no—
103C8×C53—82no—
104#88322—81no—
105#10362—531nomaximal, Sylow
106C22×C53—22no—
107C22×C53#39311yescharacteristic
108C22×C53#44711yescharacteristic
109C22×C53#44811yescharacteristic
110C2×D53—81no—
111C2×D53—82no—
112C4×C53—24no—
113C4×C53—22no—
114C4×C53—22no—
115C4×C53—42no—
116C4×C53#47411yescharacteristic
117C4×C53#47611yescharacteristic
118C4×C53#48611yescharacteristic
119C4×C53#48911yescharacteristic
120Dic53—84no—
121Dic53—81no—
122#2862—531no—
123#1962—531no—
124#1517—532no—
125#1286—532no—
126#1276D5311yescharacteristic, complemented
127C2×C53—21no—
128C2×C53—21no—
129C2×C53#202311yescharacteristic
130C2×C53#203112yes—
131D53—161no—
132#532—531no—
133C2×C4⋊Q8—1061no—
134#431C2×D5312yes—
135#423—531no—
136C2×#104—1061no—
137C2×#104—1061no—
138#391—1064no—
139#382—532no—
140#382—1061no—
141C43C2×D5311yescharacteristic
142#327—1064no—
143C53#1036211yescharacteristic, complemented, Sylow
144C2×#41—1061no—
145C22×Q8—1062no—
146C2×M16—1061no—
147C4⋊Q8—531no—
148C4⋊Q8—2122no—
149C4⋊Q8—2122no—
150#127—531no—
151#124—1061no—
152#121—1062no—
153C4×D4—1062no—
154C2×C4⋊C4—1062no—
155C2×C4⋊C4—1062no—
156C2×C42—22no—
157C2×C42D21211yescharacteristic
158C2×C42D21211yescharacteristic
159C2×C42C22×D5311yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—1062no—
162#104—2122no—
163#104—2122no—
164C22⋊C8—22no—
165C4×C8—22no—
166#95—534no—
167#95—1064no—
168#95—1062no—
169#41—2121no—
170#41—2121no—
171#41—2121no—
172#41—2121no—
173C2×Q8—1061no—
174C2×Q8—2124no—
175C2×Q8—2128no—
176C2×D4—1062no—
177C2×D4—1061no—
178C2×D4—2121no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—1064no—
182C22×C4—2121no—
183C22×C4C2×D21211yescharacteristic
184C22×C4D4×D5312yes—
185M16—2121no—
186M16—2121no—
187C2×C8—42no—
188C2×C8—2121no—
189C4⋊C4—1062no—
190C4⋊C4—2124no—
191C4⋊C4—2122no—
192C4⋊C4—2122no—
193C22⋊C4—2122no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42#1171991312yes—
198C42D42412yes—
199C42#1171994512yes—
200C42#1171994711yescharacteristic
201C42Q8×D5311yescharacteristic
202C23—2121no—
203C23#1313318711yescharacteristic
204Q8—4248no—
205Q8—2121no—
206Q8—2121no—
207D4—4242no—
208D4—4241no—
209D4—4241no—
210D4—2121no—
211D4—2121no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—4241no—
218C2×C4—4241no—
219C2×C4—2128no—
220C2×C4—2124no—
221C2×C4—2121no—
222C2×C4#1313319512yes—
223C2×C4#1313319711yescharacteristic
224C2×C4#1313320312yes—
225C2×C4#1313320511yescharacteristic
226C2×C4C2×D42411yescharacteristic
227C2×C4C2×#1171991311yescharacteristic
228C2×C4#1313324511yescharacteristic
229C2×C4#1313325011yescharacteristic
230C8—82no—
231C8—4241no—
232C22—22no—
233C22—4241no—
234C22—4242no—
235C22#1681053611yescharacteristic
236C22#1681075911yescharacteristic
237C22#1681076411yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—4244no—
243C4—4241no—
244C4#1681090411yescharacteristic
245C4#1681091411yescharacteristic
246C4#1681097011yescharacteristic
247C4#1681098311yescharacteristic
248C2—21no—
249C2—21no—
250C2—8481no—
251C2#2027585211yescharacteristic
252C2#2027588812yes—
subgroupG/H
centerC22#16810536
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#393
prime radical#16811173C22
hypercenter#1276D53
hypocenterC53#10362
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #20283093C2semidirect
4. #20272574C2semidirect
5. C53×#1276C2semidirect
22. #13133217Q8semidirect
23. #13133214Q8semidirect
126. #1276D53semidirect
143. C53#10362semidirect

Maximal subgroups

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

subgroupindex
#202830932
#202755432
#202732762
#202725742
C53×#12762
#1036253