Finite Group Databasebeta

← #27281532

Subgroups — #27281532

Order 13568.

618 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 111 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#20280095C211yesmaximal, characteristic
2#20273234C212yesmaximal
3#20273045C211yesmaximal, characteristic
4#20272601C212yesmaximal
5C53×#2274C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7#16810587C2212yes—
8#16810553C412yescomplemented
9#16810543C412yescomplemented
10#16810518—22no—
11#16810496—21no—
12#16810496—21no—
13C4×C22⋊Dic53C2211yescharacteristic
14#16810359—24no—
15#16810349—22no—
16C2×#117×C53—21no—
17C22×C4⋊C4×C53C2211yescharacteristic
18C4×C4⋊C4×C53—24no—
19C4×C22⋊C4×C53—21no—
20C4×C22⋊C4×C53C2212yes—
21C22×Dic106C2×C411yescharacteristic
22C2×#11719917—22no—
23#13133196—22no—
24#13133196—22no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53—21no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34#13133124—22no—
35#13133124—22no—
36#13133124C2×C412yes—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40#117×C53—28no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53C2×C412yes—
45C2×C4⋊C4×C53D412yes—
46C2×C22⋊C4×C53—22no—
47C2×C42×C53—22no—
48C2×C42×C53—22no—
49C2×C42×C53—22no—
50#95×C53—22no—
51#95×C53D414yes—
52C24×C53C2×D411yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—24no—
56C22×C4×C53—21no—
57C22×C4×C53—24no—
58C22×C4×C53—21no—
59C22×C4×C53—81no—
60C22×C4×C53—81no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D412yes—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53#4112yes—
67C22×Dic53—22no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C2×Dic106—22no—
71C2×Dic106—81no—
72C4⋊C4×C53—28no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—24no—
75C22⋊C4×C53—28no—
76C42×C53—28no—
77C42×C53—82no—
78C42×C53—82no—
79C22⋊Dic53—24no—
80C22⋊Dic53—41no—
81C22⋊Dic53—41no—
82C4⋊Dic53—22no—
83#11719919—44no—
84#11719919—42no—
85#11719919—82no—
86C4×Dic53—81no—
87#11719917—82no—
88C2×#88322—42no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—41no—
92C23×C53C2×C22⋊C411yescharacteristic
93C23×C53#12011yescharacteristic
94C23×C53#12311yescharacteristic
95C2×C4×C53—24no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—28no—
101C2×C4×C53—22no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—44no—
108C2×C4×C53—44no—
109C2×C4×C53—42no—
110C2×C4×C53—81no—
111C2×C4×C53#10414yes—
112C2×C4×C53C4×D412yes—
113C2×C4×C53#12112yes—
114C2×Dic53—24no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—42no—
121Dic106—82no—
122#88322—82no—
123#9408—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—44no—
132C22×C53#38611yescharacteristic
133C22×C53C2×#10412yes—
134C22×C53#45612yes—
135C4×C53—24no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—48no—
139C4×C53—42no—
140Dic53—44no—
141Dic53—81no—
142Dic53—81no—
143#2583—531no—
144#2274D5311yescharacteristic
145#1508—532no—
146#1463—531no—
147#1297—532no—
148C2×C53—22no—
149C2×C53—22no—
150C2×C53—41no—
151C2×C53#188611yescharacteristic
152C2×C53#189611yescharacteristic
153C2×C53#191311yescharacteristic
154C2×C22⋊Q8—531no—
155C2×#117—21no—
156C22×C4⋊C4C2×D5311yescharacteristic
157#407—532no—
158#398—532no—
159#395—532no—
160#389—1062no—
161#385—1061no—
162#385—1061no—
163C4×C4⋊C4—24no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—531no—
166C4×C22⋊C4C2×D5312yes—
167#328—1064no—
168#324—1062no—
169C53#940811yescharacteristic, complemented, Sylow
170C22×Q8—531no—
171C23×C4C22⋊D5311yescharacteristic
172C23×C4C22⋊D5311yescharacteristic
173C23×C4C22×D5311yescharacteristic
174C2×M16—1062no—
175C22⋊Q8—1062no—
176C22⋊Q8—1062no—
177#117—28no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—1062no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4—1062no—
185C2×C4⋊C4C4×D5312yes—
186C2×C4⋊C4D21212yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1061no—
189C2×C22⋊C4—1061no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—1061no—
194C22⋊C8—1062no—
195C22⋊C8—1062no—
196#95—22no—
197#95—532no—
198#95—1062no—
199#95—1062no—
200#95C22⋊D5314yes—
201C24C2×C22⋊D5311yescharacteristic
202C2×Q8—1062no—
203C2×Q8—4241no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—21no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—1062no—
213C22×C4—1061no—
214C22×C4—1061no—
215C22×C4#1171992112yes—
216C22×C4#1171992112yes—
217C22×C4C2×C4×D5311yescharacteristic
218C22×C4C2×D21211yescharacteristic
219C22×C4#1171994512yes—
220C22×C4C2×C22⋊D5312yes—
221M16—4242no—
222C2×C8—2122no—
223C4⋊C4—28no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—1062no—
227C4⋊C4—4242no—
228C4⋊C4—2124no—
229C4⋊C4—2122no—
230C22⋊C4—28no—
231C22⋊C4—1064no—
232C22⋊C4—2121no—
233C22⋊C4—2121no—
234C42—28no—
235C42—82no—
236C42—82no—
237C42—4241no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23C2×#1171992111yescharacteristic
242C23#1313318611yescharacteristic
243C23#1313320011yescharacteristic
244Q8—4242no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—28no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—81no—
261C2×C4—1064no—
262C2×C4—2122no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4—2122no—
268C2×C4#1313314514yes—
269C2×C4C4×C22⋊D5312yes—
270C2×C4#1313319112yes—
271C8—4242no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22#1681050211yescharacteristic
281C22C2×#1313314512yes—
282C22#1681080612yes—
283C4—22no—
284C4—22no—
285C4—24no—
286C4—48no—
287C4—42no—
288C4—4241no—
289C4—4241no—
290C4—2124no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2#2027510811yescharacteristic
295C2#2027515811yescharacteristic
296C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2274C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2274D53
hypocenterC53#9408
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
8. #16810553C4semidirect
9. #16810543C4semidirect
169. C53#9408semidirect

Maximal subgroups

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

subgroupindex
#202800952
#202732342
#202730452
#202726012
C53×#22742
#940853