Finite Group Databasebeta

← #27281826

Subgroups — #27281826

Order 13568.

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

normal subgroups only

This group has 700 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#20278355C211yesmaximal, characteristic
2#20274639C211yesmaximal, characteristic
3#20273247C212yesmaximal
4#20272567C212yesmaximal
5C53×#2310C211yesmaximal, characteristic
6C2×#13133215C412yescomplemented
7C2×#13133196C2211yescharacteristic
8C2×#13133136—22no—
9#16810587C2212yes—
10#16810542—22no—
11#16810530C412yescomplemented
12#16810496—21no—
13#16810496—21no—
14C4×C22⋊Dic53C2211yescharacteristic
15#16810359—24no—
16C23×Q8×C53C2211yescharacteristic
17C2×#117×C53—21no—
18C53×#391—24no—
19C53×#381—21no—
20C53×#381C2212yes—
21C22×Dic106C2×C411yescharacteristic
22C2×#11719917—22no—
23#13133215—44no—
24#13133196—22no—
25#13133196—22no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C4×Dic53C2×C412yes—
33#13133136—44no—
34C22⋊#88322—22no—
35C22⋊#88322—22no—
36#13133124—22no—
37#13133124—22no—
38#13133124—21no—
39C22×Q8×C53—22no—
40C22×Q8×C53—22no—
41C22×Q8×C53—44no—
42C22×Q8×C53C2×C412yes—
43C22×Q8×C53D412yes—
44C23×C4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53C2311yescharacteristic
47#117×C53—44no—
48C2×C4⋊C4×C53—22no—
49C2×C22⋊C4×C53—22no—
50C2×C42×C53—22no—
51#95×C53—24no—
52#95×C53—22no—
53#95×C53D414yes—
54C24×C53C2×D411yescharacteristic
55C2×Q8×C53—24no—
56C2×Q8×C53—24no—
57C2×Q8×C53—24no—
58C2×Q8×C53—24no—
59C2×Q8×C53—42no—
60C2×Q8×C53—42no—
61C2×Q8×C53—48no—
62C2×Q8×C53—44no—
63C2×Q8×C53—44no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—24no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—24no—
70C22×C4×C53—42no—
71C22×C4×C53—42no—
72C22×C4×C53C22⋊C412yes—
73C22×C4×C53C22⋊C412yes—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D412yes—
77C22×C4×C53#4112yes—
78C22×Dic53—22no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C2×Dic106—22no—
82C2×Dic106—42no—
83C4⋊C4×C53—44no—
84C22⋊C4×C53—44no—
85C42×C53—44no—
86C22⋊Dic53—24no—
87C22⋊Dic53—41no—
88C22⋊Dic53—41no—
89C4⋊Dic53—22no—
90#11719919—44no—
91#11719919—44no—
92#11719919—42no—
93C4×Dic53—42no—
94#11719917—44no—
95C2×#88322—42no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100C23×C53#12011yescharacteristic
101C23×C53#12311yescharacteristic
102Q8×C53—48no—
103Q8×C53—48no—
104C2×C4×C53—24no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—44no—
119C2×C4×C53—44no—
120C2×C4×C53—44no—
121C2×C4×C53#9914yes—
122C2×C4×C53C4×D412yes—
123C2×C4×C53#12112yes—
124C2×Dic53—24no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—42no—
131Dic106—82no—
132#88322—82no—
133#9462—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—44no—
142C22×C53#38611yescharacteristic
143C22×C53C2×#9912yes—
144C22×C53#45612yes—
145C4×C53—22no—
146C4×C53—22no—
147C4×C53—24no—
148C4×C53—42no—
149C4×C53—84no—
150Dic53—44no—
151Dic53—81no—
152Dic53—81no—
153#2434—531no—
154#2310D5311yescharacteristic
155#1784—531no—
156#1511—532no—
157#1284—532no—
158C2×C53—22no—
159C2×C53—22no—
160C2×C53—41no—
161C2×C53#188611yescharacteristic
162C2×C53#189811yescharacteristic
163C2×C53#191211yescharacteristic
164C23×Q8C2×D5311yescharacteristic
165C2×C4⋊Q8—532no—
166C2×C22⋊Q8—531no—
167C2×#117—21no—
168C2×#101—1062no—
169#407—532no—
170#395—1062no—
171#391—24no—
172#391—532no—
173#385—1061no—
174#385—1061no—
175#381—21no—
176#381C2×D5312yes—
177C4×C22⋊C4—531no—
178#328—1064no—
179C53#946211yescharacteristic, complemented, Sylow
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—44no—
183C22×Q8—531no—
184C22×Q8C4×D5312yes—
185C22×Q8D21212yes—
186C23×C4C22⋊D5311yescharacteristic
187C23×C4C22⋊D5311yescharacteristic
188C23×C4C22×D5311yescharacteristic
189C2×M16—1062no—
190C4⋊Q8—2124no—
191C22⋊Q8—1062no—
192C22⋊Q8—1062no—
193#117—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—1062no—
198C2×C4⋊C4—1062no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—1061no—
201C2×C22⋊C4—1061no—
202C2×C42—22no—
203C2×C42—532no—
204#101—2124no—
205C22⋊C8—1062no—
206C22⋊C8—1062no—
207#95—24no—
208#95—22no—
209#95—1061no—
210#95—1062no—
211#95—1062no—
212#95C22⋊D5314yes—
213C24C2×C22⋊D5311yescharacteristic
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—48no—
223C2×Q8—1062no—
224C2×Q8—2122no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—1062no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4#1171992112yes—
237C22×C4#1171992112yes—
238C22×C4C2×C4×D5311yescharacteristic
239C22×C4C2×D21211yescharacteristic
240C22×C4#1171994512yes—
241C22×C4C2×C22⋊D5312yes—
242M16—2124no—
243C2×C8—2122no—
244C4⋊C4—44no—
245C4⋊C4—1062no—
246C4⋊C4—2124no—
247C4⋊C4—2124no—
248C4⋊C4—2122no—
249C22⋊C4—44no—
250C22⋊C4—1064no—
251C22⋊C4—2121no—
252C22⋊C4—2121no—
253C42—44no—
254C42—2122no—
255C23—22no—
256C23—22no—
257C23—41no—
258C23C2×#1171992111yescharacteristic
259C23#1313318611yescharacteristic
260C23#1313320011yescharacteristic
261Q8—48no—
262Q8—48no—
263Q8—4242no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—1064no—
282C2×C4—2122no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2122no—
286C2×C4—2122no—
287C2×C4—2122no—
288C2×C4#1313313114yes—
289C2×C4C4×C22⋊D5312yes—
290C2×C4#1313319112yes—
291C8—4242no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#1681050211yescharacteristic
301C22C2×#1313313112yes—
302C22#1681080612yes—
303C4—22no—
304C4—24no—
305C4—22no—
306C4—42no—
307C4—84no—
308C4—4241no—
309C4—4241no—
310C4—2124no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2#2027510811yescharacteristic
315C2#2027516811yescharacteristic
316C2#2027525211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2310C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2310D53
hypocenterC53#9462
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
6. C2×#13133215C4semidirect
11. #16810530C4semidirect
179. C53#9462semidirect

Maximal subgroups

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

subgroupindex
#202783552
#202746392
#202732472
#202725672
C53×#23102
#946253