Finite Group Databasebeta

← #27305709

Subgroups — #27305709

Order 13568.

2878 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 2878 conjugacy classes of subgroups — too many to draw. Only its 427 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
1C2×#16810478C213yesmaximal
2#20272953C212yesmaximal, complemented
3#20272952C213yesmaximal, complemented
4#20272951C216yesmaximal, complemented
5C53×#1437C211yesmaximal, characteristic, complemented
6C25×D53C2211yescharacteristic
7C22×C22⋊Dic53C2213yes—
8C22×#11719921C2216yes—
9C22×#11719921C2216yes—
10C22×C4×Dic53C2213yescomplemented
11#16810478C4124yescomplemented
12C2×#13133124C2213yescomplemented
13C2×#13133124C2216yescomplemented
14C22×C22⋊C4×C53C2213yes—
15C22×C42×C53C2211yescharacteristic, complemented
16C2×#95×C53C2213yescomplemented
17C24×D53—43no—
18C24×D53—43no—
19C24×D53—46no—
20C24×D53C2×C4112yes—
21C24×D53C2312yes—
22C23×Dic53C2316yes—
23C2×C22⋊Dic53—46no—
24C2×C22⋊Dic53C2×C4112yes—
25C2×#11719921—46no—
26C2×#11719921—46no—
27C2×#11719921—46no—
28C2×#11719921—46no—
29C2×#11719921C2×C4124yes—
30C2×#11719921C2×C4124yes—
31C2×C4×Dic53—43no—
32C2×C4×Dic53—46no—
33#13133124—43no—
34#13133124—46no—
35C25×C53C2311yescharacteristic
36C23×C4×C53C2313yes—
37C23×C4×C53C2313yes—
38C2×C22⋊C4×C53—43no—
39C2×C22⋊C4×C53—43no—
40C2×C22⋊C4×C53C2×C4112yes—
41C2×C42×C53—43no—
42#95×C53—43no—
43C24×C53—23no—
44C24×C53—23no—
45C24×C53—26no—
46C24×C53C22×C416yes—
47C24×C53C2411yescharacteristic
48C23×D53—26no—
49C23×D53—26no—
50C23×D53—26no—
51C23×D53—26no—
52C23×D53—412no—
53C23×D53—412no—
54C23×D53—424no—
55C23×D53—424no—
56C23×D53—44no—
57C23×D53—424no—
58C23×D53—412no—
59C23×D53—48no—
60C23×D53—412no—
61C23×D53—44no—
62C23×D53C42116yescomplemented
63C23×D53C22×C4112yes—
64C22×C4×C53—23no—
65C22×C4×C53—26no—
66C22×C4×C53—23no—
67C22×C4×C53—26no—
68C22×C4×C53—23no—
69C22×C4×C53—23no—
70C22×C4×C53—26no—
71C22×C4×C53—26no—
72C22×C4×C53C22×C416yes—
73C22×C4×C53C22×C416yes—
74C22×Dic53—26no—
75C22×Dic53—26no—
76C22×Dic53—26no—
77C22×Dic53—26no—
78C22×Dic53—26no—
79C22×Dic53—26no—
80C22×Dic53C22×C4112yes—
81C22⋊C4×C53—412no—
82C42×C53—44no—
83C22⋊Dic53—412no—
84#11719921—424no—
85#11719921—424no—
86C4×Dic53—412no—
87C23×C53—212no—
88C23×C53—212no—
89C23×C53—26no—
90C23×C53—26no—
91C23×C53—41no—
92C23×C53—46no—
93C23×C53—43no—
94C23×C53—42no—
95C23×C53—43no—
96C23×C53—41no—
97C23×C53C2×C4214yes—
98C23×C53C23×C413yes—
99C23×C53H(5,𝔽2)13yes—
100C23×C53H(5,𝔽2)13yes—
101C23×C53H(5,𝔽2)16yes—
102C22×D53—212no—
103C22×D53—212no—
104C22×D53—212no—
105C22×D53—212no—
106C22×D53—212no—
107C22×D53—212no—
108C22×D53—448no—
109C22×D53—448no—
110C22×D53—448no—
111C22×D53—496no—
112C22×D53—42no—
113C22×D53—46no—
114C22×D53—46no—
115C22×D53—46no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—46no—
119C22×D53—42no—
120C22×D53C2×C4218yes—
121C2×C4×C53—212no—
122C2×C4×C53—212no—
123C2×C4×C53—26no—
124C2×C4×C53—26no—
125C2×C4×C53—43no—
126C2×C4×C53—46no—
127C2×C4×C53—46no—
128C2×C4×C53—43no—
129C2×C4×C53—43no—
130C2×C4×C53—43no—
131C2×Dic53—212no—
132C2×Dic53—212no—
133C2×Dic53—212no—
134C2×Dic53—46no—
135C2×Dic53—46no—
136C2×Dic53—46no—
137C2×Dic53—46no—
138#13827—531nomaximal, Sylow
139C22×C53—212no—
140C22×C53—46no—
141C22×C53—46no—
142C22×C53—412no—
143C22×C53C22×C4211yescharacteristic
144C22×C53#51816yes—
145C22×C53#51816yes—
146C22×C53#51813yes—
147C22×C53#51813yes—
148C22×C53#56011yescharacteristic
149C22×C53#56016yes—
150C22×C53#56013yes—
151C22×C53#56012yes—
152C22×C53H(3,𝔽4)13yes—
153C22×C53H(3,𝔽4)11yescharacteristic
154C2×D53—224no—
155C2×D53—464no—
156C2×D53—412no—
157C2×D53—412no—
158C2×D53—412no—
159C2×D53—412no—
160C4×C53—46no—
161C4×C53—46no—
162Dic53—412no—
163C2×#379—533no—
164#1437—532no—
165#1437—536no—
166#1437—533no—
167#1437D5311yescharacteristic, complemented
168C2×C53—44no—
169C2×C53#227313yes—
170C2×C53#237513yes—
171C2×C53#237513yes—
172C2×C53#237516yes—
173D53—48no—
174C26—531no—
175C22×C22⋊C4—533no—
176C22×C22⋊C4—536no—
177C22×C22⋊C4—536no—
178C22×C22⋊C4C2×D5313yes—
179C22×C42—533no—
180C22×C42C2×D5311yescharacteristic, complemented
181#379—5324no—
182C2×#95—536no—
183C2×#95—533no—
184C2×#95C2×D5313yescomplemented
185C53#1382711yescharacteristic, complemented, Sylow
186C25—532no—
187C25—5312no—
188C25—2123no—
189C25—2123no—
190C25—2126no—
191C25C22×D5311yescharacteristic
192C23×C4—536no—
193C23×C4C22×D5313yes—
194C23×C4C22×D5313yes—
195C2×C22⋊C4—43no—
196C2×C22⋊C4—43no—
197C2×C22⋊C4—5312no—
198C2×C22⋊C4—5324no—
199C2×C22⋊C4—5324no—
200C2×C22⋊C4—2126no—
201C2×C22⋊C4—2126no—
202C2×C22⋊C4—2126no—
203C2×C22⋊C4—2126no—
204C2×C22⋊C4—2126no—
205C2×C22⋊C4C4×D53112yes—
206C2×C42—43no—
207C2×C42—2123no—
208C2×C42—2126no—
209#95—43no—
210#95—2123no—
211#95—2126no—
212C24—23no—
213C24—23no—
214C24—26no—
215C24—5312no—
216C24—5316no—
217C24—1066no—
218C24—1066no—
219C24—1066no—
220C24—1066no—
221C24—21224no—
222C24—21224no—
223C24—21212no—
224C24—21212no—
225C24—2124no—
226C24—21224no—
227C24—21212no—
228C24—2128no—
229C24—21212no—
230C24—2124no—
231C24C2×C4×D5316yes—
232C24C23×D5311yescharacteristic
233C22×C4—23no—
234C22×C4—26no—
235C22×C4—23no—
236C22×C4—23no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—5312no—
242C22×C4—1066no—
243C22×C4—1066no—
244C22×C4—1066no—
245C22×C4—1066no—
246C22×C4—1066no—
247C22×C4—1066no—
248C22×C4C2×C4×D5316yes—
249C22×C4C2×C4×D5316yes—
250C22⋊C4—412no—
251C22⋊C4—21224no—
252C22⋊C4—21224no—
253C22⋊C4—21212no—
254C42—44no—
255C42—21212no—
256C23—212no—
257C23—212no—
258C23—26no—
259C23—26no—
260C23—41no—
261C23—46no—
262C23—43no—
263C23—42no—
264C23—43no—
265C23—41no—
266C23—538no—
267C23—10612no—
268C23—10612no—
269C23—10612no—
270C23—10612no—
271C23—10612no—
272C23—10612no—
273C23—21248no—
274C23—21248no—
275C23—21248no—
276C23—21296no—
277C23—2122no—
278C23—2126no—
279C23—2126no—
280C23—2126no—
281C23—2122no—
282C23—2122no—
283C23—2126no—
284C23—2122no—
285C23C42×D5314yes—
286C23C22×C4×D5313yes—
287C23#1313326213yes—
288C23#1313326213yes—
289C23#1313326316yes—
290C2×C4—212no—
291C2×C4—212no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—43no—
295C2×C4—46no—
296C2×C4—46no—
297C2×C4—43no—
298C2×C4—43no—
299C2×C4—43no—
300C2×C4—10612no—
301C2×C4—10612no—
302C2×C4—10612no—
303C2×C4—2126no—
304C2×C4—2126no—
305C2×C4—2126no—
306C2×C4—2126no—
307C22—212no—
308C22—46no—
309C22—46no—
310C22—412no—
311C22—10624no—
312C22—21264no—
313C22—21212no—
314C22—21212no—
315C22—21212no—
316C22—21212no—
317C22C2×C42×D5311yescharacteristic
318C22#1681109516yes—
319C22#1681109516yes—
320C22#1681109613yes—
321C22#1681109713yes—
322C22#1681136011yescharacteristic
323C22#1681136116yes—
324C22#1681136313yes—
325C22#1681136412yes—
326C22#1681136513yes—
327C22#1681136611yescharacteristic
328C4—46no—
329C4—46no—
330C4—21212no—
331C2—44no—
332C2—2128no—
333C2#2027706713yes—
334C2#2027785213yes—
335C2#2027785313yes—
336C2#2027785516yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC53×#1437C2
socle, cosocleC24×C53C24
prime radicalC25×D53C22
hypercenter#1437D53
hypocenterC53#13827
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. #20272953C2semidirect
3. #20272952C2semidirect
4. #20272951C2semidirect
5. C53×#1437C2semidirect
10. C22×C4×Dic53C22semidirect
11. #16810478C4semidirect
12. C2×#13133124C22semidirect
13. C2×#13133124C22semidirect
15. C22×C42×C53C22semidirect
16. C2×#95×C53C22semidirect
62. C23×D53C42semidirect
167. #1437D53semidirect
180. C22×C42C2×D53semidirect
184. C2×#95C2×D53semidirect
185. C53#13827semidirect

Maximal subgroups

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

subgroupindex
C2×#168104782
#202729532
#202729522
#202729512
C53×#14372
#1382753