Finite Group Databasebeta

← #27393034

Subgroups — #27393034

Order 13568.

1570 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 1570 conjugacy classes of subgroups — too many to draw. Only its 243 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
1C22×#13133201C211yesmaximal, characteristic
2#20277093C212yesmaximal, complemented
3#20275172C218yesmaximal, complemented
4#20274789C211yesmaximal, characteristic, complemented
5#20274629C212yesmaximal
6C53×#1819C211yesmaximal, characteristic, complemented
7C23×C22⋊D53C2211yescharacteristic
8C24×Dic53C2211yescharacteristic
9C2×#13133201—24no—
10C2×#13133201—24no—
11C2×#13133201C414yescomplemented
12C2×#13133201C2214yes—
13C2×#13133173C2214yescomplemented
14C22×C4⋊Dic53C412yescomplemented
15C22×#11719921C412yescomplemented
16C2×#13133131C2218yes—
17#16810484C2214yescomplemented
18#16810481—24no—
19#16810479C2212yes—
20C22⋊C4×Dic53—24no—
21#16810390C2214yescomplemented
22C22×C22⋊C4×C53C2211yescharacteristic
23C53×#379C2212yes—
24C53×#342C2214yescomplemented
25C24×D53C2×C411yescharacteristic
26C22×C22⋊D53—24no—
27C22×C22⋊D53—44no—
28C22×C22⋊D53C2×C412yes—
29C22×C22⋊D53C2312yes—
30C23×Dic53—44no—
31C23×Dic53C2×C411yescharacteristic
32C23×Dic53C2×C412yes—
33C23×Dic53C2×C412yes—
34C23×Dic53D414yes—
35C23×Dic53D414yes—
36C23×Dic53C2312yes—
37#13133201—24no—
38#13133201—28no—
39#13133201—24no—
40#13133201—28no—
41#13133201—24no—
42#13133201—24no—
43#13133173—48no—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—24no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53C2×C418yes—
48C2×#11719919—24no—
49C2×C22⋊Dic53—24no—
50C2×C22⋊Dic53C2314yes—
51C2×#11719921—22no—
52C2×#11719921—24no—
53C2×#11719921—22no—
54C2×#11719921C2×C418yes—
55C2×C4×Dic53—24no—
56#13133131—48no—
57#13133124—28no—
58C25×C53C2311yescharacteristic
59C23×C4×C53C2×C412yes—
60C2×C22⋊C4×C53—24no—
61C2×C22⋊C4×C53—24no—
62C2×C22⋊C4×C53C2×C412yes—
63C2×C22⋊C4×C53C2314yes—
64C2×C22⋊C4×C53C2312yes—
65C24×C53—24no—
66C24×C53—44no—
67C24×C53C22×C411yescharacteristic
68C24×C53C22×C411yescharacteristic
69C24×C53C2×D414yes—
70C24×C53C2411yescharacteristic
71C23×D53—22no—
72C23×D53—42no—
73C23×D53—44no—
74C23×D53C22×C412yes—
75C22×C4×C53—22no—
76C22×C4×C53—22no—
77C22×C4×C53—24no—
78C22×C4×C53—44no—
79C22×C4×C53C22×C414yes—
80C2×C22⋊D53—24no—
81C2×C22⋊D53—24no—
82C2×C22⋊D53—28no—
83C2×C22⋊D53—44no—
84C2×C22⋊D53—416no—
85C22×Dic53—28no—
86C22×Dic53—22no—
87C22×Dic53—24no—
88C22×Dic53—44no—
89C22×Dic53—44no—
90C22×Dic53—48no—
91C22×Dic53—48no—
92C22×Dic53—42no—
93C22×Dic53C22⋊C412yes—
94C22×Dic53C22⋊C414yes—
95C22×Dic53C22⋊C412yes—
96C22×Dic53C22⋊C412yes—
97C22×Dic53C22⋊C414yes—
98C22×Dic53C22⋊C412yes—
99C22×Dic53C22×C412yes—
100C22×Dic53C22×C414yes—
101C22×Dic53C2×D414yes—
102C22×Dic53C2×D412yes—
103C22×Dic53C2×D412yes—
104C22⋊C4×C53—24no—
105C22⋊C4×C53—24no—
106C22⋊C4×C53—28no—
107C22⋊C4×C53—48no—
108C22⋊Dic53—28no—
109C22⋊Dic53—48no—
110#11719921—28no—
111#11719921—28no—
112C4⋊Dic53—28no—
113C4⋊Dic53—28no—
114#11719919—48no—
115C4×Dic53—48no—
116C23×C53—22no—
117C23×C53—24no—
118C23×C53—22no—
119C23×C53—28no—
120C23×C53—28no—
121C23×C53—42no—
122C23×C53—44no—
123C23×C53—42no—
124C23×C53—416no—
125C23×C53C2×C22⋊C412yes—
126C23×C53C2×C22⋊C412yes—
127C23×C53C23×C411yescharacteristic
128C23×C53C22×D412yes—
129C23×C53H(5,𝔽2)14yes—
130C22×D53—22no—
131C22×D53—24no—
132C22×D53—44no—
133C22×D53—44no—
134C22×D53—44no—
135C22×D53—44no—
136C22×D53—88no—
137C2×C4×C53—24no—
138C2×C4×C53—44no—
139C2×C4×C53—44no—
140C2×C4×C53—44no—
141C2×C4×C53—48no—
142C2×C4×C53—44no—
143C22⋊D53—48no—
144C22⋊D53—48no—
145C2×Dic53—28no—
146C2×Dic53—28no—
147C2×Dic53—24no—
148C2×Dic53—24no—
149C2×Dic53—22no—
150C2×Dic53—24no—
151C2×Dic53—42no—
152C2×Dic53—44no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—44no—
156C2×Dic53—42no—
157C2×Dic53—44no—
158C2×Dic53—48no—
159C2×Dic53—44no—
160C2×Dic53C2×C22⋊C414yes—
161C2×Dic53C2×C22⋊C414yes—
162#21747—531nomaximal, Sylow
163C22×C53—28no—
164C22×C53—22no—
165C22×C53—24no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—24no—
170C22×C53—44no—
171C22×C53—416no—
172C22×C53—44no—
173C22×C53#41014yes—
174C22×C53C22×C22⋊C411yescharacteristic
175C22×C53#51812yes—
176C22×C53#53414yes—
177C2×D53—42no—
178C2×D53—44no—
179C2×D53—88no—
180C2×D53—84no—
181C4×C53—44no—
182C4×C53—84no—
183Dic53—24no—
184Dic53—24no—
185Dic53—42no—
186Dic53—84no—
187C22×#123—531no—
188#2280—532no—
189#1899—538no—
190#1819—531no—
191#1819D5311yescharacteristic, complemented
192#1781—532no—
193C2×C53—24no—
194C2×C53—22no—
195C2×C53—24no—
196C2×C53—42no—
197C2×C53#230811yescharacteristic
198C2×C53C2×#41012yes—
199D53—82no—
200C23×D4—531no—
201C24×C4—531no—
202C2×#123—534no—
203C2×#123—534no—
204C2×#123—1064no—
205C2×#123—1064no—
206C2×#117—534no—
207C22×C4⋊C4—532no—
208C22×C22⋊C4—532no—
209C22×C22⋊C4C2×D5311yescharacteristic
210C2×#99—538no—
211#381—534no—
212#380—1064no—
213#379—532no—
214#379C2×D5312yes—
215C4×C22⋊C4—1064no—
216#342—534no—
217#342C2×D5314yescomplemented
218C53#2174711yescharacteristic, complemented, Sylow
219C25—531no—
220C25C22×D5311yescharacteristic
221C22×D4—532no—
222C22×D4—532no—
223C22×D4—1064no—
224C22×D4—2124no—
225C23×C4—532no—
226C23×C4—532no—
227C23×C4—532no—
228C23×C4—534no—
229C23×C4—534no—
230C23×C4—531no—
231C23×C4—2124no—
232C23×C4C4×D5312yes—
233#123—1064no—
234#123—1068no—
235#123—1064no—
236#123—1068no—
237#123—1064no—
238#123—1064no—
239#117—2128no—
240C2×C4⋊C4—538no—
241C2×C4⋊C4—1064no—
242C2×C4⋊C4—1062no—
243C2×C4⋊C4—1062no—
244C2×C4⋊C4—1064no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—538no—
248C2×C22⋊C4—534no—
249C2×C22⋊C4—1062no—
250C2×C22⋊C4—1062no—
251C2×C22⋊C4—1064no—
252C2×C22⋊C4—1064no—
253C2×C22⋊C4C4×D5312yes—
254C2×C22⋊C4C22×D5314yes—
255C2×C22⋊C4C22×D5312yes—
256C2×C42—1064no—
257#99—2128no—
258#95—1068no—
259C24—24no—
260C24—44no—
261C24—532no—
262C24—1062no—
263C24—2122no—
264C24—2124no—
265C24C2×C4×D5311yescharacteristic
266C24C2×C4×D5311yescharacteristic
267C24D4×D5314yes—
268C24C23×D5311yescharacteristic
269C2×D4—1064no—
270C2×D4—1064no—
271C2×D4—1068no—
272C2×D4—2124no—
273C2×D4—21216no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—44no—
278C22×C4—534no—
279C22×C4—534no—
280C22×C4—532no—
281C22×C4—532no—
282C22×C4—532no—
283C22×C4—534no—
284C22×C4—532no—
285C22×C4—532no—
286C22×C4—534no—
287C22×C4—532no—
288C22×C4—532no—
289C22×C4—1068no—
290C22×C4—1062no—
291C22×C4—1064no—
292C22×C4—2124no—
293C22×C4—2124no—
294C22×C4—2128no—
295C22×C4—2128no—
296C22×C4—2122no—
297C22×C4C2×C4×D5314yes—
298C4⋊C4—1068no—
299C4⋊C4—1068no—
300C4⋊C4—2128no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—28no—
304C22⋊C4—48no—
305C22⋊C4—1068no—
306C22⋊C4—1068no—
307C22⋊C4—1068no—
308C22⋊C4—2128no—
309C42—2128no—
310C23—22no—
311C23—24no—
312C23—22no—
313C23—28no—
314C23—28no—
315C23—42no—
316C23—44no—
317C23—42no—
318C23—416no—
319C23—1064no—
320C23—1062no—
321C23—4248no—
322C23—2124no—
323C23—2124no—
324C23—2124no—
325C23—2124no—
326C23C22⋊C4×D5312yes—
327C23C22⋊C4×D5312yes—
328C23C22×C4×D5311yescharacteristic
329C23C2×D4×D5312yes—
330C23#1313326314yes—
331D4—2128no—
332D4—2128no—
333C2×C4—24no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—534no—
340C2×C4—534no—
341C2×C4—1068no—
342C2×C4—1064no—
343C2×C4—1064no—
344C2×C4—1068no—
345C2×C4—1062no—
346C2×C4—1064no—
347C2×C4—2122no—
348C2×C4—2124no—
349C2×C4—2122no—
350C2×C4—2122no—
351C2×C4—2124no—
352C2×C4—2122no—
353C2×C4—2128no—
354C2×C4—2124no—
355C2×C4—2124no—
356C22—28no—
357C22—24no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—24no—
362C22—22no—
363C22—44no—
364C22—416no—
365C22—44no—
366C22—4244no—
367C22—4248no—
368C22—2124no—
369C22—2122no—
370C22#1681059514yes—
371C22C2×C22⋊C4×D5311yescharacteristic
372C22#1681109712yes—
373C22#1681118414yes—
374C4—44no—
375C4—84no—
376C4—1064no—
377C4—1064no—
378C4—4244no—
379C4—2122no—
380C2—24no—
381C2—22no—
382C2—24no—
383C2—42no—
384C2—4242no—
385C2#2027729211yescharacteristic
386C2C2×#1681059512yes—
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1819C2
socleC22×C53C22×C22⋊C4
prime radicalC23×C22⋊D53C22
cosocleC24×C53C24
hypercenter#1819D53
hypocenterC53#21747
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
2. #20277093C2semidirect
3. #20275172C2semidirect
4. #20274789C2semidirect
6. C53×#1819C2semidirect
11. C2×#13133201C4semidirect
13. C2×#13133173C22semidirect
14. C22×C4⋊Dic53C4semidirect
15. C22×#11719921C4semidirect
17. #16810484C22semidirect
21. #16810390C22semidirect
24. C53×#342C22semidirect
191. #1819D53semidirect
217. #342C2×D53semidirect
218. C53#21747semidirect

Maximal subgroups

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

subgroupindex
C22×#131332012
#202770932
#202751722
#202747892
#202746292
C53×#18192
#2174753