Finite Group Databasebeta

← #27281524

Subgroups — #27281524

Order 13568.

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

normal subgroups only

This group has 726 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#20279874C211yesmaximal, characteristic
2#20277067C211yesmaximal, characteristic
3#20273235C211yesmaximal, characteristic, complemented
4#20273041C211yesmaximal, characteristic
5#20272601C211yesmaximal, characteristic, complemented
6#20272600C211yesmaximal, characteristic, complemented
7C53×#1508C211yesmaximal, characteristic, complemented
8C22×#11719921C2211yescharacteristic
9C2×C22⋊C4×D53—21no—
10#16810587C2211yescharacteristic
11#16810549C412yescomplemented
12#16810534C412yescomplemented
13#16810513—22no—
14#16810484—21no—
15#16810478—21no—
16C22⋊C4×Dic53—21no—
17C4×#11719921—22no—
18C4×C22⋊Dic53C2211yescharacteristic
19C4×C22⋊Dic53C2211yescharacteristic
20#16810470—22no—
21#16810360—22no—
22#16810346—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C53×#407C2211yescharacteristic
25C4×C22⋊C4×C53C2211yescharacteristic
26C53×#328—22no—
27C24×D53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C22×C4×D53—22no—
30C2×#11719917—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×#11719921—22no—
34C2×#11719921—22no—
35C2×#11719921—22no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921C2×C412yes—
41C2×#11719921D412yes—
42C22⋊C4×D53—28no—
43C2×C4×Dic53—22no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53—21no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133124—22no—
49#13133124—22no—
50#13133124—22no—
51#13133124C2×C412yescomplemented
52#13133124D412yes—
53C22×Q8×C53C2×C411yescharacteristic, complemented
54C23×C4×C53C2311yescharacteristic
55C2×M16×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C42×C53—21no—
65C22⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67#95×C53D412yes—
68C24×C53C2×D411yescharacteristic
69C23×D53—21no—
70C23×D53—21no—
71C23×D53—24no—
72C23×D53—42no—
73C23×D53—81no—
74C23×D53C22⋊C412yes—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—81no—
78C22×C4×C53—22no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—81no—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×Dic53—21no—
89C22×Dic53—22no—
90C22×Dic53—21no—
91C22×Dic53—22no—
92C22×Dic53—42no—
93C22×Dic53C22⋊C412yes—
94C2×C4×D53—28no—
95C2×C4×D53—82no—
96C2×C4×D53—82no—
97M16×C53—81no—
98C2×C8×C53—41no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C22⋊C4×C53—24no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C42×C53—81no—
107C22⋊Dic53—24no—
108C22⋊Dic53—42no—
109#11719921—28no—
110#11719921—22no—
111#11719921—22no—
112#11719921—22no—
113#11719921—22no—
114#11719921—44no—
115#11719921—82no—
116C4×Dic53—81no—
117C4×Dic53—82no—
118C4×Dic53—81no—
119#11719917—81no—
120C2×#88322—41no—
121C23×C53—22no—
122C23×C53—22no—
123C23×C53—41no—
124C23×C53C2×C22⋊C411yescharacteristic
125C23×C53#12011yescharacteristic
126C23×C53#12311yescharacteristic
127C22×D53—22no—
128C22×D53—22no—
129C22×D53—22no—
130C22×D53—24no—
131C22×D53—22no—
132C22×D53—42no—
133C22×D53—42no—
134C22×D53—42no—
135C22×D53—42no—
136C22×D53—88no—
137C22×D53—81no—
138C22×D53—81no—
139C22×D53#10414yes—
140Q8×C53—81no—
141Q8×C53—81no—
142C2×C4×C53—24no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×Dic53—24no—
160C2×Dic53—22no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—81no—
169C4×D53—88no—
170C8×C53—81no—
171#88322—81no—
172#9407—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—44no—
181C22×C53#38611yescharacteristic
182C22×C53C2×#10412yes—
183C22×C53#45512yes—
184C2×D53—24no—
185C2×D53—44no—
186C2×D53—44no—
187C2×D53—42no—
188C2×D53—42no—
189C2×D53—44no—
190C2×D53—82no—
191C2×D53—84no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—44no—
197C4×C53—81no—
198C4×C53—81no—
199Dic53—44no—
200Dic53—42no—
201Dic53—82no—
202#2561—531no—
203#2273—531no—
204#1508—531no—
205#1508D5311yescharacteristic, complemented
206#1462—531no—
207#1297—531no—
208#1297—531no—
209C2×C53—22no—
210C2×C53—22no—
211C2×C53—41no—
212C2×C53#188511yescharacteristic
213C2×C53#189611yescharacteristic
214C2×C53#191311yescharacteristic
215D53—44no—
216C2×C22⋊Q8C2×D5311yescharacteristic
217C22×C22⋊C4—531no—
218C22×C22⋊C4—1061no—
219#407—531no—
220#407C2×D5311yescharacteristic
221#397—532no—
222#393—532no—
223#388—1062no—
224#381—1061no—
225#379—1061no—
226C4×C22⋊C4—531no—
227C4×C22⋊C4—531no—
228C4×C22⋊C4—1062no—
229C4×C22⋊C4—1061no—
230C4×C22⋊C4—1062no—
231C4×C22⋊C4C2×D5311yescharacteristic
232#328—22no—
233#328—1062no—
234#323—1062no—
235C53#940711yescharacteristic, complemented, Sylow
236C25—531no—
237C22×Q8C4×D5311yescharacteristic, complemented
238C23×C4—531no—
239C23×C4—1062no—
240C23×C4C22×D5311yescharacteristic
241C2×M16—21no—
242C2×M16—1061no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C4×D5311yescharacteristic, complemented
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—532no—
252C2×C22⋊C4—532no—
253C2×C22⋊C4—1068no—
254C2×C22⋊C4—1062no—
255C2×C22⋊C4—1061no—
256C2×C22⋊C4—1061no—
257C2×C22⋊C4—1061no—
258C2×C22⋊C4—1061no—
259C2×C22⋊C4—1061no—
260C2×C22⋊C4—1062no—
261C2×C22⋊C4—1062no—
262C2×C22⋊C4—1061no—
263C2×C42—21no—
264C2×C42—1062no—
265C2×C42—1061no—
266C2×C42—1061no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—1061no—
270C22⋊C8—1061no—
271#95—532no—
272#95—532no—
273#95—1062no—
274#95—1062no—
275#95—1062no—
276#95C22⋊D5312yes—
277C24—532no—
278C24—1064no—
279C24—1061no—
280C24—1061no—
281C24—4241no—
282C24—2122no—
283C24D4×D5311yescharacteristic
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—81no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—81no—
291C22×C4—532no—
292C22×C4—1068no—
293C22×C4—1061no—
294C22×C4—1062no—
295C22×C4—1061no—
296C22×C4—1062no—
297C22×C4—4242no—
298C22×C4—4242no—
299C22×C4—2122no—
300C22×C4C2×C4×D5311yescharacteristic
301C22×C4#1171994511yescharacteristic
302C22×C4D4×D5311yescharacteristic
303C22×C4D4×D5311yescharacteristic
304C22×C4#1171994711yescharacteristic
305C22×C4C2×C22⋊D5311yescharacteristic
306M16—81no—
307M16—4241no—
308C2×C8—41no—
309C2×C8—2121no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C22⋊C4—24no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—1068no—
318C22⋊C4—1064no—
319C22⋊C4—1062no—
320C22⋊C4—1062no—
321C22⋊C4—1062no—
322C22⋊C4—1062no—
323C22⋊C4—4242no—
324C22⋊C4—2124no—
325C22⋊C4—2122no—
326C42—81no—
327C42—4241no—
328C42—4241no—
329C42—4242no—
330C23—22no—
331C23—22no—
332C23—41no—
333C23—534no—
334C23—1064no—
335C23—1062no—
336C23—1062no—
337C23—1062no—
338C23—1062no—
339C23—4241no—
340C23—4241no—
341C23—4248no—
342C23—2122no—
343C23—2122no—
344C23—2122no—
345C23—2122no—
346C23C22⋊C4×D5311yescharacteristic
347C23#1313318511yescharacteristic
348C23#1313319911yescharacteristic
349Q8—81no—
350Q8—81no—
351C2×C4—24no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—1064no—
365C2×C4—1062no—
366C2×C4—4248no—
367C2×C4—4241no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4#1313317711yescharacteristic
376C2×C4C4×C22⋊D5311yescharacteristic
377C2×C4#1313318811yescharacteristic
378C2×C4#1313319011yescharacteristic
379C8—81no—
380C8—4241no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—44no—
389C22—1064no—
390C22—4242no—
391C22—4244no—
392C22—2124no—
393C22—2124no—
394C22—2124no—
395C22—2122no—
396C22—2122no—
397C22#1681049911yescharacteristic
398C22#104×D5312yes—
399C22#1681079912yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—44no—
405C4—81no—
406C4—81no—
407C4—4242no—
408C4—2124no—
409C4—2122no—
410C2—22no—
411C2—22no—
412C2—41no—
413C2—2124no—
414C2#2027510011yescharacteristic
415C2#2027515511yescharacteristic
416C2#2027525411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1508C2
socleC22×C53#386
prime radicalC24×D53D4
cosocleC23×C4×C53C23
hypercenter#1508D53
hypocenterC53#9407
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
3. #20273235C2semidirect
5. #20272601C2semidirect
6. #20272600C2semidirect
7. C53×#1508C2semidirect
11. #16810549C4semidirect
12. #16810534C4semidirect
51. #13133124C2×C4semidirect
53. C22×Q8×C53C2×C4semidirect
61. C2×C4⋊C4×C53C2×C4semidirect
205. #1508D53semidirect
235. C53#9407semidirect
237. C22×Q8C4×D53semidirect
248. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202798742
#202770672
#202732352
#202730412
#202726012
#202726002
C53×#15082
#940753