Finite Group Databasebeta

← #27393784

Subgroups — #27393784

Order 13568.

1166 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811158C211yesmaximal, characteristic
2C2×#16810674C211yesmaximal, characteristic, complemented
3#382×D53C211yesmaximal, characteristic, complemented
4#20275681C211yesmaximal, characteristic, complemented
5#20275526C211yesmaximal, characteristic, complemented
6#20275524C211yesmaximal, characteristic, complemented
7#20275031C212yesmaximal
8#20275021C212yesmaximal, complemented
9#20274729C212yesmaximal, complemented
10#20274724C212yesmaximal, complemented
11C53×#1991C211yesmaximal, characteristic, complemented
12#16811158—24no—
13#16811158C418yescomplemented
14C2×#13133212—21no—
15C2×#13133205C2211yescharacteristic
16C2×#13133202—21no—
17C2×#13133197C2211yescharacteristic
18C2×#13133192C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×#13133179—21no—
21C2×#13133179C2211yescharacteristic
22C2×#13133175—21no—
23C2×C4⋊C4×D53C2212yes—
24C2×C42×D53C2211yescharacteristic
25#16810674—24no—
26C2×#13133146C2212yes—
27C2×#13133142C2212yes—
28C2×#13133140C2212yes—
29C2×#13133139C2212yes—
30C2×#13133137C2212yes—
31C2×#13133129C2212yes—
32C2×C4×#88322C2211yescharacteristic
33#16810489C2211yescharacteristic
34#16810488C2212yes—
35#16810487—21no—
36#16810487—21no—
37#95×D53—21no—
38#95×D53—21no—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×#125×C53C2211yescharacteristic
42C2×C4×C8×C53C2211yescharacteristic
43C53×#382C2211yescharacteristic
44C53×#340C2212yes—
45C53×#340C2212yes—
46C22×Dic106C2311yescharacteristic
47C22×D212—21no—
48C22×D212C2311yescharacteristic
49C22×C4×D53—21no—
50C22×C4×D53—22no—
51C22×C4×D53D411yescharacteristic
52C22×C4×D53D411yescharacteristic
53C22×C4×D53C2311yescharacteristic
54C22×#88322C2312yes—
55#13133212—28no—
56#13133205—22no—
57#13133205C2×C414yescomplemented
58#13133202—28no—
59#13133197—22no—
60#13133197C2×C414yes—
61#13133192—22no—
62#13133192C2×C414yes—
63#13133182—22no—
64#13133182C2×C414yescomplemented
65#13133179—22no—
66#13133179—28no—
67#13133179C2×C414yescomplemented
68#13133175—28no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2312yes—
71C2×#11719919—21no—
72C2×#11719919—22no—
73C4⋊C4×D53—42no—
74C4⋊C4×D53—42no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C42×D53C2×C414yes—
85C42×D53D414yes—
86#13133146—24no—
87#13133142—42no—
88#13133140—42no—
89#13133139—42no—
90#13133137—42no—
91#13133129—24no—
92C4×#88322—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—22no—
96#13133124—22no—
97C22×C8×C53C2312yes—
98#125×C53—22no—
99#125×C53C2×C414yescomplemented
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53C2312yes—
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C42×C53C2311yescharacteristic
106C4×C8×C53—22no—
107#95×C53—21no—
108#95×C53—21no—
109C23×D53—21no—
110C23×D53C2×D411yescharacteristic
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2411yescharacteristic
117C22×Dic53—21no—
118C22×Dic53—22no—
119C22×Dic53C2×D411yescharacteristic
120C22×Dic53C2×D411yescharacteristic
121C22×Dic53C2×D411yescharacteristic
122C2×D212—22no—
123C2×D212—22no—
124C2×D212—24no—
125C2×D212—21no—
126C2×D212—44no—
127C2×D212C22×C412yes—
128C2×C4×D53—24no—
129C2×C4×D53—22no—
130C2×C4×D53—22no—
131C2×C4×D53—24no—
132C2×C4×D53—42no—
133C2×C4×D53—44no—
134C2×C4×D53—44no—
135C2×C4×D53—44no—
136C2×C4×D53—44no—
137C2×C4×D53C22⋊C414yes—
138C2×C4×D53C22⋊C414yes—
139C2×C4×D53C22×C412yes—
140C2×C4×D53C2×D412yes—
141C2×Dic106—22no—
142C2×Dic106—22no—
143C2×Dic106—21no—
144C2×Dic106C22×C412yes—
145C2×C8×C53—22no—
146C2×C8×C53—22no—
147C2×C8×C53—22no—
148C4⋊C4×C53—24no—
149C4⋊C4×C53—24no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53C22×C412yes—
155C4⋊C4×C53C22×C412yes—
156C42×C53C22×C412yes—
157C42×C53C2×D412yes—
158#11719921—24no—
159#11719921—24no—
160#11719921—24no—
161#11719921—24no—
162#11719921—24no—
163C4⋊Dic53—24no—
164C4⋊Dic53—22no—
165C4⋊Dic53—22no—
166#11719919—24no—
167#11719919—44no—
168C4×Dic53—24no—
169C4×Dic53C22⋊C414yes—
170C4×Dic53C22⋊C414yes—
171C4×Dic53C22×C412yes—
172C4×Dic53C2×D412yes—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—22no—
176C23×C53C22×D411yescharacteristic
177C22×D53—21no—
178C22×D53—21no—
179C22×D53—22no—
180C22×D53—22no—
181C22×D53—41no—
182C22×D53—41no—
183C22×D53—42no—
184C22×D53—42no—
185C22×D53C2×C22⋊C412yes—
186C22×D53#13514yes—
187C2×C4×C53—22no—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—22no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53C2×C22⋊C412yes—
197C2×C4×C53C2×C22⋊C412yes—
198C2×C4×C53C23×C411yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C2×C4×C53H(5,𝔽2)12yes—
201C2×C4×C532−1+412yes—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—22no—
205C2×Dic53—22no—
206C2×Dic53—22no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—42no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53C2×C22⋊C412yes—
213C2×Dic53C2×C22⋊C412yes—
214C2×Dic53C2×C22⋊C412yes—
215D212—24no—
216D212—48no—
217D212—42no—
218C4×D53—28no—
219C4×D53—24no—
220C4×D53—24no—
221C4×D53—44no—
222C4×D53—48no—
223C4×D53—88no—
224Dic106—24no—
225Dic106—42no—
226C8×C53—42no—
227#88322—42no—
228#21861—531nomaximal, Sylow
229C22×C53C22×C22⋊C411yescharacteristic
230C22×C53#53511yescharacteristic
231C22×C53#53611yescharacteristic
232C22×C53C2×#13512yes—
233C22×C53#57512yes—
234C2×D53—22no—
235C2×D53—22no—
236C2×D53—24no—
237C2×D53—24no—
238C2×D53—42no—
239C2×D53—42no—
240C2×D53—42no—
241C2×D53—82no—
242C2×D53—82no—
243C2×D53#41614yes—
244C4×C53—22no—
245C4×C53—22no—
246C4×C53—42no—
247C4×C53—42no—
248C4×C53—82no—
249C4×C53#52012yes—
250C4×C53#53712yes—
251Dic53—22no—
252Dic53—22no—
253Dic53—22no—
254Dic53—42no—
255Dic53—82no—
256C2×#529—531no—
257C2×#431—531no—
258C2×#382—531no—
259#1991—531no—
260#1991D5311yescharacteristic, complemented
261#1958—531no—
262#1958—531no—
263#1874—532no—
264#1872—532no—
265#1805—532no—
266#1804—532no—
267C2×C53#231111yescharacteristic
268C2×C53C2×#41611yescharacteristic
269C2×C53#289611yescharacteristic
270C2×C53#321112yes—
271C2×C53#323212yes—
272D53—24no—
273D53—82no—
274#529—538no—
275#529—1064no—
276C2×#126—1061no—
277C2×#125C2×D5311yescharacteristic
278C2×#124—531no—
279C2×#123—1061no—
280C2×C22⋊Q8—531no—
281C2×#121—531no—
282C2×C4×Q8—531no—
283C2×C4×D4—531no—
284C2×C4×D4—1061no—
285C2×#117—1061no—
286C22×C4⋊C4—532no—
287C22×C42—531no—
288#431—1064no—
289C2×C4⋊C8—532no—
290C2×#103—532no—
291C2×#103—532no—
292C2×#102—532no—
293C2×#102—532no—
294C2×C22⋊C8—532no—
295C2×C4×C8—531no—
296C2×C4×C8C2×D5311yescharacteristic
297#382—532no—
298#382—531no—
299#382—1061no—
300#382—1061no—
301#382C2×D5311yescharacteristic
302C2×#95—1061no—
303C2×#95—1061no—
304#340—532no—
305#340—532no—
306#340C2×D5312yes—
307#340C2×D5312yes—
308C53#2186111yescharacteristic, complemented, Sylow
309C22×Q8—531no—
310C22×D4—531no—
311C22×D4—1061no—
312C23×C4—531no—
313C23×C4—531no—
314C23×C4—531no—
315C23×C4—1061no—
316C23×C4—1062no—
317C22×C8—532no—
318C22×C8C22×D5312yes—
319#126—1068no—
320#125—22no—
321#125C4×D5314yescomplemented
322#124—534no—
323#124—1062no—
324#123—1068no—
325C22⋊Q8—534no—
326C22⋊Q8—1062no—
327#121—534no—
328#121—1062no—
329C4×Q8—534no—
330C4×Q8—1062no—
331C4×D4—534no—
332C4×D4—1062no—
333C4×D4—1068no—
334#117—1068no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—532no—
338C2×C4⋊C4—1061no—
339C2×C4⋊C4—1061no—
340C2×C4⋊C4—1062no—
341C2×C4⋊C4—2122no—
342C2×C4⋊C4—2122no—
343C2×C4⋊C4C22×D5312yes—
344C2×C4⋊C4C22×D5311yescharacteristic
345C2×C4⋊C4C22×D5311yescharacteristic
346C2×C22⋊C4—1061no—
347C2×C22⋊C4—1061no—
348C2×C22⋊C4—1061no—
349C2×C22⋊C4—1061no—
350C2×C22⋊C4—1061no—
351C2×C42—531no—
352C2×C42—531no—
353C2×C42—531no—
354C2×C42—534no—
355C2×C42—534no—
356C2×C42—1061no—
357C2×C42C22×D5311yescharacteristic
358C4⋊C8—1064no—
359#103—2122no—
360#103—2122no—
361#102—2122no—
362#102—2122no—
363C22⋊C8—1064no—
364C4×C8—22no—
365C4×C8—1062no—
366#95—21no—
367#95—21no—
368#95—1062no—
369#95—1062no—
370#95—1061no—
371#95—1061no—
372C24—531no—
373C24—1061no—
374C2×Q8—532no—
375C2×Q8—1061no—
376C2×Q8—1062no—
377C2×Q8—1062no—
378C2×D4—532no—
379C2×D4—1064no—
380C2×D4—1061no—
381C2×D4—1062no—
382C2×D4—1062no—
383C2×D4—2124no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—531no—
388C22×C4—531no—
389C22×C4—531no—
390C22×C4—534no—
391C22×C4—532no—
392C22×C4—532no—
393C22×C4—534no—
394C22×C4—1064no—
395C22×C4—1064no—
396C22×C4—1062no—
397C22×C4—1062no—
398C22×C4—1061no—
399C22×C4—1062no—
400C22×C4—2122no—
401C22×C4—2124no—
402C22×C4—2124no—
403C22×C4—2124no—
404C22×C4—2124no—
405C22×C4D4×D5311yescharacteristic
406C22×C4D4×D5311yescharacteristic
407C22×C4C23×D5311yescharacteristic
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—1062no—
412C2×C8—1062no—
413C2×C8—1062no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—1064no—
421C4⋊C4—1064no—
422C4⋊C4—1062no—
423C4⋊C4—1062no—
424C4⋊C4—2124no—
425C4⋊C4C2×C4×D5312yes—
426C4⋊C4C2×C4×D5312yes—
427C22⋊C4—1064no—
428C22⋊C4—1064no—
429C22⋊C4—1064no—
430C22⋊C4—1064no—
431C22⋊C4—1064no—
432C42—534no—
433C42—534no—
434C42—532no—
435C42—532no—
436C42—1064no—
437C42C2×C4×D5312yes—
438C42D4×D5312yes—
439C23—532no—
440C23—534no—
441C23—1062no—
442C23—1061no—
443C23—1061no—
444C23—1062no—
445C23—2122no—
446C23—2122no—
447C23—2121no—
448C23—2121no—
449C23C2×D4×D5311yescharacteristic
450Q8—1064no—
451Q8—2122no—
452D4—1064no—
453D4—2128no—
454D4—2122no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—532no—
465C2×C4—532no—
466C2×C4—532no—
467C2×C4—1068no—
468C2×C4—1062no—
469C2×C4—1062no—
470C2×C4—1061no—
471C2×C4—1061no—
472C2×C4—1064no—
473C2×C4—1064no—
474C2×C4—1062no—
475C2×C4—4248no—
476C2×C4—2124no—
477C2×C4—2128no—
478C2×C4—2122no—
479C2×C4—2122no—
480C2×C4—2122no—
481C2×C4—2121no—
482C2×C4—2121no—
483C2×C4C22⋊C4×D5312yes—
484C2×C4C22⋊C4×D5312yes—
485C2×C4C22×C4×D5311yescharacteristic
486C2×C4C2×D4×D5311yescharacteristic
487C2×C4#1313326312yes—
488C2×C4#1313326512yes—
489C8—42no—
490C8—2122no—
491C22—534no—
492C22—1062no—
493C22—1062no—
494C22—1064no—
495C22—1064no—
496C22—4242no—
497C22—4242no—
498C22—2122no—
499C22—2122no—
500C22—2122no—
501C22C2×C22⋊C4×D5311yescharacteristic
502C22#1681119111yescharacteristic
503C22#1681119811yescharacteristic
504C22#135×D5312yes—
505C22#1681142012yes—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—82no—
511C4—1062no—
512C4—1062no—
513C4—1062no—
514C4—4242no—
515C4—2122no—
516C4#1681110512yes—
517C4#1681120312yes—
518C2—1064no—
519C2—4242no—
520C2#2027730611yescharacteristic
521C2#416×D5311yescharacteristic
522C2#2028323011yescharacteristic
523C2#2028568112yes—
524C2#2028588412yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1991C2
socleC23×C53C22×D4
prime radicalC2×#13133192C22
cosocleC22×C4×C53C24
hypercenter#1991D53
hypocenterC53#21861
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
2. C2×#16810674C2semidirect
3. #382×D53C2semidirect
4. #20275681C2semidirect
5. #20275526C2semidirect
6. #20275524C2semidirect
8. #20275021C2semidirect
9. #20274729C2semidirect
10. #20274724C2semidirect
11. C53×#1991C2semidirect
13. #16811158C4semidirect
57. #13133205C2×C4semidirect
64. #13133182C2×C4semidirect
67. #13133179C2×C4semidirect
99. #125×C53C2×C4semidirect
260. #1991D53semidirect
308. C53#21861semidirect
321. #125C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111582
C2×#168106742
#382×D532
#202756812
#202755262
#202755242
#202750312
#202750212
#202747292
#202747242
C53×#19912
#2186153