Finite Group Databasebeta

← #27418025

Subgroups — #27418025

Order 13568.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811277C211yesmaximal, characteristic
2C2×#16810908C211yesmaximal, characteristic, complemented
3C2×C4×#11719924C211yesmaximal, characteristic, complemented
4#20275904C212yesmaximal, complemented
5#20275716C212yesmaximal, complemented
6#20275675C211yesmaximal, characteristic, complemented
7#20275510C211yesmaximal, characteristic, complemented
8#20275023C212yesmaximal, complemented
9#20274907C212yesmaximal, complemented
10#20274712C211yesmaximal, characteristic, complemented
11C53×#2538C211yesmaximal, characteristic, complemented
12C22×#11719947—21no—
13C22×#11719924C2211yescharacteristic
14#16811277—24no—
15#16811277—24no—
16C2×#13133217C2211yescharacteristic
17C2×#13133201—22no—
18C2×#13133197—21no—
19C2×#13133191C2212yes—
20C2×C4×D212C2211yescharacteristic
21C22×C4⋊Dic53C2212yes—
22#16810908—24no—
23C4×#11719924—24no—
24C2×#13133148C2212yes—
25C2×#13133148C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133141C2212yes—
28C2×#13133141C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2212yes—
31C2×C22⋊#88322C2212yes—
32C2×C22⋊#88322C2212yes—
33C2×C4×#88322C2211yescharacteristic
34#16810385C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37C2×C22⋊Q8×C53C2212yes—
38C2×C4×Q8×C53C2211yescharacteristic
39C2×C4×D4×C53C2211yescharacteristic
40C53×#396—22no—
41C53×#390—21no—
42C53×#389—21no—
43C53×#384C2211yescharacteristic
44C53×#380C2212yes—
45C2×#11719947—28no—
46C2×#11719947—44no—
47C2×#11719947—44no—
48C22×Dic106—21no—
49C22×D212C2311yescharacteristic
50C22×C22⋊D53—22no—
51C23×Dic53—22no—
52C22×C4×D53—21no—
53C2×#11719924—22no—
54C2×#11719924—22no—
55C2×#11719924—22no—
56C22×#88322C2311yescharacteristic
57C22×#88322C2312yes—
58C22×#88322C2311yescharacteristic
59#13133217—22no—
60#13133217D414yes—
61#13133201—216no—
62#13133197—28no—
63#13133191—24no—
64#13133191—24no—
65C4×D212—22no—
66C4×D212D414yes—
67C2×C4⋊Dic53—24no—
68C2×C4⋊Dic53—24no—
69C2×C4⋊Dic53—22no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2312yes—
73C2×#11719921—21no—
74C2×#11719921—22no—
75#13133148—24no—
76#13133148—22no—
77C4⋊#88322—22no—
78#13133141—24no—
79#13133141—22no—
80#13133137—22no—
81#13133137—42no—
82C22⋊#88322—24no—
83C22⋊#88322—42no—
84C4×#88322—22no—
85C22×Q8×C53C2311yescharacteristic
86C22×D4×C53D411yescharacteristic
87C23×C4×C53C2312yes—
88C22⋊Q8×C53—24no—
89C22⋊Q8×C53—42no—
90C4×Q8×C53—22no—
91C4×Q8×C53—22no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2312yes—
101C2×C22⋊C4×C53—22no—
102C2×C22⋊C4×C53—22no—
103C2×C22⋊C4×C53D412yes—
104C2×C42×C53—21no—
105C2×C42×C53C2311yescharacteristic
106#95×C53—22no—
107C24×C53C2×D412yes—
108C23×D53—21no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53#4112yescomplemented
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53SD1614yescomplemented
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53C2×D412yes—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C2×C22⋊D53—28no—
132C2×C22⋊D53—44no—
133C2×C22⋊D53—44no—
134C22×Dic53—28no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—44no—
138C22×Dic53—44no—
139#11719947—432no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212C2×D412yes—
146C2×C4×D53—24no—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×Dic106—24no—
150C2×Dic106—42no—
151C2×Dic106—42no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—44no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53—81no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—44no—
163C22⋊C4×C53—82no—
164C42×C53—42no—
165C42×C53C2×D412yes—
166C42×C53C2×D412yes—
167#11719924—28no—
168#11719921—28no—
169#11719921—24no—
170C4⋊Dic53—28no—
171C4⋊Dic53—24no—
172C4⋊Dic53—21no—
173C4⋊Dic53—22no—
174C4⋊Dic53C2×D412yes—
175C4⋊Dic53C2×D414yes—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—21no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—21no—
182C2×#88322—22no—
183C2×#88322—21no—
184C2×#88322#4112yescomplemented
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53C2×SD1614yes—
192C23×C53C22×D411yescharacteristic
193C22×D53—22no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53—41no—
200Q8×C53—22no—
201Q8×C53—22no—
202Q8×C53—41no—
203Q8×C53—41no—
204D4×C53—28no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—41no—
221C2×C4×C53—42no—
222C2×C4×C53—41no—
223C2×C4×C53#12012yes—
224C2×C4×C53#12012yes—
225C2×C4×C53C2×SD1612yes—
226C2×C4×C53AGL(1,ℤ/8)12yes—
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53H(5,𝔽2)11yescharacteristic
232C2×C4×C532−1+411yescharacteristic
233C22⋊D53—416no—
234C2×Dic53—24no—
235C2×Dic53—22no—
236C2×Dic53—416no—
237C2×Dic53—42no—
238C2×Dic53—41no—
239C2×Dic53—42no—
240C2×Dic53—41no—
241D212—22no—
242D212—22no—
243D212—41no—
244D212—41no—
245C4×D53—48no—
246Dic106—48no—
247#88322—22no—
248#88322—41no—
249#88322—42no—
250#24120—531nomaximal, Sylow
251C22×C53—24no—
252C22×C53—24no—
253C22×C53—24no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53C2×#12011yescharacteristic
259C22×C53#54611yescharacteristic
260C22×C53#54711yescharacteristic
261C22×C53C22×SD1611yescharacteristic
262C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C53#57711yescharacteristic
264C22×C53#57811yescharacteristic
265C2×D53—42no—
266C2×D53—42no—
267C2×D53—42no—
268C2×D53—81no—
269C2×D53—81no—
270C2×D53—81no—
271C2×D53—81no—
272C4×C53—22no—
273C4×C53—42no—
274C4×C53—42no—
275C4×C53—82no—
276C4×C53—81no—
277C4×C53#45014yes—
278C4×C53#52511yescharacteristic
279C4×C53#53711yescharacteristic
280C4×C53#54811yescharacteristic
281C4×C53#55011yescharacteristic
282Dic53—44no—
283Dic53—42no—
284C2×#547—531no—
285C2×#475—531no—
286C2×C4×SD16—531no—
287#2538D5311yescharacteristic, complemented
288#2034—532no—
289#2000—532no—
290#1990—531no—
291#1954—531no—
292#1872—532no—
293#1853—532no—
294#1802—531no—
295C2×C53—44no—
296C2×C53#240711yescharacteristic
297C2×C53C2×#45011yescharacteristic
298C2×C53#301311yescharacteristic
299C2×C53#329811yescharacteristic
300C2×C53#333211yescharacteristic
301C2×C53#333511yescharacteristic
302C2×C53#335311yescharacteristic
303D53—82no—
304C22×#41—1061no—
305C22×SD16—531no—
306#547—1064no—
307#547—1064no—
308C2×C4⋊Q8—531no—
309C2×#123—1062no—
310C2×C22⋊Q8—1061no—
311C2×C22⋊Q8C2×D5312yes—
312C2×#121—532no—
313C2×C4×Q8C2×D5311yescharacteristic
314C2×C4×D4—531no—
315C2×C4×D4C2×D5311yescharacteristic
316C22×C4⋊C4—532no—
317#475—1064no—
318C4×SD16—1064no—
319C2×#106—532no—
320C2×#106—531no—
321C2×C4⋊C8—531no—
322C2×#103—532no—
323C2×#103—531no—
324C2×#102—531no—
325C2×#102—532no—
326C2×C22⋊C8—532no—
327C2×C22⋊C8—532no—
328C2×C4×C8—531no—
329#396—22no—
330#390—21no—
331#389—21no—
332#384C2×D5311yescharacteristic
333#380C2×D5312yes—
334#340—532no—
335#340—532no—
336#340—532no—
337C53#2412011yescharacteristic, complemented, Sylow
338C2×#41—1068no—
339C2×#41—2124no—
340C2×#41—2124no—
341C22×Q8—1061no—
342C22×Q8C22×D5311yescharacteristic
343C22×D4—531no—
344C22×D4—1062no—
345C22×D4C22⋊D5311yescharacteristic
346C23×C4—1062no—
347C23×C4—1061no—
348C23×C4C22×D5312yes—
349C2×SD16—1062no—
350C2×SD16—1062no—
351C2×SD16—1062no—
352C22×C8—531no—
353C22×C8—531no—
354C22×C8—532no—
355C4⋊Q8—534no—
356C4⋊Q8—1062no—
357#123—10616no—
358C22⋊Q8—24no—
359C22⋊Q8—42no—
360C22⋊Q8—1068no—
361#121—1064no—
362#121—1064no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—534no—
368C4×D4—1062no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—1064no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4C22⋊D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5312yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1062no—
386C2×C22⋊C4C22⋊D5312yes—
387C2×C42—21no—
388C2×C42C22×D5311yescharacteristic
389#106—1062no—
390#106—1064no—
391C4⋊C8—1062no—
392#103—1062no—
393#103—1064no—
394#102—1062no—
395#102—2122no—
396C22⋊C8—1064no—
397C22⋊C8—2122no—
398C4×C8—1062no—
399#95—22no—
400C24—1061no—
401C24C2×C22⋊D5312yes—
402#41—21232no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—1064no—
409C2×Q8—2122no—
410C2×Q8—2122no—
411C2×Q8#1171994712yescomplemented
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—532no—
415C2×D4—1068no—
416C2×D4—1061no—
417C2×D4—1061no—
418C2×D4—1061no—
419C2×D4—1061no—
420C2×D4—1061no—
421C2×D4—2124no—
422C2×D4—2124no—
423C2×D4#1171992414yescomplemented
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—1068no—
434C22×C4—1061no—
435C22×C4—1062no—
436C22×C4—1064no—
437C22×C4—2124no—
438C22×C4—2124no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4C2×C22⋊D5312yes—
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C23×D5311yescharacteristic
445SD16—1068no—
446C2×C8—532no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1062no—
451C2×C8—1062no—
452C2×C8—1061no—
453C2×C8—1062no—
454C2×C8—1061no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—81no—
464C4⋊C4—532no—
465C4⋊C4—534no—
466C4⋊C4—1068no—
467C4⋊C4—1064no—
468C4⋊C4—1061no—
469C4⋊C4—1062no—
470C22⋊C4—24no—
471C22⋊C4—44no—
472C22⋊C4—82no—
473C22⋊C4—1068no—
474C22⋊C4—1064no—
475C42—42no—
476C42D4×D5312yes—
477C42D4×D5312yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—1062no—
485C23—2121no—
486C23—2121no—
487C23—2121no—
488C23—2121no—
489C23—2121no—
490C23—2121no—
491C23C2×#1171992414yes—
492C23C22×C22⋊D5311yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497Q8—2128no—
498D4—28no—
499D4—1062no—
500D4—1062no—
501D4—21216no—
502D4—2121no—
503D4—2121no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—1062no—
523C2×C4—1064no—
524C2×C4—21216no—
525C2×C4—2121no—
526C2×C4—2122no—
527C2×C4—2121no—
528C2×C4—2122no—
529C2×C4—2128no—
530C2×C4#1313318512yes—
531C2×C4#1313318512yes—
532C2×C4C2×#1171992412yes—
533C2×C4#1313324212yes—
534C2×C4C2×D4×D5311yescharacteristic
535C2×C4C2×D4×D5311yescharacteristic
536C2×C4#41×D5311yescharacteristic
537C2×C4C2×#1171994711yescharacteristic
538C2×C4#1313326211yescharacteristic
539C2×C4#1313326411yescharacteristic
540C8—1062no—
541C8—2122no—
542C8—2121no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—4241no—
551C22—4241no—
552C22—4241no—
553C22—4241no—
554C22—2122no—
555C22—2122no—
556C22—2122no—
557C22C2×#1313318511yescharacteristic
558C22#1681126711yescharacteristic
559C22#1681127211yescharacteristic
560C22C22×#1171992411yescharacteristic
561C22C2×#1313324211yescharacteristic
562C22#1681143411yescharacteristic
563C22#1681143811yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—82no—
568C4—81no—
569C4—2122no—
570C4—2124no—
571C4#1681077314yes—
572C4#1681113311yescharacteristic
573C4#1681120211yescharacteristic
574C4#1681127911yescharacteristic
575C4#1681129111yescharacteristic
576C2—44no—
577C2—4242no—
578C2#2027805611yescharacteristic
579C2C2×#1681077311yescharacteristic
580C2#2028402711yescharacteristic
581C2#2028658011yescharacteristic
582C2#2028704211yescharacteristic
583C2#2028706611yescharacteristic
584C2#2028729311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2538C2
socleC23×C53C22×D4
prime radicalC2×#16811277C2
hypercenter#2538D53
hypocenterC53#24120
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C2×#16810908C2semidirect
3. C2×C4×#11719924C2semidirect
4. #20275904C2semidirect
5. #20275716C2semidirect
6. #20275675C2semidirect
7. #20275510C2semidirect
8. #20275023C2semidirect
9. #20274907C2semidirect
10. #20274712C2semidirect
11. C53×#2538C2semidirect
114. C2×Q8×C53#41semidirect
117. C2×D4×C53SD16semidirect
184. C2×#88322#41semidirect
287. #2538D53semidirect
337. C53#24120semidirect
411. C2×Q8#11719947semidirect
423. C2×D4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168112772
C2×#168109082
C2×C4×#117199242
#202759042
#202757162
#202756752
#202755102
#202750232
#202749072
#202747122
C53×#25382
#2412053