Finite Group Databasebeta

← #27418023

Subgroups — #27418023

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×#16810905C211yesmaximal, characteristic, complemented
2C2×C4×#11719923C211yesmaximal, characteristic, complemented
3#20279637C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275718C212yesmaximal, complemented
6#20275677C211yesmaximal, characteristic, complemented
7#20275510C211yesmaximal, characteristic, complemented
8#20275022C212yesmaximal, complemented
9#20274908C212yesmaximal, complemented
10#20274715C211yesmaximal, characteristic, complemented
11C2×C53×#547C211yesmaximal, characteristic
12C22×#11719923C2211yescharacteristic
13C2×#13133196C2212yes—
14C2×C4×Dic106C2211yescharacteristic
15#16810905—24no—
16C4×#11719923—24no—
17C2×#13133148C2212yes—
18C2×#13133148C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133140C2212yes—
21C2×#13133140C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2212yes—
24C2×C22⋊#88322C2212yes—
25C2×C22⋊#88322C2212yes—
26C2×C4×#88322C2211yescharacteristic
27#16810546—22no—
28#16810526—21no—
29#16810519—21no—
30#16810493C2211yescharacteristic
31#16810483C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C22×#41×C53—21no—
36C53×#547—24no—
37C53×#547—24no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×#123×C53—22no—
40C2×C22⋊Q8×C53—21no—
41C2×#121×C53C2212yes—
42C2×C4×D4×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2212yes—
45C22×Dic106C2311yescharacteristic
46C2×#11719923—22no—
47C2×#11719923—22no—
48C2×#11719923—22no—
49C22×#88322C2311yescharacteristic
50C22×#88322C2312yes—
51C22×#88322C2311yescharacteristic
52#13133196—24no—
53#13133196—42no—
54C4×Dic106—22no—
55C4×Dic106—22no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2312yes—
60C2×#11719919—22no—
61C2×#11719919—21no—
62C2×C22⋊Dic53—22no—
63C2×C22⋊Dic53—22no—
64C2×C4×Dic53—21no—
65#13133148—24no—
66#13133148—22no—
67C4⋊#88322—22no—
68#13133140—24no—
69#13133140—22no—
70#13133138—22no—
71#13133138—42no—
72C22⋊#88322—24no—
73C22⋊#88322—42no—
74C4×#88322—22no—
75#13133124—22no—
76C2×#41×C53—28no—
77C2×#41×C53—44no—
78C2×#41×C53—44no—
79C22×Q8×C53—21no—
80C22×D4×C53—22no—
81C22×D4×C53D411yescharacteristic
82C22×D4×C53C2311yescharacteristic
83C23×C4×C53—22no—
84C23×C4×C53—21no—
85C23×C4×C53C2312yes—
86C4⋊Q8×C53—22no—
87C4⋊Q8×C53D414yes—
88#123×C53—216no—
89C22⋊Q8×C53—28no—
90#121×C53—24no—
91#121×C53—24no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95C4×D4×C53D414yes—
96C2×C4⋊C4×C53—24no—
97C2×C4⋊C4×C53—24no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2312yes—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—22no—
105C2×C22⋊C4×C53D412yes—
106C2×C42×C53C2311yescharacteristic
107C24×C53—21no—
108C24×C53C2×D412yes—
109#41×C53—432no—
110C2×Q8×C53—24no—
111C2×Q8×C53—42no—
112C2×Q8×C53—42no—
113C2×D4×C53—28no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—44no—
122C2×D4×C53—44no—
123C2×D4×C53SD1614yescomplemented
124C2×D4×C53C2×D412yes—
125C22×C4×C53—28no—
126C22×C4×C53—21no—
127C22×C4×C53—22no—
128C22×C4×C53—24no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—44no—
136C22×C4×C53—44no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C2×Dic106—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106#4112yescomplemented
152C4⋊C4×C53—28no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53C2×D412yes—
158C4⋊C4×C53C2×D414yes—
159C22⋊C4×C53—28no—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—24no—
162C42×C53C2×D412yes—
163C42×C53C2×D412yes—
164C22⋊Dic53—44no—
165C22⋊Dic53—82no—
166#11719923—28no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53—42no—
172C4⋊Dic53—81no—
173#11719919—44no—
174#11719919—42no—
175C4×Dic53—42no—
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×C53—22no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53C2×SD1614yes—
199C23×C53C22×D411yescharacteristic
200Q8×C53—48no—
201D4×C53—22no—
202D4×C53—22no—
203D4×C53—28no—
204D4×C53—416no—
205D4×C53—41no—
206D4×C53—41no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—24no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—416no—
215C2×C4×C53—41no—
216C2×C4×C53—42no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—48no—
224C2×C4×C53#12012yes—
225C2×C4×C53#12012yes—
226C2×C4×C53C2×SD1612yes—
227C2×C4×C53AGL(1,ℤ/8)12yes—
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53H(5,𝔽2)11yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×Dic53—22no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243Dic106—22no—
244Dic106—22no—
245Dic106—41no—
246Dic106—41no—
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×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53C2×#12011yescharacteristic
266C22×C53#54611yescharacteristic
267C22×C53#54711yescharacteristic
268C22×C53C22×SD1611yescharacteristic
269C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C53#57711yescharacteristic
271C22×C53#57811yescharacteristic
272C4×C53—22no—
273C4×C53—42no—
274C4×C53—42no—
275C4×C53—44no—
276C4×C53#45014yes—
277C4×C53#52511yescharacteristic
278C4×C53#53711yescharacteristic
279C4×C53#54811yescharacteristic
280C4×C53#55011yescharacteristic
281Dic53—42no—
282Dic53—82no—
283Dic53—81no—
284C2×#547D5311yescharacteristic
285C2×#475—531no—
286C2×C4×SD16—531no—
287#2538—531no—
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—82no—
297C2×C53#240711yescharacteristic
298C2×C53C2×#45011yescharacteristic
299C2×C53#301311yescharacteristic
300C2×C53#329811yescharacteristic
301C2×C53#333211yescharacteristic
302C2×C53#333511yescharacteristic
303C2×C53#335311yescharacteristic
304C22×#41—21no—
305C22×SD16—531no—
306#547—24no—
307#547—24no—
308C2×C4⋊Q8C2×D5311yescharacteristic
309C2×#123—22no—
310C2×C22⋊Q8—21no—
311C2×C22⋊Q8—532no—
312C2×#121C2×D5312yes—
313C2×C4×Q8—531no—
314C2×C4×D4C2×D5311yescharacteristic
315C2×C4×D4C2×D5311yescharacteristic
316C22×C4⋊C4C2×D5312yes—
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—1062no—
330#390—1061no—
331#389—1061no—
332#384—531no—
333#380—532no—
334#340—532no—
335#340—532no—
336#340—532no—
337C53#2412011yescharacteristic, complemented, Sylow
338C2×#41—28no—
339C2×#41—44no—
340C2×#41—44no—
341C22×Q8—21no—
342C22×Q8—531no—
343C22×D4—22no—
344C22×D4C22⋊D5311yescharacteristic
345C22×D4C22×D5311yescharacteristic
346C23×C4—22no—
347C23×C4—21no—
348C23×C4C22×D5312yes—
349C2×SD16—1062no—
350C2×SD16—1062no—
351C2×SD16—1062no—
352C22×C8—531no—
353C22×C8—531no—
354C22×C8—532no—
355C4⋊Q8—22no—
356C4⋊Q8C22⋊D5314yes—
357#123—216no—
358C22⋊Q8—28no—
359C22⋊Q8—1064no—
360C22⋊Q8—2122no—
361#121—24no—
362#121—24no—
363C4×Q8—1062no—
364C4×Q8—1062no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4C22⋊D5314yes—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4C22⋊D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5312yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1062no—
385C2×C22⋊C4—1062no—
386C2×C22⋊C4C22⋊D5312yes—
387C2×C42—1061no—
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—1062no—
400C24—21no—
401C24C2×C22⋊D5312yes—
402#41—432no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—532no—
407C2×Q8—1061no—
408C2×Q8—1061no—
409C2×Q8—1061no—
410C2×Q8—1061no—
411C2×Q8—1061no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4#1171992314yescomplemented
423C2×D4C2×C22⋊D5312yes—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1062no—
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—28no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1062no—
463C4⋊C4—1062no—
464C4⋊C4—4241no—
465C4⋊C4—2124no—
466C4⋊C4—2122no—
467C4⋊C4—2122no—
468C4⋊C4C2×C22⋊D5312yes—
469C4⋊C4C2×C22⋊D5314yes—
470C22⋊C4—28no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—4242no—
474C22⋊C4—2124no—
475C42—2122no—
476C42C2×C22⋊D5312yes—
477C42C2×C22⋊D5312yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×#1171992314yes—
492C23C22×C22⋊D5311yescharacteristic
493Q8—48no—
494Q8—1062no—
495Q8—1062no—
496Q8—2121no—
497Q8—2121no—
498D4—28no—
499D4—22no—
500D4—22no—
501D4—416no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—416no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—1062no—
522C2×C4—2121no—
523C2×C4—2122no—
524C2×C4—2121no—
525C2×C4—2122no—
526C2×C4—2121no—
527C2×C4—2122no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4#1313318612yes—
531C2×C4#1313318612yes—
532C2×C4C2×#1171992312yes—
533C2×C4#1313324312yes—
534C2×C4C22×C22⋊D5311yescharacteristic
535C2×C4C22×C22⋊D5311yescharacteristic
536C2×C4#41×D5311yescharacteristic
537C2×C4#41×D5311yescharacteristic
538C2×C4#1313326311yescharacteristic
539C2×C4#1313326411yescharacteristic
540C8—1062no—
541C8—2122no—
542C8—2121no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1313318611yescharacteristic
558C22#1681126111yescharacteristic
559C22#1681127211yescharacteristic
560C22C22×#1171992311yescharacteristic
561C22C2×#1313324311yescharacteristic
562C22#1681143111yescharacteristic
563C22#1681143811yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—44no—
568C4—4242no—
569C4—4241no—
570C4—2122no—
571C4#1681077414yes—
572C4#1681113011yescharacteristic
573C4#1681120111yescharacteristic
574C4#1681128011yescharacteristic
575C4#1681129211yescharacteristic
576C2—44no—
577C2—82no—
578C2#2027805411yescharacteristic
579C2C2×#1681077411yescharacteristic
580C2#2028403011yescharacteristic
581C2#2028657411yescharacteristic
582C2#2028703611yescharacteristic
583C2#2028707211yescharacteristic
584C2#2028729911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC23×C53C22×D4
hypercenterC2×#547D53
hypocenterC53#24120
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16810905C2semidirect
2. C2×C4×#11719923C2semidirect
3. #20279637C2semidirect
4. #20275901C2semidirect
5. #20275718C2semidirect
6. #20275677C2semidirect
7. #20275510C2semidirect
8. #20275022C2semidirect
9. #20274908C2semidirect
10. #20274715C2semidirect
123. C2×D4×C53SD16semidirect
151. C2×Dic106#41semidirect
184. C2×#88322#41semidirect
337. C53#24120semidirect
422. C2×D4#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168109052
C2×C4×#117199232
#202796372
#202759012
#202757182
#202756772
#202755102
#202750222
#202749082
#202747152
C2×C53×#5472
#2412053