Finite Group Databasebeta

← #27399553

Subgroups — #27399553

Order 13568.

1490 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811276C211yesmaximal, characteristic
2C2×#16810688C211yesmaximal, characteristic, complemented
3C2×#16810618C212yesmaximal, complemented
4#20277662C211yesmaximal, characteristic, complemented
5#20275518C211yesmaximal, characteristic, complemented
6#20275135C212yesmaximal, complemented
7#20275024C212yesmaximal, complemented
8#20274716C211yesmaximal, characteristic, complemented
9#20274654C212yesmaximal, complemented
10C4×#13133137C211yesmaximal, characteristic, complemented
11C53×#1981C211yesmaximal, characteristic, complemented
12C22×#11719949C2212yes—
13C22×C8×D53C2212yes—
14#16811276—24no—
15#16811276C418yescomplemented
16C2×#13133202—22no—
17C2×#13133193C2212yes—
18C2×#13133192—21no—
19C2×C4×D212C2211yescharacteristic
20C2×C4×D212C2211yescharacteristic
21C2×#13133175C2212yes—
22C2×C4⋊C4×D53—21no—
23#16810688—24no—
24#16810618—28no—
25C2×#13133141C2212yes—
26C2×#13133141C2212yes—
27C2×#13133139C2212yes—
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133129C2212yes—
31C2×C4×#88322C2211yescharacteristic
32#16810525—21no—
33#16810515—22no—
34#16810493C2211yescharacteristic
35#16810480C2212yes—
36C4×C4⋊Dic53—21no—
37#16810386C2212yes—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40C2×C4⋊Q8×C53C2211yescharacteristic
41C2×C4⋊C8×C53C2211yescharacteristic
42C2×#103×C53C2212yes—
43C2×#103×C53C2212yes—
44C4×C4⋊C4×C53C2211yescharacteristic
45C2×#11719949—24no—
46C2×#11719949—24no—
47C2×#11719949—24no—
48C2×#11719949—42no—
49C2×#11719949—42no—
50C2×#11719949C2×C418yes—
51C22×D212—22no—
52C22×D212D411yescharacteristic
53C22×D212C2311yescharacteristic
54C22×C4×D53—21no—
55C22×C4×D53—22no—
56C22×C4×D53C2312yes—
57C22×#88322C2311yescharacteristic
58C22×#88322C2311yescharacteristic
59C2×C8×D53—24no—
60C2×C8×D53—24no—
61C2×C8×D53—42no—
62#13133202—216no—
63#13133193—24no—
64#13133193C2×C418yes—
65#13133192—28no—
66C4×D212—22no—
67C4×D212C2×C414yes—
68C4×D212C2×C414yes—
69C4×D212D414yes—
70#13133175—24no—
71#13133175—42no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53D411yescharacteristic
75C2×C4⋊Dic53C2312yes—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—22no—
78C4⋊C4×D53—28no—
79C2×#11719921—22no—
80C2×#11719921—22no—
81C2×#11719921—22no—
82C2×#11719921—21no—
83C2×#11719921D412yes—
84C2×C4×Dic53—22no—
85#13133141—24no—
86#13133141—42no—
87#13133139—24no—
88#13133137—22no—
89#13133137D414yescomplemented
90#13133129—24no—
91C4×#88322—22no—
92#13133124—22no—
93C22×Q8×C53C2312yes—
94C22×C8×C53C2312yes—
95C4⋊Q8×C53—22no—
96C4⋊Q8×C53C2×C414yes—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C42×C53—21no—
102C2×C42×C53C2311yescharacteristic
103C4⋊C8×C53—22no—
104#103×C53—24no—
105#103×C53—42no—
106#95×C53—21no—
107C23×D53—21no—
108C23×D53C2×D412yes—
109C2×Q8×C53—22no—
110C2×Q8×C53—22no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—22no—
114C2×Q8×C53C22×C414yes—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2411yescharacteristic
121C22×Dic53—21no—
122C22×Dic53—22no—
123C22×Dic53C2×D412yes—
124#11719949—216no—
125#11719949—48no—
126#11719949—48no—
127#11719949—48no—
128C2×D212—22no—
129C2×D212—22no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—28no—
134C2×D212—44no—
135C2×D212—44no—
136C2×D212C22⋊C414yes—
137C2×D212C22×C412yes—
138C2×D212C2×D412yes—
139C2×D212#4112yes—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—24no—
145C2×C4×D53—28no—
146C2×C4×D53—44no—
147C2×C4×D53—44no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53C22×C414yes—
151C2×C4×D53C2×D414yes—
152C2×C8×C53—22no—
153C2×C8×C53—22no—
154C2×C8×C53—22no—
155C4⋊C4×C53—24no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53C2×D412yes—
161C42×C53—42no—
162C42×C53C22×C412yes—
163C42×C53C2×D412yes—
164#11719921—28no—
165#11719921—24no—
166#11719921—44no—
167#11719921—82no—
168#11719921C22⋊C418yes—
169C4⋊Dic53—21no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—42no—
173C4⋊Dic53—81no—
174C4⋊Dic53C22⋊C414yes—
175C4⋊Dic53C22×C412yes—
176#11719919—28no—
177C4×Dic53—44no—
178C2×#88322—21no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322C2×D412yes—
184C8×D53—48no—
185C23×C53C22×D411yescharacteristic
186C22×D53—22no—
187C22×D53—22no—
188C22×D53—22no—
189C22×D53—22no—
190C22×D53—22no—
191C22×D53—22no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53C2×C22⋊C414yes—
200Q8×C53—24no—
201Q8×C53—24no—
202Q8×C53—42no—
203Q8×C53—42no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—24no—
207C2×C4×C53—22no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53C2×C22⋊C412yes—
216C2×C4×C53C2×C22⋊C412yes—
217C2×C4×C53#13512yes—
218C2×C4×C53#13512yes—
219C2×C4×C53C23×C411yescharacteristic
220C2×C4×C53C22×D411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53H(5,𝔽2)11yescharacteristic
224C2×C4×C53H(5,𝔽2)11yescharacteristic
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53C2×C22⋊C414yes—
234D212—22no—
235D212—22no—
236D212—22no—
237D212—28no—
238D212—416no—
239D212C4×D414yes—
240C4×D53—24no—
241C4×D53—24no—
242C4×D53—24no—
243C4×D53—24no—
244C4×D53—416no—
245C4×D53—48no—
246C8×C53—42no—
247#88322—22no—
248#88322—41no—
249#22417—531nomaximal, Sylow
250C22×C53C22×C22⋊C411yescharacteristic
251C22×C53D4211yescharacteristic
252C22×C53#54611yescharacteristic
253C22×C53C2×#13511yescharacteristic
254C22×C53C2×#13511yescharacteristic
255C22×C53#57711yescharacteristic
256C22×C53#57811yescharacteristic
257C2×D53—24no—
258C2×D53—24no—
259C2×D53—24no—
260C2×D53—24no—
261C2×D53—42no—
262C2×D53—42no—
263C2×D53—42no—
264C2×D53—42no—
265C2×D53—42no—
266C2×D53—42no—
267C2×D53—42no—
268C2×D53—42no—
269C2×D53—42no—
270C2×D53—42no—
271C4×C53—22no—
272C4×C53—44no—
273C4×C53—42no—
274C4×C53—81no—
275C4×C53#41614yescomplemented
276C4×C53C2×C4×D411yescharacteristic
277C4×C53#51811yescharacteristic
278C4×C53D4211yescharacteristic
279C4×C53#54611yescharacteristic
280Dic53—24no—
281Dic53—42no—
282Dic53—82no—
283C2×#547—531no—
284C2×C8×D4—531no—
285C2×#416—532no—
286#2354—531no—
287#1981D5311yescharacteristic, complemented
288#1956—531no—
289#1892—532no—
290#1873—532no—
291#1802—531no—
292#1788—532no—
293C4×#102—531no—
294C2×C53D4×C22⋊C411yescharacteristic
295C2×C53C2×#41611yescharacteristic
296C2×C53#289811yescharacteristic
297C2×C53#328311yescharacteristic
298C2×C53#328411yescharacteristic
299C2×C53#330211yescharacteristic
300C2×C53#330311yescharacteristic
301D53—44no—
302D53—44no—
303C22×#41—532no—
304C23×C8—532no—
305#547—538no—
306#547—1064no—
307C2×C4⋊Q8C2×D5311yescharacteristic
308C2×#123—1062no—
309C2×C22⋊Q8—532no—
310C2×#121—1061no—
311C2×C4×D4—531no—
312C2×C4×D4—531no—
313C2×#117—532no—
314C22×C4⋊C4—1061no—
315C8×D4—1064no—
316C2×C4⋊C8C2×D5311yescharacteristic
317#416—1068no—
318C2×#103—532no—
319C2×#103—532no—
320C2×#103C2×D5312yes—
321C2×#103C2×D5312yes—
322C2×#102—531no—
323C2×#102—532no—
324C2×#102—531no—
325C2×C22⋊C8—532no—
326C2×C4×C8—531no—
327#390—1061no—
328#388—1062no—
329#384—531no—
330#380—532no—
331C4×C4⋊C4—1061no—
332C4×C4⋊C4C2×D5311yescharacteristic
333#340—532no—
334#340—531no—
335#340—531no—
336C53#2241711yescharacteristic, complemented, Sylow
337C2×#41—538no—
338C2×#41—1064no—
339C2×#41—1064no—
340C2×#41—1064no—
341C2×#41—2122no—
342C2×#41—2122no—
343C22×Q8C22×D5312yes—
344C22×D4—531no—
345C22×D4—531no—
346C22×D4—1062no—
347C23×C4—532no—
348C23×C4—1062no—
349C23×C4—1061no—
350C22×C8—531no—
351C22×C8—531no—
352C22×C8—1064no—
353C22×C8—1064no—
354C22×C8—2122no—
355C22×C8C22×D5312yes—
356C4⋊Q8—22no—
357C4⋊Q8C4×D5314yes—
358#123—10616no—
359C22⋊Q8—538no—
360C22⋊Q8—1064no—
361#121—1068no—
362C4×D4—534no—
363C4×D4—534no—
364C4×D4—534no—
365C4×D4—1062no—
366#117—1064no—
367#117—2122no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—532no—
373C2×C4⋊C4—1068no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C22⋊C4—532no—
380C2×C22⋊C4—1062no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4—1062no—
384C2×C42—21no—
385C2×C42—1062no—
386C2×C42C22×D5311yescharacteristic
387C4⋊C8—22no—
388#103—24no—
389#103—42no—
390#103—1064no—
391#103—2122no—
392#102—534no—
393#102—1062no—
394#102—1064no—
395C22⋊C8—1064no—
396C4×C8—1062no—
397#95—21no—
398#95—1062no—
399C24—532no—
400C24—1061no—
401#41—10616no—
402#41—2128no—
403#41—2128no—
404#41—2128no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8C2×C4×D5314yes—
411C2×D4—534no—
412C2×D4—532no—
413C2×D4—532no—
414C2×D4—532no—
415C2×D4—1068no—
416C2×D4—1062no—
417C2×D4—1062no—
418C2×D4—1061no—
419C2×D4—1061no—
420C2×D4—1061no—
421C2×D4—2124no—
422C2×D4—2124no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—532no—
427C22×C4—534no—
428C22×C4—534no—
429C22×C4—1068no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1061no—
435C22×C4—1064no—
436C22×C4—1062no—
437C22×C4—2124no—
438C22×C4—2124no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4D4×D5311yescharacteristic
442C22×C4D4×D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—532no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—2128no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—534no—
460C4⋊C4—532no—
461C4⋊C4—1068no—
462C4⋊C4—1061no—
463C4⋊C4—1062no—
464C4⋊C4—1062no—
465C4⋊C4—4241no—
466C4⋊C4—2122no—
467C4⋊C4D4×D5312yes—
468C22⋊C4—538no—
469C22⋊C4—1068no—
470C22⋊C4—1064no—
471C22⋊C4—4242no—
472C22⋊C4—2124no—
473C42—42no—
474C42—2124no—
475C42C2×C4×D5312yes—
476C42D4×D5312yes—
477C23—534no—
478C23—1062no—
479C23—1062no—
480C23—1062no—
481C23—1062no—
482C23—1062no—
483C23—1062no—
484C23—1062no—
485C23—1062no—
486C23—1062no—
487C23—2121no—
488C23—2121no—
489C23—2121no—
490C23—2121no—
491C23C2×D4×D5311yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—42no—
495Q8—42no—
496D4—534no—
497D4—1062no—
498D4—1062no—
499D4—1068no—
500D4—1062no—
501D4—21216no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—534no—
514C2×C4—1062no—
515C2×C4—1062no—
516C2×C4—1064no—
517C2×C4—1064no—
518C2×C4—1064no—
519C2×C4—1064no—
520C2×C4—1062no—
521C2×C4—21216no—
522C2×C4—2122no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2128no—
528C2×C4C22⋊C4×D5312yes—
529C2×C4C22⋊C4×D5312yes—
530C2×C4#1313323812yes—
531C2×C4#1313324012yes—
532C2×C4C22×C4×D5311yescharacteristic
533C2×C4C2×D4×D5311yescharacteristic
534C2×C4C2×D4×D5311yescharacteristic
535C2×C4#41×D5311yescharacteristic
536C2×C4#1313326211yescharacteristic
537C2×C4#1313326311yescharacteristic
538C8—42no—
539C8—1062no—
540C8—2121no—
541C22—1064no—
542C22—1064no—
543C22—1064no—
544C22—1064no—
545C22—2122no—
546C22—2122no—
547C22—2122no—
548C22—2122no—
549C22—2122no—
550C22—2122no—
551C22—2122no—
552C22—2122no—
553C22—2122no—
554C22—2122no—
555C22C2×C22⋊C4×D5311yescharacteristic
556C22#1681125711yescharacteristic
557C22#1681127011yescharacteristic
558C22C2×#1313323811yescharacteristic
559C22C2×#1313324011yescharacteristic
560C22#1681143211yescharacteristic
561C22#1681143911yescharacteristic
562C4—22no—
563C4—44no—
564C4—42no—
565C4—81no—
566C4—1064no—
567C4—4242no—
568C4—2122no—
569C4#1681061814yescomplemented
570C4C4×D4×D5311yescharacteristic
571C4#1681109511yescharacteristic
572C4#1681125811yescharacteristic
573C4#1681126611yescharacteristic
574C2—2124no—
575C2—2124no—
576C2#2027753611yescharacteristic
577C2C2×#1681061811yescharacteristic
578C2#2028324711yescharacteristic
579C2#2028643211yescharacteristic
580C2#2028644011yescharacteristic
581C2#2028663511yescharacteristic
582C2#2028664611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1981C2
socleC23×C53C22×D4
prime radicalC2×#16811276C2
cosocleC22×C4×C53C24
hypercenter#1981D53
hypocenterC53#22417
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
2. C2×#16810688C2semidirect
3. C2×#16810618C2semidirect
4. #20277662C2semidirect
5. #20275518C2semidirect
6. #20275135C2semidirect
7. #20275024C2semidirect
8. #20274716C2semidirect
9. #20274654C2semidirect
10. C4×#13133137C2semidirect
11. C53×#1981C2semidirect
15. #16811276C4semidirect
89. #13133137D4semidirect
275. C4×C53#416semidirect
287. #1981D53semidirect
336. C53#22417semidirect
569. C4#16810618semidirect

Maximal subgroups

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

subgroupindex
C2×#168112762
C2×#168106882
C2×#168106182
#202776622
#202755182
#202751352
#202750242
#202747162
#202746542
C4×#131331372
C53×#19812
#2241753