Finite Group Databasebeta

← #27389756

Subgroups — #27389756

Order 13568.

1520 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133182C211yesmaximal, characteristic, complemented
2C2×#16810764C211yesmaximal, characteristic
3C2×#16810722C212yesmaximal, complemented
4#20277221C211yesmaximal, characteristic
5#20275071C212yesmaximal, complemented
6#20275032C211yesmaximal, characteristic
7#20274966C212yesmaximal, complemented
8#20274649C211yesmaximal, characteristic, complemented
9#20274544C212yesmaximal, complemented
10#20274479C211yesmaximal, characteristic, complemented
11C53×#1852C211yesmaximal, characteristic, complemented
12C23×Dic106C2211yescharacteristic
13C22×#11719913C2212yes—
14C2×#13133191C2211yescharacteristic
15C2×#13133182—28no—
16C2×#13133182—21no—
17C2×#13133182—21no—
18C2×#13133182C2212yes—
19C2×#13133182C2212yes—
20C2×#13133179C2211yescharacteristic
21C2×#13133177—21no—
22C22×#11719919—21no—
23C22×C4×Dic53—21no—
24C22×C4×Dic53C2211yescharacteristic
25#16810764C418yescomplemented
26#16810722—28no—
27C2×#13133142C2212yes—
28C2×#13133140C2212yes—
29C2×#13133140C2212yes—
30C2×#13133137C2212yes—
31C2×C22⋊#88322C2211yescharacteristic
32C2×#13133127C2212yes—
33#16810524C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810512—21no—
36#16810500—21no—
37C4⋊C4×Dic53C2211yescharacteristic
38#16810470—21no—
39#16810385C2212yes—
40#16810377C2212yes—
41C22×C4⋊C4×C53C2211yescharacteristic
42C2×#107×C53C2212yes—
43C2×C22⋊C8×C53C2211yescharacteristic
44C53×#380C2211yescharacteristic
45C53×#340C2212yes—
46C22×Dic106—21no—
47C22×Dic106—22no—
48C22×Dic106—24no—
49C22×Dic106—24no—
50C22×Dic106C2×C414yescomplemented
51C22×Dic106C2312yes—
52C22×D212C2311yescharacteristic
53C22×C22⋊D53—21no—
54C23×Dic53—21no—
55C23×Dic53D411yescharacteristic
56C23×Dic53D411yescharacteristic
57C22×C4×D53—21no—
58C2×#11719913—24no—
59C2×#11719913—24no—
60C2×#11719913C2×C418yescomplemented
61C22×#88322C2312yes—
62#13133191—22no—
63#13133191C2×C414yes—
64#13133182—28no—
65#13133182—28no—
66#13133182—24no—
67#13133182—24no—
68#13133182—24no—
69#13133182D418yescomplemented
70#13133179—24no—
71#13133177—44no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—28no—
74C2×#11719919—22no—
75C2×#11719919—22no—
76C2×#11719919—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×C4×Dic53—28no—
82C2×C4×Dic53—24no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133142C2×C418yescomplemented
91#13133140—24no—
92#13133140—24no—
93#13133137—24no—
94C22⋊#88322—22no—
95#13133127—24no—
96#13133124—21no—
97#13133124—21no—
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C22×C8×C53C2312yes—
101C2×C4⋊C4×C53—24no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2312yes—
106C2×C22⋊C4×C53—21no—
107#107×C53—24no—
108C22⋊C8×C53C2×C414yescomplemented
109#95×C53—21no—
110C24×C53C2×D411yescharacteristic
111C23×D53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—24no—
115C22×C4×C53—22no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53#4112yes—
123C22×C4×C53C2411yescharacteristic
124C2×C22⋊D53—24no—
125C2×C22⋊D53—42no—
126C2×C22⋊D53—42no—
127C22×Dic53—24no—
128C22×Dic53—24no—
129C22×Dic53—24no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—41no—
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C2×D212—22no—
143C2×D212—22no—
144C2×D212—21no—
145C2×D212C22×C412yes—
146C2×C4×D53—42no—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×Dic106—24no—
150C2×Dic106—24no—
151C2×Dic106—216no—
152C2×Dic106—24no—
153C2×Dic106—22no—
154C2×Dic106—22no—
155C2×Dic106—22no—
156C2×Dic106—216no—
157C2×Dic106C22×C414yes—
158C2×Dic106C2×D414yes—
159C2×Dic106#4114yes—
160C2×C8×C53—22no—
161C2×C8×C53—22no—
162C2×C8×C53C22×C414yes—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53C2×D414yes—
168C22⋊C4×C53—42no—
169#11719921—24no—
170#11719921—42no—
171#11719921—42no—
172#11719921—42no—
173C4⋊Dic53—21no—
174C4⋊Dic53C22×C412yes—
175#11719919—28no—
176#11719919—28no—
177#11719919—42no—
178C4×Dic53—28no—
179C4×Dic53—22no—
180C4×Dic53—28no—
181C4×Dic53—22no—
182C4×Dic53—22no—
183C4×Dic53—42no—
184C4×Dic53C2×D414yes—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188#11719913—216no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53C4×D412yes—
194C23×C53AGL(1,ℤ/8)12yes—
195C23×C53#13712yes—
196C23×C53C22×D411yescharacteristic
197C22×D53—22no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—48no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53C4×D412yes—
219C2×C4×C53C23×C411yescharacteristic
220C2×C4×C53C22×D412yes—
221C2×C4×C53C2×#4112yes—
222C2×C4×C53C2×#4111yescharacteristic
223C22⋊D53—48no—
224C2×Dic53—216no—
225C2×Dic53—28no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—21no—
229C2×Dic53—24no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—22no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—44no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53#12012yes—
243C2×Dic53#12012yes—
244C2×Dic53#13712yes—
245D212—24no—
246D212—42no—
247C4×D53—84no—
248Dic106—216no—
249Dic106—24no—
250Dic106—24no—
251Dic106—24no—
252Dic106C4×D418yescomplemented
253C8×C53—24no—
254#88322—42no—
255#21454—531nomaximal, Sylow
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53#44014yescomplemented
263C22×C53C2×C4×D411yescharacteristic
264C22×C53C2×#12011yescharacteristic
265C22×C53#52511yescharacteristic
266C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C53C2×#13711yescharacteristic
268C22×C53C2×#13711yescharacteristic
269C22×C53#57511yescharacteristic
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—42no—
273C2×D53—81no—
274C2×D53—81no—
275C2×D53—81no—
276C2×D53—81no—
277C4×C53—22no—
278C4×C53—44no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53C2×C4×D412yes—
282C4×C53#52512yes—
283Dic53—22no—
284Dic53—22no—
285Dic53—22no—
286Dic53—28no—
287Dic53—42no—
288Dic53—41no—
289C22×C4×Q8—531no—
290C2×#448—531no—
291C2×#440—532no—
292C4×#121—531no—
293#1880—532no—
294#1874—531no—
295#1863—532no—
296#1852D5311yescharacteristic, complemented
297#1787—531no—
298#1763—532no—
299#1749—531no—
300C2×C53—24no—
301C2×C53C4×#12011yescharacteristic
302C2×C53C2×#44011yescharacteristic
303C2×C53C4×#13711yescharacteristic
304C2×C53#300411yescharacteristic
305C2×C53#300611yescharacteristic
306C2×C53#311211yescharacteristic
307C2×C53#311711yescharacteristic
308D53—82no—
309C23×Q8—531no—
310C22×SD16—532no—
311C2×#121—531no—
312C2×C4×Q8—532no—
313C2×C4×Q8—532no—
314C2×C4×Q8—1061no—
315C2×C4×Q8—1061no—
316C2×C4×Q8—1068no—
317C2×C4×D4—531no—
318C2×C4×D4—1061no—
319C22×C4⋊C4—1061no—
320C22×C4⋊C4C2×D5311yescharacteristic
321C22×C42—531no—
322C22×C42—1061no—
323#448—538no—
324#440—1068no—
325C2×#107C2×D5312yes—
326C2×#103—532no—
327C2×#103—532no—
328C2×#103—532no—
329C2×#102—532no—
330C2×C22⋊C8—531no—
331C2×C22⋊C8C2×D5311yescharacteristic
332C2×#97—532no—
333#390—531no—
334#389—531no—
335#388—1061no—
336#386—1061no—
337#380C2×D5311yescharacteristic
338C4×C4⋊C4—531no—
339C4×C22⋊C4—1061no—
340#340—532no—
341#340C2×D5312yes—
342#336—532no—
343C53#2145411yescharacteristic, complemented, Sylow
344C22×Q8—534no—
345C22×Q8—532no—
346C22×Q8—1064no—
347C22×Q8—1064no—
348C22×Q8—1061no—
349C22×Q8—1062no—
350C22×D4—531no—
351C22×D4—1061no—
352C23×C4—21no—
353C23×C4—531no—
354C23×C4—531no—
355C23×C4—1061no—
356C23×C4—1061no—
357C23×C4C22×D5311yescharacteristic
358C2×SD16—538no—
359C2×SD16—1064no—
360C2×SD16—1064no—
361C22×C8—532no—
362C22×C8C22×D5312yes—
363#121—534no—
364#121—1062no—
365C4×Q8—538no—
366C4×Q8—1064no—
367C4×Q8—1064no—
368C4×Q8—1068no—
369C4×Q8—1068no—
370C4×Q8—1064no—
371C4×D4—1064no—
372C4×D4—2124no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1068no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5312yes—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C42—531no—
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—531no—
392C2×C42—1064no—
393C2×C42—1062no—
394C2×C42—1062no—
395C2×C42—1068no—
396C2×C42—1061no—
397#107—24no—
398#103—538no—
399#103—1064no—
400#103—1064no—
401#102—1064no—
402C22⋊C8—1062no—
403C22⋊C8C4×D5314yescomplemented
404#97—1064no—
405#95—21no—
406#95—1061no—
407#95—1061no—
408C24—1061no—
409C24D4×D5311yescharacteristic
410C2×Q8—534no—
411C2×Q8—534no—
412C2×Q8—534no—
413C2×Q8—10616no—
414C2×Q8—1064no—
415C2×Q8—1064no—
416C2×Q8—10616no—
417C2×Q8—1064no—
418C2×Q8—1062no—
419C2×Q8—1062no—
420C2×Q8—1062no—
421C2×D4—532no—
422C2×D4—1064no—
423C2×D4—1062no—
424C2×D4—1062no—
425C2×D4—1061no—
426C2×D4—2122no—
427C2×D4—2122no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1064no—
444C22×C4—1064no—
445C22×C4—1064no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1061no—
450C22×C4—2122no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4—2121no—
454C22×C4C2×C4×D5312yes—
455C22×C4D4×D5311yescharacteristic
456C22×C4#1171994912yes—
457C22×C4C23×D5311yescharacteristic
458SD16—10616no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8C2×C4×D5314yes—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—532no—
470C4⋊C4—1068no—
471C4⋊C4—1068no—
472C4⋊C4—1061no—
473C4⋊C4—2122no—
474C4⋊C4D4×D5314yes—
475C22⋊C4—42no—
476C22⋊C4—1064no—
477C22⋊C4—2122no—
478C22⋊C4—2122no—
479C22⋊C4—2122no—
480C42—534no—
481C42—1062no—
482C42—1062no—
483C42—1068no—
484C42—1068no—
485C42—1062no—
486C42—2122no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—1062no—
492C23—2121no—
493C23—2121no—
494C23—2121no—
495C23—2121no—
496C23—2121no—
497C23—2121no—
498C23#1313317912yes—
499C23#1313324112yes—
500C23#1313324912yes—
501C23C2×D4×D5311yescharacteristic
502Q8—538no—
503Q8—10616no—
504Q8—1064no—
505Q8—1064no—
506Q8—1064no—
507D4—1064no—
508D4—2128no—
509D4—2122no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—532no—
525C2×C4—532no—
526C2×C4—532no—
527C2×C4—1068no—
528C2×C4—10616no—
529C2×C4—1064no—
530C2×C4—1061no—
531C2×C4—1064no—
532C2×C4—1064no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1062no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—4244no—
543C2×C4—2124no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4#1313317912yes—
547C2×C4C22×C4×D5311yescharacteristic
548C2×C4C2×D4×D5312yes—
549C2×C4#41×D5312yes—
550C2×C4C2×#1171994911yescharacteristic
551C8—24no—
552C8—2122no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—4241no—
560C22—4241no—
561C22—4241no—
562C22—4241no—
563C22—2122no—
564C22—2122no—
565C22—2122no—
566C22#1681072214yescomplemented
567C22C2×#1313317911yescharacteristic
568C22#120×D5311yescharacteristic
569C22#1681113111yescharacteristic
570C22C2×#1313324111yescharacteristic
571C22#137×D5311yescharacteristic
572C22C2×#1313324911yescharacteristic
573C22#1681141911yescharacteristic
574C4—22no—
575C4—44no—
576C4—81no—
577C4—81no—
578C4—1062no—
579C4—1062no—
580C4—1062no—
581C4—1068no—
582C4—2121no—
583C4—2122no—
584C4C4×D4×D5312yes—
585C4#1681113612yes—
586C2—24no—
587C2—4242no—
588C2#2027721611yescharacteristic
589C2C2×#1681072211yescharacteristic
590C2#2028353311yescharacteristic
591C2#2028395011yescharacteristic
592C2#2028396211yescharacteristic
593C2#2028483411yescharacteristic
594C2#2028488911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenter#1852D53
hypocenterC53#21454
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. C22×#13133182C2semidirect
3. C2×#16810722C2semidirect
5. #20275071C2semidirect
7. #20274966C2semidirect
8. #20274649C2semidirect
9. #20274544C2semidirect
10. #20274479C2semidirect
11. C53×#1852C2semidirect
25. #16810764C4semidirect
50. C22×Dic106C2×C4semidirect
60. C2×#11719913C2×C4semidirect
69. #13133182D4semidirect
90. #13133142C2×C4semidirect
108. C22⋊C8×C53C2×C4semidirect
252. Dic106C4×D4semidirect
262. C22×C53#440semidirect
296. #1852D53semidirect
343. C53#21454semidirect
403. C22⋊C8C4×D53semidirect
566. C22#16810722semidirect

Maximal subgroups

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

subgroupindex
C22×#131331822
C2×#168107642
C2×#168107222
#202772212
#202750712
#202750322
#202749662
#202746492
#202745442
#202744792
C53×#18522
#2145453