Finite Group Databasebeta

← #33055756

Subgroups — #33055756

Order 18688.

666 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
1#22618896C211yesmaximal, characteristic, complemented
2#22618894C211yesmaximal, characteristic, complemented
3C4×#14556612C211yesmaximal, characteristic
4#22608843C211yesmaximal, characteristic, complemented
5#22608842C211yesmaximal, characteristic, complemented
6#22608838C211yesmaximal, characteristic, complemented
7#22608837C211yesmaximal, characteristic, complemented
8#22608835C211yesmaximal, characteristic, complemented
9#22608834C211yesmaximal, characteristic, complemented
10#22608832C211yesmaximal, characteristic, complemented
11#22608678C211yesmaximal, characteristic, complemented
12#22608312C211yesmaximal, characteristic, complemented
13#22608310C211yesmaximal, characteristic, complemented
14#322×D73C211yesmaximal, characteristic, complemented
15C73×#1592C211yesmaximal, characteristic, complemented
16C2×#14556612C411yescharacteristic
17C4×#11924029C2211yescharacteristic
18C4×#11924027C2211yescharacteristic
19#17988718C2211yescharacteristic
20#17988717C2211yescharacteristic
21#17988714C2211yescharacteristic
22C4×#11924018C2211yescharacteristic
23#17988679C411yescharacteristic
24C4×#11924017C2211yescharacteristic
25SD16×Dic73C411yescharacteristic
26#17988674C411yescharacteristic
27C4×#11924013C2211yescharacteristic
28#17988667C411yescharacteristic
29#17988666C2211yescharacteristic
30#17988665C2211yescharacteristic
31#17988664C2211yescharacteristic
32#17988661—21no—
33#17988660—21no—
34#17988657—21no—
35#17988656—21no—
36#17988639—21no—
37#17988639C2211yescharacteristic
38#17988639C2211yescharacteristic
39#17988637—21no—
40#17988622C411yescharacteristic
41#17988582C411yescharacteristic
42#17988580C411yescharacteristic
43#17988539—21no—
44M16×Dic73—21no—
45C4×C8×D73C2211yescharacteristic
46C4×C8×D73C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988340C2211yescharacteristic
49#17988337C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52Q8⋊#10596831C2211yescharacteristic
53#17988317C2211yescharacteristic
54#17988316C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59#17988308C2211yescharacteristic
60#446×C73C2211yescharacteristic
61C4×SD16×C73C2211yescharacteristic
62#436×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×#11924029C2×C411yescharacteristic
68C2×#11924027C2×C411yescharacteristic
69#14556612—24no—
70C2×#11924018C2×C411yescharacteristic
71C2×#11924017C2×C411yescharacteristic
72C2×#11924013C2×C411yescharacteristic
73C2×#11924012—21no—
74C2×#11924011—21no—
75C2×#11924011—21no—
76C2×C8×D73—21no—
77C2×C8×D73C2×C411yescharacteristic
78C2×C8×D73C2×C411yescharacteristic
79C2×C8×D73D411yescharacteristic
80C2×C8×D73D411yescharacteristic
81Q8×Dic73C2×C411yescharacteristic
82C4×Dic146C2311yescharacteristic
83#14556556C2×C411yescharacteristic
84C4×D292—21no—
85C4×D292C2311yescharacteristic
86D4×Dic73C2×C411yescharacteristic
87#14556553—21no—
88#14556553C2×C411yescharacteristic
89C4×C22⋊D73—21no—
90#14556551—21no—
91#14556549—21no—
92#14556549C2×C411yescharacteristic
93#14556549C2×C411yescharacteristic
94#14556547—21no—
95C2×C4×Dic73—21no—
96C42×D73—21no—
97C42×D73C2311yescharacteristic
98#14556524C2×C411yescharacteristic
99#14556524C2×C411yescharacteristic
100#14556523C2×C411yescharacteristic
101C4⋊#10596831—21no—
102C4⋊#10596831C2311yescharacteristic
103C4⋊#10596831C2311yescharacteristic
104#14556520—21no—
105#14556520—21no—
106#14556516C2×C411yescharacteristic
107#14556515C2×C411yescharacteristic
108#14556514C2×C411yescharacteristic
109#14556513C2×C411yescharacteristic
110#14556512C2×C411yescharacteristic
111#14556511C2×C411yescharacteristic
112C22⋊#10596831—21no—
113#14556503—21no—
114#14556503—21no—
115#14556503—21no—
116#14556503—21no—
117#14556502—21no—
118#14556501—21no—
119#14556501—21no—
120C4×#10596831C2311yescharacteristic
121C4×#10596831C2311yescharacteristic
122C4×#10596831C2311yescharacteristic
123C8×Dic73—21no—
124C8×Dic73—21no—
125C8×Dic73—21no—
126C8×Dic73C2×C411yescharacteristic
127C8×Dic73C2×C411yescharacteristic
128C8×Dic73D411yescharacteristic
129C8×Dic73D411yescharacteristic
130C2×SD16×C73C2×C411yescharacteristic
131C2×M16×C73—21no—
132C4×Q8×C73C2311yescharacteristic
133C4×D4×C73C2311yescharacteristic
134#106×C73C2×C411yescharacteristic
135C4⋊C8×C73—21no—
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#103×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73—21no—
141#97×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145#11924029—22no—
146#11924029—22no—
147#11924027—22no—
148#11924027—22no—
149C2×Dic146C22×C411yescharacteristic
150C2×D292—21no—
151C2×D292C22×C411yescharacteristic
152C2×C22⋊D73—21no—
153C22×Dic73—21no—
154C22×Dic73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73—21no—
158C2×C4×D73C22×C411yescharacteristic
159C2×C4×D73C2×D411yescharacteristic
160C2×C4×D73#4111yescharacteristic
161#11924018—22no—
162#11924017—22no—
163#11924013—22no—
164#11924012—42no—
165#11924011—42no—
166#11924011—42no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C22×C411yescharacteristic
172C2×#10596831C2×D411yescharacteristic
173C2×#10596831C2×D411yescharacteristic
174C8×D73—22no—
175C8×D73—41no—
176C8×D73—41no—
177C8×D73—42no—
178C8×D73M1614yescomplemented
179C4⋊Dic73—21no—
180C4⋊Dic73C22×C411yescharacteristic
181C4⋊Dic73C22×C411yescharacteristic
182#11924007—21no—
183#11924007—21no—
184C22⋊Dic73—21no—
185#11924005—21no—
186#11924005—21no—
187#11924005—21no—
188#11924005—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73C22×C411yescharacteristic
195C4×Dic73C2×D411yescharacteristic
196C4×Dic73#4111yescharacteristic
197C2×Q8×C73C22×C411yescharacteristic
198C2×D4×C73C22×C411yescharacteristic
199C22×C4×C73—21no—
200SD16×C73—22no—
201M16×C73—42no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73C22×C411yescharacteristic
206C2×C8×C73C22×C411yescharacteristic
207C2×C8×C73C2×D411yescharacteristic
208C2×C8×C73C2×D411yescharacteristic
209C4⋊C4×C73—21no—
210C4⋊C4×C73C22×C411yescharacteristic
211C4⋊C4×C73C22×C411yescharacteristic
212C22⋊C4×C73—21no—
213C42×C73—21no—
214C42×C73C2411yescharacteristic
215C23×C73—21no—
216C22×D73—21no—
217C22×D73C4×D411yescharacteristic
218Q8×C73—21no—
219Q8×C73—21no—
220D4×C73—21no—
221D4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73C23×C411yescharacteristic
228C2×C4×C73C22×D411yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C22⋊D73—42no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73C4×D411yescharacteristic
237C2×Dic73C4×D411yescharacteristic
238C2×Dic73C4×D411yescharacteristic
239D292—21no—
240D292—21no—
241D292—42no—
242C4×D73—21no—
243C4×D73—21no—
244C4×D73—22no—
245C4×D73—41no—
246C4×D73—41no—
247C4×D73—42no—
248C4×D73—42no—
249C4×D73C2×M1612yes—
250C4×D73#13112yes—
251Dic146—21no—
252Dic146—21no—
253C8×C73—21no—
254C8×C73—21no—
255C8×C73—21no—
256C8×C73—41no—
257C8×C73—41no—
258C8×C73—41no—
259C8×C73C2×M1612yes—
260#10596831—21no—
261#10596831—21no—
262#10596831—21no—
263#10596831—41no—
264#10596831—41no—
265#10596831—41no—
266#10596831C2×M1612yes—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73C2×C4×D411yescharacteristic
271C2×D73—21no—
272C2×D73—21no—
273C2×D73—41no—
274C2×D73—41no—
275C2×D73—41no—
276C2×D73#43612yes—
277C4×C73—21no—
278C4×C73—41no—
279C4×C73—41no—
280C4×C73—41no—
281C4×C73C22×M1611yescharacteristic
282C4×C73C2×#13111yescharacteristic
283C4×C73#57511yescharacteristic
284C4×C73#57511yescharacteristic
285Dic73—21no—
286Dic73—41no—
287Dic73—41no—
288Dic73—41no—
289Dic73#43612yes—
290Dic73#44412yes—
291#17890—731nomaximal, Sylow
292C2×C73—81no—
293C2×C73C2×#43611yescharacteristic
294C2×C73#294311yescharacteristic
295C2×C73C2×#44411yescharacteristic
296D73—22no—
297D73—81no—
298#2965—731no—
299#2965—731no—
300C4×#135—731no—
301#1594—731no—
302#1594—731no—
303#1593—731no—
304#1593—731no—
305#1592—731no—
306#1592—731no—
307#1592—731no—
308#1592D7311yescharacteristic, complemented
309#1564—731no—
310#1476—731no—
311#1476—731no—
312C2×#322—731no—
313C73#1789011yescharacteristic, complemented, Sylow
314C2×#135—731no—
315C4×#41—731no—
316C4×#41—731no—
317#446—731no—
318#446—731no—
319#446—731no—
320#446C2×D7311yescharacteristic
321C4×Q16—731no—
322C4×Q16—731no—
323C4×SD16—731no—
324C4×SD16—731no—
325C4×SD16—731no—
326C4×SD16C2×D7311yescharacteristic
327C4×D8—731no—
328C4×D8—731no—
329#436—731no—
330#436—731no—
331#436—731no—
332#436C2×D7311yescharacteristic
333#435—1461no—
334#435—1461no—
335#435—1461no—
336#435—1461no—
337#431—731no—
338#431—731no—
339#431—1461no—
340#431—1461no—
341#427—731no—
342#416—731no—
343#416—731no—
344C4×M16—1461no—
345C4×M16—1461no—
346C2×C4×C8—731no—
347C2×C4×C8—731no—
348C2×C4×C8—731no—
349#335—731no—
350#335C2×D7311yescharacteristic
351#334—731no—
352#334—731no—
353Q8⋊C8—731no—
354Q8⋊C8—731no—
355Q8⋊C8—731no—
356Q8⋊C8C2×D7311yescharacteristic
357D4⋊C8—731no—
358D4⋊C8—731no—
359D4⋊C8—731no—
360D4⋊C8C2×D7311yescharacteristic
361#322—731no—
362#322—731no—
363#322—731no—
364#322C2×D7311yescharacteristic
365C2×#41—731no—
366C2×#41—731no—
367#135—1464no—
368C2×Q16—731no—
369C2×SD16—731no—
370C2×SD16C4×D7311yescharacteristic
371C2×D8—731no—
372C2×M16—21no—
373C2×M16—1461no—
374C2×M16—1461no—
375C2×M16—1461no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8—731no—
380C22×C8—1461no—
381C4×Q8—731no—
382C4×Q8—731no—
383C4×Q8—731no—
384C4×Q8C22×D7311yescharacteristic
385C4×D4—731no—
386C4×D4—731no—
387C4×D4—731no—
388C4×D4—1461no—
389C4×D4—1461no—
390C4×D4—1461no—
391C4×D4—1461no—
392C4×D4C22×D7311yescharacteristic
393#117—731no—
394#117—731no—
395#117—1461no—
396#117—1461no—
397C2×C42—731no—
398C2×C42—1461no—
399C2×C42—1461no—
400#107—731no—
401#107—731no—
402#106—731no—
403#106C4×D7311yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—731no—
406C4⋊C8—731no—
407C4⋊C8—1461no—
408C4⋊C8—1461no—
409C4⋊C8—1461no—
410C4⋊C8C22×D7311yescharacteristic
411C4⋊C8C22×D7311yescharacteristic
412#103—731no—
413#103—731no—
414#103—731no—
415#103C4×D7311yescharacteristic
416#102—731no—
417#102—731no—
418#102—731no—
419#102C4×D7311yescharacteristic
420C22⋊C8—21no—
421C22⋊C8—1461no—
422C22⋊C8—1461no—
423C22⋊C8—1461no—
424C22⋊C8—1461no—
425C22⋊C8—1461no—
426#97—21no—
427#97—1461no—
428#97—1461no—
429#97—1461no—
430C4×C8—731no—
431C4×C8—731no—
432C4×C8—731no—
433C4×C8—731no—
434C4×C8—731no—
435C4×C8—731no—
436C4×C8—731no—
437C4×C8—1461no—
438C4×C8—1461no—
439C4×C8—1461no—
440C4×C8C22×D7311yescharacteristic
441C4×C8C22×D7311yescharacteristic
442C4×C8C22×D7311yescharacteristic
443#41—1462no—
444#41—1462no—
445#41—1462no—
446#41—1462no—
447C2×Q8—731no—
448C2×Q8C2×C4×D7311yescharacteristic
449C2×D4—731no—
450C2×D4—1461no—
451C2×D4—1461no—
452C2×D4C2×C4×D7311yescharacteristic
453C22×C4—21no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462Q16—1462no—
463SD16—22no—
464SD16—1462no—
465D8—1462no—
466M16—42no—
467M16—2922no—
468M16—2922no—
469M16—2922no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—2921no—
474C2×C8—2921no—
475C2×C8—2922no—
476C2×C8—731no—
477C2×C8—734no—
478C2×C8—731no—
479C2×C8—731no—
480C2×C8—731no—
481C2×C8—1462no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8C2×C4×D7311yescharacteristic
486C2×C8C2×C4×D7311yescharacteristic
487C2×C8D4×D7311yescharacteristic
488C2×C8D4×D7311yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—731no—
491C4⋊C4—731no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4C2×C4×D7311yescharacteristic
496C4⋊C4C2×C4×D7311yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—1461no—
499C22⋊C4—1461no—
500C22⋊C4—1461no—
501C22⋊C4—1461no—
502C22⋊C4—1461no—
503C42—21no—
504C42—731no—
505C42—731no—
506C42—731no—
507C42—1461no—
508C42—1461no—
509C42—1461no—
510C42—1461no—
511C42—1461no—
512C42C23×D7311yescharacteristic
513C23—21no—
514C23—731no—
515C23—1461no—
516Q8—21no—
517Q8—21no—
518Q8—1461no—
519Q8—1461no—
520D4—21no—
521D4—21no—
522D4—2922no—
523D4—2922no—
524D4—1461no—
525D4—1461no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—2922no—
534C2×C4—2922no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—731no—
538C2×C4—731no—
539C2×C4—732no—
540C2×C4—732no—
541C2×C4—731no—
542C2×C4—1462no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4C22×C4×D7311yescharacteristic
549C2×C4C2×D4×D7311yescharacteristic
550C2×C4#41×D7311yescharacteristic
551C8—21no—
552C8—21no—
553C8—21no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—2921no—
558C8—2921no—
559C8—2921no—
560C8—732no—
561C8—1461no—
562C8—1461no—
563C8—1461no—
564C8M16×D7312yes—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—2921no—
569C22—2921no—
570C22—2921no—
571C22—732no—
572C22—1461no—
573C22—1461no—
574C22C4×D4×D7311yescharacteristic
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—2921no—
580C4—2921no—
581C4—2921no—
582C4—732no—
583C4—732no—
584C4—1461no—
585C4C2×M16×D7311yescharacteristic
586C4#131×D7311yescharacteristic
587C4#1798938311yescharacteristic
588C4#1798938611yescharacteristic
589C2—81no—
590C2—5841no—
591C2—1462no—
592C2#436×D7311yescharacteristic
593C2#2261872611yescharacteristic
594C2#444×D7311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1592C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556612C4
cosocleC42×C73C24
hypercenter#1592D73
hypocenterC73#17890
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1592

Complements

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

NH ≅ G/Ntype
1. #22618896C2semidirect
2. #22618894C2semidirect
4. #22608843C2semidirect
5. #22608842C2semidirect
6. #22608838C2semidirect
7. #22608837C2semidirect
8. #22608835C2semidirect
9. #22608834C2semidirect
10. #22608832C2semidirect
11. #22608678C2semidirect
12. #22608312C2semidirect
13. #22608310C2semidirect
14. #322×D73C2semidirect
15. C73×#1592C2semidirect
178. C8×D73M16semidirect
308. #1592D73semidirect
313. C73#17890semidirect

Maximal subgroups

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

subgroupindex
#226188962
#226188942
C4×#145566122
#226088432
#226088422
#226088382
#226088372
#226088352
#226088342
#226088322
#226086782
#226083122
#226083102
#322×D732
C73×#15922
#1789073