Finite Group Databasebeta

← #33104510

Subgroups — #33104510

Order 18688.

808 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618846C211yesmaximal, characteristic
2#22618844C211yesmaximal, characteristic
3C2×#17988706C211yesmaximal, characteristic
4#22618541C211yesmaximal, characteristic, complemented
5#22618540C211yesmaximal, characteristic, complemented
6#22610720C211yesmaximal, characteristic, complemented
7#22610718C211yesmaximal, characteristic, complemented
8#22610173C211yesmaximal, characteristic, complemented
9#22610172C211yesmaximal, characteristic, complemented
10#22610170C211yesmaximal, characteristic, complemented
11#22610105C211yesmaximal, characteristic, complemented
12#22609984C211yesmaximal, characteristic, complemented
13#22609983C211yesmaximal, characteristic, complemented
14#22609814C211yesmaximal, characteristic, complemented
15C73×#1861C211yesmaximal, characteristic, complemented
16C2×#14556611C2211yescharacteristic
17#17989354C2211yescharacteristic
18#17989353C2211yescharacteristic
19#17989353C2211yescharacteristic
20C2×#14556599C2211yescharacteristic
21C2×#14556598C2211yescharacteristic
22#17988707—22no—
23#17988706—22no—
24#17988706C412yescomplemented
25#17988706C412yescomplemented
26#17988706C412yescomplemented
27#17988706C412yescomplemented
28#17988702—22no—
29#17988701—22no—
30#17988636C2211yescharacteristic
31#17988635—22no—
32#17988635C2211yescharacteristic
33#17988635C2211yescharacteristic
34#17988634—22no—
35C2×#14556526—22no—
36C2×#14556526C2211yescharacteristic
37C2×#14556526C2211yescharacteristic
38C2×#14556526C2211yescharacteristic
39C2×#14556526C2211yescharacteristic
40C2×#14556526C2211yescharacteristic
41C2×#14556526C2211yescharacteristic
42#108×D73—22no—
43#17988623C2211yescharacteristic
44#17988611C2211yescharacteristic
45#17988605C2211yescharacteristic
46#17988604C2211yescharacteristic
47#17988603C2211yescharacteristic
48C2×#14556519C2211yescharacteristic
49#17988545C2211yescharacteristic
50#17988543C2211yescharacteristic
51C2×#14556502C2211yescharacteristic
52#17988359C2211yescharacteristic
53#17988358C2211yescharacteristic
54#17988357C2211yescharacteristic
55#17988347C2211yescharacteristic
56#568×C73C2211yescharacteristic
57#430×C73C2211yescharacteristic
58#423×C73C2211yescharacteristic
59#421×C73C2211yescharacteristic
60C2×#104×C73C2211yescharacteristic
61#344×C73C2211yescharacteristic
62#339×C73C2211yescharacteristic
63C2×#11924028C2311yescharacteristic
64#14556611C2×C412yes—
65#14556611C2×C412yes—
66#14556611D412yes—
67#14556611D412yes—
68C2×#11924018Q811yescharacteristic
69C2×#11924016Q811yescharacteristic
70C2×#11924015Q811yescharacteristic
71C2×#11924013Q811yescharacteristic
72#14556599—21no—
73#14556599—21no—
74#14556599—22no—
75#14556599—41no—
76#14556599C2×C412yes—
77#14556599C2×C412yes—
78#14556598—21no—
79#14556598—21no—
80#14556598—22no—
81#14556598—41no—
82#14556598C2×C412yes—
83#14556598C2×C412yes—
84#14556597—22no—
85#14556597—41no—
86C2×#11924012—21no—
87C2×#11924012—22no—
88C2×#11924012C2311yescharacteristic
89C2×#11924012C2311yescharacteristic
90C2×#11924012C2311yescharacteristic
91C2×#11924012C2311yescharacteristic
92C2×#11924011—21no—
93M16×D73—22no—
94M16×D73—22no—
95C22×#10596831—21no—
96C22×#10596831C2311yescharacteristic
97C22×#10596831C2311yescharacteristic
98C2×C8×D73—21no—
99#14556550C2311yescharacteristic
100#14556527—22no—
101#14556526—22no—
102#14556526—21no—
103#14556526—21no—
104#14556526—21no—
105#14556526—21no—
106#14556526—21no—
107#14556526—21no—
108#14556526—21no—
109#14556526—21no—
110#14556526—22no—
111#14556526C2×C412yes—
112#14556526C2×C412yes—
113#14556526D412yes—
114#14556526D412yes—
115#14556524—21no—
116#14556522—21no—
117C4⋊#10596831—41no—
118#14556519—22no—
119#14556519C2×C412yes—
120#14556519C2×C412yes—
121#14556518—22no—
122#14556517—22no—
123#14556502—21no—
124#14556502C2×C411yescharacteristic
125#14556502C2×C411yescharacteristic
126#14556502C2×C411yescharacteristic
127#14556502C2×C411yescharacteristic
128#14556501—21no—
129C4×#10596831—21no—
130C8×Dic73—21no—
131C2×#41×C73C2311yescharacteristic
132#131×C73—22no—
133#131×C73—41no—
134C2×M16×C73—22no—
135C2×M16×C73C2311yescharacteristic
136C2×M16×C73C2311yescharacteristic
137C2×M16×C73C2311yescharacteristic
138C2×M16×C73C2311yescharacteristic
139#117×C73C2311yescharacteristic
140C2×C42×C73C2311yescharacteristic
141#108×C73—22no—
142C4⋊C8×C73—41no—
143#104×C73—22no—
144#104×C73C2×C412yes—
145#104×C73C2×C412yes—
146#11924028—21no—
147#11924028—21no—
148#11924028C22×C412yes—
149#11924028C2×D412yes—
150C2×Dic146C2×Q811yescharacteristic
151C2×D292C2×Q811yescharacteristic
152C2×C22⋊D73—21no—
153C2×C4×D73—21no—
154#11924018C4⋊C412yes—
155#11924018C4⋊C412yes—
156#11924016C4⋊C412yes—
157#11924016C4⋊C412yes—
158#11924015C4⋊C412yes—
159#11924015C4⋊C412yes—
160#11924013C4⋊C412yes—
161#11924013C4⋊C412yes—
162#11924012—21no—
163#11924012—21no—
164#11924012—22no—
165#11924012—22no—
166#11924012—21no—
167#11924012—21no—
168#11924012—21no—
169#11924012—21no—
170#11924012—42no—
171#11924012C22×C412yes—
172#11924012C2×D412yes—
173#11924011—22no—
174#11924011—22no—
175#11924011—42no—
176#11924011—42no—
177C2×#10596831—21no—
178C2×#10596831—21no—
179C2×#10596831—22no—
180C2×#10596831—22no—
181C2×#10596831—21no—
182C2×#10596831—21no—
183C2×#10596831—42no—
184C2×#10596831C22×C411yescharacteristic
185C2×#10596831C22×C411yescharacteristic
186C2×#10596831C22×C411yescharacteristic
187C2×#10596831C22×C411yescharacteristic
188C2×#10596831C2×D411yescharacteristic
189C2×#10596831C2×D411yescharacteristic
190C2×#10596831C2×D411yescharacteristic
191C2×#10596831C2×D411yescharacteristic
192C2×#10596831C2×Q811yescharacteristic
193C2×#10596831C2×Q811yescharacteristic
194C2×#10596831#4111yescharacteristic
195C2×#10596831#4111yescharacteristic
196C8×D73—22no—
197C8×D73—22no—
198C8×D73—42no—
199C8×D73—42no—
200C4⋊Dic73—21no—
201C22⋊Dic73—21no—
202C4×Dic73—21no—
203#41×C73—21no—
204#41×C73—21no—
205#41×C73C22×C412yes—
206#41×C73C2×D412yes—
207C2×Q8×C73C2×Q811yescharacteristic
208C2×D4×C73—21no—
209C2×D4×C73C2×Q811yescharacteristic
210C22×C4×C73—21no—
211C22×C4×C73—21no—
212C22×C4×C73C2411yescharacteristic
213M16×C73—21no—
214M16×C73—21no—
215M16×C73—21no—
216M16×C73—21no—
217M16×C73—21no—
218M16×C73—21no—
219M16×C73—42no—
220M16×C73C22×C412yes—
221M16×C73C2×D412yes—
222C2×C8×C73—21no—
223C2×C8×C73—21no—
224C2×C8×C73—21no—
225C2×C8×C73—21no—
226C2×C8×C73—42no—
227C4⋊C4×C73—21no—
228C22⋊C4×C73—21no—
229C42×C73—21no—
230C42×C73—21no—
231C42×C73C22×C411yescharacteristic
232C42×C73C22×C411yescharacteristic
233C23×C73—21no—
234C23×C732−1+411yescharacteristic
235C22×D73—21no—
236Q8×C73—21no—
237Q8×C73C2×C4⋊C412yes—
238D4×C73—22no—
239D4×C73—22no—
240D4×C73—21no—
241D4×C73C2×C4⋊C412yes—
242C2×C4×C73—21no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—41no—
248C2×C4×C73—81no—
249C2×C4×C73C23×C411yescharacteristic
250C2×C4×C73C22×D411yescharacteristic
251C2×C4×C73C22×D411yescharacteristic
252C2×C4×C73C22×Q811yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73H(5,𝔽2)11yescharacteristic
255C22⋊D73—22no—
256C22⋊D73—22no—
257C2×Dic73—21no—
258C2×Dic73—41no—
259D292—21no—
260D292C2×C4⋊C412yes—
261C4×D73—22no—
262C4×D73—22no—
263Dic146—21no—
264Dic146C2×C4⋊C412yes—
265C8×C73—22no—
266C8×C73—22no—
267C8×C73—41no—
268C8×C73—41no—
269#10596831—22no—
270#10596831—22no—
271#10596831—41no—
272#10596831—41no—
273#10596831C2×C4⋊C411yescharacteristic
274#10596831C2×C4⋊C411yescharacteristic
275#10596831C2×C4⋊C411yescharacteristic
276#10596831C2×C4⋊C411yescharacteristic
277#10596831C4×D411yescharacteristic
278#10596831C4×D411yescharacteristic
279#10596831C4×D411yescharacteristic
280#10596831C4×D411yescharacteristic
281C22×C73—22no—
282C22×C73—41no—
283C22×C73—81no—
284C22×C73#52011yescharacteristic
285C22×C73#54711yescharacteristic
286C22×C73D4×Q811yescharacteristic
287C2×D73—22no—
288C2×D73—81no—
289C4×C73—22no—
290C4×C73—41no—
291C4×C73—41no—
292C4×C73—81no—
293C4×C73C22×C4⋊C411yescharacteristic
294C4×C73C2×C4×D411yescharacteristic
295C4×C73D4211yescharacteristic
296C4×C73#55411yescharacteristic
297Dic73—22no—
298Dic73—81no—
299#22695—731nomaximal, Sylow
300C2×C73—21no—
301C2×C73—21no—
302C2×C73—21no—
303C2×C73—42no—
304C2×C73D4×C4⋊C411yescharacteristic
305D73—42no—
306#2957—731no—
307#2957—731no—
308C2×#444—731no—
309#2916—731no—
310#2916—731no—
311#1953—731no—
312#1953—731no—
313#1861—731no—
314#1861—731no—
315#1861—731no—
316#1861D7311yescharacteristic, complemented
317#1849—731no—
318#1814—731no—
319#1814—731no—
320#1773—731no—
321C73#2269511yescharacteristic, complemented, Sylow
322C2×#135—731no—
323#568—731no—
324#568—731no—
325#568—731no—
326#568C2×D7311yescharacteristic
327C2×#131—731no—
328C2×#131—731no—
329#444—732no—
330#444—732no—
331#444—732no—
332#444—732no—
333#444—1462no—
334#444—1462no—
335#443—1462no—
336#443—1462no—
337#430—731no—
338#430—731no—
339#430—731no—
340#430—1462no—
341#430—1462no—
342#430C2×D7311yescharacteristic
343C2×#108—731no—
344C2×#108—731no—
345C2×#108—731no—
346C2×#108—731no—
347C2×#108—731no—
348C2×#108—731no—
349C2×#108—1462no—
350C2×#108—1462no—
351#427—731no—
352#423—731no—
353#423C2×D7311yescharacteristic
354#421—731no—
355#421—731no—
356#421—731no—
357#421C2×D7311yescharacteristic
358C2×#104—731no—
359C2×#104C2×D7311yescharacteristic
360#405—731no—
361#405—731no—
362C2×#97—731no—
363#344—731no—
364#344—731no—
365#344—731no—
366#344C2×D7311yescharacteristic
367#339—731no—
368#339C2×D7311yescharacteristic
369C2×#41—731no—
370C2×#41C22×D7311yescharacteristic
371#135—732no—
372#135—732no—
373#135—732no—
374#135—732no—
375C2×Q16—731no—
376C2×SD16—731no—
377C2×SD16—731no—
378C2×D8—731no—
379#131—22no—
380#131—41no—
381#131—2921no—
382#131—2921no—
383#131—2921no—
384#131—732no—
385#131—732no—
386#131—732no—
387#131—732no—
388#131—1462no—
389#131—1462no—
390#131—1462no—
391#131—1461no—
392#131—1461no—
393#131—1461no—
394#131—1461no—
395C2×M16—22no—
396C2×M16—731no—
397C2×M16—731no—
398C2×M16—731no—
399C2×M16—731no—
400C2×M16—1461no—
401C2×M16—1462no—
402C2×M16—1462no—
403C2×M16—1462no—
404C2×M16—1461no—
405C2×M16C22×D7311yescharacteristic
406C2×M16C22×D7311yescharacteristic
407C2×M16C22×D7311yescharacteristic
408C2×M16C22×D7311yescharacteristic
409C22×C8—731no—
410C22×C8—731no—
411C22×C8—1461no—
412C22×C8—1461no—
413#117—731no—
414#117C22×D7311yescharacteristic
415C2×C42C22×D7311yescharacteristic
416#108—22no—
417#108—732no—
418#108—732no—
419#108—732no—
420#108—732no—
421#108—1462no—
422#108—1461no—
423#108—1461no—
424#108—1461no—
425#108—1461no—
426#108—1461no—
427#108—1461no—
428#108—1461no—
429#108—1461no—
430#108—1462no—
431#108—1462no—
432#107—1461no—
433#106—1461no—
434C4⋊C8—41no—
435C4⋊C8—2921no—
436#104—22no—
437#104—732no—
438#104—732no—
439#104—1462no—
440#104—1462no—
441#104—1462no—
442#104C4×D7312yes—
443#104C4×D7312yes—
444#97—731no—
445#97—731no—
446#97—731no—
447#97—731no—
448#97—1461no—
449#97—1461no—
450C4×C8—1461no—
451C4×C8—1461no—
452#41—21no—
453#41—21no—
454#41—732no—
455#41—732no—
456#41—1461no—
457#41—1461no—
458#41C2×C4×D7312yes—
459#41D4×D7312yes—
460C2×Q8—731no—
461C2×Q8Q8×D7311yescharacteristic
462C2×D4—21no—
463C2×D4—731no—
464C2×D4—1461no—
465C2×D4Q8×D7311yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—1461no—
469C22×C4C23×D7311yescharacteristic
470Q16—732no—
471Q16—732no—
472SD16—732no—
473SD16—732no—
474SD16—732no—
475SD16—732no—
476D8—732no—
477D8—732no—
478M16—21no—
479M16—21no—
480M16—21no—
481M16—21no—
482M16—21no—
483M16—21no—
484M16—42no—
485M16—2922no—
486M16—2922no—
487M16—2922no—
488M16—732no—
489M16—732no—
490M16—1462no—
491M16—1462no—
492M16—1461no—
493M16—1461no—
494M16—1461no—
495M16—1461no—
496M16—1462no—
497M16—1462no—
498M16—1461no—
499M16—1461no—
500M16C2×C4×D7312yes—
501M16D4×D7312yes—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—42no—
507C2×C8—2922no—
508C2×C8—2922no—
509C2×C8—2922no—
510C2×C8—731no—
511C2×C8—731no—
512C2×C8—731no—
513C2×C8—731no—
514C2×C8—731no—
515C2×C8—731no—
516C2×C8—731no—
517C2×C8—731no—
518C2×C8—731no—
519C2×C8—731no—
520C2×C8—731no—
521C2×C8—731no—
522C2×C8—1462no—
523C2×C8—1462no—
524C2×C8—1461no—
525C2×C8—1461no—
526C2×C8—1462no—
527C2×C8—1462no—
528C2×C8—1461no—
529C2×C8—1461no—
530C4⋊C4—21no—
531C4⋊C4—1461no—
532C22⋊C4—21no—
533C22⋊C4—1461no—
534C42—21no—
535C42—21no—
536C42—1461no—
537C42C2×C4×D7311yescharacteristic
538C42C2×C4×D7311yescharacteristic
539C23—21no—
540C23—1461no—
541C23#1455663811yescharacteristic
542Q8—21no—
543Q8—732no—
544Q8—1461no—
545Q8C4⋊C4×D7312yes—
546D4—22no—
547D4—22no—
548D4—21no—
549D4—732no—
550D4—1462no—
551D4—1462no—
552D4—1461no—
553D4C4⋊C4×D7312yes—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—81no—
561C2×C4—2921no—
562C2×C4—1461no—
563C2×C4—1462no—
564C2×C4—1462no—
565C2×C4C22×C4×D7311yescharacteristic
566C2×C4C2×D4×D7311yescharacteristic
567C2×C4C2×D4×D7311yescharacteristic
568C2×C4C2×Q8×D7311yescharacteristic
569C2×C4#41×D7311yescharacteristic
570C2×C4#1455663611yescharacteristic
571C8—22no—
572C8—22no—
573C8—41no—
574C8—41no—
575C8—2921no—
576C8—2921no—
577C8—731no—
578C8—731no—
579C8—731no—
580C8—731no—
581C8—731no—
582C8—731no—
583C8—731no—
584C8—731no—
585C8—1462no—
586C8—1462no—
587C22—22no—
588C22—41no—
589C22—81no—
590C22—5841no—
591C22—1462no—
592C22#1798906611yescharacteristic
593C22#1798923511yescharacteristic
594C22#1798925011yescharacteristic
595C4—22no—
596C4—41no—
597C4—41no—
598C4—81no—
599C4—5841no—
600C4—1462no—
601C4C2×C4⋊C4×D7311yescharacteristic
602C4C4×D4×D7311yescharacteristic
603C4#1798921911yescharacteristic
604C4#1798929111yescharacteristic
605C2—21no—
606C2—21no—
607C2—21no—
608C2—42no—
609C2—2922no—
610C2#2261282911yescharacteristic
subgroupG/H
centerC4#17989219
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1861C2
socleC2×C73D4×C4⋊C4
prime radicalC2×#14556611C22
cosocleC22×C4×C73C24
hypercenter#1861D73
hypocenterC73#22695
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1861

Complements

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

NH ≅ G/Ntype
4. #22618541C2semidirect
5. #22618540C2semidirect
6. #22610720C2semidirect
7. #22610718C2semidirect
8. #22610173C2semidirect
9. #22610172C2semidirect
10. #22610170C2semidirect
11. #22610105C2semidirect
12. #22609984C2semidirect
13. #22609983C2semidirect
14. #22609814C2semidirect
15. C73×#1861C2semidirect
24. #17988706C4semidirect
25. #17988706C4semidirect
26. #17988706C4semidirect
27. #17988706C4semidirect
316. #1861D73semidirect
321. C73#22695semidirect

Maximal subgroups

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

subgroupindex
#226188462
#226188442
C2×#179887062
#226185412
#226185402
#226107202
#226107182
#226101732
#226101722
#226101702
#226101052
#226099842
#226099832
#226098142
C73×#18612
#2269573