Finite Group Databasebeta

← #33144909

Subgroups — #33144909

Order 18688.

678 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620471C211yesmaximal, characteristic, complemented
2#22619793C211yesmaximal, characteristic
3#22619782C211yesmaximal, characteristic
4#22611444C211yesmaximal, characteristic, complemented
5#22611439C211yesmaximal, characteristic, complemented
6#22611279C211yesmaximal, characteristic, complemented
7#22611274C211yesmaximal, characteristic, complemented
8#22610709C211yesmaximal, characteristic, complemented
9#22610685C211yesmaximal, characteristic, complemented
10#22610164C211yesmaximal, characteristic
11#22610157C211yesmaximal, characteristic
12#22610096C211yesmaximal, characteristic, complemented
13#22609995C211yesmaximal, characteristic
14#22609773C211yesmaximal, characteristic
15C73×#2912C211yesmaximal, characteristic
16#17989215C2211yescharacteristic
17#17989215C2211yescharacteristic
18#17988994—21no—
19#17988994—21no—
20#17988991—21no—
21#17988988—21no—
22#17988894—21no—
23#17988894—21no—
24#17988887—21no—
25#17988887—21no—
26#17988882—21no—
27#17988877—21no—
28#17988872—21no—
29#17988867—21no—
30C2×#14556527C2211yescharacteristic
31#17988630C2211yescharacteristic
32#17988630C2211yescharacteristic
33#17988627C2211yescharacteristic
34#17988627C2211yescharacteristic
35C2×#14556525C2211yescharacteristic
36C2×#14556524C2211yescharacteristic
37C2×#14556522C2211yescharacteristic
38#17988614C2211yescharacteristic
39#17988614C2211yescharacteristic
40C2×#14556518C2211yescharacteristic
41#17988588C2211yescharacteristic
42#17988588C2211yescharacteristic
43#17988583C2211yescharacteristic
44#17988542C2211yescharacteristic
45C2×#14556501C2211yescharacteristic
46#17988456C2211yescharacteristic
47#17988452C2211yescharacteristic
48#17988356C2211yescharacteristic
49#17988356C2211yescharacteristic
50#17988355C2211yescharacteristic
51#17988355C2211yescharacteristic
52#17988351C2211yescharacteristic
53#17988348C2211yescharacteristic
54#17988346C2211yescharacteristic
55#17988346C2211yescharacteristic
56C2×#131×C73C2211yescharacteristic
57#520×C73C2211yescharacteristic
58#520×C73C2211yescharacteristic
59#428×C73—21no—
60#428×C73—21no—
61#428×C73C2211yescharacteristic
62#428×C73C2211yescharacteristic
63#427×C73—21no—
64#427×C73C2211yescharacteristic
65C2×#106×C73—21no—
66C2×#106×C73C2211yescharacteristic
67C2×#11924012C2311yescharacteristic
68C2×#11924012C2311yescharacteristic
69C2×#11924012C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71#14556591—21no—
72#14556591—21no—
73#14556590—21no—
74#14556590—21no—
75#14556583—21no—
76#14556583—21no—
77#14556582—21no—
78#14556582—21no—
79#14556568—41no—
80#14556568—41no—
81#14556550C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×#11924007—41no—
85C2×C4×Dic73C2311yescharacteristic
86#14556527—22no—
87#14556525—21no—
88#14556525—21no—
89#14556524—21no—
90#14556524—21no—
91#14556524—21no—
92#14556524—21no—
93#14556524—21no—
94#14556524—21no—
95#14556522—21no—
96#14556522—21no—
97#14556522—21no—
98#14556522—21no—
99#14556522—21no—
100#14556522—21no—
101C4⋊#10596831—21no—
102C4⋊#10596831—21no—
103C4⋊#10596831—21no—
104C4⋊#10596831—21no—
105#14556518Q814yescomplemented
106#14556515—21no—
107#14556515—21no—
108#14556515D411yescharacteristic
109#14556515D411yescharacteristic
110#14556512—21no—
111#14556512—21no—
112#14556512D411yescharacteristic
113#14556512D411yescharacteristic
114#14556501—21no—
115#14556501—22no—
116C8×Dic73—21no—
117#14556498—41no—
118C2×#41×C73C2311yescharacteristic
119#131×C73—22no—
120#131×C73Q814yescomplemented
121C2×M16×C73—21no—
122C2×M16×C73C2311yescharacteristic
123C22×C8×C73—21no—
124C22×C8×C73C2311yescharacteristic
125C4×Q8×C73—21no—
126C4×Q8×C73—21no—
127C4×D4×C73—21no—
128C4×D4×C73—21no—
129C4×D4×C73—41no—
130C4×D4×C73—41no—
131#117×C73—21no—
132#117×C73—21no—
133#117×C73C2311yescharacteristic
134#117×C73C2311yescharacteristic
135C2×C4⋊C4×C73—21no—
136C2×C4⋊C4×C73—21no—
137C2×C4⋊C4×C73C2311yescharacteristic
138C2×C4⋊C4×C73C2311yescharacteristic
139#107×C73—21no—
140#107×C73—21no—
141#107×C73D411yescharacteristic
142#107×C73D411yescharacteristic
143#106×C73—21no—
144#106×C73—21no—
145#106×C73—21no—
146#106×C73—21no—
147#106×C73D411yescharacteristic
148#106×C73D411yescharacteristic
149C2×Dic146—41no—
150C2×Dic146—41no—
151C22×Dic73—21no—
152C22×Dic73—21no—
153C22×Dic73—21no—
154#11924012—21no—
155#11924012—21no—
156#11924012—21no—
157#11924012—21no—
158#11924012—21no—
159#11924012C2×Q812yes—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831C2×D411yescharacteristic
166C2×#10596831C2×D411yescharacteristic
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—41no—
172C4⋊Dic73—41no—
173C4⋊Dic73C2×D411yescharacteristic
174C4⋊Dic73C2×D411yescharacteristic
175#11924007—41no—
176#11924007—41no—
177#11924007—41no—
178#11924007—41no—
179#11924007—82no—
180C22⋊Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73C2×Q812yes—
184#41×C73—21no—
185#41×C73—21no—
186#41×C73C2×Q812yes—
187#41×C73C2×Q812yes—
188C2×Q8×C73C2×D411yescharacteristic
189C2×D4×C73—21no—
190C2×D4×C73C2×D411yescharacteristic
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73—41no—
195C22×C4×C73—41no—
196C22×C4×C73C2411yescharacteristic
197M16×C73—21no—
198M16×C73—24no—
199M16×C73C2×Q812yes—
200C2×C8×C73—21no—
201C2×C8×C73—24no—
202C2×C8×C73C2×D411yescharacteristic
203C2×C8×C73C2×D411yescharacteristic
204C2×C8×C73C2×Q812yes—
205C2×C8×C73#4112yes—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—41no—
211C4⋊C4×C73—41no—
212C4⋊C4×C73C2×D411yescharacteristic
213C4⋊C4×C73C2×D411yescharacteristic
214C4⋊C4×C73C2×D411yescharacteristic
215C4⋊C4×C73C2×D411yescharacteristic
216C22⋊C4×C73—21no—
217C22⋊C4×C73—21no—
218C22⋊C4×C73—41no—
219C22⋊C4×C73—41no—
220C42×C73—21no—
221C42×C73—21no—
222C42×C73—41no—
223C42×C73—41no—
224C23×C73—21no—
225C23×C73H(5,𝔽2)11yescharacteristic
226Q8×C73—21no—
227Q8×C73C4⋊Q812yes—
228D4×C73—22no—
229D4×C73—22no—
230D4×C73—21no—
231D4×C73C4⋊Q812yes—
232C2×C4×C73—21no—
233C2×C4×C73—22no—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—81no—
242C2×C4×C73—81no—
243C2×C4×C73—161no—
244C2×C4×C73—161no—
245C2×C4×C73C22×D411yescharacteristic
246C2×C4×C73C22×D411yescharacteristic
247C2×C4×C73C22×Q811yescharacteristic
248C2×C4×C73C22×Q811yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C732−1+411yescharacteristic
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—22no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—81no—
259C2×Dic73—81no—
260Dic146—161no—
261Dic146—161no—
262C8×C73—22no—
263C8×C73C22⋊Q812yes—
264C8×C73C22⋊Q812yes—
265#10596831—21no—
266#10596831—21no—
267#10596831—22no—
268#10596831—41no—
269#10596831—41no—
270C22×C73—21no—
271C22×C73—21no—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73#54411yescharacteristic
276C22×C73D4211yescharacteristic
277C22×C73#55411yescharacteristic
278C4×C73—22no—
279C4×C73—41no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73—81no—
284C4×C73—81no—
285C4×C73C2×C22⋊Q811yescharacteristic
286C4×C73C2×C4⋊Q811yescharacteristic
287C4×C73#54711yescharacteristic
288C4×C73D4×Q811yescharacteristic
289Dic73—42no—
290Dic73—41no—
291Dic73—41no—
292Dic73—81no—
293Dic73—81no—
294#26084—731nomaximal, Sylow
295C2×C73—21no—
296C2×C73—21no—
297C2×C73—42no—
298C2×C73#244011yescharacteristic
299C2×C73#328811yescharacteristic
300C2×C73#328911yescharacteristic
301#3164—731no—
302#3084—731no—
303#3083—731no—
304#2912D7311yescharacteristic
305#2090—731no—
306#2089—731no—
307#2061—731no—
308#2060—731no—
309#1951—731no—
310#1946—731no—
311#1860—731no—
312#1859—731no—
313#1847—731no—
314#1816—731no—
315#1765—731no—
316C73#2608411yescharacteristic, complemented, Sylow
317C2×#131C2×D7311yescharacteristic
318#544—731no—
319#544—731no—
320#520C2×D7311yescharacteristic
321#520C2×D7311yescharacteristic
322#501—1461no—
323#501—1461no—
324#500—1461no—
325#499—1461no—
326#479—1461no—
327#479—1461no—
328#478—1461no—
329#478—1461no—
330#477—1461no—
331#476—1461no—
332#475—1461no—
333#474—1461no—
334C2×#108—731no—
335#428—21no—
336#428—21no—
337#428—731no—
338#428—731no—
339#428—731no—
340#428—731no—
341#428C2×D7311yescharacteristic
342#428C2×D7311yescharacteristic
343#427—21no—
344#427C2×D7311yescharacteristic
345C2×#107—731no—
346C2×#107—731no—
347C2×#106—21no—
348C2×#106—731no—
349C2×#106C2×D7311yescharacteristic
350#424—731no—
351#424—731no—
352C2×#104—731no—
353#417—731no—
354#417—731no—
355#416—731no—
356#405—731no—
357C2×#97—731no—
358#384—731no—
359#382—731no—
360#343—731no—
361#343—731no—
362#343—731no—
363#343—731no—
364#341—731no—
365#339—731no—
366#339—731no—
367#339—731no—
368C2×#41C22×D7311yescharacteristic
369#131—22no—
370#131Dic14614yescomplemented
371C2×M16—21no—
372C2×M16—731no—
373C2×M16—731no—
374C2×M16—731no—
375C2×M16C22×D7311yescharacteristic
376C22×C8—21no—
377C22×C8—731no—
378C22×C8C22×D7311yescharacteristic
379C4⋊Q8—1461no—
380C4⋊Q8—1461no—
381C4⋊Q8—1461no—
382C4⋊Q8—1461no—
383#125—1461no—
384#125—1461no—
385#125—1461no—
386#125—1461no—
387C22⋊Q8—2921no—
388C22⋊Q8—2921no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—41no—
394C4×D4—41no—
395#117—21no—
396#117—21no—
397#117—731no—
398#117C22×D7311yescharacteristic
399#117C22×D7311yescharacteristic
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—2921no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4C22×D7311yescharacteristic
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C42—731no—
408#108—1462no—
409#107—21no—
410#107—21no—
411#107—1461no—
412#107—1461no—
413#107—1461no—
414#107—1461no—
415#107—1461no—
416#107—1461no—
417#107—1461no—
418#107—1461no—
419#107C22⋊D7311yescharacteristic
420#107C22⋊D7311yescharacteristic
421#106—21no—
422#106—21no—
423#106—21no—
424#106—21no—
425#106—1461no—
426#106—1461no—
427#106—1461no—
428#106—1461no—
429#106—1461no—
430#106—1461no—
431#106C22⋊D7311yescharacteristic
432#106C22⋊D7311yescharacteristic
433C4⋊C8—1461no—
434C4⋊C8—1461no—
435C4⋊C8—1461no—
436C4⋊C8—1461no—
437#104—734no—
438#103—731no—
439#103—731no—
440#103—1461no—
441#103—1461no—
442#102—731no—
443#102—731no—
444#102—1461no—
445#102—1461no—
446#97—1461no—
447#97—1462no—
448C4×C8—1461no—
449#95—2921no—
450#41—21no—
451#41—21no—
452#41Q8×D7312yes—
453#41C2×Dic14612yes—
454C2×Q8—2921no—
455C2×Q8—2921no—
456C2×Q8D4×D7311yescharacteristic
457C2×D4—21no—
458C2×D4D4×D7311yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4C23×D7311yescharacteristic
468M16—21no—
469M16—24no—
470M16—732no—
471M16—1461no—
472M16—1461no—
473M16—1461no—
474M16—1461no—
475M16—1461no—
476M16C2×Dic14612yes—
477C2×C8—21no—
478C2×C8—24no—
479C2×C8—731no—
480C2×C8—731no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8C2×C22⋊D7311yescharacteristic
487C2×C8C2×C22⋊D7311yescharacteristic
488C2×C8C2×Dic14612yes—
489C2×C8#1192402812yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—2921no—
497C4⋊C4—2921no—
498C4⋊C4—2921no—
499C4⋊C4—2921no—
500C4⋊C4—2921no—
501C4⋊C4—2921no—
502C4⋊C4—5842no—
503C4⋊C4—731no—
504C4⋊C4—731no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4C2×C22⋊D7311yescharacteristic
510C4⋊C4C2×C22⋊D7311yescharacteristic
511C4⋊C4C2×C22⋊D7311yescharacteristic
512C4⋊C4C2×C22⋊D7311yescharacteristic
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—1461no—
518C42—21no—
519C42—21no—
520C42—41no—
521C42—41no—
522C42—732no—
523C42—1461no—
524C42—1461no—
525C23—21no—
526C23#1455663611yescharacteristic
527Q8—21no—
528Q8—11681no—
529Q8—11681no—
530Q8#1455659012yes—
531D4—22no—
532D4—22no—
533D4—21no—
534D4#1455659012yes—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—161no—
547C2×C4—161no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—5841no—
553C2×C4—5841no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1462no—
557C2×C4C2×D4×D7311yescharacteristic
558C2×C4C22×C22⋊D7311yescharacteristic
559C2×C4C2×Q8×D7311yescharacteristic
560C2×C4C22×Dic14611yescharacteristic
561C2×C4C2×#1192402811yescharacteristic
562C2×C4#1455663911yescharacteristic
563C8—22no—
564C8—2921no—
565C8—2921no—
566C8—1461no—
567C8—1461no—
568C8—1462no—
569C8#1455657012yes—
570C8#1455657012yes—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22#1798921811yescharacteristic
577C22D4×C22⋊D7311yescharacteristic
578C22#1798929211yescharacteristic
579C4—22no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—2922no—
587C4—2921no—
588C4—2921no—
589C4—5841no—
590C4—5841no—
591C4C2×#1455657011yescharacteristic
592C4C2×#1455659011yescharacteristic
593C4#1798923811yescharacteristic
594C4Q8×C22⋊D7311yescharacteristic
595C2—21no—
596C2—21no—
597C2—42no—
598C2#2261362711yescharacteristic
599C2#2262169311yescharacteristic
600C2#2262170411yescharacteristic
subgroupG/H
centerC22D4×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2912C2
socleC22×C73D42
prime radicalC2×#41×C73C23
hypercenter#2912D73
hypocenterC73#26084
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2912

Complements

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

NH ≅ G/Ntype
1. #22620471C2semidirect
4. #22611444C2semidirect
5. #22611439C2semidirect
6. #22611279C2semidirect
7. #22611274C2semidirect
8. #22610709C2semidirect
9. #22610685C2semidirect
12. #22610096C2semidirect
105. #14556518Q8semidirect
120. #131×C73Q8semidirect
316. C73#26084semidirect
370. #131Dic146semidirect

Maximal subgroups

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

subgroupindex
#226204712
#226197932
#226197822
#226114442
#226114392
#226112792
#226112742
#226107092
#226106852
#226101642
#226101572
#226100962
#226099952
#226097732
C73×#29122
#2608473