Finite Group Databasebeta

← #33068428

Subgroups — #33068428

Order 18688.

618 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620513C211yesmaximal, characteristic, complemented
2#22619719C211yesmaximal, characteristic
3#41×#10596831C211yesmaximal, characteristic
4#22609391C211yesmaximal, characteristic, complemented
5#22609387C211yesmaximal, characteristic, complemented
6#22609384C211yesmaximal, characteristic, complemented
7#22609380C211yesmaximal, characteristic, complemented
8#22609333C211yesmaximal, characteristic, complemented
9#22609328C211yesmaximal, characteristic, complemented
10#22609299C211yesmaximal, characteristic, complemented
11#22609294C211yesmaximal, characteristic, complemented
12C2×#17988339C211yesmaximal, characteristic, complemented
13#22608589C211yesmaximal, characteristic, complemented
14#22608588C211yesmaximal, characteristic, complemented
15C73×#3076C211yesmaximal, characteristic
16#17989144C2211yescharacteristic
17#17988990C2211yescharacteristic
18#17988966C2211yescharacteristic
19#17988939C2211yescharacteristic
20#17988937C2211yescharacteristic
21#17988894—21no—
22#17988887—21no—
23#17988882C2211yescharacteristic
24#17988877C2211yescharacteristic
25#17988872C2211yescharacteristic
26#17988867C2211yescharacteristic
27#17988826—21no—
28#17988826—21no—
29Q8×#10596831C2211yescharacteristic
30Q8×#10596831C2211yescharacteristic
31C4×#11924018—21no—
32C4×#11924018—21no—
33D4×#10596831—21no—
34D4×#10596831—21no—
35D4×#10596831C2211yescharacteristic
36D4×#10596831C2211yescharacteristic
37#17988640—21no—
38#17988640C2211yescharacteristic
39#17988623—21no—
40C2×C4⋊#10596831C2211yescharacteristic
41#17988583—21no—
42C2×C4×#10596831—21no—
43C2×C4×#10596831C2211yescharacteristic
44#17988339C2211yescharacteristic
45#17988339C2211yescharacteristic
46#17988339C2211yescharacteristic
47#17988339C2211yescharacteristic
48#17988336C2211yescharacteristic
49#17988336C2211yescharacteristic
50#17988335C2211yescharacteristic
51#17988335C2211yescharacteristic
52#17988333C2211yescharacteristic
53#17988333C2211yescharacteristic
54#17988332C2211yescharacteristic
55#17988332C2211yescharacteristic
56#533×C73C2211yescharacteristic
57C4×#41×C73C2211yescharacteristic
58#479×C73—21no—
59#478×C73—21no—
60#477×C73C2211yescharacteristic
61#476×C73C2211yescharacteristic
62#475×C73C2211yescharacteristic
63#474×C73C2211yescharacteristic
64#427×C73—21no—
65C2×C4⋊C8×C73C2211yescharacteristic
66#416×C73—21no—
67C2×#11924018—21no—
68C2×#11924018—21no—
69C22×#10596831—21no—
70C22×#10596831—21no—
71C22×#10596831—21no—
72C22×#10596831D411yescharacteristic
73C22×#10596831D411yescharacteristic
74#14556591C2311yescharacteristic
75#14556591C2311yescharacteristic
76#14556583—21no—
77#14556570—21no—
78#14556570—21no—
79C4×Dic146—21no—
80C4×Dic146—21no—
81#14556550—21no—
82#14556524—21no—
83#14556524—21no—
84#14556524—21no—
85#14556524—21no—
86#14556522—21no—
87#14556522—21no—
88C4⋊#10596831—21no—
89C4⋊#10596831—21no—
90C4⋊#10596831—21no—
91C4⋊#10596831C2311yescharacteristic
92C4⋊#10596831C2311yescharacteristic
93C4⋊#10596831C2311yescharacteristic
94C4⋊#10596831C2311yescharacteristic
95#14556515—21no—
96#14556515—21no—
97#14556515—21no—
98#14556515—21no—
99#14556514—21no—
100#14556514—21no—
101#14556512—21no—
102#14556512—21no—
103C22⋊#10596831—21no—
104C22⋊#10596831—21no—
105C22⋊#10596831—21no—
106C4×#10596831—21no—
107C4×#10596831—21no—
108C4×#10596831—21no—
109C4×#10596831D411yescharacteristic
110C4×#10596831D411yescharacteristic
111C4×#10596831C2311yescharacteristic
112C4×#10596831C2311yescharacteristic
113C2×#41×C73D411yescharacteristic
114C22×C8×C73—21no—
115C4⋊Q8×C73C2311yescharacteristic
116C4⋊Q8×C73C2311yescharacteristic
117#125×C73—21no—
118C22⋊Q8×C73—21no—
119C22⋊Q8×C73—21no—
120C4×Q8×C73—21no—
121C4×Q8×C73—21no—
122C4×Q8×C73D411yescharacteristic
123C4×Q8×C73C2311yescharacteristic
124C4×D4×C73—21no—
125C4×D4×C73—21no—
126C4×D4×C73D411yescharacteristic
127C4×D4×C73C2311yescharacteristic
128#117×C73—21no—
129#117×C73—21no—
130#117×C73D411yescharacteristic
131C2×C42×C73—21no—
132C2×C42×C73C2311yescharacteristic
133#107×C73—21no—
134#107×C73—21no—
135#106×C73—21no—
136#106×C73—21no—
137C4⋊C8×C73—21no—
138C4⋊C8×C73C2311yescharacteristic
139C4⋊C8×C73C2311yescharacteristic
140#103×C73—21no—
141#103×C73—21no—
142#102×C73—21no—
143#102×C73—21no—
144C2×Dic146—21no—
145C2×Dic146—21no—
146#11924018—42no—
147#11924018—42no—
148C2×#10596831—22no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—41no—
154C2×#10596831—42no—
155C2×#10596831—42no—
156C2×#10596831—42no—
157C2×#10596831C2×D411yescharacteristic
158C2×#10596831C2×D411yescharacteristic
159C2×#10596831C2×D411yescharacteristic
160C2×#10596831C2×D411yescharacteristic
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165#11924007—41no—
166#11924007—41no—
167C22⋊Dic73—41no—
168C4×Dic73—41no—
169#41×C73—22no—
170#41×C73Q1614yescomplemented
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73C2×D411yescharacteristic
174C2×D4×C73—21no—
175C2×D4×C73C2×D411yescharacteristic
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73#4111yescharacteristic
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—42no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—41no—
192C4⋊C4×C73—41no—
193C4⋊C4×C73C2×D411yescharacteristic
194C4⋊C4×C73C2×D411yescharacteristic
195C22⋊C4×C73—21no—
196C22⋊C4×C73—21no—
197C22⋊C4×C73—21no—
198C22⋊C4×C73—41no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—41no—
203C42×C73C2×D411yescharacteristic
204C42×C73C2×D411yescharacteristic
205C42×C73#4111yescharacteristic
206C42×C73C2411yescharacteristic
207C23×C73—21no—
208C23×C73#12111yescharacteristic
209Q8×C73—21no—
210Q8×C73—41no—
211Q8×C73—41no—
212Q8×C73—81no—
213Q8×C73C2×Q1612yes—
214D4×C73—24no—
215D4×C73—21no—
216D4×C73C2×Q1612yes—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—24no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—41no—
224C2×C4×C73—42no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73#12111yescharacteristic
230C2×C4×C73#12111yescharacteristic
231C2×C4×C73#12111yescharacteristic
232C2×C4×C73C2×Q1612yes—
233C2×C4×C73#13512yes—
234C2×C4×C73C22×D411yescharacteristic
235C2×C4×C73C22×D411yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×Dic73—41no—
238C2×Dic73—41no—
239Dic146—41no—
240Dic146—41no—
241Dic146—81no—
242C8×C73—41no—
243C8×C73—41no—
244#10596831—21no—
245#10596831—21no—
246#10596831—21no—
247#10596831—41no—
248#10596831—41no—
249#10596831—41no—
250#10596831—41no—
251#10596831#13512yes—
252C22×C73—21no—
253C22×C73—21no—
254C22×C73—22no—
255C22×C73—41no—
256C22×C73—41no—
257C22×C73#46712yes—
258C22×C73C2×#12111yescharacteristic
259C4×C73—21no—
260C4×C73—22no—
261C4×C73—21no—
262C4×C73—21no—
263C4×C73—41no—
264C4×C73—41no—
265C4×C73—81no—
266C4×C73—81no—
267C4×C73#46712yes—
268C4×C73C22×Q1611yescharacteristic
269C4×C73C2×#13511yescharacteristic
270C4×C73C2×#13511yescharacteristic
271C4×C73#57511yescharacteristic
272Dic73—81no—
273Dic73—81no—
274#19126—731nomaximal, Sylow
275C2×C73—22no—
276C2×C73—42no—
277C2×C73#303511yescharacteristic
278C2×C73C2×#46711yescharacteristic
279C2×C73#306711yescharacteristic
280#3170—731no—
281#3076—731no—
282#3076D7311yescharacteristic
283C8×#41—731no—
284#1684—731no—
285#1684—731no—
286#1683—731no—
287#1683—731no—
288#1676—731no—
289#1675—731no—
290#1670—731no—
291#1669—731no—
292C2×#335—731no—
293#1542—731no—
294#1542—731no—
295C73#1912611yescharacteristic, complemented, Sylow
296#533—731no—
297#533C2×D7311yescharacteristic
298C4×#41C2×D7311yescharacteristic
299#500—731no—
300#494—731no—
301#487—731no—
302#487—731no—
303#479—21no—
304#479—1461no—
305#478—21no—
306#478—1461no—
307#477—731no—
308#477C2×D7311yescharacteristic
309#476—731no—
310#476C2×D7311yescharacteristic
311#475—731no—
312#475C2×D7311yescharacteristic
313#474—731no—
314#474C2×D7311yescharacteristic
315#467—1461no—
316#467—1461no—
317C8×Q8—731no—
318C8×Q8—731no—
319C4×Q16—1461no—
320C4×Q16—1461no—
321C8×D4—731no—
322C8×D4—731no—
323C8×D4—1461no—
324C8×D4—1461no—
325#431—731no—
326#431—1461no—
327#427—21no—
328#427—1461no—
329C2×C4⋊C8—731no—
330C2×C4⋊C8C2×D7311yescharacteristic
331#416—21no—
332#416—1461no—
333C2×C4×C8—731no—
334C2×C4×C8—1461no—
335#335—731no—
336#335—731no—
337#335—731no—
338#335—731no—
339#333—731no—
340#333—731no—
341#333—731no—
342#333—731no—
343#332—731no—
344#332—731no—
345#332—731no—
346#332—731no—
347C2×#41C22⋊D7311yescharacteristic
348C2×Q16—1461no—
349C2×Q16—1461no—
350C22×C8—21no—
351C22×C8—731no—
352C22×C8—731no—
353C22×C8—1461no—
354C22×C8—1461no—
355C22×C8—1461no—
356C4⋊Q8—731no—
357C4⋊Q8—731no—
358C4⋊Q8C22×D7311yescharacteristic
359C4⋊Q8C22×D7311yescharacteristic
360#125—21no—
361#125—1461no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—1461no—
365C22⋊Q8—1461no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—1461no—
369C4×Q8—1461no—
370C4×Q8C22⋊D7311yescharacteristic
371C4×Q8C22×D7311yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C22⋊D7311yescharacteristic
375C4×D4C22×D7311yescharacteristic
376#117—21no—
377#117—21no—
378#117—1461no—
379#117C22⋊D7311yescharacteristic
380C2×C42—21no—
381C2×C42C22×D7311yescharacteristic
382#107—21no—
383#107—21no—
384#107—1461no—
385#107—1461no—
386#107—1461no—
387#107—1461no—
388#106—21no—
389#106—21no—
390#106—1461no—
391#106—1461no—
392C4⋊C8—21no—
393C4⋊C8—731no—
394C4⋊C8—731no—
395C4⋊C8—731no—
396C4⋊C8—731no—
397C4⋊C8—1461no—
398C4⋊C8—1461no—
399C4⋊C8—1461no—
400C4⋊C8C22×D7311yescharacteristic
401C4⋊C8C22×D7311yescharacteristic
402#103—21no—
403#103—21no—
404#103—1461no—
405#103—1461no—
406#103—1461no—
407#103—1461no—
408#103—1461no—
409#103—1461no—
410#102—21no—
411#102—21no—
412#102—1461no—
413#102—1461no—
414C22⋊C8—1461no—
415C22⋊C8—1461no—
416C22⋊C8—1461no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—1461no—
422C4×C8—1461no—
423C4×C8—1461no—
424#41—22no—
425#41#1192401814yescomplemented
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—1461no—
429C2×Q8—1461no—
430C2×Q8C2×C22⋊D7311yescharacteristic
431C2×D4—21no—
432C2×D4C2×C22⋊D7311yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4D4×D7311yescharacteristic
437C22×C4C2×C22⋊D7311yescharacteristic
438C22×C4#1192402711yescharacteristic
439Q16—2922no—
440Q16—2922no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—42no—
444C2×C8—2922no—
445C2×C8—2922no—
446C2×C8—2922no—
447C2×C8—2921no—
448C2×C8—731no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—731no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1462no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—2921no—
466C4⋊C4—2921no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4C2×C22⋊D7311yescharacteristic
472C4⋊C4C2×C22⋊D7311yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—41no—
477C22⋊C4—2921no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—2921no—
483C42D4×D7311yescharacteristic
484C42C2×C22⋊D7311yescharacteristic
485C42#1192402711yescharacteristic
486C42C23×D7311yescharacteristic
487C23—21no—
488C23#1455656311yescharacteristic
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493Q8—2921no—
494Q8—2921no—
495Q8—5841no—
496Q8C2×#1192401812yes—
497D4—24no—
498D4—21no—
499D4C2×#1192401812yes—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—2921no—
513C2×C4—2921no—
514C2×C4#1455656311yescharacteristic
515C2×C4#1455656311yescharacteristic
516C2×C4#1455656311yescharacteristic
517C2×C4C2×#1192401812yes—
518C2×C4#1455661112yes—
519C2×C4C2×D4×D7311yescharacteristic
520C2×C4C22×C22⋊D7311yescharacteristic
521C2×C4C2×#1192402711yescharacteristic
522C8—41no—
523C8—41no—
524C8—2921no—
525C8—2921no—
526C8—2921no—
527C8—2921no—
528C8—732no—
529C8—1461no—
530C8—1461no—
531C8—1461no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—41no—
536C22—41no—
537C22#1798882612yes—
538C22C2×#1455656311yescharacteristic
539C4—21no—
540C4—21no—
541C4—22no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—5841no—
548C4—5841no—
549C4#1798882612yes—
550C4C22×#1192401811yescharacteristic
551C4#135×D7311yescharacteristic
552C4C2×#1455661111yescharacteristic
553C4#1798938411yescharacteristic
554C2—22no—
555C2—42no—
556C2#2261938211yescharacteristic
557C2C2×#1798882611yescharacteristic
558C2#2261964811yescharacteristic
subgroupG/H
centerC2×C4#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3076C2
socleC22×C73C2×#121
prime radicalC2×#41×C73D4
hypercenter#3076D73
hypocenterC73#19126
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #22620513C2semidirect
4. #22609391C2semidirect
5. #22609387C2semidirect
6. #22609384C2semidirect
7. #22609380C2semidirect
8. #22609333C2semidirect
9. #22609328C2semidirect
10. #22609299C2semidirect
11. #22609294C2semidirect
12. C2×#17988339C2semidirect
13. #22608589C2semidirect
14. #22608588C2semidirect
170. #41×C73Q16semidirect
295. C73#19126semidirect
425. #41#11924018semidirect

Maximal subgroups

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

subgroupindex
#226205132
#226197192
#41×#105968312
#226093912
#226093872
#226093842
#226093802
#226093332
#226093282
#226092992
#226092942
C2×#179883392
#226085892
#226085882
C73×#30762
#1912673