Finite Group Databasebeta

← #33057778

Subgroups — #33057778

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620495C211yesmaximal, characteristic
2#22618935C211yesmaximal, characteristic, complemented
3#22608821C211yesmaximal, characteristic, complemented
4#22608819C211yesmaximal, characteristic, complemented
5#22608781C211yesmaximal, characteristic, complemented
6#22608779C211yesmaximal, characteristic, complemented
7#22608760C211yesmaximal, characteristic, complemented
8#22608758C211yesmaximal, characteristic, complemented
9C8×#11924015C211yesmaximal, characteristic, complemented
10C8×#11924016C211yesmaximal, characteristic, complemented
11#22608677C211yesmaximal, characteristic, complemented
12#22608339C211yesmaximal, characteristic, complemented
13#22608337C211yesmaximal, characteristic, complemented
14M16×#10596831C211yesmaximal, characteristic, complemented
15C73×#2969C211yesmaximal, characteristic, complemented
16#17989121C2211yescharacteristic
17#17988986C411yescharacteristic
18#17988969C411yescharacteristic
19#17988944C411yescharacteristic
20#17988941C411yescharacteristic
21#17988826C411yescharacteristic
22#17988821C411yescharacteristic
23#17988818C411yescharacteristic
24#17988814C411yescharacteristic
25#17988717C2211yescharacteristic
26C8×Dic146C2211yescharacteristic
27C4×#11924018C2211yescharacteristic
28C4×#11924015C2211yescharacteristic
29C4×#11924016C2211yescharacteristic
30C4×#11924013C2211yescharacteristic
31#17988666C2211yescharacteristic
32#17988663—21no—
33#17988656—21no—
34C8×D292C2211yescharacteristic
35#17988649—21no—
36#17988640C2211yescharacteristic
37#17988640C2211yescharacteristic
38#17988544—21no—
39C2×C4×#10596831C2211yescharacteristic
40#17988339C2211yescharacteristic
41#17988339C2211yescharacteristic
42#17988337C2211yescharacteristic
43#17988337C2211yescharacteristic
44Q8⋊#10596831C2211yescharacteristic
45Q8⋊#10596831C2211yescharacteristic
46#17988317C2211yescharacteristic
47#17988317C2211yescharacteristic
48D4⋊#10596831C2211yescharacteristic
49D4⋊#10596831C2211yescharacteristic
50#17988314C2211yescharacteristic
51#17988314C2211yescharacteristic
52#17988308C2211yescharacteristic
53#17988308C2211yescharacteristic
54C8×#10596831C2211yescharacteristic
55C8×#10596831C2211yescharacteristic
56#529×C73C2211yescharacteristic
57#446×C73C2211yescharacteristic
58C8×Q8×C73C2211yescharacteristic
59#436×C73—21no—
60#436×C73C2211yescharacteristic
61#435×C73—21no—
62C8×D4×C73C2211yescharacteristic
63#433×C73—21no—
64#431×C73C2211yescharacteristic
65#405×C73—21no—
66C4×M16×C73C2211yescharacteristic
67C2×#11924018C2×C411yescharacteristic
68C2×#11924016C2×C411yescharacteristic
69C2×#11924015C2×C411yescharacteristic
70C2×#11924013C2×C411yescharacteristic
71C2×#11924011—21no—
72C22×#10596831C2×C411yescharacteristic
73C22×#10596831C2×C411yescharacteristic
74C2×C8×D73—21no—
75#14556585—21no—
76#14556583C2×C411yescharacteristic
77#14556579C2×C411yescharacteristic
78#14556574—21no—
79#14556570C2×C411yescharacteristic
80#14556565C2×C411yescharacteristic
81C4×Dic146C2311yescharacteristic
82C4×D292C2311yescharacteristic
83C4×C22⋊D73—21no—
84#14556548—21no—
85#14556524C2×C411yescharacteristic
86#14556524C2×C411yescharacteristic
87#14556522C2×C411yescharacteristic
88#14556522C2×C411yescharacteristic
89C4⋊#10596831—21no—
90C4⋊#10596831C2311yescharacteristic
91C4⋊#10596831C2311yescharacteristic
92#14556520—21no—
93#14556520—21no—
94#14556515C2×C411yescharacteristic
95#14556515C2×C411yescharacteristic
96#14556514C2×C411yescharacteristic
97#14556514C2×C411yescharacteristic
98#14556512C2×C411yescharacteristic
99#14556512C2×C411yescharacteristic
100#14556511C2×C411yescharacteristic
101#14556511C2×C411yescharacteristic
102C22⋊#10596831—21no—
103C22⋊#10596831—21no—
104#14556503—21no—
105#14556503—21no—
106#14556501—21no—
107C4×#10596831—21no—
108C4×#10596831C2×C411yescharacteristic
109C4×#10596831C2×C411yescharacteristic
110C4×#10596831C2311yescharacteristic
111C4×#10596831C2311yescharacteristic
112C4×#10596831C2311yescharacteristic
113C4×#10596831C2311yescharacteristic
114C8×Dic73—21no—
115C2×M16×C73—21no—
116C2×M16×C73D411yescharacteristic
117C2×M16×C73D411yescharacteristic
118C22×C8×C73—21no—
119#126×C73—21no—
120#125×C73C2×C411yescharacteristic
121#124×C73C2×C411yescharacteristic
122#123×C73—21no—
123C22⋊Q8×C73C2×C411yescharacteristic
124#121×C73C2×C411yescharacteristic
125C4×Q8×C73C2311yescharacteristic
126C4×D4×C73—21no—
127C4×D4×C73C2311yescharacteristic
128#117×C73—21no—
129C2×C42×C73C2311yescharacteristic
130C4⋊C8×C73—21no—
131C4⋊C8×C73—21no—
132C4⋊C8×C73C2311yescharacteristic
133C4⋊C8×C73C2311yescharacteristic
134C22⋊C8×C73—21no—
135C22⋊C8×C73—21no—
136C22⋊C8×C73—21no—
137#97×C73—21no—
138#97×C73D411yescharacteristic
139#97×C73D411yescharacteristic
140C4×C8×C73—21no—
141C4×C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C2×Dic146C22×C411yescharacteristic
145C2×D292C22×C411yescharacteristic
146C2×C22⋊D73—21no—
147C2×C4×D73—21no—
148#11924018—41no—
149#11924016—22no—
150#11924015—22no—
151#11924013—41no—
152#11924011—42no—
153C2×#10596831—22no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—41no—
159C2×#10596831C22×C411yescharacteristic
160C2×#10596831C22×C411yescharacteristic
161C2×#10596831C22×C411yescharacteristic
162C2×#10596831C22×C411yescharacteristic
163C8×D73—42no—
164C4⋊Dic73C22×C411yescharacteristic
165C4⋊Dic73C22×C411yescharacteristic
166#11924007—21no—
167#11924007—21no—
168#11924007—21no—
169#11924007—21no—
170C22⋊Dic73—21no—
171#11924005—21no—
172#11924005—21no—
173#11924005—21no—
174#11924005—21no—
175C4×Dic73—21no—
176C2×Q8×C73C22×C411yescharacteristic
177C2×D4×C73—21no—
178C2×D4×C73C22×C411yescharacteristic
179C22×C4×C73—21no—
180C22×C4×C73C22×C411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73#4111yescharacteristic
183M16×C73—22no—
184M16×C73—22no—
185M16×C73—42no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—42no—
189C2×C8×C73C2×D411yescharacteristic
190C2×C8×C73C2×D411yescharacteristic
191C2×C8×C73C2×D411yescharacteristic
192C2×C8×C73C2×D411yescharacteristic
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73C22×C411yescharacteristic
198C4⋊C4×C73C22×C411yescharacteristic
199C22⋊C4×C73—21no—
200C22⋊C4×C73—21no—
201C22⋊C4×C73—21no—
202C22⋊C4×C73—21no—
203C22⋊C4×C73—21no—
204C42×C73—21no—
205C42×C73C22×C411yescharacteristic
206C42×C73C2×D411yescharacteristic
207C42×C73#4111yescharacteristic
208C42×C73C2411yescharacteristic
209C23×C73—21no—
210C23×C73C4×D411yescharacteristic
211C22×D73—21no—
212Q8×C73—21no—
213Q8×C73—21no—
214D4×C73—21no—
215D4×C73—21no—
216D4×C73—81no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73C4×D411yescharacteristic
228C2×C4×C73C4×D411yescharacteristic
229C2×C4×C73C4×D411yescharacteristic
230C2×C4×C73C23×C411yescharacteristic
231C2×C4×C73C22×D411yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C22⋊D73—81no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237D292—21no—
238D292—21no—
239C4×D73—41no—
240C4×D73—41no—
241Dic146—21no—
242Dic146—21no—
243C8×C73—21no—
244C8×C73—21no—
245C8×C73—41no—
246C8×C73—41no—
247C8×C73#13512yes—
248C8×C73#13512yes—
249#10596831—21no—
250#10596831—21no—
251#10596831—21no—
252#10596831—41no—
253#10596831—41no—
254#10596831—41no—
255#10596831—41no—
256#10596831#13112yes—
257C22×C73—21no—
258C22×C73—21no—
259C22×C73—21no—
260C22×C73—41no—
261C22×C73—41no—
262C22×C73—41no—
263C22×C73C2×C4×D411yescharacteristic
264C2×D73—41no—
265C2×D73—41no—
266C2×D73—41no—
267C4×C73—21no—
268C4×C73—21no—
269C4×C73—21no—
270C4×C73—21no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73C2×#13111yescharacteristic
275C4×C73#56811yescharacteristic
276C4×C73C2×#13511yescharacteristic
277C4×C73C2×#13511yescharacteristic
278Dic73—41no—
279Dic73—41no—
280Dic73—41no—
281#18070—731nomaximal, Sylow
282C2×C73—41no—
283C2×C73—81no—
284C2×C73#293511yescharacteristic
285C2×C73C4×#13511yescharacteristic
286C2×C73#295711yescharacteristic
287D73—81no—
288#3168—731no—
289#2969—731no—
290#2969D7311yescharacteristic, complemented
291#1590—731no—
292#1590—731no—
293#1584—731no—
294#1584—731no—
295#1581—731no—
296#1581—731no—
297C8×SD16—731no—
298C8×SD16—731no—
299#1564—731no—
300#1482—731no—
301#1482—731no—
302C8×M16—731no—
303C73#1807011yescharacteristic, complemented, Sylow
304#529—731no—
305#529C2×D7311yescharacteristic
306#499—731no—
307#495—731no—
308#488—731no—
309#488—731no—
310#467—731no—
311#466—731no—
312#465—731no—
313#465—731no—
314#446—731no—
315#446C2×D7311yescharacteristic
316C8×Q8—731no—
317C8×Q8C2×D7311yescharacteristic
318C4×Q16—731no—
319C4×SD16—731no—
320C4×SD16—731no—
321C4×D8—731no—
322#436—21no—
323#436—731no—
324#436—1461no—
325#436C2×D7311yescharacteristic
326#435—21no—
327#435—1461no—
328C8×D4—731no—
329C8×D4C2×D7311yescharacteristic
330#433—21no—
331#433—1461no—
332#431—731no—
333#431—731no—
334#431C2×D7311yescharacteristic
335#405—21no—
336#405—1461no—
337C4×M16C2×D7311yescharacteristic
338C2×C4×C8—731no—
339#335—731no—
340#335—731no—
341#334—731no—
342#334—731no—
343Q8⋊C8—731no—
344Q8⋊C8—731no—
345Q8⋊C8—731no—
346Q8⋊C8—731no—
347D4⋊C8—731no—
348D4⋊C8—731no—
349D4⋊C8—731no—
350D4⋊C8—731no—
351#322—731no—
352#322—731no—
353C82—731no—
354C82—731no—
355C2×Q16—731no—
356C2×SD16—731no—
357C2×SD16—731no—
358C2×D8—731no—
359C2×M16—21no—
360C2×M16—1461no—
361C2×M16C22⋊D7311yescharacteristic
362C2×M16C22⋊D7311yescharacteristic
363C22×C8—21no—
364C22×C8—731no—
365C22×C8—731no—
366C22×C8—1461no—
367#126—21no—
368#126—1461no—
369#125—731no—
370#125C4×D7311yescharacteristic
371#124—731no—
372#124C4×D7311yescharacteristic
373#123—21no—
374#123—1461no—
375C22⋊Q8—731no—
376C22⋊Q8C4×D7311yescharacteristic
377#121—731no—
378#121C4×D7311yescharacteristic
379C4×Q8—731no—
380C4×Q8C22×D7311yescharacteristic
381C4×D4—21no—
382C4×D4—731no—
383C4×D4—1461no—
384C4×D4C22×D7311yescharacteristic
385#117—21no—
386#117—1461no—
387C2×C42C22×D7311yescharacteristic
388#107—731no—
389#107—731no—
390#106—731no—
391#106—731no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—731no—
395C4⋊C8—731no—
396C4⋊C8—1461no—
397C4⋊C8—1461no—
398C4⋊C8—1461no—
399C4⋊C8C22×D7311yescharacteristic
400C4⋊C8C22×D7311yescharacteristic
401#103—731no—
402#103—731no—
403#103—731no—
404#103—731no—
405#102—731no—
406#102—731no—
407#102—731no—
408#102—731no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412C22⋊C8—1461no—
413C22⋊C8—1461no—
414C22⋊C8—1461no—
415C22⋊C8—1461no—
416#97—21no—
417#97—1461no—
418#97C22⋊D7311yescharacteristic
419#97C22⋊D7311yescharacteristic
420C4×C8—21no—
421C4×C8—21no—
422C4×C8—731no—
423C4×C8—731no—
424C4×C8—731no—
425C4×C8—731no—
426C4×C8—731no—
427C4×C8—731no—
428C4×C8—1461no—
429C4×C8—1461no—
430C4×C8C22×D7311yescharacteristic
431C4×C8C22×D7311yescharacteristic
432C2×Q8—731no—
433C2×Q8C2×C4×D7311yescharacteristic
434C2×D4—21no—
435C2×D4—731no—
436C2×D4—1461no—
437C2×D4C2×C4×D7311yescharacteristic
438C22×C4—21no—
439C22×C4—1461no—
440C22×C4C2×C4×D7311yescharacteristic
441C22×C4C2×C22⋊D7311yescharacteristic
442C22×C4#1192402811yescharacteristic
443Q16—2921no—
444SD16—1462no—
445SD16—1462no—
446D8—2921no—
447M16—22no—
448M16—22no—
449M16—42no—
450M16—2922no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—42no—
454C2×C8—2922no—
455C2×C8—2921no—
456C2×C8—731no—
457C2×C8—731no—
458C2×C8—731no—
459C2×C8—731no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1462no—
465C2×C8C2×C22⋊D7311yescharacteristic
466C2×C8C2×C22⋊D7311yescharacteristic
467C2×C8C2×C22⋊D7311yescharacteristic
468C2×C8C2×C22⋊D7311yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—731no—
474C4⋊C4—731no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4C2×C4×D7311yescharacteristic
480C4⋊C4C2×C4×D7311yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—1461no—
487C22⋊C4—1461no—
488C22⋊C4—1461no—
489C22⋊C4—1461no—
490C22⋊C4—1461no—
491C42—21no—
492C42—1461no—
493C42C2×C4×D7311yescharacteristic
494C42C2×C22⋊D7311yescharacteristic
495C42#1192402811yescharacteristic
496C42C23×D7311yescharacteristic
497C23—21no—
498C23—1461no—
499C23C4×C22⋊D7311yescharacteristic
500Q8—21no—
501Q8—21no—
502Q8—1461no—
503Q8—1461no—
504D4—21no—
505D4—21no—
506D4—81no—
507D4—5841no—
508D4—1461no—
509D4—1461no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4C4×C22⋊D7311yescharacteristic
526C2×C4C4×C22⋊D7311yescharacteristic
527C2×C4C4×C22⋊D7311yescharacteristic
528C2×C4C22×C4×D7311yescharacteristic
529C2×C4C22×C22⋊D7311yescharacteristic
530C2×C4C2×#1192402811yescharacteristic
531C8—21no—
532C8—21no—
533C8—41no—
534C8—41no—
535C8—2921no—
536C8—2921no—
537C8—2921no—
538C8—2921no—
539C8—732no—
540C8—1461no—
541C8—1461no—
542C8—1461no—
543C8#1455661112yes—
544C8#1455661112yes—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—2921no—
552C22—2921no—
553C22—2921no—
554C22C2×C4×C22⋊D7311yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—2921no—
563C4—2921no—
564C4—2921no—
565C4#131×D7311yescharacteristic
566C4#1798935211yescharacteristic
567C4C2×#1455661111yescharacteristic
568C4C2×#1455661111yescharacteristic
569C2—41no—
570C2—81no—
571C2—5841no—
572C2#2261867311yescharacteristic
573C2C4×#1455661111yescharacteristic
574C2#2261884611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#2969C2
socleC22×C73C2×C4×D4
prime radical#17988821C4
cosocleC42×C73C24
hypercenter#2969D73
hypocenterC73#18070
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. #22618935C2semidirect
3. #22608821C2semidirect
4. #22608819C2semidirect
5. #22608781C2semidirect
6. #22608779C2semidirect
7. #22608760C2semidirect
8. #22608758C2semidirect
9. C8×#11924015C2semidirect
10. C8×#11924016C2semidirect
11. #22608677C2semidirect
12. #22608339C2semidirect
13. #22608337C2semidirect
14. M16×#10596831C2semidirect
15. C73×#2969C2semidirect
290. #2969D73semidirect
303. C73#18070semidirect

Maximal subgroups

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

subgroupindex
#226204952
#226189352
#226088212
#226088192
#226087812
#226087792
#226087602
#226087582
C8×#119240152
C8×#119240162
#226086772
#226083392
#226083372
M16×#105968312
C73×#29692
#1807073