Finite Group Databasebeta

← #33057322

Subgroups — #33057322

Order 18688.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619940C211yesmaximal, characteristic
2#22619032C211yesmaximal, characteristic, complemented
3#22618965C211yesmaximal, characteristic
4#22608835C211yesmaximal, characteristic, complemented
5#22608823C211yesmaximal, characteristic, complemented
6#22608811C211yesmaximal, characteristic, complemented
7#22608790C211yesmaximal, characteristic, complemented
8#22608776C211yesmaximal, characteristic, complemented
9#22608769C211yesmaximal, characteristic, complemented
10#22608745C211yesmaximal, characteristic, complemented
11C8×#11924017C211yesmaximal, characteristic, complemented
12#22608711C211yesmaximal, characteristic, complemented
13#22608405C211yesmaximal, characteristic, complemented
14#22608323C211yesmaximal, characteristic
15C73×#1454C211yesmaximal, characteristic, complemented
16#17989144C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18#17988918C411yescharacteristic
19#17988918C411yescharacteristic
20#17988878C411yescharacteristic
21#17988878C411yescharacteristic
22#17988793C411yescharacteristic
23#17988793C411yescharacteristic
24#17988740C411yescharacteristic
25#17988740C411yescharacteristic
26#17988717C2211yescharacteristic, complemented
27#17988717C2211yescharacteristic, complemented
28#17988714—21no—
29C8×Dic146C2211yescharacteristic, complemented
30C8×Dic146C2211yescharacteristic, complemented
31#17988711—21no—
32C4×#11924017C2211yescharacteristic
33C4×#11924017C2211yescharacteristic
34C4×#11924017C2211yescharacteristic
35C4×#11924017C2211yescharacteristic
36#17988666C2211yescharacteristic
37#17988666C2211yescharacteristic
38#17988657—21no—
39#17988657—21no—
40C8×D292C2211yescharacteristic
41C8×D292C2211yescharacteristic
42#17988645—21no—
43#17988645—21no—
44#17988644C2211yescharacteristic
45#17988640C2211yescharacteristic
46#17988549—21no—
47C22⋊C8×D73—21no—
48#17988340C2211yescharacteristic, complemented
49#17988340C2211yescharacteristic, complemented
50#17988338C2211yescharacteristic, complemented
51#17988338C2211yescharacteristic, complemented
52#17988319C2211yescharacteristic, complemented
53#17988319C2211yescharacteristic, complemented
54#17988319C2211yescharacteristic, complemented
55#17988319C2211yescharacteristic, complemented
56#17988316C2211yescharacteristic
57#17988316C2211yescharacteristic
58#17988316C2211yescharacteristic
59#17988316C2211yescharacteristic
60#431×C73C2211yescharacteristic
61#431×C73C2211yescharacteristic
62#431×C73C2211yescharacteristic
63#322×C73C2211yescharacteristic, complemented
64#322×C73C2211yescharacteristic, complemented
65#322×C73C2211yescharacteristic, complemented
66C82×C73C2211yescharacteristic, complemented
67C22×D292C2×C411yescharacteristic
68C22×C4×D73—21no—
69C2×#11924017C2×C411yescharacteristic
70C2×#11924017C2×C411yescharacteristic
71C2×#11924017C2×C411yescharacteristic
72C2×#11924017C2×C411yescharacteristic
73C2×#11924011—21no—
74C2×#11924011—21no—
75C2×C8×D73—21no—
76C2×C8×D73—21no—
77#14556591C2×C411yescharacteristic
78#14556591C2×C411yescharacteristic
79#14556583—21no—
80#14556570—21no—
81#14556570C2×C411yescharacteristic
82#14556570C2×C411yescharacteristic
83C4×Dic146—21no—
84C4×Dic146C2311yescharacteristic
85C4×Dic146C2311yescharacteristic
86C4×D292—21no—
87C4×D292—21no—
88C4×D292C2×C411yescharacteristic
89C4×D292C2×C411yescharacteristic
90C4×D292C2311yescharacteristic
91C4×D292C2311yescharacteristic
92#14556550—21no—
93C2×C4⋊Dic73C2×C411yescharacteristic
94C2×#11924005—21no—
95#14556523C2×C411yescharacteristic
96#14556523C2×C411yescharacteristic
97#14556523C2×C411yescharacteristic
98#14556523C2×C411yescharacteristic
99C4⋊#10596831C2311yescharacteristic
100C4⋊#10596831C2311yescharacteristic
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556520—21no—
107#14556516C2×C411yescharacteristic
108#14556516C2×C411yescharacteristic
109#14556516C2×C411yescharacteristic
110#14556516C2×C411yescharacteristic
111#14556513C2×C411yescharacteristic
112#14556513C2×C411yescharacteristic
113#14556513C2×C411yescharacteristic
114#14556513C2×C411yescharacteristic
115C22⋊#10596831—21no—
116C22⋊#10596831—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556503—21no—
121#14556502—21no—
122#14556501—21no—
123#14556501—21no—
124C4×#10596831—21no—
125C8×Dic73—21no—
126C8×Dic73—21no—
127C2×C42×C73C2311yescharacteristic
128C4⋊C8×C73—21no—
129C4⋊C8×C73C2×C411yescharacteristic
130C4⋊C8×C73C2×C411yescharacteristic
131C4⋊C8×C73D411yescharacteristic
132C4⋊C8×C73D411yescharacteristic
133C22⋊C8×C73—21no—
134C22⋊C8×C73C2×C411yescharacteristic
135C22⋊C8×C73C2×C411yescharacteristic
136C22⋊C8×C73D411yescharacteristic
137C22⋊C8×C73D411yescharacteristic
138C4×C8×C73C2311yescharacteristic
139C4×C8×C73C2311yescharacteristic
140C4×C8×C73C2311yescharacteristic
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C23×D73—21no—
145C2×Dic146—21no—
146C2×Dic146C22×C411yescharacteristic
147C2×Dic146C22×C411yescharacteristic
148C2×D292—22no—
149C2×D292—21no—
150C2×D292—41no—
151C2×D292C22×C411yescharacteristic
152C2×D292C22×C411yescharacteristic
153C22×Dic73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73—21no—
158C2×C4×D73—81no—
159C2×C4×D73—81no—
160#11924017—22no—
161#11924017—22no—
162#11924017—41no—
163#11924017—41no—
164#11924011—81no—
165#11924011—81no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C8×D73—81no—
171C8×D73—81no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73—21no—
176C4⋊Dic73C22×C411yescharacteristic
177C4⋊Dic73C22×C411yescharacteristic
178C4⋊Dic73C22×C411yescharacteristic
179C4⋊Dic73C22×C411yescharacteristic
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184C22⋊Dic73—21no—
185C22⋊Dic73—21no—
186#11924005—21no—
187#11924005—21no—
188#11924005—21no—
189#11924005—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C22×C4×C73C22×C411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73#4111yescharacteristic
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73C22×C411yescharacteristic
198C2×C8×C73C22×C411yescharacteristic
199C2×C8×C73C22×C411yescharacteristic
200C2×C8×C73C22×C411yescharacteristic
201C2×C8×C73C2×D411yescharacteristic
202C2×C8×C73C2×D411yescharacteristic
203C2×C8×C73C2×D411yescharacteristic
204C2×C8×C73C2×D411yescharacteristic
205C42×C73C22×C411yescharacteristic
206C42×C73C2×D411yescharacteristic
207C42×C73#4111yescharacteristic
208C42×C73C2411yescharacteristic
209C23×C73C4×D411yescharacteristic
210C22×D73—21no—
211C22×D73—21no—
212C22×D73—42no—
213C22×D73—42no—
214C22×D73—81no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—41no—
220C2×C4×C73C4×D411yescharacteristic
221C2×C4×C73C4×D411yescharacteristic
222C2×C4×C73C4×D411yescharacteristic
223C2×C4×C73C23×C411yescharacteristic
224C2×C4×C73C22×D411yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—81no—
233D292—21no—
234D292—21no—
235D292—21no—
236D292—42no—
237D292#13112yes—
238C4×D73—41no—
239C4×D73—41no—
240C4×D73—41no—
241C4×D73—41no—
242C4×D73—81no—
243C4×D73—81no—
244Dic146—21no—
245Dic146—21no—
246Dic146—21no—
247Dic146—21no—
248Dic146—42no—
249C8×C73—21no—
250C8×C73—21no—
251C8×C73—21no—
252C8×C73—21no—
253C8×C73—21no—
254C8×C73—21no—
255C8×C73—21no—
256C8×C73—21no—
257C8×C73—41no—
258C8×C73—41no—
259#10596831—41no—
260#10596831—41no—
261#10596831—41no—
262#10596831—41no—
263C22×C73—21no—
264C22×C73—21no—
265C22×C73—21no—
266C22×C73C2×C4×D411yescharacteristic
267C2×D73—22no—
268C2×D73—41no—
269C2×D73—41no—
270C2×D73—41no—
271C2×D73—41no—
272C2×D73—41no—
273C2×D73—84no—
274C4×C73—21no—
275C4×C73—21no—
276C4×C73—21no—
277C4×C73—21no—
278C4×C73C2×#13111yescharacteristic
279C4×C73#56811yescharacteristic
280C4×C73#57711yescharacteristic
281C4×C73#57711yescharacteristic
282Dic73—41no—
283Dic73—41no—
284Dic73—41no—
285Dic73—41no—
286Dic73—41no—
287Dic73—41no—
288#18028—731nomaximal, Sylow
289C2×C73—41no—
290C2×C73#293311yescharacteristic
291C2×C73#294911yescharacteristic
292C2×C73#295511yescharacteristic
293D73—42no—
294D73—81no—
295#3101—731no—
296#2986—731no—
297#2974—731no—
298#1592—731no—
299#1590—731no—
300#1588—731no—
301#1585—731no—
302#1583—731no—
303#1582—731no—
304SD16⋊C8—731no—
305C8×SD16—731no—
306#1572—731no—
307#1497—731no—
308#1478—731no—
309#1454D7311yescharacteristic, complemented
310C73#1802811yescharacteristic, complemented, Sylow
311#533—731no—
312C2×C4×D4—731no—
313#483—731no—
314#483—731no—
315#476—731no—
316#476—731no—
317#461—731no—
318#461—731no—
319#450—731no—
320#450—731no—
321#446—731no—
322#446—731no—
323#446—1461no—
324C8×Q8—731no—
325C8×Q8—731no—
326C8×Q8—1461no—
327C4×SD16—731no—
328C4×SD16—731no—
329C4×SD16—731no—
330C4×SD16—731no—
331#436—731no—
332#436—731no—
333#435—1461no—
334#435—1461no—
335C8×D4—731no—
336C8×D4—731no—
337#433—1461no—
338#433—1461no—
339#432—731no—
340#431—731no—
341#431C2×D7311yescharacteristic
342#431C2×D7311yescharacteristic
343#431C2×D7311yescharacteristic
344#407—1461no—
345C2×C22⋊C8—1461no—
346#335—731no—
347#335—731no—
348#334—731no—
349#334—731no—
350Q8⋊C8—731no—
351Q8⋊C8—731no—
352Q8⋊C8—731no—
353Q8⋊C8—731no—
354D4⋊C8—731no—
355D4⋊C8—731no—
356D4⋊C8—731no—
357D4⋊C8—731no—
358#322C2×D7311yescharacteristic, complemented
359#322C2×D7311yescharacteristic, complemented
360#322C2×D7311yescharacteristic, complemented
361C82C2×D7311yescharacteristic, complemented
362C22×D4—731no—
363C23×C4—1461no—
364C2×SD16—731no—
365C2×SD16—731no—
366C2×SD16—731no—
367C2×SD16—731no—
368C2×M16—1461no—
369C2×M16—1461no—
370C22×C8—1461no—
371C22×C8—1461no—
372C4⋊Q8—731no—
373C4⋊Q8—731no—
374#125—1461no—
375C22⋊Q8—731no—
376C22⋊Q8—731no—
377C22⋊Q8—1461no—
378C4×Q8—731no—
379C4×Q8—731no—
380C4×Q8—1461no—
381C4×D4—731no—
382C4×D4—731no—
383C4×D4—731no—
384C4×D4—731no—
385C4×D4—1461no—
386C4×D4—1461no—
387#117—1461no—
388C2×C4⋊C4—731no—
389C2×C22⋊C4—1461no—
390C2×C42C22×D7311yescharacteristic
391#106—731no—
392#106—731no—
393#106—731no—
394#106—731no—
395C4⋊C8—21no—
396C4⋊C8—731no—
397C4⋊C8—731no—
398C4⋊C8—731no—
399C4⋊C8—731no—
400C4⋊C8—1461no—
401C4⋊C8—1461no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8C4×D7311yescharacteristic
405C4⋊C8C4×D7311yescharacteristic
406C4⋊C8D29211yescharacteristic
407C4⋊C8D29211yescharacteristic
408#103—731no—
409#103—731no—
410#103—731no—
411#103—731no—
412#102—731no—
413#102—731no—
414#102—731no—
415#102—731no—
416C22⋊C8—21no—
417C22⋊C8—1461no—
418C22⋊C8—1461no—
419C22⋊C8—1461no—
420C22⋊C8—1461no—
421C22⋊C8—1461no—
422C22⋊C8—1461no—
423C22⋊C8C4×D7311yescharacteristic
424C22⋊C8C4×D7311yescharacteristic
425C22⋊C8D29211yescharacteristic
426C22⋊C8D29211yescharacteristic
427#97—1461no—
428#97—1461no—
429#97—1461no—
430C4×C8—1461no—
431C4×C8—1461no—
432C4×C8—1461no—
433C4×C8C22×D7311yescharacteristic
434C4×C8C22×D7311yescharacteristic
435C4×C8C22×D7311yescharacteristic
436C4×C8C22×D7311yescharacteristic
437C4×C8C22×D7311yescharacteristic
438C4×C8C22×D7311yescharacteristic
439C24—1461no—
440C2×Q8—731no—
441C2×Q8—731no—
442C2×Q8—1461no—
443C2×D4—2921no—
444C2×D4—731no—
445C2×D4—731no—
446C2×D4—1462no—
447C2×D4—1461no—
448C22×C4—5841no—
449C22×C4—5841no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4C2×C4×D7311yescharacteristic
456C22×C4C2×D29211yescharacteristic
457C22×C4#1192402811yescharacteristic
458SD16—2921no—
459SD16—2921no—
460SD16—1462no—
461SD16—1462no—
462M16—5841no—
463M16—5841no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—5841no—
467C2×C8—5841no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8C2×C4×D7311yescharacteristic
473C2×C8C2×C4×D7311yescharacteristic
474C2×C8C2×C4×D7311yescharacteristic
475C2×C8C2×C4×D7311yescharacteristic
476C2×C8C2×D29211yescharacteristic
477C2×C8C2×D29211yescharacteristic
478C2×C8C2×D29211yescharacteristic
479C2×C8C2×D29211yescharacteristic
480C4⋊C4—731no—
481C4⋊C4—731no—
482C4⋊C4—731no—
483C4⋊C4—731no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C22⋊C4—1461no—
493C22⋊C4—1461no—
494C22⋊C4—1461no—
495C22⋊C4—1461no—
496C22⋊C4—1461no—
497C22⋊C4—1461no—
498C42—1461no—
499C42—1461no—
500C42C2×C4×D7311yescharacteristic
501C42C2×D29211yescharacteristic
502C42#1192402811yescharacteristic
503C42C23×D7311yescharacteristic
504C23—2922no—
505C23—2922no—
506C23—5841no—
507C23—1461no—
508C23—1461no—
509C23C4×D29211yescharacteristic
510Q8—2922no—
511Q8—1461no—
512Q8—1461no—
513Q8—1461no—
514Q8—1461no—
515D4—2922no—
516D4—732no—
517D4—1461no—
518D4—1461no—
519D4—1461no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—5841no—
530C2×C4—5841no—
531C2×C4—5841no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4C4×D29211yescharacteristic
539C2×C4C4×D29211yescharacteristic
540C2×C4C4×D29211yescharacteristic
541C2×C4C22×C4×D7311yescharacteristic
542C2×C4C22×D29211yescharacteristic
543C2×C4C2×#1192402811yescharacteristic
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—2921no—
555C8—2921no—
556C8—2921no—
557C8—2921no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—2921no—
562C22—2921no—
563C22—2921no—
564C22—2921no—
565C22—2921no—
566C22—5844no—
567C22—1462no—
568C22C2×C4×D29211yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—2921no—
574C4—2921no—
575C4—2921no—
576C4—2921no—
577C4—2921no—
578C4—2921no—
579C4#131×D7311yescharacteristic
580C4#1798935211yescharacteristic
581C4#1798939911yescharacteristic
582C4#1798939911yescharacteristic
583C2—41no—
584C2—2922no—
585C2—5841no—
586C2#2261865911yescharacteristic
587C2#2261879211yescharacteristic
588C2#2261883511yescharacteristic
subgroupG/H
centerC2×C4C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1454C2
socleC22×C73C2×C4×D4
prime radicalC22×D292C2×C4
cosocleC42×C73C24
hypercenter#1454D73
hypocenterC73#18028
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
2. #22619032C2semidirect
4. #22608835C2semidirect
5. #22608823C2semidirect
6. #22608811C2semidirect
7. #22608790C2semidirect
8. #22608776C2semidirect
9. #22608769C2semidirect
10. #22608745C2semidirect
11. C8×#11924017C2semidirect
12. #22608711C2semidirect
13. #22608405C2semidirect
15. C73×#1454C2semidirect
26. #17988717C22semidirect
27. #17988717C22semidirect
29. C8×Dic146C22semidirect
30. C8×Dic146C22semidirect
48. #17988340C22semidirect
49. #17988340C22semidirect
50. #17988338C22semidirect
51. #17988338C22semidirect
52. #17988319C22semidirect
53. #17988319C22semidirect
54. #17988319C22semidirect
55. #17988319C22semidirect
63. #322×C73C22semidirect
64. #322×C73C22semidirect
65. #322×C73C22semidirect
66. C82×C73C22semidirect
309. #1454D73semidirect
310. C73#18028semidirect
358. #322C2×D73semidirect
359. #322C2×D73semidirect
360. #322C2×D73semidirect
361. C82C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199402
#226190322
#226189652
#226088352
#226088232
#226088112
#226087902
#226087762
#226087692
#226087452
C8×#119240172
#226087112
#226084052
#226083232
C73×#14542
#1802873