Finite Group Databasebeta

← #33057316

Subgroups — #33057316

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#22618968C211yesmaximal, characteristic
4#22608832C211yesmaximal, characteristic, complemented
5#22608820C211yesmaximal, characteristic, complemented
6#22608808C211yesmaximal, characteristic, complemented
7#22608787C211yesmaximal, characteristic, complemented
8#22608773C211yesmaximal, characteristic, complemented
9#22608766C211yesmaximal, characteristic, complemented
10#22608742C211yesmaximal, characteristic, complemented
11#22608730C211yesmaximal, characteristic, complemented
12#22608403C211yesmaximal, characteristic, complemented
13#22608321C211yesmaximal, characteristic
14#22608231C211yesmaximal, characteristic, complemented
15C73×#1572C211yesmaximal, characteristic, complemented
16#17989144C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18#17988918C411yescharacteristic
19#17988918C411yescharacteristic
20#17988878C411yescharacteristic
21#17988878C411yescharacteristic
22#17988793C411yescharacteristic
23#17988793C411yescharacteristic
24#17988740C411yescharacteristic
25#17988740C411yescharacteristic
26#17988717—21no—
27#17988714C2211yescharacteristic, complemented
28#17988714C2211yescharacteristic, complemented
29C8×Dic146—21no—
30#17988711C2211yescharacteristic, complemented
31#17988711C2211yescharacteristic, complemented
32C4×#11924017C2211yescharacteristic
33C4×#11924017C2211yescharacteristic
34C4×#11924017C2211yescharacteristic
35C4×#11924017C2211yescharacteristic
36#17988664C2211yescharacteristic
37#17988664C2211yescharacteristic
38#17988660—21no—
39#17988660—21no—
40#17988652C2211yescharacteristic
41#17988652C2211yescharacteristic
42#17988645—21no—
43#17988645—21no—
44#17988640C2211yescharacteristic
45#17988640C2211yescharacteristic
46#17988549—21no—
47C22⋊C8×D73—21no—
48#17988317C2211yescharacteristic, complemented
49#17988317C2211yescharacteristic, complemented
50#17988317C2211yescharacteristic, complemented
51#17988317C2211yescharacteristic, complemented
52#17988314C2211yescharacteristic
53#17988314C2211yescharacteristic
54#17988314C2211yescharacteristic
55#17988314C2211yescharacteristic
56#17988309C2211yescharacteristic, complemented
57#17988309C2211yescharacteristic, complemented
58#17988308C2211yescharacteristic, complemented
59C8×#10596831C2211yescharacteristic, complemented
60#432×C73C2211yescharacteristic
61#431×C73C2211yescharacteristic
62#431×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic, complemented
64#335×C73C2211yescharacteristic, complemented
65#334×C73C2211yescharacteristic, complemented
66#334×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⋊#10596831—21no—
100C4⋊#10596831D411yescharacteristic
101C4⋊#10596831D411yescharacteristic
102#14556520—21no—
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556516C2×C411yescharacteristic
107#14556516C2×C411yescharacteristic
108#14556516C2×C411yescharacteristic
109#14556516C2×C411yescharacteristic
110#14556513C2×C411yescharacteristic
111#14556513C2×C411yescharacteristic
112#14556513C2×C411yescharacteristic
113#14556513C2×C411yescharacteristic
114C22⋊#10596831—21no—
115C22⋊#10596831D411yescharacteristic
116C22⋊#10596831D411yescharacteristic
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556503—21no—
121#14556501—21no—
122#14556501—21no—
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C8×Dic73—21no—
128C8×Dic73—21no—
129C2×C42×C73C2311yescharacteristic
130C4⋊C8×C73C2×C411yescharacteristic
131C4⋊C8×C73C2×C411yescharacteristic
132C4⋊C8×C73C2311yescharacteristic
133C4⋊C8×C73C2311yescharacteristic
134C4⋊C8×C73C2311yescharacteristic
135C4⋊C8×C73C2311yescharacteristic
136C22⋊C8×C73—21no—
137C22⋊C8×C73—21no—
138C22⋊C8×C73C2×C411yescharacteristic
139C22⋊C8×C73C2×C411yescharacteristic
140#97×C73—21no—
141C4×C8×C73—21no—
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×#10596831C2×D411yescharacteristic
169C2×#10596831C2×D411yescharacteristic
170C2×#10596831C2×D411yescharacteristic
171C2×#10596831C2×D411yescharacteristic
172C8×D73—81no—
173C8×D73—81no—
174C4⋊Dic73—21no—
175C4⋊Dic73—21no—
176C4⋊Dic73—21no—
177C4⋊Dic73—21no—
178C4⋊Dic73C22×C411yescharacteristic
179C4⋊Dic73C22×C411yescharacteristic
180C4⋊Dic73C22×C411yescharacteristic
181C4⋊Dic73C22×C411yescharacteristic
182#11924007—21no—
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186C22⋊Dic73—21no—
187C22⋊Dic73—21no—
188#11924005—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C22×C4×C73C22×C411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73#4111yescharacteristic
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73C22×C411yescharacteristic
202C2×C8×C73C22×C411yescharacteristic
203C2×C8×C73C22×C411yescharacteristic
204C2×C8×C73C22×C411yescharacteristic
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—41no—
254C8×C73—41no—
255C8×C73—41no—
256C8×C73—41no—
257#10596831—21no—
258#10596831—21no—
259#10596831—21no—
260#10596831—21no—
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#1572D7311yescharacteristic, complemented
307#1497—731no—
308#1478—731no—
309#1454—731no—
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#432C2×D7311yescharacteristic
340#431—731no—
341#431—731no—
342#431C2×D7311yescharacteristic
343#431C2×D7311yescharacteristic
344#407—1461no—
345C2×C22⋊C8—1461no—
346#335C2×D7311yescharacteristic, complemented
347#335C2×D7311yescharacteristic, complemented
348#334C2×D7311yescharacteristic, complemented
349#334C2×D7311yescharacteristic, complemented
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#322—731no—
359#322—731no—
360#322—731no—
361C82—731no—
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—731no—
396C4⋊C8—731no—
397C4⋊C8—1461no—
398C4⋊C8—1461no—
399C4⋊C8—1461no—
400C4⋊C8—1461no—
401C4⋊C8—1461no—
402C4⋊C8C4×D7311yescharacteristic
403C4⋊C8C4×D7311yescharacteristic
404C4⋊C8C22×D7311yescharacteristic
405C4⋊C8C22×D7311yescharacteristic
406C4⋊C8C22×D7311yescharacteristic
407C4⋊C8C22×D7311yescharacteristic
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—21no—
418C22⋊C8—731no—
419C22⋊C8—731no—
420C22⋊C8—1461no—
421C22⋊C8—1461no—
422C22⋊C8—1461no—
423C22⋊C8—1461no—
424C22⋊C8—1461no—
425C22⋊C8C4×D7311yescharacteristic
426C22⋊C8C4×D7311yescharacteristic
427#97—21no—
428#97—1461no—
429#97—1461no—
430C4×C8—21no—
431C4×C8—731no—
432C4×C8—731no—
433C4×C8—731no—
434C4×C8—731no—
435C4×C8—1461no—
436C4×C8—1461no—
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×C4D4×D7311yescharacteristic
457C22×C4#1192402911yescharacteristic
458SD16—2921no—
459SD16—2921no—
460SD16—1462no—
461SD16—1462no—
462M16—5841no—
463M16—5841no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—5841no—
469C2×C8—5841no—
470C2×C8—731no—
471C2×C8—731no—
472C2×C8—731no—
473C2×C8—731no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8C2×C4×D7311yescharacteristic
477C2×C8C2×C4×D7311yescharacteristic
478C2×C8C2×C4×D7311yescharacteristic
479C2×C8C2×C4×D7311yescharacteristic
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
501C42D4×D7311yescharacteristic
502C42#1192402911yescharacteristic
503C42C23×D7311yescharacteristic
504C23—2922no—
505C23—2922no—
506C23—5841no—
507C23—1461no—
508C23—1461no—
509C23#1455655311yescharacteristic
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×C4#1455655311yescharacteristic
539C2×C4#1455655311yescharacteristic
540C2×C4#1455655311yescharacteristic
541C2×C4C22×C4×D7311yescharacteristic
542C2×C4C2×D4×D7311yescharacteristic
543C2×C4C2×#1192402911yescharacteristic
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—2921no—
553C8—2921no—
554C8—1461no—
555C8—1461no—
556C8—1461no—
557C8—1461no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—2921no—
562C22—2921no—
563C22—2921no—
564C22—2921no—
565C22—2921no—
566C22—5844no—
567C22—1462no—
568C22C2×#1455655311yescharacteristic
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#1798939411yescharacteristic
582C4#1798939411yescharacteristic
583C2—41no—
584C2—2922no—
585C2—5841no—
586C2#2261865511yescharacteristic
587C2#2261878911yescharacteristic
588C2#2261883311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1572C2
socleC22×C73C2×C4×D4
prime radicalC22×D292C2×C4
cosocleC42×C73C24
hypercenter#1572D73
hypocenterC73#18028
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
2. #22619032C2semidirect
4. #22608832C2semidirect
5. #22608820C2semidirect
6. #22608808C2semidirect
7. #22608787C2semidirect
8. #22608773C2semidirect
9. #22608766C2semidirect
10. #22608742C2semidirect
11. #22608730C2semidirect
12. #22608403C2semidirect
14. #22608231C2semidirect
15. C73×#1572C2semidirect
27. #17988714C22semidirect
28. #17988714C22semidirect
30. #17988711C22semidirect
31. #17988711C22semidirect
48. #17988317C22semidirect
49. #17988317C22semidirect
50. #17988317C22semidirect
51. #17988317C22semidirect
56. #17988309C22semidirect
57. #17988309C22semidirect
58. #17988308C22semidirect
59. C8×#10596831C22semidirect
63. #335×C73C22semidirect
64. #335×C73C22semidirect
65. #334×C73C22semidirect
66. #334×C73C22semidirect
306. #1572D73semidirect
310. C73#18028semidirect
346. #335C2×D73semidirect
347. #335C2×D73semidirect
348. #334C2×D73semidirect
349. #334C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199402
#226190322
#226189682
#226088322
#226088202
#226088082
#226087872
#226087732
#226087662
#226087422
#226087302
#226084032
#226083212
#226082312
C73×#15722
#1802873