Finite Group Databasebeta

← #33056147

Subgroups — #33056147

Order 18688.

620 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619955C211yesmaximal, characteristic
2#22619083C211yesmaximal, characteristic, complemented
3#22619041C211yesmaximal, characteristic, complemented
4#22608809C211yesmaximal, characteristic, complemented
5#22608786C211yesmaximal, characteristic, complemented
6#22608762C211yesmaximal, characteristic, complemented
7#22608746C211yesmaximal, characteristic, complemented
8#22608746C211yesmaximal, characteristic, complemented
9Q16×#10596831C211yesmaximal, characteristic, complemented
10#22608725C211yesmaximal, characteristic, complemented
11#22608704C211yesmaximal, characteristic, complemented
12#22608369C211yesmaximal, characteristic, complemented
13#22608343C211yesmaximal, characteristic, complemented
14#22608227C211yesmaximal, characteristic, complemented
15C73×#1584C211yesmaximal, characteristic, complemented
16#17989124C2211yescharacteristic
17C4×#11924029C2211yescharacteristic
18#17988927C411yescharacteristic
19#17988914C411yescharacteristic
20#17988896C411yescharacteristic
21#17988890C411yescharacteristic
22#17988811C411yescharacteristic
23#17988805C411yescharacteristic
24#17988748C411yescharacteristic
25#17988735C411yescharacteristic
26Q8×#10596831C2211yescharacteristic
27Q8×#10596831C2211yescharacteristic
28C4×#11924015C2211yescharacteristic
29C4×#11924015C2211yescharacteristic
30C4×D584C2211yescharacteristic
31#17988666C2211yescharacteristic
32#17988664—21no—
33#17988660—21no—
34C8×D292—21no—
35C8×D292C2211yescharacteristic
36#17988652—21no—
37#17988652C2211yescharacteristic
38#17988652C2211yescharacteristic
39#17988649—21no—
40#17988638—21no—
41#17988638C2211yescharacteristic
42#17988638C2211yescharacteristic
43#17988612—21no—
44#17988609C2211yescharacteristic
45C4⋊C8×D73—21no—
46C4⋊C8×D73C2211yescharacteristic
47C4×#11924011C2211yescharacteristic
48#17988340C2211yescharacteristic
49#17988339C2211yescharacteristic
50#17988339C2211yescharacteristic
51Q8⋊#10596831C2211yescharacteristic
52Q8⋊#10596831C2211yescharacteristic
53#17988316C2211yescharacteristic
54#17988316C2211yescharacteristic
55#17988314C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60#446×C73C2211yescharacteristic
61C8×Q8×C73C2211yescharacteristic
62C4×Q16×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65Q8⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×#11924029C2×C411yescharacteristic
68C2×#11924015C2×C411yescharacteristic
69C2×#11924015C2×C411yescharacteristic
70C2×D584C2×C411yescharacteristic
71C2×#11924011—21no—
72C2×#11924011—21no—
73C2×#11924011—21no—
74C2×C8×D73—21no—
75C2×C8×D73—21no—
76C2×C8×D73—21no—
77C2×C8×D73—21no—
78#14556586—21no—
79#14556582C2×C411yescharacteristic
80#14556580C2×C411yescharacteristic
81#14556576—21no—
82#14556567C2×C411yescharacteristic
83#14556566C2×C411yescharacteristic
84Q8×Dic73C2×C411yescharacteristic
85C4×D292—21no—
86C4×D292—21no—
87C4×D292C2311yescharacteristic
88C4×D292C2311yescharacteristic
89#14556553—21no—
90#14556553C2×C411yescharacteristic
91#14556549—21no—
92#14556549C2×C411yescharacteristic
93#14556548—21no—
94C42×D73—21no—
95C42×D73C2311yescharacteristic
96#14556525C2×C411yescharacteristic
97#14556522C2×C411yescharacteristic
98#14556522C2×C411yescharacteristic
99C4⋊#10596831—21no—
100C4⋊#10596831—21no—
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520C2×C411yescharacteristic
106#14556520C2×C411yescharacteristic
107#14556520D411yescharacteristic
108#14556520D411yescharacteristic
109#14556515C2×C411yescharacteristic
110#14556515C2×C411yescharacteristic
111#14556513C2×C411yescharacteristic
112#14556513C2×C411yescharacteristic
113#14556511C2×C411yescharacteristic
114#14556511C2×C411yescharacteristic
115#14556503—21no—
116#14556503—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503C2×C411yescharacteristic
120#14556503C2×C411yescharacteristic
121#14556503D411yescharacteristic
122#14556503D411yescharacteristic
123#14556501—21no—
124C4×#10596831—21no—
125C4×#10596831—21no—
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
128C4×#10596831C2311yescharacteristic
129C2×Q16×C73C2×C411yescharacteristic
130C4×Q8×C73C2311yescharacteristic
131C4×Q8×C73C2311yescharacteristic
132#107×C73C2×C411yescharacteristic
133C4⋊C8×C73—21no—
134C4⋊C8×C73—21no—
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#103×C73C2×C411yescharacteristic
138#103×C73C2×C411yescharacteristic
139#97×C73—21no—
140C4×C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144#11924029—22no—
145#11924029—22no—
146C2×D292—21no—
147C2×D292—21no—
148C2×D292C22×C411yescharacteristic
149C2×D292C22×C411yescharacteristic
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154C2×C4×D73C22×C411yescharacteristic
155C2×C4×D73C2×D411yescharacteristic
156C2×C4×D73#4111yescharacteristic
157#11924015—22no—
158#11924015—22no—
159D584—22no—
160#11924011—42no—
161#11924011—81no—
162#11924011—81no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831C22×C411yescharacteristic
167C2×#10596831C22×C411yescharacteristic
168C2×#10596831C2×D411yescharacteristic
169C2×#10596831C2×D411yescharacteristic
170C8×D73—81no—
171C8×D73—81no—
172C8×D73—81no—
173C8×D73—81no—
174C4⋊Dic73—21no—
175C4⋊Dic73—21no—
176C4⋊Dic73C22×C411yescharacteristic
177C4⋊Dic73C22×C411yescharacteristic
178#11924007—21no—
179#11924007—21no—
180#11924005—21no—
181#11924005—21no—
182#11924005—21no—
183#11924005—21no—
184#11924005—21no—
185#11924005—21no—
186#11924005—21no—
187#11924005—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73C22×C411yescharacteristic
191C4×Dic73C2×D411yescharacteristic
192C4×Dic73#4111yescharacteristic
193C2×Q8×C73C22×C411yescharacteristic
194C2×Q8×C73C22×C411yescharacteristic
195Q16×C73—22no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73C22×C411yescharacteristic
200C2×C8×C73C22×C411yescharacteristic
201C2×C8×C73C2×D411yescharacteristic
202C2×C8×C73C2×D411yescharacteristic
203C4⋊C4×C73—21no—
204C4⋊C4×C73—21no—
205C4⋊C4×C73C22×C411yescharacteristic
206C4⋊C4×C73C22×C411yescharacteristic
207C42×C73—21no—
208C42×C73—21no—
209C42×C73C2411yescharacteristic
210C22×D73—21no—
211C22×D73—21no—
212C22×D73C4×D411yescharacteristic
213Q8×C73—21no—
214Q8×C73—21no—
215Q8×C73—21no—
216Q8×C73#13112yes—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73C23×C411yescharacteristic
222C2×C4×C73C22×D411yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73C4×D411yescharacteristic
227C2×Dic73C4×D411yescharacteristic
228C2×Dic73C4×D411yescharacteristic
229D292—24no—
230D292—21no—
231D292—21no—
232D292—21no—
233D292—81no—
234D292#13112yes—
235C4×D73—24no—
236C4×D73—21no—
237C4×D73—21no—
238C4×D73—21no—
239C4×D73—21no—
240C4×D73—41no—
241C4×D73—41no—
242C4×D73—41no—
243C4×D73—41no—
244C4×D73—41no—
245C4×D73—81no—
246C8×C73—21no—
247C8×C73—21no—
248C8×C73—21no—
249C8×C73—21no—
250C8×C73—41no—
251C8×C73—41no—
252C8×C73—41no—
253#10596831—22no—
254#10596831—21no—
255#10596831—21no—
256#10596831—21no—
257#10596831—21no—
258#10596831—41no—
259#10596831—41no—
260C22×C73C2×C4×D411yescharacteristic
261C2×D73—21no—
262C2×D73—22no—
263C2×D73—21no—
264C2×D73—21no—
265C2×D73—41no—
266C2×D73—41no—
267C2×D73—41no—
268C2×D73—41no—
269C2×D73—41no—
270C4×C73—22no—
271C4×C73—21no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73—41no—
275C4×C73C2×#13111yescharacteristic
276C4×C73C2×#13111yescharacteristic
277C4×C73#57611yescharacteristic
278C4×C73#57811yescharacteristic
279Dic73—21no—
280Dic73—21no—
281Dic73—21no—
282Dic73—41no—
283Dic73—41no—
284#17926—731nomaximal, Sylow
285C2×C73#292911yescharacteristic
286C2×C73#295111yescharacteristic
287C2×C73#295611yescharacteristic
288D73—41no—
289D73—42no—
290D73—81no—
291#3102—731no—
292#2993—731no—
293#2988—731no—
294#1588—731no—
295#1585—731no—
296#1584D7311yescharacteristic, complemented
297#1581—731no—
298SD16⋊C8—731no—
299SD16⋊C8—731no—
300C8×Q16—731no—
301C8×D8—731no—
302#1571—731no—
303#1489—731no—
304#1483—731no—
305#1453—731no—
306C73#1792611yescharacteristic, complemented, Sylow
307#529—731no—
308C4×#41—731no—
309#485—731no—
310#482—731no—
311#479—731no—
312#478—731no—
313#464—731no—
314#463—731no—
315#452—731no—
316#449—731no—
317#446C2×D7311yescharacteristic
318C8×Q8—731no—
319C8×Q8—731no—
320C8×Q8C2×D7311yescharacteristic
321C4×Q16C2×D7311yescharacteristic
322C4×SD16—731no—
323C4×SD16—731no—
324C4×D8—731no—
325#436—731no—
326#436—1461no—
327#435—1461no—
328C8×D4—731no—
329C8×D4—731no—
330C8×D4—731no—
331C8×D4—1461no—
332C8×D4—1461no—
333#433—1461no—
334#431—731no—
335#431—731no—
336#431—1461no—
337#424—1461no—
338#423—731no—
339C2×C4⋊C8—731no—
340C2×C4⋊C8—1461no—
341C4×M16—731no—
342#335—731no—
343#335—731no—
344#335—731no—
345#335C2×D7311yescharacteristic
346Q8⋊C8—731no—
347Q8⋊C8—731no—
348Q8⋊C8C2×D7311yescharacteristic
349Q8⋊C8C2×D7311yescharacteristic
350D4⋊C8—731no—
351D4⋊C8—731no—
352D4⋊C8—731no—
353D4⋊C8—731no—
354#322—731no—
355#322—731no—
356#322C2×D7311yescharacteristic
357C82—731no—
358C2×#41—731no—
359C2×Q16C4×D7311yescharacteristic
360C2×SD16—731no—
361C2×SD16—731no—
362C2×D8—731no—
363C2×M16—1461no—
364C2×M16—1461no—
365C2×M16—1461no—
366C22×C8—1461no—
367C22×C8—1461no—
368C22×C8—1461no—
369C22×C8—1461no—
370#126—1461no—
371#125—731no—
372#124—731no—
373#123—1461no—
374C22⋊Q8—731no—
375#121—731no—
376C4×Q8—731no—
377C4×Q8C22×D7311yescharacteristic
378C4×Q8C22×D7311yescharacteristic
379C4×D4—731no—
380C4×D4—731no—
381C4×D4—731no—
382C4×D4—1461no—
383C4×D4—1461no—
384C4×D4—1461no—
385#117—731no—
386#117—1461no—
387#117—1461no—
388C2×C42—731no—
389C2×C42—1461no—
390#107—731no—
391#107C4×D7311yescharacteristic
392#106—731no—
393#106—731no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—731no—
397C4⋊C8—731no—
398C4⋊C8—731no—
399C4⋊C8—731no—
400C4⋊C8—731no—
401C4⋊C8—731no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405C4⋊C8—1461no—
406C4⋊C8C22×D7311yescharacteristic
407C4⋊C8C22×D7311yescharacteristic
408#103—731no—
409#103—731no—
410#103C4×D7311yescharacteristic
411#103C4×D7311yescharacteristic
412#102—731no—
413#102—731no—
414#102—731no—
415#102—731no—
416C22⋊C8—731no—
417C22⋊C8—731no—
418C22⋊C8—731no—
419C22⋊C8—731no—
420C22⋊C8—1461no—
421C22⋊C8—1461no—
422C22⋊C8—1461no—
423C22⋊C8—1461no—
424#97—21no—
425#97—1461no—
426C4×C8—21no—
427C4×C8—731no—
428C4×C8—731no—
429C4×C8—731no—
430C4×C8—1461no—
431C4×C8—1461no—
432C4×C8C22×D7311yescharacteristic
433C4×C8C22×D7311yescharacteristic
434C4×C8C22×D7311yescharacteristic
435#41—1462no—
436#41—1462no—
437C2×Q8C2×C4×D7311yescharacteristic
438C2×Q8C2×C4×D7311yescharacteristic
439C2×D4—731no—
440C2×D4—731no—
441C2×D4—1461no—
442C2×D4—1461no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—731no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1461no—
449C22×C4—1461no—
450Q16—22no—
451SD16—1462no—
452SD16—1462no—
453D8—1462no—
454M16—2922no—
455M16—5841no—
456M16—5841no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—5841no—
461C2×C8—5841no—
462C2×C8—5841no—
463C2×C8—5841no—
464C2×C8—731no—
465C2×C8—731no—
466C2×C8—731no—
467C2×C8—731no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8C2×C4×D7311yescharacteristic
472C2×C8C2×C4×D7311yescharacteristic
473C2×C8D4×D7311yescharacteristic
474C2×C8D4×D7311yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—731no—
478C4⋊C4—731no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4C2×C4×D7311yescharacteristic
484C4⋊C4C2×C4×D7311yescharacteristic
485C22⋊C4—1461no—
486C22⋊C4—1461no—
487C22⋊C4—1461no—
488C22⋊C4—1461no—
489C22⋊C4—1461no—
490C22⋊C4—1461no—
491C22⋊C4—1461no—
492C22⋊C4—1461no—
493C42—21no—
494C42—21no—
495C42—731no—
496C42—731no—
497C42—731no—
498C42—1461no—
499C42—1461no—
500C42C23×D7311yescharacteristic
501C23—731no—
502C23—1461no—
503C23—1461no—
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8#1455659912yes—
508D4—5841no—
509D4—732no—
510D4—1464no—
511D4—1461no—
512D4—1461no—
513D4—1461no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—5841no—
524C2×C4—731no—
525C2×C4—731no—
526C2×C4—731no—
527C2×C4—1464no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4C22×C4×D7311yescharacteristic
535C2×C4C2×D4×D7311yescharacteristic
536C2×C4#41×D7311yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—41no—
542C8—41no—
543C8—41no—
544C8—2921no—
545C8—2921no—
546C8—1462no—
547C8—1461no—
548C8—1461no—
549C8—1461no—
550C8—1461no—
551C22—2921no—
552C22—2921no—
553C22—2921no—
554C22—2921no—
555C22—2921no—
556C22—1461no—
557C22—1461no—
558C22—1462no—
559C22—1461no—
560C22C4×D4×D7311yescharacteristic
561C4—21no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—2921no—
567C4—2921no—
568C4—1461no—
569C4—1461no—
570C4—1461no—
571C4#131×D7311yescharacteristic
572C4C2×#1455659911yescharacteristic
573C4#1798938911yescharacteristic
574C4#1798940011yescharacteristic
575C2—2922no—
576C2—2921no—
577C2—5841no—
578C2#2261862211yescharacteristic
579C2#2261880611yescharacteristic
580C2#2261883911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1584C2
socleC22×C73C2×C4×D4
prime radical#17988735C4
cosocleC42×C73C24
hypercenter#1584D73
hypocenterC73#17926
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
2. #22619083C2semidirect
3. #22619041C2semidirect
4. #22608809C2semidirect
5. #22608786C2semidirect
6. #22608762C2semidirect
7. #22608746C2semidirect
8. #22608746C2semidirect
9. Q16×#10596831C2semidirect
10. #22608725C2semidirect
11. #22608704C2semidirect
12. #22608369C2semidirect
13. #22608343C2semidirect
14. #22608227C2semidirect
15. C73×#1584C2semidirect
296. #1584D73semidirect
306. C73#17926semidirect

Maximal subgroups

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

subgroupindex
#226199552
#226190832
#226190412
#226088092
#226087862
#226087622
#226087462
Q16×#105968312
#226087252
#226087042
#226083692
#226083432
#226082272
C73×#15842
#1792673