Finite Group Databasebeta

← #33245130

Subgroups — #33245130

Order 18688.

2070 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989200C211yesmaximal, characteristic
2#22612734C214yesmaximal, complemented
3#22612705C214yesmaximal
4#22612650C211yesmaximal, characteristic, complemented
5#22612614C212yesmaximal
6#22612578C212yesmaximal, complemented
7#22612539C211yesmaximal, characteristic
8#22612498C211yesmaximal, characteristic, complemented
9#125×Dic73C211yesmaximal, characteristic, complemented
10#22612490C214yesmaximal, complemented
11#22612481C214yesmaximal
12#22612461C212yesmaximal
13C4×#14556556C212yesmaximal, complemented
14#22612270C211yesmaximal, characteristic
15C73×#2330C211yesmaximal, characteristic, complemented
16#17989200C4116yescomplemented
17C2×#14556591C2211yescharacteristic
18C2×#14556586C2214yes—
19C2×#14556582C2214yes—
20C2×#14556571C2212yes—
21C2×C4×Dic146C2212yes—
22C2×#14556556C2212yes—
23C2×#14556556C2212yes—
24C2×#14556548C2211yescharacteristic
25C2×#14556548C2211yescharacteristic
26C2×C42×D73C2211yescharacteristic
27#17988493C2212yes—
28#17988492C2212yes—
29#17988486C2212yes—
30#17988485C2214yes—
31#17988483C2214yes—
32#17988479C2214yes—
33#17988478C2214yes—
34#17988477C2214yes—
35#17988474C2214yes—
36#17988473C2214yes—
37#17988472C2214yes—
38#17988472C2214yes—
39#17988472C2214yes—
40#17988470C2214yes—
41#17988468C2214yes—
42#17988468C2214yes—
43#17988458C2214yes—
44#17988454C2214yes—
45#17988453C2211yescharacteristic
46#17988452C2214yes—
47#17988444C2212yes—
48C4⋊C4×Dic73C2214yes—
49C4⋊C4×Dic73C2214yes—
50C4⋊C4×Dic73C2214yes—
51C4⋊C4×Dic73C2212yes—
52C4⋊C4×Dic73C2212yes—
53C4×#11924007C2214yes—
54#17988438C2214yes—
55#17988438C2212yes—
56#17988438C2212yes—
57#17988438C2214yes—
58#17988434C2212yes—
59#17988434C2214yes—
60#17988434C2212yes—
61#17988433C2211yescharacteristic
62C42×Dic73C2214yes—
63C42×Dic73C2211yescharacteristic
64C42×Dic73C2211yescharacteristic
65C2×#125×C73C2211yescharacteristic
66#389×C73C2214yes—
67#389×C73C2212yes—
68#387×C73C2214yes—
69#384×C73C2211yescharacteristic
70C4×C4⋊C4×C73C2212yes—
71#376×C73C2211yescharacteristic
72C22×Dic146C2312yes—
73C22×C4×D73C2311yescharacteristic
74C22×C4×D73C2311yescharacteristic
75C22×C4×D73C2311yescharacteristic
76#14556591C2×C418yescomplemented
77#14556586C2×C4132yes—
78#14556582C2×C4132yescomplemented
79#14556571C2×C4116yes—
80C4×Dic146—28no—
81#14556556—28no—
82#14556556C2×C4116yescomplemented
83#14556548—24no—
84#14556548C2×C418yes—
85C2×C4⋊Dic73C2312yes—
86C2×C4⋊Dic73C2314yes—
87C2×C4⋊Dic73C2314yes—
88C2×C4⋊Dic73C2314yes—
89C2×#11924007C2314yes—
90C2×#11924007C2312yes—
91C2×#11924007C2314yes—
92C2×#11924007C2312yes—
93C2×#11924007C2314yes—
94C2×#11924005C2312yes—
95C2×#11924005C2314yes—
96C2×#11924005C2312yes—
97C2×#11924005C2314yes—
98C2×C4×Dic73C2312yes—
99C2×C4×Dic73C2311yescharacteristic
100C2×C4×Dic73C2314yes—
101C2×C4×Dic73C2312yes—
102C2×C4×Dic73C2314yes—
103C2×C4×Dic73C2314yes—
104C2×C4×Dic73C2311yescharacteristic
105C2×C4×Dic73C2311yescharacteristic
106C2×C4×Dic73C2312yes—
107C2×C4×Dic73C2312yes—
108C2×C4×Dic73C2314yes—
109C2×C4×Dic73C2311yescharacteristic
110C2×C4×Dic73C2314yes—
111C2×C4×Dic73C2311yescharacteristic
112C42×D73—24no—
113#14556498C2312yes—
114#14556498C2312yes—
115#14556498C2314yes—
116#14556498C2314yes—
117#14556498C2314yes—
118#14556498C2314yes—
119#14556498C2314yes—
120#14556498C2314yes—
121#14556498C2314yes—
122#14556498C2314yes—
123#14556498C2314yes—
124#125×C73C2×C418yescomplemented
125C2×C4⋊C4×C73C2312yes—
126C2×C4⋊C4×C73C2314yes—
127C2×C4⋊C4×C73C2314yes—
128C2×C4⋊C4×C73C2312yes—
129C2×C4⋊C4×C73C2314yes—
130C2×C42×C73C2311yescharacteristic
131C2×C42×C73C2314yes—
132C2×C42×C73C2311yescharacteristic
133C2×C42×C73C2311yescharacteristic
134#95×C73C2314yes—
135#95×C73C2314yes—
136#95×C73C2312yes—
137#95×C73C2312yes—
138C23×D73C2411yescharacteristic
139C2×Dic146—24no—
140C2×Dic146—24no—
141C2×Dic146C22×C418yes—
142C22×Dic73C2412yes—
143C22×Dic73C2414yes—
144C22×Dic73C2412yes—
145C22×Dic73C2414yes—
146C22×Dic73C2411yescharacteristic
147C22×Dic73C2411yescharacteristic
148C22×Dic73C2411yescharacteristic
149C2×C4×D73—22no—
150C2×C4×D73—22no—
151C2×C4×D73—22no—
152C2×C4×D73—22no—
153C2×C4×D73—24no—
154C2×C4×D73—42no—
155C2×C4×D73C22×C414yes—
156C4⋊Dic73—24no—
157C4⋊Dic73—44no—
158C4⋊Dic73C22×C4116yes—
159C4⋊Dic73C22×C4116yes—
160#11924007—24no—
161#11924007—28no—
162#11924007—28no—
163#11924007C22×C418yes—
164#11924007C22×C4116yes—
165#11924005—24no—
166#11924005—44no—
167#11924005C22×C418yes—
168#11924005C22×C4116yes—
169C4×Dic73—22no—
170C4×Dic73—28no—
171C4×Dic73—22no—
172C4×Dic73—22no—
173C4×Dic73—28no—
174C4×Dic73—24no—
175C4×Dic73—28no—
176C4×Dic73—24no—
177C4×Dic73—42no—
178C4×Dic73—44no—
179C4×Dic73—42no—
180C4×Dic73C22×C414yes—
181C4×Dic73C22×C4116yes—
182C4×Dic73#4114yes—
183C22×C4×C73C2411yescharacteristic
184C22×C4×C73C2411yescharacteristic
185C22×C4×C73C2414yes—
186C22×C4×C73C2414yes—
187C22×C4×C73C2412yes—
188C22×C4×C73C2412yes—
189C22×C4×C73C2411yescharacteristic
190C4⋊C4×C73—24no—
191C4⋊C4×C73—44no—
192C4⋊C4×C73—44no—
193C4⋊C4×C73C22×C418yes—
194C4⋊C4×C73C22×C4116yes—
195C42×C73—22no—
196C42×C73—44no—
197C42×C73C22×C414yes—
198C42×C73#4114yes—
199C23×C73C2511yescharacteristic
200C22×D73—21no—
201C22×D73—21no—
202C22×D73—22no—
203C22×D73—22no—
204C22×D73C23×C412yes—
205C2×C4×C73—24no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—24no—
210C2×C4×C73—22no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—24no—
214C2×C4×C73—22no—
215C2×C4×C73—24no—
216C2×C4×C73—24no—
217C2×C4×C73—22no—
218C2×C4×C73C23×C414yes—
219C2×C4×C73C23×C418yes—
220C2×C4×C73C23×C412yes—
221C2×C4×C73C2×#4112yes—
222C2×C4×C73C2×#4114yes—
223C2×C4×C73C2×#4112yes—
224C2×Dic73—21no—
225C2×Dic73—24no—
226C2×Dic73—24no—
227C2×Dic73—21no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—24no—
233C2×Dic73—22no—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—22no—
237C2×Dic73C23×C414yes—
238C2×Dic73C23×C418yes—
239C2×Dic73C23×C412yes—
240C2×Dic73C2×#4112yes—
241C2×Dic73C2×#4112yes—
242C2×Dic732−1+414yes—
243C4×D73—28no—
244C4×D73—44no—
245C4×D73—44no—
246Dic146—216no—
247C22×C73C24×C411yescharacteristic
248C22×C73C22×#4111yescharacteristic
249C22×C73C22×#4111yescharacteristic
250C22×C73C2×2−1+412yes—
251C22×C73#58512yes—
252C2×D73—22no—
253C2×D73—24no—
254C2×D73—41no—
255C2×D73—41no—
256C2×D73—42no—
257C4×C73—28no—
258C4×C73—24no—
259C4×C73—24no—
260C4×C73—41no—
261C4×C73—41no—
262C4×C73—44no—
263C4×C73C4×#4114yes—
264Dic73—28no—
265Dic73—22no—
266Dic73—24no—
267Dic73—22no—
268Dic73—42no—
269Dic73—44no—
270Dic73#54114yescomplemented
271C4×#541—731nomaximal, Sylow
272C2×C73C2×C4×#4111yescharacteristic
273C2×C73C4×2−1+412yes—
274C2×C73C2×#54111yescharacteristic
275C2×C73#345511yescharacteristic
276C2×C73#346112yes—
277D73—42no—
278C2×#541—731no—
279#2336—734no—
280#2334—734no—
281#2330D7311yescharacteristic, complemented
282#2327—731no—
283#2323—732no—
284#2320—732no—
285#2314—731no—
286C4×C4⋊Q8—731no—
287C4×#125—734no—
288C4×#125—731no—
289C4×#126—734no—
290C4×C22⋊Q8—732no—
291C42×Q8—732no—
292C4×#117—731no—
293C73C4×#54111yescharacteristic, complemented, Sylow
294#541—7316no—
295C2×C4⋊Q8—731no—
296C2×#126—734no—
297C2×#125—734no—
298C2×#125C2×D7311yescharacteristic
299C2×C22⋊Q8—732no—
300C2×C4×Q8—732no—
301C2×C4×Q8—732no—
302C2×C4×Q8—732no—
303C2×#117—731no—
304C2×#117—731no—
305C22×C42—731no—
306#391—732no—
307#391—732no—
308#389—732no—
309#389—734no—
310#389—734no—
311#389C2×D7314yes—
312#389C2×D7312yes—
313#388—734no—
314#388—734no—
315#388—734no—
316#387—734no—
317#387—734no—
318#387—734no—
319#387—734no—
320#387—734no—
321#387—734no—
322#387—734no—
323#387—734no—
324#387C2×D7314yes—
325#385—734no—
326#384C2×D7311yescharacteristic
327#383—734no—
328#382—734no—
329#382—731no—
330#380—732no—
331C4×C4⋊C4—734no—
332C4×C4⋊C4—734no—
333C4×C4⋊C4—734no—
334C4×C4⋊C4—732no—
335C4×C4⋊C4—732no—
336C4×C4⋊C4—734no—
337C4×C4⋊C4—734no—
338C4×C4⋊C4—732no—
339C4×C4⋊C4—734no—
340C4×C4⋊C4—732no—
341C4×C4⋊C4C2×D7312yes—
342C4×C22⋊C4—732no—
343C4×C22⋊C4—734no—
344C4×C22⋊C4—732no—
345#376—731no—
346#376C2×D7311yescharacteristic
347C43—734no—
348C43—731no—
349C43—731no—
350C22×Q8—732no—
351C23×C4—731no—
352C23×C4—731no—
353C23×C4—731no—
354C4⋊Q8—738no—
355#126—7332no—
356#125—7332no—
357#125C4×D7318yescomplemented
358C22⋊Q8—7316no—
359C4×Q8—7316no—
360C4×Q8—1468no—
361C4×Q8—1468no—
362#117—738no—
363#117—1464no—
364C2×C4⋊C4—734no—
365C2×C4⋊C4—732no—
366C2×C4⋊C4—734no—
367C2×C4⋊C4—734no—
368C2×C4⋊C4—734no—
369C2×C4⋊C4—732no—
370C2×C4⋊C4—732no—
371C2×C4⋊C4—734no—
372C2×C4⋊C4—734no—
373C2×C4⋊C4C22×D7312yes—
374C2×C4⋊C4C22×D7314yes—
375C2×C4⋊C4C22×D7314yes—
376C2×C4⋊C4C22×D7312yes—
377C2×C4⋊C4C22×D7314yes—
378C2×C22⋊C4—734no—
379C2×C22⋊C4—732no—
380C2×C22⋊C4—734no—
381C2×C22⋊C4—732no—
382C2×C42—732no—
383C2×C42—732no—
384C2×C42—734no—
385C2×C42—731no—
386C2×C42—731no—
387C2×C42—734no—
388C2×C42—731no—
389C2×C42—734no—
390C2×C42—734no—
391C2×C42—732no—
392C2×C42—734no—
393C2×C42—732no—
394C2×C42—731no—
395C2×C42—731no—
396C2×C42—1464no—
397C2×C42C22×D7311yescharacteristic
398C2×C42C22×D7314yes—
399C2×C42C22×D7311yescharacteristic
400C2×C42C22×D7311yescharacteristic
401#95—732no—
402#95—732no—
403#95—734no—
404#95—734no—
405#95—734no—
406#95—734no—
407#95—734no—
408#95—734no—
409#95—734no—
410#95—734no—
411#95—734no—
412#95C22×D7312yes—
413#95C22×D7312yes—
414#95C22×D7314yes—
415#95C22×D7314yes—
416C24—731no—
417C2×Q8—738no—
418C2×Q8—1464no—
419C2×Q8—1464no—
420C22×C4—2922no—
421C22×C4—731no—
422C22×C4—732no—
423C22×C4—731no—
424C22×C4—734no—
425C22×C4—731no—
426C22×C4—734no—
427C22×C4—732no—
428C22×C4—734no—
429C22×C4—1462no—
430C22×C4—1462no—
431C22×C4—1464no—
432C22×C4—1462no—
433C22×C4—1462no—
434C22×C4C23×D7311yescharacteristic
435C22×C4C23×D7311yescharacteristic
436C22×C4C23×D7314yes—
437C22×C4C23×D7314yes—
438C22×C4C23×D7312yes—
439C22×C4C23×D7312yes—
440C22×C4C23×D7311yescharacteristic
441C4⋊C4—24no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—2924no—
445C4⋊C4—738no—
446C4⋊C4—7316no—
447C4⋊C4—7316no—
448C4⋊C4—7316no—
449C4⋊C4—1468no—
450C4⋊C4—1464no—
451C4⋊C4—1468no—
452C4⋊C4—1464no—
453C4⋊C4C2×C4×D7318yes—
454C4⋊C4C2×C4×D73116yes—
455C22⋊C4—2924no—
456C22⋊C4—738no—
457C22⋊C4—7316no—
458C22⋊C4—1464no—
459C42—22no—
460C42—44no—
461C42—2922no—
462C42—2922no—
463C42—2924no—
464C42—734no—
465C42—7316no—
466C42—734no—
467C42—1468no—
468C42—1464no—
469C42—1464no—
470C42—1462no—
471C42—1462no—
472C42—1468no—
473C42—1462no—
474C42—1468no—
475C42C2×C4×D7314yes—
476C42#1192402914yes—
477C23—732no—
478C23—1461no—
479C23—1461no—
480C23—1462no—
481C23—1462no—
482C23C24×D7311yescharacteristic
483Q8—14616no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—2924no—
498C2×C4—2924no—
499C2×C4—732no—
500C2×C4—734no—
501C2×C4—734no—
502C2×C4—738no—
503C2×C4—732no—
504C2×C4—732no—
505C2×C4—1462no—
506C2×C4—1464no—
507C2×C4—1462no—
508C2×C4—1464no—
509C2×C4—1464no—
510C2×C4—1462no—
511C2×C4—1461no—
512C2×C4—1464no—
513C2×C4—1464no—
514C2×C4—1461no—
515C2×C4—1462no—
516C2×C4—1462no—
517C2×C4—1462no—
518C2×C4—1468no—
519C2×C4C22×C4×D7318yes—
520C2×C4C22×C4×D7314yes—
521C2×C4C22×C4×D7312yes—
522C2×C4#41×D7312yes—
523C2×C4C2×#1192402914yes—
524C2×C4C2×#1192402912yes—
525C22—2921no—
526C22—2921no—
527C22—2922no—
528C22—1462no—
529C22—1464no—
530C22C23×C4×D7311yescharacteristic
531C22C2×#41×D7311yescharacteristic
532C22C22×#1192402911yescharacteristic
533C222−1+4×D7312yes—
534C22#1798942212yes—
535C4—24no—
536C4—28no—
537C4—24no—
538C4—41no—
539C4—41no—
540C4—44no—
541C4—2922no—
542C4—2924no—
543C4—734no—
544C4—1462no—
545C4—1462no—
546C4—1468no—
547C4—1464no—
548C4C4×#1192402914yes—
549C2—2922no—
550C2C2×C4×#1192402911yescharacteristic
551C2#2262314212yes—
552C2#541×D7311yescharacteristic
553C2#2262331511yescharacteristic
554C2#2262336512yes—
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2330C2
socle, cosocleC23×C73C25
prime radicalC23×D73C24
hypercenter#2330D73
hypocenterC73C4×#541
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #22612734C2semidirect
4. #22612650C2semidirect
6. #22612578C2semidirect
8. #22612498C2semidirect
9. #125×Dic73C2semidirect
10. #22612490C2semidirect
13. C4×#14556556C2semidirect
15. C73×#2330C2semidirect
16. #17989200C4semidirect
76. #14556591C2×C4semidirect
78. #14556582C2×C4semidirect
82. #14556556C2×C4semidirect
124. #125×C73C2×C4semidirect
270. Dic73#541semidirect
281. #2330D73semidirect
293. C73C4×#541semidirect
357. #125C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179892002
#226127342
#226127052
#226126502
#226126142
#226125782
#226125392
#226124982
#125×Dic732
#226124902
#226124812
#226124612
C4×#145565562
#226122702
C73×#23302
C4×#54173