Finite Group Databasebeta

← #31016305

Subgroups — #31016305

Order 17152.

2094 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D67C211yesmaximal, characteristic
2#22563306C214yesmaximal, complemented
3#22563300C214yesmaximal, complemented
4#22563083C214yesmaximal
5#22563082C214yesmaximal
6#22562949C212yesmaximal, complemented
7#22562946C212yesmaximal, complemented
8#22562824C212yesmaximal
9#125×Dic67C211yesmaximal, characteristic, complemented
10C4×#14524035C211yesmaximal, characteristic, complemented
11#22562764C211yesmaximal, characteristic, complemented
12#22562615C211yesmaximal, characteristic
13#22562563C212yesmaximal
14C4×#14524000C211yesmaximal, characteristic
15C4×#125×C67C211yesmaximal, characteristic, complemented
16C2×#14524035C2211yescharacteristic
17C2×#14524034C2212yes—
18C2×#14524033C2214yes—
19#125×D67C4116yescomplemented
20C2×#14524001C2212yes—
21C2×#14524000C2211yescharacteristic
22C2×#14524000C2211yescharacteristic
23C2×C4⋊C4×D67C2212yes—
24C2×C4⋊C4×D67C2214yes—
25C2×C42×D67C2211yescharacteristic
26#16863295C2212yes—
27#16863294C2212yes—
28#16863294C2214yes—
29#16863293C2214yes—
30#16863293C2214yes—
31#16863292C2212yes—
32#16863292C2214yes—
33#16863291C2214yes—
34#16863291C2214yes—
35#16863282C2214yes—
36#16863281C2214yes—
37#16863280C2214yes—
38#16863280C2214yes—
39#16863279C2214yes—
40#16863278C2214yes—
41#16863278C2214yes—
42#16863265C2212yes—
43#16863262C2211yescharacteristic
44#16863255C2214yes—
45#16863255C2214yes—
46#16863253C2214yes—
47C4⋊C4×Dic67C2212yes—
48C4⋊C4×Dic67C2212yes—
49C4⋊C4×Dic67C2214yes—
50C4×Dic67⋊C4C2214yes—
51C4×Dic67⋊C4C2214yes—
52C4×Dic67⋊C4C2212yes—
53C4×Dic67⋊C4C2212yes—
54C4×C4⋊Dic67C2212yes—
55#16863247C2214yes—
56#16863247C2214yes—
57#16863247C2214yes—
58C4×(C2×D67)⋊C4C2214yes—
59C4×(C2×D67)⋊C4C2212yes—
60C4×(C2×D67)⋊C4C2212yes—
61#16863243C2214yes—
62#16863242C2211yescharacteristic
63#16863242C2211yescharacteristic
64C42×Dic67C2211yescharacteristic
65C2×#125×C67C2211yescharacteristic
66#389×C67C2212yes—
67#387×C67C2214yes—
68#382×C67C2211yescharacteristic
69C4×C4⋊C4×C67C2212yes—
70C4×C4⋊C4×C67C2214yes—
71C43×C67C2211yescharacteristic
72C22×C4×D67C2314yes—
73C22×C4×D67C2312yes—
74C22×C4×D67C2311yescharacteristic
75#14524035C2×C418yes—
76#14524034C2×C4116yes—
77#14524033C2×C4132yes—
78#14524001—28no—
79#14524000—24no—
80#14524000Q818yes—
81C2×C4⋊Dic67C2312yes—
82C2×C4⋊Dic67C2314yes—
83C2×C4⋊Dic67C2312yes—
84C2×C4⋊Dic67C2314yes—
85C2×Dic67⋊C4C2312yes—
86C2×Dic67⋊C4C2314yes—
87C2×Dic67⋊C4C2314yes—
88C2×Dic67⋊C4C2314yes—
89C2×Dic67⋊C4C2314yes—
90C2×Dic67⋊C4C2314yes—
91C2×Dic67⋊C4C2314yes—
92C2×Dic67⋊C4C2314yes—
93C2×Dic67⋊C4C2311yescharacteristic
94C2×Dic67⋊C4C2311yescharacteristic
95C4⋊C4×D67C2×C4132yes—
96C4⋊C4×D67C2×C4116yes—
97C2×(C2×D67)⋊C4C2314yes—
98C2×(C2×D67)⋊C4C2312yes—
99C2×(C2×D67)⋊C4C2311yescharacteristic
100C2×(C2×D67)⋊C4C2311yescharacteristic
101C2×C4×Dic67C2311yescharacteristic
102C2×C4×Dic67C2314yes—
103C2×C4×Dic67C2312yes—
104C2×C4×Dic67C2312yes—
105C2×C4×Dic67C2314yes—
106C2×C4×Dic67C2314yes—
107C2×C4×Dic67C2314yes—
108C2×C4×Dic67C2311yescharacteristic
109C2×C4×Dic67C2312yes—
110C2×C4×Dic67C2311yescharacteristic
111C42×D67C2×C418yes—
112C67⋊#95C2312yes—
113C67⋊#95C2312yes—
114C67⋊#95C2312yes—
115C67⋊#95C2312yes—
116C67⋊#95C2314yes—
117C67⋊#95C2314yes—
118C67⋊#95C2314yes—
119C67⋊#95C2314yes—
120C67⋊#95C2314yes—
121C67⋊#95C2314yes—
122C67⋊#95C2314yes—
123#125×C67C2×C418yes—
124C2×C4⋊C4×C67C2312yes—
125C2×C4⋊C4×C67C2314yes—
126C2×C4⋊C4×C67C2314yes—
127C2×C4⋊C4×C67C2312yes—
128C2×C42×C67C2311yescharacteristic
129C2×C42×C67C2312yes—
130C2×C42×C67C2312yes—
131C2×C42×C67C2314yes—
132C2×C42×C67C2311yescharacteristic
133C2×C42×C67C2311yescharacteristic
134#95×C67C2314yes—
135#95×C67C2314yes—
136#95×C67C2314yes—
137C23×D67C2411yescharacteristic
138C22×Dic67C2412yes—
139C22×Dic67C2414yes—
140C22×Dic67C2414yes—
141C22×Dic67C2412yes—
142C22×Dic67C2411yescharacteristic
143C22×Dic67C2411yescharacteristic
144C22×Dic67C2411yescharacteristic
145C2×C4×D67—44no—
146C2×C4×D67—44no—
147C2×C4×D67—42no—
148C2×C4×D67—42no—
149C2×C4×D67C22×C418yes—
150C2×C4×D67C22×C4116yes—
151C2×C4×D67C22×C414yes—
152C2×C4×D67C2×Q814yes—
153C2×C4×D67#4114yes—
154C4⋊Dic67—24no—
155C4⋊Dic67—44no—
156C4⋊Dic67C22×C4116yes—
157C4⋊Dic67C22×C418yes—
158Dic67⋊C4—24no—
159Dic67⋊C4—44no—
160Dic67⋊C4—44no—
161Dic67⋊C4—44no—
162Dic67⋊C4—44no—
163Dic67⋊C4C22×C4116yes—
164Dic67⋊C4C22×C4116yes—
165Dic67⋊C4C22×C4116yes—
166Dic67⋊C4C2×Q814yes—
167Dic67⋊C4C2×Q814yes—
168(C2×D67)⋊C4—24no—
169(C2×D67)⋊C4—28no—
170(C2×D67)⋊C4C2×Q814yes—
171(C2×D67)⋊C4C2×Q814yes—
172C4×Dic67—22no—
173C4×Dic67—28no—
174C4×Dic67—44no—
175C4×Dic67—44no—
176C4×Dic67—42no—
177C4×Dic67—42no—
178C4×Dic67—44no—
179C4×Dic67C22×C418yes—
180C4×Dic67C22×C414yes—
181C4×Dic67C2×Q814yes—
182C22×C4×C67C2412yes—
183C22×C4×C67C2414yes—
184C22×C4×C67C2412yes—
185C22×C4×C67C2414yes—
186C22×C4×C67C2411yescharacteristic
187C22×C4×C67C2411yescharacteristic
188C22×C4×C67C2411yescharacteristic
189C4⋊C4×C67—24no—
190C4⋊C4×C67—44no—
191C4⋊C4×C67C22×C4116yes—
192C4⋊C4×C67C22×C418yes—
193C42×C67—22no—
194C42×C67—42no—
195C42×C67—44no—
196C42×C67—42no—
197C42×C67C22×C414yes—
198C42×C67C2×Q814yes—
199C23×C67C2511yescharacteristic
200C22×D67—22no—
201C22×D67—22no—
202C22×D67C23×C412yes—
203C22×D67C22×Q812yes—
204C22×D67C2×#4112yes—
205C2×C4×C67—22no—
206C2×C4×C67—22no—
207C2×C4×C67—22no—
208C2×C4×C67—22no—
209C2×C4×C67—24no—
210C2×C4×C67—24no—
211C2×C4×C67—22no—
212C2×C4×C67—24no—
213C2×C4×C67—22no—
214C2×C4×C67—24no—
215C2×C4×C67—22no—
216C2×C4×C67—24no—
217C2×C4×C67C23×C414yes—
218C2×C4×C67C23×C418yes—
219C2×C4×C67C23×C412yes—
220C2×C4×C67C22×Q812yes—
221C2×C4×C67C22×Q812yes—
222C2×C4×C67C22×Q812yes—
223C2×C4×C67C2×#4112yes—
224C2×Dic67—22no—
225C2×Dic67—22no—
226C2×Dic67—24no—
227C2×Dic67—24no—
228C2×Dic67—22no—
229C2×Dic67—22no—
230C2×Dic67—22no—
231C2×Dic67—24no—
232C2×Dic67—24no—
233C2×Dic67—22no—
234C2×Dic67—22no—
235C2×Dic67—24no—
236C2×Dic67C23×C414yes—
237C2×Dic67C23×C418yes—
238C2×Dic67C23×C412yes—
239C2×Dic67C22×Q812yes—
240C2×Dic67C22×Q812yes—
241C2×Dic67C22×Q812yes—
242C2×Dic67C2×#4112yes—
243C4×D67—48no—
244C4×D67—416no—
245C4×D67C4×Q8116yes—
246C22×C67C24×C411yescharacteristic
247C22×C67C23×Q811yescharacteristic
248C22×C67C22×#4111yescharacteristic
249C22×C67#58512yes—
250C22×C67#58512yes—
251C2×D67—24no—
252C2×D67—22no—
253C2×D67—22no—
254C2×D67—24no—
255C2×D67C2×C4×Q814yes—
256C4×C67—24no—
257C4×C67—28no—
258C4×C67—22no—
259C4×C67—22no—
260C4×C67—42no—
261C4×C67—44no—
262C4×C67C2×C4×Q814yes—
263Dic67—28no—
264Dic67—24no—
265Dic67—22no—
266Dic67—22no—
267Dic67—42no—
268Dic67—44no—
269Dic67C2×C4×Q814yes—
270#34217—671nomaximal, Sylow
271C2×C67C22×C4×Q811yescharacteristic
272C2×C67#342812yes—
273C2×C67#350411yescharacteristic
274C2×C67#350411yescharacteristic
275C2×C67#351212yes—
276D67—24no—
277C22×#125—671no—
278#2370—674no—
279#2370—674no—
280#2345—674no—
281#2345—674no—
282#2330—672no—
283#2330—672no—
284#2316—672no—
285C4×#125—671no—
286C4×#125—671no—
287C4×#125—671no—
288C4×#125D6711yescharacteristic, complemented
289#2285—671no—
290#2274—672no—
291C4×#117—671no—
292C67#3421711yescharacteristic, complemented, Sylow
293C2×#125—6716no—
294C2×#125—672no—
295C2×#125—674no—
296C2×#125—671no—
297C2×#125C2×D6711yescharacteristic
298C2×#117—672no—
299C2×#117—671no—
300C2×#117—671no—
301C22×C4⋊C4—672no—
302C22×C4⋊C4—674no—
303C22×C42—671no—
304#389—672no—
305#389—672no—
306#389—674no—
307#389—674no—
308#389—674no—
309#389—672no—
310#389—674no—
311#389—674no—
312#389—674no—
313#389C2×D6712yes—
314#387—674no—
315#387—674no—
316#387—674no—
317#387—674no—
318#387—674no—
319#387—674no—
320#387—674no—
321#387C2×D6714yes—
322#384—672no—
323#382—671no—
324#382C2×D6711yescharacteristic
325#380—674no—
326#380—674no—
327#380—674no—
328C4×C4⋊C4—672no—
329C4×C4⋊C4—672no—
330C4×C4⋊C4—674no—
331C4×C4⋊C4—672no—
332C4×C4⋊C4—674no—
333C4×C4⋊C4—672no—
334C4×C4⋊C4—672no—
335C4×C4⋊C4—674no—
336C4×C4⋊C4—674no—
337C4×C4⋊C4—674no—
338C4×C4⋊C4—674no—
339C4×C4⋊C4C2×D6712yes—
340C4×C4⋊C4C2×D6714yes—
341C4×C22⋊C4—672no—
342C4×C22⋊C4—672no—
343C4×C22⋊C4—674no—
344C4×C22⋊C4—674no—
345#376—671no—
346#376—671no—
347C43—671no—
348C43C2×D6711yescharacteristic
349C23×C4—674no—
350C23×C4—672no—
351C23×C4—671no—
352#125—6732no—
353#125—6716no—
354#125—678no—
355#125C4×D6718yes—
356#117—678no—
357#117—1344no—
358#117—1348no—
359C2×C4⋊C4—674no—
360C2×C4⋊C4—674no—
361C2×C4⋊C4—674no—
362C2×C4⋊C4—674no—
363C2×C4⋊C4—674no—
364C2×C4⋊C4—674no—
365C2×C4⋊C4—674no—
366C2×C4⋊C4—674no—
367C2×C4⋊C4—672no—
368C2×C4⋊C4—674no—
369C2×C4⋊C4—671no—
370C2×C4⋊C4—671no—
371C2×C4⋊C4—6732no—
372C2×C4⋊C4—6716no—
373C2×C4⋊C4—672no—
374C2×C4⋊C4—672no—
375C2×C4⋊C4C22×D6712yes—
376C2×C4⋊C4C22×D6714yes—
377C2×C4⋊C4C22×D6714yes—
378C2×C4⋊C4C22×D6712yes—
379C2×C22⋊C4—671no—
380C2×C22⋊C4—671no—
381C2×C22⋊C4—672no—
382C2×C22⋊C4—674no—
383C2×C42—672no—
384C2×C42—672no—
385C2×C42—674no—
386C2×C42—674no—
387C2×C42—674no—
388C2×C42—671no—
389C2×C42—671no—
390C2×C42—672no—
391C2×C42—678no—
392C2×C42—671no—
393C2×C42—674no—
394C2×C42C22×D6711yescharacteristic
395C2×C42C22×D6712yes—
396C2×C42C22×D6712yes—
397C2×C42C22×D6714yes—
398C2×C42C22×D6711yescharacteristic
399C2×C42C22×D6711yescharacteristic
400#95—672no—
401#95—672no—
402#95—672no—
403#95—672no—
404#95—674no—
405#95—674no—
406#95—674no—
407#95—674no—
408#95—674no—
409#95—674no—
410#95—674no—
411#95C22×D6714yes—
412#95C22×D6714yes—
413#95C22×D6714yes—
414C24—671no—
415C22×C4—2682no—
416C22×C4—2682no—
417C22×C4—2684no—
418C22×C4—2684no—
419C22×C4—678no—
420C22×C4—6716no—
421C22×C4—671no—
422C22×C4—674no—
423C22×C4—674no—
424C22×C4—674no—
425C22×C4—671no—
426C22×C4—671no—
427C22×C4—672no—
428C22×C4—674no—
429C22×C4—674no—
430C22×C4—672no—
431C22×C4C23×D6712yes—
432C22×C4C23×D6714yes—
433C22×C4C23×D6712yes—
434C22×C4C23×D6714yes—
435C22×C4C23×D6711yescharacteristic
436C22×C4C23×D6711yescharacteristic
437C22×C4C23×D6711yescharacteristic
438C4⋊C4—24no—
439C4⋊C4—44no—
440C4⋊C4—2684no—
441C4⋊C4—2684no—
442C4⋊C4—2684no—
443C4⋊C4—2684no—
444C4⋊C4—2684no—
445C4⋊C4—6716no—
446C4⋊C4—6716no—
447C4⋊C4—6716no—
448C4⋊C4—6716no—
449C4⋊C4—678no—
450C4⋊C4—674no—
451C4⋊C4—674no—
452C4⋊C4—1344no—
453C4⋊C4—1344no—
454C4⋊C4C2×C4×D67116yes—
455C4⋊C4C2×C4×D6718yes—
456C22⋊C4—674no—
457C22⋊C4—674no—
458C22⋊C4—1344no—
459C22⋊C4—1348no—
460C42—22no—
461C42—42no—
462C42—44no—
463C42—42no—
464C42—2684no—
465C42—2682no—
466C42—2682no—
467C42—2684no—
468C42—2684no—
469C42—678no—
470C42—674no—
471C42—674no—
472C42—1348no—
473C42—1342no—
474C42C2×C4×D6714yes—
475C42Q8×D6714yes—
476C23—672no—
477C23—672no—
478C23—672no—
479C23—1342no—
480C23—1342no—
481C23C24×D6711yescharacteristic
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—2688no—
495C2×C4—26816no—
496C2×C4—672no—
497C2×C4—672no—
498C2×C4—672no—
499C2×C4—6716no—
500C2×C4—674no—
501C2×C4—678no—
502C2×C4—672no—
503C2×C4—672no—
504C2×C4—1344no—
505C2×C4—1342no—
506C2×C4—1344no—
507C2×C4—1342no—
508C2×C4—1342no—
509C2×C4—1344no—
510C2×C4—1342no—
511C2×C4—1342no—
512C2×C4—1344no—
513C2×C4—1344no—
514C2×C4—1342no—
515C2×C4—1342no—
516C2×C4C22×C4×D6714yes—
517C2×C4C22×C4×D6718yes—
518C2×C4C22×C4×D6712yes—
519C2×C4C2×Q8×D6712yes—
520C2×C4C2×Q8×D6712yes—
521C2×C4C2×Q8×D6712yes—
522C2×C4#41×D6712yes—
523C22—674no—
524C22—1344no—
525C22—1344no—
526C22—1342no—
527C22—1342no—
528C22C23×C4×D6711yescharacteristic
529C22C22×Q8×D6711yescharacteristic
530C22C2×#41×D6711yescharacteristic
531C22#1686423212yes—
532C22#1686423212yes—
533C4—24no—
534C4—28no—
535C4—22no—
536C4—22no—
537C4—42no—
538C4—44no—
539C4—2684no—
540C4—2682no—
541C4—674no—
542C4—1344no—
543C4—1348no—
544C4—1342no—
545C4—1342no—
546C4C4×Q8×D6714yes—
547C2—1344no—
548C2C2×C4×Q8×D6711yescharacteristic
549C2#2257340812yes—
550C2#2257403011yescharacteristic
551C2#2257403311yescharacteristic
552C2#2257412412yes—
subgroupG/H
center, FrattiniC23C24×D67
commutatorC22×C67C24×C4
FittingC4×#125×C67C2
socle, cosocleC23×C67C25
prime radicalC23×D67C24
hypercenterC4×#125D67
hypocenterC67#34217
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
2. #22563306C2semidirect
3. #22563300C2semidirect
6. #22562949C2semidirect
7. #22562946C2semidirect
9. #125×Dic67C2semidirect
10. C4×#14524035C2semidirect
11. #22562764C2semidirect
15. C4×#125×C67C2semidirect
19. #125×D67C4semidirect
288. C4×#125D67semidirect
292. C67#34217semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D672
#225633062
#225633002
#225630832
#225630822
#225629492
#225629462
#225628242
#125×Dic672
C4×#145240352
#225627642
#225626152
#225625632
C4×#145240002
C4×#125×C672
#3421767