Finite Group Databasebeta

← #31720375

Subgroups — #31720375

Order 17152.

1962 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22573506C211yesmaximal, characteristic
2#22569222C218yesmaximal, complemented
3#22569202C218yesmaximal, complemented
4#22569150C211yesmaximal, characteristic, complemented
5#22568928C211yesmaximal, characteristic, complemented
6#22568926C211yesmaximal, characteristic, complemented
7#22568905C212yesmaximal, complemented
8#22568902C212yesmaximal, complemented
9#22568859C212yesmaximal, complemented
10#22568841C211yesmaximal, characteristic, complemented
11#22568636C212yesmaximal, complemented
12C2×#97×D67C211yesmaximal, characteristic, complemented
13C67×#2961C211yesmaximal, characteristic, complemented
14C2×#14524051C2212yes—
15C2×#14524049C2212yes—
16C22×C8×D67C2211yescharacteristic
17C22×C8×D67C2211yescharacteristic
18#16863951C412yes—
19#16863947C412yes—
20#16863946C2211yescharacteristic
21#16863933C2214yes—
22#16863932C414yes—
23#16863931C414yes—
24#16863909C414yes—
25#16863908C2212yes—
26C4×#11761629C2214yes—
27#16863864C2212yes—
28C2×C42×D67C2211yescharacteristic
29#16863527C2218yescomplemented
30#16863469C2218yes—
31#16863469C2218yescomplemented
32#16863466C2218yes—
33#16863466C2218yescomplemented
34C8×D268C2218yes—
35#16863461C2218yes—
36#16863456C2218yescomplemented
37#16863452C2214yes—
38#16863452C2218yescomplemented
39#16863448C2214yes—
40#16863423C2212yes—
41#16863420C2211yescharacteristic
42#16863364C2212yes—
43#16863362C2212yes—
44#16863359C2212yes—
45#16863353C2214yescomplemented
46#16863352C2212yes—
47#16863350C2214yescomplemented
48M16×Dic67C2212yes—
49C4×#11761611C2214yescomplemented
50C2×#14523954C2211yescharacteristic
51C2×#14523953C2211yescharacteristic
52C2×#14523953C2211yescharacteristic
53#97×D67C2214yes—
54#97×D67C2214yes—
55#97×D67C2214yescomplemented
56#533×C67C2211yescharacteristic
57#446×C67C2218yescomplemented
58#424×C67C2212yes—
59#423×C67C2211yescharacteristic
60#405×C67C2212yes—
61C2×#97×C67C2211yescharacteristic
62C2×#11761629C2×C414yes—
63C2×#11761628C2312yes—
64C22×C4×D67C2×C412yes—
65C22×C4×D67C2311yescharacteristic
66#14524051—28no—
67#14524049—28no—
68C2×#11761612C2312yes—
69C2×#11761611C2314yes—
70C2×#11761611C2314yes—
71C22×C67⋊C8C2311yescharacteristic
72C22×C67⋊C8C2311yescharacteristic
73C2×C8×D67—24no—
74C2×C8×D67—24no—
75C2×C8×D67C2314yes—
76C2×C8×D67C2314yes—
77C2×C8×D67C2314yes—
78C2×C8×D67C2314yes—
79#14524043C2×C411yescharacteristic
80#14524040C2×C411yescharacteristic
81#14524038C2×C418yes—
82#14524034C2×C414yes—
83#14524032C2×C414yes—
84#14524031C2×C412yes—
85#14524028C2×C418yes—
86#14524027C2×C414yes—
87#14524023C2×C414yes—
88#14524019C2×C414yes—
89#14524018C2×C414yes—
90#14524018C2×C414yes—
91#14524017C2×C414yes—
92#14524013C2×C414yes—
93Q8×Dic67C2×C414yes—
94#14524008C2×C414yes—
95C4×D268C2318yes—
96C4×D268C2314yes—
97#14524005C2×C414yes—
98#14524005C2×C418yes—
99#14524005C2×C414yes—
100(C22⋊D67)⋊C4C2×C418yes—
101#14524001C2×C414yes—
102#14524001C2×C418yes—
103#14524000C2314yes—
104C2×C4×Dic67C2×C412yes—
105C2×C4×Dic67C2311yescharacteristic
106C42×D67C2×C412yes—
107C42×D67C2×C412yes—
108C42×D67C2312yes—
109C42×D67C2312yes—
110C42×D67C2314yes—
111C4⋊(C67⋊C8)C2314yes—
112C4⋊(C67⋊C8)C2318yes—
113(C2×D67)⋊C8C2318yes—
114(C2×D67)⋊C8C2318yes—
115(C2×D67)⋊C8C2318yes—
116#14523954C2314yes—
117#14523954C2314yes—
118#14523953C2312yes—
119#14523953C2312yes—
120#14523953C2312yes—
121#14523953C2312yes—
122#14523953C2314yes—
123C4×C67⋊C8C2314yes—
124C8×Dic67C2314yes—
125C2×M16×C67C2312yes—
126C22×C8×C67C2311yescharacteristic
127C22×C8×C67C2311yescharacteristic
128C4⋊Q8×C67C2×C412yes—
129#125×C67C2×C412yes—
130C22⋊Q8×C67C2×C414yes—
131C4×Q8×C67C2314yes—
132#117×C67C2312yes—
133C2×C42×C67C2311yescharacteristic
134C4⋊C8×C67C2314yes—
135C4⋊C8×C67C2318yes—
136#97×C67C2314yes—
137#97×C67C2314yes—
138C4×C8×C67C2314yes—
139C23×D67C22×C411yescharacteristic
140#11761629—48no—
141#11761628—24no—
142#11761628—24no—
143C2×Dic134C22×C412yes—
144C2×D268C22×C412yes—
145C2×D268C22×C418yes—
146C2×D268C22×C414yes—
147C2×C22⋊D67C22×C414yes—
148C22×Dic67C22×C411yescharacteristic
149C22×Dic67C22×C412yes—
150C2×C4×D67—22no—
151C2×C4×D67—22no—
152C2×C4×D67—44no—
153C2×C4×D67C22×C412yes—
154C2×C4×D67C22×C412yes—
155C2×C4×D67C22×C412yes—
156C2×C4×D67C22×C412yes—
157C2×C4×D67C22×C412yes—
158C2×C4×D67C22×C418yes—
159C2×C4×D67C2412yes—
160C2×C4×D67C2414yes—
161#11761612—24no—
162#11761611—28no—
163#11761611—28no—
164C2×C67⋊C8—22no—
165C2×C67⋊C8#4114yescomplemented
166C2×C67⋊C8C2414yes—
167C2×C67⋊C8C2412yes—
168C2×C67⋊C8C2412yes—
169C8×D67—28no—
170C8×D67—28no—
171C8×D67—28no—
172C8×D67—28no—
173C4⋊Dic67C22×C414yes—
174C4⋊Dic67C22×C418yes—
175Dic67⋊C4C22×C418yes—
176(C2×D67)⋊C4C22×C418yes—
177(C2×D67)⋊C4C22×C418yes—
178(C2×D67)⋊C4C22×C418yes—
179(C2×D67)⋊C4C22×C418yes—
180C4×Dic67C22×C412yes—
181C4×Dic67C22×C412yes—
182C4×Dic67C22×C412yes—
183C4×Dic67C22×C412yes—
184C4×Dic67C22×C412yes—
185C4×Dic67C22×C418yes—
186C4×Dic67C2411yescharacteristic
187C4×Dic67C2411yescharacteristic
188C2×Q8×C67C22×C414yes—
189C22×C4×C67C22×C412yes—
190C22×C4×C67C2411yescharacteristic
191M16×C67—24no—
192C2×C8×C67—22no—
193C2×C8×C67#4114yescomplemented
194C2×C8×C67C2412yes—
195C2×C8×C67C2414yes—
196C2×C8×C67C2412yes—
197C4⋊C4×C67C22×C414yes—
198C4⋊C4×C67C22×C418yes—
199C4⋊C4×C67C22×C414yes—
200C22⋊C4×C67C22×C414yes—
201C42×C67C22×C411yescharacteristic
202C42×C67C22×C411yescharacteristic
203C42×C67C2412yes—
204C42×C67C2414yes—
205C23×C67C23×C411yescharacteristic
206C22×D67—22no—
207C22×D67—24no—
208C22×D67C23×C414yes—
209C22×D67C23×C412yes—
210Q8×C67—44no—
211C2×C4×C67—42no—
212C2×C4×C67C23×C412yes—
213C2×C4×C67C23×C412yes—
214C2×C4×C67C23×C418yes—
215C2×C4×C67C23×C411yescharacteristic
216C2×C4×C67C2×#4112yes—
217C2×C4×C67C2×#4112yes—
218C2×C4×C67C2511yescharacteristic
219C22⋊D67—28no—
220C2×Dic67—22no—
221C2×Dic67—42no—
222C2×Dic67C23×C412yes—
223C2×Dic67C23×C412yes—
224C2×Dic67C23×C412yes—
225C2×Dic67C23×C414yes—
226D268—216no—
227D268—24no—
228D268—44no—
229C4×D67—24no—
230C4×D67—24no—
231C4×D67—24no—
232C4×D67—22no—
233C4×D67—22no—
234C4×D67—24no—
235C4×D67—24no—
236C4×D67—42no—
237C4×D67—42no—
238C4×D67—48no—
239Dic134—24no—
240C8×C67—22no—
241C8×C67—24no—
242C8×C67C2×#4114yes—
243C67⋊C8—22no—
244C67⋊C8—24no—
245C67⋊C8C2×#4114yes—
246C22×C67—22no—
247C22×C67C4×#4112yes—
248C22×C67C24×C411yescharacteristic
249C2×D67—28no—
250C2×D67—22no—
251C2×D67—22no—
252C2×D67—24no—
253C2×D67—28no—
254C2×D67#56814yes—
255C4×C67—22no—
256C4×C67—22no—
257C4×C67—28no—
258C4×C67C4×#4112yes—
259C4×C67C22×#4111yescharacteristic
260C4×C67C22×#4111yescharacteristic
261Dic67—24no—
262Dic67—22no—
263Dic67—22no—
264Dic67—22no—
265#60207—671nomaximal, Sylow
266C2×C67—22no—
267C2×C67C2×C4×#4111yescharacteristic
268C2×C67C2×#56812yes—
269D67—24no—
270D67—44no—
271#3444—671no—
272#2970—678no—
273#2968—678no—
274#2961—671no—
275#2961D6711yescharacteristic, complemented
276#2932—671no—
277#2932—671no—
278#2929—672no—
279#2929—672no—
280#2923—672no—
281#2920—671no—
282#2875—672no—
283C22×#97—671no—
284C67#6020711yescharacteristic, complemented, Sylow
285C2×#131—672no—
286C2×#131—672no—
287C23×C8—671no—
288C23×C8—671no—
289#533—672no—
290#533C2×D6711yescharacteristic
291#532—672no—
292#532—671no—
293#529—674no—
294#529—674no—
295#529—674no—
296#525—674no—
297#525—672no—
298C4×#41—674no—
299C4×#41—672no—
300C22×C42—671no—
301#446—678no—
302#446C2×D6718yescomplemented
303#435—678no—
304#435—678no—
305#435—678no—
306#435—678no—
307C8×D4—678no—
308C8×D4—678no—
309#433—678no—
310#432—674no—
311#432—678no—
312#431—674no—
313#424—672no—
314#424C2×D6712yes—
315#423—671no—
316#423C2×D6711yescharacteristic
317#408—672no—
318#408—672no—
319#407—672no—
320#405—672no—
321#405—674no—
322#405—674no—
323#405C2×D6712yes—
324C4×M16—672no—
325C4×M16—674no—
326C2×#97—671no—
327C2×#97—671no—
328C2×#97—671no—
329C2×#97—674no—
330C2×#97—674no—
331C2×#97—674no—
332C2×#97C2×D6711yescharacteristic
333C2×#41—674no—
334C2×#41—672no—
335C23×C4—672no—
336C23×C4—671no—
337#131—1348no—
338#131—1348no—
339C2×M16—672no—
340C2×M16—674no—
341C2×M16—674no—
342C2×M16C22×D6712yes—
343C22×C8—671no—
344C22×C8—671no—
345C22×C8—674no—
346C22×C8—674no—
347C22×C8—674no—
348C22×C8—674no—
349C22×C8—1344no—
350C22×C8—1344no—
351C22×C8C22×D6711yescharacteristic
352C22×C8C22×D6711yescharacteristic
353C4⋊Q8—671no—
354C4⋊Q8C4×D6712yes—
355#127—671no—
356#126—678no—
357#125—674no—
358#125C4×D6712yes—
359#124—674no—
360#124—672no—
361#123—678no—
362#123—674no—
363C22⋊Q8—674no—
364C22⋊Q8—674no—
365C22⋊Q8C4×D6714yes—
366#121—674no—
367#121—674no—
368#121—674no—
369#120—674no—
370C4×Q8—674no—
371C4×Q8—674no—
372C4×Q8C22×D6714yes—
373C4×D4—678no—
374C4×D4—674no—
375C4×D4—674no—
376C4×D4—678no—
377C4×D4—678no—
378C4×D4—674no—
379#117—674no—
380#117—678no—
381#117—674no—
382#117C22×D6712yes—
383C2×C42—672no—
384C2×C42—671no—
385C2×C42—672no—
386C2×C42—672no—
387C2×C42—674no—
388C2×C42—672no—
389C2×C42—672no—
390C2×C42C22×D6711yescharacteristic
391C4⋊C8—674no—
392C4⋊C8—678no—
393C4⋊C8C22×D6714yes—
394C4⋊C8C22×D6718yes—
395C22⋊C8—678no—
396C22⋊C8—678no—
397C22⋊C8—678no—
398#97—672no—
399#97—672no—
400#97—672no—
401#97—672no—
402#97—674no—
403#97—674no—
404#97—674no—
405#97C22×D6714yes—
406#97C22×D6714yes—
407C4×C8—674no—
408C4×C8—674no—
409C4×C8C22×D6714yes—
410C24—671no—
411#41—2688no—
412#41—1344no—
413#41—1344no—
414C2×Q8—672no—
415C2×Q8C2×C4×D6714yes—
416C2×D4—674no—
417C2×D4—674no—
418C2×D4—678no—
419C2×D4—672no—
420C22×C4—2684no—
421C22×C4—672no—
422C22×C4—672no—
423C22×C4—672no—
424C22×C4—678no—
425C22×C4—671no—
426C22×C4—672no—
427C22×C4—672no—
428C22×C4—672no—
429C22×C4—672no—
430C22×C4—674no—
431C22×C4—1342no—
432C22×C4—1342no—
433C22×C4C2×C4×D6712yes—
434C22×C4C23×D6711yescharacteristic
435M16—24no—
436M16—1348no—
437M16—1344no—
438M16—1348no—
439C2×C8—22no—
440C2×C8—674no—
441C2×C8—672no—
442C2×C8—674no—
443C2×C8—672no—
444C2×C8—1348no—
445C2×C8—1348no—
446C2×C8—1342no—
447C2×C8—1348no—
448C2×C8—1348no—
449C2×C8#1176162914yescomplemented
450C2×C8C23×D6714yes—
451C2×C8C23×D6712yes—
452C2×C8C23×D6712yes—
453C4⋊C4—674no—
454C4⋊C4—678no—
455C4⋊C4—678no—
456C4⋊C4C2×C4×D6714yes—
457C4⋊C4C2×C4×D6718yes—
458C4⋊C4C2×C4×D6714yes—
459C22⋊C4—678no—
460C22⋊C4—678no—
461C22⋊C4—678no—
462C22⋊C4—678no—
463C22⋊C4C2×C4×D6714yes—
464C42—672no—
465C42—672no—
466C42—672no—
467C42—672no—
468C42—672no—
469C42—678no—
470C42—671no—
471C42—671no—
472C42C2×C4×D6711yescharacteristic
473C42C2×C4×D6711yescharacteristic
474C42C23×D6714yes—
475C42C23×D6712yes—
476C23—672no—
477C23—674no—
478C23—1344no—
479C23—1342no—
480C23C22×C4×D6711yescharacteristic
481Q8—44no—
482Q8—1344no—
483D4—2684no—
484D4—13416no—
485D4—1348no—
486D4—1344no—
487C2×C4—42no—
488C2×C4—2682no—
489C2×C4—2682no—
490C2×C4—2688no—
491C2×C4—2682no—
492C2×C4—672no—
493C2×C4—672no—
494C2×C4—672no—
495C2×C4—674no—
496C2×C4—1344no—
497C2×C4—1344no—
498C2×C4—1342no—
499C2×C4—1344no—
500C2×C4—1342no—
501C2×C4—1344no—
502C2×C4—1342no—
503C2×C4—1344no—
504C2×C4C22×C4×D6712yes—
505C2×C4C22×C4×D6712yes—
506C2×C4C22×C4×D6718yes—
507C2×C4C22×C4×D6711yescharacteristic
508C2×C4#41×D6712yes—
509C2×C4C2×#1176162912yes—
510C2×C4C24×D6711yescharacteristic
511C8—22no—
512C8—24no—
513C8—674no—
514C8—1342no—
515C8—1344no—
516C8C2×#1176162914yes—
517C22—22no—
518C22—674no—
519C22—1348no—
520C22—1348no—
521C22—1344no—
522C22—1342no—
523C22—1342no—
524C22C4×#1176162912yes—
525C22C23×C4×D6711yescharacteristic
526C4—22no—
527C4—22no—
528C4—28no—
529C4—1342no—
530C4—1342no—
531C4—1342no—
532C4—1344no—
533C4C4×#1176162912yes—
534C4C2×#41×D6711yescharacteristic
535C4C22×#1176162911yescharacteristic
536C2—22no—
537C2—2684no—
538C2—1344no—
539C2C2×C4×#1176162911yescharacteristic
540C2#568×D6712yes—
subgroupG/H
center, FrattiniC2×C4C24×D67
commutator, socleC22×C67C24×C4
FittingC67×#2961C2
prime radical#22573506C2
cosocleC2×C4×C67C25
hypercenter#2961D67
hypocenterC67#60207
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #22569222C2semidirect
3. #22569202C2semidirect
4. #22569150C2semidirect
5. #22568928C2semidirect
6. #22568926C2semidirect
7. #22568905C2semidirect
8. #22568902C2semidirect
9. #22568859C2semidirect
10. #22568841C2semidirect
11. #22568636C2semidirect
12. C2×#97×D67C2semidirect
13. C67×#2961C2semidirect
29. #16863527C22semidirect
31. #16863469C22semidirect
33. #16863466C22semidirect
36. #16863456C22semidirect
38. #16863452C22semidirect
45. #16863353C22semidirect
47. #16863350C22semidirect
49. C4×#11761611C22semidirect
55. #97×D67C22semidirect
57. #446×C67C22semidirect
165. C2×C67⋊C8#41semidirect
193. C2×C8×C67#41semidirect
275. #2961D67semidirect
284. C67#60207semidirect
302. #446C2×D67semidirect
449. C2×C8#11761629semidirect

Maximal subgroups

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

subgroupindex
#225735062
#225692222
#225692022
#225691502
#225689282
#225689262
#225689052
#225689022
#225688592
#225688412
#225686362
C2×#97×D672
C67×#29612
#6020767