Finite Group Databasebeta

← #17019727

Subgroups — #17019727

Order 4352.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539727C211yesmaximal, characteristic, complemented
2#14538572C211yesmaximal, characteristic, complemented
3#14538264C211yesmaximal, characteristic
4#14528200C212yesmaximal, complemented
5#14528193C212yesmaximal, complemented
6#14528132C212yesmaximal, complemented
7#14528125C212yesmaximal, complemented
8#14527555C211yesmaximal, characteristic, complemented
9#14527472C211yesmaximal, characteristic, complemented
10C2×#11762552C211yesmaximal, characteristic, complemented
11C17×#2988C211yesmaximal, characteristic, complemented
12C2×#10587635—21no—
13#11763366C2211yescharacteristic
14C2×#10587560C2211yescharacteristic
15#11763182C2212yes—
16#11763156C2212yes—
17#11763116—21no—
18#11763109—21no—
19#11763104C2211yescharacteristic, complemented
20#11763099C2211yescharacteristic, complemented
21#11763094C2211yescharacteristic, complemented
22#11763089C2211yescharacteristic, complemented
23#11763010—21no—
24#11763010—21no—
25#11763009—21no—
26#11763009C2212yes—
27#11763004—21no—
28#11763004C2212yes—
29#11762962—22no—
30#11762845—21no—
31#11762833C2211yescharacteristic
32C2×C4⋊#3622C2211yescharacteristic
33#11762813—21no—
34#11762805—21no—
35#11762560C2212yescomplemented
36#11762560C2212yescomplemented
37#11762557C2212yescomplemented
38#11762556C2212yescomplemented
39#11762552C2212yes—
40#11762552C2212yes—
41C4×#41×C17C2211yescharacteristic
42C8×Q8×C17C2212yescomplemented
43C8×D4×C17—22no—
44C8×D4×C17C2212yescomplemented
45C17×#431—21no—
46C17×#431C2211yescharacteristic
47C2×C17×C4⋊C8—21no—
48C2×C17×C4⋊C8C2211yescharacteristic
49C2×#41×C17D411yescharacteristic
50C22×D68—21no—
51C22×D68—21no—
52C22×D68—21no—
53C22×C8×C17—21no—
54C22×C8×C17—22no—
55C4×Q8×C17D411yescharacteristic
56C4×Q8×C17C2311yescharacteristic
57C4×D4×C17—21no—
58C4×D4×C17—21no—
59C4×D4×C17D411yescharacteristic
60C4×D4×C17C2311yescharacteristic
61C17×#117—21no—
62C17×#117D411yescharacteristic
63C2×C42×C17—21no—
64C2×C42×C17C2311yescharacteristic
65#10587635—44no—
66C2×#63230—22no—
67C2×#63228—22no—
68#10587612—21no—
69#10587610—21no—
70#10587610—21no—
71C2×#63223—21no—
72C22×#3622—21no—
73C2×#63219—22no—
74C2×#63219—22no—
75#10587560—21no—
76#10587560—21no—
77#10587560—21no—
78#10587560C2312yes—
79#10587556—21no—
80#10587555C2311yescharacteristic
81#10587555C2311yescharacteristic
82C4×Dic34—21no—
83C4×Dic34—21no—
84C17×C4⋊C8—22no—
85C17×C4⋊C8—21no—
86C17×C4⋊C8D411yescharacteristic
87C17×C4⋊C8D411yescharacteristic
88C17×C4⋊C8C2312yes—
89C17×C22⋊C8—22no—
90C17×C22⋊C8D412yes—
91C4×C8×C17—22no—
92C4×C8×C17C2312yes—
93#10587506—21no—
94#10587506—21no—
95#10587503—21no—
96#10587503—21no—
97#10587492—22no—
98#10587492—22no—
99#10587480—22no—
100#10587480—22no—
101#10587479—21no—
102#10587479—21no—
103#10587478—21no—
104#10587478—21no—
105C4⋊#3622—21no—
106C4⋊#3622—21no—
107C4⋊#3622C2312yes—
108C4⋊#3622C2311yescharacteristic
109C4⋊#3622C2311yescharacteristic
110C23×D17—41no—
111#41×C17—22no—
112#41×C17SD1614yescomplemented
113C2×Q8×C17C2×D411yescharacteristic
114C2×D4×C17—21no—
115C2×D4×C17C2×D411yescharacteristic
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17C2×D411yescharacteristic
120C22×C4×C17C2×D411yescharacteristic
121C22×C4×C17#4111yescharacteristic
122C2×D68—22no—
123C2×D68—22no—
124C2×D68—22no—
125C2×D68—42no—
126C2×D68—42no—
127C2×D68—41no—
128C2×D68—41no—
129C2×D68—41no—
130C2×D68—81no—
131C2×D68—81no—
132C2×D68—82no—
133C2×Dic34—21no—
134C2×Dic34—21no—
135C2×C8×C17—22no—
136C2×C8×C17—82no—
137C2×C8×C17—81no—
138C2×C8×C17C2×D412yes—
139C2×C8×C17C2×D412yes—
140C4⋊C4×C17—21no—
141C4⋊C4×C17—21no—
142C4⋊C4×C17C2×D411yescharacteristic
143C4⋊C4×C17C2×D411yescharacteristic
144C22⋊C4×C17—21no—
145C22⋊C4×C17—21no—
146C22⋊C4×C17—21no—
147C42×C17—21no—
148C42×C17—21no—
149C42×C17—21no—
150C42×C17C2×D411yescharacteristic
151C42×C17C2×D411yescharacteristic
152C42×C17#4111yescharacteristic
153C42×C17C2411yescharacteristic
154C22⋊Dic17—41no—
155#63230—44no—
156#63228—44no—
157C4⋊Dic17—21no—
158C4⋊Dic17—21no—
159C4⋊Dic17—21no—
160C4⋊Dic17—21no—
161#63225—41no—
162#63225—41no—
163C4×Dic17—41no—
164#63223—42no—
165C2×#3622—21no—
166C2×#3622—21no—
167C2×#3622—22no—
168C2×#3622—42no—
169#63219—82no—
170#63219—82no—
171#19409—171nomaximal, Sylow
172C23×C17—21no—
173C23×C17#12111yescharacteristic
174C22×D17—42no—
175C22×D17—82no—
176C22×D17—82no—
177C22×D17—82no—
178Q8×C17—21no—
179Q8×C17C2×SD1612yes—
180D4×C17—24no—
181D4×C17—21no—
182D4×C17C2×SD1612yes—
183C2×C4×C17—21no—
184C2×C4×C17—21no—
185C2×C4×C17—21no—
186C2×C4×C17—24no—
187C2×C4×C17—21no—
188C2×C4×C17—21no—
189C2×C4×C17—41no—
190C2×C4×C17—41no—
191C2×C4×C17—41no—
192C2×C4×C17—81no—
193C2×C4×C17#12111yescharacteristic
194C2×C4×C17#12111yescharacteristic
195C2×C4×C17#12111yescharacteristic
196C2×C4×C17C2×SD1612yes—
197C2×C4×C17#13512yes—
198C2×C4×C17C22×D411yescharacteristic
199C2×C4×C17C22×D411yescharacteristic
200C2×C4×C17C2×#4111yescharacteristic
201C2×Dic17—41no—
202C2×Dic17—41no—
203D68—42no—
204D68—42no—
205D68—42no—
206D68—42no—
207D68—82no—
208D68—82no—
209D68—82no—
210D68—82no—
211Dic34—41no—
212Dic34—41no—
213Dic34—81no—
214C8×C17—22no—
215C8×C17—22no—
216C8×C17—42no—
217#3622—42no—
218#3622—41no—
219#3622—41no—
220#3208—171no—
221#3076—171no—
222#3038—171no—
223#2988D1711yescharacteristic, complemented
224#1678—172no—
225#1677—172no—
226#1666—172no—
227#1665—172no—
228#1570—171no—
229#1550—171no—
230C2×#331—171no—
231C22×C17—21no—
232C22×C17—21no—
233C22×C17—22no—
234C22×C17—41no—
235C22×C17—41no—
236C22×C17#46512yes—
237C22×C17C2×#12111yescharacteristic
238C2×D17—82no—
239C2×D17—82no—
240C2×D17—82no—
241C2×D17—164no—
242C4×C17—21no—
243C4×C17—22no—
244C4×C17—21no—
245C4×C17—21no—
246C4×C17—41no—
247C4×C17—41no—
248C4×C17#46512yes—
249C4×C17C22×SD1611yescharacteristic
250C4×C17C2×#13511yescharacteristic
251C4×C17#57611yescharacteristic
252C4×C17#57711yescharacteristic
253Dic17—81no—
254Dic17—81no—
255C2×AGL(1,ℤ/8)—341no—
256#533—171no—
257C2×#127—171no—
258C4×#41C2×D1711yescharacteristic
259#492—172no—
260#486—172no—
261#479—341no—
262#478—341no—
263#477—171no—
264#476—171no—
265#475—171no—
266#474—171no—
267#460—172no—
268#460—341no—
269#460—341no—
270#460—341no—
271#459—172no—
272#459—341no—
273#450—342no—
274C8×Q8C2×D1712yescomplemented
275C8×D4—22no—
276C8×D4C2×D1712yescomplemented
277#431—21no—
278#431C2×D1711yescharacteristic
279#427—341no—
280#423—171no—
281C2×C4⋊C8—21no—
282C2×C4⋊C8—171no—
283C2×C4⋊C8C2×D1711yescharacteristic
284#418—341no—
285#416—341no—
286#334—172no—
287#334—172no—
288#333—172no—
289#332—172no—
290#331—172no—
291#331—172no—
292C2×C17—22no—
293C2×C17—42no—
294C2×C17#304511yescharacteristic
295C2×C17C2×#46511yescharacteristic
296C2×C17#306811yescharacteristic
297D17—162no—
298C2×#41D6811yescharacteristic
299C22×D4—341no—
300C22×D4—341no—
301C22×D4—341no—
302AGL(1,ℤ/8)—684no—
303C2×SD16—342no—
304C2×SD16—342no—
305C2×SD16—342no—
306C2×D8—342no—
307C2×M16—341no—
308C22×C8—21no—
309C22×C8—22no—
310C22×C8—341no—
311C4⋊Q8—171no—
312C4⋊Q8—171no—
313#127—172no—
314#127—341no—
315#127—341no—
316#127—341no—
317#125—341no—
318C22⋊Q8—341no—
319C22⋊Q8—341no—
320C4×Q8—341no—
321C4×Q8—341no—
322C4×Q8D6811yescharacteristic
323C4×Q8C22×D1711yescharacteristic
324C4×D4—21no—
325C4×D4—21no—
326C4×D4D6811yescharacteristic
327C4×D4C22×D1711yescharacteristic
328#117—21no—
329#117—341no—
330#117D6811yescharacteristic
331C2×C42—21no—
332C2×C42C22×D1711yescharacteristic
333#107—341no—
334#107—341no—
335#106—341no—
336#106—341no—
337C4⋊C8—22no—
338C4⋊C8—21no—
339C4⋊C8—171no—
340C4⋊C8—171no—
341C4⋊C8—172no—
342C4⋊C8—341no—
343C4⋊C8—341no—
344C4⋊C8C22⋊D1711yescharacteristic
345C4⋊C8C22⋊D1711yescharacteristic
346C4⋊C8C22×D1712yes—
347#103—341no—
348#103—341no—
349#102—341no—
350#102—341no—
351#102—342no—
352#102—342no—
353#102—342no—
354#102—342no—
355C22⋊C8—22no—
356C22⋊C8C22⋊D1712yes—
357C4×C8—22no—
358C4×C8C22×D1712yes—
359C17#1940911yescharacteristic, complemented, Sylow
360C24—681no—
361#41—22no—
362#41#6321914yescomplemented
363C2×Q8—341no—
364C2×Q8—341no—
365C2×Q8C2×D6811yescharacteristic
366C2×D4—21no—
367C2×D4—342no—
368C2×D4—342no—
369C2×D4—342no—
370C2×D4—682no—
371C2×D4—682no—
372C2×D4—681no—
373C2×D4—681no—
374C2×D4—681no—
375C2×D4—1361no—
376C2×D4—1361no—
377C2×D4—1362no—
378C2×D4C2×D6811yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4C2×D6811yescharacteristic
383C22×C4#6325211yescharacteristic
384C22×C4C2×C22⋊D1711yescharacteristic
385SD16—684no—
386SD16—1362no—
387SD16—1362no—
388D8—684no—
389M16—682no—
390C2×C8—22no—
391C2×C8—82no—
392C2×C8—81no—
393C2×C8—342no—
394C2×C8—341no—
395C2×C8—341no—
396C2×C8—682no—
397C2×C8C2×C22⋊D1712yes—
398C2×C8C2×C22⋊D1712yes—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—341no—
402C4⋊C4—341no—
403C4⋊C4—341no—
404C4⋊C4—341no—
405C4⋊C4—681no—
406C4⋊C4—681no—
407C4⋊C4C2×D6811yescharacteristic
408C4⋊C4C2×D6811yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—681no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—681no—
417C42C2×D6811yescharacteristic
418C42#6325211yescharacteristic
419C42C2×C22⋊D1711yescharacteristic
420C42C23×D1711yescharacteristic
421C23—21no—
422C23—682no—
423C23—1362no—
424C23—1362no—
425C23—1362no—
426C23#1058761611yescharacteristic
427Q8—21no—
428Q8—681no—
429Q8—681no—
430Q8—1361no—
431Q8C2×#6321912yes—
432D4—24no—
433D4—21no—
434D4—682no—
435D4—682no—
436D4—682no—
437D4—682no—
438D4—1362no—
439D4—1362no—
440D4—1362no—
441D4—1362no—
442D4C2×#6321912yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—681no—
454C2×C4—681no—
455C2×C4C2×#6321912yes—
456C2×C4#1058759012yes—
457C2×C4#1058761611yescharacteristic
458C2×C4#1058761611yescharacteristic
459C2×C4#1058761611yescharacteristic
460C2×C4C22×D6811yescharacteristic
461C2×C4C2×#6325211yescharacteristic
462C2×C4C22×C22⋊D1711yescharacteristic
463C8—22no—
464C8—22no—
465C8—42no—
466C8—682no—
467C8—681no—
468C8—681no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—2724no—
475C22—1362no—
476C22—1362no—
477C22—1362no—
478C22#1176303812yes—
479C22C2×#1058761611yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—1361no—
487C4—1361no—
488C4#1176303812yes—
489C4C22×#6321911yescharacteristic
490C4C2×#1058759011yescharacteristic
491C4#1176361211yescharacteristic
492C4#1176361711yescharacteristic
493C2—22no—
494C2—42no—
495C2—2722no—
496C2#1453832311yescharacteristic
497C2C2×#1176303811yescharacteristic
498C2#1453851211yescharacteristic
subgroupG/H
centerC22C2×#10587616
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#2988C2
socleC22×C17C2×#121
prime radical#14538264C2
hypercenter#2988D17
hypocenterC17#19409
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #14539727C2semidirect
2. #14538572C2semidirect
4. #14528200C2semidirect
5. #14528193C2semidirect
6. #14528132C2semidirect
7. #14528125C2semidirect
8. #14527555C2semidirect
9. #14527472C2semidirect
10. C2×#11762552C2semidirect
11. C17×#2988C2semidirect
19. #11763104C22semidirect
20. #11763099C22semidirect
21. #11763094C22semidirect
22. #11763089C22semidirect
35. #11762560C22semidirect
36. #11762560C22semidirect
37. #11762557C22semidirect
38. #11762556C22semidirect
42. C8×Q8×C17C22semidirect
44. C8×D4×C17C22semidirect
112. #41×C17SD16semidirect
223. #2988D17semidirect
274. C8×Q8C2×D17semidirect
276. C8×D4C2×D17semidirect
359. C17#19409semidirect
362. #41#63219semidirect

Maximal subgroups

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

subgroupindex
#145397272
#145385722
#145382642
#145282002
#145281932
#145281322
#145281252
#145275552
#145274722
C2×#117625522
C17×#29882
#1940917