Finite Group Databasebeta

← #17189260

Subgroups — #17189260

Order 4352.

558 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587590C211yesmaximal, characteristic, complemented
2#14530989C211yesmaximal, characteristic, complemented
3#14530972C211yesmaximal, characteristic, complemented
4#14530945C211yesmaximal, characteristic, complemented
5#14530938C211yesmaximal, characteristic, complemented
6#14530843C211yesmaximal, characteristic, complemented
7#14530836C211yesmaximal, characteristic, complemented
8#14530722C211yesmaximal, characteristic, complemented
9#14530705C211yesmaximal, characteristic
10C4×#10587500C211yesmaximal, characteristic, complemented
11C4×#10587499C211yesmaximal, characteristic, complemented
12C4×#10587498C211yesmaximal, characteristic, complemented
13C4×#10587497C211yesmaximal, characteristic, complemented
14#14530583C211yesmaximal, characteristic, complemented
15C17×#3168C211yesmaximal, characteristic, complemented
16C2×#10587590C2211yescharacteristic
17C4×#63252—21no—
18C2×#10587500C2211yescharacteristic
19C2×#10587498C2211yescharacteristic
20C2×#10587499C2211yescharacteristic
21C2×#10587497C2211yescharacteristic
22C4×Dic68C2211yescharacteristic
23C4×Dic68C2211yescharacteristic
24C4×#63219—21no—
25C4×#63219—21no—
26C4×D136C2211yescharacteristic
27C4×D136C2211yescharacteristic
28#11762847C2211yescharacteristic
29#11762806—21no—
30#11762636C2211yescharacteristic
31#11762636C2211yescharacteristic
32#11762634C2211yescharacteristic
33#11762634C2211yescharacteristic
34C4⋊#63214C2211yescharacteristic
35C4⋊#63214C2211yescharacteristic
36#11762613C2211yescharacteristic
37#11762613C2211yescharacteristic
38#11762612C2211yescharacteristic
39#11762612C2211yescharacteristic
40#11762610C2211yescharacteristic
41#11762610C2211yescharacteristic
42#11762609C2211yescharacteristic
43#11762609C2211yescharacteristic
44C22⋊#63214C2211yescharacteristic
45C22⋊#63214C2211yescharacteristic
46C4×#63214C2211yescharacteristic
47C4×#63214C2211yescharacteristic
48C17×#529—21no—
49C17×#499C2211yescharacteristic
50C17×#495C2211yescharacteristic
51C17×#488—21no—
52C17×#467C2211yescharacteristic
53C17×#466C2211yescharacteristic
54C17×#465—21no—
55C4×Q16×C17C2211yescharacteristic
56C4×SD16×C17—21no—
57C4×D8×C17C2211yescharacteristic
58C2×C4×C8×C17C2211yescharacteristic
59C2×#63252—21no—
60C2×Q16×C17C2311yescharacteristic
61C2×SD16×C17—21no—
62C2×D8×C17C2311yescharacteristic
63C22×C8×C17D411yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C17×#126—41no—
66C17×#125—21no—
67C17×#124—21no—
68C17×#123—41no—
69C17×C22⋊Q8—21no—
70C17×#121—21no—
71C4×Q8×C17—21no—
72C4×D4×C17—21no—
73C4×D4×C17—41no—
74C17×#117—41no—
75C2×C42×C17D411yescharacteristic
76C4×C22⋊D17—41no—
77C4×C22⋊D17—41no—
78#10587612—21no—
79C2×Dic68C2311yescharacteristic
80#10587590—22no—
81#10587590—22no—
82C2×D136C2311yescharacteristic
83C2×#63219—21no—
84C4×D68—21no—
85C4×D68—21no—
86#10587558—41no—
87C42×D17—41no—
88C4×Dic34—21no—
89C4×Dic34—21no—
90C17×#107C2311yescharacteristic
91C17×#107C2311yescharacteristic
92C17×#106—21no—
93C17×#103—21no—
94C17×#103—21no—
95C17×#102—21no—
96C17×#102—21no—
97C4×C8×C17D411yescharacteristic
98C4×C8×C17D411yescharacteristic
99C4×C8×C17C2311yescharacteristic
100C4×C8×C17C2311yescharacteristic
101#10587500—22no—
102#10587499—22no—
103#10587498—22no—
104#10587497—22no—
105#10587492—21no—
106#10587492—21no—
107#10587489C2311yescharacteristic
108#10587489C2311yescharacteristic
109#10587488—21no—
110#10587487—21no—
111#10587487—21no—
112C2×#63214C2311yescharacteristic
113C2×#63214C2311yescharacteristic
114C2×#63214C2311yescharacteristic
115C2×#63214C2311yescharacteristic
116C2×Q8×C17—21no—
117C2×D4×C17—21no—
118C2×D4×C17—41no—
119C22×C4×C17—41no—
120C22×C4×C17D811yescharacteristic
121C22×C4×C17D811yescharacteristic
122C22×C4×C17C2×D411yescharacteristic
123C2×C22⋊D17—41no—
124#63252—24no—
125#63252—42no—
126C2×D68—21no—
127C2×C4×D17—41no—
128C2×C4×D17—41no—
129C2×C4×D17—41no—
130C2×Dic34—21no—
131Q16×C17—21no—
132Q16×C17—21no—
133SD16×C17—42no—
134D8×C17—21no—
135D8×C17—21no—
136C2×C8×C17—21no—
137C2×C8×C17—21no—
138C2×C8×C17—41no—
139C2×C8×C17C2×D411yescharacteristic
140C2×C8×C17C2×D411yescharacteristic
141C2×C8×C17C2×D411yescharacteristic
142C2×C8×C17C2411yescharacteristic
143C4⋊C4×C17—21no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—41no—
146C4⋊C4×C17—41no—
147C4⋊C4×C17—41no—
148C4⋊C4×C17—41no—
149C22⋊C4×C17—41no—
150C22⋊C4×C17—41no—
151C22⋊C4×C17—41no—
152C22⋊C4×C17—41no—
153C22⋊C4×C17—41no—
154C42×C17—41no—
155C42×C17D811yescharacteristic
156C42×C17D811yescharacteristic
157C42×C17C2×D411yescharacteristic
158C42×C17C2×D411yescharacteristic
159C22⋊Dic17—41no—
160#63227—41no—
161#63227—41no—
162C4⋊Dic17—21no—
163C4⋊Dic17—21no—
164#63225—41no—
165#63225—41no—
166C4×Dic17—41no—
167C4×Dic17—41no—
168C4×Dic17—41no—
169Dic68—21no—
170Dic68#4112yes—
171D136—21no—
172D136#4112yes—
173#63219—24no—
174#63214—21no—
175#63214—21no—
176#63214—21no—
177#63214—21no—
178#33662—171nomaximal, Sylow
179C23×C17—41no—
180C23×C17C2×D811yescharacteristic
181C22×D17—41no—
182Q8×C17—41no—
183Q8×C17—41no—
184D4×C17—41no—
185D4×C17—41no—
186D4×C17—82no—
187C2×C4×C17—21no—
188C2×C4×C17—21no—
189C2×C4×C17—22no—
190C2×C4×C17—41no—
191C2×C4×C17—41no—
192C2×C4×C17—41no—
193C2×C4×C17—41no—
194C2×C4×C17—82no—
195C2×C4×C17C2×D811yescharacteristic
196C2×C4×C17C2×D811yescharacteristic
197C2×C4×C17C2×D811yescharacteristic
198C2×C4×C17C2×D811yescharacteristic
199C2×C4×C17C2×D811yescharacteristic
200C2×C4×C17C22×D411yescharacteristic
201C22⋊D17—44no—
202C2×Dic17—41no—
203C2×Dic17—41no—
204C2×Dic17—41no—
205D68—22no—
206D68—41no—
207C4×D17—44no—
208C4×D17—82no—
209C4×D17—82no—
210Dic34—22no—
211Dic34—41no—
212C8×C17—21no—
213C8×C17—21no—
214C8×C17—21no—
215C8×C17C2×#4111yescharacteristic
216C8×C17C2×#4111yescharacteristic
217#3168D1711yescharacteristic, complemented
218C4×#135—171no—
219#2237—171no—
220#2234—171no—
221#2229—171no—
222#2228—171no—
223#2211—171no—
224#2210—171no—
225#2189—171no—
226#2186—171no—
227C4×Q32—171no—
228C4×SD32—171no—
229C4×SD32—171no—
230C4×D16—171no—
231#2162—171no—
232C22×C17—21no—
233C22×C17—21no—
234C22×C17—21no—
235C22×C17—81no—
236C22×C17—81no—
237C22×C17—81no—
238C22×C17C22×D811yescharacteristic
239C2×D17—42no—
240C2×D17—81no—
241C2×D17—81no—
242C4×C17—21no—
243C4×C17—21no—
244C4×C17—21no—
245C4×C17—21no—
246C4×C17—81no—
247C4×C17—81no—
248C4×C17—81no—
249C4×C17#50812yes—
250C4×C17#52511yescharacteristic
251C4×C17#57511yescharacteristic
252Dic17—42no—
253Dic17—81no—
254Dic17—81no—
255C2×#135—171no—
256#529—21no—
257C4×#41—341no—
258C2×Q32—171no—
259C2×SD32—171no—
260C2×SD32—171no—
261C2×D16—171no—
262#499C2×D1711yescharacteristic
263#495C2×D1711yescharacteristic
264#488—21no—
265#467C2×D1711yescharacteristic
266#466C2×D1711yescharacteristic
267#465—21no—
268C4×Q16—171no—
269C4×Q16—171no—
270C4×Q16C2×D1711yescharacteristic
271C4×SD16—21no—
272C4×SD16—341no—
273C4×SD16—341no—
274C4×D8—171no—
275C4×D8—171no—
276C4×D8C2×D1711yescharacteristic
277#427—171no—
278#416—341no—
279C2×C4×C8C2×D1711yescharacteristic
280#367—171no—
281#367—171no—
282#366—171no—
283#366—171no—
284C4⋊C16—171no—
285C4⋊C16—171no—
286#358—171no—
287#358—171no—
288#358—171no—
289#358—171no—
290#357—171no—
291#357—171no—
292#357—171no—
293#357—171no—
294C22⋊C16—171no—
295C22⋊C16—171no—
296C4×C16—171no—
297C4×C16—171no—
298C2×C17—41no—
299C2×C17—161no—
300C2×C17#310011yescharacteristic
301C2×C17C2×#50811yescharacteristic
302C2×C17#341411yescharacteristic
303D17—82no—
304C2×#41—341no—
305#135—342no—
306#135—342no—
307C2×Q16—171no—
308C2×Q16C22×D1711yescharacteristic
309C2×SD16—21no—
310C2×SD16—341no—
311C2×D8—171no—
312C2×D8C22×D1711yescharacteristic
313C22×C8C22⋊D1711yescharacteristic
314C22×C8C22×D1711yescharacteristic
315#126—41no—
316#125—21no—
317#124—21no—
318#123—41no—
319C22⋊Q8—21no—
320#121—21no—
321C4×Q8—21no—
322C4×Q8—341no—
323C4×Q8—341no—
324C4×D4—21no—
325C4×D4—41no—
326C4×D4—341no—
327C4×D4—341no—
328C4×D4—681no—
329C4×D4—681no—
330#117—41no—
331#117—341no—
332#117—681no—
333C2×C42—681no—
334C2×C42C22⋊D1711yescharacteristic
335Q32—342no—
336SD32—342no—
337SD32—342no—
338D16—342no—
339C2×C16—171no—
340C2×C16—171no—
341C2×C16—171no—
342C2×C16—171no—
343#107—171no—
344#107—171no—
345#107C22×D1711yescharacteristic
346#107C22×D1711yescharacteristic
347#106—21no—
348#106—341no—
349#103—21no—
350#103—21no—
351#103—341no—
352#103—341no—
353#102—21no—
354#102—21no—
355#102—341no—
356#102—341no—
357C4×C8C22⋊D1711yescharacteristic
358C4×C8C22⋊D1711yescharacteristic
359C4×C8C22×D1711yescharacteristic
360C4×C8C22×D1711yescharacteristic
361C17#3366211yescharacteristic, complemented, Sylow
362#41—344no—
363#41—682no—
364C2×Q8—21no—
365C2×Q8—341no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—341no—
369C2×D4—681no—
370C22×C4—41no—
371C22×C4—681no—
372C22×C4—681no—
373C22×C4—681no—
374C22×C4#6322811yescharacteristic
375C22×C4#6322811yescharacteristic
376C22×C4C2×C22⋊D1711yescharacteristic
377Q16—21no—
378Q16—21no—
379Q16—172no—
380Q16—341no—
381SD16—42no—
382SD16—344no—
383D8—21no—
384D8—21no—
385D8—172no—
386D8—341no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—41no—
390C2×C8C2×C22⋊D1711yescharacteristic
391C2×C8C2×C22⋊D1711yescharacteristic
392C2×C8C2×C22⋊D1711yescharacteristic
393C2×C8C23×D1711yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—341no—
401C4⋊C4—341no—
402C4⋊C4—681no—
403C4⋊C4—681no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—681no—
410C22⋊C4—681no—
411C22⋊C4—681no—
412C42—41no—
413C42—681no—
414C42—681no—
415C42—681no—
416C42#6322811yescharacteristic
417C42#6322811yescharacteristic
418C42C2×C22⋊D1711yescharacteristic
419C42C2×C22⋊D1711yescharacteristic
420C16—341no—
421C16—341no—
422C16—341no—
423C16—341no—
424C23—41no—
425C23—681no—
426C23C2×#6322811yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—342no—
430Q8—681no—
431D4—41no—
432D4—41no—
433D4—82no—
434D4—342no—
435D4—684no—
436D4—681no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—82no—
445C2×C4—684no—
446C2×C4—681no—
447C2×C4—681no—
448C2×C4—681no—
449C2×C4—1362no—
450C2×C4—1362no—
451C2×C4C2×#6322811yescharacteristic
452C2×C4C2×#6322811yescharacteristic
453C2×C4C2×#6322811yescharacteristic
454C2×C4C2×#6322811yescharacteristic
455C2×C4C2×#6322811yescharacteristic
456C2×C4C22×C22⋊D1711yescharacteristic
457C8—21no—
458C8—21no—
459C8—21no—
460C8#41×D1711yescharacteristic
461C8#41×D1711yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—682no—
469C22—1361no—
470C22—1361no—
471C22C22×#6322811yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—682no—
480C4—1361no—
481C4—1361no—
482C4#1176323912yes—
483C4#1176331611yescharacteristic
484C4#1176360711yescharacteristic
485C2—41no—
486C2—161no—
487C2—1362no—
488C2#1453878011yescharacteristic
489C2C2×#1176323911yescharacteristic
490C2#1454191811yescharacteristic
subgroupG/H
centerC2×C4C2×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#3168C2
socleC22×C17C22×D8
prime radical#14530705C2
hypercenter#3168D17
hypocenterC17#33662
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. C4×#10587590C2semidirect
2. #14530989C2semidirect
3. #14530972C2semidirect
4. #14530945C2semidirect
5. #14530938C2semidirect
6. #14530843C2semidirect
7. #14530836C2semidirect
8. #14530722C2semidirect
10. C4×#10587500C2semidirect
11. C4×#10587499C2semidirect
12. C4×#10587498C2semidirect
13. C4×#10587497C2semidirect
14. #14530583C2semidirect
15. C17×#3168C2semidirect
217. #3168D17semidirect
361. C17#33662semidirect

Maximal subgroups

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

subgroupindex
C4×#105875902
#145309892
#145309722
#145309452
#145309382
#145308432
#145308362
#145307222
#145307052
C4×#105875002
C4×#105874992
C4×#105874982
C4×#105874972
#145305832
C17×#31682
#3366217