Finite Group Databasebeta

← #17004393

Subgroups — #17004393

Order 4352.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538864C211yesmaximal, characteristic
2#14537929C211yesmaximal, characteristic
3#14537865C211yesmaximal, characteristic, complemented
4#14527659C211yesmaximal, characteristic, complemented
5#14527652C211yesmaximal, characteristic, complemented
6#14527631C211yesmaximal, characteristic, complemented
7#14527624C211yesmaximal, characteristic, complemented
8#14527600C211yesmaximal, characteristic, complemented
9#14527599C211yesmaximal, characteristic, complemented
10#14527588C211yesmaximal, characteristic, complemented
11#14527581C211yesmaximal, characteristic, complemented
12#14527575C211yesmaximal, characteristic, complemented
13#14527193C211yesmaximal, characteristic
14#14527084C211yesmaximal, characteristic, complemented
15C17×#1487C211yesmaximal, characteristic, complemented
16#11763343C2211yescharacteristic
17#11763139C411yescharacteristic
18#11763122C411yescharacteristic
19#11763116C411yescharacteristic
20#11763116C411yescharacteristic
21#11763031C411yescharacteristic
22#11763031C411yescharacteristic
23#11762965C411yescharacteristic
24#11762948C411yescharacteristic
25#11762936C2211yescharacteristic
26Q8×#3622—21no—
27Q8×#3622—21no—
28Q8×#3622C2211yescharacteristic
29Q8×#3622C2211yescharacteristic
30Q8×#3622C2211yescharacteristic
31Q8×#3622C2211yescharacteristic
32#11762933C2211yescharacteristic
33C4×#63231C2211yescharacteristic
34C4×#63231C2211yescharacteristic
35C4×#63230C2211yescharacteristic
36C4×#63230C2211yescharacteristic
37#11762886C2211yescharacteristic
38#11762885—21no—
39#11762878—21no—
40#11762874C2211yescharacteristic
41#11762869—21no—
42#11762862—21no—
43#11762862—21no—
44#11762862C2211yescharacteristic
45#11762862C2211yescharacteristic
46#11762862C2211yescharacteristic
47#11762834—21no—
48#11762561C2211yescharacteristic
49#11762561C2211yescharacteristic
50#11762559C2211yescharacteristic
51#11762559C2211yescharacteristic
52#11762541C2211yescharacteristic
53#11762541C2211yescharacteristic
54#11762538C2211yescharacteristic
55#11762538C2211yescharacteristic
56#11762531C2211yescharacteristic
57#11762531C2211yescharacteristic
58#11762530C2211yescharacteristic
59C8×#3622C2211yescharacteristic
60C2×C4×Q8×C17C2211yescharacteristic
61C17×#431C2211yescharacteristic
62C17×#423C2211yescharacteristic
63C17×Q8⋊C8C2211yescharacteristic
64C17×Q8⋊C8C2211yescharacteristic
65C17×Q8⋊C8C2211yescharacteristic
66C17×Q8⋊C8C2211yescharacteristic
67C22×Q8×C17C2×C411yescharacteristic
68C2×M16×C17—21no—
69C4×Q8×C17—21no—
70C4×Q8×C17—21no—
71C4×Q8×C17C2×C411yescharacteristic
72C4×Q8×C17C2×C411yescharacteristic
73C4×Q8×C17C2311yescharacteristic
74C4×Q8×C17C2311yescharacteristic
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17C2×C411yescharacteristic
77C2×C42×C17—21no—
78C2×C42×C17C2311yescharacteristic
79C2×#63231C2×C411yescharacteristic
80C2×#63231C2×C411yescharacteristic
81C2×#63230C2×C411yescharacteristic
82C2×#63230C2×C411yescharacteristic
83#10587616C2×C411yescharacteristic
84#10587615—21no—
85C4×C22⋊D17—21no—
86#10587610C2×C411yescharacteristic
87C2×#63218—21no—
88C2×C8×D17—21no—
89#10587562—21no—
90#10587561C2×C411yescharacteristic
91C4×D68C2311yescharacteristic
92#10587558—21no—
93#10587556C2×C411yescharacteristic
94C4×Dic34C2311yescharacteristic
95C17×C4⋊C8—21no—
96C17×C4⋊C8—21no—
97C17×C4⋊C8C2311yescharacteristic
98C17×C4⋊C8C2311yescharacteristic
99C17×C22⋊C8—21no—
100C4×C8×C17C2311yescharacteristic
101C4×C8×C17C2311yescharacteristic
102#10587506C2×C411yescharacteristic
103#10587506C2×C411yescharacteristic
104#10587506C2×C411yescharacteristic
105#10587506C2×C411yescharacteristic
106C22⋊#3622—21no—
107C22⋊#3622C2×C411yescharacteristic
108C22⋊#3622C2×C411yescharacteristic
109C22⋊#3622D411yescharacteristic
110C22⋊#3622D411yescharacteristic
111#10587491—21no—
112#10587491—21no—
113#10587486—21no—
114#10587485—21no—
115#10587485—21no—
116C8×Dic17—21no—
117#10587481C2×C411yescharacteristic
118#10587481C2×C411yescharacteristic
119#10587480C2×C411yescharacteristic
120#10587480C2×C411yescharacteristic
121#10587479C2×C411yescharacteristic
122#10587479C2×C411yescharacteristic
123#10587478C2×C411yescharacteristic
124#10587478C2×C411yescharacteristic
125C4⋊#3622—21no—
126C4⋊#3622—21no—
127C4⋊#3622—21no—
128C4⋊#3622—21no—
129C4⋊#3622C2×C411yescharacteristic
130C4⋊#3622C2×C411yescharacteristic
131C4⋊#3622D411yescharacteristic
132C4⋊#3622D411yescharacteristic
133C4⋊#3622C2311yescharacteristic
134C4⋊#3622C2311yescharacteristic
135C4×#3622—21no—
136C4×#3622—21no—
137C4×#3622—21no—
138C4×#3622—21no—
139C4×#3622—21no—
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C2×Q8×C17—22no—
145C2×Q8×C17—22no—
146C2×Q8×C17—21no—
147C2×Q8×C17C22×C411yescharacteristic
148C2×Q8×C17C22×C411yescharacteristic
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17C22×C411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17#4111yescharacteristic
154C2×C22⋊D17—21no—
155C2×D68C22×C411yescharacteristic
156C2×C4×D17—21no—
157C2×Dic34C22×C411yescharacteristic
158M16×C17—81no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17C22×C411yescharacteristic
169C4⋊C4×C17C22×C411yescharacteristic
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17C22×C411yescharacteristic
175C42×C17C2×D411yescharacteristic
176C42×C17#4111yescharacteristic
177C42×C17C2411yescharacteristic
178C22⋊Dic17—21no—
179#63231—22no—
180#63231—22no—
181#63230—22no—
182#63230—22no—
183#63227—21no—
184#63227—21no—
185#63227—21no—
186#63227—21no—
187C4⋊Dic17C22×C411yescharacteristic
188C4⋊Dic17C22×C411yescharacteristic
189#63225—21no—
190#63225—21no—
191#63225—21no—
192#63225—21no—
193C4×Dic17—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622C22×C411yescharacteristic
197C2×#3622C22×C411yescharacteristic
198C2×#3622C22×C411yescharacteristic
199C2×#3622C22×C411yescharacteristic
200C2×#3622C2×D411yescharacteristic
201C2×#3622C2×D411yescharacteristic
202C2×#3622C2×D411yescharacteristic
203C2×#3622C2×D411yescharacteristic
204#63218—81no—
205C8×D17—81no—
206#17919—171nomaximal, Sylow
207C23×C17C4×D411yescharacteristic
208C22×D17—21no—
209Q8×C17—24no—
210Q8×C17—21no—
211Q8×C17—21no—
212Q8×C17#13112yes—
213Q8×C17#13112yes—
214C2×C4×C17—24no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—41no—
224C2×C4×C17—81no—
225C2×C4×C17C4×D411yescharacteristic
226C2×C4×C17C4×D411yescharacteristic
227C2×C4×C17C4×D411yescharacteristic
228C2×C4×C17C23×C411yescharacteristic
229C2×C4×C17C22×D411yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C22⋊D17—81no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235D68—21no—
236D68—21no—
237C4×D17—41no—
238C4×D17—41no—
239Dic34—21no—
240Dic34—21no—
241C8×C17—41no—
242C8×C17—41no—
243C8×C17—41no—
244C8×C17—41no—
245#3622—21no—
246#3622—21no—
247#3622—21no—
248#3622—21no—
249#3622—21no—
250#3622—21no—
251#3622—21no—
252#3622—21no—
253#3622—41no—
254#3622—41no—
255#3108—171no—
256#2993—171no—
257#2982—171no—
258#1588—171no—
259#1587—171no—
260#1584—171no—
261#1583—171no—
262#1579—171no—
263SD16⋊C8—171no—
264C8×Q16—171no—
265C8×SD16—171no—
266#1574—171no—
267#1487D1711yescharacteristic, complemented
268#1482—171no—
269#1454—171no—
270C22×C17—21no—
271C22×C17—21no—
272C22×C17—21no—
273C22×C17C2×C4×D411yescharacteristic
274C2×D17—41no—
275C2×D17—41no—
276C2×D17—41no—
277C4×C17—22no—
278C4×C17—22no—
279C4×C17—21no—
280C4×C17—21no—
281C4×C17—21no—
282C4×C17—21no—
283C4×C17—41no—
284C4×C17—41no—
285C4×C17C2×#13111yescharacteristic
286C4×C17C2×#13111yescharacteristic
287C4×C17#57711yescharacteristic
288C4×C17#57811yescharacteristic
289Dic17—41no—
290Dic17—41no—
291Dic17—41no—
292#529—171no—
293C2×C4×Q8C2×D1711yescharacteristic
294#483—171no—
295#480—171no—
296#479—171no—
297#479—171no—
298#464—171no—
299#464—171no—
300#451—171no—
301#448—171no—
302#446—171no—
303C8×Q8—171no—
304C8×Q8—171no—
305C8×Q8—171no—
306C8×Q8—171no—
307C8×Q8—171no—
308C8×Q8—341no—
309C8×Q8—341no—
310C4×Q16—171no—
311C4×Q16—171no—
312C4×SD16—171no—
313C4×SD16—171no—
314#436—171no—
315#436—341no—
316#435—341no—
317C8×D4—171no—
318#433—341no—
319#431—171no—
320#431—171no—
321#431—171no—
322#431—341no—
323#431—341no—
324#431C2×D1711yescharacteristic
325#424—341no—
326#423C2×D1711yescharacteristic
327#335—171no—
328#335—171no—
329#334—171no—
330#334—171no—
331Q8⋊C8—171no—
332Q8⋊C8—171no—
333Q8⋊C8C2×D1711yescharacteristic
334Q8⋊C8C2×D1711yescharacteristic
335Q8⋊C8C2×D1711yescharacteristic
336Q8⋊C8C2×D1711yescharacteristic
337D4⋊C8—171no—
338D4⋊C8—171no—
339#322—171no—
340#322—171no—
341#322—171no—
342C82—171no—
343C2×C17—41no—
344C2×C17#292911yescharacteristic
345C2×C17#294711yescharacteristic
346C2×C17#295711yescharacteristic
347D17—81no—
348C22×Q8C4×D1711yescharacteristic
349C2×Q16—171no—
350C2×Q16—171no—
351C2×SD16—171no—
352C2×SD16—171no—
353C2×M16—21no—
354C2×M16—341no—
355C22×C8—341no—
356#126—341no—
357#125—171no—
358#124—171no—
359#123—341no—
360C22⋊Q8—171no—
361#121—171no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—171no—
365C4×Q8C4×D1711yescharacteristic
366C4×Q8C4×D1711yescharacteristic
367C4×Q8C22×D1711yescharacteristic
368C4×Q8C22×D1711yescharacteristic
369C4×D4—171no—
370C4×D4—341no—
371#117—341no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4C4×D1711yescharacteristic
374C2×C42—21no—
375C2×C42C22×D1711yescharacteristic
376#107—171no—
377#107—171no—
378#106—171no—
379#106—171no—
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—171no—
383C4⋊C8—171no—
384C4⋊C8—171no—
385C4⋊C8—171no—
386C4⋊C8—171no—
387C4⋊C8—171no—
388C4⋊C8—341no—
389C4⋊C8—341no—
390C4⋊C8—341no—
391C4⋊C8—341no—
392C4⋊C8—341no—
393C4⋊C8—341no—
394C4⋊C8C22×D1711yescharacteristic
395C4⋊C8C22×D1711yescharacteristic
396#103—171no—
397#103—171no—
398#103—171no—
399#103—171no—
400#103—171no—
401#103—171no—
402#102—171no—
403#102—171no—
404C22⋊C8—21no—
405C22⋊C8—171no—
406C22⋊C8—171no—
407C22⋊C8—171no—
408C22⋊C8—171no—
409C22⋊C8—341no—
410C22⋊C8—341no—
411C22⋊C8—341no—
412#97—341no—
413C4×C8—171no—
414C4×C8—171no—
415C4×C8—171no—
416C4×C8—171no—
417C4×C8—341no—
418C4×C8—341no—
419C4×C8—341no—
420C4×C8—341no—
421C4×C8—341no—
422C4×C8—341no—
423C4×C8C22×D1711yescharacteristic
424C4×C8C22×D1711yescharacteristic
425C17#1791911yescharacteristic, complemented, Sylow
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—171no—
430C2×Q8C2×C4×D1711yescharacteristic
431C2×Q8C2×C4×D1711yescharacteristic
432C2×D4—171no—
433C2×D4—341no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—341no—
437C22×C4C2×C4×D1711yescharacteristic
438C22×C4D4×D1711yescharacteristic
439C22×C4#6325411yescharacteristic
440Q16—342no—
441Q16—342no—
442SD16—342no—
443SD16—342no—
444M16—81no—
445M16—1361no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—171no—
451C2×C8—171no—
452C2×C8—171no—
453C2×C8—171no—
454C2×C8—171no—
455C2×C8—171no—
456C2×C8—171no—
457C2×C8—171no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—1361no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—171no—
467C4⋊C4—171no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4C2×C4×D1711yescharacteristic
473C4⋊C4C2×C4×D1711yescharacteristic
474C22⋊C4—341no—
475C22⋊C4—341no—
476C22⋊C4—341no—
477C22⋊C4—341no—
478C22⋊C4—341no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—341no—
484C42C2×C4×D1711yescharacteristic
485C42D4×D1711yescharacteristic
486C42#6325411yescharacteristic
487C42C23×D1711yescharacteristic
488C23—341no—
489C23#1058756711yescharacteristic
490Q8—24no—
491Q8—21no—
492Q8—21no—
493Q8—341no—
494Q8—341no—
495Q8#1058759312yes—
496Q8#1058759312yes—
497D4—341no—
498D4—341no—
499D4—1361no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—81no—
511C2×C4—341no—
512C2×C4—341no—
513C2×C4—341no—
514C2×C4—681no—
515C2×C4—681no—
516C2×C4#1058756711yescharacteristic
517C2×C4#1058756711yescharacteristic
518C2×C4#1058756711yescharacteristic
519C2×C4C22×C4×D1711yescharacteristic
520C2×C4C2×D4×D1711yescharacteristic
521C2×C4C2×#6325411yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—341no—
527C8—341no—
528C8—341no—
529C8—341no—
530C8—341no—
531C8—341no—
532C8—341no—
533C8—341no—
534C8—681no—
535C8—681no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—681no—
540C22—681no—
541C22—681no—
542C22C2×#1058756711yescharacteristic
543C4—22no—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—681no—
552C4—681no—
553C4—681no—
554C4C2×#1058759311yescharacteristic
555C4C2×#1058759311yescharacteristic
556C4#1176361511yescharacteristic
557C4#1176362211yescharacteristic
558C2—41no—
559C2—1361no—
560C2#1453747411yescharacteristic
561C2#1453762111yescharacteristic
562C2#1453769711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1487C2
socleC22×C17C2×C4×D4
prime radical#10587616C2×C4
cosocleC42×C17C24
hypercenter#1487D17
hypocenterC17#17919
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
3. #14537865C2semidirect
4. #14527659C2semidirect
5. #14527652C2semidirect
6. #14527631C2semidirect
7. #14527624C2semidirect
8. #14527600C2semidirect
9. #14527599C2semidirect
10. #14527588C2semidirect
11. #14527581C2semidirect
12. #14527575C2semidirect
14. #14527084C2semidirect
15. C17×#1487C2semidirect
267. #1487D17semidirect
425. C17#17919semidirect

Maximal subgroups

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

subgroupindex
#145388642
#145379292
#145378652
#145276592
#145276522
#145276312
#145276242
#145276002
#145275992
#145275882
#145275812
#145275752
#145271932
#145270842
C17×#14872
#1791917