Finite Group Databasebeta

← #17005158

Subgroups — #17005158

Order 4352.

566 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539390C211yesmaximal, characteristic
2#14537935C211yesmaximal, characteristic
3#14537878C211yesmaximal, characteristic, complemented
4#14527664C211yesmaximal, characteristic, complemented
5#14527657C211yesmaximal, characteristic, complemented
6#14527657C211yesmaximal, characteristic, complemented
7#14527643C211yesmaximal, characteristic, complemented
8#14527636C211yesmaximal, characteristic, complemented
9#14527636C211yesmaximal, characteristic, complemented
10#14527629C211yesmaximal, characteristic, complemented
11#14527622C211yesmaximal, characteristic, complemented
12#14527551C211yesmaximal, characteristic, complemented
13#14527245C211yesmaximal, characteristic
14#14527240C211yesmaximal, characteristic, complemented
15C17×#1451C211yesmaximal, characteristic, complemented
16#11763366C2211yescharacteristic
17#11763343C2211yescharacteristic
18#11763218C411yescharacteristic
19#11763203C411yescharacteristic
20#11763178C411yescharacteristic
21#11763169C411yescharacteristic
22#11763067C411yescharacteristic
23#11763067C411yescharacteristic
24#11763055C411yescharacteristic
25#11763055C411yescharacteristic
26#11762939C2211yescharacteristic
27#11762939C2211yescharacteristic
28#11762939C2211yescharacteristic
29#11762938—21no—
30#11762938—21no—
31C8×Dic34C2211yescharacteristic
32C8×Dic34C2211yescharacteristic
33C8×Dic34C2211yescharacteristic
34C4×Dic68C2211yescharacteristic
35C4×Dic68C2211yescharacteristic
36C4×#63219C2211yescharacteristic
37C4×#63219C2211yescharacteristic
38#11762888C2211yescharacteristic
39#11762884—21no—
40#11762883—21no—
41C8×D68C2211yescharacteristic
42#11762869—21no—
43#11762867—21no—
44#11762867—21no—
45#11762866C2211yescharacteristic
46#11762834—21no—
47#11762833C2211yescharacteristic
48#11762562C2211yescharacteristic
49#11762562C2211yescharacteristic
50#11762560C2211yescharacteristic
51#11762560C2211yescharacteristic
52#11762541C2211yescharacteristic
53#11762541C2211yescharacteristic
54#11762541C2211yescharacteristic
55#11762541C2211yescharacteristic
56#11762541C2211yescharacteristic
57#11762541C2211yescharacteristic
58#11762538C2211yescharacteristic
59#11762538C2211yescharacteristic
60C17×#431C2211yescharacteristic
61C4×M16×C17C2211yescharacteristic
62C2×C4×C8×C17C2211yescharacteristic
63C17×#322C2211yescharacteristic
64C17×#322C2211yescharacteristic
65C17×#322C2211yescharacteristic
66C17×#322C2211yescharacteristic
67C2×M16×C17C2×C411yescharacteristic
68C2×M16×C17C2×C411yescharacteristic
69C22×C8×C17—21no—
70C2×C42×C17C2311yescharacteristic
71#10587616C2×C411yescharacteristic
72#10587615—21no—
73C4×C22⋊D17—21no—
74#10587612—21no—
75#10587610—21no—
76#10587610C2×C411yescharacteristic
77#10587610C2×C411yescharacteristic
78#10587610C2×C411yescharacteristic
79C2×#63223—21no—
80C2×Dic68C2×C411yescharacteristic
81C2×Dic68C2×C411yescharacteristic
82C2×#63219C2×C411yescharacteristic
83C2×#63219C2×C411yescharacteristic
84C2×#63218—21no—
85C2×C8×D17—21no—
86#10587562—21no—
87#10587561C2×C411yescharacteristic
88C4×D68C2311yescharacteristic
89#10587558—21no—
90#10587556—21no—
91#10587556C2×C411yescharacteristic
92#10587555C2×C411yescharacteristic
93#10587555C2×C411yescharacteristic
94C4×Dic34—21no—
95C4×Dic34C2311yescharacteristic
96C4×Dic34C2311yescharacteristic
97C4×Dic34C2311yescharacteristic
98C17×C4⋊C8D411yescharacteristic
99C17×C4⋊C8D411yescharacteristic
100C17×C22⋊C8D411yescharacteristic
101C17×C22⋊C8D411yescharacteristic
102C17×#97C2×C411yescharacteristic
103C17×#97C2×C411yescharacteristic
104C4×C8×C17—21no—
105C4×C8×C17C2311yescharacteristic
106C4×C8×C17C2311yescharacteristic
107C4×C8×C17C2311yescharacteristic
108C4×C8×C17C2311yescharacteristic
109C4×C8×C17C2311yescharacteristic
110C4×C8×C17C2311yescharacteristic
111C22⋊#3622—21no—
112#10587492C2×C411yescharacteristic
113#10587492C2×C411yescharacteristic
114#10587491—21no—
115#10587491—21no—
116#10587489C2×C411yescharacteristic
117#10587489C2×C411yescharacteristic
118#10587488C2×C411yescharacteristic
119#10587488C2×C411yescharacteristic
120#10587487C2×C411yescharacteristic
121#10587487C2×C411yescharacteristic
122#10587487C2×C411yescharacteristic
123#10587487C2×C411yescharacteristic
124#10587487C2×C411yescharacteristic
125#10587487C2×C411yescharacteristic
126#10587486—21no—
127#10587486—21no—
128#10587486—21no—
129#10587485—21no—
130#10587485—21no—
131#10587485—21no—
132#10587485—21no—
133#10587485—21no—
134#10587485—21no—
135C8×Dic17—21no—
136C8×Dic17—21no—
137C8×Dic17—21no—
138C4⋊#3622—21no—
139C4⋊#3622C2311yescharacteristic
140C4⋊#3622C2311yescharacteristic
141C4⋊#3622C2311yescharacteristic
142C4⋊#3622C2311yescharacteristic
143#10587474—21no—
144C22×C4×C17C22×C411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17#4111yescharacteristic
147C2×C22⋊D17—21no—
148C2×D68C22×C411yescharacteristic
149C2×C4×D17—21no—
150C2×Dic34—21no—
151C2×Dic34C22×C411yescharacteristic
152C2×Dic34C22×C411yescharacteristic
153C2×Dic34C22×C411yescharacteristic
154M16×C17—41no—
155M16×C17—41no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—42no—
159C2×C8×C17C22×C411yescharacteristic
160C2×C8×C17C22×C411yescharacteristic
161C2×C8×C17C22×C411yescharacteristic
162C2×C8×C17C22×C411yescharacteristic
163C2×C8×C17C2×D411yescharacteristic
164C2×C8×C17C2×D411yescharacteristic
165C2×C8×C17C2×D411yescharacteristic
166C2×C8×C17C2×D411yescharacteristic
167C42×C17C22×C411yescharacteristic
168C42×C17C2×D411yescharacteristic
169C42×C17#4111yescharacteristic
170C42×C17C2411yescharacteristic
171C22⋊Dic17—21no—
172C22⋊Dic17—21no—
173C22⋊Dic17—21no—
174#63227—21no—
175#63227—21no—
176#63227—21no—
177#63227—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17C22×C411yescharacteristic
182C4⋊Dic17C22×C411yescharacteristic
183C4⋊Dic17C22×C411yescharacteristic
184C4⋊Dic17C22×C411yescharacteristic
185#63225—21no—
186#63225—21no—
187#63225—21no—
188#63225—21no—
189#63225—21no—
190#63225—21no—
191#63225—21no—
192#63225—21no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196#63223—81no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201Dic68—41no—
202Dic68—41no—
203#63219—41no—
204#63219—41no—
205#63218—81no—
206C8×D17—81no—
207#17988—171nomaximal, Sylow
208C23×C17C4×D411yescharacteristic
209C22×D17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—41no—
215C2×C4×C17C4×D411yescharacteristic
216C2×C4×C17C4×D411yescharacteristic
217C2×C4×C17C4×D411yescharacteristic
218C2×C4×C17C23×C411yescharacteristic
219C2×C4×C17C22×D411yescharacteristic
220C2×C4×C17C2×#4111yescharacteristic
221C22⋊D17—81no—
222C2×Dic17—21no—
223C2×Dic17—21no—
224C2×Dic17—21no—
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229D68—21no—
230D68—21no—
231C4×D17—41no—
232C4×D17—41no—
233Dic34—21no—
234Dic34—21no—
235Dic34—21no—
236Dic34—21no—
237Dic34—21no—
238Dic34—21no—
239Dic34—81no—
240C8×C17—22no—
241C8×C17—21no—
242C8×C17—21no—
243C8×C17—21no—
244C8×C17—21no—
245C8×C17—21no—
246C8×C17—21no—
247C8×C17—21no—
248C8×C17—21no—
249C8×C17—41no—
250#3622—41no—
251#3622—41no—
252#3622—41no—
253#3622—41no—
254#3172—171no—
255#2993—171no—
256#2985—171no—
257#1588—171no—
258#1587—171no—
259#1587—171no—
260#1585—171no—
261#1584—171no—
262#1584—171no—
263#1583—171no—
264#1582—171no—
265#1569—171no—
266#1494—171no—
267#1493—171no—
268#1451D1711yescharacteristic, complemented
269C22×C17—21no—
270C22×C17—21no—
271C22×C17—21no—
272C22×C17C2×C4×D411yescharacteristic
273C2×D17—41no—
274C2×D17—41no—
275C2×D17—41no—
276C4×C17—21no—
277C4×C17—21no—
278C4×C17—21no—
279C4×C17—21no—
280C4×C17#56811yescharacteristic
281C4×C17#56811yescharacteristic
282C4×C17#57711yescharacteristic
283C4×C17#57811yescharacteristic
284Dic17—41no—
285Dic17—41no—
286Dic17—41no—
287Dic17—41no—
288Dic17—41no—
289Dic17—41no—
290Dic17—41no—
291#533—171no—
292#529—171no—
293#501—171no—
294#497—171no—
295#491—171no—
296#489—171no—
297#470—171no—
298#470—171no—
299#468—171no—
300#468—171no—
301#446—171no—
302#446—171no—
303#446—171no—
304#446—341no—
305#446—341no—
306C8×Q8—171no—
307C8×Q8—171no—
308C8×Q8—171no—
309C4×Q16—171no—
310C4×Q16—171no—
311C4×SD16—171no—
312C4×SD16—171no—
313#436—171no—
314#436—341no—
315#435—341no—
316C8×D4—171no—
317#433—341no—
318#433—341no—
319#433—341no—
320#432—171no—
321#431C2×D1711yescharacteristic
322#424—341no—
323#423—171no—
324C4×M16C2×D1711yescharacteristic
325C2×C4×C8C2×D1711yescharacteristic
326#335—171no—
327#335—171no—
328#334—171no—
329#334—171no—
330Q8⋊C8—171no—
331Q8⋊C8—171no—
332Q8⋊C8—171no—
333Q8⋊C8—171no—
334Q8⋊C8—171no—
335Q8⋊C8—171no—
336D4⋊C8—171no—
337D4⋊C8—171no—
338#322C2×D1711yescharacteristic
339#322C2×D1711yescharacteristic
340#322C2×D1711yescharacteristic
341#322C2×D1711yescharacteristic
342C2×C17—41no—
343C2×C17#293211yescharacteristic
344C2×C17#294711yescharacteristic
345C2×C17#295611yescharacteristic
346D17—81no—
347C2×Q16—171no—
348C2×Q16—171no—
349C2×SD16—171no—
350C2×SD16—171no—
351C2×M16—341no—
352C2×M16—341no—
353C2×M16C4×D1711yescharacteristic
354C2×M16C4×D1711yescharacteristic
355C22×C8—21no—
356C22×C8—341no—
357C4⋊Q8—171no—
358C4⋊Q8—171no—
359#126—341no—
360#125—171no—
361#125—341no—
362#124—171no—
363#123—341no—
364C22⋊Q8—171no—
365C22⋊Q8—171no—
366C22⋊Q8—171no—
367C22⋊Q8—341no—
368#121—171no—
369C4×Q8—171no—
370C4×Q8—171no—
371C4×Q8—171no—
372C4×Q8—341no—
373C4×D4—171no—
374C4×D4—341no—
375#117—341no—
376#117—341no—
377C2×C42C22×D1711yescharacteristic
378#107—171no—
379#107—171no—
380#106—171no—
381#106—171no—
382C4⋊C8—171no—
383C4⋊C8—171no—
384C4⋊C8—171no—
385C4⋊C8—171no—
386C4⋊C8—341no—
387C4⋊C8—341no—
388C4⋊C8—341no—
389C4⋊C8—341no—
390C4⋊C8—341no—
391C4⋊C8—341no—
392C4⋊C8—341no—
393C4⋊C8D6811yescharacteristic
394C4⋊C8D6811yescharacteristic
395#103—171no—
396#103—171no—
397#103—171no—
398#103—171no—
399#103—171no—
400#103—171no—
401#102—171no—
402#102—171no—
403C22⋊C8—341no—
404C22⋊C8—341no—
405C22⋊C8—341no—
406C22⋊C8D6811yescharacteristic
407C22⋊C8D6811yescharacteristic
408#97—341no—
409#97—341no—
410#97—341no—
411#97—341no—
412#97C4×D1711yescharacteristic
413#97C4×D1711yescharacteristic
414C4×C8—21no—
415C4×C8—341no—
416C4×C8—341no—
417C4×C8—341no—
418C4×C8C22×D1711yescharacteristic
419C4×C8C22×D1711yescharacteristic
420C4×C8C22×D1711yescharacteristic
421C4×C8C22×D1711yescharacteristic
422C4×C8C22×D1711yescharacteristic
423C4×C8C22×D1711yescharacteristic
424C17#1798811yescharacteristic, complemented, Sylow
425C2×Q8—171no—
426C2×Q8—171no—
427C2×Q8—171no—
428C2×Q8—341no—
429C2×D4—171no—
430C2×D4—341no—
431C22×C4—341no—
432C22×C4C2×C4×D1711yescharacteristic
433C22×C4C2×D6811yescharacteristic
434C22×C4#6325211yescharacteristic
435Q16—681no—
436Q16—681no—
437SD16—681no—
438SD16—681no—
439M16—41no—
440M16—41no—
441M16—1361no—
442M16—1361no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—42no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—1361no—
451C2×C8C2×C4×D1711yescharacteristic
452C2×C8C2×C4×D1711yescharacteristic
453C2×C8C2×C4×D1711yescharacteristic
454C2×C8C2×C4×D1711yescharacteristic
455C2×C8C2×D6811yescharacteristic
456C2×C8C2×D6811yescharacteristic
457C2×C8C2×D6811yescharacteristic
458C2×C8C2×D6811yescharacteristic
459C4⋊C4—171no—
460C4⋊C4—171no—
461C4⋊C4—171no—
462C4⋊C4—171no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C22⋊C4—341no—
475C22⋊C4—341no—
476C22⋊C4—341no—
477C22⋊C4—341no—
478C22⋊C4—341no—
479C22⋊C4—341no—
480C22⋊C4—341no—
481C42—341no—
482C42—341no—
483C42—341no—
484C42C2×C4×D1711yescharacteristic
485C42C2×D6811yescharacteristic
486C42#6325211yescharacteristic
487C42C23×D1711yescharacteristic
488C23—341no—
489C23C4×D6811yescharacteristic
490Q8—341no—
491Q8—341no—
492Q8—341no—
493Q8—341no—
494Q8—341no—
495Q8—341no—
496Q8—1361no—
497D4—341no—
498D4—341no—
499D4—1361no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—341no—
506C2×C4—341no—
507C2×C4—341no—
508C2×C4—341no—
509C2×C4—341no—
510C2×C4—341no—
511C2×C4—341no—
512C2×C4—681no—
513C2×C4—681no—
514C2×C4C4×D6811yescharacteristic
515C2×C4C4×D6811yescharacteristic
516C2×C4C4×D6811yescharacteristic
517C2×C4C22×C4×D1711yescharacteristic
518C2×C4C22×D6811yescharacteristic
519C2×C4C2×#6325211yescharacteristic
520C8—22no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—681no—
531C8—681no—
532C8—681no—
533C8—681no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—681no—
538C22—681no—
539C22—681no—
540C22C2×C4×D6811yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—681no—
546C4—681no—
547C4—681no—
548C4—681no—
549C4—681no—
550C4—681no—
551C4—681no—
552C4#1176357511yescharacteristic
553C4#1176357511yescharacteristic
554C4#1176362111yescharacteristic
555C4#1176362611yescharacteristic
556C2—41no—
557C2—1361no—
558C2#1453750311yescharacteristic
559C2#1453762911yescharacteristic
560C2#1453769511yescharacteristic
subgroupG/H
centerC2×C4C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1451C2
socleC22×C17C2×C4×D4
prime radical#10587616C2×C4
cosocleC42×C17C24
hypercenter#1451D17
hypocenterC17#17988
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1451

Complements

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

NH ≅ G/Ntype
3. #14537878C2semidirect
4. #14527664C2semidirect
5. #14527657C2semidirect
6. #14527657C2semidirect
7. #14527643C2semidirect
8. #14527636C2semidirect
9. #14527636C2semidirect
10. #14527629C2semidirect
11. #14527622C2semidirect
12. #14527551C2semidirect
14. #14527240C2semidirect
15. C17×#1451C2semidirect
268. #1451D17semidirect
424. C17#17988semidirect

Maximal subgroups

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

subgroupindex
#145393902
#145379352
#145378782
#145276642
#145276572
#145276432
#145276362
#145276292
#145276222
#145275512
#145272452
#145272402
C17×#14512
#1798817