Finite Group Databasebeta

← #17005156

Subgroups — #17005156

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#14537933C211yesmaximal, characteristic
2#14537879C211yesmaximal, characteristic, complemented
3#14527663C211yesmaximal, characteristic, complemented
4#14527656C211yesmaximal, characteristic, complemented
5#14527656C211yesmaximal, characteristic, complemented
6#14527642C211yesmaximal, characteristic, complemented
7#14527635C211yesmaximal, characteristic, complemented
8#14527635C211yesmaximal, characteristic, complemented
9#14527628C211yesmaximal, characteristic, complemented
10#14527621C211yesmaximal, characteristic, complemented
11#14527551C211yesmaximal, characteristic, complemented
12#14527244C211yesmaximal, characteristic
13#14527239C211yesmaximal, characteristic, complemented
14#14527071C211yesmaximal, characteristic, complemented
15C17×#3172C211yesmaximal, characteristic
16#11762940—21no—
17#11762940—21no—
18#11762940C2211yescharacteristic
19#11762940C2211yescharacteristic
20#11762940C2211yescharacteristic
21Q8×#3622C2211yescharacteristic
22Q8×#3622C2211yescharacteristic
23Q8×#3622C2211yescharacteristic
24#11762887—21no—
25#11762887C2211yescharacteristic
26#11762881—21no—
27D4×#3622C2211yescharacteristic
28#11762870—21no—
29#11762870—21no—
30#11762870—21no—
31#11762866C2211yescharacteristic
32#11762862C2211yescharacteristic
33#11762836—21no—
34#11762833C2211yescharacteristic
35C2×C4×#3622C2211yescharacteristic
36#11762562C2211yescharacteristic
37#11762562C2211yescharacteristic
38#11762560C2211yescharacteristic
39#11762560C2211yescharacteristic
40Q8⋊#3622C2211yescharacteristic
41Q8⋊#3622C2211yescharacteristic
42Q8⋊#3622C2211yescharacteristic
43Q8⋊#3622C2211yescharacteristic
44Q8⋊#3622C2211yescharacteristic
45Q8⋊#3622C2211yescharacteristic
46D4⋊#3622C2211yescharacteristic
47D4⋊#3622C2211yescharacteristic
48#11762531C2211yescharacteristic
49#11762531C2211yescharacteristic
50#11762531C2211yescharacteristic
51#11762531C2211yescharacteristic
52C17×#533C2211yescharacteristic
53C17×#529C2211yescharacteristic
54C17×#501C411yescharacteristic
55C17×#497C411yescharacteristic
56C17×#491C411yescharacteristic
57C17×#489C411yescharacteristic
58C17×#470C411yescharacteristic
59C17×#470C411yescharacteristic
60C17×#468C411yescharacteristic
61C17×#468C411yescharacteristic
62C4×Q16×C17C2211yescharacteristic
63C4×Q16×C17C2211yescharacteristic
64C4×SD16×C17C2211yescharacteristic
65C4×SD16×C17C2211yescharacteristic
66C4×M16×C17C2211yescharacteristic
67C2×Q16×C17C2×C411yescharacteristic
68C2×Q16×C17C2×C411yescharacteristic
69C2×SD16×C17C2×C411yescharacteristic
70C2×SD16×C17C2×C411yescharacteristic
71C2×M16×C17C2×C411yescharacteristic
72C2×M16×C17C2×C411yescharacteristic
73C17×C4⋊Q8C2×C411yescharacteristic
74C17×C4⋊Q8C2×C411yescharacteristic
75C17×#126—21no—
76C17×#125—21no—
77C17×#125C2×C411yescharacteristic
78C17×#124C2×C411yescharacteristic
79C17×#123—21no—
80C17×C22⋊Q8—21no—
81C17×C22⋊Q8C2×C411yescharacteristic
82C17×C22⋊Q8C2×C411yescharacteristic
83C17×C22⋊Q8C2×C411yescharacteristic
84C17×#121C2×C411yescharacteristic
85C4×Q8×C17—21no—
86C4×Q8×C17C2311yescharacteristic
87C4×Q8×C17C2311yescharacteristic
88C4×Q8×C17C2311yescharacteristic
89C4×D4×C17—21no—
90C4×D4×C17C2311yescharacteristic
91C17×#117—21no—
92C17×#117—21no—
93C2×C42×C17C2311yescharacteristic
94C2×#63223—21no—
95C2×#63223—21no—
96C22×#3622—21no—
97C22×#3622—21no—
98C17×#107C2×C411yescharacteristic
99C17×#107C2×C411yescharacteristic
100C17×#106C2×C411yescharacteristic
101C17×#106C2×C411yescharacteristic
102C17×#103C2×C411yescharacteristic
103C17×#103C2×C411yescharacteristic
104C17×#103C2×C411yescharacteristic
105C17×#103C2×C411yescharacteristic
106C17×#103C2×C411yescharacteristic
107C17×#103C2×C411yescharacteristic
108C17×#102C2×C411yescharacteristic
109C17×#102C2×C411yescharacteristic
110C17×#97C2×C411yescharacteristic
111C17×#97C2×C411yescharacteristic
112C4×C8×C17C2311yescharacteristic
113C4×C8×C17C2311yescharacteristic
114C22⋊#3622—21no—
115C22⋊#3622—21no—
116C22⋊#3622—21no—
117C22⋊#3622D411yescharacteristic
118C22⋊#3622D411yescharacteristic
119C4⋊#3622—21no—
120C4⋊#3622—21no—
121C4⋊#3622—21no—
122C4⋊#3622—21no—
123C4⋊#3622—21no—
124C4⋊#3622—21no—
125C4⋊#3622—21no—
126C4⋊#3622D411yescharacteristic
127C4⋊#3622D411yescharacteristic
128C4⋊#3622C2311yescharacteristic
129C4⋊#3622C2311yescharacteristic
130C4⋊#3622C2311yescharacteristic
131C4⋊#3622C2311yescharacteristic
132#10587474—21no—
133#10587474—21no—
134#10587474—21no—
135#10587474—21no—
136C4×#3622—21no—
137C4×#3622—21no—
138C4×#3622—21no—
139C4×#3622—21no—
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C2×Q8×C17—21no—
145C2×Q8×C17C22×C411yescharacteristic
146C2×Q8×C17C22×C411yescharacteristic
147C2×Q8×C17C22×C411yescharacteristic
148C2×D4×C17—21no—
149C2×D4×C17C22×C411yescharacteristic
150C22×C4×C17—21no—
151C22×C4×C17C22×C411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17#4111yescharacteristic
154Q16×C17—41no—
155Q16×C17—41no—
156SD16×C17—41no—
157SD16×C17—41no—
158M16×C17—41no—
159M16×C17—41no—
160C2×C8×C17C22×C411yescharacteristic
161C2×C8×C17C22×C411yescharacteristic
162C2×C8×C17C22×C411yescharacteristic
163C2×C8×C17C22×C411yescharacteristic
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17C22×C411yescharacteristic
176C4⋊C4×C17C22×C411yescharacteristic
177C4⋊C4×C17C22×C411yescharacteristic
178C4⋊C4×C17C22×C411yescharacteristic
179C22⋊C4×C17—21no—
180C22⋊C4×C17—21no—
181C22⋊C4×C17—21no—
182C22⋊C4×C17—21no—
183C22⋊C4×C17—21no—
184C22⋊C4×C17—21no—
185C22⋊C4×C17—21no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—21no—
189C42×C17C22×C411yescharacteristic
190C42×C17C2×D411yescharacteristic
191C42×C17#4111yescharacteristic
192C42×C17C2411yescharacteristic
193#63223—81no—
194#63223—81no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—42no—
202C2×#3622—81no—
203C2×#3622C2×D411yescharacteristic
204C2×#3622C2×D411yescharacteristic
205C2×#3622C2×D411yescharacteristic
206C2×#3622C2×D411yescharacteristic
207#17988—171nomaximal, Sylow
208C23×C17—21no—
209C23×C17C4×D411yescharacteristic
210Q8×C17—21no—
211Q8×C17—21no—
212Q8×C17—21no—
213Q8×C17—21no—
214Q8×C17—21no—
215Q8×C17—21no—
216Q8×C17—81no—
217D4×C17—21no—
218D4×C17—21no—
219D4×C17—81no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17C4×D411yescharacteristic
235C2×C4×C17C4×D411yescharacteristic
236C2×C4×C17C4×D411yescharacteristic
237C2×C4×C17C23×C411yescharacteristic
238C2×C4×C17C22×D411yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C8×C17—21no—
241C8×C17—21no—
242C8×C17—21no—
243C8×C17—21no—
244#3622—22no—
245#3622—21no—
246#3622—21no—
247#3622—21no—
248#3622—21no—
249#3622—41no—
250#3622—41no—
251#3622—41no—
252#3622—41no—
253#3622—41no—
254#3172D1711yescharacteristic
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#1451—171no—
269C22×C17—21no—
270C22×C17—21no—
271C22×C17—21no—
272C22×C17—41no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17C2×C4×D411yescharacteristic
276C4×C17—21no—
277C4×C17—21no—
278C4×C17—21no—
279C4×C17—21no—
280C4×C17—41no—
281C4×C17—41no—
282C4×C17—41no—
283C4×C17—41no—
284C4×C17—41no—
285C4×C17—41no—
286C4×C17—41no—
287C4×C17#56811yescharacteristic
288C4×C17#56811yescharacteristic
289C4×C17#57711yescharacteristic
290C4×C17#57811yescharacteristic
291#533C2×D1711yescharacteristic
292#529C2×D1711yescharacteristic
293#501Dic1711yescharacteristic
294#497Dic1711yescharacteristic
295#491Dic1711yescharacteristic
296#489Dic1711yescharacteristic
297#470Dic1711yescharacteristic
298#470Dic1711yescharacteristic
299#468Dic1711yescharacteristic
300#468Dic1711yescharacteristic
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×Q16C2×D1711yescharacteristic
310C4×Q16C2×D1711yescharacteristic
311C4×SD16C2×D1711yescharacteristic
312C4×SD16C2×D1711yescharacteristic
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#431—171no—
322#424—341no—
323#423—171no—
324C4×M16C2×D1711yescharacteristic
325C2×C4×C8—171no—
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#322—171no—
339#322—171no—
340#322—171no—
341#322—171no—
342C2×C17—41no—
343C2×C17—81no—
344C2×C17#293211yescharacteristic
345C2×C17#294711yescharacteristic
346C2×C17#295611yescharacteristic
347C2×Q16C2×Dic1711yescharacteristic
348C2×Q16C2×Dic1711yescharacteristic
349C2×SD16C2×Dic1711yescharacteristic
350C2×SD16C2×Dic1711yescharacteristic
351C2×M16—341no—
352C2×M16—341no—
353C2×M16C2×Dic1711yescharacteristic
354C2×M16C2×Dic1711yescharacteristic
355C22×C8—341no—
356C22×C8—341no—
357C4⋊Q8C2×Dic1711yescharacteristic
358C4⋊Q8C2×Dic1711yescharacteristic
359#126—21no—
360#125—21no—
361#125C2×Dic1711yescharacteristic
362#124C2×Dic1711yescharacteristic
363#123—21no—
364C22⋊Q8—21no—
365C22⋊Q8C2×Dic1711yescharacteristic
366C22⋊Q8C2×Dic1711yescharacteristic
367C22⋊Q8C2×Dic1711yescharacteristic
368#121C2×Dic1711yescharacteristic
369C4×Q8—21no—
370C4×Q8C22×D1711yescharacteristic
371C4×Q8C22×D1711yescharacteristic
372C4×Q8C22×D1711yescharacteristic
373C4×D4—21no—
374C4×D4C22×D1711yescharacteristic
375#117—21no—
376#117—21no—
377C2×C42C22×D1711yescharacteristic
378#107C2×Dic1711yescharacteristic
379#107C2×Dic1711yescharacteristic
380#106C2×Dic1711yescharacteristic
381#106C2×Dic1711yescharacteristic
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⋊C8—341no—
395#103C2×Dic1711yescharacteristic
396#103C2×Dic1711yescharacteristic
397#103C2×Dic1711yescharacteristic
398#103C2×Dic1711yescharacteristic
399#103C2×Dic1711yescharacteristic
400#103C2×Dic1711yescharacteristic
401#102C2×Dic1711yescharacteristic
402#102C2×Dic1711yescharacteristic
403C22⋊C8—171no—
404C22⋊C8—171no—
405C22⋊C8—341no—
406C22⋊C8—341no—
407C22⋊C8—341no—
408#97—341no—
409#97—341no—
410#97—341no—
411#97—341no—
412#97C2×Dic1711yescharacteristic
413#97C2×Dic1711yescharacteristic
414C4×C8—171no—
415C4×C8—171no—
416C4×C8—171no—
417C4×C8—171no—
418C4×C8—341no—
419C4×C8—341no—
420C4×C8—341no—
421C4×C8—341no—
422C4×C8C22×D1711yescharacteristic
423C4×C8C22×D1711yescharacteristic
424C17#1798811yescharacteristic, complemented, Sylow
425C2×Q8—21no—
426C2×Q8C22×Dic1711yescharacteristic
427C2×Q8C22×Dic1711yescharacteristic
428C2×Q8C22×Dic1711yescharacteristic
429C2×D4—21no—
430C2×D4C22×Dic1711yescharacteristic
431C22×C4—21no—
432C22×C4D4×D1711yescharacteristic
433C22×C4#6325411yescharacteristic
434C22×C4C22×Dic1711yescharacteristic
435Q16—41no—
436Q16—41no—
437SD16—41no—
438SD16—41no—
439M16—41no—
440M16—41no—
441M16—1361no—
442M16—1361no—
443C2×C8—171no—
444C2×C8—171no—
445C2×C8—171no—
446C2×C8—171no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—682no—
454C2×C8—1361no—
455C2×C8C22×Dic1711yescharacteristic
456C2×C8C22×Dic1711yescharacteristic
457C2×C8C22×Dic1711yescharacteristic
458C2×C8C22×Dic1711yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4C22×Dic1711yescharacteristic
471C4⋊C4C22×Dic1711yescharacteristic
472C4⋊C4C22×Dic1711yescharacteristic
473C4⋊C4C22×Dic1711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42D4×D1711yescharacteristic
485C42#6325411yescharacteristic
486C42C22×Dic1711yescharacteristic
487C42C23×D1711yescharacteristic
488C23—21no—
489C23D4×Dic1711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—81no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
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—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4D4×Dic1711yescharacteristic
515C2×C4D4×Dic1711yescharacteristic
516C2×C4D4×Dic1711yescharacteristic
517C2×C4C2×D4×D1711yescharacteristic
518C2×C4C2×#6325411yescharacteristic
519C2×C4C23×Dic1711yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—342no—
525C8—341no—
526C8—341no—
527C8—341no—
528C8—341no—
529C8—681no—
530C8—681no—
531C8—681no—
532C8—681no—
533C8—681no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22C2×D4×Dic1711yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4#1176357611yescharacteristic
553C4#1176357611yescharacteristic
554C4#1176361811yescharacteristic
555C4#1176362511yescharacteristic
556C2—41no—
557C2—81no—
558C2#1453750211yescharacteristic
559C2#1453762811yescharacteristic
560C2#1453769411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#3172C2
socleC22×C17C2×C4×D4
prime radicalC17×#121C2×C4
cosocleC42×C17C24
hypercenter#3172D17
hypocenterC17#17988
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3172

Complements

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

NH ≅ G/Ntype
2. #14537879C2semidirect
3. #14527663C2semidirect
4. #14527656C2semidirect
5. #14527656C2semidirect
6. #14527642C2semidirect
7. #14527635C2semidirect
8. #14527635C2semidirect
9. #14527628C2semidirect
10. #14527621C2semidirect
11. #14527551C2semidirect
13. #14527239C2semidirect
14. #14527071C2semidirect
424. C17#17988semidirect

Maximal subgroups

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

subgroupindex
#145379332
#145378792
#145276632
#145276562
#145276422
#145276352
#145276282
#145276212
#145275512
#145272442
#145272392
#145270712
C17×#31722
#1798817