Finite Group Databasebeta

← #17005450

Subgroups — #17005450

Order 4352.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538864C211yesmaximal, characteristic
2#14537788C211yesmaximal, characteristic
3#14527691C211yesmaximal, characteristic, complemented
4#14527674C211yesmaximal, characteristic, complemented
5#14527655C211yesmaximal, characteristic, complemented
6#14527639C211yesmaximal, characteristic, complemented
7#14527632C211yesmaximal, characteristic, complemented
8#14527627C211yesmaximal, characteristic, complemented
9#14527594C211yesmaximal, characteristic, complemented
10C8×#63231C211yesmaximal, characteristic, complemented
11#14527575C211yesmaximal, characteristic, complemented
12#14527247C211yesmaximal, characteristic
13#14527179C211yesmaximal, characteristic, complemented
14#14527081C211yesmaximal, characteristic, complemented
15Q8×M16×C17C211yesmaximal, characteristic, complemented
16#11763343C2211yescharacteristic
17#11763139C411yescharacteristic
18#11763122C411yescharacteristic
19#11763116C411yescharacteristic
20#11763116C411yescharacteristic
21#11763031C411yescharacteristic
22#11763031C411yescharacteristic
23#11762965C411yescharacteristic
24#11762948C411yescharacteristic
25#11762939C2211yescharacteristic
26C8×Dic34C2211yescharacteristic
27C4×#63231C2211yescharacteristic
28C4×#63231C2211yescharacteristic
29C4×#63230C2211yescharacteristic
30C4×#63230C2211yescharacteristic
31#11762888C2211yescharacteristic
32#11762885—21no—
33#11762878—21no—
34C8×D68C2211yescharacteristic
35#11762871—21no—
36#11762862C2211yescharacteristic
37#11762862C2211yescharacteristic
38#11762862C2211yescharacteristic
39#11762766—21no—
40#11762561C2211yescharacteristic
41#11762561C2211yescharacteristic
42#11762559C2211yescharacteristic
43#11762559C2211yescharacteristic
44Q8⋊#3622C2211yescharacteristic
45Q8⋊#3622C2211yescharacteristic
46Q8⋊#3622C2211yescharacteristic
47Q8⋊#3622C2211yescharacteristic
48#11762539C2211yescharacteristic
49#11762539C2211yescharacteristic
50#11762536C2211yescharacteristic
51#11762536C2211yescharacteristic
52#11762531C2211yescharacteristic
53#11762531C2211yescharacteristic
54#11762530C2211yescharacteristic
55C8×#3622C2211yescharacteristic
56C2×C4×Q8×C17C2211yescharacteristic
57C17×#446—21no—
58C17×#446—21no—
59C17×#446C2211yescharacteristic
60C17×#446C2211yescharacteristic
61C8×Q8×C17C2211yescharacteristic
62C8×Q8×C17C2211yescharacteristic
63C17×#423—21no—
64C17×#423C2211yescharacteristic
65C4×M16×C17—21no—
66C4×M16×C17C2211yescharacteristic
67C22×Q8×C17C2×C411yescharacteristic
68C2×M16×C17—21no—
69C2×M16×C17D411yescharacteristic
70C2×M16×C17D411yescharacteristic
71C4×Q8×C17—21no—
72C4×Q8×C17—21no—
73C4×Q8×C17C2×C411yescharacteristic
74C4×Q8×C17C2×C411yescharacteristic
75C4×Q8×C17C2311yescharacteristic
76C4×Q8×C17C2311yescharacteristic
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17C2×C411yescharacteristic
79C2×C42×C17—21no—
80C2×C42×C17C2311yescharacteristic
81C2×#63231C2×C411yescharacteristic
82C2×#63231C2×C411yescharacteristic
83C2×#63230C2×C411yescharacteristic
84C2×#63230C2×C411yescharacteristic
85#10587616C2×C411yescharacteristic
86#10587615—21no—
87C4×C22⋊D17—21no—
88#10587610C2×C411yescharacteristic
89C2×#63218—21no—
90C2×C8×D17—21no—
91#10587562—21no—
92#10587561C2×C411yescharacteristic
93C4×D68C2311yescharacteristic
94#10587558—21no—
95#10587556C2×C411yescharacteristic
96C4×Dic34C2311yescharacteristic
97C17×C4⋊C8—21no—
98C17×C4⋊C8—21no—
99C17×C4⋊C8—21no—
100C17×C4⋊C8—21no—
101C17×C4⋊C8—21no—
102C17×C4⋊C8C2311yescharacteristic
103C17×C4⋊C8C2311yescharacteristic
104C17×#97—21no—
105C17×#97—21no—
106C17×#97D411yescharacteristic
107C17×#97D411yescharacteristic
108C4×C8×C17—21no—
109C4×C8×C17—21no—
110C4×C8×C17C2311yescharacteristic
111C4×C8×C17C2311yescharacteristic
112#10587506C2×C411yescharacteristic
113#10587506C2×C411yescharacteristic
114#10587506C2×C411yescharacteristic
115#10587506C2×C411yescharacteristic
116C22⋊#3622—21no—
117C22⋊#3622—21no—
118C22⋊#3622C2×C411yescharacteristic
119C22⋊#3622C2×C411yescharacteristic
120#10587491—21no—
121#10587491—21no—
122#10587486—21no—
123#10587485—21no—
124#10587485—21no—
125C8×Dic17—21no—
126#10587481C2×C411yescharacteristic
127#10587481C2×C411yescharacteristic
128#10587480C2×C411yescharacteristic
129#10587480C2×C411yescharacteristic
130#10587479C2×C411yescharacteristic
131#10587479C2×C411yescharacteristic
132#10587478C2×C411yescharacteristic
133#10587478C2×C411yescharacteristic
134C4⋊#3622—21no—
135C4⋊#3622C2×C411yescharacteristic
136C4⋊#3622C2×C411yescharacteristic
137C4⋊#3622C2311yescharacteristic
138C4⋊#3622C2311yescharacteristic
139C4×#3622—21no—
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C2×Q8×C17—21no—
145C2×Q8×C17—22no—
146C2×Q8×C17—41no—
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—22no—
159M16×C17—41no—
160M16×C17—42no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17C2×D411yescharacteristic
164C2×C8×C17C2×D411yescharacteristic
165C2×C8×C17C2×D411yescharacteristic
166C2×C8×C17C2×D411yescharacteristic
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17C22×C411yescharacteristic
173C4⋊C4×C17C22×C411yescharacteristic
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17C22×C411yescharacteristic
179C42×C17C2×D411yescharacteristic
180C42×C17#4111yescharacteristic
181C42×C17C2411yescharacteristic
182C22⋊Dic17—21no—
183#63231—22no—
184#63231—41no—
185#63230—22no—
186#63230—41no—
187#63227—21no—
188#63227—21no—
189#63227—21no—
190#63227—21no—
191C4⋊Dic17C22×C411yescharacteristic
192C4⋊Dic17C22×C411yescharacteristic
193#63225—21no—
194#63225—21no—
195#63225—21no—
196#63225—21no—
197C4×Dic17—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622C22×C411yescharacteristic
203C2×#3622C22×C411yescharacteristic
204C2×#3622C22×C411yescharacteristic
205C2×#3622C22×C411yescharacteristic
206#63218—42no—
207C8×D17—42no—
208#18012—171nomaximal, Sylow
209C23×C17C4×D411yescharacteristic
210C22×D17—21no—
211Q8×C17—21no—
212Q8×C17—21no—
213Q8×C17—21no—
214Q8×C17—42no—
215Q8×C17#13112yes—
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—21no—
224C2×C4×C17—41no—
225C2×C4×C17—42no—
226C2×C4×C17—81no—
227C2×C4×C17C4×D411yescharacteristic
228C2×C4×C17C4×D411yescharacteristic
229C2×C4×C17C4×D411yescharacteristic
230C2×C4×C17C23×C411yescharacteristic
231C2×C4×C17C22×D411yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C22⋊D17—81no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237D68—21no—
238D68—21no—
239C4×D17—41no—
240C4×D17—41no—
241Dic34—21no—
242Dic34—21no—
243C8×C17—21no—
244C8×C17—21no—
245C8×C17—21no—
246C8×C17—41no—
247C8×C17—41no—
248C8×C17#13712yes—
249#3622—21no—
250#3622—21no—
251#3622—21no—
252#3622—21no—
253#3622—41no—
254#3622—41no—
255#3622—41no—
256#3622—41no—
257#3108—171no—
258#2969—171no—
259Q8×M16D1711yescharacteristic, complemented
260#1593—171no—
261#1590—171no—
262#1587—171no—
263#1585—171no—
264#1584—171no—
265#1583—171no—
266SD16⋊C8—171no—
267C8×Q16—171no—
268#1574—171no—
269#1495—171no—
270#1479—171no—
271#1453—171no—
272C22×C17—21no—
273C22×C17—21no—
274C22×C17—21no—
275C22×C17C2×C4×D411yescharacteristic
276C2×D17—41no—
277C2×D17—41no—
278C2×D17—41no—
279C4×C17—21no—
280C4×C17—22no—
281C4×C17—21no—
282C4×C17—21no—
283C4×C17—21no—
284C4×C17—41no—
285C4×C17—41no—
286C4×C17—41no—
287C4×C17C2×#13111yescharacteristic
288C4×C17#56811yescharacteristic
289C4×C17C2×#13711yescharacteristic
290C4×C17#57511yescharacteristic
291Dic17—41no—
292Dic17—41no—
293Dic17—41no—
294#529—171no—
295C2×C4×Q8C2×D1711yescharacteristic
296#483—171no—
297#480—171no—
298#479—171no—
299#479—171no—
300#464—171no—
301#464—171no—
302#451—171no—
303#448—171no—
304#446—21no—
305#446—21no—
306#446—171no—
307#446C2×D1711yescharacteristic
308#446C2×D1711yescharacteristic
309C8×Q8—171no—
310C8×Q8C2×D1711yescharacteristic
311C8×Q8C2×D1711yescharacteristic
312C4×Q16—171no—
313C4×Q16—171no—
314C4×SD16—171no—
315C4×SD16—171no—
316#436—171no—
317#436—341no—
318#435—341no—
319C8×D4—171no—
320#433—341no—
321#431—171no—
322#431—171no—
323#431—171no—
324#423—21no—
325#423C2×D1711yescharacteristic
326#405—341no—
327C4×M16—21no—
328C4×M16C2×D1711yescharacteristic
329#335—171no—
330#335—171no—
331#334—171no—
332#334—171no—
333Q8⋊C8—171no—
334Q8⋊C8—171no—
335Q8⋊C8—171no—
336Q8⋊C8—171no—
337Q8⋊C8—171no—
338Q8⋊C8—171no—
339D4⋊C8—171no—
340D4⋊C8—171no—
341#322—171no—
342#322—171no—
343#322—171no—
344C82—171no—
345C2×C17—41no—
346C2×C17#293311yescharacteristic
347C2×C17C4×#13711yescharacteristic
348C2×C17#295711yescharacteristic
349D17—81no—
350C22×Q8C4×D1711yescharacteristic
351C2×Q16—171no—
352C2×Q16—171no—
353C2×SD16—171no—
354C2×SD16—171no—
355C2×M16—21no—
356C2×M16—341no—
357C2×M16C22⋊D1711yescharacteristic
358C2×M16C22⋊D1711yescharacteristic
359C22×C8—341no—
360#126—341no—
361#125—171no—
362#124—171no—
363#123—341no—
364C22⋊Q8—171no—
365#121—171no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—171no—
369C4×Q8C4×D1711yescharacteristic
370C4×Q8C4×D1711yescharacteristic
371C4×Q8C22×D1711yescharacteristic
372C4×Q8C22×D1711yescharacteristic
373C4×D4—171no—
374C4×D4—341no—
375#117—341no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4C4×D1711yescharacteristic
378C2×C42—21no—
379C2×C42C22×D1711yescharacteristic
380#107—171no—
381#107—171no—
382#106—171no—
383#106—171no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—171no—
390C4⋊C8—171no—
391C4⋊C8—171no—
392C4⋊C8—171no—
393C4⋊C8—341no—
394C4⋊C8—341no—
395C4⋊C8—341no—
396C4⋊C8C22×D1711yescharacteristic
397C4⋊C8C22×D1711yescharacteristic
398#103—171no—
399#103—171no—
400#103—171no—
401#103—171no—
402#103—171no—
403#103—171no—
404#102—171no—
405#102—171no—
406C22⋊C8—171no—
407C22⋊C8—171no—
408C22⋊C8—341no—
409C22⋊C8—341no—
410C22⋊C8—341no—
411C22⋊C8—341no—
412#97—21no—
413#97—21no—
414#97—341no—
415#97C22⋊D1711yescharacteristic
416#97C22⋊D1711yescharacteristic
417C4×C8—21no—
418C4×C8—21no—
419C4×C8—171no—
420C4×C8—171no—
421C4×C8—171no—
422C4×C8—171no—
423C4×C8—341no—
424C4×C8—341no—
425C4×C8C22×D1711yescharacteristic
426C4×C8C22×D1711yescharacteristic
427C17#1801211yescharacteristic, complemented, Sylow
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8—171no—
432C2×Q8C2×C4×D1711yescharacteristic
433C2×Q8C2×C4×D1711yescharacteristic
434C2×D4—171no—
435C2×D4—341no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—341no—
439C22×C4C2×C4×D1711yescharacteristic
440C22×C4#6325211yescharacteristic
441C22×C4C2×C22⋊D1711yescharacteristic
442Q16—342no—
443Q16—681no—
444SD16—342no—
445SD16—681no—
446M16—22no—
447M16—41no—
448M16—42no—
449M16—682no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—171no—
453C2×C8—171no—
454C2×C8—171no—
455C2×C8—171no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—682no—
461C2×C8C2×C22⋊D1711yescharacteristic
462C2×C8C2×C22⋊D1711yescharacteristic
463C2×C8C2×C22⋊D1711yescharacteristic
464C2×C8C2×C22⋊D1711yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—171no—
471C4⋊C4—171no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4C2×C4×D1711yescharacteristic
477C4⋊C4C2×C4×D1711yescharacteristic
478C22⋊C4—341no—
479C22⋊C4—341no—
480C22⋊C4—341no—
481C22⋊C4—341no—
482C22⋊C4—341no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—341no—
488C42C2×C4×D1711yescharacteristic
489C42#6325211yescharacteristic
490C42C2×C22⋊D1711yescharacteristic
491C42C23×D1711yescharacteristic
492C23—341no—
493C23C4×C22⋊D1711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8—341no—
499Q8—341no—
500Q8#1058759312yes—
501D4—341no—
502D4—341no—
503D4—1361no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—81no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—681no—
519C2×C4—681no—
520C2×C4C4×C22⋊D1711yescharacteristic
521C2×C4C4×C22⋊D1711yescharacteristic
522C2×C4C4×C22⋊D1711yescharacteristic
523C2×C4C22×C4×D1711yescharacteristic
524C2×C4C2×#6325211yescharacteristic
525C2×C4C22×C22⋊D1711yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—341no—
532C8—341no—
533C8—341no—
534C8—341no—
535C8—681no—
536C8—681no—
537C8—681no—
538C8—681no—
539C8#1058762812yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—681no—
544C22—681no—
545C22—681no—
546C22C2×C4×C22⋊D1711yescharacteristic
547C4—21no—
548C4—22no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—681no—
556C4—681no—
557C4—681no—
558C4C2×#1058759311yescharacteristic
559C4#1176357511yescharacteristic
560C4C2×#1058762811yescharacteristic
561C4#1176360711yescharacteristic
562C2—41no—
563C2—1361no—
564C2#1453750911yescharacteristic
565C2C4×#1058762811yescharacteristic
566C2#1453769911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingQ8×M16×C17C2
socleC22×C17C2×C4×D4
prime radical#10587616C2×C4
cosocleC42×C17C24
hypercenterQ8×M16D17
hypocenterC17#18012
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. #14527691C2semidirect
4. #14527674C2semidirect
5. #14527655C2semidirect
6. #14527639C2semidirect
7. #14527632C2semidirect
8. #14527627C2semidirect
9. #14527594C2semidirect
10. C8×#63231C2semidirect
11. #14527575C2semidirect
13. #14527179C2semidirect
14. #14527081C2semidirect
15. Q8×M16×C17C2semidirect
259. Q8×M16D17semidirect
427. C17#18012semidirect

Maximal subgroups

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

subgroupindex
#145388642
#145377882
#145276912
#145276742
#145276552
#145276392
#145276322
#145276272
#145275942
C8×#632312
#145275752
#145272472
#145271792
#145270812
Q8×M16×C172
#1801217