Finite Group Databasebeta

← #17097258

Subgroups — #17097258

Order 4352.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763117C211yesmaximal, characteristic, complemented
2C2×#11762851C211yesmaximal, characteristic
3#14532508C211yesmaximal, characteristic
4#14531809C211yesmaximal, characteristic
5#14530331C211yesmaximal, characteristic, complemented
6#14530277C211yesmaximal, characteristic, complemented
7#14530178C211yesmaximal, characteristic, complemented
8#14530123C211yesmaximal, characteristic, complemented
9#14529554C211yesmaximal, characteristic, complemented
10#14529363C211yesmaximal, characteristic, complemented
11#14529217C211yesmaximal, characteristic
12#14529089C211yesmaximal, characteristic
13#14529055C211yesmaximal, characteristic, complemented
14#14528985C211yesmaximal, characteristic
15C17×#1851C211yesmaximal, characteristic
16C2×#10587556C2211yescharacteristic
17C2×#10587610C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20C22×C4⋊Dic17C2211yescharacteristic
21#11763117—24no—
22#11762851—24no—
23C2×#10587489C2211yescharacteristic
24C2×#10587489C2211yescharacteristic
25C2×#10587489C2211yescharacteristic
26C2×#10587488C2211yescharacteristic
27C2×#10587488C2211yescharacteristic
28C2×#10587488C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587487C2211yescharacteristic
33#11762751—21no—
34#11762743—21no—
35#11762715C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762703—21no—
40#11762697—21no—
41#11762691—21no—
42#11762681—21no—
43#11762675C2211yescharacteristic
44#11762669C2211yescharacteristic
45#11762667—21no—
46C4×C4⋊Dic17C2211yescharacteristic
47#11762660—21no—
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762572C2211yescharacteristic
54C22×M16×C17C2211yescharacteristic
55C2×C17×C4⋊C8C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×C22⋊C8C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C17×#336C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C22×Dic34C2311yescharacteristic
64C2×M16×C17—22no—
65C2×M16×C17—22no—
66C2×M16×C17—22no—
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C2×C22⋊C4×C17D411yescharacteristic
72C2×C22⋊C4×C17D411yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78#10587612—22no—
79#10587612—41no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—22no—
82C2×C4⋊Dic17—41no—
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90#10587610—22no—
91#10587610Q814yes—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102#10587556—22no—
103#10587556—41no—
104C4×Dic34—22no—
105C4×Dic34—41no—
106C17×C4⋊C8—22no—
107C17×C4⋊C8—22no—
108C17×C22⋊C8—22no—
109C17×#95D411yescharacteristic
110C17×#95D411yescharacteristic
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587489—22no—
121#10587489—22no—
122#10587489—22no—
123#10587488—22no—
124#10587488—41no—
125#10587488D414yes—
126#10587487—22no—
127#10587487—22no—
128#10587487—22no—
129#10587487—41no—
130C24×C17C2×D411yescharacteristic
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×Q812yes—
141C22×C4×C17#4112yes—
142C22×C4×C17C2411yescharacteristic
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—42no—
152C22×Dic17—42no—
153C22×Dic17—81no—
154C2×Dic34—21no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34C2×Q812yes—
160M16×C17—28no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17C2×D412yes—
172C2×C8×C17#4112yes—
173C22⋊C4×C17—41no—
174C22⋊C4×C17—41no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—21no—
179C22⋊Dic17—24no—
180C22⋊Dic17—42no—
181C22⋊Dic17—42no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—42no—
194C4⋊Dic17C2×D412yes—
195C4⋊Dic17C2×D412yes—
196C4⋊Dic17C2×Q812yes—
197#63225—24no—
198#63225—42no—
199#63225—42no—
200#63225—42no—
201#63225—42no—
202#63225—81no—
203#63225—81no—
204C4×Dic17—42no—
205C4×Dic17—42no—
206C4×Dic17—42no—
207#26367—171nomaximal, Sylow
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17C22⋊Q812yes—
215C23×C17C22×D411yescharacteristic
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
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—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17C22⋊Q812yes—
235C2×C4×C17C22×D411yescharacteristic
236C2×C4×C17C22×Q811yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17H(5,𝔽2)11yescharacteristic
240C2×C4×C172−1+411yescharacteristic
241C2×Dic17—22no—
242C2×Dic17—22no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—84no—
266Dic34—22no—
267Dic34—22no—
268Dic34—41no—
269Dic34—41no—
270C8×C17—22no—
271C8×C17—22no—
272C8×C17—41no—
273C8×C17—41no—
274C2×#479—171no—
275C2×#428—171no—
276#2442—171no—
277#2362—171no—
278#2097—171no—
279#2087—171no—
280#2068—171no—
281#2059—171no—
282#1949—171no—
283#1919—171no—
284#1894—171no—
285#1872—171no—
286#1865—171no—
287#1854—171no—
288#1851D1711yescharacteristic
289C22×C17—22no—
290C22×C17—22no—
291C22×C17—22no—
292C22×C17—41no—
293C22×C17—41no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17C2×C22⋊Q811yescharacteristic
297C22×C17D4211yescharacteristic
298C22×C17#54711yescharacteristic
299C22×C17#57511yescharacteristic
300C22×C17#57511yescharacteristic
301C22×C17#57711yescharacteristic
302C22×C17#57811yescharacteristic
303C4×C17—22no—
304C4×C17—41no—
305C4×C17—41no—
306C4×C17—41no—
307C4×C17—41no—
308C4×C17#52511yescharacteristic
309C4×C17#53911yescharacteristic
310C4×C17D4×Q811yescharacteristic
311C4×C17#55411yescharacteristic
312Dic17—42no—
313Dic17—42no—
314Dic17—42no—
315Dic17—42no—
316Dic17—81no—
317Dic17—81no—
318Dic17—81no—
319Dic17—81no—
320C22×M16C2×D1711yescharacteristic
321C2×#125—171no—
322C2×C22⋊Q8—171no—
323C2×C4×Q8—171no—
324C2×#117—171no—
325C22×C4⋊C4—171no—
326#479—344no—
327#428—344no—
328C2×#107—171no—
329C2×#107—171no—
330C2×#107—171no—
331C2×#106—171no—
332C2×#106—171no—
333C2×#106—171no—
334C2×C4⋊C8C2×D1711yescharacteristic
335C2×C4⋊C8C2×D1711yescharacteristic
336C2×#103—171no—
337C2×#103—171no—
338C2×#103—171no—
339C2×#103—171no—
340C2×C22⋊C8C2×D1711yescharacteristic
341#400—341no—
342#398—341no—
343#391—171no—
344#389—171no—
345#389—171no—
346#389—171no—
347#388—341no—
348#388—341no—
349#387—341no—
350#385—341no—
351#382—171no—
352#380—171no—
353#380—341no—
354C4×C4⋊C4—171no—
355C4×C4⋊C4—341no—
356C4×C22⋊C4C2×D1711yescharacteristic
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#336C2×D1711yescharacteristic
364#336C2×D1711yescharacteristic
365C2×C17—42no—
366C2×C17#244911yescharacteristic
367C2×C17#308011yescharacteristic
368C2×C17#308211yescharacteristic
369C2×C17#326911yescharacteristic
370C2×C17#327511yescharacteristic
371C2×C17#334911yescharacteristic
372C2×C17#335111yescharacteristic
373C22×Q8—171no—
374C23×C4—341no—
375C23×C4C22×D1711yescharacteristic
376C2×M16—22no—
377C2×M16—22no—
378C2×M16—22no—
379C22×C8C22×D1711yescharacteristic
380C22×C8C22×D1711yescharacteristic
381C22×C8C22×D1711yescharacteristic
382C22×C8C22×D1711yescharacteristic
383#125—342no—
384#125—681no—
385C22⋊Q8—174no—
386C22⋊Q8—342no—
387C4×Q8—342no—
388C4×Q8—681no—
389#117—342no—
390#117—681no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—342no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4—681no—
408C2×C22⋊C4—341no—
409C2×C22⋊C4—341no—
410C2×C22⋊C4—341no—
411C2×C22⋊C4D6811yescharacteristic
412C2×C22⋊C4D6811yescharacteristic
413C2×C42—341no—
414C2×C42—341no—
415C2×C42—341no—
416C2×C42C22×D1711yescharacteristic
417C2×C42C22×D1711yescharacteristic
418#107—342no—
419#107—342no—
420#107—342no—
421#106—174no—
422#106—342no—
423#106—681no—
424C4⋊C8—22no—
425C4⋊C8—22no—
426#103—342no—
427#103—342no—
428#103—342no—
429#103—681no—
430C22⋊C8—22no—
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95D6811yescharacteristic
441#95D6811yescharacteristic
442C17#2636711yescharacteristic, complemented, Sylow
443C24C2×D6811yescharacteristic
444C2×Q8—172no—
445C2×Q8—341no—
446C2×Q8—341no—
447C2×Q8—341no—
448C2×Q8—341no—
449C2×Q8—341no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—682no—
463C22×C4—682no—
464C22×C4—1361no—
465C22×C4C2×D6811yescharacteristic
466C22×C4C2×D6811yescharacteristic
467C22×C4C2×D6811yescharacteristic
468C22×C4C2×D6811yescharacteristic
469C22×C4C2×D6811yescharacteristic
470C22×C4#6325412yes—
471C22×C4Q8×D1712yes—
472C22×C4C23×D1711yescharacteristic
473M16—28no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8D4×D1712yes—
485C2×C8#6325412yes—
486C4⋊C4—172no—
487C4⋊C4—172no—
488C4⋊C4—172no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—344no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—682no—
502C4⋊C4—682no—
503C4⋊C4—682no—
504C4⋊C4—682no—
505C4⋊C4—682no—
506C4⋊C4—1361no—
507C4⋊C4—1361no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—344no—
511C22⋊C4—682no—
512C22⋊C4—682no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—682no—
518C42—682no—
519C42—682no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1058758312yes—
527C23C22×D6811yescharacteristic
528Q8—342no—
529Q8—342no—
530Q8—681no—
531Q8—681no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—342no—
551C2×C4—342no—
552C2×C4—342no—
553C2×C4—342no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—1364no—
575C2×C4#1058758312yes—
576C2×C4C2×D4×D1711yescharacteristic
577C2×C4C2×#6325411yescharacteristic
578C2×C4C2×#6325411yescharacteristic
579C2×C4C2×Q8×D1711yescharacteristic
580C2×C4#1058769611yescharacteristic
581C2×C4#1058769711yescharacteristic
582C8—22no—
583C8—22no—
584C8—41no—
585C8—41no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1058758311yescharacteristic
594C22D4×D6811yescharacteristic
595C22#1176346311yescharacteristic
596C22#1176360911yescharacteristic
597C22#1176360911yescharacteristic
598C22#1176362111yescharacteristic
599C22#1176362611yescharacteristic
600C4—22no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—682no—
606C4—682no—
607C4—682no—
608C4—682no—
609C4—1361no—
610C4—1361no—
611C4—1361no—
612C4—1361no—
613C4#1176331511yescharacteristic
614C4#1176340311yescharacteristic
615C4#1176347211yescharacteristic
616C4#1176351711yescharacteristic
617C2—42no—
618C2#1453259111yescharacteristic
619C2#1453860711yescharacteristic
620C2#1453862711yescharacteristic
621C2#1454034411yescharacteristic
622C2#1454042011yescharacteristic
623C2#1454131611yescharacteristic
624C2#1454134611yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1851C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1851D17
hypocenterC17#26367
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. C2×#11763117C2semidirect
5. #14530331C2semidirect
6. #14530277C2semidirect
7. #14530178C2semidirect
8. #14530123C2semidirect
9. #14529554C2semidirect
10. #14529363C2semidirect
13. #14529055C2semidirect
442. C17#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#117631172
C2×#117628512
#145325082
#145318092
#145303312
#145302772
#145301782
#145301232
#145295542
#145293632
#145292172
#145290892
#145290552
#145289852
C17×#18512
#2636717