Finite Group Databasebeta

← #17075708

Subgroups — #17075708

Order 4352.

738 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763113C211yesmaximal, characteristic, complemented
2C2×#11762870C211yesmaximal, characteristic
3#14531466C211yesmaximal, characteristic
4#14530220C211yesmaximal, characteristic
5#14530210C211yesmaximal, characteristic
6#14529987C211yesmaximal, characteristic
7#14529987C211yesmaximal, characteristic
8#14529742C211yesmaximal, characteristic, complemented
9#14529540C211yesmaximal, characteristic, complemented
10#14529519C211yesmaximal, characteristic, complemented
11#14529514C211yesmaximal, characteristic, complemented
12#14529473C211yesmaximal, characteristic, complemented
13#14529459C211yesmaximal, characteristic, complemented
14#14528786C211yesmaximal, characteristic, complemented
15C17×#2629C211yesmaximal, characteristic
16C2×#10587610C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18#11763113—24no—
19#11762870—24no—
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×C22⋊#3622C2211yescharacteristic
31C2×C22⋊#3622C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762715C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762704—21no—
37#11762703—21no—
38#11762697—21no—
39#11762690—21no—
40#11762669C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50C2×C17×#125C2211yescharacteristic
51C2×C17×#117C2211yescharacteristic
52C17×#398—21no—
53C17×#398—21no—
54C17×#396—21no—
55C17×#396—21no—
56C17×#389C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#384C2211yescharacteristic
59C17×#380C2211yescharacteristic
60C17×#380C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C22×Dic34C2311yescharacteristic
63C17×#125—22no—
64C17×#125—22no—
65C17×#117—22no—
66C17×#117—22no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17D411yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17D411yescharacteristic
82C2×C22⋊C4×C17D411yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90#10587610—22no—
91#10587610—22no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C4×Dic34—22no—
102C4×Dic34—22no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95—21no—
106C17×#95—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112C22⋊#3622—22no—
113C22⋊#3622—22no—
114#10587481—22no—
115#10587481—22no—
116#10587481—22no—
117#10587481—22no—
118#10587479—22no—
119#10587478—22no—
120#10587478—22no—
121#10587478—22no—
122C4⋊#3622—22no—
123C4⋊#3622D414yes—
124#10587474—22no—
125C4×#3622—22no—
126C24×C17C2×D411yescharacteristic
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2411yescharacteristic
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C2×Dic34—21no—
148C2×Dic34—21no—
149C2×Dic34—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34#4112yes—
153C4⋊C4×C17—22no—
154C4⋊C4×C17—22no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—42no—
164C4⋊C4×C17—42no—
165C4⋊C4×C17—81no—
166C4⋊C4×C17—81no—
167C4⋊C4×C17—81no—
168C4⋊C4×C17—81no—
169C22⋊C4×C17—22no—
170C22⋊C4×C17—22no—
171C22⋊C4×C17—81no—
172C22⋊C4×C17—81no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17C2×D412yes—
176C42×C17#4112yes—
177C22⋊Dic17—24no—
178C22⋊Dic17—81no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17#4112yes—
185#63225—24no—
186#63225—42no—
187#63225—81no—
188C4×Dic17—42no—
189C4×Dic17—81no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622C2×D412yes—
201C2×#3622C2×D412yes—
202#24715—171nomaximal, Sylow
203C23×C17—21no—
204C23×C17—21no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—21no—
208C23×C17—21no—
209C23×C17—21no—
210C23×C17C22×D411yescharacteristic
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
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—21no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17#12112yes—
241C2×C4×C17#12112yes—
242C2×C4×C17C22×D411yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C172−1+411yescharacteristic
247C2×C4×C172−1+411yescharacteristic
248C2×Dic17—22no—
249C2×Dic17—22no—
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—
260Dic34—22no—
261Dic34—22no—
262Dic34—41no—
263Dic34—41no—
264#3622—22no—
265#3622—22no—
266#3622—41no—
267#3622—41no—
268C2×#479—171no—
269C2×#433—171no—
270#2629D1711yescharacteristic
271#2323—171no—
272#2077—171no—
273#2075—171no—
274#2036—171no—
275#2036—171no—
276#1989—171no—
277#1947—171no—
278#1943—171no—
279#1942—171no—
280#1935—171no—
281#1933—171no—
282#1803—171no—
283C22×C17—41no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17—41no—
287C22×C17—41no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17C2×#12111yescharacteristic
291C22×C17#52511yescharacteristic
292C22×C17#53611yescharacteristic
293C22×C17#57611yescharacteristic
294C22×C17#57711yescharacteristic
295C22×C17#57811yescharacteristic
296C22×C17#57811yescharacteristic
297C4×C17—22no—
298C4×C17—22no—
299C4×C17—41no—
300C4×C17—41no—
301C4×C17—41no—
302C4×C17—81no—
303C4×C17—81no—
304C4×C17—81no—
305C4×C17—81no—
306C4×C17#54711yescharacteristic
307C4×C17#54811yescharacteristic
308C4×C17#55211yescharacteristic
309C4×C17#55611yescharacteristic
310Dic17—42no—
311Dic17—42no—
312Dic17—81no—
313Dic17—81no—
314C2×#125C2×D1711yescharacteristic
315C2×C22⋊Q8—171no—
316C2×C4×Q8—171no—
317C2×#117C2×D1711yescharacteristic
318#479—344no—
319#433—344no—
320C2×#107—171no—
321C2×#107—171no—
322C2×#107—171no—
323C2×#106—171no—
324C2×C4⋊C8—171no—
325C2×C4⋊C8—171no—
326C2×#103—171no—
327C2×#103—171no—
328C2×#103—171no—
329C2×#103—171no—
330C2×C22⋊C8—171no—
331C2×C22⋊C8—171no—
332C2×#97—171no—
333C2×C4×C8—171no—
334#398—21no—
335#398—21no—
336#396—21no—
337#396—21no—
338#391—171no—
339#389—171no—
340#389—341no—
341#389C2×D1711yescharacteristic
342#389C2×D1711yescharacteristic
343#388—341no—
344#388—341no—
345#387—341no—
346#384C2×D1711yescharacteristic
347#380—171no—
348#380C2×D1711yescharacteristic
349#380C2×D1711yescharacteristic
350C4×C4⋊C4—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359C2×C17—81no—
360C2×C17#241511yescharacteristic
361C2×C17#306111yescharacteristic
362C2×C17#306311yescharacteristic
363C2×C17#311911yescharacteristic
364C2×C17#312711yescharacteristic
365C2×C17#324211yescharacteristic
366C2×C17#324311yescharacteristic
367C22×Q8—171no—
368C23×C4C22×D1711yescharacteristic
369C22×C8—171no—
370C22×C8—171no—
371C22×C8—171no—
372C22×C8—171no—
373#125—22no—
374#125—22no—
375C22⋊Q8—342no—
376C22⋊Q8—342no—
377C4×Q8—342no—
378C4×Q8—342no—
379#117—22no—
380#117—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4C22⋊D1711yescharacteristic
394C2×C4⋊C4C22⋊D1711yescharacteristic
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C4⋊C4C22×D1711yescharacteristic
397C2×C4⋊C4C22×D1711yescharacteristic
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—341no—
402C2×C22⋊C4—341no—
403C2×C22⋊C4C22⋊D1711yescharacteristic
404C2×C22⋊C4C22⋊D1711yescharacteristic
405C2×C42—341no—
406C2×C42—341no—
407C2×C42C22×D1711yescharacteristic
408C2×C42C22×D1711yescharacteristic
409#107—342no—
410#107—342no—
411#107—342no—
412#106—342no—
413C4⋊C8—174no—
414C4⋊C8—342no—
415#103—342no—
416#103—342no—
417#103—342no—
418#103—342no—
419C22⋊C8—342no—
420C22⋊C8—342no—
421#97—342no—
422C4×C8—342no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432C17#2471511yescharacteristic, complemented, Sylow
433C24C2×C22⋊D1711yescharacteristic
434C2×Q8—172no—
435C2×Q8—341no—
436C2×Q8—341no—
437C2×Q8—341no—
438C2×Q8—341no—
439C2×Q8—341no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4C2×C22⋊D1711yescharacteristic
455C22×C4C2×C22⋊D1711yescharacteristic
456C22×C4C2×C22⋊D1711yescharacteristic
457C22×C4C2×C22⋊D1711yescharacteristic
458C22×C4C2×C22⋊D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460C2×C8—172no—
461C2×C8—172no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—172no—
489C4⋊C4—344no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—682no—
496C4⋊C4—1361no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4—344no—
502C22⋊C4—1361no—
503C42—21no—
504C42—21no—
505C42—682no—
506C42—1361no—
507C42D4×D1712yes—
508C42#6325412yes—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C22×C22⋊D1711yescharacteristic
517Q8—342no—
518Q8—342no—
519Q8—681no—
520Q8—681no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—342no—
551C2×C4—342no—
552C2×C4—681no—
553C2×C4—681no—
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#1058762512yes—
563C2×C4#1058762512yes—
564C2×C4C2×D4×D1711yescharacteristic
565C2×C4C2×#6325411yescharacteristic
566C2×C4#1058769411yescharacteristic
567C2×C4#1058769411yescharacteristic
568C2×C4#41×D1711yescharacteristic
569C2×C4#41×D1711yescharacteristic
570C8—342no—
571C8—342no—
572C8—681no—
573C8—681no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1058762511yescharacteristic
582C22#1176331611yescharacteristic
583C22#1176338311yescharacteristic
584C22#1176361211yescharacteristic
585C22#1176362011yescharacteristic
586C22#1176362411yescharacteristic
587C22#1176362411yescharacteristic
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—682no—
598C4—682no—
599C4—1361no—
600C4—1361no—
601C4#1176346111yescharacteristic
602C4#1176346911yescharacteristic
603C4#1176349311yescharacteristic
604C4#1176353011yescharacteristic
605C2—81no—
606C2#1453221411yescharacteristic
607C2#1453845111yescharacteristic
608C2#1453847711yescharacteristic
609C2#1453897711yescharacteristic
610C2#1453907911yescharacteristic
611C2#1454005211yescharacteristic
612C2#1454006711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2629C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2629D17
hypocenterC17#24715
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2629

Complements

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

NH ≅ G/Ntype
1. C2×#11763113C2semidirect
8. #14529742C2semidirect
9. #14529540C2semidirect
10. #14529519C2semidirect
11. #14529514C2semidirect
12. #14529473C2semidirect
13. #14529459C2semidirect
14. #14528786C2semidirect
432. C17#24715semidirect

Maximal subgroups

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

subgroupindex
C2×#117631132
C2×#117628702
#145314662
#145302202
#145302102
#145299872
#145297422
#145295402
#145295192
#145295142
#145294732
#145294592
#145287862
C17×#26292
#2471517