Finite Group Databasebeta

← #17064338

Subgroups — #17064338

Order 4352.

796 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587632C211yesmaximal, characteristic
2#14531188C211yesmaximal, characteristic
3#14530344C211yesmaximal, characteristic, complemented
4#14530325C211yesmaximal, characteristic, complemented
5#14530264C211yesmaximal, characteristic, complemented
6#14530222C211yesmaximal, characteristic
7#14530219C211yesmaximal, characteristic
8#14530195C211yesmaximal, characteristic, complemented
9#14529979C211yesmaximal, characteristic
10#14529978C211yesmaximal, characteristic
11C4×#10587479C211yesmaximal, characteristic, complemented
12C4×#10587488C211yesmaximal, characteristic, complemented
13C4×#10587506C211yesmaximal, characteristic, complemented
14C4×#10587491C211yesmaximal, characteristic
15C4×C17×#103C211yesmaximal, characteristic, complemented
16C2×#10587632C2211yescharacteristic
17C2×#10587558—21no—
18C2×C42×D17C2211yescharacteristic
19C2×#10587488C2211yescharacteristic
20C2×#10587488C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587491C2211yescharacteristic
26C2×#10587491C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28#11762716C2211yescharacteristic
29#11762705C2211yescharacteristic
30#11762690—21no—
31#11762680—21no—
32#11762678C2211yescharacteristic
33#11762678C2211yescharacteristic
34#11762673—21no—
35#11762668—21no—
36#11762666C2211yescharacteristic
37#11762666C2211yescharacteristic
38#11762666C2211yescharacteristic
39C4×#63225—21no—
40C4×#63225—21no—
41C4×C4⋊Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43C4×#63227—21no—
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C2×C17×#103C2211yescharacteristic
54C2×C17×#103C2211yescharacteristic
55C2×C4×C8×C17C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C22×C4×D17D411yescharacteristic
61C22×C4×D17D411yescharacteristic
62C22×C4×D17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C4×Q8×C17—22no—
66C4×Q8×C17—22no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C42×C17—21no—
72C2×C42×C17—21no—
73C2×C42×C17C2311yescharacteristic
74#10587632—22no—
75#10587632—22no—
76C2×#63227—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×#63227—21no—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17D411yescharacteristic
93C2×C4×Dic17D411yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97#10587558—44no—
98C42×D17—22no—
99C42×D17—22no—
100C17×#103—22no—
101C17×#103—22no—
102C4×C8×C17—22no—
103C17×#95—21no—
104#10587506—22no—
105#10587506—22no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587491—22no—
115#10587491—22no—
116#10587488—22no—
117#10587488—22no—
118#10587479—22no—
119#10587479—22no—
120C4×#3622—22no—
121C23×D17C2×D411yescharacteristic
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17#4112yes—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2411yescharacteristic
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C2×C4×D17—22no—
143C2×C4×D17—22no—
144C2×C4×D17—22no—
145C2×C4×D17—21no—
146C2×C4×D17—21no—
147C2×C4×D17—21no—
148C2×C4×D17—21no—
149C2×C4×D17—21no—
150C2×C4×D17—21no—
151C2×C4×D17—22no—
152C2×C4×D17—41no—
153C2×C4×D17SD1614yes—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17#4112yes—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—42no—
167C42×C17—42no—
168C42×C17—42no—
169C42×C17#4112yes—
170C42×C17#4112yes—
171#63227—24no—
172#63227—42no—
173#63227—42no—
174#63227—42no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17#4112yes—
183#63225—24no—
184#63225—42no—
185#63225—42no—
186#63225—42no—
187#63225—42no—
188#63225—42no—
189C4×Dic17—22no—
190C4×Dic17—22no—
191C4×Dic17—21no—
192C4×Dic17—21no—
193C4×Dic17—42no—
194C4×Dic17—42no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622#4112yes—
201#23756—171nomaximal, Sylow
202C23×C17C22×D411yescharacteristic
203C22×D17—21no—
204C22×D17—21no—
205C22×D17—21no—
206C22×D17—21no—
207C22×D17—21no—
208C22×D17—21no—
209C22×D17C2×SD1612yes—
210Q8×C17—22no—
211Q8×C17—22no—
212Q8×C17—41no—
213Q8×C17—41no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—22no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17#12312yes—
229C2×C4×C17#12312yes—
230C2×C4×C17C2×SD1612yes—
231C2×C4×C17#13712yes—
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—22no—
245C2×Dic17—21no—
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×Dic17C2×SD1612yes—
258C2×Dic17#13512yes—
259C4×D17—28no—
260C4×D17—44no—
261C4×D17—41no—
262C4×D17—41no—
263C4×D17—41no—
264C4×D17—41no—
265C8×C17—22no—
266C8×C17—41no—
267#3622—22no—
268#3622—41no—
269C4×C22⋊Q8—171no—
270#2285—171no—
271#2099—171no—
272#2096—171no—
273#2084—171no—
274#2077—171no—
275#2077—171no—
276#2072—171no—
277#2035—171no—
278#2035—171no—
279C4×#106—171no—
280C4×#106—171no—
281C4×#103—171no—
282C4×#103D1711yescharacteristic, complemented
283C4×C22⋊C8—171no—
284C22×C17C2×#12311yescharacteristic
285C22×C17#52511yescharacteristic
286C22×C17#53211yescharacteristic
287C22×C17C22×SD1611yescharacteristic
288C22×C17C2×#13511yescharacteristic
289C22×C17C2×#13711yescharacteristic
290C22×C17#57511yescharacteristic
291C2×D17—22no—
292C2×D17—22no—
293C2×D17—22no—
294C2×D17—41no—
295C2×D17—41no—
296C2×D17—41no—
297C2×D17—41no—
298C4×C17—22no—
299C4×C17—42no—
300C4×C17—81no—
301C4×C17—81no—
302C4×C17—81no—
303C4×C17#48314yes—
304C4×C17#52911yescharacteristic
305C4×C17#52911yescharacteristic
306C4×C17#52911yescharacteristic
307C4×C17#52911yescharacteristic
308Dic17—22no—
309Dic17—22no—
310Dic17—42no—
311Dic17—41no—
312Dic17—81no—
313Dic17—81no—
314Dic17—81no—
315C2×C22⋊Q8—171no—
316C2×C4×Q8C2×D1711yescharacteristic
317C2×#117—341no—
318C22×C42—171no—
319C2×#106—171no—
320C2×#106—171no—
321C2×#106—171no—
322C2×#106—171no—
323C2×#103—171no—
324C2×#103—171no—
325C2×#103C2×D1711yescharacteristic
326C2×#103C2×D1711yescharacteristic
327C2×C22⋊C8—171no—
328C2×C22⋊C8—171no—
329C2×C4×C8—171no—
330C2×C4×C8C2×D1711yescharacteristic
331#391—171no—
332#389—171no—
333#387—341no—
334#385—341no—
335#384—171no—
336#384—171no—
337#382—341no—
338#380—171no—
339#380—171no—
340#380—171no—
341#380—341no—
342C4×C4⋊C4—171no—
343C4×C4⋊C4—171no—
344C4×C4⋊C4—341no—
345C4×C4⋊C4—341no—
346C4×C4⋊C4C2×D1711yescharacteristic
347C4×C22⋊C4—341no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340C2×D1711yescharacteristic
355#340C2×D1711yescharacteristic
356#336—171no—
357#336—171no—
358C2×C17#239511yescharacteristic
359C2×C17C2×#48311yescharacteristic
360C2×C17#309111yescharacteristic
361C2×C17#310311yescharacteristic
362C2×C17#310711yescharacteristic
363C2×C17#316711yescharacteristic
364C2×C17#317211yescharacteristic
365D17—42no—
366C22×Q8C22×D1711yescharacteristic
367C23×C4—171no—
368C23×C4—171no—
369C23×C4—171no—
370C22×C8—171no—
371C22×C8—171no—
372C22×C8C22×D1711yescharacteristic
373C22×C8C22×D1711yescharacteristic
374C22⋊Q8—342no—
375C22⋊Q8—342no—
376C4×Q8—22no—
377C4×Q8—22no—
378#117—684no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4C22×D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C22⋊C4—341no—
394C2×C22⋊C4—341no—
395C2×C22⋊C4—341no—
396C2×C22⋊C4—341no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—171no—
400C2×C42—171no—
401C2×C42—171no—
402C2×C42—342no—
403C2×C42—342no—
404C2×C42—341no—
405C2×C42—341no—
406C2×C42C22×D1711yescharacteristic
407#106—342no—
408#106—342no—
409#106—342no—
410#106—342no—
411#103—22no—
412#103—22no—
413#103—342no—
414#103—342no—
415C22⋊C8—342no—
416C22⋊C8—342no—
417C4×C8—22no—
418C4×C8—342no—
419#95—21no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428C17#2375611yescharacteristic, complemented, Sylow
429C24—171no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8#6325212yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—171no—
441C22×C4—171no—
442C22×C4—174no—
443C22×C4—171no—
444C22×C4—342no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—342no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—681no—
459C22×C4D4×D1711yescharacteristic
460C22×C4D4×D1711yescharacteristic
461C22×C4C23×D1711yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—172no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8#6325212yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—172no—
482C4⋊C4—344no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—682no—
491C4⋊C4—682no—
492C4⋊C4—682no—
493C4⋊C4—682no—
494C4⋊C4—682no—
495C22⋊C4—344no—
496C22⋊C4—682no—
497C22⋊C4—682no—
498C22⋊C4—682no—
499C42—42no—
500C42—42no—
501C42—341no—
502C42—341no—
503C42—342no—
504C42—342no—
505C42—682no—
506C42—682no—
507C42#6325212yes—
508C42#6325412yes—
509C23—172no—
510C23—341no—
511C23—341no—
512C23—341no—
513C23—341no—
514C23—341no—
515C23—341no—
516C23C2×D4×D1711yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—172no—
536C2×C4—172no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—348no—
542C2×C4—341no—
543C2×C4—342no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—684no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
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#1058756912yes—
563C2×C4#1058756912yes—
564C2×C4SD16×D1712yes—
565C2×C4#1058760412yes—
566C2×C4C2×#6325211yescharacteristic
567C2×C4C2×#6325211yescharacteristic
568C2×C4C2×#6325211yescharacteristic
569C2×C4C2×#6325411yescharacteristic
570C2×C4#41×D1711yescharacteristic
571C2×C4#41×D1711yescharacteristic
572C8—22no—
573C8—41no—
574C8—342no—
575C8—681no—
576C22—342no—
577C22—342no—
578C22—342no—
579C22—681no—
580C22—681no—
581C22—681no—
582C22—681no—
583C22C2×#1058756911yescharacteristic
584C22#1176331811yescharacteristic
585C22#1176335611yescharacteristic
586C22C2×SD16×D1711yescharacteristic
587C22#135×D1711yescharacteristic
588C22C2×#1058760411yescharacteristic
589C22#1176360611yescharacteristic
590C4—22no—
591C4—42no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—342no—
596C4—342no—
597C4—682no—
598C4—681no—
599C4—1361no—
600C4—1361no—
601C4—1361no—
602C4#1176313814yes—
603C4#1176333711yescharacteristic
604C4#1176333911yescharacteristic
605C4#1176333911yescharacteristic
606C4#1176334311yescharacteristic
607C2—682no—
608C2#1453204511yescharacteristic
609C2C2×#1176313811yescharacteristic
610C2#1453869511yescharacteristic
611C2#1453881311yescharacteristic
612C2#1453885311yescharacteristic
613C2#1453934011yescharacteristic
614C2#1453938811yescharacteristic
subgroupG/H
centerC22×C4D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC4×C17×#103C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
hypercenterC4×#103D17
hypocenterC17#23756
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. #14530344C2semidirect
4. #14530325C2semidirect
5. #14530264C2semidirect
8. #14530195C2semidirect
11. C4×#10587479C2semidirect
12. C4×#10587488C2semidirect
13. C4×#10587506C2semidirect
15. C4×C17×#103C2semidirect
282. C4×#103D17semidirect
428. C17#23756semidirect

Maximal subgroups

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

subgroupindex
C4×#105876322
#145311882
#145303442
#145303252
#145302642
#145302222
#145302192
#145301952
#145299792
#145299782
C4×#105874792
C4×#105874882
C4×#105875062
C4×#105874912
C4×C17×#1032
#2375617