Finite Group Databasebeta

← #17064373

Subgroups — #17064373

Order 4352.

752 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531182C211yesmaximal, characteristic
2#14530352C211yesmaximal, characteristic, complemented
3#14530332C211yesmaximal, characteristic, complemented
4#14530265C211yesmaximal, characteristic, complemented
5#14530220C211yesmaximal, characteristic
6#14530210C211yesmaximal, characteristic
7#14530202C211yesmaximal, characteristic, complemented
8#14529984C211yesmaximal, characteristic
9#14529977C211yesmaximal, characteristic
10C4×#10587478C211yesmaximal, characteristic, complemented
11C4×#10587479C211yesmaximal, characteristic, complemented
12C4×#10587506C211yesmaximal, characteristic, complemented
13C4×#10587506C211yesmaximal, characteristic, complemented
14C4×C22⋊#3622C211yesmaximal, characteristic
15C4×C17×C22⋊Q8C211yesmaximal, characteristic
16C2×#10587478C2211yescharacteristic
17C2×#10587478C2211yescharacteristic
18C2×#10587479C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×#10587506C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×C4×#3622C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28#11762677—21no—
29#11762675C2211yescharacteristic
30#11762675C2211yescharacteristic
31#11762672—21no—
32#11762669C2211yescharacteristic
33#11762669C2211yescharacteristic
34C4×C4⋊Dic17C2211yescharacteristic
35C4×C4⋊Dic17C2211yescharacteristic
36C4×C22⋊Dic17—21no—
37#11762655—21no—
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×C22⋊Q8C2211yescharacteristic
49C2×C4×Q8×C17C2211yescharacteristic
50C22×C42×C17C2211yescharacteristic
51C17×#391C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#387—21no—
54C17×#385—21no—
55C17×#380C2211yescharacteristic
56C4×C4⋊C4×C17—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17—21no—
59C22×Q8×C17C2311yescharacteristic
60C23×C4×C17D411yescharacteristic
61C23×C4×C17D411yescharacteristic
62C23×C4×C17C2311yescharacteristic
63C17×C22⋊Q8—22no—
64C17×C22⋊Q8—22no—
65C4×Q8×C17—22no—
66C4×Q8×C17—22no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C42×C17—22no—
76C2×C42×C17—22no—
77C2×C42×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17D411yescharacteristic
80C2×C42×C17D411yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C17×#95—21no—
96C17×#95—21no—
97C17×#95—21no—
98C17×#95—21no—
99C17×#95—21no—
100#10587506—22no—
101#10587506—22no—
102#10587506—22no—
103#10587506—22no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112C22⋊#3622—22no—
113C22⋊#3622—22no—
114#10587479—22no—
115#10587479—22no—
116#10587478—22no—
117#10587478—22no—
118C4×#3622—22no—
119C4×#3622—22no—
120C24×C17C2×D411yescharacteristic
121C2×Q8×C17—21no—
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—21no—
126C2×Q8×C17#4112yes—
127C22×C4×C17—22no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—41no—
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2411yescharacteristic
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C4⋊C4×C17—24no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—42no—
160C4⋊C4×C17—42no—
161C4⋊C4×C17#4112yes—
162C22⋊C4×C17—24no—
163C22⋊C4×C17—42no—
164C42×C17—22no—
165C42×C17—22no—
166C42×C17—21no—
167C42×C17—21no—
168C42×C17—42no—
169C42×C17—42no—
170C42×C17#4112yes—
171C42×C17#4112yes—
172C22⋊Dic17—42no—
173C22⋊Dic17—42no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4×Dic17—42no—
183C4×Dic17—42no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622#4112yes—
195C2×#3622#4112yes—
196#23759—171nomaximal, Sylow
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17#13512yes—
204C23×C17C22×D411yescharacteristic
205Q8×C17—22no—
206Q8×C17—22no—
207Q8×C17—41no—
208Q8×C17—41no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
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—44no—
223C2×C4×C17—41no—
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—44no—
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#12312yes—
239C2×C4×C17#12312yes—
240C2×C4×C17#13512yes—
241C2×C4×C17#13712yes—
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
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—
260#3622—22no—
261#3622—22no—
262#3622—41no—
263#3622—41no—
264C4×C22⋊Q8D1711yescharacteristic
265#2283—171no—
266#2101—171no—
267#2097—171no—
268#2085—171no—
269#2077—171no—
270#2075—171no—
271#2074—171no—
272#2036—171no—
273#2035—171no—
274C4×#107—171no—
275C4×#106—171no—
276C4×#103—171no—
277C4×#103—171no—
278C4×C22⋊C8—171no—
279C22×C17—22no—
280C22×C17—22no—
281C22×C17—22no—
282C22×C17—41no—
283C22×C17—41no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17C2×#12311yescharacteristic
287C22×C17#52511yescharacteristic
288C22×C17#53211yescharacteristic
289C22×C17C2×#13511yescharacteristic
290C22×C17C2×#13511yescharacteristic
291C22×C17C2×#13711yescharacteristic
292C22×C17#57511yescharacteristic
293C4×C17—22no—
294C4×C17—41no—
295C4×C17—41no—
296C4×C17—42no—
297C4×C17—41no—
298C4×C17—42no—
299C4×C17—81no—
300C4×C17—81no—
301C4×C17#48514yes—
302C4×C17#52911yescharacteristic
303C4×C17#52911yescharacteristic
304C4×C17#52911yescharacteristic
305C4×C17#52911yescharacteristic
306Dic17—81no—
307Dic17—81no—
308Dic17—81no—
309Dic17—81no—
310C2×C22⋊Q8C2×D1711yescharacteristic
311C2×C4×Q8C2×D1711yescharacteristic
312C22×C42C2×D1711yescharacteristic
313C2×#107—171no—
314C2×#107—171no—
315C2×#106—171no—
316C2×#106—171no—
317C2×#103—171no—
318C2×#103—171no—
319C2×#103—171no—
320C2×#103—171no—
321C2×C22⋊C8—171no—
322C2×C22⋊C8—171no—
323C2×C4×C8—171no—
324C2×C4×C8—171no—
325#391C2×D1711yescharacteristic
326#389C2×D1711yescharacteristic
327#387—21no—
328#385—21no—
329#383—341no—
330#382—171no—
331#382—171no—
332#381—341no—
333#380—171no—
334#380—171no—
335#380C2×D1711yescharacteristic
336C4×C4⋊C4—21no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4C2×D1711yescharacteristic
340C4×C22⋊C4—21no—
341C4×C22⋊C4—341no—
342#376—341no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#336—171no—
352#336—171no—
353C2×C17—42no—
354C2×C17#239511yescharacteristic
355C2×C17C2×#48511yescharacteristic
356C2×C17#309111yescharacteristic
357C2×C17#310211yescharacteristic
358C2×C17#310811yescharacteristic
359C2×C17#316811yescharacteristic
360C2×C17#317211yescharacteristic
361C22×Q8C22×D1711yescharacteristic
362C23×C4C22⋊D1711yescharacteristic
363C23×C4C22⋊D1711yescharacteristic
364C23×C4C22×D1711yescharacteristic
365C22×C8—171no—
366C22×C8—171no—
367C22×C8—171no—
368C22×C8—171no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C4×Q8—22no—
372C4×Q8—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C4⋊C4C22×D1711yescharacteristic
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—341no—
392C2×C42—341no—
393C2×C42C22⋊D1711yescharacteristic
394C2×C42C22⋊D1711yescharacteristic
395C2×C42C22×D1711yescharacteristic
396C2×C42C22×D1711yescharacteristic
397#107—342no—
398#107—342no—
399#106—342no—
400#106—342no—
401#103—342no—
402#103—342no—
403#103—342no—
404#103—342no—
405C22⋊C8—342no—
406C22⋊C8—342no—
407C4×C8—342no—
408C4×C8—342no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422C17#2375911yescharacteristic, complemented, Sylow
423C24C2×C22⋊D1711yescharacteristic
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#6325212yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—341no—
450C22×C4C2×C22⋊D1711yescharacteristic
451C22×C4C2×C22⋊D1711yescharacteristic
452C22×C4C2×C22⋊D1711yescharacteristic
453C22×C4C2×C22⋊D1711yescharacteristic
454C22×C4C2×C22⋊D1711yescharacteristic
455C22×C4C23×D1711yescharacteristic
456C2×C8—172no—
457C2×C8—172no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4#6325212yes—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—682no—
488C22⋊C4—682no—
489C42—21no—
490C42—21no—
491C42—22no—
492C42—22no—
493C42—42no—
494C42—42no—
495C42—682no—
496C42—682no—
497C42#6325412yes—
498C42#6325412yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#1058763612yes—
506C23C22×C22⋊D1711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
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—44no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4#1058762112yes—
553C2×C4#1058762112yes—
554C2×C4#1058762812yes—
555C2×C4#1058763612yes—
556C2×C4C2×#6325211yescharacteristic
557C2×C4C2×#6325211yescharacteristic
558C2×C4C2×#6325411yescharacteristic
559C2×C4C2×#6325411yescharacteristic
560C2×C4#41×D1711yescharacteristic
561C2×C4#41×D1711yescharacteristic
562C8—342no—
563C8—342no—
564C8—681no—
565C8—681no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C2×#1058762111yescharacteristic
574C22#1176332011yescharacteristic
575C22#1176335511yescharacteristic
576C22C2×#1058763611yescharacteristic
577C22C2×#1058763611yescharacteristic
578C22C2×#1058762811yescharacteristic
579C22#1176360711yescharacteristic
580C4—22no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—1361no—
589C4—1361no—
590C4—1361no—
591C4—1361no—
592C4#1176314714yes—
593C4#1176333911yescharacteristic
594C4#1176333911yescharacteristic
595C4#1176334111yescharacteristic
596C4#1176334111yescharacteristic
597C2—42no—
598C2#1453204311yescharacteristic
599C2C2×#1176314711yescharacteristic
600C2#1453869311yescharacteristic
601C2#1453880211yescharacteristic
602C2#1453886411yescharacteristic
603C2#1453934811yescharacteristic
604C2#1453938411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC4×C17×C22⋊Q8C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenterC4×C22⋊Q8D17
hypocenterC17#23759
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #14530352C2semidirect
3. #14530332C2semidirect
4. #14530265C2semidirect
7. #14530202C2semidirect
10. C4×#10587478C2semidirect
11. C4×#10587479C2semidirect
12. C4×#10587506C2semidirect
13. C4×#10587506C2semidirect
422. C17#23759semidirect

Maximal subgroups

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

subgroupindex
#145311822
#145303522
#145303322
#145302652
#145302202
#145302102
#145302022
#145299842
#145299772
C4×#105874782
C4×#105874792
C4×#105875062
C4×C22⋊#36222
C4×C17×C22⋊Q82
#2375917