Finite Group Databasebeta

← #17134095

Subgroups — #17134095

Order 4352.

690 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14534759C211yesmaximal, characteristic, complemented
2#14530346C211yesmaximal, characteristic, complemented
3#14530341C211yesmaximal, characteristic, complemented
4#14530020C211yesmaximal, characteristic
5#14530020C211yesmaximal, characteristic
6#14529484C211yesmaximal, characteristic
7#14529477C211yesmaximal, characteristic, complemented
8#14529470C211yesmaximal, characteristic
9#14529463C211yesmaximal, characteristic, complemented
10#14528799C211yesmaximal, characteristic, complemented
11#14528794C211yesmaximal, characteristic
12#14528789C211yesmaximal, characteristic, complemented
13#14528784C211yesmaximal, characteristic
14#14528641C211yesmaximal, characteristic, complemented
15C17×#2326C211yesmaximal, characteristic
16C2×C4⋊#3622C2211yescharacteristic
17C2×C4⋊#3622C2211yescharacteristic
18C2×#10587506C2211yescharacteristic
19C2×#10587506C2211yescharacteristic
20C2×#10587506C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587474C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28#11762751—21no—
29#11762751—21no—
30#11762743—21no—
31#11762743—21no—
32#11762707C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762679C2211yescharacteristic
37#11762675C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×#124C2211yescharacteristic
49C2×C4×Q8×C17C2211yescharacteristic
50C2×C4×D4×C17C2211yescharacteristic
51C17×#391C2211yescharacteristic
52C17×#390C2211yescharacteristic
53C17×#388—21no—
54C17×#388—21no—
55C17×#385—21no—
56C17×#385—21no—
57C17×#384C2211yescharacteristic
58C17×#376C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C17×#124—22no—
63C17×#124—22no—
64C4×Q8×C17—22no—
65C4×Q8×C17—22no—
66C4×D4×C17—22no—
67C4×D4×C17—22no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17C2311yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C17×#95—21no—
97C17×#95—21no—
98C17×#95D411yescharacteristic
99C17×#95D411yescharacteristic
100C17×#95D411yescharacteristic
101C17×#95D411yescharacteristic
102#10587506—22no—
103#10587506—22no—
104#10587506—41no—
105#10587506—41no—
106#10587503—22no—
107#10587503—22no—
108#10587503—41no—
109#10587503—41no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116C4⋊#3622—22no—
117C4⋊#3622—22no—
118#10587474—41no—
119C4×#3622—41no—
120C24×C17—21no—
121C2×Q8×C17—21no—
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—21no—
126C2×Q8×C17#4112yes—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17#4112yes—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—42no—
137C22×C4×C17—81no—
138C22×C4×C17—81no—
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2411yescharacteristic
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—42no—
155C4⋊C4×C17—81no—
156C4⋊C4×C17—81no—
157C22⋊C4×C17—24no—
158C22⋊C4×C17—24no—
159C22⋊C4×C17—42no—
160C22⋊C4×C17—81no—
161C22⋊C4×C17—81no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—21no—
166C42×C17—42no—
167C42×C17#4112yes—
168C42×C17#4112yes—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—81no—
178C4⋊Dic17—81no—
179#63225—81no—
180#63225—81no—
181#63225—81no—
182#63225—81no—
183C2×#3622—21no—
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—21no—
195#29134—171nomaximal, Sylow
196C23×C17—22no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17C22×D411yescharacteristic
204Q8×C17—22no—
205Q8×C17—22no—
206Q8×C17—41no—
207Q8×C17—41no—
208D4×C17—22no—
209D4×C17—22no—
210D4×C17—41no—
211D4×C17—41no—
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—22no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
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—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—84no—
238C2×C4×C17#12412yes—
239C2×C4×C17#12412yes—
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17H(5,𝔽2)11yescharacteristic
245C2×C4×C172−1+411yescharacteristic
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—
258#3622—41no—
259#3622—41no—
260#3622—41no—
261#3622—41no—
262#2647—171no—
263#2326D1711yescharacteristic
264#2100—171no—
265#2099—171no—
266#2042—171no—
267#2042—171no—
268#1936—171no—
269#1935—171no—
270#1934—171no—
271#1933—171no—
272#1805—171no—
273#1804—171no—
274#1803—171no—
275#1802—171no—
276#1768—171no—
277C22×C17—42no—
278C22×C17—42no—
279C22×C17—42no—
280C22×C17—81no—
281C22×C17—81no—
282C22×C17—81no—
283C22×C17—81no—
284C22×C17#52511yescharacteristic
285C22×C17C2×#12411yescharacteristic
286C22×C17#53711yescharacteristic
287C22×C17#57611yescharacteristic
288C22×C17#57711yescharacteristic
289C22×C17#57711yescharacteristic
290C22×C17#57811yescharacteristic
291C4×C17—22no—
292C4×C17—22no—
293C4×C17—42no—
294C4×C17—41no—
295C4×C17—41no—
296C4×C17—41no—
297C4×C17—41no—
298C4×C17—81no—
299C4×C17—81no—
300C4×C17#55211yescharacteristic
301C4×C17#55211yescharacteristic
302C4×C17#55311yescharacteristic
303C4×C17#55311yescharacteristic
304Dic17—81no—
305Dic17—81no—
306Dic17—81no—
307Dic17—81no—
308C2×#124C2×D1711yescharacteristic
309C2×C4×Q8C2×D1711yescharacteristic
310C2×C4×D4C2×D1711yescharacteristic
311C2×C4⋊C8—171no—
312C2×C4⋊C8—171no—
313C2×#103—171no—
314C2×#103—171no—
315C2×#103—171no—
316C2×#103—171no—
317C2×#102—171no—
318C2×#102—171no—
319C2×#102—171no—
320C2×#102—171no—
321C2×#97—171no—
322C2×C4×C8—171no—
323#400—341no—
324#400—341no—
325#398—341no—
326#398—341no—
327#391C2×D1711yescharacteristic
328#390C2×D1711yescharacteristic
329#389—171no—
330#389—171no—
331#389—171no—
332#389—171no—
333#388—21no—
334#388—21no—
335#385—21no—
336#385—21no—
337#384—171no—
338#384C2×D1711yescharacteristic
339#382—171no—
340#376C2×D1711yescharacteristic
341#340—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#336—171no—
350#336—171no—
351C2×C17—82no—
352C2×C17#256211yescharacteristic
353C2×C17#311911yescharacteristic
354C2×C17#312011yescharacteristic
355C2×C17#313611yescharacteristic
356C2×C17#313711yescharacteristic
357C2×C17#326211yescharacteristic
358C2×C17#326311yescharacteristic
359C22×Q8C22×D1711yescharacteristic
360C22×D4C22×D1711yescharacteristic
361C23×C4—21no—
362C22×C8—171no—
363C22×C8—171no—
364C22×C8—171no—
365C22×C8—171no—
366#124—22no—
367#124—22no—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×D4—22no—
371C4×D4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C42—21no—
393C2×C42C22×D1711yescharacteristic
394C2×C42C22×D1711yescharacteristic
395C2×C42C22×D1711yescharacteristic
396C4⋊C8—342no—
397C4⋊C8—342no—
398#103—342no—
399#103—342no—
400#103—681no—
401#103—681no—
402#102—342no—
403#102—342no—
404#102—681no—
405#102—681no—
406#97—681no—
407C4×C8—681no—
408#95—21no—
409#95—21no—
410#95—341no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95C22⋊D1711yescharacteristic
417#95C22⋊D1711yescharacteristic
418#95C22⋊D1711yescharacteristic
419#95C22⋊D1711yescharacteristic
420C17#2913411yescharacteristic, complemented, Sylow
421C24—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#6325212yes—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4#6325212yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—341no—
441C22×C4—341no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4C2×C22⋊D1711yescharacteristic
445C22×C4C2×C22⋊D1711yescharacteristic
446C22×C4C2×C22⋊D1711yescharacteristic
447C22×C4C2×C22⋊D1711yescharacteristic
448C22×C4C2×C22⋊D1711yescharacteristic
449C22×C4C2×C22⋊D1711yescharacteristic
450C22×C4C23×D1711yescharacteristic
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—341no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—1361no—
479C4⋊C4—1361no—
480C4⋊C4—1361no—
481C4⋊C4—1361no—
482C4⋊C4—1361no—
483C4⋊C4—1361no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42#6325412yes—
495C42#6325412yes—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C22×C22⋊D1711yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—41no—
507Q8—41no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
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—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—681no—
539C2×C4—681no—
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#1058762212yes—
551C2×C4#1058762212yes—
552C2×C4C2×#6325211yescharacteristic
553C2×C4C2×#6325211yescharacteristic
554C2×C4C2×#6325411yescharacteristic
555C2×C4C2×#6325411yescharacteristic
556C2×C4#1058769611yescharacteristic
557C2×C4#1058769711yescharacteristic
558C8—681no—
559C8—681no—
560C8—681no—
561C8—681no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22#1176332011yescharacteristic
570C22C2×#1058762211yescharacteristic
571C22#1176338711yescharacteristic
572C22#1176361211yescharacteristic
573C22#1176361711yescharacteristic
574C22#1176362011yescharacteristic
575C22#1176362411yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—1361no—
586C4—1361no—
587C4—1361no—
588C4—1361no—
589C4#1176349911yescharacteristic
590C4#1176349911yescharacteristic
591C4#1176350511yescharacteristic
592C4#1176350511yescharacteristic
593C2—82no—
594C2#1453395011yescharacteristic
595C2#1453898311yescharacteristic
596C2#1453899811yescharacteristic
597C2#1453913811yescharacteristic
598C2#1453914911yescharacteristic
599C2#1454024011yescharacteristic
600C2#1454024911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2326C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2326D17
hypocenterC17#29134
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. #14534759C2semidirect
2. #14530346C2semidirect
3. #14530341C2semidirect
7. #14529477C2semidirect
9. #14529463C2semidirect
10. #14528799C2semidirect
12. #14528789C2semidirect
14. #14528641C2semidirect
420. C17#29134semidirect

Maximal subgroups

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

subgroupindex
#145347592
#145303462
#145303412
#145300202
#145294842
#145294772
#145294702
#145294632
#145287992
#145287942
#145287892
#145287842
#145286412
C17×#23262
#2913417