Finite Group Databasebeta

← #17102949

Subgroups — #17102949

Order 4352.

704 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#14533322C211yesmaximal, characteristic, complemented
2#14530197C211yesmaximal, characteristic
3#14530158C211yesmaximal, characteristic
4#14530020C211yesmaximal, characteristic
5#14530020C211yesmaximal, characteristic
6#14529463C211yesmaximal, characteristic, complemented
7#14529415C211yesmaximal, characteristic, complemented
8#14529369C211yesmaximal, characteristic, complemented
9#14529362C211yesmaximal, characteristic, complemented
10#14529343C211yesmaximal, characteristic, complemented
11#14529052C211yesmaximal, characteristic, complemented
12#14529045C211yesmaximal, characteristic
13#14528799C211yesmaximal, characteristic, complemented
14C4×#10587503C211yesmaximal, characteristic
15C17×#2367C211yesmaximal, characteristic
16C2×#10587479C2211yescharacteristic
17C2×#10587479C2211yescharacteristic
18C2×C4⋊#3622C2211yescharacteristic
19C2×#10587506C2211yescharacteristic
20C2×#10587506C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28#11762750—21no—
29#11762740—21no—
30#11762707C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762690—21no—
35#11762675C2211yescharacteristic
36#11762673—21no—
37C4×C4⋊Dic17C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762564C2211yescharacteristic
45#11762564C2211yescharacteristic
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—
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17—21no—
59C22×Q8×C17C2311yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C17×#124—22no—
63C17×#124—22no—
64C4×Q8×C17—22no—
65C4×Q8×C17—41no—
66C4×D4×C17—22no—
67C4×D4×C17—41no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17D411yescharacteristic
70C2×C4⋊C4×C17D411yescharacteristic
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×C17—21no—
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×C4×Dic17—21no—
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C17×#95—21no—
97C17×#95—21no—
98C17×#95—21no—
99C17×#95D411yescharacteristic
100C17×#95D411yescharacteristic
101#10587506—22no—
102#10587506—22no—
103#10587506—22no—
104#10587506—41no—
105#10587503—22no—
106#10587503—22no—
107#10587503—22no—
108#10587503—41no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587479—22no—
117#10587479—22no—
118C4⋊#3622—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—42no—
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—22no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—41no—
155C4⋊C4×C17—42no—
156C4⋊C4×C17#4112yes—
157C22⋊C4×C17—24no—
158C22⋊C4×C17—24no—
159C22⋊C4×C17—42no—
160C22⋊C4×C17—42no—
161C22⋊C4×C17—81no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—21no—
166C42×C17—21no—
167C42×C17—21no—
168C42×C17—42no—
169C42×C17—81no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17#4112yes—
178#63225—42no—
179#63225—81no—
180#63225—81no—
181#63225—81no—
182C4×Dic17—42no—
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#26760—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—22no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—22no—
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—21no—
230C2×C4×C17—21no—
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—84no—
239C2×C4×C17#13512yes—
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—22no—
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#2508—171no—
263#2367D1711yescharacteristic
264#2073—171no—
265#2065—171no—
266#2042—171no—
267#2042—171no—
268#1933—171no—
269#1927—171no—
270#1920—171no—
271#1919—171no—
272#1916—171no—
273#1865—171no—
274#1864—171no—
275#1805—171no—
276C4×#102—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×C17#54611yescharacteristic
286C22×C17#54811yescharacteristic
287C22×C17C2×#13511yescharacteristic
288C22×C17#57511yescharacteristic
289C22×C17#57711yescharacteristic
290C22×C17#57711yescharacteristic
291C4×C17—22no—
292C4×C17—41no—
293C4×C17—41no—
294C4×C17—42no—
295C4×C17—41no—
296C4×C17—42no—
297C4×C17—41no—
298C4×C17—41no—
299C4×C17—81no—
300C4×C17#52911yescharacteristic
301C4×C17#53911yescharacteristic
302C4×C17#55211yescharacteristic
303C4×C17#55311yescharacteristic
304Dic17—42no—
305Dic17—81no—
306Dic17—81no—
307Dic17—81no—
308C2×#124C2×D1711yescharacteristic
309C2×C4×Q8C2×D1711yescharacteristic
310C2×C4×D4C2×D1711yescharacteristic
311C2×#106—171no—
312C2×#106—171no—
313C2×C4⋊C8—171no—
314C2×#103—171no—
315C2×#103—171no—
316C2×#103—171no—
317C2×#103—171no—
318C2×#102—171no—
319C2×#102—171no—
320C2×#102—171no—
321C2×#102—171no—
322C2×C4×C8—171no—
323#400—341no—
324#398—341no—
325#391C2×D1711yescharacteristic
326#390C2×D1711yescharacteristic
327#389—171no—
328#389—171no—
329#389—171no—
330#389—171no—
331#388—21no—
332#388—21no—
333#387—341no—
334#385—21no—
335#382—171no—
336#382—341no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4C2×D1711yescharacteristic
339C4×C4⋊C4C2×D1711yescharacteristic
340C4×C22⋊C4—21no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#336—171no—
348#336—171no—
349#336—171no—
350#336—171no—
351C2×C17—82no—
352C2×C17#246011yescharacteristic
353C2×C17#310211yescharacteristic
354C2×C17#312511yescharacteristic
355C2×C17#329611yescharacteristic
356C2×C17#332111yescharacteristic
357C2×C17#336011yescharacteristic
358C2×C17#337111yescharacteristic
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—41no—
370C4×D4—22no—
371C4×D4—41no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4C22⋊D1711yescharacteristic
382C2×C4⋊C4C22⋊D1711yescharacteristic
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—341no—
393C2×C42C22×D1711yescharacteristic
394C2×C42C22×D1711yescharacteristic
395C2×C42C22×D1711yescharacteristic
396#106—342no—
397#106—342no—
398C4⋊C8—681no—
399#103—342no—
400#103—342no—
401#103—342no—
402#103—681no—
403#102—342no—
404#102—342no—
405#102—342no—
406#102—681no—
407C4×C8—681no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95C22⋊D1711yescharacteristic
419#95C22⋊D1711yescharacteristic
420C17#2676011yescharacteristic, 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—42no—
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—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—172no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—682no—
478C4⋊C4—1361no—
479C4⋊C4—1361no—
480C4⋊C4—1361no—
481C4⋊C4#6325412yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42—81no—
495C42—682no—
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—22no—
517C2×C4—21no—
518C2×C4—21no—
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—21no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—84no—
539C2×C4—342no—
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#1058763612yes—
552C2×C4C2×#6325211yescharacteristic
553C2×C4C2×#6325211yescharacteristic
554C2×C4C2×#6325411yescharacteristic
555C2×C4#1058769111yescharacteristic
556C2×C4#41×D1711yescharacteristic
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
570C22#1176345311yescharacteristic
571C22#1176346611yescharacteristic
572C22C2×#1058763611yescharacteristic
573C22#1176360711yescharacteristic
574C22#1176361711yescharacteristic
575C22#1176362011yescharacteristic
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—682no—
586C4—1361no—
587C4—1361no—
588C4—1361no—
589C4#1176333911yescharacteristic
590C4#1176340111yescharacteristic
591C4#1176349911yescharacteristic
592C4#1176350811yescharacteristic
593C2—82no—
594C2#1453273011yescharacteristic
595C2#1453880211yescharacteristic
596C2#1453906111yescharacteristic
597C2#1454061911yescharacteristic
598C2#1454097011yescharacteristic
599C2#1454142911yescharacteristic
600C2#1454154611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2367C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2367D17
hypocenterC17#26760
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #14533322C2semidirect
6. #14529463C2semidirect
7. #14529415C2semidirect
8. #14529369C2semidirect
9. #14529362C2semidirect
10. #14529343C2semidirect
11. #14529052C2semidirect
13. #14528799C2semidirect
420. C17#26760semidirect

Maximal subgroups

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

subgroupindex
#145333222
#145301972
#145301582
#145300202
#145294632
#145294152
#145293692
#145293622
#145293432
#145290522
#145290452
#145287992
C4×#105875032
C17×#23672
#2676017