Finite Group Databasebeta

← #17107575

Subgroups — #17107575

Order 4352.

740 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533392C211yesmaximal, characteristic, complemented
2#14530305C211yesmaximal, characteristic
3#14530101C211yesmaximal, characteristic
4#14530020C211yesmaximal, characteristic
5#14529878C211yesmaximal, characteristic
6#14529644C211yesmaximal, characteristic, complemented
7#14529418C211yesmaximal, characteristic, complemented
8#14529415C211yesmaximal, characteristic, complemented
9#14529376C211yesmaximal, characteristic, complemented
10#14529343C211yesmaximal, characteristic, complemented
11#14529125C211yesmaximal, characteristic, complemented
12#14529118C211yesmaximal, characteristic
13#14528799C211yesmaximal, characteristic, complemented
14C4×#10587503C211yesmaximal, characteristic
15C17×#2367C211yesmaximal, characteristic
16C22×#63231C2211yescharacteristic
17C22×#63229C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30C17⋊#401—21no—
31#11762747—21no—
32#11762715C2211yescharacteristic
33#11762715C2211yescharacteristic
34#11762715C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762690—21no—
37#11762676—21no—
38#11762675C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
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—
62C22×Dic34C2311yescharacteristic
63C17×#124—22no—
64C17×#124—22no—
65C4×Q8×C17—22no—
66C4×Q8×C17—41no—
67C4×D4×C17—22no—
68C4×D4×C17—41no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17D411yescharacteristic
71C2×C4⋊C4×C17D411yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C42×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×#63231—22no—
85C2×#63231—22no—
86C2×#63231—41no—
87C2×#63229—22no—
88C2×#63229—41no—
89C2×#63229—41no—
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95D411yescharacteristic
103C17×#95D411yescharacteristic
104#10587506—22no—
105#10587506—41no—
106#10587506—41no—
107#10587503—22no—
108#10587503—22no—
109#10587503—41no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587481—22no—
121#10587481—41no—
122#10587479—41no—
123#10587478—22no—
124C4⋊#3622—41no—
125C4×#3622—41no—
126C24×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17#4112yes—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17#4112yes—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17—81no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2411yescharacteristic
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34—21no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—41no—
167C4⋊C4×C17—42no—
168C4⋊C4×C17#4112yes—
169C22⋊C4×C17—24no—
170C22⋊C4×C17—24no—
171C22⋊C4×C17—42no—
172C22⋊C4×C17—42no—
173C22⋊C4×C17—81no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17—21no—
180C42×C17—42no—
181C42×C17—81no—
182#63231—44no—
183#63229—44no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190#63225—42no—
191C4×Dic17—42no—
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×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622#4112yes—
204#27070—171nomaximal, Sylow
205C23×C17—22no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17—41no—
211C23×C17—41no—
212C23×C17C22×D411yescharacteristic
213Q8×C17—22no—
214Q8×C17—22no—
215Q8×C17—41no—
216Q8×C17—41no—
217D4×C17—22no—
218D4×C17—22no—
219D4×C17—41no—
220D4×C17—41no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—22no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—22no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—84no—
248C2×C4×C17#13512yes—
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C17H(5,𝔽2)11yescharacteristic
254C2×C4×C17H(5,𝔽2)11yescharacteristic
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267Dic34—41no—
268Dic34—41no—
269Dic34—41no—
270Dic34—41no—
271#3622—22no—
272#3622—41no—
273#3622—41no—
274#3622—41no—
275#2514—171no—
276#2367D1711yescharacteristic
277#2092—171no—
278#2055—171no—
279#2042—171no—
280#2015—171no—
281#1969—171no—
282#1927—171no—
283#1927—171no—
284#1921—171no—
285#1916—171no—
286#1878—171no—
287#1877—171no—
288#1805—171no—
289C4×#102—171no—
290C22×C17—42no—
291C22×C17—42no—
292C22×C17—42no—
293C22×C17—81no—
294C22×C17—81no—
295C22×C17—81no—
296C22×C17—81no—
297C22×C17#52511yescharacteristic
298C22×C17#54611yescharacteristic
299C22×C17#55011yescharacteristic
300C22×C17C2×#13511yescharacteristic
301C22×C17#57511yescharacteristic
302C22×C17#57711yescharacteristic
303C22×C17#57811yescharacteristic
304C4×C17—22no—
305C4×C17—41no—
306C4×C17—41no—
307C4×C17—42no—
308C4×C17—41no—
309C4×C17—42no—
310C4×C17—41no—
311C4×C17—41no—
312C4×C17—81no—
313C4×C17#52911yescharacteristic
314C4×C17#53811yescharacteristic
315C4×C17#55111yescharacteristic
316C4×C17#55311yescharacteristic
317Dic17—81no—
318Dic17—81no—
319Dic17—81no—
320Dic17—81no—
321C22×Q16—171no—
322C22×SD16—171no—
323C2×#124C2×D1711yescharacteristic
324C2×C4×Q8C2×D1711yescharacteristic
325C2×C4×D4C2×D1711yescharacteristic
326C2×#107—171no—
327C2×#106—171no—
328C2×C4⋊C8—171no—
329C2×#103—171no—
330C2×#103—171no—
331C2×#103—171no—
332C2×#103—171no—
333C2×#103—171no—
334C2×#102—171no—
335C2×#102—171no—
336C2×#102—171no—
337C2×C4×C8—171no—
338#401—341no—
339#400—341no—
340#391—171no—
341#391—171no—
342#391—171no—
343#391C2×D1711yescharacteristic
344#390C2×D1711yescharacteristic
345#389—171no—
346#388—21no—
347#388—21no—
348#387—341no—
349#385—21no—
350#383—341no—
351#382—171no—
352C4×C4⋊C4—171no—
353C4×C4⋊C4C2×D1711yescharacteristic
354C4×C4⋊C4C2×D1711yescharacteristic
355C4×C22⋊C4—21no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#336—171no—
361#336—171no—
362#336—171no—
363#336—171no—
364C2×C17—82no—
365C2×C17#246711yescharacteristic
366C2×C17#310211yescharacteristic
367C2×C17#312011yescharacteristic
368C2×C17#329611yescharacteristic
369C2×C17#331311yescharacteristic
370C2×C17#339311yescharacteristic
371C2×C17#340111yescharacteristic
372C22×Q8—171no—
373C22×Q8C22×D1711yescharacteristic
374C22×D4C22×D1711yescharacteristic
375C23×C4—21no—
376C2×Q16—342no—
377C2×Q16—342no—
378C2×Q16—681no—
379C2×SD16—342no—
380C2×SD16—681no—
381C2×SD16—681no—
382C22×C8—171no—
383C22×C8—171no—
384C22×C8—171no—
385C22×C8—171no—
386#124—22no—
387#124—22no—
388C4×Q8—22no—
389C4×Q8—41no—
390C4×D4—22no—
391C4×D4—41no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4C22⋊D1711yescharacteristic
398C2×C4⋊C4C22⋊D1711yescharacteristic
399C2×C4⋊C4C22×D1711yescharacteristic
400C2×C4⋊C4C22×D1711yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—341no—
409C2×C42C22×D1711yescharacteristic
410C2×C42C22×D1711yescharacteristic
411C2×C42C22×D1711yescharacteristic
412#107—342no—
413#106—681no—
414C4⋊C8—681no—
415#103—342no—
416#103—342no—
417#103—681no—
418#103—681no—
419#103—681no—
420#102—342no—
421#102—342no—
422#102—681no—
423C4×C8—681no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95C22⋊D1711yescharacteristic
438#95C22⋊D1711yescharacteristic
439C17#2707011yescharacteristic, complemented, Sylow
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—341no—
447C2×Q8—341no—
448C2×Q8—341no—
449C2×Q8—341no—
450C2×Q8—341no—
451C2×Q8—341no—
452C2×Q8#6325212yes—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4#6325212yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—341no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4C2×C22⋊D1711yescharacteristic
470C22×C4C2×C22⋊D1711yescharacteristic
471C22×C4C2×C22⋊D1711yescharacteristic
472C22×C4C2×C22⋊D1711yescharacteristic
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C23×D1711yescharacteristic
476Q16—684no—
477SD16—684no—
478C2×C8—172no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C2×C8—341no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—682no—
503C4⋊C4#6325412yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—81no—
517C42—682no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×C22⋊D1711yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—681no—
531Q8—681no—
532Q8—681no—
533Q8—681no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—84no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4#1058763612yes—
578C2×C4C2×#6325211yescharacteristic
579C2×C4C2×#6325211yescharacteristic
580C2×C4C2×#6325411yescharacteristic
581C2×C4#1058769111yescharacteristic
582C2×C4#41×D1711yescharacteristic
583C2×C4#1058769611yescharacteristic
584C8—342no—
585C8—681no—
586C8—681no—
587C8—681no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22#1176332011yescharacteristic
596C22#1176345311yescharacteristic
597C22#1176347811yescharacteristic
598C22C2×#1058763611yescharacteristic
599C22#1176360711yescharacteristic
600C22#1176362011yescharacteristic
601C22#1176362411yescharacteristic
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—1361no—
612C4—1361no—
613C4—1361no—
614C4—1361no—
615C4#1176333911yescharacteristic
616C4#1176339211yescharacteristic
617C4#1176348811yescharacteristic
618C4#1176350511yescharacteristic
619C2—82no—
620C2#1453282911yescharacteristic
621C2#1453880211yescharacteristic
622C2#1453899811yescharacteristic
623C2#1454061911yescharacteristic
624C2#1454085011yescharacteristic
625C2#1454176811yescharacteristic
626C2#1454184411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2367C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2367D17
hypocenterC17#27070
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. #14533392C2semidirect
6. #14529644C2semidirect
7. #14529418C2semidirect
8. #14529415C2semidirect
9. #14529376C2semidirect
10. #14529343C2semidirect
11. #14529125C2semidirect
13. #14528799C2semidirect
439. C17#27070semidirect

Maximal subgroups

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

subgroupindex
#145333922
#145303052
#145301012
#145300202
#145298782
#145296442
#145294182
#145294152
#145293762
#145293432
#145291252
#145291182
#145287992
C4×#105875032
C17×#23672
#2707017