Finite Group Databasebeta

← #17137418

Subgroups — #17137418

Order 4352.

746 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#14537166C211yesmaximal, characteristic, complemented
2#14530286C211yesmaximal, characteristic, complemented
3#14530176C211yesmaximal, characteristic, complemented
4#14530172C211yesmaximal, characteristic
5#14530101C211yesmaximal, characteristic
6#14529637C211yesmaximal, characteristic
7#14529514C211yesmaximal, characteristic, complemented
8#14529470C211yesmaximal, characteristic
9#14529459C211yesmaximal, characteristic, complemented
10#14529425C211yesmaximal, characteristic, complemented
11#14529362C211yesmaximal, characteristic, complemented
12#14529118C211yesmaximal, characteristic
13#14529031C211yesmaximal, characteristic
14C4×C4⋊#3622C211yesmaximal, characteristic, complemented
15C17×#2293C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×#10587478C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587481C2211yescharacteristic
23C2×#10587481C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×C4×#3622C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30#11762715C2211yescharacteristic
31#11762715C2211yescharacteristic
32#11762715C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762693—21no—
37#11762693—21no—
38#11762693—21no—
39#11762677—21no—
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C4×D4×C17C2211yescharacteristic
49C2×C17×#117—21no—
50C17×#390C2211yescharacteristic
51C17×#390C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#388—21no—
55C17×#388—21no—
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C22⋊C4×C17—21no—
58C43×C17C2211yescharacteristic
59C22×D4×C17C2311yescharacteristic
60C23×C4×C17—21no—
61C22×Dic34C2311yescharacteristic
62C4×D4×C17—22no—
63C4×D4×C17—22no—
64C17×#117—44no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C22⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17D411yescharacteristic
77C2×C42×C17D411yescharacteristic
78C2×C42×C17D411yescharacteristic
79C2×C42×C17D411yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×#63229—22no—
84C2×#63229—22no—
85C2×#63229—41no—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C17×#95—21no—
97C17×#95—21no—
98C17×#95—21no—
99#10587503—22no—
100#10587503—22no—
101#10587503—41no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587481—22no—
112#10587481—22no—
113#10587481—41no—
114#10587479—22no—
115#10587478—22no—
116#10587478—41no—
117C4⋊#3622—22no—
118C4⋊#3622—22no—
119C4×#3622—41no—
120C4×#3622—41no—
121C24×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17#4112yes—
128C22×C4×C17—24no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—42no—
133C22×C4×C17—42no—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C2×Dic34—21no—
146C2×Dic34—21no—
147C2×Dic34—21no—
148C2×Dic34—21no—
149C2×Dic34—21no—
150C2×Dic34—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—42no—
157C4⋊C4×C17—42no—
158C4⋊C4×C17#4112yes—
159C22⋊C4×C17—24no—
160C22⋊C4×C17—42no—
161C22⋊C4×C17—42no—
162C22⋊C4×C17—42no—
163C22⋊C4×C17—42no—
164C42×C17—22no—
165C42×C17—22no—
166C42×C17—22no—
167C42×C17—22no—
168C42×C17—21no—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—42no—
175#63229—44no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182#63225—42no—
183#63225—42no—
184#63225—42no—
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—
195C2×#3622#4112yes—
196C2×#3622#4112yes—
197#29433—171nomaximal, Sylow
198C23×C17—22no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17C22×D411yescharacteristic
206D4×C17—22no—
207D4×C17—22no—
208D4×C17—41no—
209D4×C17—41no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
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—21no—
219C2×C4×C17—22no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—48no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17#13512yes—
235C2×C4×C17#13512yes—
236C2×C4×C17AGL(1,ℤ/8)12yes—
237C2×C4×C17#13712yes—
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17H(5,𝔽2)11yescharacteristic
243C2×C4×C172−1+411yescharacteristic
244C2×Dic17—41no—
245C2×Dic17—41no—
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—
256Dic34—41no—
257Dic34—41no—
258Dic34—41no—
259Dic34—41no—
260#3622—22no—
261#3622—22no—
262#3622—41no—
263#3622—41no—
264#2863—171no—
265#2293D1711yescharacteristic
266#2088—171no—
267#2068—171no—
268#2067—171no—
269#2055—171no—
270#1968—171no—
271#1942—171no—
272#1934—171no—
273#1933—171no—
274#1928—171no—
275#1919—171no—
276#1877—171no—
277#1862—171no—
278C4×C4⋊C8—171no—
279C22×C17—42no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—81no—
283C22×C17—81no—
284C22×C17—81no—
285C22×C17—81no—
286C22×C17#52511yescharacteristic
287C22×C17#53211yescharacteristic
288C22×C17#53711yescharacteristic
289C22×C17C2×#13511yescharacteristic
290C22×C17C2×#13511yescharacteristic
291C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C17C2×#13711yescharacteristic
293C4×C17—22no—
294C4×C17—22no—
295C4×C17—22no—
296C4×C17—22no—
297C4×C17—42no—
298C4×C17—41no—
299C4×C17—41no—
300C4×C17—81no—
301C4×C17—81no—
302C4×C17#55111yescharacteristic
303C4×C17#55211yescharacteristic
304C4×C17#55311yescharacteristic
305C4×C17#55611yescharacteristic
306Dic17—81no—
307Dic17—81no—
308Dic17—81no—
309Dic17—81no—
310C22×SD16—171no—
311C2×C4×D4C2×D1711yescharacteristic
312C2×#117—21no—
313C2×#107—171no—
314C2×#107—171no—
315C2×#106—171no—
316C2×C4⋊C8—171no—
317C2×C4⋊C8—171no—
318C2×#103—171no—
319C2×#103—171no—
320C2×#103—171no—
321C2×#102—171no—
322C2×#102—171no—
323C2×#102—171no—
324C2×C4×C8—171no—
325C2×C4×C8—171no—
326#391—171no—
327#391—171no—
328#391—171no—
329#390C2×D1711yescharacteristic
330#390C2×D1711yescharacteristic
331#389—171no—
332#389—171no—
333#389—171no—
334#389C2×D1711yescharacteristic
335#389C2×D1711yescharacteristic
336#388—21no—
337#388—21no—
338#387—341no—
339#387—341no—
340#387—341no—
341#383—341no—
342C4×C4⋊C4C2×D1711yescharacteristic
343C4×C22⋊C4—21no—
344C43C2×D1711yescharacteristic
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—82no—
354C2×C17#260111yescharacteristic
355C2×C17#310511yescharacteristic
356C2×C17#310711yescharacteristic
357C2×C17#316811yescharacteristic
358C2×C17#318211yescharacteristic
359C2×C17#324711yescharacteristic
360C2×C17#324811yescharacteristic
361C22×Q8—171no—
362C22×D4C22×D1711yescharacteristic
363C23×C4—21no—
364C2×SD16—342no—
365C2×SD16—342no—
366C2×SD16—681no—
367C22×C8—171no—
368C22×C8—171no—
369C22×C8—171no—
370C22×C8—171no—
371C4×D4—22no—
372C4×D4—22no—
373#117—44no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
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⋊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×C42C22⋊D1711yescharacteristic
392C2×C42C22⋊D1711yescharacteristic
393C2×C42C22⋊D1711yescharacteristic
394C2×C42C22⋊D1711yescharacteristic
395C2×C42C22×D1711yescharacteristic
396C2×C42C22×D1711yescharacteristic
397C2×C42C22×D1711yescharacteristic
398#107—342no—
399#107—681no—
400#106—342no—
401C4⋊C8—342no—
402C4⋊C8—342no—
403#103—342no—
404#103—342no—
405#103—681no—
406#102—342no—
407#102—342no—
408#102—681no—
409C4×C8—681no—
410C4×C8—681no—
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—
422#95—341no—
423C17#2943311yescharacteristic, complemented, Sylow
424C24—21no—
425C2×Q8—341no—
426C2×Q8—341no—
427C2×Q8—341no—
428C2×Q8—341no—
429C2×Q8—341no—
430C2×Q8—341no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4#6325212yes—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4C2×C22⋊D1711yescharacteristic
448C22×C4C2×C22⋊D1711yescharacteristic
449C22×C4C2×C22⋊D1711yescharacteristic
450C22×C4C2×C22⋊D1711yescharacteristic
451C22×C4C2×C22⋊D1711yescharacteristic
452C22×C4C2×C22⋊D1711yescharacteristic
453C22×C4C23×D1711yescharacteristic
454SD16—684no—
455C2×C8—172no—
456C2×C8—172no—
457C2×C8—341no—
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—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—682no—
481C4⋊C4—682no—
482C4⋊C4—682no—
483C4⋊C4#6325212yes—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—42no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C22×C22⋊D1711yescharacteristic
508Q8—681no—
509Q8—681no—
510Q8—681no—
511Q8—681no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
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—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—41no—
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#1058761812yes—
553C2×C4#1058763612yes—
554C2×C4#1058763612yes—
555C2×C4#1058763712yes—
556C2×C4C2×#6325211yescharacteristic
557C2×C4C2×#6325211yescharacteristic
558C2×C4#41×D1711yescharacteristic
559C2×C4#41×D1711yescharacteristic
560C2×C4#1058769611yescharacteristic
561C2×C4#1058769711yescharacteristic
562C8—342no—
563C8—342no—
564C8—681no—
565C8—681no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22#1176332011yescharacteristic
574C22#1176335511yescharacteristic
575C22#1176338711yescharacteristic
576C22C2×#1058763611yescharacteristic
577C22C2×#1058763611yescharacteristic
578C22C2×#1058761811yescharacteristic
579C22C2×#1058763711yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—1361no—
590C4—1361no—
591C4—1361no—
592C4—1361no—
593C4#1176348311yescharacteristic
594C4#1176349211yescharacteristic
595C4#1176350311yescharacteristic
596C4#1176352711yescharacteristic
597C2—82no—
598C2#1453438111yescharacteristic
599C2#1453883111yescharacteristic
600C2#1453885511yescharacteristic
601C2#1453934811yescharacteristic
602C2#1453949811yescharacteristic
603C2#1454010411yescharacteristic
604C2#1454012111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2293C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2293D17
hypocenterC17#29433
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. #14537166C2semidirect
2. #14530286C2semidirect
3. #14530176C2semidirect
7. #14529514C2semidirect
9. #14529459C2semidirect
10. #14529425C2semidirect
11. #14529362C2semidirect
14. C4×C4⋊#3622C2semidirect
423. C17#29433semidirect

Maximal subgroups

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

subgroupindex
#145371662
#145302862
#145301762
#145301722
#145301012
#145296372
#145295142
#145294702
#145294592
#145294252
#145293622
#145291182
#145290312
C4×C4⋊#36222
C17×#22932
#2943317