Finite Group Databasebeta

← #17052091

Subgroups — #17052091

Order 4352.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762852C211yesmaximal, characteristic
2#14531659C211yesmaximal, characteristic
3#14529524C211yesmaximal, characteristic, complemented
4#14529509C211yesmaximal, characteristic, complemented
5#14529376C211yesmaximal, characteristic, complemented
6#14529366C211yesmaximal, characteristic, complemented
7#14529359C211yesmaximal, characteristic, complemented
8#14529355C211yesmaximal, characteristic, complemented
9#14528879C211yesmaximal, characteristic
10#14528868C211yesmaximal, characteristic
11#14528815C211yesmaximal, characteristic
12#14528682C211yesmaximal, characteristic, complemented
13C4×#10587478C211yesmaximal, characteristic, complemented
14#14528568C211yesmaximal, characteristic
15C17×#2347C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C22×C4⋊Dic17C2211yescharacteristic
18#11762852C418yescomplemented
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×#10587474C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28#11762707C2211yescharacteristic
29#11762707C2211yescharacteristic
30#11762692—21no—
31#11762692—21no—
32#11762669C2211yescharacteristic
33#11762667—21no—
34C4×C4⋊Dic17C2211yescharacteristic
35C4×C4⋊Dic17C2211yescharacteristic
36C22⋊C4×Dic17—21no—
37#11762571C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C17×#117C2211yescharacteristic
50C17×#389C2211yescharacteristic
51C17×#389C2211yescharacteristic
52C17×#388—21no—
53C17×#388—21no—
54C17×#383—21no—
55C17×#380C2211yescharacteristic
56C4×C4⋊C4×C17—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C4×C22⋊C4×C17C2211yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×Dic17—21no—
62C17×#117—22no—
63C17×#117C2×C414yes—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17C2311yescharacteristic
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×C17D411yescharacteristic
73C2×C22⋊C4×C17D411yescharacteristic
74C2×C42×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17—21no—
77C2×C42×C17C2311yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×C22⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—22no—
82C2×C4⋊Dic17C2×C414yes—
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×#63225—21no—
88C2×#63225—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×#63223—22no—
92C2×#63223—22no—
93C2×#63223C2×C414yes—
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95D411yescharacteristic
104C17×#95D411yescharacteristic
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110C22⋊#3622—22no—
111#10587479—22no—
112#10587479C2×C414yes—
113#10587479C2×C414yes—
114#10587478—22no—
115#10587478C2×C414yes—
116#10587478C2×C414yes—
117#10587474—22no—
118C4×#3622—22no—
119C24×C17C2×D411yescharacteristic
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17C22×C412yes—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17#4112yes—
135C22×C4×C17C2411yescharacteristic
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—24no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—81no—
142C22×Dic17—81no—
143C4⋊C4×C17—21no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—21no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—42no—
150C4⋊C4×C17—42no—
151C4⋊C4×C17C22×C412yes—
152C4⋊C4×C17C22×C412yes—
153C22⋊C4×C17—24no—
154C22⋊C4×C17—22no—
155C22⋊C4×C17—41no—
156C22⋊C4×C17—81no—
157C42×C17—24no—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—21no—
161C42×C17—21no—
162C42×C17—81no—
163C42×C17—81no—
164C22⋊Dic17—42no—
165C4⋊Dic17—24no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17C22×C412yes—
173C4⋊Dic17C22×C412yes—
174#63225—81no—
175#63225—81no—
176C4×Dic17—81no—
177C4×Dic17—81no—
178#63223—28no—
179C2×#3622—21no—
180C2×#3622—21no—
181C2×#3622—21no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622C22×C412yes—
188C2×#3622C22×C412yes—
189C2×#3622#4112yes—
190C2×#3622#4112yes—
191#22623—171nomaximal, Sylow
192C23×C17—21no—
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17C4×D412yes—
199C23×C17C22×D411yescharacteristic
200C2×C4×C17—21no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—22no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
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—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
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×C17C4×D412yes—
230C2×C4×C17#13712yes—
231C2×C4×C17C23×C411yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C172−1+411yescharacteristic
236C2×C4×C172−1+411yescharacteristic
237C2×Dic17—22no—
238C2×Dic17—22no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—84no—
250#3622—22no—
251#3622—41no—
252#3622—41no—
253#3622#11714yes—
254C2×#428—171no—
255#2347D1711yescharacteristic
256#2345—171no—
257#1944—171no—
258#1941—171no—
259#1921—171no—
260#1920—171no—
261#1919—171no—
262#1918—171no—
263#1825—171no—
264#1823—171no—
265#1809—171no—
266#1776—171no—
267C4×#107—171no—
268#1753—171no—
269C22×C17—22no—
270C22×C17—22no—
271C22×C17—22no—
272C22×C17—41no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17C2×C4×D411yescharacteristic
277C22×C17#54711yescharacteristic
278C22×C17#54711yescharacteristic
279C22×C17C2×#13711yescharacteristic
280C22×C17#57511yescharacteristic
281C22×C17#57611yescharacteristic
282C22×C17#57711yescharacteristic
283C4×C17—22no—
284C4×C17—22no—
285C4×C17—41no—
286C4×C17—41no—
287C4×C17—41no—
288C4×C17—42no—
289C4×C17—42no—
290C4×C17—81no—
291C4×C17—81no—
292C4×C17C2×#11711yescharacteristic
293C4×C17#51911yescharacteristic
294C4×C17#55211yescharacteristic
295C4×C17#55211yescharacteristic
296Dic17—42no—
297Dic17—42no—
298Dic17—81no—
299Dic17—81no—
300C22×M16—171no—
301C2×#117C2×D1711yescharacteristic
302C22×C4⋊C4—171no—
303#428—178no—
304C2×#107—171no—
305C2×#107—171no—
306C2×#107—171no—
307C2×#106—171no—
308C2×#106—171no—
309C2×#106—171no—
310C2×C22⋊C8—171no—
311C2×#97—171no—
312C2×C4×C8—171no—
313#389—171no—
314#389—171no—
315#389C2×D1711yescharacteristic
316#389C2×D1711yescharacteristic
317#388—21no—
318#388—21no—
319#387—341no—
320#387—341no—
321#383—21no—
322#380—171no—
323#380—341no—
324#380C2×D1711yescharacteristic
325C4×C4⋊C4—21no—
326C4×C4⋊C4—171no—
327C4×C4⋊C4—171no—
328C4×C4⋊C4C2×D1711yescharacteristic
329C4×C4⋊C4C2×D1711yescharacteristic
330C4×C22⋊C4—341no—
331C4×C22⋊C4C2×D1711yescharacteristic
332#340—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#336—171no—
343#336—171no—
344C2×C17—42no—
345C2×C17#234411yescharacteristic
346C2×C17C4×#13711yescharacteristic
347C2×C17#294611yescharacteristic
348C2×C17#334611yescharacteristic
349C2×C17#335011yescharacteristic
350C2×C17#335511yescharacteristic
351C2×C17#335611yescharacteristic
352C23×C4—341no—
353C23×C4C22×D1711yescharacteristic
354C2×M16—174no—
355C2×M16—342no—
356C2×M16—342no—
357C22×C8—171no—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8—171no—
361#117—22no—
362#117C4×D1714yes—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—174no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—342no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4C22×D1711yescharacteristic
375C2×C4⋊C4C22×D1711yescharacteristic
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—341no—
381C2×C22⋊C4C22⋊D1711yescharacteristic
382C2×C22⋊C4C22⋊D1711yescharacteristic
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—341no—
387C2×C42—341no—
388C2×C42C22×D1711yescharacteristic
389C2×C42C22×D1711yescharacteristic
390#107—174no—
391#107—174no—
392#107—342no—
393#106—174no—
394#106—174no—
395#106—342no—
396C22⋊C8—342no—
397#97—342no—
398C4×C8—342no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95C22⋊D1711yescharacteristic
410#95C22⋊D1711yescharacteristic
411C17#2262311yescharacteristic, complemented, Sylow
412C24C2×C22⋊D1711yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—341no—
422C22×C4—341no—
423C22×C4—344no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—1361no—
427C22×C4—1361no—
428C22×C4C2×C4×D1712yes—
429C22×C4#6325212yes—
430C22×C4C2×C22⋊D1711yescharacteristic
431C22×C4C2×C22⋊D1711yescharacteristic
432C22×C4C2×C22⋊D1711yescharacteristic
433C22×C4C2×C22⋊D1711yescharacteristic
434C22×C4C2×C22⋊D1711yescharacteristic
435C22×C4C23×D1711yescharacteristic
436M16—348no—
437C2×C8—172no—
438C2×C8—172no—
439C2×C8—172no—
440C2×C8—172no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—172no—
458C4⋊C4—172no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—344no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—1361no—
467C4⋊C4—1361no—
468C4⋊C4C2×C4×D1712yes—
469C4⋊C4C2×C4×D1712yes—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—41no—
473C22⋊C4—81no—
474C22⋊C4—682no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—24no—
480C42—81no—
481C42—81no—
482C42—1361no—
483C42—1361no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23C4×C22⋊D1712yes—
491C23C22×C22⋊D1711yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—342no—
522C2×C4—342no—
523C2×C4—681no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—1364no—
534C2×C4C4×C22⋊D1712yes—
535C2×C4#1058763712yes—
536C2×C4C22×C4×D1711yescharacteristic
537C2×C4C2×#6325211yescharacteristic
538C2×C4#41×D1711yescharacteristic
539C2×C4#41×D1711yescharacteristic
540C2×C4#1058769711yescharacteristic
541C2×C4#1058769711yescharacteristic
542C8—174no—
543C8—342no—
544C8—681no—
545C8—681no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×C4×C22⋊D1711yescharacteristic
554C22#1176345811yescharacteristic
555C22#1176345811yescharacteristic
556C22C2×#1058763711yescharacteristic
557C22#1176360711yescharacteristic
558C22#1176361211yescharacteristic
559C22#1176361711yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—682no—
570C4—682no—
571C4—1361no—
572C4—1361no—
573C4#117×D1711yescharacteristic
574C4#1176328611yescharacteristic
575C4#1176349211yescharacteristic
576C4#1176349211yescharacteristic
577C2—42no—
578C2#1453165411yescharacteristic
579C2C4×#1058763711yescharacteristic
580C2#1453761211yescharacteristic
581C2#1454126911yescharacteristic
582C2#1454132911yescharacteristic
583C2#1454138411yescharacteristic
584C2#1454139311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2347C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#2347D17
hypocenterC17#22623
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
3. #14529524C2semidirect
4. #14529509C2semidirect
5. #14529376C2semidirect
6. #14529366C2semidirect
7. #14529359C2semidirect
8. #14529355C2semidirect
12. #14528682C2semidirect
13. C4×#10587478C2semidirect
18. #11762852C4semidirect
411. C17#22623semidirect

Maximal subgroups

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

subgroupindex
C2×#117628522
#145316592
#145295242
#145295092
#145293762
#145293662
#145293592
#145293552
#145288792
#145288682
#145288152
#145286822
C4×#105874782
#145285682
C17×#23472
#2262317