Finite Group Databasebeta

← #17107035

Subgroups — #17107035

Order 4352.

728 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#14533206C211yesmaximal, characteristic
2#14530192C211yesmaximal, characteristic, complemented
3#14530165C211yesmaximal, characteristic, complemented
4#14529977C211yesmaximal, characteristic
5#14529977C211yesmaximal, characteristic
6#14529547C211yesmaximal, characteristic, complemented
7#14529509C211yesmaximal, characteristic, complemented
8#14529392C211yesmaximal, characteristic
9#14529366C211yesmaximal, characteristic, complemented
10#14529362C211yesmaximal, characteristic, complemented
11#14529052C211yesmaximal, characteristic, complemented
12#14528977C211yesmaximal, characteristic
13#14528874C211yesmaximal, characteristic
14C4×#10587506C211yesmaximal, characteristic, complemented
15C17×#2360C211yesmaximal, characteristic
16C2×#10587565—21no—
17C2×#10587479C2211yescharacteristic
18C2×#10587479C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×C22⋊#3622C2211yescharacteristic
27C2×C22⋊#3622C2211yescharacteristic
28C2×C4×#3622C2211yescharacteristic
29#11762749—21no—
30#11762734—21no—
31#11762707C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762694—21no—
34#11762679C2211yescharacteristic
35#11762669C2211yescharacteristic
36#11762669C2211yescharacteristic
37C4×C4⋊Dic17C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×C22⋊Q8C2211yescharacteristic
49C2×C4×Q8×C17C2211yescharacteristic
50C22×C4⋊C4×C17C2211yescharacteristic
51C17×#391C2211yescharacteristic
52C17×#389—21no—
53C17×#389C2211yescharacteristic
54C17×#387—21no—
55C17×#385—21no—
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C22⋊C4×C17—21no—
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C23×C4×C17—21no—
61C23×C4×C17C2311yescharacteristic
62C17×C22⋊Q8—22no—
63C17×C22⋊Q8—22no—
64C4×Q8×C17—22no—
65C4×Q8×C17—41no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—22no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—41no—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17D411yescharacteristic
77C2×C22⋊C4×C17D411yescharacteristic
78C2×C42×C17—21no—
79C2×C42×C17—21no—
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×#63225—21no—
91C2×#63225—21no—
92C2×C4×Dic17—21no—
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97#10587565—44no—
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95D411yescharacteristic
103C17×#95D411yescharacteristic
104#10587506—22no—
105#10587506—22no—
106#10587506—22no—
107#10587506—41no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113C22⋊#3622—22no—
114C22⋊#3622—41no—
115#10587479—22no—
116#10587479—22no—
117#10587479—22no—
118#10587479—41no—
119C4⋊#3622—22no—
120C4×#3622—41no—
121C24×C17C2×D411yescharacteristic
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17#4112yes—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—42no—
139C22×C4×C17—42no—
140C22×C4×C17—81no—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2411yescharacteristic
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C4⋊C4×C17—24no—
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—81no—
159C4⋊C4×C17#4112yes—
160C4⋊C4×C17#4112yes—
161C22⋊C4×C17—24no—
162C22⋊C4×C17—41no—
163C22⋊C4×C17SD1614yes—
164C42×C17—21no—
165C42×C17—21no—
166C42×C17—21no—
167C42×C17—21no—
168C42×C17—42no—
169C42×C17—81no—
170C22⋊Dic17—42no—
171C22⋊Dic17—42no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—42no—
181C4⋊Dic17—42no—
182#63225—42no—
183#63225—42no—
184C4×Dic17—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—21no—
196C2×#3622#4112yes—
197#27034—171nomaximal, Sylow
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17—21no—
204C23×C17C2×SD1612yes—
205C23×C17C22×D411yescharacteristic
206Q8×C17—22no—
207Q8×C17—22no—
208Q8×C17—41no—
209Q8×C17—41no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
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—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—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—84no—
240C2×C4×C17C2×SD1612yes—
241C2×C4×C17C2×SD1612yes—
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17H(5,𝔽2)11yescharacteristic
247C2×C4×C172−1+411yescharacteristic
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—
258C2×Dic17—41no—
259C2×Dic17—41no—
260#3622—22no—
261#3622—41no—
262#3622—41no—
263#3622—41no—
264#2496—171no—
265#2360D1711yescharacteristic
266#2072—171no—
267#2066—171no—
268#2035—171no—
269#2035—171no—
270#1948—171no—
271#1941—171no—
272#1923—171no—
273#1920—171no—
274#1919—171no—
275#1865—171no—
276#1853—171no—
277#1824—171no—
278C4×#103—171no—
279C22×C17—22no—
280C22×C17—22no—
281C22×C17—22no—
282C22×C17—41no—
283C22×C17—41no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17#52511yescharacteristic
287C22×C17#54611yescharacteristic
288C22×C17#54711yescharacteristic
289C22×C17C22×SD1611yescharacteristic
290C22×C17#57511yescharacteristic
291C22×C17#57711yescharacteristic
292C22×C17#57811yescharacteristic
293C4×C17—22no—
294C4×C17—22no—
295C4×C17—41no—
296C4×C17—42no—
297C4×C17—41no—
298C4×C17—42no—
299C4×C17—41no—
300C4×C17—42no—
301C4×C17—81no—
302C4×C17#52911yescharacteristic
303C4×C17#53911yescharacteristic
304C4×C17#55311yescharacteristic
305C4×C17#55611yescharacteristic
306Dic17—81no—
307Dic17—81no—
308Dic17—81no—
309Dic17—81no—
310C2×#126—341no—
311C2×C22⋊Q8C2×D1711yescharacteristic
312C2×C4×Q8C2×D1711yescharacteristic
313C22×C4⋊C4C2×D1711yescharacteristic
314C2×#106—171no—
315C2×#106—171no—
316C2×#106—171no—
317C2×#106—171no—
318C2×C4⋊C8—171no—
319C2×#103—171no—
320C2×#103—171no—
321C2×#103—171no—
322C2×#103—171no—
323C2×C22⋊C8—171no—
324C2×C22⋊C8—171no—
325C2×C4×C8—171no—
326#400—341no—
327#397—341no—
328#391C2×D1711yescharacteristic
329#389—21no—
330#389—171no—
331#389—171no—
332#389C2×D1711yescharacteristic
333#387—21no—
334#387—341no—
335#385—21no—
336#384—171no—
337#380—171no—
338#380—171no—
339C4×C4⋊C4—171no—
340C4×C4⋊C4C2×D1711yescharacteristic
341C4×C22⋊C4—21no—
342C4×C22⋊C4C2×D1711yescharacteristic
343#340—171no—
344#340—171no—
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—42no—
354C2×C17#246611yescharacteristic
355C2×C17#310311yescharacteristic
356C2×C17#312011yescharacteristic
357C2×C17#329811yescharacteristic
358C2×C17#331111yescharacteristic
359C2×C17#333511yescharacteristic
360C2×C17#335211yescharacteristic
361C22×Q8C22×D1711yescharacteristic
362C23×C4—21no—
363C23×C4C22×D1711yescharacteristic
364C22×C8—171no—
365C22×C8—171no—
366C22×C8—171no—
367C22×C8—171no—
368#126—684no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C4×Q8—22no—
372C4×Q8—41no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4C22×D1711yescharacteristic
388C2×C4⋊C4C22×D1711yescharacteristic
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C22⋊C4C22⋊D1711yescharacteristic
394C2×C22⋊C4C22⋊D1711yescharacteristic
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—341no—
398C2×C42C22×D1711yescharacteristic
399C2×C42C22×D1711yescharacteristic
400#106—342no—
401#106—342no—
402#106—342no—
403#106—681no—
404C4⋊C8—342no—
405#103—342no—
406#103—342no—
407#103—342no—
408#103—681no—
409C22⋊C8—342no—
410C22⋊C8—681no—
411C4×C8—681no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95C22⋊D1711yescharacteristic
422#95C22⋊D1711yescharacteristic
423C17#2703411yescharacteristic, complemented, Sylow
424C24C2×C22⋊D1711yescharacteristic
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8#6325212yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4C2×C22⋊D1711yescharacteristic
449C22×C4C2×C22⋊D1711yescharacteristic
450C22×C4C2×C22⋊D1711yescharacteristic
451C22×C4C2×C22⋊D1711yescharacteristic
452C22×C4C2×C22⋊D1711yescharacteristic
453C22×C4C23×D1711yescharacteristic
454C2×C8—172no—
455C2×C8—341no—
456C2×C8—341no—
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—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—81no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—682no—
483C4⋊C4—682no—
484C4⋊C4—682no—
485C4⋊C4—682no—
486C4⋊C4#6325212yes—
487C4⋊C4#6325412yes—
488C22⋊C4—24no—
489C22⋊C4—41no—
490C22⋊C4—682no—
491C22⋊C4—682no—
492C22⋊C4#6323014yes—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—81no—
499C42—682no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23C2×#6323012yes—
507C23C22×C22⋊D1711yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—41no—
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—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
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—681no—
553C2×C4—681no—
554C2×C4C2×#6323012yes—
555C2×C4C2×#6323012yes—
556C2×C4C2×#6325211yescharacteristic
557C2×C4C2×#6325211yescharacteristic
558C2×C4C2×#6325411yescharacteristic
559C2×C4#1058769111yescharacteristic
560C2×C4#41×D1711yescharacteristic
561C2×C4#1058769711yescharacteristic
562C8—342no—
563C8—681no—
564C8—681no—
565C8—681no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#1176332011yescharacteristic
574C22#1176345311yescharacteristic
575C22#1176345811yescharacteristic
576C22C22×#6323011yescharacteristic
577C22#1176360711yescharacteristic
578C22#1176362011yescharacteristic
579C22#1176362411yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—42no—
588C4—81no—
589C4—1361no—
590C4—1361no—
591C4—1361no—
592C4—1361no—
593C4#1176333911yescharacteristic
594C4#1176340111yescharacteristic
595C4#1176350911yescharacteristic
596C4#1176352911yescharacteristic
597C2—42no—
598C2#1453281411yescharacteristic
599C2#1453881511yescharacteristic
600C2#1453899811yescharacteristic
601C2#1454064911yescharacteristic
602C2#1454082011yescharacteristic
603C2#1454113511yescharacteristic
604C2#1454135311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2360C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2360D17
hypocenterC17#27034
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #14530192C2semidirect
3. #14530165C2semidirect
6. #14529547C2semidirect
7. #14529509C2semidirect
9. #14529366C2semidirect
10. #14529362C2semidirect
11. #14529052C2semidirect
14. C4×#10587506C2semidirect
423. C17#27034semidirect

Maximal subgroups

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

subgroupindex
#145332062
#145301922
#145301652
#145299772
#145295472
#145295092
#145293922
#145293662
#145293622
#145290522
#145289772
#145288742
C4×#105875062
C17×#23602
#2703417