Finite Group Databasebeta

← #17093201

Subgroups — #17093201

Order 4352.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763219C211yesmaximal, characteristic, complemented
2C2×#11762849C211yesmaximal, characteristic
3#14532475C211yesmaximal, characteristic
4#14530266C211yesmaximal, characteristic, complemented
5#14530265C211yesmaximal, characteristic, complemented
6#14530119C211yesmaximal, characteristic, complemented
7#14530112C211yesmaximal, characteristic, complemented
8#14529519C211yesmaximal, characteristic, complemented
9#14529514C211yesmaximal, characteristic, complemented
10#14528981C211yesmaximal, characteristic
11#14528972C211yesmaximal, characteristic
12#14528946C211yesmaximal, characteristic, complemented
13#14528815C211yesmaximal, characteristic
14#14528568C211yesmaximal, characteristic
15C17×#2442C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19#11763219—24no—
20#11762849—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×#10587479C2211yescharacteristic
29C2×#10587479C2211yescharacteristic
30C2×#10587479C2211yescharacteristic
31C2×C22⋊#3622C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762732—21no—
35#11762721—21no—
36#11762713—21no—
37#11762707C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762696—21no—
40#11762675C2211yescharacteristic
41#11762669C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×#125C2211yescharacteristic
51C22×C4⋊C4×C17C2211yescharacteristic
52C17×#400—21no—
53C17×#398—21no—
54C17×#389C2211yescharacteristic
55C17×#389C2211yescharacteristic
56C17×#385—21no—
57C17×#382C2211yescharacteristic
58C17×#380—21no—
59C17×#380C2211yescharacteristic
60C4×C22⋊C4×C17C2211yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17C2311yescharacteristic
63C22×Dic34—21no—
64C17×#125—22no—
65C17×#125—41no—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—41no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17D411yescharacteristic
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84#10587612—22no—
85#10587612—41no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×#63225—21no—
92C2×#63225—21no—
93C2×C4×Dic17—21no—
94C2×#63223—22no—
95C2×#63223—22no—
96C2×#63223Q814yescomplemented
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587555—22no—
102#10587555—41no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95—21no—
106C17×#95—21no—
107C17×#95—21no—
108C17×#95D411yescharacteristic
109C17×#95D411yescharacteristic
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115C22⋊#3622Q814yescomplemented
116#10587479—22no—
117#10587479—22no—
118#10587479—22no—
119#10587479—22no—
120#10587478—22no—
121#10587478—22no—
122#10587478—22no—
123#10587478—41no—
124#10587478—41no—
125#10587478D414yes—
126#10587474—22no—
127C4×#3622—22no—
128C24×C17C2×D411yescharacteristic
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—42no—
138C22×C4×C17—42no—
139C22×C4×C17—81no—
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×Q812yes—
146C22×C4×C17C2×Q812yes—
147C22×C4×C17C2411yescharacteristic
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C2×Dic34—42no—
153C2×Dic34—42no—
154C2×Dic34—81no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—42no—
163C4⋊C4×C17—42no—
164C4⋊C4×C17—42no—
165C4⋊C4×C17—81no—
166C4⋊C4×C17—81no—
167C4⋊C4×C17C2×D412yes—
168C22⋊C4×C17—41no—
169C22⋊C4×C17—41no—
170C22⋊C4×C17—42no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—21no—
175C22⋊Dic17—42no—
176C22⋊Dic17—81no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—42no—
185C4⋊Dic17C2×D412yes—
186#63225—42no—
187#63225—81no—
188C4×Dic17—42no—
189#63223—28no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622C2×D412yes—
197C2×#3622C2×Q812yes—
198C2×#3622C2×Q812yes—
199C2×#3622C2×Q812yes—
200C2×#3622C2×Q812yes—
201C2×#3622#4112yes—
202#26079—171nomaximal, Sylow
203C23×C17—21no—
204C23×C17—21no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—21no—
208C23×C17—21no—
209C23×C17C4⋊Q812yes—
210C23×C17C22×D411yescharacteristic
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
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—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
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—41no—
240C2×C4×C17—41no—
241C2×C4×C17—84no—
242C2×C4×C17C4⋊Q812yes—
243C2×C4×C17C22×D411yescharacteristic
244C2×C4×C17C22×Q811yescharacteristic
245C2×C4×C17C22×Q811yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17H(5,𝔽2)11yescharacteristic
248C2×C4×C172−1+411yescharacteristic
249C2×Dic17—22no—
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—
260C2×Dic17—41no—
261Dic34—84no—
262#3622—22no—
263#3622—22no—
264#3622—22no—
265#3622C22⋊Q814yes—
266C2×#501—171no—
267C2×#428—171no—
268#2442D1711yescharacteristic
269#2439—171no—
270#2085—171no—
271#2085—171no—
272#2059—171no—
273#2057—171no—
274#1943—171no—
275#1942—171no—
276#1854—171no—
277#1852—171no—
278#1846—171no—
279#1809—171no—
280#1753—171no—
281C22×C17—22no—
282C22×C17—22no—
283C22×C17—22no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17—41no—
287C22×C17—41no—
288C22×C17C2×C4⋊Q811yescharacteristic
289C22×C17D4211yescharacteristic
290C22×C17#54711yescharacteristic
291C22×C17#57511yescharacteristic
292C22×C17#57511yescharacteristic
293C22×C17#57611yescharacteristic
294C22×C17#57811yescharacteristic
295C4×C17—22no—
296C4×C17—41no—
297C4×C17—41no—
298C4×C17—41no—
299C4×C17—42no—
300C4×C17—41no—
301C4×C17—81no—
302C4×C17—81no—
303C4×C17—81no—
304C4×C17C2×C22⋊Q811yescharacteristic
305C4×C17#54411yescharacteristic
306C4×C17D4×Q811yescharacteristic
307C4×C17#55411yescharacteristic
308Dic17—42no—
309Dic17—81no—
310Dic17—81no—
311Dic17—81no—
312C22×M16—171no—
313C2×C4⋊Q8—171no—
314C2×#125C2×D1711yescharacteristic
315C2×#117—171no—
316C22×C4⋊C4C2×D1711yescharacteristic
317#501—344no—
318#428—344no—
319C2×#107—171no—
320C2×#107—171no—
321C2×#107—171no—
322C2×#107—171no—
323C2×#107—171no—
324C2×#107—171no—
325C2×#106—171no—
326C2×#106—171no—
327C2×#106—171no—
328C2×#106—171no—
329C2×C22⋊C8—171no—
330C2×#97—171no—
331C2×C4×C8—171no—
332#400—21no—
333#398—21no—
334#396—341no—
335#393—341no—
336#391—341no—
337#389—171no—
338#389—171no—
339#389C2×D1711yescharacteristic
340#389C2×D1711yescharacteristic
341#387—341no—
342#385—21no—
343#382—171no—
344#382C2×D1711yescharacteristic
345#380—21no—
346#380—171no—
347#380C2×D1711yescharacteristic
348C4×C22⋊C4C2×D1711yescharacteristic
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#336—171no—
356#336—171no—
357C2×C17—42no—
358C2×C17#244011yescharacteristic
359C2×C17#316411yescharacteristic
360C2×C17#316611yescharacteristic
361C2×C17#327311yescharacteristic
362C2×C17#327711yescharacteristic
363C2×C17#334711yescharacteristic
364C2×C17#334911yescharacteristic
365C22×Q8—341no—
366C23×C4—21no—
367C23×C4C22×D1711yescharacteristic
368C2×M16—174no—
369C2×M16—342no—
370C2×M16—342no—
371C22×C8—171no—
372C22×C8—171no—
373C22×C8—171no—
374C22×C8—171no—
375C4⋊Q8—342no—
376C4⋊Q8—681no—
377#125—22no—
378#125—41no—
379#117—342no—
380#117—681no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—41no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C4⋊C4C22×D1711yescharacteristic
397C2×C4⋊C4C22×D1711yescharacteristic
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—341no—
401C2×C22⋊C4—341no—
402C2×C22⋊C4C22⋊D1711yescharacteristic
403C2×C22⋊C4C22⋊D1711yescharacteristic
404C2×C42—341no—
405C2×C42C22×D1711yescharacteristic
406C2×C42C22×D1711yescharacteristic
407#107—174no—
408#107—342no—
409#107—342no—
410#107—342no—
411#107—681no—
412#107—681no—
413#106—342no—
414#106—342no—
415#106—342no—
416#106—342no—
417C22⋊C8—174no—
418#97—342no—
419C4×C8—342no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95C22⋊D1711yescharacteristic
431#95C22⋊D1711yescharacteristic
432C17#2607911yescharacteristic, complemented, Sylow
433C24C2×C22⋊D1711yescharacteristic
434C2×Q8—682no—
435C2×Q8—682no—
436C2×Q8—1361no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—341no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4Q8×D1712yes—
453C22×C4Q8×D1712yes—
454C22×C4C2×C22⋊D1711yescharacteristic
455C22×C4C2×C22⋊D1711yescharacteristic
456C22×C4C2×C22⋊D1711yescharacteristic
457C22×C4C2×C22⋊D1711yescharacteristic
458C22×C4C2×C22⋊D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460M16—348no—
461C2×C8—172no—
462C2×C8—172no—
463C2×C8—172no—
464C2×C8—172no—
465C2×C8—172no—
466C2×C8—172no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8—341no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—172no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—682no—
494C4⋊C4—682no—
495C4⋊C4—1361no—
496C4⋊C4D4×D1712yes—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—42no—
500C22⋊C4—682no—
501C22⋊C4—1361no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—682no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1058763012yes—
514C23C22×C22⋊D1711yescharacteristic
515Q8—1364no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
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—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—84no—
547C2×C4—342no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4#1058763012yes—
560C2×C4C2×D4×D1711yescharacteristic
561C2×C4#1058769111yescharacteristic
562C2×C4C2×Q8×D1711yescharacteristic
563C2×C4C2×Q8×D1711yescharacteristic
564C2×C4#41×D1711yescharacteristic
565C2×C4#1058769711yescharacteristic
566C8—174no—
567C8—342no—
568C8—342no—
569C8—342no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1058763011yescharacteristic
578C22D4×C22⋊D1711yescharacteristic
579C22#1176345811yescharacteristic
580C22#1176360711yescharacteristic
581C22#1176360711yescharacteristic
582C22#1176361211yescharacteristic
583C22#1176362411yescharacteristic
584C4—22no—
585C4—41no—
586C4—41no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—682no—
594C4—1361no—
595C4—1361no—
596C4—1361no—
597C4C22⋊Q8×D1711yescharacteristic
598C4#1176343811yescharacteristic
599C4#1176347211yescharacteristic
600C4#1176351311yescharacteristic
601C2—42no—
602C2#1453248311yescharacteristic
603C2#1453931811yescharacteristic
604C2#1453933311yescharacteristic
605C2#1454039111yescharacteristic
606C2#1454043711yescharacteristic
607C2#1454127811yescharacteristic
608C2#1454131411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2442C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2442D17
hypocenterC17#26079
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C2×#11763219C2semidirect
4. #14530266C2semidirect
5. #14530265C2semidirect
6. #14530119C2semidirect
7. #14530112C2semidirect
8. #14529519C2semidirect
9. #14529514C2semidirect
12. #14528946C2semidirect
96. C2×#63223Q8semidirect
115. C22⋊#3622Q8semidirect
432. C17#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#117632192
C2×#117628492
#145324752
#145302662
#145302652
#145301192
#145301122
#145295192
#145295142
#145289812
#145289722
#145289462
#145288152
#145285682
C17×#24422
#2607917