Finite Group Databasebeta

← #17068636

Subgroups — #17068636

Order 4352.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763096C211yesmaximal, characteristic, complemented
2C2×#11763011C211yesmaximal, characteristic, complemented
3C4×#10587616C211yesmaximal, characteristic
4#14529977C212yesmaximal, complemented
5#14529785C212yesmaximal
6#14529733C211yesmaximal, characteristic, complemented
7#14529708C211yesmaximal, characteristic, complemented
8#14529139C212yesmaximal, complemented
9#14529052C212yesmaximal, complemented
10#14528764C211yesmaximal, characteristic, complemented
11C2×C17×#533C211yesmaximal, characteristic, complemented
12C22×#63230C2212yes—
13C2×#10587616C2211yescharacteristic
14C2×C4×D68C2211yescharacteristic
15C2×C4×C22⋊D17—21no—
16#11763096—24no—
17#11763011—24no—
18C2×#10587479C2212yes—
19C2×C4⋊#3622C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587506C2212yes—
22C2×#10587506C2212yes—
23C2×#10587506C2212yes—
24C2×#10587480C2212yes—
25C2×C22⋊#3622C2212yes—
26#11762711C2211yescharacteristic
27#11762707C2211yescharacteristic
28#11762700—21no—
29#11762688—21no—
30#11762669C2211yescharacteristic
31C4×C4⋊Dic17C2211yescharacteristic
32C4×#63227—21no—
33#11762571C2212yes—
34#11762571C2212yes—
35#11762564C2212yes—
36C17×#533—24no—
37C17×#533—24no—
38C2×C17×C4⋊Q8C2211yescharacteristic
39C2×C17×C4⋊Q8C2211yescharacteristic
40C2×C17×#125—21no—
41C2×C17×C22⋊Q8—21no—
42C2×C17×C22⋊Q8C2212yes—
43C2×C4×Q8×C17—21no—
44C2×C4×Q8×C17C2212yes—
45C2×C17×#117—21no—
46C22×C42×C17C2211yescharacteristic
47C22×Q8×C17—21no—
48C22×Q8×C17C2312yes—
49C23×C4×C17D411yescharacteristic
50C23×C4×C17D411yescharacteristic
51C23×C4×C17C2311yescharacteristic
52C22×C22⋊D17—21no—
53C22×D68C2311yescharacteristic
54C22×C4×D17—21no—
55C17×C4⋊Q8—22no—
56C17×C4⋊Q8—22no—
57C17×C4⋊Q8D414yescomplemented
58C17×C4⋊Q8D414yescomplemented
59C17×#125—28no—
60C17×C22⋊Q8—28no—
61C17×C22⋊Q8—24no—
62C17×C22⋊Q8—24no—
63C4×Q8×C17—28no—
64C4×Q8×C17—24no—
65C4×Q8×C17—24no—
66C17×#117—28no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—22no—
72C2×C4⋊C4×C17C2312yes—
73C2×C22⋊C4×C17—22no—
74C2×C42×C17—22no—
75C2×C42×C17—22no—
76C2×C42×C17—22no—
77C2×C42×C17D411yescharacteristic
78C2×C42×C17D411yescharacteristic
79C2×C42×C17C2311yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C22⋊Dic17—21no—
82C2×#63230—24no—
83C2×#63230—24no—
84C2×#63230—42no—
85#10587616—22no—
86#10587616—22no—
87C4×C22⋊D17—44no—
88C2×#63227—21no—
89C2×#63227—21no—
90C2×#63227—21no—
91C2×#63227—21no—
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C22×#3622C2312yes—
98C22×#3622C2312yes—
99C4×D68—22no—
100C4×D68—22no—
101#10587506—24no—
102#10587506—24no—
103#10587506—42no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107C22⋊#3622—42no—
108#10587480—24no—
109#10587479—24no—
110C4⋊#3622—22no—
111C4⋊#3622—22no—
112C24×C17C2×D411yescharacteristic
113C23×D17—21no—
114C2×Q8×C17—24no—
115C2×Q8×C17—22no—
116C2×Q8×C17—22no—
117C2×Q8×C17—22no—
118C2×Q8×C17—22no—
119C2×Q8×C17—22no—
120C2×Q8×C17—42no—
121C2×Q8×C17—42no—
122C2×Q8×C17C2×D414yes—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
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—41no—
135C22×C4×C17—41no—
136C22×C4×C17SD1614yescomplemented
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2411yescharacteristic
143C2×C22⋊D17—24no—
144C2×C22⋊D17—42no—
145C2×C22⋊D17—42no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C2×D68—21no—
150C2×D68—21no—
151C2×D68—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68#4112yes—
155C2×C4×D17—42no—
156C2×C4×D17—42no—
157C2×C4×D17—42no—
158C4⋊C4×C17—28no—
159C4⋊C4×C17—28no—
160C4⋊C4×C17—24no—
161C4⋊C4×C17—24no—
162C4⋊C4×C17—24no—
163C4⋊C4×C17—22no—
164C4⋊C4×C17C2×D414yes—
165C22⋊C4×C17—28no—
166C42×C17—28no—
167C42×C17—22no—
168C42×C17—22no—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17C2×D412yes—
172C42×C17C2×D412yes—
173C22⋊Dic17—42no—
174#63230—48no—
175#63227—24no—
176#63227—42no—
177#63227—42no—
178#63227—42no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17#4112yes—
185#63225—42no—
186C4×Dic17—42no—
187C2×#3622—22no—
188C2×#3622—22no—
189C2×#3622—22no—
190C2×#3622—22no—
191C2×#3622—22no—
192C2×#3622—22no—
193#24151—171nomaximal, Sylow
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17C2×SD1612yes—
201C23×C17C22×D411yescharacteristic
202C22×D17—22no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C22×D17—41no—
209Q8×C17—24no—
210Q8×C17—24no—
211Q8×C17—48no—
212Q8×C17—42no—
213Q8×C17—42no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—28no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—24no—
222C2×C4×C17—24no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—44no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17#12012yes—
237C2×C4×C17#12012yes—
238C2×C4×C17C2×SD1612yes—
239C2×C4×C17C2×SD1612yes—
240C2×C4×C17#13712yes—
241C2×C4×C17C22×D411yescharacteristic
242C2×C4×C17C22×D411yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17H(5,𝔽2)11yescharacteristic
246C2×C4×C172−1+411yescharacteristic
247C22⋊D17—48no—
248C2×Dic17—22no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257D68—22no—
258D68—22no—
259D68—41no—
260D68—41no—
261C4×D17—84no—
262#3622—42no—
263#3622—42no—
264C2×#533D1711yescharacteristic, complemented
265C2×#476—171no—
266C2×#461—171no—
267C4×#121—171no—
268#2035—172no—
269#2000—172no—
270#1987—171no—
271#1981—171no—
272#1880—172no—
273#1865—172no—
274#1797—171no—
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17C2×#12011yescharacteristic
283C22×C17#52511yescharacteristic
284C22×C17#53711yescharacteristic
285C22×C17C22×SD1611yescharacteristic
286C22×C17C2×#13711yescharacteristic
287C22×C17#57511yescharacteristic
288C22×C17#57511yescharacteristic
289C2×D17—42no—
290C2×D17—42no—
291C2×D17—42no—
292C2×D17—81no—
293C2×D17—81no—
294C2×D17—81no—
295C2×D17—81no—
296C4×C17—22no—
297C4×C17—22no—
298C4×C17—41no—
299C4×C17—44no—
300C4×C17—41no—
301C4×C17—44no—
302C4×C17#44814yes—
303C4×C17#54611yescharacteristic
304C4×C17#54711yescharacteristic
305C4×C17#54811yescharacteristic
306C4×C17#55011yescharacteristic
307Dic17—42no—
308Dic17—81no—
309Dic17—81no—
310C22×SD16—172no—
311#533—24no—
312#533—24no—
313C2×C4⋊Q8C2×D1711yescharacteristic
314C2×C4⋊Q8C2×D1711yescharacteristic
315C2×#125—21no—
316C2×C22⋊Q8—21no—
317C2×C22⋊Q8C2×D1712yes—
318C2×#121—171no—
319C2×C4×Q8—21no—
320C2×C4×Q8C2×D1712yes—
321C2×C4×D4—171no—
322C2×C4×D4—341no—
323C2×#117—21no—
324C22×C42C2×D1711yescharacteristic
325#476—344no—
326#461—344no—
327C2×#106—172no—
328C2×C4⋊C8—171no—
329C2×C4⋊C8—171no—
330C2×#103—172no—
331C2×#103—172no—
332C2×#103—172no—
333C2×#102—172no—
334C2×C22⋊C8—172no—
335#390—171no—
336#389—171no—
337#388—341no—
338#386—341no—
339#380—171no—
340C4×C4⋊C4—171no—
341C4×C22⋊C4—341no—
342#340—172no—
343#340—172no—
344#336—172no—
345C2×C17—42no—
346C2×C17#240711yescharacteristic
347C2×C17C2×#44811yescharacteristic
348C2×C17#300511yescharacteristic
349C2×C17#310111yescharacteristic
350C2×C17#311611yescharacteristic
351C2×C17#324611yescharacteristic
352C2×C17#325211yescharacteristic
353D17—82no—
354C22×Q8—21no—
355C22×Q8C22×D1712yes—
356C22×D4—171no—
357C22×D4—341no—
358C23×C4—341no—
359C23×C4C22⋊D1711yescharacteristic
360C23×C4C22⋊D1711yescharacteristic
361C23×C4C22×D1711yescharacteristic
362C2×SD16—344no—
363C2×SD16—344no—
364C2×SD16—682no—
365C22×C8—172no—
366C22×C8—172no—
367C4⋊Q8—22no—
368C4⋊Q8—22no—
369C4⋊Q8C22⋊D1714yescomplemented
370C4⋊Q8C22⋊D1714yescomplemented
371#125—28no—
372C22⋊Q8—28no—
373C22⋊Q8—24no—
374C22⋊Q8—24no—
375#121—342no—
376#121—342no—
377C4×Q8—28no—
378C4×Q8—24no—
379C4×Q8—24no—
380C4×D4—342no—
381C4×D4—342no—
382C4×D4—684no—
383#117—28no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4C22×D1712yes—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—341no—
396C2×C22⋊C4—341no—
397C2×C22⋊C4—341no—
398C2×C22⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—341no—
404C2×C42C22⋊D1711yescharacteristic
405C2×C42C22⋊D1711yescharacteristic
406C2×C42C22×D1711yescharacteristic
407C2×C42C22×D1711yescharacteristic
408#106—344no—
409C4⋊C8—342no—
410C4⋊C8—342no—
411#103—344no—
412#103—344no—
413#103—682no—
414#102—344no—
415C22⋊C8—682no—
416#95—341no—
417#95—341no—
418#95—341no—
419C17#2415111yescharacteristic, complemented, Sylow
420C24—341no—
421C24C2×C22⋊D1711yescharacteristic
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8C2×C22⋊D1714yes—
431C2×D4—172no—
432C2×D4—344no—
433C2×D4—341no—
434C2×D4—341no—
435C2×D4—341no—
436C2×D4—341no—
437C2×D4—341no—
438C2×D4—682no—
439C2×D4—682no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—682no—
457C22×C4—682no—
458C22×C4—682no—
459C22×C4#6323014yescomplemented
460C22×C4C2×C22⋊D1711yescharacteristic
461C22×C4C2×C22⋊D1711yescharacteristic
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C23×D1711yescharacteristic
466SD16—688no—
467C2×C8—342no—
468C2×C8—342no—
469C2×C8—342no—
470C2×C8—342no—
471C2×C8—342no—
472C2×C8—342no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—172no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—682no—
486C4⋊C4C2×C22⋊D1714yes—
487C22⋊C4—28no—
488C22⋊C4—344no—
489C22⋊C4—682no—
490C22⋊C4—682no—
491C22⋊C4—682no—
492C22⋊C4—682no—
493C42—28no—
494C42—21no—
495C42—21no—
496C42—22no—
497C42—22no—
498C42—682no—
499C42C2×C22⋊D1712yes—
500C42C2×C22⋊D1712yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—342no—
508C23—681no—
509C23—681no—
510C23—681no—
511C23—681no—
512C23—681no—
513C23—681no—
514C23C2×#6323012yes—
515C23C22×C22⋊D1711yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—48no—
519Q8—42no—
520Q8—42no—
521D4—342no—
522D4—342no—
523D4—688no—
524D4—681no—
525D4—681no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—342no—
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—1364no—
558C2×C4C2×#6323012yes—
559C2×C4C2×#6323012yes—
560C2×C4#1058762812yes—
561C2×C4#1058763912yes—
562C2×C4#1058763912yes—
563C2×C4#41×D1711yescharacteristic
564C2×C4#41×D1711yescharacteristic
565C2×C4#1058769611yescharacteristic
566C2×C4#1058769711yescharacteristic
567C2×C4C22×C22⋊D1711yescharacteristic
568C2×C4C22×C22⋊D1711yescharacteristic
569C8—682no—
570C8—682no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—682no—
579C22—682no—
580C22—682no—
581C22—1361no—
582C22—1361no—
583C22—1361no—
584C22—1361no—
585C22C2×#1058763911yescharacteristic
586C22#1176331611yescharacteristic
587C22#1176338711yescharacteristic
588C22C22×#6323011yescharacteristic
589C22C2×#1058762811yescharacteristic
590C22#1176360711yescharacteristic
591C22#1176360711yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—44no—
596C4—41no—
597C4—44no—
598C4—682no—
599C4—1361no—
600C4—1361no—
601C4#1176294814yes—
602C4#1176344711yescharacteristic
603C4#1176345811yescharacteristic
604C4#1176346611yescharacteristic
605C4#1176347811yescharacteristic
606C2—42no—
607C2—1362no—
608C2#1453212311yescharacteristic
609C2C2×#1176294811yescharacteristic
610C2#1453802611yescharacteristic
611C2#1453878811yescharacteristic
612C2#1453894011yescharacteristic
613C2#1454009611yescharacteristic
614C2#1454015111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#533C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenterC2×#533D17
hypocenterC17#24151
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#11763096C2semidirect
2. C2×#11763011C2semidirect
4. #14529977C2semidirect
6. #14529733C2semidirect
7. #14529708C2semidirect
8. #14529139C2semidirect
9. #14529052C2semidirect
10. #14528764C2semidirect
11. C2×C17×#533C2semidirect
57. C17×C4⋊Q8D4semidirect
58. C17×C4⋊Q8D4semidirect
136. C22×C4×C17SD16semidirect
264. C2×#533D17semidirect
369. C4⋊Q8C22⋊D17semidirect
370. C4⋊Q8C22⋊D17semidirect
419. C17#24151semidirect
459. C22×C4#63230semidirect

Maximal subgroups

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

subgroupindex
C2×#117630962
C2×#117630112
C4×#105876162
#145299772
#145297852
#145297332
#145297082
#145291392
#145290522
#145287642
C2×C17×#5332
#2415117