Finite Group Databasebeta

← #17070777

Subgroups — #17070777

Order 4352.

1042 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763106C211yesmaximal, characteristic
2C2×C4×#63230C211yesmaximal, characteristic
3#14534142C211yesmaximal, characteristic
4#14530119C211yesmaximal, characteristic, complemented
5#14530114C211yesmaximal, characteristic, complemented
6#14529927C211yesmaximal, characteristic
7#14529917C211yesmaximal, characteristic
8#14529750C211yesmaximal, characteristic, complemented
9#14529721C211yesmaximal, characteristic, complemented
10#14529139C211yesmaximal, characteristic
11#14529139C211yesmaximal, characteristic
12#14529052C211yesmaximal, characteristic, complemented
13#14529038C211yesmaximal, characteristic, complemented
14#14528799C211yesmaximal, characteristic, complemented
15C2×C17×#557C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18#11763106—24no—
19C4×#63230—24no—
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31#11762744—21no—
32#11762720—21no—
33#11762711C2211yescharacteristic
34#11762711C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762680—21no—
38#11762680—21no—
39#11762675C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C17×#557—24no—
51C17×#557—24no—
52C2×C17×C4⋊Q8—21no—
53C2×C17×C4⋊Q8C2211yescharacteristic
54C2×C17×#125—21no—
55C2×C17×#125—21no—
56C2×C17×#125C2211yescharacteristic
57C2×C17×#125C2211yescharacteristic
58C2×C4×Q8×C17—21no—
59C2×C4×Q8×C17C2211yescharacteristic
60C2×C4×Q8×C17C2211yescharacteristic
61C2×C4×Q8×C17C2211yescharacteristic
62C2×C4×Q8×C17C2211yescharacteristic
63C22×Q8×C17—21no—
64C22×Q8×C17D411yescharacteristic
65C22×Q8×C17C2311yescharacteristic
66C22×D68C2311yescharacteristic
67C22×C4×D17—21no—
68C17×C4⋊Q8—28no—
69C17×C4⋊Q8—22no—
70C17×C4⋊Q8—22no—
71C17×#125—28no—
72C17×#125—28no—
73C17×#125—22no—
74C17×#125—22no—
75C17×#125—22no—
76C17×#125D414yes—
77C4×Q8×C17—28no—
78C4×Q8×C17—22no—
79C4×Q8×C17—22no—
80C4×Q8×C17—22no—
81C4×Q8×C17—22no—
82C4×Q8×C17—22no—
83C4×Q8×C17—22no—
84C4×Q8×C17—22no—
85C4×Q8×C17D414yes—
86C2×C4⋊C4×C17—21no—
87C2×C4⋊C4×C17—21no—
88C2×C4⋊C4×C17—21no—
89C2×C4⋊C4×C17—21no—
90C2×C4⋊C4×C17—21no—
91C2×C4⋊C4×C17—21no—
92C2×C4⋊C4×C17—21no—
93C2×C4⋊C4×C17—21no—
94C2×C4⋊C4×C17D411yescharacteristic
95C2×C4⋊C4×C17D411yescharacteristic
96C2×C4⋊C4×C17D411yescharacteristic
97C2×C4⋊C4×C17C2311yescharacteristic
98C2×C4⋊C4×C17C2311yescharacteristic
99C2×C4⋊C4×C17C2311yescharacteristic
100C2×C42×C17—21no—
101C2×C42×C17—21no—
102C2×C42×C17—21no—
103C2×C42×C17C2311yescharacteristic
104C2×C42×C17C2311yescharacteristic
105C2×C42×C17C2311yescharacteristic
106C2×#63230—22no—
107C2×#63230—22no—
108C2×#63230Q814yescomplemented
109C2×#63227—21no—
110C2×#63227—21no—
111C2×#63227—21no—
112C2×#63227—21no—
113C2×#63227—21no—
114C2×C4⋊Dic17C2311yescharacteristic
115C2×C4⋊Dic17C2311yescharacteristic
116C2×C4⋊Dic17C2311yescharacteristic
117C2×#63225—21no—
118C2×#63225—21no—
119C22×#3622C2311yescharacteristic
120C22×#3622C2311yescharacteristic
121C22×#3622C2311yescharacteristic
122C22×#3622C2311yescharacteristic
123C4×D68—22no—
124C4×D68—22no—
125#10587506—22no—
126#10587506—22no—
127#10587506—22no—
128#10587502—21no—
129#10587502—21no—
130#10587502—21no—
131#10587502—21no—
132#10587480—22no—
133#10587480—22no—
134#10587480—22no—
135#10587479—22no—
136#10587479—22no—
137#10587478—22no—
138C4⋊#3622—22no—
139C4×#3622—22no—
140C23×D17—21no—
141C2×Q8×C17—24no—
142C2×Q8×C17—22no—
143C2×Q8×C17—22no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17—41no—
148C2×Q8×C17—42no—
149C2×Q8×C17—42no—
150C2×Q8×C17C2×D412yes—
151C2×Q8×C17C2×Q812yes—
152C2×Q8×C17#4112yes—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2411yescharacteristic
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—21no—
167C2×D68—21no—
168C2×D68—21no—
169C2×D68—21no—
170C2×D68—21no—
171C2×D68—21no—
172C2×D68C2×Q812yes—
173C2×C4×D17—42no—
174C2×C4×D17—81no—
175C2×C4×D17—81no—
176C4⋊C4×C17—24no—
177C4⋊C4×C17—24no—
178C4⋊C4×C17—24no—
179C4⋊C4×C17—24no—
180C4⋊C4×C17—24no—
181C4⋊C4×C17—22no—
182C4⋊C4×C17—22no—
183C4⋊C4×C17—24no—
184C4⋊C4×C17—24no—
185C4⋊C4×C17—24no—
186C4⋊C4×C17—22no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17C2×D412yes—
191C4⋊C4×C17C2×D412yes—
192C4⋊C4×C17C2×D412yes—
193C42×C17—24no—
194C42×C17—24no—
195C42×C17—24no—
196C42×C17—21no—
197C42×C17—21no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17C2×D412yes—
201C42×C17C2×D412yes—
202#63230—28no—
203#63227—42no—
204#63227—42no—
205#63227—42no—
206#63227—81no—
207#63227—81no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—21no—
213C4⋊Dic17—21no—
214#63225—81no—
215#63225—81no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622C2×Q812yes—
228#24334—171nomaximal, Sylow
229C23×C17C22×D411yescharacteristic
230C22×D17—22no—
231C22×D17—41no—
232C22×D17—41no—
233C22×D17—41no—
234C22×D17—41no—
235C22×D17—41no—
236C22×D17—41no—
237Q8×C17—22no—
238Q8×C17—22no—
239Q8×C17—22no—
240Q8×C17—44no—
241Q8×C17—48no—
242Q8×C17C22⋊Q814yes—
243C2×C4×C17—22no—
244C2×C4×C17—21no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—22no—
255C2×C4×C17—21no—
256C2×C4×C17—22no—
257C2×C4×C17—22no—
258C2×C4×C17—21no—
259C2×C4×C17—21no—
260C2×C4×C17—21no—
261C2×C4×C17—21no—
262C2×C4×C17—21no—
263C2×C4×C17—21no—
264C2×C4×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17—41no—
268C2×C4×C17—41no—
269C2×C4×C17—41no—
270C2×C4×C17#12012yes—
271C2×C4×C17#12012yes—
272C2×C4×C17#13512yes—
273C2×C4×C17C22×D411yescharacteristic
274C2×C4×C17C22×D411yescharacteristic
275C2×C4×C17C22×Q811yescharacteristic
276C2×C4×C17C2×#4111yescharacteristic
277C2×C4×C172−1+411yescharacteristic
278C2×C4×C172−1+411yescharacteristic
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288D68—22no—
289D68—22no—
290D68—41no—
291D68—41no—
292C4×D17—84no—
293#3622—22no—
294#3622—41no—
295#3622—41no—
296#3622—41no—
297C2×#557D1711yescharacteristic, complemented
298C2×#478—171no—
299C2×C4×SD16—171no—
300#2580—171no—
301#2059—171no—
302#2058—171no—
303#2025—171no—
304#2023—171no—
305#1991—171no—
306#1984—171no—
307#1880—171no—
308#1880—171no—
309#1865—171no—
310#1863—171no—
311#1805—171no—
312C22×C17C2×#12011yescharacteristic
313C22×C17#54811yescharacteristic
314C22×C17D4×Q811yescharacteristic
315C22×C17C2×#13511yescharacteristic
316C22×C17#57511yescharacteristic
317C22×C17#57711yescharacteristic
318C22×C17#57711yescharacteristic
319C2×D17—42no—
320C2×D17—42no—
321C2×D17—42no—
322C2×D17—81no—
323C2×D17—81no—
324C2×D17—81no—
325C2×D17—81no—
326C4×C17—24no—
327C4×C17—22no—
328C4×C17—22no—
329C4×C17—22no—
330C4×C17—41no—
331C4×C17—42no—
332C4×C17—42no—
333C4×C17—41no—
334C4×C17—41no—
335C4×C17—42no—
336C4×C17C2×C22⋊Q811yescharacteristic
337C4×C17#53611yescharacteristic
338C4×C17#54811yescharacteristic
339C4×C17D4×Q811yescharacteristic
340Dic17—81no—
341Dic17—81no—
342Dic17—81no—
343C22×SD16—171no—
344#557—24no—
345#557—24no—
346C2×C4⋊Q8—21no—
347C2×C4⋊Q8C2×D1711yescharacteristic
348C2×#125—21no—
349C2×#125—21no—
350C2×#125C2×D1711yescharacteristic
351C2×#125C2×D1711yescharacteristic
352C2×C4×Q8—21no—
353C2×C4×Q8C2×D1711yescharacteristic
354C2×C4×Q8C2×D1711yescharacteristic
355C2×C4×Q8C2×D1711yescharacteristic
356C2×C4×Q8C2×D1711yescharacteristic
357C2×C4×D4—171no—
358#478—344no—
359C4×SD16—344no—
360C2×#107—171no—
361C2×#106—171no—
362C2×#106—171no—
363C2×C4⋊C8—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#102—171no—
368C2×#102—171no—
369C2×#102—171no—
370C2×C4×C8—171no—
371#399—341no—
372#393—341no—
373#390—171no—
374#390—171no—
375#389—171no—
376#389—171no—
377#385—341no—
378#385—341no—
379#382—171no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#340—171no—
386#336—171no—
387#336—171no—
388#336—171no—
389#336—171no—
390C2×C17#240911yescharacteristic
391C2×C17#300211yescharacteristic
392C2×C17#301011yescharacteristic
393C2×C17#336511yescharacteristic
394C2×C17#337111yescharacteristic
395C2×C17#338111yescharacteristic
396C2×C17#338811yescharacteristic
397D17—82no—
398C22×Q8—21no—
399C22×Q8C22⋊D1711yescharacteristic
400C22×Q8C22×D1711yescharacteristic
401C22×D4—171no—
402C23×C4—341no—
403C2×SD16—174no—
404C2×SD16—342no—
405C2×SD16—342no—
406C22×C8—171no—
407C22×C8—171no—
408C22×C8—171no—
409C22×C8—171no—
410C4⋊Q8—28no—
411C4⋊Q8—22no—
412C4⋊Q8—22no—
413#125—28no—
414#125—28no—
415#125—22no—
416#125—22no—
417#125—22no—
418#125C22⋊D1714yes—
419C4×Q8—28no—
420C4×Q8—22no—
421C4×Q8—22no—
422C4×Q8—22no—
423C4×Q8—22no—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×Q8—22no—
427C4×Q8C22⋊D1714yes—
428C4×D4—342no—
429C4×D4—342no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—171no—
439C2×C4⋊C4—171no—
440C2×C4⋊C4—171no—
441C2×C4⋊C4—341no—
442C2×C4⋊C4—341no—
443C2×C4⋊C4C22⋊D1711yescharacteristic
444C2×C4⋊C4C22⋊D1711yescharacteristic
445C2×C4⋊C4C22⋊D1711yescharacteristic
446C2×C4⋊C4C22×D1711yescharacteristic
447C2×C4⋊C4C22×D1711yescharacteristic
448C2×C4⋊C4C22×D1711yescharacteristic
449C2×C22⋊C4—341no—
450C2×C22⋊C4—341no—
451C2×C22⋊C4—341no—
452C2×C22⋊C4—341no—
453C2×C22⋊C4—341no—
454C2×C42—21no—
455C2×C42—21no—
456C2×C42—21no—
457C2×C42C22×D1711yescharacteristic
458C2×C42C22×D1711yescharacteristic
459C2×C42C22×D1711yescharacteristic
460#107—342no—
461#106—342no—
462#106—342no—
463C4⋊C8—342no—
464#103—342no—
465#103—342no—
466#103—342no—
467#102—342no—
468#102—342no—
469#102—342no—
470C4×C8—342no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475C17#2433411yescharacteristic, complemented, Sylow
476C24—341no—
477C2×Q8—24no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—22no—
482C2×Q8—22no—
483C2×Q8—41no—
484C2×Q8—42no—
485C2×Q8—42no—
486C2×Q8Q8×D1712yes—
487C2×Q8#6325612yes—
488C2×Q8C2×C22⋊D1712yes—
489C2×D4—172no—
490C2×D4—341no—
491C2×D4—341no—
492C2×D4—341no—
493C2×D4—341no—
494C2×D4—341no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—682no—
503C22×C4—1361no—
504C22×C4—1361no—
505C22×C4C2×C22⋊D1711yescharacteristic
506C22×C4C2×C22⋊D1711yescharacteristic
507C22×C4C2×C22⋊D1711yescharacteristic
508C22×C4C2×C22⋊D1711yescharacteristic
509C22×C4C2×C22⋊D1711yescharacteristic
510C22×C4C2×C22⋊D1711yescharacteristic
511C22×C4C23×D1711yescharacteristic
512SD16—348no—
513C2×C8—172no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—341no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—341no—
543C4⋊C4—341no—
544C4⋊C4—341no—
545C4⋊C4—1361no—
546C4⋊C4—1361no—
547C4⋊C4C2×C22⋊D1712yes—
548C4⋊C4C2×C22⋊D1712yes—
549C4⋊C4C2×C22⋊D1712yes—
550C22⋊C4—682no—
551C22⋊C4—682no—
552C22⋊C4—682no—
553C22⋊C4—1361no—
554C22⋊C4—1361no—
555C42—24no—
556C42—24no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—24no—
562C42C2×C22⋊D1712yes—
563C42C2×C22⋊D1712yes—
564C23—342no—
565C23—681no—
566C23—681no—
567C23—681no—
568C23—681no—
569C23—681no—
570C23—681no—
571C23C22×C22⋊D1711yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—22no—
575Q8—44no—
576Q8—48no—
577Q8#1058763214yes—
578D4—342no—
579D4—342no—
580D4—681no—
581D4—681no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—1364no—
619C2×C4#1058763612yes—
620C2×C4#1058763912yes—
621C2×C4#1058763912yes—
622C2×C4C2×Q8×D1711yescharacteristic
623C2×C4C2×#6325611yescharacteristic
624C2×C4#1058769411yescharacteristic
625C2×C4#1058769411yescharacteristic
626C2×C4C22×C22⋊D1711yescharacteristic
627C2×C4C22×C22⋊D1711yescharacteristic
628C8—342no—
629C8—681no—
630C8—681no—
631C8—681no—
632C22—682no—
633C22—682no—
634C22—682no—
635C22—1361no—
636C22—1361no—
637C22—1361no—
638C22—1361no—
639C22C2×#1058763911yescharacteristic
640C22#1176347011yescharacteristic
641C22Q8×C22⋊D1711yescharacteristic
642C22C2×#1058763611yescharacteristic
643C22#1176360711yescharacteristic
644C22#1176361711yescharacteristic
645C22#1176361711yescharacteristic
646C4—22no—
647C4—22no—
648C4—24no—
649C4—22no—
650C4—41no—
651C4—42no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—42no—
656C4—1361no—
657C4—1361no—
658C4—1361no—
659C4C2×#1058763211yescharacteristic
660C4#1176338311yescharacteristic
661C4#1176347011yescharacteristic
662C4Q8×C22⋊D1711yescharacteristic
663C2—1362no—
664C2#1453214311yescharacteristic
665C2#1453800211yescharacteristic
666C2#1453807011yescharacteristic
667C2#1454148011yescharacteristic
668C2#1454155211yescharacteristic
669C2#1454165511yescharacteristic
670C2#1454172311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#557C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenterC2×#557D17
hypocenterC17#24334
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #14530119C2semidirect
5. #14530114C2semidirect
8. #14529750C2semidirect
9. #14529721C2semidirect
12. #14529052C2semidirect
13. #14529038C2semidirect
14. #14528799C2semidirect
15. C2×C17×#557C2semidirect
108. C2×#63230Q8semidirect
297. C2×#557D17semidirect
475. C17#24334semidirect

Maximal subgroups

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

subgroupindex
C2×#117631062
C2×C4×#632302
#145341422
#145301192
#145301142
#145299272
#145299172
#145297502
#145297212
#145291392
#145290522
#145290382
#145287992
C2×C17×#5572
#2433417