Finite Group Databasebeta

← #17090579

Subgroups — #17090579

Order 4352.

1238 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763168C211yesmaximal, characteristic, complemented
2C2×#11763159C211yesmaximal, characteristic, complemented
3#14534605C211yesmaximal, characteristic, complemented
4#14530027C212yesmaximal, complemented
5#14530020C212yesmaximal, complemented
6#14529605C211yesmaximal, characteristic, complemented
7#14529596C211yesmaximal, characteristic, complemented
8#14528724C212yesmaximal
9#14528719C212yesmaximal, complemented
10#14528550C211yesmaximal, characteristic, complemented
11C2×C17×#536C211yesmaximal, characteristic
12C2×#10587555C2211yescharacteristic
13#11763168—24no—
14#11763159—24no—
15C2×#10587506C2212yes—
16C2×#10587506C2212yes—
17C2×#10587506C2212yes—
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587503C2212yes—
22C2×#10587503C2212yes—
23C2×C22⋊#3622C2212yes—
24C2×#10587474C2211yescharacteristic
25C2×C4×#3622C2211yescharacteristic
26#11762748—21no—
27#11762745—21no—
28#11762729—21no—
29#11762721—21no—
30#11762707C2212yes—
31#11762675C2211yescharacteristic
32#11762669C2212yes—
33#11762564C2212yes—
34C22×#41×C17C2211yescharacteristic
35C23×Q8×C17C2211yescharacteristic
36C17×#536—24no—
37C17×#536—24no—
38C2×C17×C4⋊Q8C2211yescharacteristic
39C2×C17×C4⋊Q8C2211yescharacteristic
40C2×C17×#124C2211yescharacteristic
41C2×C17×#124C2211yescharacteristic
42C2×C17×#123—21no—
43C2×C17×#123—21no—
44C2×C17×C22⋊Q8—21no—
45C2×C17×C22⋊Q8—21no—
46C2×C17×#117C2211yescharacteristic
47C2×#41×C17—22no—
48C2×#41×C17—22no—
49C2×#41×C17—24no—
50C2×#41×C17—42no—
51C22×Q8×C17—22no—
52C22×Q8×C17—22no—
53C22×Q8×C17—21no—
54C22×Q8×C17—24no—
55C22×Q8×C17—42no—
56C22×Q8×C17C2311yescharacteristic
57C22×Q8×C17C2311yescharacteristic
58C22×Q8×C17C2311yescharacteristic
59C22×D4×C17—21no—
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17—21no—
63C23×C4×C17C2311yescharacteristic
64C22×Dic34—21no—
65C17×C4⋊Q8—22no—
66C17×C4⋊Q8—22no—
67C17×C4⋊Q8—22no—
68C17×C4⋊Q8D414yes—
69C17×#124—22no—
70C17×#124—22no—
71C17×#124—22no—
72C17×#124D414yes—
73C17×#123—28no—
74C17×#123—28no—
75C17×C22⋊Q8—28no—
76C17×C22⋊Q8—28no—
77C17×#117—22no—
78C17×#117—22no—
79C2×C4⋊C4×C17—21no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17—21no—
83C2×C4⋊C4×C17D411yescharacteristic
84C2×C4⋊C4×C17D411yescharacteristic
85C2×C22⋊C4×C17—21no—
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17—21no—
89C2×C22⋊C4×C17D411yescharacteristic
90C2×C22⋊C4×C17D411yescharacteristic
91C2×C42×C17C2311yescharacteristic
92C2×C42×C17C2311yescharacteristic
93C2×C22⋊Dic17—22no—
94C2×C4⋊Dic17—21no—
95C2×C4⋊Dic17C2312yes—
96C2×C4⋊Dic17C2312yes—
97C2×#63225—22no—
98C22×#3622C2312yes—
99C22×#3622C2312yes—
100#10587555—22no—
101#10587555—22no—
102#10587506—24no—
103#10587506—24no—
104#10587506—24no—
105#10587506—42no—
106#10587506—42no—
107#10587506—42no—
108#10587503—24no—
109#10587503—42no—
110#10587502—22no—
111#10587502—21no—
112#10587502—22no—
113#10587502—21no—
114C22⋊#3622—24no—
115#10587474—22no—
116C4×#3622—22no—
117C24×C17—21no—
118C24×C17C2×D411yescharacteristic
119#41×C17—48no—
120#41×C17—48no—
121C2×Q8×C17—24no—
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—22no—
126C2×Q8×C17—22no—
127C2×Q8×C17—22no—
128C2×Q8×C17—22no—
129C2×Q8×C17—22no—
130C2×Q8×C17—22no—
131C2×Q8×C17—48no—
132C2×Q8×C17—48no—
133C2×Q8×C17—42no—
134C2×Q8×C17—42no—
135C2×Q8×C17C2×D412yes—
136C2×Q8×C17C2×D412yes—
137C2×Q8×C17C2×D412yes—
138C2×D4×C17—24no—
139C2×D4×C17—21no—
140C2×D4×C17—22no—
141C2×D4×C17—22no—
142C2×D4×C17—42no—
143C2×D4×C17—42no—
144C2×D4×C17C2×D412yes—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—22no—
148C22×C4×C17—24no—
149C22×C4×C17—24no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—42no—
154C22×C4×C17—42no—
155C22×C4×C17—42no—
156C22×C4×C17—42no—
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17#4114yes—
163C22×C4×C17C2411yescharacteristic
164C22×Dic17—22no—
165C22×Dic17—22no—
166C2×Dic34—42no—
167C2×Dic34—82no—
168C4⋊C4×C17—24no—
169C4⋊C4×C17—24no—
170C4⋊C4×C17—22no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—24no—
173C4⋊C4×C17—24no—
174C22⋊C4×C17—24no—
175C22⋊C4×C17—24no—
176C22⋊C4×C17—24no—
177C22⋊C4×C17—24no—
178C22⋊C4×C17—22no—
179C22⋊C4×C17—22no—
180C42×C17—21no—
181C42×C17—21no—
182C42×C17C2×D412yes—
183C42×C17C2×D412yes—
184C22⋊Dic17—82no—
185C4⋊Dic17—22no—
186C4⋊Dic17—22no—
187C4⋊Dic17—22no—
188C4⋊Dic17—22no—
189C4⋊Dic17—42no—
190#63225—82no—
191C2×#3622—22no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622—22no—
196C2×#3622—22no—
197#25866—171nomaximal, Sylow
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—22no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17#12412yes—
208C23×C17C22×D411yescharacteristic
209Q8×C17—24no—
210Q8×C17—24no—
211Q8×C17—24no—
212Q8×C17—48no—
213Q8×C17—42no—
214Q8×C17—42no—
215Q8×C17—42no—
216D4×C17—24no—
217D4×C17—48no—
218D4×C17—42no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—24no—
224C2×C4×C17—24no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—48no—
233C2×C4×C17—48no—
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#12012yes—
241C2×C4×C17#12012yes—
242C2×C4×C17#12412yes—
243C2×C4×C17C22×D411yescharacteristic
244C2×C4×C17C22×D411yescharacteristic
245C2×C4×C17C2×#4112yes—
246C2×C4×C17H(5,𝔽2)12yes—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—42no—
251C2×Dic17—42no—
252C2×Dic17—42no—
253Dic34—84no—
254#3622—42no—
255#3622—42no—
256C2×#536D1711yescharacteristic
257C2×#489—171no—
258C2×#487—171no—
259#2628—171no—
260#2043—172no—
261#2042—172no—
262#1960—171no—
263#1958—171no—
264#1788—172no—
265#1787—172no—
266#1750—171no—
267C22×C17—24no—
268C22×C17—22no—
269C22×C17—42no—
270C22×C17—41no—
271C22×C17—41no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—42no—
275C22×C17—82no—
276C22×C17—82no—
277C22×C17C2×#12011yescharacteristic
278C22×C17C2×#12411yescharacteristic
279C22×C17#53511yescharacteristic
280C22×C17#57712yes—
281C22×C17#57812yes—
282C4×C17—22no—
283C4×C17—22no—
284C4×C17—22no—
285C4×C17—42no—
286C4×C17—42no—
287C4×C17—42no—
288C4×C17—41no—
289C4×C17—41no—
290C4×C17#54612yes—
291C4×C17#54612yes—
292Dic17—82no—
293Dic17—82no—
294C22×#41C2×D1711yescharacteristic
295C23×Q8C2×D1711yescharacteristic
296#536—24no—
297#536—24no—
298C2×C4⋊Q8—171no—
299C2×C4⋊Q8C2×D1711yescharacteristic
300C2×C4⋊Q8C2×D1711yescharacteristic
301C2×#124C2×D1711yescharacteristic
302C2×#124C2×D1711yescharacteristic
303C2×#123—21no—
304C2×#123—21no—
305C2×C22⋊Q8—21no—
306C2×C22⋊Q8—21no—
307C2×#117C2×D1711yescharacteristic
308#489—344no—
309#487—344no—
310C2×#103—172no—
311C2×#103—172no—
312C2×#103—172no—
313C2×#103—172no—
314C2×#103—172no—
315C2×#103—172no—
316C2×#102—172no—
317C2×#102—172no—
318C2×C22⋊C8—172no—
319C2×#97—171no—
320C2×C4×C8—171no—
321#400—341no—
322#399—341no—
323#395—341no—
324#393—341no—
325#389—172no—
326#382—171no—
327#380—172no—
328#336—172no—
329C2×C17—42no—
330C2×C17—82no—
331C2×C17#243511yescharacteristic
332C2×C17#301011yescharacteristic
333C2×C17#301111yescharacteristic
334C2×C17#313712yes—
335C2×C17#322512yes—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—24no—
339C2×#41—42no—
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—21no—
343C22×Q8—24no—
344C22×Q8—42no—
345C22×Q8—341no—
346C22×Q8C22×D1711yescharacteristic
347C22×Q8C22×D1711yescharacteristic
348C22×Q8C22×D1711yescharacteristic
349C22×D4—21no—
350C22×D4C22×D1711yescharacteristic
351C23×C4—21no—
352C23×C4—21no—
353C23×C4C22×D1711yescharacteristic
354C22×C8—172no—
355C22×C8—172no—
356C4⋊Q8—22no—
357C4⋊Q8—22no—
358C4⋊Q8—22no—
359C4⋊Q8—342no—
360C4⋊Q8—342no—
361C4⋊Q8C22⋊D1714yes—
362#124—22no—
363#124—22no—
364#124—22no—
365#124C22⋊D1714yes—
366#123—28no—
367#123—28no—
368C22⋊Q8—28no—
369C22⋊Q8—28no—
370#117—22no—
371#117—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—172no—
377C2×C4⋊C4—172no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—342no—
380C2×C4⋊C4C22⋊D1711yescharacteristic
381C2×C4⋊C4C22⋊D1711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—342no—
387C2×C22⋊C4C22⋊D1711yescharacteristic
388C2×C22⋊C4C22⋊D1711yescharacteristic
389C2×C42C22×D1711yescharacteristic
390C2×C42C22×D1711yescharacteristic
391#103—344no—
392#103—344no—
393#103—344no—
394#103—682no—
395#103—682no—
396#103—682no—
397#102—344no—
398#102—682no—
399C22⋊C8—344no—
400#97—342no—
401C4×C8—342no—
402#95—342no—
403#95—341no—
404#95—341no—
405#95—342no—
406C17#2586611yescharacteristic, complemented, Sylow
407C24—21no—
408C24C2×C22⋊D1711yescharacteristic
409#41—48no—
410#41—48no—
411C2×Q8—24no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—682no—
426C2×Q8—1362no—
427C2×Q8C2×C22⋊D1712yes—
428C2×Q8C2×C22⋊D1712yes—
429C2×Q8C2×C22⋊D1712yes—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4C2×C22⋊D1712yes—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4#6325414yes—
452C22×C4C2×C22⋊D1711yescharacteristic
453C22×C4C2×C22⋊D1711yescharacteristic
454C22×C4C2×C22⋊D1711yescharacteristic
455C22×C4C2×C22⋊D1711yescharacteristic
456C22×C4C2×C22⋊D1711yescharacteristic
457C22×C4C23×D1711yescharacteristic
458C2×C8—342no—
459C2×C8—342no—
460C2×C8—342no—
461C2×C8—342no—
462C2×C8—342no—
463C2×C8—342no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—342no—
471C4⋊C4—342no—
472C4⋊C4—342no—
473C4⋊C4—342no—
474C4⋊C4—682no—
475C4⋊C4—1362no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—1362no—
483C42—21no—
484C42—21no—
485C42C2×C22⋊D1712yes—
486C42C2×C22⋊D1712yes—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—21no—
491C23—22no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23#1058762212yes—
497C23C22×C22⋊D1711yescharacteristic
498Q8—24no—
499Q8—24no—
500Q8—24no—
501Q8—48no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—1364no—
506D4—24no—
507D4—48no—
508D4—42no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—682no—
531C2×C4—682no—
532C2×C4—682no—
533C2×C4—682no—
534C2×C4—682no—
535C2×C4—682no—
536C2×C4#1058762212yes—
537C2×C4#1058763912yes—
538C2×C4#1058763912yes—
539C2×C4C2×#6325412yes—
540C2×C4#1058769112yes—
541C2×C4C22×C22⋊D1711yescharacteristic
542C2×C4C22×C22⋊D1711yescharacteristic
543C8—682no—
544C8—682no—
545C22—22no—
546C22—24no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22C2×#1058763911yescharacteristic
556C22C2×#1058762211yescharacteristic
557C22#1176337611yescharacteristic
558C22#1176361712yes—
559C22#1176362412yes—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—1362no—
569C4—1362no—
570C4#1176345312yes—
571C4#1176345312yes—
572C2—42no—
573C2—82no—
574C2#1453243111yescharacteristic
575C2#1453807011yescharacteristic
576C2#1453807911yescharacteristic
577C2#1453914912yes—
578C2#1453989312yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#536C2
socleC23×C17C22×D4
prime radicalC22×#41×C17C22
hypercenterC2×#536D17
hypocenterC17#25866
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#11763168C2semidirect
2. C2×#11763159C2semidirect
3. #14534605C2semidirect
4. #14530027C2semidirect
5. #14530020C2semidirect
6. #14529605C2semidirect
7. #14529596C2semidirect
9. #14528719C2semidirect
10. #14528550C2semidirect
406. C17#25866semidirect

Maximal subgroups

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

subgroupindex
C2×#117631682
C2×#117631592
#145346052
#145300272
#145300202
#145296052
#145295962
#145287242
#145287192
#145285502
C2×C17×#5362
#2586617