Finite Group Databasebeta

← #17107425

Subgroups — #17107425

Order 4352.

680 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533217C211yesmaximal, characteristic
2#14530192C211yesmaximal, characteristic, complemented
3#14530179C211yesmaximal, characteristic, complemented
4#14529977C211yesmaximal, characteristic
5#14529977C211yesmaximal, characteristic
6#14529547C211yesmaximal, characteristic, complemented
7#14529524C211yesmaximal, characteristic, complemented
8#14529399C211yesmaximal, characteristic
9#14529369C211yesmaximal, characteristic, complemented
10#14529366C211yesmaximal, characteristic, complemented
11#14529052C211yesmaximal, characteristic, complemented
12#14528983C211yesmaximal, characteristic
13#14528879C211yesmaximal, characteristic
14#14528614C211yesmaximal, characteristic, complemented
15C17×#2362C211yesmaximal, characteristic
16C2×#10587478C2211yescharacteristic
17C2×#10587479C2211yescharacteristic
18C2×#10587479C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×C22⋊#3622C2211yescharacteristic
27C2×#10587474C2211yescharacteristic
28#11762751—21no—
29#11762732—21no—
30#11762707C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762702—21no—
33#11762696—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
50C2×C17×#117C2211yescharacteristic
51C17×#391C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#388—21no—
54C17×#388—21no—
55C17×#387—21no—
56C4×C4⋊C4×C17—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C17×C22⋊Q8—22no—
62C17×C22⋊Q8—22no—
63C4×Q8×C17—22no—
64C4×Q8×C17—41no—
65C17×#117—22no—
66C17×#117—41no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17D411yescharacteristic
76C2×C22⋊C4×C17D411yescharacteristic
77C2×C42×C17—21no—
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
97C17×#95—21no—
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95D411yescharacteristic
102C17×#95D411yescharacteristic
103#10587506—22no—
104#10587506—22no—
105#10587506—22no—
106#10587506—41no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112C22⋊#3622—22no—
113C22⋊#3622—41no—
114#10587479—22no—
115#10587479—22no—
116#10587479—22no—
117#10587478—41no—
118C4⋊#3622—22no—
119#10587474—41no—
120C24×C17C2×D411yescharacteristic
121C2×Q8×C17—21no—
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—21no—
126C2×Q8×C17#4112yes—
127C22×C4×C17—21no—
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×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2411yescharacteristic
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C4⋊C4×C17—24no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—42no—
153C4⋊C4×C17—42no—
154C4⋊C4×C17#4112yes—
155C4⋊C4×C17#4112yes—
156C22⋊C4×C17—24no—
157C22⋊C4×C17—22no—
158C22⋊C4×C17—41no—
159C22⋊C4×C17—42no—
160C42×C17—21no—
161C42×C17—21no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17—42no—
165C42×C17—42no—
166C42×C17—42no—
167C22⋊Dic17—81no—
168C22⋊Dic17—81no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—81no—
178C4⋊Dic17—81no—
179#63225—42no—
180#63225—81no—
181C4×Dic17—81no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
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#4112yes—
194#27060—171nomaximal, Sylow
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17C22×D411yescharacteristic
203Q8×C17—22no—
204Q8×C17—22no—
205Q8×C17—41no—
206Q8×C17—41no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—22no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
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#13712yes—
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17H(5,𝔽2)11yescharacteristic
243C2×C4×C172−1+411yescharacteristic
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256#3622—22no—
257#3622—41no—
258#3622—41no—
259#3622—41no—
260#2497—171no—
261#2362D1711yescharacteristic
262#2072—171no—
263#2068—171no—
264#2035—171no—
265#2035—171no—
266#1948—171no—
267#1944—171no—
268#1924—171no—
269#1920—171no—
270#1920—171no—
271#1865—171no—
272#1854—171no—
273#1825—171no—
274#1763—171no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17—41no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17#52511yescharacteristic
283C22×C17#54611yescharacteristic
284C22×C17#54711yescharacteristic
285C22×C17C2×#13711yescharacteristic
286C22×C17#57511yescharacteristic
287C22×C17#57711yescharacteristic
288C22×C17#57711yescharacteristic
289C4×C17—22no—
290C4×C17—41no—
291C4×C17—42no—
292C4×C17—41no—
293C4×C17—42no—
294C4×C17—41no—
295C4×C17—42no—
296C4×C17—41no—
297C4×C17—81no—
298C4×C17#52911yescharacteristic
299C4×C17#53911yescharacteristic
300C4×C17#55311yescharacteristic
301C4×C17#55611yescharacteristic
302Dic17—81no—
303Dic17—81no—
304Dic17—81no—
305Dic17—81no—
306C2×C22⋊Q8C2×D1711yescharacteristic
307C2×C4×Q8C2×D1711yescharacteristic
308C2×#117C2×D1711yescharacteristic
309C2×#107—171no—
310C2×#106—171no—
311C2×#106—171no—
312C2×#106—171no—
313C2×C4⋊C8—171no—
314C2×#103—171no—
315C2×#103—171no—
316C2×#103—171no—
317C2×#103—171no—
318C2×C22⋊C8—171no—
319C2×C22⋊C8—171no—
320C2×#97—171no—
321#400—341no—
322#396—341no—
323#391C2×D1711yescharacteristic
324#389—171no—
325#389—171no—
326#389C2×D1711yescharacteristic
327#388—21no—
328#388—21no—
329#388—341no—
330#387—21no—
331#387—341no—
332#384—171no—
333#380—171no—
334#380—171no—
335C4×C4⋊C4—21no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4C2×D1711yescharacteristic
338C4×C22⋊C4C2×D1711yescharacteristic
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#336—171no—
348#336—171no—
349C2×C17—81no—
350C2×C17#246611yescharacteristic
351C2×C17#311611yescharacteristic
352C2×C17#312611yescharacteristic
353C2×C17#332111yescharacteristic
354C2×C17#333111yescharacteristic
355C2×C17#334411yescharacteristic
356C2×C17#335011yescharacteristic
357C22×Q8C22×D1711yescharacteristic
358C23×C4C22×D1711yescharacteristic
359C22×C8—171no—
360C22×C8—171no—
361C22×C8—171no—
362C22×C8—171no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C4×Q8—22no—
366C4×Q8—41no—
367#117—22no—
368#117—41no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C4⋊C4C22×D1711yescharacteristic
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C22⋊C4C22⋊D1711yescharacteristic
388C2×C22⋊C4C22⋊D1711yescharacteristic
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—341no—
393C2×C42C22×D1711yescharacteristic
394C2×C42C22×D1711yescharacteristic
395#107—681no—
396#106—342no—
397#106—342no—
398#106—342no—
399C4⋊C8—342no—
400#103—342no—
401#103—342no—
402#103—342no—
403#103—681no—
404C22⋊C8—342no—
405C22⋊C8—681no—
406#97—681no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95C22⋊D1711yescharacteristic
417#95C22⋊D1711yescharacteristic
418C17#2706011yescharacteristic, complemented, Sylow
419C24C2×C22⋊D1711yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8#6325212yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—341no—
437C22×C4—341no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4C2×C22⋊D1711yescharacteristic
441C22×C4C2×C22⋊D1711yescharacteristic
442C22×C4C2×C22⋊D1711yescharacteristic
443C22×C4C2×C22⋊D1711yescharacteristic
444C22×C4C2×C22⋊D1711yescharacteristic
445C22×C4C23×D1711yescharacteristic
446C2×C8—172no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—341no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—682no—
474C4⋊C4—1361no—
475C4⋊C4—1361no—
476C4⋊C4—1361no—
477C4⋊C4#6325212yes—
478C4⋊C4#6325412yes—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—41no—
482C22⋊C4—42no—
483C22⋊C4—1361no—
484C22⋊C4—1361no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—1361no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C22×C22⋊D1711yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4#1058763712yes—
548C2×C4C2×#6325211yescharacteristic
549C2×C4C2×#6325211yescharacteristic
550C2×C4C2×#6325411yescharacteristic
551C2×C4#1058769111yescharacteristic
552C2×C4#41×D1711yescharacteristic
553C2×C4#1058769711yescharacteristic
554C8—342no—
555C8—681no—
556C8—681no—
557C8—681no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22#1176332011yescharacteristic
566C22#1176345311yescharacteristic
567C22#1176345811yescharacteristic
568C22C2×#1058763711yescharacteristic
569C22#1176360711yescharacteristic
570C22#1176361711yescharacteristic
571C22#1176361711yescharacteristic
572C4—22no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—81no—
581C4—1361no—
582C4—1361no—
583C4—1361no—
584C4—1361no—
585C4#1176333911yescharacteristic
586C4#1176340111yescharacteristic
587C4#1176350911yescharacteristic
588C4#1176352911yescharacteristic
589C2—81no—
590C2#1453281411yescharacteristic
591C2#1453894611yescharacteristic
592C2#1453907511yescharacteristic
593C2#1454097011yescharacteristic
594C2#1454110011yescharacteristic
595C2#1454123911yescharacteristic
596C2#1454132911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2362C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2362D17
hypocenterC17#27060
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #14530192C2semidirect
3. #14530179C2semidirect
6. #14529547C2semidirect
7. #14529524C2semidirect
9. #14529369C2semidirect
10. #14529366C2semidirect
11. #14529052C2semidirect
14. #14528614C2semidirect
418. C17#27060semidirect

Maximal subgroups

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

subgroupindex
#145332172
#145301922
#145301792
#145299772
#145295472
#145295242
#145293992
#145293692
#145293662
#145290522
#145289832
#145288792
#145286142
C17×#23622
#2706017