Finite Group Databasebeta

← #17107426

Subgroups — #17107426

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#14533213C211yesmaximal, characteristic
2#14531815C211yesmaximal, characteristic
3#14530195C211yesmaximal, characteristic, complemented
4#14530177C211yesmaximal, characteristic, complemented
5#14529977C211yesmaximal, characteristic
6#14529544C211yesmaximal, characteristic, complemented
7#14529528C211yesmaximal, characteristic, complemented
8#14529400C211yesmaximal, characteristic
9#14529372C211yesmaximal, characteristic, complemented
10#14529368C211yesmaximal, characteristic, complemented
11#14529054C211yesmaximal, characteristic, complemented
12#14528985C211yesmaximal, characteristic
13#14528876C211yesmaximal, characteristic
14#14528616C211yesmaximal, characteristic, complemented
15C17×#2035C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×#10587612C2211yescharacteristic
18C2×#10587489C2211yescharacteristic
19C2×#10587488C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×#10587486C2211yescharacteristic
26#11762749—21no—
27#11762732—21no—
28#11762716C2211yescharacteristic
29#11762708C2211yescharacteristic
30#11762708C2211yescharacteristic
31#11762705C2211yescharacteristic
32#11762702—21no—
33#11762702—21no—
34#11762697—21no—
35#11762695—21no—
36#11762691—21no—
37#11762678C2211yescharacteristic
38#11762669C2211yescharacteristic
39C4⋊C4×Dic17—21no—
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4×C22⋊Dic17C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×C22⋊Q8C2211yescharacteristic
53C2×C17×#106C2211yescharacteristic
54C2×C17×#103C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×C22⋊C8C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C17×C22⋊Q8—22no—
64C17×C22⋊Q8—22no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C22⋊C4×C17—21no—
70C2×C22⋊C4×C17—21no—
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17D411yescharacteristic
74C2×C22⋊Dic17D411yescharacteristic
75Q8×Dic17—22no—
76Q8×Dic17—41no—
77#10587612—22no—
78#10587612—41no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17C2311yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C17×#106—22no—
98C17×#103—22no—
99C17×#103—22no—
100C17×C22⋊C8—22no—
101C17×#95—21no—
102#10587506—22no—
103#10587506—41no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502D411yescharacteristic
113#10587502D411yescharacteristic
114C22⋊#3622—41no—
115#10587489—41no—
116#10587488—22no—
117#10587486—41no—
118#10587485—22no—
119#10587479—22no—
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×C17C2411yescharacteristic
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17C2×D411yescharacteristic
144C22×Dic17C2×D411yescharacteristic
145C22×Dic17C2×D411yescharacteristic
146C22×Dic17C2×D411yescharacteristic
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17#4112yes—
153C4⋊C4×C17—24no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17#4112yes—
160C22⋊C4×C17—24no—
161C22⋊C4×C17—81no—
162C22⋊Dic17—22no—
163C22⋊Dic17—41no—
164C22⋊Dic17—42no—
165C22⋊Dic17—81no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—42no—
174C4⋊Dic17—42no—
175C4⋊Dic17—81no—
176C4⋊Dic17#4112yes—
177#63225—42no—
178#63225—81no—
179#63225—81no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—21no—
183C4×Dic17—21no—
184C4×Dic17—42no—
185C4×Dic17—42no—
186C4×Dic17—42no—
187C4×Dic17—81no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
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—22no—
208C2×C4×C17—22no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×C4×C17H(5,𝔽2)11yescharacteristic
231C2×C4×C172−1+411yescharacteristic
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—22no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
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#13712yes—
256C8×C17—22no—
257C8×C17—41no—
258#3622—41no—
259#3622—41no—
260#2497—171no—
261#2362—171no—
262#2072—171no—
263#2068—171no—
264#2035—171no—
265#2035D1711yescharacteristic
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—42no—
290C4×C17—41no—
291C4×C17—42no—
292C4×C17—81no—
293C4×C17—81no—
294C4×C17#52911yescharacteristic
295C4×C17#53911yescharacteristic
296C4×C17#55311yescharacteristic
297C4×C17#55611yescharacteristic
298Dic17—22no—
299Dic17—41no—
300Dic17—42no—
301Dic17—41no—
302Dic17—41no—
303Dic17—81no—
304Dic17—81no—
305Dic17—81no—
306C2×C22⋊Q8C2×D1711yescharacteristic
307C2×C4×Q8—171no—
308C2×#117—171no—
309C2×#107—171no—
310C2×#106—171no—
311C2×#106—171no—
312C2×#106C2×D1711yescharacteristic
313C2×C4⋊C8—171no—
314C2×#103—171no—
315C2×#103—171no—
316C2×#103C2×D1711yescharacteristic
317C2×#103C2×D1711yescharacteristic
318C2×C22⋊C8—171no—
319C2×C22⋊C8C2×D1711yescharacteristic
320C2×#97—171no—
321#400—341no—
322#396—341no—
323#391—171no—
324#389—171no—
325#389—171no—
326#389—171no—
327#388—341no—
328#388—341no—
329#388—341no—
330#387—341no—
331#387—341no—
332#384—171no—
333#380—171no—
334#380C2×D1711yescharacteristic
335C4×C4⋊C4—171no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—341no—
338C4×C22⋊C4—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340C2×D1711yescharacteristic
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×C8C22×D1711yescharacteristic
362C22×C8C22×D1711yescharacteristic
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C4×Q8—342no—
366C4×Q8—681no—
367#117—342no—
368#117—681no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
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—171no—
386C2×C22⋊C4—171no—
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C42—171no—
390C2×C42—171no—
391C2×C42—341no—
392C2×C42—341no—
393C2×C42—341no—
394C2×C42—341no—
395#107—681no—
396#106—22no—
397#106—342no—
398#106—342no—
399C4⋊C8—342no—
400#103—22no—
401#103—22no—
402#103—342no—
403#103—681no—
404C22⋊C8—22no—
405C22⋊C8—681no—
406#97—681no—
407#95—21no—
408#95—171no—
409#95—171no—
410#95—341no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418C17#2706011yescharacteristic, complemented, Sylow
419C24D4×D1711yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8#6325412yes—
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—171no—
437C22×C4—171no—
438C22×C4—171no—
439C22×C4—171no—
440C22×C4—341no—
441C22×C4—341no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4D4×D1711yescharacteristic
445C22×C4C23×D1711yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8#6325412yes—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—172no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—682no—
473C4⋊C4—682no—
474C4⋊C4—682no—
475C4⋊C4—1361no—
476C4⋊C4—1361no—
477C4⋊C4—1361no—
478C4⋊C4#6325412yes—
479C22⋊C4—24no—
480C22⋊C4—81no—
481C22⋊C4—342no—
482C22⋊C4—681no—
483C22⋊C4—682no—
484C22⋊C4—1361no—
485C42—341no—
486C42—341no—
487C42—341no—
488C42—341no—
489C42—682no—
490C42—682no—
491C42—682no—
492C42—1361no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×D4×D1711yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—172no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—342no—
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—681no—
548C2×C4C2×#6325411yescharacteristic
549C2×C4C2×#6325411yescharacteristic
550C2×C4C2×#6325411yescharacteristic
551C2×C4#41×D1711yescharacteristic
552C2×C4#1058769611yescharacteristic
553C2×C4#1058769711yescharacteristic
554C8—22no—
555C8—41no—
556C8—681no—
557C8—681no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22#1176331511yescharacteristic
566C22#1176345111yescharacteristic
567C22#1176345911yescharacteristic
568C22#137×D1711yescharacteristic
569C22#1176360611yescharacteristic
570C22#1176361511yescharacteristic
571C22#1176361811yescharacteristic
572C4—42no—
573C4—41no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—342no—
578C4—681no—
579C4—681no—
580C4—682no—
581C4—681no—
582C4—1361no—
583C4—1361no—
584C4—1361no—
585C4#1176334011yescharacteristic
586C4#1176340311yescharacteristic
587C4#1176351111yescharacteristic
588C4#1176352811yescharacteristic
589C2—81no—
590C2#1453281511yescharacteristic
591C2#1453893911yescharacteristic
592C2#1453906911yescharacteristic
593C2#1454096811yescharacteristic
594C2#1454109811yescharacteristic
595C2#1454123411yescharacteristic
596C2#1454133511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2035C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2035D17
hypocenterC17#27060
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #14530195C2semidirect
4. #14530177C2semidirect
6. #14529544C2semidirect
7. #14529528C2semidirect
9. #14529372C2semidirect
10. #14529368C2semidirect
11. #14529054C2semidirect
14. #14528616C2semidirect
418. C17#27060semidirect

Maximal subgroups

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

subgroupindex
#145332132
#145318152
#145301952
#145301772
#145299772
#145295442
#145295282
#145294002
#145293722
#145293682
#145290542
#145289852
#145288762
#145286162
C17×#20352
#2706017