Finite Group Databasebeta

← #17092394

Subgroups — #17092394

Order 4352.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532509C211yesmaximal, characteristic
2#14531669C211yesmaximal, characteristic
3#14530357C211yesmaximal, characteristic, complemented
4#14530288C211yesmaximal, characteristic, complemented
5#14530284C211yesmaximal, characteristic, complemented
6#14530121C211yesmaximal, characteristic, complemented
7#14529740C211yesmaximal, characteristic, complemented
8#14529551C211yesmaximal, characteristic, complemented
9#14529365C211yesmaximal, characteristic, complemented
10#14529361C211yesmaximal, characteristic, complemented
11#14528979C211yesmaximal, characteristic
12#14528966C211yesmaximal, characteristic
13#14528815C211yesmaximal, characteristic
14#14528505C211yesmaximal, characteristic
15C17×#1825C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587576C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587488C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25#11762751—21no—
26#11762741—21no—
27#11762708C2211yescharacteristic
28#11762708C2211yescharacteristic
29#11762708C2211yescharacteristic
30#11762705C2211yescharacteristic
31#11762702—21no—
32#11762702—21no—
33#11762684—21no—
34#11762675—21no—
35#11762674C2211yescharacteristic
36#11762669C2211yescharacteristic
37#11762667—21no—
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4×C22⋊Dic17C2211yescharacteristic
41#11762655—21no—
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#117C2211yescharacteristic
53C2×C17×C22⋊C8C2211yescharacteristic
54C17×#380C2211yescharacteristic
55C17×#340C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C17×#117—22no—
64C17×#117Q814yes—
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C22⋊C4×C17—21no—
70C2×C22⋊C4×C17—21no—
71C2×C42×C17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17D411yescharacteristic
74C2×C22⋊Dic17D411yescharacteristic
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—22no—
81C2×C4⋊Dic17—41no—
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×#63225—21no—
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C2×#63223—22no—
93C2×#63223—22no—
94C2×#63223—41no—
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97#10587576—22no—
98#10587576—41no—
99C17×C22⋊C8—22no—
100C17×#95—21no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502D411yescharacteristic
111#10587502D411yescharacteristic
112#10587488—22no—
113#10587488—41no—
114#10587485—22no—
115#10587485—41no—
116#10587479—22no—
117#10587478—41no—
118C24×C17C2×D411yescharacteristic
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×Q812yes—
129C22×C4×C17#4112yes—
130C22×C4×C17C2411yescharacteristic
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—42no—
136C22×Dic17—42no—
137C22×Dic17—81no—
138C22×Dic17C2×D411yescharacteristic
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C2×C8×C17—21no—
143C2×C8×C17—21no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17C2×Q812yes—
155C4⋊C4×C17C2×Q812yes—
156C22⋊C4×C17—24no—
157C22⋊C4×C17—42no—
158C42×C17—24no—
159C22⋊Dic17—41no—
160C22⋊Dic17—41no—
161C22⋊Dic17—81no—
162C4⋊Dic17—21no—
163C4⋊Dic17—21no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—42no—
170C4⋊Dic17—42no—
171C4⋊Dic17—81no—
172C4⋊Dic17—81no—
173C4⋊Dic17—81no—
174C4⋊Dic17#4112yes—
175#63225—42no—
176#63225—81no—
177C4×Dic17—21no—
178C4×Dic17—21no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—42no—
182C4×Dic17—42no—
183#63223—44no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622#4112yes—
190#26019—171nomaximal, Sylow
191C23×C17—21no—
192C23×C17—21no—
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17C22⋊Q812yes—
198C23×C17C22×D411yescharacteristic
199C2×C4×C17—21no—
200C2×C4×C17—21no—
201C2×C4×C17—22no—
202C2×C4×C17—22no—
203C2×C4×C17—22no—
204C2×C4×C17—22no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17C22⋊Q812yes—
218C2×C4×C17C22×Q811yescharacteristic
219C2×C4×C17C2×#4111yescharacteristic
220C2×C4×C17C2×#4111yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C17H(5,𝔽2)11yescharacteristic
223C2×C4×C172−1+411yescharacteristic
224C2×Dic17—22no—
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—84no—
249C8×C17—41no—
250C8×C17—41no—
251#3622—22no—
252#3622—41no—
253#2442—171no—
254#2346—171no—
255#2101—171no—
256#2088—171no—
257#2088—171no—
258#2059—171no—
259#1989—171no—
260#1949—171no—
261#1919—171no—
262#1919—171no—
263#1853—171no—
264#1851—171no—
265#1825D1711yescharacteristic
266#1809—171no—
267#1736—171no—
268C22×C17—22no—
269C22×C17—22no—
270C22×C17—22no—
271C22×C17—41no—
272C22×C17—41no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17C2×C22⋊Q811yescharacteristic
276C22×C17#54611yescharacteristic
277C22×C17#54711yescharacteristic
278C22×C17#57511yescharacteristic
279C22×C17#57511yescharacteristic
280C22×C17#57711yescharacteristic
281C22×C17#57711yescharacteristic
282C4×C17—22no—
283C4×C17—42no—
284C4×C17—42no—
285C4×C17—81no—
286C4×C17—81no—
287C4×C17#53311yescharacteristic
288C4×C17#54411yescharacteristic
289C4×C17#55111yescharacteristic
290C4×C17#55211yescharacteristic
291Dic17—41no—
292Dic17—41no—
293Dic17—41no—
294Dic17—41no—
295Dic17—42no—
296Dic17—81no—
297Dic17—81no—
298Dic17—81no—
299C22×M16—171no—
300C2×#125—171no—
301C2×#117C2×D1711yescharacteristic
302C22×C4⋊C4—171no—
303C2×#107—171no—
304C2×#106—171no—
305C2×#106—171no—
306C2×#106—171no—
307C2×C4⋊C8—171no—
308C2×C4⋊C8—171no—
309C2×C22⋊C8C2×D1711yescharacteristic
310#400—341no—
311#398—341no—
312#389—171no—
313#389—171no—
314#389—171no—
315#389—171no—
316#388—341no—
317#388—341no—
318#385—341no—
319#382—171no—
320#382—341no—
321#380—171no—
322#380—341no—
323#380C2×D1711yescharacteristic
324C4×C4⋊C4—171no—
325C4×C4⋊C4—171no—
326C4×C22⋊C4—171no—
327#376—341no—
328#340—171no—
329#340—171no—
330#340—171no—
331#340—171no—
332#340—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340C2×D1711yescharacteristic
337#340C2×D1711yescharacteristic
338#340C2×D1711yescharacteristic
339#340C2×D1711yescharacteristic
340#336—171no—
341#336—171no—
342C2×C17—42no—
343C2×C17#243811yescharacteristic
344C2×C17#308011yescharacteristic
345C2×C17#308411yescharacteristic
346C2×C17#332111yescharacteristic
347C2×C17#332311yescharacteristic
348C2×C17#334811yescharacteristic
349C2×C17#335111yescharacteristic
350C23×C4—341no—
351C23×C4C22×D1711yescharacteristic
352C2×M16—342no—
353C2×M16—342no—
354C2×M16—681no—
355C22×C8—171no—
356C22×C8—171no—
357C22×C8C22×D1711yescharacteristic
358C22×C8C22×D1711yescharacteristic
359#125—342no—
360#125—681no—
361#117—22no—
362#117Dic3414yes—
363C2×C4⋊C4—171no—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—342no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4—341no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—681no—
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—171no—
383C2×C22⋊C4—171no—
384C2×C22⋊C4—341no—
385C2×C42—21no—
386C2×C42—171no—
387C2×C42—171no—
388C2×C42—341no—
389C2×C42—341no—
390#107—681no—
391#106—342no—
392#106—342no—
393#106—681no—
394C4⋊C8—342no—
395C4⋊C8—681no—
396C22⋊C8—22no—
397#95—21no—
398#95—171no—
399#95—171no—
400#95—341no—
401#95—341no—
402#95—341no—
403#95—341no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409C17#2601911yescharacteristic, complemented, Sylow
410C24D4×D1711yescharacteristic
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—171no—
420C22×C4—171no—
421C22×C4—171no—
422C22×C4—171no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—682no—
428C22×C4—682no—
429C22×C4—1361no—
430C22×C4C2×Dic3412yes—
431C22×C4D4×D1711yescharacteristic
432C22×C4#6325412yes—
433C22×C4C23×D1711yescharacteristic
434M16—684no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—172no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—172no—
454C4⋊C4—341no—
455C4⋊C4—341no—
456C4⋊C4—341no—
457C4⋊C4—341no—
458C4⋊C4—341no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—682no—
462C4⋊C4—682no—
463C4⋊C4—682no—
464C4⋊C4—1361no—
465C4⋊C4—1361no—
466C4⋊C4—1361no—
467C4⋊C4—1361no—
468C4⋊C4C2×Dic3412yes—
469C4⋊C4C2×Dic3412yes—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—681no—
473C22⋊C4—681no—
474C22⋊C4—1361no—
475C42—24no—
476C42—341no—
477C42—341no—
478C42—341no—
479C42—341no—
480C42—682no—
481C42—682no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1058756412yes—
489C23C2×D4×D1711yescharacteristic
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—341no—
509C2×C4—341no—
510C2×C4—341no—
511C2×C4—341no—
512C2×C4—341no—
513C2×C4—341no—
514C2×C4—341no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—342no—
521C2×C4—681no—
522C2×C4—681no—
523C2×C4—681no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—1364no—
533C2×C4#1058756412yes—
534C2×C4C22×Dic3411yescharacteristic
535C2×C4C2×#6325411yescharacteristic
536C2×C4#41×D1711yescharacteristic
537C2×C4#41×D1711yescharacteristic
538C2×C4#1058769611yescharacteristic
539C2×C4#1058769711yescharacteristic
540C8—41no—
541C8—41no—
542C8—342no—
543C8—681no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1058756411yescharacteristic
552C22#1176345111yescharacteristic
553C22#1176345711yescharacteristic
554C22#1176360511yescharacteristic
555C22#1176360811yescharacteristic
556C22#1176361611yescharacteristic
557C22#1176361611yescharacteristic
558C4—22no—
559C4—42no—
560C4—42no—
561C4—81no—
562C4—81no—
563C4—681no—
564C4—681no—
565C4—681no—
566C4—682no—
567C4—681no—
568C4—1361no—
569C4—1361no—
570C4—1361no—
571C4#1176336211yescharacteristic
572C4#1176343711yescharacteristic
573C4#1176348211yescharacteristic
574C4#1176349611yescharacteristic
575C2—42no—
576C2#1453246211yescharacteristic
577C2#1453860411yescharacteristic
578C2#1453864411yescharacteristic
579C2#1454097311yescharacteristic
580C2#1454099811yescharacteristic
581C2#1454129211yescharacteristic
582C2#1454133711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1825C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1825D17
hypocenterC17#26019
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
3. #14530357C2semidirect
4. #14530288C2semidirect
5. #14530284C2semidirect
6. #14530121C2semidirect
7. #14529740C2semidirect
8. #14529551C2semidirect
9. #14529365C2semidirect
10. #14529361C2semidirect
409. C17#26019semidirect

Maximal subgroups

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

subgroupindex
#145325092
#145316692
#145303572
#145302882
#145302842
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C17×#18252
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