Finite Group Databasebeta

← #17092402

Subgroups — #17092402

Order 4352.

786 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533812C211yesmaximal, characteristic, complemented
2#14531732C211yesmaximal, characteristic
3#14530168C211yesmaximal, characteristic
4#14529920C211yesmaximal, characteristic
5#14529918C211yesmaximal, characteristic
6#14529735C211yesmaximal, characteristic, complemented
7#14529734C211yesmaximal, characteristic, complemented
8#14529539C211yesmaximal, characteristic, complemented
9#14529499C211yesmaximal, characteristic, complemented
10#14529357C211yesmaximal, characteristic, complemented
11#14529352C211yesmaximal, characteristic, complemented
12#14528910C211yesmaximal, characteristic, complemented
13#14528782C211yesmaximal, characteristic
14#102×Dic17C211yesmaximal, characteristic
15C17×#2067C211yesmaximal, characteristic
16C2×#10587555C2211yescharacteristic
17C2×#10587574—21no—
18C2×#10587574C2211yescharacteristic
19C2×D4×Dic17C2211yescharacteristic
20C2×#10587563—21no—
21C2×#10587478C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762730—21no—
28#11762729—21no—
29#11762712C2211yescharacteristic
30#11762708C2211yescharacteristic
31#11762706—21no—
32#11762703—21no—
33#11762697—21no—
34#11762678C2211yescharacteristic
35#11762678C2211yescharacteristic
36#11762667—21no—
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4×#63225C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41#11762572C2211yescharacteristic
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762564C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#107C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C17×#390C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×D4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C22×Dic34—21no—
62C22×Dic34—21no—
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
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×C42×C17C2311yescharacteristic
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75D4×Dic17—22no—
76D4×Dic17Q814yescomplemented
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225D411yescharacteristic
89C2×#63225D411yescharacteristic
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17C2311yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97#10587574—22no—
98#10587574—41no—
99#10587574—44no—
100#10587563—44no—
101#10587555—22no—
102#10587555—41no—
103C17×#107—22no—
104C17×C4⋊C8—41no—
105C17×#102—22no—
106C17×#102—41no—
107C17×#95—21no—
108#10587503—22no—
109#10587503—41no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502D411yescharacteristic
115#10587502D411yescharacteristic
116#10587485—22no—
117#10587485—41no—
118C8×Dic17—22no—
119#10587478—22no—
120C24×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17C2×Q812yes—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2411yescharacteristic
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—24no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—42no—
139C22×Dic17—42no—
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144C2×Dic34—42no—
145C2×Dic34—42no—
146C2×Dic34—42no—
147C2×Dic34—42no—
148C2×Dic34—81no—
149C2×Dic34—81no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—42no—
162C4⋊C4×C17#4112yes—
163C22⋊C4×C17—42no—
164C22⋊C4×C17—81no—
165C42×C17—21no—
166C42×C17—21no—
167C22⋊Dic17—24no—
168C22⋊Dic17—42no—
169C22⋊Dic17—81no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—42no—
177C4⋊Dic17—42no—
178C4⋊Dic17C2×Q812yes—
179C4⋊Dic17#4112yes—
180#63225—22no—
181#63225—42no—
182#63225—42no—
183#63225—42no—
184#63225—81no—
185#63225—81no—
186#63225D814yes—
187C4×Dic17—21no—
188C4×Dic17—21no—
189C4×Dic17—42no—
190C4×Dic17—42no—
191C4×Dic17—42no—
192C4×Dic17C2×Q812yes—
193C4×Dic17#4112yes—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200#26020—171nomaximal, Sylow
201C23×C17—22no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17C22×D411yescharacteristic
209D4×C17—22no—
210D4×C17—22no—
211D4×C17—41no—
212D4×C17—41no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—22no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
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×C17—41no—
227C2×C4×C17C2×D812yes—
228C2×C4×C17C22×Q811yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17H(5,𝔽2)11yescharacteristic
233C2×C4×C172−1+411yescharacteristic
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—22no—
239C2×Dic17—22no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—48no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17C22⋊Q812yes—
255C2×Dic17C22⋊Q812yes—
256C2×Dic17C2×D812yes—
257C2×Dic17C2×D812yes—
258C2×Dic17#13712yes—
259Dic34—84no—
260Dic34—84no—
261C8×C17—41no—
262C8×C17—41no—
263#3622—41no—
264#3622—41no—
265#2551—171no—
266#2354—171no—
267#2067—171no—
268#2067D1711yescharacteristic
269#2023—171no—
270#2023—171no—
271#1988—171no—
272#1988—171no—
273#1947—171no—
274#1939—171no—
275#1918—171no—
276#1918—171no—
277#1834—171no—
278#1802—171no—
279C4×#102—171no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—42no—
283C22×C17—81no—
284C22×C17—81no—
285C22×C17—81no—
286C22×C17—81no—
287C22×C17C2×C22⋊Q811yescharacteristic
288C22×C17#54611yescharacteristic
289C22×C17#54811yescharacteristic
290C22×C17C22×D811yescharacteristic
291C22×C17C2×#13711yescharacteristic
292C22×C17#57711yescharacteristic
293C22×C17#57711yescharacteristic
294C4×C17—22no—
295C4×C17—41no—
296C4×C17—42no—
297C4×C17—81no—
298C4×C17—81no—
299C4×C17#53311yescharacteristic
300C4×C17#54411yescharacteristic
301C4×C17#55511yescharacteristic
302C4×C17#55611yescharacteristic
303Dic17—22no—
304Dic17—22no—
305Dic17—41no—
306Dic17—42no—
307Dic17—42no—
308Dic17—81no—
309Dic17—81no—
310Dic17#47414yes—
311C2×C4⋊Q8—171no—
312C2×C4⋊Q8—171no—
313C2×C4⋊Q8—341no—
314C2×C4×D4—171no—
315C2×#117—341no—
316C2×#107—171no—
317C2×#107C2×D1711yescharacteristic
318C2×C4⋊C8—171no—
319C2×C4⋊C8—171no—
320C2×C4⋊C8C2×D1711yescharacteristic
321C2×#102—171no—
322C2×#102—171no—
323C2×#102C2×D1711yescharacteristic
324C2×#102C2×D1711yescharacteristic
325C2×C4×C8—171no—
326#395—341no—
327#395—341no—
328#390—171no—
329#390C2×D1711yescharacteristic
330#389—171no—
331#389—341no—
332#389C2×D1711yescharacteristic
333#388—341no—
334#388—341no—
335#384—171no—
336#384—171no—
337#380—341no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340C4×C4⋊C4—171no—
341C4×C4⋊C4—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340C2×D1711yescharacteristic
352#336—171no—
353#336—171no—
354C2×C17—82no—
355C2×C17#243911yescharacteristic
356C2×C17C2×#47411yescharacteristic
357C2×C17#308411yescharacteristic
358C2×C17#329411yescharacteristic
359C2×C17#332911yescharacteristic
360C2×C17#335811yescharacteristic
361C2×C17#336911yescharacteristic
362C22×Q8—341no—
363C22×Q8—341no—
364C22×D4C22×D1711yescharacteristic
365C23×C4—341no—
366C22×C8—171no—
367C22×C8—171no—
368C22×C8C22×D1711yescharacteristic
369C22×C8C22×D1711yescharacteristic
370C4⋊Q8—342no—
371C4⋊Q8—342no—
372C4⋊Q8—681no—
373C4⋊Q8—681no—
374C4⋊Q8—684no—
375C4×D4—174no—
376C4×D4—342no—
377#117—684no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C4⋊C4C22×D1711yescharacteristic
394C2×C4⋊C4C22×D1711yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—341no—
398C2×C22⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C42—171no—
401C2×C42—171no—
402C2×C42—341no—
403C2×C42—341no—
404C2×C42—341no—
405C2×C42C22×D1711yescharacteristic
406#107—22no—
407#107—342no—
408C4⋊C8—41no—
409C4⋊C8—342no—
410C4⋊C8—681no—
411#102—22no—
412#102—41no—
413#102—342no—
414#102—681no—
415C4×C8—342no—
416#95—21no—
417#95—171no—
418#95—171no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95—341no—
423C17#2602011yescharacteristic, complemented, Sylow
424C24—21no—
425C2×Q8—682no—
426C2×Q8—682no—
427C2×Q8—682no—
428C2×Q8—682no—
429C2×Q8—1361no—
430C2×Q8—1361no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D1712yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—171no—
441C22×C4—171no—
442C22×C4—171no—
443C22×C4—171no—
444C22×C4—344no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—682no—
450C22×C4—682no—
451C22×C4D4×D1711yescharacteristic
452C22×C4D4×D1711yescharacteristic
453C22×C4C23×D1711yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—174no—
473C4⋊C4—172no—
474C4⋊C4—172no—
475C4⋊C4—342no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—682no—
483C4⋊C4—682no—
484C4⋊C4—682no—
485C4⋊C4—682no—
486C4⋊C4—682no—
487C4⋊C4—1361no—
488C4⋊C4—1361no—
489C4⋊C4#6325412yes—
490C22⋊C4—42no—
491C22⋊C4—81no—
492C22⋊C4—344no—
493C22⋊C4—682no—
494C22⋊C4—1361no—
495C42—21no—
496C42—21no—
497C42—172no—
498C42—172no—
499C42—341no—
500C42—341no—
501C42—682no—
502C42—682no—
503C42—682no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D1711yescharacteristic
512Q8—1364no—
513Q8—1364no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
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—172no—
533C2×C4—172no—
534C2×C4—172no—
535C2×C4—172no—
536C2×C4—172no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—342no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—342no—
546C2×C4—688no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4D8×D1712yes—
558C2×C4C2×#6325411yescharacteristic
559C2×C4#1058769111yescharacteristic
560C2×C4C2×Q8×D1711yescharacteristic
561C2×C4#41×D1711yescharacteristic
562C2×C4#41×D1711yescharacteristic
563C2×C4#1058769711yescharacteristic
564C8—41no—
565C8—41no—
566C8—681no—
567C8—681no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22C22⋊Q8×D1711yescharacteristic
576C22#1176344811yescharacteristic
577C22#1176346511yescharacteristic
578C22C2×D8×D1711yescharacteristic
579C22#137×D1711yescharacteristic
580C22#1176361611yescharacteristic
581C22#1176361911yescharacteristic
582C4—22no—
583C4—41no—
584C4—42no—
585C4—81no—
586C4—81no—
587C4—174no—
588C4—342no—
589C4—342no—
590C4—681no—
591C4—682no—
592C4—682no—
593C4—1361no—
594C4—1361no—
595C4#1176336111yescharacteristic
596C4#1176343811yescharacteristic
597C4#1176352311yescharacteristic
598C4#1176352611yescharacteristic
599C2—82no—
600C2#1453247011yescharacteristic
601C2D17×#47411yescharacteristic
602C2#1453864711yescharacteristic
603C2#1454059211yescharacteristic
604C2#1454107311yescharacteristic
605C2#1454141411yescharacteristic
606C2#1454151911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2067C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2067D17
hypocenterC17#26020
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
1. #14533812C2semidirect
6. #14529735C2semidirect
7. #14529734C2semidirect
8. #14529539C2semidirect
9. #14529499C2semidirect
10. #14529357C2semidirect
11. #14529352C2semidirect
12. #14528910C2semidirect
76. D4×Dic17Q8semidirect
423. C17#26020semidirect

Maximal subgroups

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

subgroupindex
#145338122
#145317322
#145301682
#145299202
#145299182
#145297352
#145297342
#145295392
#145294992
#145293572
#145293522
#145289102
#145287822
#102×Dic172
C17×#20672
#2602017