Finite Group Databasebeta

← #17106930

Subgroups — #17106930

Order 4352.

690 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532656C211yesmaximal, characteristic
2#14530169C211yesmaximal, characteristic
3#14530122C211yesmaximal, characteristic, complemented
4#14530119C211yesmaximal, characteristic, complemented
5#14529927C211yesmaximal, characteristic
6#14529741C211yesmaximal, characteristic, complemented
7#14529524C211yesmaximal, characteristic, complemented
8#14529411C211yesmaximal, characteristic
9#14529404C211yesmaximal, characteristic
10#14529362C211yesmaximal, characteristic, complemented
11#14529355C211yesmaximal, characteristic, complemented
12#14529350C211yesmaximal, characteristic, complemented
13#14529044C211yesmaximal, characteristic
14#14528612C211yesmaximal, characteristic
15C17×#2370C211yesmaximal, characteristic, complemented
16C2×#10587556C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×#10587480C2211yescharacteristic
24C2×#10587480C2211yescharacteristic
25C2×#10587480C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587474C2211yescharacteristic
28#11762744—21no—
29#11762733—21no—
30#11762711C2211yescharacteristic
31#11762711C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762680—21no—
35#11762668—21no—
36#11762571C2211yescharacteristic
37#11762571C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762564C2211yescharacteristic
45#11762564C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×#125C2211yescharacteristic
49C17×#389—21no—
50C17×#389C2211yescharacteristic
51C17×#389C2211yescharacteristic
52C17×#387—21no—
53C17×#387—21no—
54C4×C4⋊C4×C17—21no—
55C4×C4⋊C4×C17C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C22×D68C2311yescharacteristic
60C22×C4×D17—21no—
61C17×#125—22no—
62C17×#125—22no—
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17D411yescharacteristic
68C2×C4⋊C4×C17D411yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C42×C17—21no—
74C2×C42×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×#63227—21no—
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C4×D68—22no—
96C4×D68—41no—
97#10587556—22no—
98#10587556—41no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95D411yescharacteristic
105C17×#95D411yescharacteristic
106#10587502—21no—
107#10587502—21no—
108#10587480—22no—
109#10587480—22no—
110#10587480—41no—
111#10587480Q814yes—
112#10587479—22no—
113#10587479—22no—
114#10587478—22no—
115#10587478—41no—
116C4⋊#3622—41no—
117#10587474—41no—
118C23×D17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17C2×D411yescharacteristic
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2411yescharacteristic
130C22×Dic17—21no—
131C22×Dic17—21no—
132C22×Dic17—21no—
133C2×D68—21no—
134C2×D68—21no—
135C2×D68—21no—
136C2×D68—21no—
137C2×D68—21no—
138C2×D68C2×Q812yes—
139C2×C4×D17—42no—
140C2×C4×D17—42no—
141C2×C4×D17—81no—
142C4⋊C4×C17—24no—
143C4⋊C4×C17—24no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—21no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—41no—
150C4⋊C4×C17—41no—
151C4⋊C4×C17—42no—
152C4⋊C4×C17—81no—
153C4⋊C4×C17C2×Q812yes—
154C4⋊C4×C17#4112yes—
155C4⋊C4×C17#4112yes—
156C42×C17—21no—
157C42×C17—21no—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—21no—
161C42×C17—21no—
162C42×C17—42no—
163C42×C17—81no—
164C42×C17—81no—
165#63227—42no—
166#63227—42no—
167#63227—81no—
168#63227—81no—
169#63227—81no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17#4112yes—
176#63225—42no—
177#63225—42no—
178#63225—81no—
179#63225—81no—
180C2×#3622—21no—
181C2×#3622—21no—
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×#3622C2×Q812yes—
192#27027—171nomaximal, Sylow
193C23×C17C22×D411yescharacteristic
194C22×D17—22no—
195C22×D17—41no—
196C22×D17—41no—
197C22×D17—41no—
198C22×D17—41no—
199C22×D17—41no—
200C22×D17—41no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—22no—
204C2×C4×C17—22no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—22no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
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—41no—
223C2×C4×C17—41no—
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×C17C22×Q811yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17H(5,𝔽2)11yescharacteristic
236C2×C4×C172−1+411yescharacteristic
237C2×Dic17—22no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246D68—22no—
247D68—22no—
248D68—41no—
249D68—41no—
250C4×D17—84no—
251#3622—22no—
252#3622—41no—
253#3622—41no—
254#3622—41no—
255#2454—171no—
256#2370D1711yescharacteristic, complemented
257#2067—171no—
258#2059—171no—
259#2059—171no—
260#2025—171no—
261#1989—171no—
262#1944—171no—
263#1926—171no—
264#1925—171no—
265#1919—171no—
266#1918—171no—
267#1917—171no—
268#1864—171no—
269#1762—171no—
270C22×C17#52511yescharacteristic
271C22×C17#54611yescharacteristic
272C22×C17D4×Q811yescharacteristic
273C22×C17#57511yescharacteristic
274C22×C17#57511yescharacteristic
275C22×C17#57611yescharacteristic
276C22×C17#57711yescharacteristic
277C2×D17—42no—
278C2×D17—42no—
279C2×D17—42no—
280C2×D17—81no—
281C2×D17—81no—
282C2×D17—81no—
283C2×D17—81no—
284C4×C17—41no—
285C4×C17—42no—
286C4×C17—41no—
287C4×C17—42no—
288C4×C17—42no—
289C4×C17—41no—
290C4×C17—41no—
291C4×C17—41no—
292C4×C17—41no—
293C4×C17—81no—
294C4×C17#53311yescharacteristic
295C4×C17#54211yescharacteristic
296C4×C17#55411yescharacteristic
297C4×C17#55611yescharacteristic
298Dic17—42no—
299Dic17—81no—
300Dic17—81no—
301C2×#125—171no—
302C2×#125C2×D1711yescharacteristic
303C2×C4×D4—171no—
304C2×#107—171no—
305C2×#107—171no—
306C2×#106—171no—
307C2×#106—171no—
308C2×C4⋊C8—171no—
309C2×#102—171no—
310C2×#102—171no—
311C2×#102—171no—
312C2×#102—171no—
313C2×#97—171no—
314#399—341no—
315#396—341no—
316#390—171no—
317#390—171no—
318#389—21no—
319#389—171no—
320#389—171no—
321#389C2×D1711yescharacteristic
322#389C2×D1711yescharacteristic
323#387—21no—
324#387—21no—
325#385—341no—
326#380—341no—
327C4×C4⋊C4—21no—
328C4×C4⋊C4C2×D1711yescharacteristic
329C4×C4⋊C4C2×D1711yescharacteristic
330C4×C4⋊C4C2×D1711yescharacteristic
331C4×C4⋊C4C2×D1711yescharacteristic
332#340—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#336—171no—
341#336—171no—
342#336—171no—
343#336—171no—
344C2×C17#246511yescharacteristic
345C2×C17#311111yescharacteristic
346C2×C17#311911yescharacteristic
347C2×C17#330911yescharacteristic
348C2×C17#332311yescharacteristic
349C2×C17#338811yescharacteristic
350C2×C17#338911yescharacteristic
351D17—82no—
352C22×D4—171no—
353C23×C4—341no—
354C22×C8—171no—
355C22×C8—171no—
356C22×C8—171no—
357C22×C8—171no—
358#125—22no—
359#125—22no—
360#125—342no—
361#125—681no—
362C4×D4—342no—
363C4×D4—681no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4C22⋊D1711yescharacteristic
376C2×C4⋊C4C22⋊D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C22⋊C4—341no—
382C2×C22⋊C4—341no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C22⋊C4—341no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42C22×D1711yescharacteristic
390C2×C42C22×D1711yescharacteristic
391C2×C42C22×D1711yescharacteristic
392#107—342no—
393#107—681no—
394#106—342no—
395#106—342no—
396C4⋊C8—681no—
397#102—174no—
398#102—342no—
399#102—342no—
400#102—681no—
401#97—681no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—341no—
408#95—341no—
409#95C22⋊D1711yescharacteristic
410#95C22⋊D1711yescharacteristic
411C17#2702711yescharacteristic, complemented, Sylow
412C24—341no—
413C2×D4—172no—
414C2×D4—341no—
415C2×D4—341no—
416C2×D4—341no—
417C2×D4—341no—
418C2×D4—341no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—341no—
426C22×C4—682no—
427C22×C4—682no—
428C22×C4—1361no—
429C22×C4C2×C22⋊D1711yescharacteristic
430C22×C4C2×C22⋊D1711yescharacteristic
431C22×C4C2×C22⋊D1711yescharacteristic
432C22×C4C2×C22⋊D1711yescharacteristic
433C22×C4C2×C22⋊D1711yescharacteristic
434C22×C4C2×C22⋊D1711yescharacteristic
435C22×C4C23×D1711yescharacteristic
436C2×C8—172no—
437C2×C8—341no—
438C2×C8—341no—
439C2×C8—341no—
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—81no—
459C4⋊C4—172no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—682no—
466C4⋊C4—682no—
467C4⋊C4—1361no—
468C4⋊C4—1361no—
469C4⋊C4#6325212yes—
470C4⋊C4#6325212yes—
471C4⋊C4Q8×D1712yes—
472C22⋊C4—682no—
473C22⋊C4—682no—
474C22⋊C4—1361no—
475C22⋊C4—1361no—
476C22⋊C4—1361no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—81no—
485C42—81no—
486C23—342no—
487C23—681no—
488C23—681no—
489C23—681no—
490C23—681no—
491C23—681no—
492C23—681no—
493C23C22×C22⋊D1711yescharacteristic
494D4—342no—
495D4—342no—
496D4—681no—
497D4—681no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
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—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—342no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—1364no—
538C2×C4C2×#6325211yescharacteristic
539C2×C4C2×#6325211yescharacteristic
540C2×C4C2×Q8×D1711yescharacteristic
541C2×C4#1058769411yescharacteristic
542C2×C4#41×D1711yescharacteristic
543C2×C4#1058769611yescharacteristic
544C8—342no—
545C8—681no—
546C8—681no—
547C8—681no—
548C22—682no—
549C22—682no—
550C22—682no—
551C22—1361no—
552C22—1361no—
553C22—1361no—
554C22—1361no—
555C22#1176332011yescharacteristic
556C22#1176344711yescharacteristic
557C22Q8×C22⋊D1711yescharacteristic
558C22#1176360711yescharacteristic
559C22#1176360711yescharacteristic
560C22#1176361211yescharacteristic
561C22#1176361711yescharacteristic
562C4—41no—
563C4—42no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—682no—
573C4—1361no—
574C4—1361no—
575C4#1176336311yescharacteristic
576C4#1176343111yescharacteristic
577C4#1176351911yescharacteristic
578C4#1176353111yescharacteristic
579C2—1362no—
580C2#1453279911yescharacteristic
581C2#1453889311yescharacteristic
582C2#1453898311yescharacteristic
583C2#1454078411yescharacteristic
584C2#1454099411yescharacteristic
585C2#1454172311yescharacteristic
586C2#1454173811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2370C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenter#2370D17
hypocenterC17#27027
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
3. #14530122C2semidirect
4. #14530119C2semidirect
6. #14529741C2semidirect
7. #14529524C2semidirect
10. #14529362C2semidirect
11. #14529355C2semidirect
12. #14529350C2semidirect
15. C17×#2370C2semidirect
256. #2370D17semidirect
411. C17#27027semidirect

Maximal subgroups

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

subgroupindex
#145326562
#145301692
#145301222
#145301192
#145299272
#145297412
#145295242
#145294112
#145294042
#145293622
#145293552
#145293502
#145290442
#145286122
C17×#23702
#2702717