Finite Group Databasebeta

← #17106915

Subgroups — #17106915

Order 4352.

726 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532634C211yesmaximal, characteristic
2#14530155C211yesmaximal, characteristic
3#14530119C211yesmaximal, characteristic, complemented
4#14530117C211yesmaximal, characteristic, complemented
5#14529927C211yesmaximal, characteristic
6#14529736C211yesmaximal, characteristic, complemented
7#14529519C211yesmaximal, characteristic, complemented
8#14529411C211yesmaximal, characteristic
9#14529404C211yesmaximal, characteristic
10#14529369C211yesmaximal, characteristic, complemented
11#14529369C211yesmaximal, characteristic, complemented
12#14529350C211yesmaximal, characteristic, complemented
13#14529044C211yesmaximal, characteristic
14#14528612C211yesmaximal, characteristic
15C17×#2370C211yesmaximal, characteristic, complemented
16C2×#10587555C2211yescharacteristic
17C2×#10587582—21no—
18C2×C4×D68C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587480C2211yescharacteristic
25C2×#10587480C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29#11762731—21no—
30#11762722—21no—
31#11762711C2211yescharacteristic
32#11762711C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762683—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—
61C22×Dic34—21no—
62C17×#125—22no—
63C17×#125—22no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17D411yescharacteristic
69C2×C4⋊C4×C17D411yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C42×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17—21no—
77C2×C42×C17C2311yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×C42×C17C2311yescharacteristic
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×#63227—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×#63225—21no—
90C2×#63225—21no—
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95#10587582—44no—
96C4×D68—22no—
97C4×D68—41no—
98#10587555—22no—
99#10587555—41no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95D411yescharacteristic
106C17×#95D411yescharacteristic
107#10587502—21no—
108#10587502—21no—
109#10587480—22no—
110#10587480—22no—
111#10587480—41no—
112#10587480Q814yescomplemented
113#10587479—22no—
114#10587479—22no—
115#10587479—41no—
116#10587478—22no—
117C4⋊#3622—41no—
118#10587474—41no—
119C23×D17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2411yescharacteristic
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C2×D68—21no—
135C2×D68—21no—
136C2×D68—21no—
137C2×D68—21no—
138C2×D68—21no—
139C2×D68C2×Q812yes—
140C2×C4×D17—42no—
141C2×C4×D17—42no—
142C2×C4×D17—81no—
143C2×Dic34—42no—
144C2×Dic34—42no—
145C2×Dic34—81no—
146C4⋊C4×C17—24no—
147C4⋊C4×C17—24no—
148C4⋊C4×C17—22no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—41no—
155C4⋊C4×C17—42no—
156C4⋊C4×C17—81no—
157C4⋊C4×C17C2×Q812yes—
158C4⋊C4×C17#4112yes—
159C4⋊C4×C17#4112yes—
160C42×C17—21no—
161C42×C17—21no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—21no—
166C42×C17—42no—
167C42×C17—81no—
168C42×C17—81no—
169#63227—42no—
170#63227—42no—
171#63227—42no—
172#63227—42no—
173#63227—81no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—42no—
180C4⋊Dic17#4112yes—
181#63225—42no—
182#63225—42no—
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—21no—
194C2×#3622C2×Q812yes—
195#27026—171nomaximal, Sylow
196C23×C17C22×D411yescharacteristic
197C22×D17—22no—
198C22×D17—41no—
199C22×D17—41no—
200C22×D17—41no—
201C22×D17—41no—
202C22×D17—41no—
203C22×D17—41no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—22no—
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—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×C17AGL(1,ℤ/8)12yes—
234C2×C4×C17C22×Q811yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17H(5,𝔽2)11yescharacteristic
239C2×C4×C172−1+411yescharacteristic
240C2×Dic17—22no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249D68—22no—
250D68—22no—
251D68—41no—
252D68—41no—
253C4×D17—84no—
254Dic34—84no—
255#3622—22no—
256#3622—41no—
257#3622—41no—
258#3622—41no—
259#2452—171no—
260#2370D1711yescharacteristic, complemented
261#2065—171no—
262#2059—171no—
263#2058—171no—
264#2025—171no—
265#1988—171no—
266#1943—171no—
267#1926—171no—
268#1925—171no—
269#1920—171no—
270#1920—171no—
271#1917—171no—
272#1864—171no—
273#1762—171no—
274C22×C17#52511yescharacteristic
275C22×C17#54611yescharacteristic
276C22×C17D4×Q811yescharacteristic
277C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C17#57511yescharacteristic
279C22×C17#57711yescharacteristic
280C22×C17#57711yescharacteristic
281C2×D17—42no—
282C2×D17—42no—
283C2×D17—42no—
284C2×D17—81no—
285C2×D17—81no—
286C2×D17—81no—
287C2×D17—81no—
288C4×C17—22no—
289C4×C17—41no—
290C4×C17—42no—
291C4×C17—41no—
292C4×C17—42no—
293C4×C17—42no—
294C4×C17—41no—
295C4×C17—41no—
296C4×C17—41no—
297C4×C17—81no—
298C4×C17#53311yescharacteristic
299C4×C17#54211yescharacteristic
300C4×C17#55411yescharacteristic
301C4×C17#55611yescharacteristic
302Dic17—42no—
303Dic17—81no—
304Dic17—81no—
305C2×C4⋊Q8—171no—
306C2×#125C2×D1711yescharacteristic
307C2×C22⋊Q8—341no—
308C2×C4×D4—171no—
309C2×#107—171no—
310C2×#106—171no—
311C2×#106—171no—
312C2×#106—171no—
313C2×C4⋊C8—171no—
314C2×#102—171no—
315C2×#102—171no—
316C2×#102—171no—
317C2×#102—171no—
318C2×#97—171no—
319#396—341no—
320#393—341no—
321#390—171no—
322#390—171no—
323#389—21no—
324#389—171no—
325#389—171no—
326#389C2×D1711yescharacteristic
327#389C2×D1711yescharacteristic
328#387—21no—
329#387—21no—
330#385—341no—
331C4×C4⋊C4—21no—
332C4×C4⋊C4C2×D1711yescharacteristic
333C4×C4⋊C4C2×D1711yescharacteristic
334C4×C4⋊C4C2×D1711yescharacteristic
335C4×C4⋊C4C2×D1711yescharacteristic
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#336—171no—
345#336—171no—
346#336—171no—
347#336—171no—
348C2×C17#246511yescharacteristic
349C2×C17#311511yescharacteristic
350C2×C17#312511yescharacteristic
351C2×C17#331011yescharacteristic
352C2×C17#332611yescharacteristic
353C2×C17#338411yescharacteristic
354C2×C17#338811yescharacteristic
355D17—82no—
356C22×Q8—341no—
357C22×D4—171no—
358C23×C4—341no—
359C22×C8—171no—
360C22×C8—171no—
361C22×C8—171no—
362C22×C8—171no—
363C4⋊Q8—342no—
364C4⋊Q8—681no—
365#125—22no—
366#125—22no—
367C22⋊Q8—684no—
368C4×D4—342no—
369C4×D4—681no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
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×C4⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C22⋊C4—341no—
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42C22×D1711yescharacteristic
395C2×C42C22×D1711yescharacteristic
396C2×C42C22×D1711yescharacteristic
397#107—342no—
398#106—342no—
399#106—342no—
400#106—681no—
401C4⋊C8—681no—
402#102—174no—
403#102—342no—
404#102—342no—
405#102—681no—
406#97—681no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—341no—
413#95—341no—
414#95C22⋊D1711yescharacteristic
415#95C22⋊D1711yescharacteristic
416C17#2702611yescharacteristic, complemented, Sylow
417C24—341no—
418C2×Q8—682no—
419C2×Q8—682no—
420C2×Q8—1361no—
421C2×D4—172no—
422C2×D4—341no—
423C2×D4—341no—
424C2×D4—341no—
425C2×D4—341no—
426C2×D4—341no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—341no—
434C22×C4—682no—
435C22×C4—682no—
436C22×C4—1361no—
437C22×C4C2×C22⋊D1711yescharacteristic
438C22×C4C2×C22⋊D1711yescharacteristic
439C22×C4C2×C22⋊D1711yescharacteristic
440C22×C4C2×C22⋊D1711yescharacteristic
441C22×C4C2×C22⋊D1711yescharacteristic
442C22×C4C2×C22⋊D1711yescharacteristic
443C22×C4C23×D1711yescharacteristic
444C2×C8—172no—
445C2×C8—341no—
446C2×C8—341no—
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—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—172no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—682no—
474C4⋊C4—682no—
475C4⋊C4—682no—
476C4⋊C4#6325212yes—
477C4⋊C4#6325212yes—
478C4⋊C4Q8×D1712yes—
479C22⋊C4—682no—
480C22⋊C4—682no—
481C22⋊C4—682no—
482C22⋊C4—682no—
483C22⋊C4—1361no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—42no—
491C42—81no—
492C42—81no—
493C23—342no—
494C23—681no—
495C23—681no—
496C23—681no—
497C23—681no—
498C23—681no—
499C23—681no—
500C23C22×C22⋊D1711yescharacteristic
501Q8—1364no—
502D4—342no—
503D4—342no—
504D4—681no—
505D4—681no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
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—342no—
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—1364no—
545C2×C4#1058763512yes—
546C2×C4C2×#6325211yescharacteristic
547C2×C4C2×#6325211yescharacteristic
548C2×C4C2×Q8×D1711yescharacteristic
549C2×C4#1058769411yescharacteristic
550C2×C4#41×D1711yescharacteristic
551C2×C4#1058769611yescharacteristic
552C8—342no—
553C8—681no—
554C8—681no—
555C8—681no—
556C22—682no—
557C22—682no—
558C22—682no—
559C22—1361no—
560C22—1361no—
561C22—1361no—
562C22—1361no—
563C22#1176332011yescharacteristic
564C22#1176344711yescharacteristic
565C22Q8×C22⋊D1711yescharacteristic
566C22C2×#1058763511yescharacteristic
567C22#1176360711yescharacteristic
568C22#1176361711yescharacteristic
569C22#1176362011yescharacteristic
570C4—22no—
571C4—41no—
572C4—42no—
573C4—42no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4—682no—
581C4—1361no—
582C4—1361no—
583C4#1176336311yescharacteristic
584C4#1176343111yescharacteristic
585C4#1176351911yescharacteristic
586C4#1176353111yescharacteristic
587C2—1362no—
588C2#1453279911yescharacteristic
589C2#1453893511yescharacteristic
590C2#1453906111yescharacteristic
591C2#1454079911yescharacteristic
592C2#1454103111yescharacteristic
593C2#1454168311yescharacteristic
594C2#1454172311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2370C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenter#2370D17
hypocenterC17#27026
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. #14530119C2semidirect
4. #14530117C2semidirect
6. #14529736C2semidirect
7. #14529519C2semidirect
10. #14529369C2semidirect
11. #14529369C2semidirect
12. #14529350C2semidirect
15. C17×#2370C2semidirect
112. #10587480Q8semidirect
260. #2370D17semidirect
416. C17#27026semidirect

Maximal subgroups

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

subgroupindex
#145326342
#145301552
#145301192
#145301172
#145299272
#145297362
#145295192
#145294112
#145294042
#145293692
#145293502
#145290442
#145286122
C17×#23702
#2702617