Finite Group Databasebeta

← #17106926

Subgroups — #17106926

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#14532636C211yesmaximal, characteristic
2#14531882C211yesmaximal, characteristic, complemented
3#14530158C211yesmaximal, characteristic
4#14530120C211yesmaximal, characteristic, complemented
5#14530118C211yesmaximal, characteristic, complemented
6#14529928C211yesmaximal, characteristic
7#14529736C211yesmaximal, characteristic, complemented
8#14529523C211yesmaximal, characteristic, complemented
9#14529413C211yesmaximal, characteristic
10#14529372C211yesmaximal, characteristic, complemented
11#14529368C211yesmaximal, characteristic, complemented
12#14529351C211yesmaximal, characteristic, complemented
13#14529047C211yesmaximal, characteristic
14#14528611C211yesmaximal, characteristic
15C17×#1925C211yesmaximal, characteristic
16C2×#10587555C2211yescharacteristic
17C2×#10587576C2211yescharacteristic
18C2×#10587564—21no—
19C2×D4×Dic17C2211yescharacteristic
20C2×#10587488C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587486C2211yescharacteristic
27#11762732—21no—
28#11762721—21no—
29#11762712C2211yescharacteristic
30#11762708C2211yescharacteristic
31#11762708C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762706—21no—
34#11762705C2211yescharacteristic
35#11762693—21no—
36#11762692—21no—
37#11762684—21no—
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4×#63225C2211yescharacteristic
41C4×C4⋊Dic17—21no—
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#107C2211yescharacteristic
53C2×C17×#102C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C17×#390C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C22×D4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C22×Dic34—21no—
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C22⋊C4×C17—21no—
68C2×C22⋊C4×C17—21no—
69C2×C42×C17—21no—
70C2×C42×C17C2311yescharacteristic
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74D4×Dic17—22no—
75D4×Dic17—41no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225D411yescharacteristic
88C2×#63225D411yescharacteristic
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17C2311yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95#10587576—22no—
96#10587576—22no—
97#10587564—44no—
98#10587555—22no—
99#10587555—41no—
100C17×#107—22no—
101C17×#102—22no—
102C17×#102Q814yescomplemented
103C17×#95—21no—
104#10587503—22no—
105#10587503—41no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502D411yescharacteristic
113#10587502D411yescharacteristic
114#10587488—22no—
115#10587488—41no—
116#10587486—41no—
117#10587479—22no—
118C4⋊#3622—41no—
119C24×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17C2×Q812yes—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2411yescharacteristic
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—42no—
137C22×Dic17—42no—
138C22×Dic17—81no—
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C2×Dic34—42no—
144C2×Dic34—42no—
145C2×Dic34—81no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17C2×Q812yes—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17C2×Q812yes—
157C4⋊C4×C17#4112yes—
158C22⋊C4×C17—42no—
159C22⋊C4×C17—42no—
160C42×C17—21no—
161C42×C17—21no—
162C42×C17—42no—
163C22⋊Dic17—42no—
164C22⋊Dic17—42no—
165C22⋊Dic17—81no—
166C4⋊Dic17—24no—
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—42no—
176C4⋊Dic17#4112yes—
177C4⋊Dic17#4112yes—
178#63225—24no—
179#63225—22no—
180#63225—41no—
181#63225—42no—
182#63225—81no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—21no—
186C4×Dic17—21no—
187C4×Dic17—81no—
188C4×Dic17—81no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195#27026—171nomaximal, Sylow
196C23×C17—22no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17C22×D411yescharacteristic
204D4×C17—22no—
205D4×C17—22no—
206D4×C17—41no—
207D4×C17—41no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—22no—
211C2×C4×C17—21no—
212C2×C4×C17—22no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
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×C17AGL(1,ℤ/8)12yes—
223C2×C4×C17C22×Q811yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17C2×#4111yescharacteristic
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17H(5,𝔽2)11yescharacteristic
228C2×C4×C172−1+411yescharacteristic
229C2×Dic17—22no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—22no—
234C2×Dic17—21no—
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—41no—
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—84no—
254Dic34—84no—
255C8×C17—22no—
256C8×C17—41no—
257#3622—41no—
258#3622—41no—
259#2452—171no—
260#2370—171no—
261#2065—171no—
262#2059—171no—
263#2058—171no—
264#2025—171no—
265#1988—171no—
266#1943—171no—
267#1926—171no—
268#1925D1711yescharacteristic
269#1920—171no—
270#1920—171no—
271#1917—171no—
272#1864—171no—
273#1762—171no—
274C22×C17—42no—
275C22×C17—42no—
276C22×C17—42no—
277C22×C17—81no—
278C22×C17—81no—
279C22×C17—81no—
280C22×C17—81no—
281C22×C17#52511yescharacteristic
282C22×C17#54611yescharacteristic
283C22×C17D4×Q811yescharacteristic
284C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C17#57511yescharacteristic
286C22×C17#57711yescharacteristic
287C22×C17#57711yescharacteristic
288C4×C17—22no—
289C4×C17—42no—
290C4×C17—42no—
291C4×C17—41no—
292C4×C17—81no—
293C4×C17#53311yescharacteristic
294C4×C17#54211yescharacteristic
295C4×C17#55411yescharacteristic
296C4×C17#55611yescharacteristic
297Dic17—41no—
298Dic17—41no—
299Dic17—42no—
300Dic17—42no—
301Dic17—41no—
302Dic17—41no—
303Dic17—81no—
304Dic17—81no—
305C2×C4⋊Q8—171no—
306C2×#125—171no—
307C2×C22⋊Q8—341no—
308C2×C4×D4—171no—
309C2×#107C2×D1711yescharacteristic
310C2×#106—171no—
311C2×#106—171no—
312C2×#106—171no—
313C2×C4⋊C8—171no—
314C2×#102—171no—
315C2×#102—171no—
316C2×#102C2×D1711yescharacteristic
317C2×#102C2×D1711yescharacteristic
318C2×#97—171no—
319#396—341no—
320#393—341no—
321#390—171no—
322#390C2×D1711yescharacteristic
323#389—171no—
324#389—171no—
325#389—171no—
326#389—171no—
327#389—341no—
328#387—341no—
329#387—341no—
330#385—341no—
331C4×C4⋊C4—171no—
332C4×C4⋊C4—171no—
333C4×C4⋊C4—171no—
334C4×C4⋊C4—341no—
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#340C2×D1711yescharacteristic
344#336—171no—
345#336—171no—
346#336—171no—
347#336C2×D1711yescharacteristic
348C2×C17—82no—
349C2×C17#246511yescharacteristic
350C2×C17#311511yescharacteristic
351C2×C17#312511yescharacteristic
352C2×C17#331011yescharacteristic
353C2×C17#332611yescharacteristic
354C2×C17#338411yescharacteristic
355C2×C17#338811yescharacteristic
356C22×Q8—341no—
357C22×D4C22×D1711yescharacteristic
358C23×C4—341no—
359C22×C8—171no—
360C22×C8—171no—
361C22×C8C22×D1711yescharacteristic
362C22×C8C22×D1711yescharacteristic
363C4⋊Q8—342no—
364C4⋊Q8—681no—
365#125—342no—
366#125—342no—
367C22⋊Q8—684no—
368C4×D4—342no—
369C4×D4—681no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—341no—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C42—21no—
392C2×C42—171no—
393C2×C42—171no—
394C2×C42—341no—
395C2×C42—341no—
396C2×C42C22×D1711yescharacteristic
397#107—22no—
398#106—342no—
399#106—342no—
400#106—681no—
401C4⋊C8—681no—
402#102—22no—
403#102—342no—
404#102—681no—
405#102Dic3414yescomplemented
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—
416C17#2702611yescharacteristic, complemented, Sylow
417C24—21no—
418C2×Q8—682no—
419C2×Q8—682no—
420C2×Q8—1361no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C2×Dic3412yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—171no—
431C22×C4—171no—
432C22×C4—171no—
433C22×C4—171no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4—341no—
438C22×C4—682no—
439C22×C4—682no—
440C22×C4—1361no—
441C22×C4D4×D1711yescharacteristic
442C22×C4D4×D1711yescharacteristic
443C22×C4C23×D1711yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8C2×Dic3412yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—172no—
461C4⋊C4—172no—
462C4⋊C4—344no—
463C4⋊C4—344no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—342no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—681no—
472C4⋊C4—682no—
473C4⋊C4—682no—
474C4⋊C4—682no—
475C4⋊C4—682no—
476C4⋊C4—1361no—
477C4⋊C4C2×Dic3412yes—
478C4⋊C4#6325412yes—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—682no—
482C22⋊C4—682no—
483C22⋊C4—1361no—
484C42—21no—
485C42—21no—
486C42—42no—
487C42—341no—
488C42—341no—
489C42—341no—
490C42—341no—
491C42—1361no—
492C42—1361no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23C2×D4×D1711yescharacteristic
501Q8—1364no—
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—342no—
523C2×C4—341no—
524C2×C4—341no—
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—342no—
534C2×C4—681no—
535C2×C4—681no—
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#1058759712yes—
546C2×C4C22×Dic3411yescharacteristic
547C2×C4C2×#6325411yescharacteristic
548C2×C4#41×D1711yescharacteristic
549C2×C4#41×D1711yescharacteristic
550C2×C4#1058769611yescharacteristic
551C2×C4#1058769711yescharacteristic
552C8—22no—
553C8—41no—
554C8—681no—
555C8—681no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22#1176331911yescharacteristic
564C22#1176345111yescharacteristic
565C22D4×Dic3411yescharacteristic
566C22C2×#1058759711yescharacteristic
567C22#1176360511yescharacteristic
568C22#1176361611yescharacteristic
569C22#1176361811yescharacteristic
570C4—22no—
571C4—42no—
572C4—42no—
573C4—41no—
574C4—81no—
575C4—681no—
576C4—682no—
577C4—681no—
578C4—682no—
579C4—681no—
580C4—681no—
581C4—1361no—
582C4—1361no—
583C4#1176336211yescharacteristic
584C4#1176342511yescharacteristic
585C4#1176351611yescharacteristic
586C4#1176352811yescharacteristic
587C2—82no—
588C2#1453281011yescharacteristic
589C2#1453893211yescharacteristic
590C2#1453906011yescharacteristic
591C2#1454080311yescharacteristic
592C2#1454103511yescharacteristic
593C2#1454168111yescharacteristic
594C2#1454173411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1925C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#1925D17
hypocenterC17#27026
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
2. #14531882C2semidirect
4. #14530120C2semidirect
5. #14530118C2semidirect
7. #14529736C2semidirect
8. #14529523C2semidirect
10. #14529372C2semidirect
11. #14529368C2semidirect
12. #14529351C2semidirect
102. C17×#102Q8semidirect
405. #102Dic34semidirect
416. C17#27026semidirect

Maximal subgroups

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

subgroupindex
#145326362
#145318822
#145301582
#145301202
#145301182
#145299282
#145297362
#145295232
#145294132
#145293722
#145293682
#145293512
#145290472
#145286112
C17×#19252
#2702617