Finite Group Databasebeta

← #17106928

Subgroups — #17106928

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#14532630C211yesmaximal, characteristic
2#14531882C211yesmaximal, characteristic, complemented
3#14530157C211yesmaximal, characteristic
4#14530121C211yesmaximal, characteristic, complemented
5#14530114C211yesmaximal, characteristic, complemented
6#14529929C211yesmaximal, characteristic
7#14529521C211yesmaximal, characteristic, complemented
8#14529409C211yesmaximal, characteristic
9#14529404C211yesmaximal, characteristic
10#14529372C211yesmaximal, characteristic, complemented
11#14529368C211yesmaximal, characteristic, complemented
12#14529345C211yesmaximal, characteristic, complemented
13#14529046C211yesmaximal, characteristic
14#14528608C211yesmaximal, characteristic
15C17×#1988C211yesmaximal, characteristic, complemented
16C2×#10587576C2211yescharacteristic
17C2×#10587632—21no—
18C2×#10587579C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587488C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587492C2211yescharacteristic
24C2×#10587492C2211yescharacteristic
25C2×#10587480C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28#11762731—21no—
29#11762720—21no—
30#11762711C2211yescharacteristic
31#11762710C2211yescharacteristic
32#11762708C2211yescharacteristic
33#11762708C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762706—21no—
36#11762705C2211yescharacteristic
37#11762693—21no—
38#11762692—21no—
39#11762680—21no—
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4×#63225C2211yescharacteristic
43C4×C4⋊Dic17—21no—
44#11762572C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×C4⋊Q8C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C4×C4⋊C4×C17C2211yescharacteristic
55C17×#340C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×Q8×C17—21no—
60C22×D68C2311yescharacteristic
61C22×C4×D17—21no—
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C17×C4⋊Q8—22no—
65C17×C4⋊Q8—41no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C42×C17—21no—
72C2×C42×C17C2311yescharacteristic
73#10587632—44no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×#63227—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225D411yescharacteristic
89C2×#63225D411yescharacteristic
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17C2311yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96#10587579—22no—
97#10587579—41no—
98#10587576—22no—
99#10587576—22no—
100C17×C4⋊C8—41no—
101C17×#95—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502D411yescharacteristic
109#10587502D411yescharacteristic
110#10587492—22no—
111#10587492—41no—
112#10587488—22no—
113#10587486—41no—
114#10587480—22no—
115#10587480Q814yescomplemented
116#10587479—22no—
117#10587479—41no—
118#10587478—22no—
119C23×D17—21no—
120C2×Q8×C17—42no—
121C2×Q8×C17—42no—
122C2×Q8×C17—81no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2411yescharacteristic
130C22×Dic17—21no—
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17C2×D411yescharacteristic
134C22×Dic17C2×D411yescharacteristic
135C22×Dic17C2×D411yescharacteristic
136C22×Dic17C2×D411yescharacteristic
137C2×D68—21no—
138C2×D68—21no—
139C2×D68—21no—
140C2×D68—21no—
141C2×D68—21no—
142C2×D68C2×Q812yes—
143C2×C4×D17—42no—
144C2×C4×D17—42no—
145C2×C4×D17—81no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—42no—
158C4⋊C4×C17C2×Q812yes—
159C4⋊C4×C17#4112yes—
160C4⋊C4×C17#4112yes—
161C42×C17—21no—
162C42×C17—21no—
163C42×C17—42no—
164#63227—42no—
165#63227—42no—
166#63227—42no—
167#63227—42no—
168#63227—81no—
169C4⋊Dic17—24no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—42no—
176C4⋊Dic17—42no—
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×#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—
204Q8×C17—84no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—21no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
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—21no—
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—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250D68—22no—
251D68—22no—
252D68—41no—
253D68—41no—
254C4×D17—84no—
255C8×C17—41no—
256C8×C17—41no—
257#3622—22no—
258#3622—41no—
259#2452—171no—
260#2370—171no—
261#2065—171no—
262#2059—171no—
263#2058—171no—
264#2025—171no—
265#1988D1711yescharacteristic, complemented
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—42no—
290C4×C17—42no—
291C4×C17—42no—
292C4×C17—41no—
293C4×C17—81no—
294C4×C17#53311yescharacteristic
295C4×C17#54211yescharacteristic
296C4×C17#55411yescharacteristic
297C4×C17#55611yescharacteristic
298Dic17—41no—
299Dic17—41no—
300Dic17—42no—
301Dic17—41no—
302Dic17—41no—
303Dic17—81no—
304Dic17—81no—
305C2×C4⋊Q8C2×D1711yescharacteristic
306C2×#125—171no—
307C2×C22⋊Q8—341no—
308C2×C4×D4—171no—
309C2×#107—171no—
310C2×#106—171no—
311C2×#106—171no—
312C2×#106—171no—
313C2×C4⋊C8C2×D1711yescharacteristic
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—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#340C2×D1711yescharacteristic
341#340C2×D1711yescharacteristic
342#340C2×D1711yescharacteristic
343#340C2×D1711yescharacteristic
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—21no—
357C22×D4—171no—
358C23×C4—341no—
359C22×C8—171no—
360C22×C8—171no—
361C22×C8C22×D1711yescharacteristic
362C22×C8C22×D1711yescharacteristic
363C4⋊Q8—22no—
364C4⋊Q8—41no—
365#125—342no—
366#125—342no—
367C22⋊Q8—684no—
368C4×D4—342no—
369C4×D4—681no—
370C2×C4⋊C4—21no—
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⋊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—171no—
393C2×C42—171no—
394C2×C42—341no—
395C2×C42—341no—
396C2×C42C22×D1711yescharacteristic
397#107—342no—
398#106—342no—
399#106—342no—
400#106—681no—
401C4⋊C8—41no—
402#102—174no—
403#102—342no—
404#102—342no—
405#102—681no—
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—341no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—81no—
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—171no—
432C22×C4—171no—
433C22×C4—171no—
434C22×C4—171no—
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—21no—
450C2×C8—172no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8—341no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—42no—
462C4⋊C4—172no—
463C4⋊C4—344no—
464C4⋊C4—344no—
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—1361no—
476C4⋊C4#6325212yes—
477C4⋊C4Q8×D1712yes—
478C4⋊C4#6325612yes—
479C22⋊C4—682no—
480C22⋊C4—682no—
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—342no—
494C23—681no—
495C23—681no—
496C23—681no—
497C23—681no—
498C23—681no—
499C23—681no—
500C23C2×D4×D1711yescharacteristic
501Q8—84no—
502D4—342no—
503D4—342no—
504D4—681no—
505D4—681no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—342no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—341no—
535C2×C4—341no—
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×C4C2×#6325211yescharacteristic
547C2×C4#1058769111yescharacteristic
548C2×C4C2×Q8×D1711yescharacteristic
549C2×C4C2×#6325611yescharacteristic
550C2×C4#41×D1711yescharacteristic
551C2×C4#1058769711yescharacteristic
552C8—41no—
553C8—41no—
554C8—342no—
555C8—681no—
556C22—682no—
557C22—682no—
558C22—682no—
559C22—1361no—
560C22—1361no—
561C22—1361no—
562C22—1361no—
563C22#1176332211yescharacteristic
564C22#1176344911yescharacteristic
565C22#1176347211yescharacteristic
566C22C2×#1058759711yescharacteristic
567C22#1176360811yescharacteristic
568C22#1176361511yescharacteristic
569C22#1176361811yescharacteristic
570C4—22no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—41no—
575C4—81no—
576C4—681no—
577C4—681no—
578C4—682no—
579C4—681no—
580C4—681no—
581C4—1361no—
582C4—1361no—
583C4#1176336311yescharacteristic
584C4#1176342311yescharacteristic
585C4#1176352011yescharacteristic
586C4#1176353211yescharacteristic
587C2—1362no—
588C2#1453281211yescharacteristic
589C2#1453893711yescharacteristic
590C2#1453906811yescharacteristic
591C2#1454080111yescharacteristic
592C2#1454103311yescharacteristic
593C2#1454168711yescharacteristic
594C2#1454173611yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1988C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenter#1988D17
hypocenterC17#27026
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. #14531882C2semidirect
4. #14530121C2semidirect
5. #14530114C2semidirect
7. #14529521C2semidirect
10. #14529372C2semidirect
11. #14529368C2semidirect
12. #14529345C2semidirect
15. C17×#1988C2semidirect
115. #10587480Q8semidirect
265. #1988D17semidirect
416. C17#27026semidirect

Maximal subgroups

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

subgroupindex
#145326302
#145318822
#145301572
#145301212
#145301142
#145299292
#145295212
#145294092
#145294042
#145293722
#145293682
#145293452
#145290462
#145286082
C17×#19882
#2702617