Finite Group Databasebeta

← #17005752

Subgroups — #17005752

Order 4352.

656 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537799C211yesmaximal, characteristic, complemented
2C4×#10587636C211yesmaximal, characteristic
3#14527679C212yesmaximal, complemented
4#14527620C212yesmaximal, complemented
5#14527618C212yesmaximal, complemented
6C8×#63228C212yesmaximal, complemented
7#14527545C211yesmaximal, characteristic, complemented
8#14527218C211yesmaximal, characteristic, complemented
9#14527216C211yesmaximal, characteristic, complemented
10M16×#3622C211yesmaximal, characteristic, complemented
11C17×#2970C211yesmaximal, characteristic, complemented
12C2×#10587636C411yescharacteristic
13C4×#63252C2211yescharacteristic
14#11762939C2212yes—
15C4×#63231C411yescharacteristic
16C4×#63231C2211yescharacteristic
17C4×#63230C411yescharacteristic
18C4×#63230C2211yescharacteristic
19C4×#63229C411yescharacteristic
20C4×#63229C2211yescharacteristic
21C4×#63228C411yescharacteristic
22C4×#63228C2211yescharacteristic
23#11762878—22no—
24C8×D68C2212yes—
25#11762866C2211yescharacteristic
26#11762864—21no—
27#11762862C2211yescharacteristic
28#11762845C411yescharacteristic
29#11762805C411yescharacteristic
30#11762803C411yescharacteristic
31#97×D17—21no—
32C2×C4×#3622C2211yescharacteristic
33#11762561C2212yes—
34#11762559C2212yes—
35Q8⋊#3622C2212yes—
36#11762539C2212yes—
37D4⋊#3622C2212yes—
38#11762536C2212yes—
39#11762530C2212yes—
40C8×#3622C2212yes—
41C4×#41×C17C2211yescharacteristic
42C17×#446C2212yes—
43C17×#435—22no—
44C8×D4×C17C2212yes—
45C17×#432—21no—
46C17×#432C2211yescharacteristic
47C4×M16×C17C2211yescharacteristic
48C2×C17×#97—21no—
49C2×#41×C17C2×C411yescharacteristic
50C2×#63252C2×C411yescharacteristic
51C2×M16×C17D412yes—
52C22×C8×C17—22no—
53C4×Q8×C17C2×C411yescharacteristic
54C4×Q8×C17C2311yescharacteristic
55C4×D4×C17—21no—
56C4×D4×C17—21no—
57C4×D4×C17C2×C411yescharacteristic
58C4×D4×C17C2311yescharacteristic
59C17×#117—21no—
60C17×#117C2×C411yescharacteristic
61C2×C42×C17—21no—
62C2×C42×C17C2311yescharacteristic
63#10587636—24no—
64C2×#63231C2×C411yescharacteristic
65C2×#63230C2×C411yescharacteristic
66C2×#63229C2×C411yescharacteristic
67C2×#63228C2×C411yescharacteristic
68C4×C22⋊D17—21no—
69C4×C22⋊D17—21no—
70#10587612C2×C411yescharacteristic
71C22×#3622C2×C411yescharacteristic
72C22×#3622C2×C411yescharacteristic
73C2×C8×D17—22no—
74C4×D68C2×C411yescharacteristic
75C4×D68C2311yescharacteristic
76#10587558—21no—
77C42×D17—21no—
78C4×Dic34C2×C411yescharacteristic
79C4×Dic34C2311yescharacteristic
80C17×C4⋊C8—22no—
81C17×C4⋊C8C2312yes—
82C17×C22⋊C8—21no—
83C17×C22⋊C8—22no—
84C17×#97—22no—
85C17×#97—21no—
86C17×#97D411yescharacteristic
87C17×#97D411yescharacteristic
88C4×C8×C17C2312yes—
89#10587506C2×C411yescharacteristic
90#10587506C2×C411yescharacteristic
91#10587503C2×C411yescharacteristic
92#10587503C2×C411yescharacteristic
93C22⋊#3622—21no—
94C22⋊#3622—21no—
95#10587491—22no—
96#10587486—22no—
97#10587485—22no—
98#10587481C2×C411yescharacteristic
99#10587481C2×C411yescharacteristic
100#10587480C2×C411yescharacteristic
101#10587480C2×C411yescharacteristic
102#10587479C2×C411yescharacteristic
103#10587479C2×C411yescharacteristic
104#10587478C2×C411yescharacteristic
105#10587478C2×C411yescharacteristic
106C4⋊#3622—21no—
107C4⋊#3622C2312yes—
108#10587474—21no—
109C4×#3622C2×C411yescharacteristic
110C4×#3622C2×C411yescharacteristic
111C4×#3622C2312yes—
112C4×#3622C2311yescharacteristic
113C4×#3622C2311yescharacteristic
114#41×C17—22no—
115#41×C17—41no—
116C2×Q8×C17C22×C411yescharacteristic
117C2×D4×C17—21no—
118C2×D4×C17C22×C411yescharacteristic
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17C22×C411yescharacteristic
123C22×C4×C17C2×D411yescharacteristic
124C22×C4×C17#4111yescharacteristic
125C2×C22⋊D17—21no—
126#63252—22no—
127#63252—41no—
128C2×D68C22×C411yescharacteristic
129C2×C4×D17—21no—
130C2×C4×D17—21no—
131C2×C4×D17—21no—
132C2×Dic34C22×C411yescharacteristic
133M16×C17—24no—
134C2×C8×C17—22no—
135C2×C8×C17—44no—
136C2×C8×C17C2×D412yes—
137C2×C8×C17C2×D412yes—
138C4⋊C4×C17—21no—
139C4⋊C4×C17—21no—
140C4⋊C4×C17C22×C411yescharacteristic
141C4⋊C4×C17C22×C411yescharacteristic
142C22⋊C4×C17—21no—
143C22⋊C4×C17—21no—
144C22⋊C4×C17—21no—
145C42×C17—21no—
146C42×C17—21no—
147C42×C17—21no—
148C42×C17C22×C411yescharacteristic
149C42×C17C2×D411yescharacteristic
150C42×C17#4111yescharacteristic
151C42×C17C2411yescharacteristic
152C22⋊Dic17—21no—
153#63231—22no—
154#63230—22no—
155#63229—22no—
156#63228M1614yescomplemented
157#63227—21no—
158#63227—21no—
159C4⋊Dic17C22×C411yescharacteristic
160C4⋊Dic17C22×C411yescharacteristic
161#63225—21no—
162#63225—21no—
163C4×Dic17—21no—
164C4×Dic17—21no—
165C4×Dic17—21no—
166C2×#3622—22no—
167C2×#3622—22no—
168C2×#3622—22no—
169C2×#3622—41no—
170C2×#3622C22×C411yescharacteristic
171C2×#3622C22×C411yescharacteristic
172C2×#3622C22×C411yescharacteristic
173C2×#3622C22×C411yescharacteristic
174C8×D17—44no—
175#18035—171nomaximal, Sylow
176C23×C17—21no—
177C23×C17C4×D411yescharacteristic
178C22×D17—21no—
179Q8×C17—21no—
180Q8×C17—21no—
181D4×C17—21no—
182D4×C17—42no—
183D4×C17C2×M1612yes—
184C2×C4×C17—21no—
185C2×C4×C17—21no—
186C2×C4×C17—22no—
187C2×C4×C17—21no—
188C2×C4×C17—21no—
189C2×C4×C17—21no—
190C2×C4×C17—41no—
191C2×C4×C17—42no—
192C2×C4×C17—42no—
193C2×C4×C17—81no—
194C2×C4×C17C4×D411yescharacteristic
195C2×C4×C17C4×D411yescharacteristic
196C2×C4×C17C4×D411yescharacteristic
197C2×C4×C17C23×C411yescharacteristic
198C2×C4×C17C22×D411yescharacteristic
199C2×C4×C17C2×#4111yescharacteristic
200C22⋊D17—42no—
201C2×Dic17—21no—
202C2×Dic17—21no—
203C2×Dic17—21no—
204D68—21no—
205D68C2×M1612yes—
206C4×D17—42no—
207C4×D17—42no—
208C4×D17—81no—
209Dic34—21no—
210Dic34—21no—
211C8×C17—22no—
212C8×C17—42no—
213C8×C17#13514yescomplemented
214#3622—21no—
215#3622—21no—
216#3622—21no—
217#3622—42no—
218#3622—42no—
219#3622C2×M1612yes—
220#2970—171no—
221#2970D1711yescharacteristic, complemented
222C4×#135—171no—
223#1591—172no—
224#1582—172no—
225#1582—172no—
226C8×D8—172no—
227#1568—171no—
228#1488—171no—
229#1488—171no—
230C8×M16—171no—
231C22×C17—21no—
232C22×C17—22no—
233C22×C17—21no—
234C22×C17—21no—
235C22×C17—41no—
236C22×C17—41no—
237C22×C17C2×C4×D411yescharacteristic
238C2×D17—22no—
239C2×D17—41no—
240C2×D17—41no—
241C4×C17—21no—
242C4×C17—21no—
243C4×C17—21no—
244C4×C17—21no—
245C4×C17—41no—
246C4×C17—41no—
247C4×C17—41no—
248C4×C17C22×M1611yescharacteristic
249C4×C17#56811yescharacteristic
250C4×C17C2×#13512yes—
251Dic17—41no—
252Dic17—41no—
253Dic17—41no—
254C2×#135—171no—
255C4×#41—171no—
256C4×#41C2×D1711yescharacteristic
257#446—172no—
258#446C2×D1712yes—
259C4×Q16—171no—
260C4×Q16—171no—
261C4×SD16—171no—
262C4×SD16—171no—
263C4×SD16—171no—
264C4×SD16—171no—
265C4×D8—171no—
266C4×D8—171no—
267#435—22no—
268#435—342no—
269C8×D4—172no—
270C8×D4C2×D1712yes—
271#432—21no—
272#432—171no—
273#432—341no—
274#432C2×D1711yescharacteristic
275#431—171no—
276#427—171no—
277#416—171no—
278#416—171no—
279C4×M16C2×D1711yescharacteristic
280C2×#97—21no—
281C2×#97—341no—
282C2×C4×C8—171no—
283#335—172no—
284#334—172no—
285Q8⋊C8—172no—
286Q8⋊C8—172no—
287D4⋊C8—172no—
288D4⋊C8—172no—
289#322—172no—
290C82—172no—
291C2×C17—41no—
292C2×C17—42no—
293C2×C17D4×M1611yescharacteristic
294C2×C17C4×#13511yescharacteristic
295C2×C17#295711yescharacteristic
296D17—42no—
297C2×#41—171no—
298C2×#41C4×D1711yescharacteristic
299#135—344no—
300C2×Q16—171no—
301C2×SD16—171no—
302C2×SD16—171no—
303C2×D8—171no—
304C2×M16C22⋊D1712yes—
305C22×C8—22no—
306C22×C8—171no—
307C22×C8—171no—
308C22×C8—342no—
309C4×Q8—171no—
310C4×Q8—171no—
311C4×Q8C4×D1711yescharacteristic
312C4×Q8C22×D1711yescharacteristic
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—171no—
316C4×D4—171no—
317C4×D4—341no—
318C4×D4—341no—
319C4×D4C4×D1711yescharacteristic
320C4×D4C22×D1711yescharacteristic
321#117—21no—
322#117—171no—
323#117—341no—
324#117C4×D1711yescharacteristic
325C2×C42—21no—
326C2×C42—341no—
327C2×C42C22×D1711yescharacteristic
328#107—171no—
329#107—171no—
330#106—171no—
331#106—171no—
332C4⋊C8—22no—
333C4⋊C8—172no—
334C4⋊C8—342no—
335C4⋊C8—341no—
336C4⋊C8C22×D1712yes—
337#103—171no—
338#103—171no—
339#103—171no—
340#103—171no—
341#102—171no—
342#102—171no—
343#102—171no—
344#102—171no—
345C22⋊C8—21no—
346C22⋊C8—22no—
347C22⋊C8—341no—
348C22⋊C8—341no—
349C22⋊C8—342no—
350#97—22no—
351#97—21no—
352#97—342no—
353#97—341no—
354#97C22⋊D1711yescharacteristic
355#97C22⋊D1711yescharacteristic
356C4×C8—171no—
357C4×C8—171no—
358C4×C8—171no—
359C4×C8—171no—
360C4×C8—172no—
361C4×C8C22×D1712yes—
362C17#1803511yescharacteristic, complemented, Sylow
363#41—22no—
364#41—41no—
365#41—342no—
366#41—681no—
367C2×Q8—171no—
368C2×Q8C2×C4×D1711yescharacteristic
369C2×D4—21no—
370C2×D4—171no—
371C2×D4—341no—
372C2×D4C2×C4×D1711yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—341no—
377C22×C4—341no—
378C22×C4—341no—
379C22×C4C2×C4×D1711yescharacteristic
380C22×C4#6325211yescharacteristic
381C22×C4C2×C22⋊D1711yescharacteristic
382Q16—342no—
383SD16—342no—
384SD16—342no—
385D8—174no—
386M16—24no—
387C2×C8—22no—
388C2×C8—44no—
389C2×C8—171no—
390C2×C8—171no—
391C2×C8—171no—
392C2×C8—171no—
393C2×C8—342no—
394C2×C8—342no—
395C2×C8—342no—
396C2×C8—681no—
397C2×C8—684no—
398C2×C8C2×C22⋊D1712yes—
399C2×C8C2×C22⋊D1712yes—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—171no—
403C4⋊C4—171no—
404C4⋊C4—341no—
405C4⋊C4—341no—
406C4⋊C4C2×C4×D1711yescharacteristic
407C4⋊C4C2×C4×D1711yescharacteristic
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—341no—
412C22⋊C4—341no—
413C22⋊C4—341no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—341no—
418C42—341no—
419C42—341no—
420C42C2×C4×D1711yescharacteristic
421C42#6325211yescharacteristic
422C42C2×C22⋊D1711yescharacteristic
423C42C23×D1711yescharacteristic
424C23—21no—
425C23—341no—
426C23C4×C22⋊D1711yescharacteristic
427Q8—21no—
428Q8—21no—
429Q8—341no—
430Q8—341no—
431D4—21no—
432D4—42no—
433D4—172no—
434D4—341no—
435D4—682no—
436D4M16×D1712yes—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—341no—
448C2×C4—341no—
449C2×C4—341no—
450C2×C4—682no—
451C2×C4—682no—
452C2×C4—1361no—
453C2×C4C4×C22⋊D1711yescharacteristic
454C2×C4C4×C22⋊D1711yescharacteristic
455C2×C4C4×C22⋊D1711yescharacteristic
456C2×C4C22×C4×D1711yescharacteristic
457C2×C4C2×#6325211yescharacteristic
458C2×C4C22×C22⋊D1711yescharacteristic
459C8—22no—
460C8—42no—
461C8—172no—
462C8—341no—
463C8—341no—
464C8—341no—
465C8—682no—
466C8—682no—
467C8#1058763614yescomplemented
468C22—21no—
469C22—22no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—342no—
475C22—681no—
476C22—681no—
477C22C2×C4×C22⋊D1711yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—681no—
486C4—681no—
487C4—681no—
488C4C2×M16×D1711yescharacteristic
489C4#1176357511yescharacteristic
490C4C2×#1058763612yes—
491C2—41no—
492C2—42no—
493C2—682no—
494C2M16×C22⋊D1711yescharacteristic
495C2C4×#1058763611yescharacteristic
496C2#1453769911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#2970C2
socleC22×C17C2×C4×D4
prime radicalC2×#10587636C4
cosocleC42×C17C24
hypercenter#2970D17
hypocenterC17#18035
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2970

Complements

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

NH ≅ G/Ntype
1. #14537799C2semidirect
3. #14527679C2semidirect
4. #14527620C2semidirect
5. #14527618C2semidirect
6. C8×#63228C2semidirect
7. #14527545C2semidirect
8. #14527218C2semidirect
9. #14527216C2semidirect
10. M16×#3622C2semidirect
11. C17×#2970C2semidirect
156. #63228M16semidirect
213. C8×C17#135semidirect
221. #2970D17semidirect
362. C17#18035semidirect
467. C8#10587636semidirect

Maximal subgroups

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

subgroupindex
#145377992
C4×#105876362
#145276792
#145276202
#145276182
C8×#632282
#145275452
#145272182
#145272162
M16×#36222
C17×#29702
#1803517