Finite Group Databasebeta

← #17189254

Subgroups — #17189254

Order 4352.

610 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539367C211yesmaximal, characteristic, complemented
2C4×D8×D17C211yesmaximal, characteristic
3#14530983C211yesmaximal, characteristic, complemented
4#14530980C211yesmaximal, characteristic, complemented
5#14530930C211yesmaximal, characteristic, complemented
6#14530929C211yesmaximal, characteristic, complemented
7#14530828C211yesmaximal, characteristic, complemented
8#14530827C211yesmaximal, characteristic, complemented
9#14530714C211yesmaximal, characteristic
10#14530711C211yesmaximal, characteristic
11C4×#10587471C211yesmaximal, characteristic, complemented
12C4×#10587499C211yesmaximal, characteristic, complemented
13C4×#10587498C211yesmaximal, characteristic, complemented
14#14530581C211yesmaximal, characteristic, complemented
15C4×C17×SD32C211yesmaximal, characteristic, complemented
16C2×D8×D17C2211yescharacteristic
17#11763363—21no—
18C4×D4×D17—21no—
19C2×#10587471C2211yescharacteristic
20C2×#10587498C2211yescharacteristic
21C2×#10587499C2211yescharacteristic
22#11763213C2211yescharacteristic, complemented
23#11763187C2211yescharacteristic, complemented
24#11763158—21no—
25#11763051C2211yescharacteristic
26#11763049—21no—
27#11763047C2211yescharacteristic
28C4×Dic68C2211yescharacteristic, complemented
29C4×#63231—21no—
30C4×D136C2211yescharacteristic
31D8×Dic17C2211yescharacteristic
32#11762891C2211yescharacteristic
33C4×#63228—21no—
34#11762889—21no—
35#107×D17C2211yescharacteristic
36#102×D17—21no—
37C4×C8×D17C2211yescharacteristic
38#11762637C2211yescharacteristic, complemented
39#11762636C2211yescharacteristic, complemented
40#11762636C2211yescharacteristic, complemented
41#11762627C2211yescharacteristic, complemented
42#11762627C2211yescharacteristic, complemented
43#11762614C2211yescharacteristic, complemented
44#11762613C2211yescharacteristic, complemented
45#11762612C2211yescharacteristic, complemented
46#11762611C2211yescharacteristic
47#11762610C2211yescharacteristic
48#11762609C2211yescharacteristic
49#11762588C2211yescharacteristic
50#11762588C2211yescharacteristic
51C4×#63214C2211yescharacteristic, complemented
52C2×C17×SD32C2211yescharacteristic
53C4×Q16×C17C2211yescharacteristic, complemented
54C4×D8×C17C2211yescharacteristic
55C17×#367C2211yescharacteristic, complemented
56C17×#358C2211yescharacteristic, complemented
57C17×#357C2211yescharacteristic
58C4×C16×C17C2211yescharacteristic, complemented
59C2×D4×D17—21no—
60C22×C4×D17—41no—
61C2×Q16×C17C2311yescharacteristic
62C2×D8×C17C2311yescharacteristic
63C4×Q8×C17—21no—
64C4×D4×C17—21no—
65#10587632—21no—
66#10587630—21no—
67C2×#63231—21no—
68D4×Dic17—21no—
69C2×#63228—21no—
70C4×C22⋊D17—41no—
71D8×D17—22no—
72D8×D17—22no—
73C2×Dic68C2311yescharacteristic
74C2×D136C2311yescharacteristic
75C2×C8×D17D411yescharacteristic
76C2×C8×D17C2311yescharacteristic
77#10587583—21no—
78#10587582—41no—
79#10587579—21no—
80C4⋊C4×D17—21no—
81#10587575—41no—
82#10587574—21no—
83#10587567—41no—
84C22⋊C4×D17—41no—
85C4×D68—21no—
86#10587558—41no—
87C42×D17D411yescharacteristic
88C4×Dic34—21no—
89C4×Dic34—41no—
90C17×SD32—22no—
91C2×C16×C17C2311yescharacteristic
92C2×C16×C17C2311yescharacteristic
93C17×#107C2311yescharacteristic
94C17×#107C2311yescharacteristic
95C17×#103—21no—
96C17×#102—21no—
97C4×C8×C17C2311yescharacteristic
98#10587506—21no—
99#10587503—21no—
100#10587499—22no—
101#10587498—22no—
102#10587492—21no—
103#10587489C2311yescharacteristic
104#10587489C2311yescharacteristic
105#10587487—21no—
106C8×Dic17D411yescharacteristic
107C8×Dic17C2311yescharacteristic
108C2×#63214C2311yescharacteristic
109C2×#63214C2311yescharacteristic
110#10587481—21no—
111#10587480—21no—
112#10587478—21no—
113#10587478—21no—
114C4×#3622D411yescharacteristic
115#10587471—22no—
116C23×D17—41no—
117C2×Q8×C17—21no—
118C2×D4×C17—21no—
119C22×C4×C17—41no—
120C2×C22⋊D17—41no—
121C22×Dic17—41no—
122D4×D17—24no—
123D4×D17—42no—
124C2×D68—21no—
125C2×C4×D17—41no—
126C2×C4×D17—41no—
127C2×C4×D17—41no—
128C2×C4×D17—84no—
129C2×C4×D17D811yescharacteristic
130C2×C4×D17D811yescharacteristic
131C2×C4×D17C2×D411yescharacteristic
132C2×Dic34—21no—
133C2×Dic34—41no—
134Q16×C17—21no—
135Q16×C17—21no—
136D8×C17—21no—
137D8×C17#4112yes—
138C2×C8×C17C2×D411yescharacteristic
139C2×C8×C17C2411yescharacteristic
140C4⋊C4×C17—21no—
141C4⋊C4×C17—21no—
142C4⋊C4×C17—41no—
143C22⋊C4×C17—41no—
144C42×C17—41no—
145C42×C17C2×D411yescharacteristic
146C22⋊Dic17—41no—
147#63231—42no—
148#63228—24no—
149#63227—41no—
150#63227—41no—
151#63227—41no—
152#63227—41no—
153C4⋊Dic17—21no—
154C4⋊Dic17—21no—
155C4⋊Dic17—41no—
156#63225—41no—
157#63225—41no—
158#63225—41no—
159#63225—41no—
160#63225—41no—
161#63225—41no—
162C4×Dic17—41no—
163C4×Dic17D811yescharacteristic
164C4×Dic17D811yescharacteristic
165C4×Dic17C2×D411yescharacteristic
166C2×#3622C2×D411yescharacteristic
167C2×#3622C2×D411yescharacteristic
168Dic68—21no—
169Dic68—21no—
170D136—21no—
171D136#4112yes—
172C8×D17—21no—
173C8×D17—21no—
174C8×D17—41no—
175C16×C17—21no—
176C16×C17—21no—
177#63214—21no—
178#63214—21no—
179#33661—171nomaximal, Sylow
180C23×C17—41no—
181C22×D17—44no—
182C22×D17—41no—
183C22×D17—82no—
184C22×D17—82no—
185C22×D17C2×D811yescharacteristic
186Q8×C17—41no—
187Q8×C17—41no—
188D4×C17—22no—
189D4×C17—41no—
190C2×C4×C17—41no—
191C2×C4×C17—41no—
192C2×C4×C17—41no—
193C2×C4×C17—82no—
194C2×C4×C17C2×D811yescharacteristic
195C2×C4×C17C2×D811yescharacteristic
196C2×C4×C17C22×D411yescharacteristic
197C22⋊D17—44no—
198C2×Dic17—41no—
199C2×Dic17—41no—
200C2×Dic17—41no—
201C2×Dic17—82no—
202C2×Dic17C2×D811yescharacteristic
203C2×Dic17C2×D811yescharacteristic
204C2×Dic17C2×D811yescharacteristic
205D68—22no—
206D68—41no—
207C4×D17—21no—
208C4×D17—21no—
209C4×D17—41no—
210C4×D17—82no—
211C4×D17—82no—
212C4×D17—82no—
213C4×D17SD3214yescomplemented
214Dic34—41no—
215Dic34—41no—
216Dic34—82no—
217C8×C17—21no—
218C8×C17C2×#4111yescharacteristic
219C8×C17C2×#4111yescharacteristic
220#3622—21no—
221#3622—21no—
222#3170—171no—
223C2×C4×D8—171no—
224#2235—171no—
225#2235—171no—
226#2226—171no—
227#2226—171no—
228#2208—171no—
229#2208—171no—
230#2187—171no—
231#2187—171no—
232C4×SD32—171no—
233C4×SD32—171no—
234C4×SD32—171no—
235C4×SD32D1711yescharacteristic, complemented
236#2162—171no—
237C22×C17—42no—
238C22×C17—81no—
239C22×C17—81no—
240C22×C17C22×D811yescharacteristic
241C2×D17—21no—
242C2×D17—21no—
243C2×D17—42no—
244C2×D17—88no—
245C2×D17—81no—
246C2×D17—81no—
247C2×D17C2×SD3212yes—
248C4×C17—21no—
249C4×C17—81no—
250C4×C17—81no—
251C4×C17—81no—
252C4×C17C2×SD3212yes—
253C4×C17#52511yescharacteristic
254C4×C17#57511yescharacteristic
255Dic17—21no—
256Dic17—21no—
257Dic17—81no—
258Dic17—81no—
259Dic17—81no—
260Dic17C2×SD3212yes—
261C22×D8—171no—
262#533—341no—
263C2×C4×D4—341no—
264C2×SD32—171no—
265C2×SD32—171no—
266C2×SD32—171no—
267C2×SD32C2×D1711yescharacteristic
268#500—171no—
269#494—171no—
270#487—341no—
271#467—171no—
272#467—171no—
273#467—341no—
274C4×Q16—171no—
275C4×Q16—341no—
276C4×Q16C2×D1711yescharacteristic, complemented
277C4×D8—171no—
278C4×D8—171no—
279C4×D8—171no—
280C4×D8—341no—
281C4×D8—341no—
282C4×D8C2×D1711yescharacteristic
283C2×#107—171no—
284C2×#102—341no—
285C2×C4×C8—171no—
286#367—171no—
287#367—171no—
288#367—171no—
289#367C2×D1711yescharacteristic, complemented
290C4⋊C16—171no—
291C4⋊C16—171no—
292#358—171no—
293#358—171no—
294#358—171no—
295#358C2×D1711yescharacteristic, complemented
296#357—171no—
297#357—171no—
298#357—171no—
299#357C2×D1711yescharacteristic
300C22⋊C16—171no—
301C22⋊C16—171no—
302C4×C16—171no—
303C4×C16C2×D1711yescharacteristic, complemented
304C2×C17—82no—
305C2×C17#310011yescharacteristic
306C2×C17C22×SD3211yescharacteristic
307C2×C17#341411yescharacteristic
308D17—22no—
309D17—82no—
310C22×D4—341no—
311C23×C4—681no—
312C2×Q16—171no—
313C2×Q16—341no—
314C2×Q16C22×D1711yescharacteristic
315C2×D8—171no—
316C2×D8—341no—
317C2×D8—342no—
318C2×D8—342no—
319C2×D8C22×D1711yescharacteristic
320C22×C8—171no—
321C22×C8—171no—
322C4⋊Q8—341no—
323C4⋊Q8—341no—
324#125—681no—
325C22⋊Q8—341no—
326C22⋊Q8—341no—
327C22⋊Q8—681no—
328C4×Q8—21no—
329C4×Q8—341no—
330C4×Q8—681no—
331C4×D4—21no—
332C4×D4—341no—
333C4×D4—341no—
334C4×D4—341no—
335C4×D4—681no—
336C4×D4—681no—
337#117—681no—
338C2×C4⋊C4—341no—
339C2×C22⋊C4—681no—
340C2×C42—171no—
341SD32—22no—
342SD32—342no—
343SD32—342no—
344SD32—342no—
345C2×C16—171no—
346C2×C16—171no—
347C2×C16C22×D1711yescharacteristic
348C2×C16C22×D1711yescharacteristic
349#107—171no—
350#107—171no—
351#107—341no—
352#107—341no—
353#107C22×D1711yescharacteristic
354#107C22×D1711yescharacteristic
355#103—21no—
356#103—341no—
357#103—341no—
358#103—341no—
359#102—21no—
360#102—341no—
361#102—341no—
362#102—341no—
363C4×C8—171no—
364C4×C8—171no—
365C4×C8—171no—
366C4×C8C22×D1711yescharacteristic
367C17#3366111yescharacteristic, complemented, Sylow
368C24—681no—
369C2×Q8—21no—
370C2×Q8—341no—
371C2×Q8—681no—
372C2×D4—21no—
373C2×D4—344no—
374C2×D4—341no—
375C2×D4—681no—
376C2×D4—682no—
377C22×C4—41no—
378C22×C4—171no—
379C22×C4—171no—
380C22×C4—171no—
381C22×C4—681no—
382C22×C4—681no—
383C22×C4—681no—
384C22×C4—681no—
385C22×C4—1364no—
386Q16—21no—
387Q16—21no—
388Q16—341no—
389Q16—341no—
390Q16—682no—
391D8—21no—
392D8—172no—
393D8—344no—
394D8—341no—
395D8#6325212yes—
396C2×C8—171no—
397C2×C8—171no—
398C2×C8—341no—
399C2×C8—341no—
400C2×C8—681no—
401C2×C8D4×D1711yescharacteristic
402C2×C8C23×D1711yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—341no—
407C4⋊C4—341no—
408C4⋊C4—681no—
409C4⋊C4—681no—
410C4⋊C4—681no—
411C4⋊C4—681no—
412C4⋊C4—681no—
413C4⋊C4—681no—
414C4⋊C4—681no—
415C22⋊C4—41no—
416C22⋊C4—681no—
417C22⋊C4—681no—
418C22⋊C4—681no—
419C22⋊C4—681no—
420C22⋊C4—681no—
421C42—41no—
422C42—171no—
423C42—171no—
424C42—171no—
425C42—681no—
426C42D4×D1711yescharacteristic
427C16—21no—
428C16—21no—
429C16—341no—
430C16—341no—
431C23—41no—
432C23—171no—
433C23—684no—
434C23—681no—
435C23—1362no—
436C23—1362no—
437Q8—41no—
438Q8—41no—
439Q8—681no—
440Q8—681no—
441Q8—1362no—
442D4—22no—
443D4—41no—
444D4—342no—
445D4—684no—
446D4—681no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—82no—
451C2×C4—171no—
452C2×C4—174no—
453C2×C4—171no—
454C2×C4—171no—
455C2×C4—341no—
456C2×C4—341no—
457C2×C4—681no—
458C2×C4—681no—
459C2×C4—681no—
460C2×C4—681no—
461C2×C4—1362no—
462C2×C4—1362no—
463C2×C4—1362no—
464C2×C4—1362no—
465C2×C4D8×D1711yescharacteristic
466C2×C4D8×D1711yescharacteristic
467C2×C4C2×D4×D1711yescharacteristic
468C8—21no—
469C8—341no—
470C8—341no—
471C8C2×#6325211yescharacteristic
472C8#41×D1711yescharacteristic
473C22—42no—
474C22—81no—
475C22—81no—
476C22—172no—
477C22—341no—
478C22—341no—
479C22—682no—
480C22—1368no—
481C22—1361no—
482C22—1361no—
483C22C2×D8×D1711yescharacteristic
484C4—21no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—172no—
489C4—341no—
490C4—341no—
491C4—1361no—
492C4—1361no—
493C4—1361no—
494C4SD32×D1712yes—
495C4#1176331811yescharacteristic
496C4#1176360611yescharacteristic
497C2—82no—
498C2—342no—
499C2—1362no—
500C2#1453878211yescharacteristic
501C2C2×SD32×D1711yescharacteristic
502C2#1454191711yescharacteristic
subgroupG/H
centerC2×C4D8×D17
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC4×C17×SD32C2
socleC22×C17C22×D8
prime radicalC2×D8×D17C22
hypercenterC4×SD32D17
hypocenterC17#33661
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ C4×SD32

Complements

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

NH ≅ G/Ntype
1. #14539367C2semidirect
3. #14530983C2semidirect
4. #14530980C2semidirect
5. #14530930C2semidirect
6. #14530929C2semidirect
7. #14530828C2semidirect
8. #14530827C2semidirect
11. C4×#10587471C2semidirect
12. C4×#10587499C2semidirect
13. C4×#10587498C2semidirect
14. #14530581C2semidirect
15. C4×C17×SD32C2semidirect
22. #11763213C22semidirect
23. #11763187C22semidirect
28. C4×Dic68C22semidirect
38. #11762637C22semidirect
39. #11762636C22semidirect
40. #11762636C22semidirect
41. #11762627C22semidirect
42. #11762627C22semidirect
43. #11762614C22semidirect
44. #11762613C22semidirect
45. #11762612C22semidirect
51. C4×#63214C22semidirect
53. C4×Q16×C17C22semidirect
55. C17×#367C22semidirect
56. C17×#358C22semidirect
58. C4×C16×C17C22semidirect
213. C4×D17SD32semidirect
235. C4×SD32D17semidirect
276. C4×Q16C2×D17semidirect
289. #367C2×D17semidirect
295. #358C2×D17semidirect
303. C4×C16C2×D17semidirect
367. C17#33661semidirect

Maximal subgroups

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

subgroupindex
#145393672
C4×D8×D172
#145309832
#145309802
#145309302
#145309292
#145308282
#145308272
#145307142
#145307112
C4×#105874712
C4×#105874992
C4×#105874982
#145305812
C4×C17×SD322
#3366117