Finite Group Databasebeta

← #16951920

Subgroups — #16951920

Order 4352.

630 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537248C211yesmaximal, characteristic
2#14530459C211yesmaximal, characteristic, complemented
3#14530438C211yesmaximal, characteristic, complemented
4#14528759C211yesmaximal, characteristic
5#14526913C211yesmaximal, characteristic, complemented
6#14526905C211yesmaximal, characteristic, complemented
7C17×#3011C211yesmaximal, characteristic
8#11763287—21no—
9#11762826—21no—
10#11762818—21no—
11#11762793C2211yescharacteristic
12#11762783—22no—
13#11762783—22no—
14#11762783—21no—
15#11762783C2211yescharacteristic
16#11762783C2211yescharacteristic
17#11762783C2211yescharacteristic
18#11762604—22no—
19#11762599—22no—
20C2×C17×2−1+4C2211yescharacteristic
21C2×C17×#137C411yescharacteristic, complemented
22C22×Q16×C17C411yescharacteristic, complemented
23C17×#536C2211yescharacteristic
24C17×#452—21no—
25C17×#452—21no—
26C17×#451—21no—
27C17×#451—21no—
28C17×#419C411yescharacteristic, complemented
29C17×#416C411yescharacteristic, complemented
30C17×#408C2211yescharacteristic
31C17×2−1+4—42no—
32C17×2−1+4—42no—
33C2×#41×C17—21no—
34C2×#41×C17—21no—
35C2×#41×C17—21no—
36C2×#41×C17C2×C411yescharacteristic
37C2×#41×C17D411yescharacteristic
38C2×#41×C17D411yescharacteristic
39C2×#41×C17C2311yescharacteristic
40C22×Q8×C17—21no—
41C22×Q8×C17C2×C411yescharacteristic
42C22×Q8×C17C2×C411yescharacteristic
43C22×Q8×C17D411yescharacteristic
44C22×Q8×C17D411yescharacteristic
45C17×#137—42no—
46C2×Q16×C17—21no—
47C2×Q16×C17—21no—
48C2×Q16×C17—21no—
49C2×Q16×C17—42no—
50C2×SD16×C17—21no—
51C2×M16×C17C2×C411yescharacteristic
52C22×C8×C17C2×C411yescharacteristic
53C17×C4⋊Q8—21no—
54C17×#124—21no—
55C17×#123—21no—
56C17×#123—21no—
57C17×C22⋊Q8—21no—
58C17×C22⋊Q8—21no—
59C17×#117C2×C411yescharacteristic
60Q8×Dic17—42no—
61Q8×Dic17—42no—
62#10587612—24no—
63#10587612—21no—
64#10587612—21no—
65#10587612—21no—
66C2×#63223—21no—
67C17×#103—21no—
68C17×#103—21no—
69C17×#103—21no—
70C17×#102—21no—
71C17×C22⋊C8—21no—
72C17×C22⋊C8—21no—
73#10587508—42no—
74#10587507—21no—
75#10587506—42no—
76#10587505—24no—
77#10587505—21no—
78#10587505—21no—
79#10587505—21no—
80#10587505—21no—
81#10587505D412yes—
82#10587505D412yes—
83#10587504—21no—
84#41×C17—22no—
85#41×C17—22no—
86#41×C17—41no—
87#41×C17—41no—
88#41×C17—41no—
89#41×C17—41no—
90#41×C17—41no—
91#41×C17—41no—
92#41×C17—44no—
93#41×C17—44no—
94#41×C17—44no—
95C2×Q8×C17—22no—
96C2×Q8×C17—22no—
97C2×Q8×C17—21no—
98C2×Q8×C17—21no—
99C2×Q8×C17—21no—
100C2×Q8×C17—22no—
101C2×Q8×C17—22no—
102C2×Q8×C17—21no—
103C2×Q8×C17—21no—
104C2×Q8×C17—41no—
105C2×Q8×C17—41no—
106C2×Q8×C17—41no—
107C2×Q8×C17—41no—
108C2×Q8×C17—44no—
109C2×Q8×C17C22⋊C411yescharacteristic
110C2×Q8×C17C22⋊C411yescharacteristic
111C2×Q8×C17C22⋊C411yescharacteristic
112C2×Q8×C17#4111yescharacteristic
113C2×D4×C17—21no—
114C2×D4×C17—21no—
115C2×D4×C17—21no—
116C2×D4×C17C22⋊C411yescharacteristic
117C2×D4×C17C2×D411yescharacteristic
118C2×D4×C17C2×D411yescharacteristic
119C2×D4×C17#4111yescharacteristic
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17C22×C411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130Q16×C17—42no—
131Q16×C17—42no—
132Q16×C17—42no—
133SD16×C17—42no—
134M16×C17—41no—
135C2×C8×C17—21no—
136C2×C8×C17—21no—
137C2×C8×C17—41no—
138C4⋊C4×C17—21no—
139C4⋊C4×C17—41no—
140C4⋊C4×C17—41no—
141C22⋊C4×C17—21no—
142C22⋊C4×C17—41no—
143C22⋊C4×C17—41no—
144C42×C17—21no—
145C22⋊Dic17—22no—
146C22⋊Dic17—22no—
147C22⋊Dic17—41no—
148C4⋊Dic17—44no—
149C4⋊Dic17—41no—
150C4⋊Dic17—41no—
151C4⋊Dic17—41no—
152C4×Dic17—44no—
153C4×Dic17—41no—
154C4×Dic17—41no—
155C4×Dic17—41no—
156#63223—41no—
157#63223—41no—
158C2×#3622—41no—
159#13101—171nomaximal, Sylow
160C23×C17—21no—
161C23×C17C4×D411yescharacteristic
162C23×C17#12011yescharacteristic
163C23×C17#12111yescharacteristic
164Q8×C17—24no—
165Q8×C17—21no—
166Q8×C17—21no—
167Q8×C17—21no—
168Q8×C17—21no—
169Q8×C17—41no—
170Q8×C17—42no—
171Q8×C17—42no—
172Q8×C17—41no—
173Q8×C17—41no—
174Q8×C17—81no—
175Q8×C17#9912yes—
176Q8×C17#9912yes—
177D4×C17—21no—
178D4×C17—21no—
179D4×C17—41no—
180D4×C17—41no—
181D4×C17—41no—
182D4×C17—42no—
183D4×C17—42no—
184D4×C17—42no—
185C2×C4×C17—24no—
186C2×C4×C17—21no—
187C2×C4×C17—21no—
188C2×C4×C17—21no—
189C2×C4×C17—21no—
190C2×C4×C17—21no—
191C2×C4×C17—21no—
192C2×C4×C17—21no—
193C2×C4×C17—21no—
194C2×C4×C17—21no—
195C2×C4×C17—42no—
196C2×C4×C17—42no—
197C2×C4×C17—42no—
198C2×C4×C17—42no—
199C2×C4×C17—42no—
200C2×C4×C17—41no—
201C2×C4×C17—41no—
202C2×C4×C17—81no—
203C2×C4×C17C2×C22⋊C411yescharacteristic
204C2×C4×C17C4×D411yescharacteristic
205C2×C4×C17#12111yescharacteristic
206C2×C4×C17#12311yescharacteristic
207C2×Dic17—42no—
208C2×Dic17—42no—
209C2×Dic17—81no—
210C8×C17—41no—
211C8×C17—41no—
212#3622—81no—
213#3011D1711yescharacteristic
214#2884—171no—
215#2129—171no—
216#2123—171no—
217#1796—171no—
218#1395—171no—
219#1391—171no—
220C22×C17—21no—
221C22×C17—21no—
222C22×C17—21no—
223C22×C17—41no—
224C22×C17—41no—
225C22×C17—41no—
226C22×C17—41no—
227C22×C17—81no—
228C22×C17#38611yescharacteristic
229C4×C17—21no—
230C4×C17—22no—
231C4×C17—22no—
232C4×C17—21no—
233C4×C17—21no—
234C4×C17—41no—
235C4×C17—81no—
236C4×C17—81no—
237C4×C17C2×#9911yescharacteristic
238C4×C17C2×#9911yescharacteristic
239Dic17—82no—
240Dic17—82no—
241Dic17—161no—
242C2×2−1+4C2×D1711yescharacteristic
243C2×#137Dic1711yescharacteristic, complemented
244C22×Q16Dic1711yescharacteristic, complemented
245#536C2×D1711yescharacteristic
246#519—341no—
247#452—21no—
248#452—21no—
249#451—21no—
250#451—21no—
251#421—341no—
252#419—341no—
253#419Dic1711yescharacteristic, complemented
254#416Dic1711yescharacteristic, complemented
255#413—171no—
256#410—171no—
257#410—171no—
258#410—171no—
259#410—342no—
260#410—342no—
261#410—341no—
262#408C2×D1711yescharacteristic
263#354—342no—
264#352—342no—
265C2×C17—21no—
266C2×C17—41no—
267C2×C17—41no—
268C2×C17—41no—
269C2×C17—81no—
270C2×C17#189811yescharacteristic
2712−1+4—42no—
2722−1+4—42no—
273C2×#41—21no—
274C2×#41—21no—
275C2×#41—21no—
276C2×#41C2×Dic1711yescharacteristic
277C2×#41C22⋊D1711yescharacteristic
278C2×#41C22⋊D1711yescharacteristic
279C2×#41C22×D1711yescharacteristic
280C22×Q8—21no—
281C22×Q8C2×Dic1711yescharacteristic
282C22×Q8C2×Dic1711yescharacteristic
283C22×Q8C22⋊D1711yescharacteristic
284C22×Q8C22⋊D1711yescharacteristic
285#137—42no—
286C2×Q16—21no—
287C2×Q16—21no—
288C2×Q16—21no—
289C2×Q16—42no—
290C2×SD16—21no—
291C2×M16—341no—
292C2×M16C2×Dic1711yescharacteristic
293C22×C8C2×Dic1711yescharacteristic
294C4⋊Q8—21no—
295#124—21no—
296#123—21no—
297#123—21no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C4×Q8—682no—
301C4×Q8—682no—
302#117—344no—
303#117—341no—
304#117—341no—
305#117—341no—
306#117C2×Dic1711yescharacteristic
307#104—682no—
308#103—21no—
309#103—21no—
310#103—21no—
311#103—682no—
312#102—21no—
313#101—341no—
314#100—341no—
315#99—172no—
316#99—172no—
317#99—344no—
318#99—341no—
319#99—341no—
320#99—341no—
321#99—341no—
322C22⋊C8—21no—
323C22⋊C8—21no—
324C17#1310111yescharacteristic, complemented, Sylow
325#41—22no—
326#41—22no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—44no—
334#41—44no—
335#41—44no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—44no—
350C2×Q8C22⋊Dic1711yescharacteristic
351C2×Q8C22⋊Dic1711yescharacteristic
352C2×Q8C22⋊Dic1711yescharacteristic
353C2×Q8#6325411yescharacteristic
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4C22⋊Dic1711yescharacteristic
358C2×D4D4×D1711yescharacteristic
359C2×D4D4×D1711yescharacteristic
360C2×D4#6325411yescharacteristic
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4C22×Dic1711yescharacteristic
369C22×C4C2×C22⋊D1711yescharacteristic
370C22×C4C2×C22⋊D1711yescharacteristic
371Q16—42no—
372Q16—42no—
373Q16—42no—
374SD16—42no—
375M16—41no—
376M16—681no—
377M16—681no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—41no—
381C2×C8—681no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—684no—
386C4⋊C4—681no—
387C4⋊C4—681no—
388C4⋊C4—681no—
389C22⋊C4—21no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—342no—
393C22⋊C4—342no—
394C22⋊C4—681no—
395C42—21no—
396C42—684no—
397C42—681no—
398C42—681no—
399C42—681no—
400C23—21no—
401C23D4×Dic1711yescharacteristic
402C23#1058762311yescharacteristic
403C23#1058762511yescharacteristic
404Q8—24no—
405Q8—21no—
406Q8—21no—
407Q8—21no—
408Q8—21no—
409Q8—41no—
410Q8—42no—
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—81no—
415Q8#1058750512yes—
416Q8#1058750512yes—
417D4—21no—
418D4—21no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—42no—
423D4—42no—
424D4—42no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—682no—
444C2×C4—682no—
445C2×C4—1361no—
446C2×C4D4×Dic1711yescharacteristic
447C2×C4#1058762111yescharacteristic
448C2×C4#1058762511yescharacteristic
449C2×C4C2×C22⋊Dic1711yescharacteristic
450C8—41no—
451C8—41no—
452C8—1361no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—81no—
461C22#1176268911yescharacteristic
462C4—21no—
463C4—22no—
464C4—22no—
465C4—21no—
466C4—21no—
467C4—41no—
468C4—81no—
469C4—81no—
470C4—2721no—
471C4—1362no—
472C4—1362no—
473C4C2×#1058750511yescharacteristic
474C4C2×#1058750511yescharacteristic
475C2—21no—
476C2—41no—
477C2—41no—
478C2—41no—
479C2—81no—
480C2#1452923811yescharacteristic
subgroupG/H
centerC2#14529238
commutatorC22×C4×C17C22×C4
FrattiniC2×#41C22×D17
FittingC17×#3011C2
socleC2×C17#1898
prime radicalC2×C17×2−1+4C22
cosocleC2×#41×C17C23
hypercenter#3011D17
hypocenterC17#13101
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #3011

Complements

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

NH ≅ G/Ntype
2. #14530459C2semidirect
3. #14530438C2semidirect
5. #14526913C2semidirect
6. #14526905C2semidirect
21. C2×C17×#137C4semidirect
22. C22×Q16×C17C4semidirect
28. C17×#419C4semidirect
29. C17×#416C4semidirect
243. C2×#137Dic17semidirect
244. C22×Q16Dic17semidirect
253. #419Dic17semidirect
254. #416Dic17semidirect
324. C17#13101semidirect

Maximal subgroups

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

subgroupindex
#145372482
#145304592
#145304382
#145287592
#145269132
#145269052
C17×#30112
#1310117