Finite Group Databasebeta

← #17017220

Subgroups — #17017220

Order 4352.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539158C211yesmaximal, characteristic
2#14538823C211yesmaximal, characteristic, complemented
3Q8×#63223C211yesmaximal, characteristic, complemented
4#14528350C212yesmaximal, complemented
5#14528322C212yesmaximal, complemented
6#14528226C212yesmaximal, complemented
7#14528198C212yesmaximal, complemented
8#14527534C211yesmaximal, characteristic, complemented
9#14527472C211yesmaximal, characteristic, complemented
10#14527452C211yesmaximal, characteristic
11C2×C17×#460C211yesmaximal, characteristic
12#11763366C2211yescharacteristic
13#11763173—22no—
14#11763155C2212yes—
15#11763155C2212yes—
16#11763144—21no—
17#11763104C2211yescharacteristic, complemented
18#11763104C2211yescharacteristic, complemented
19#11763019C2211yescharacteristic, complemented
20#11763019C2211yescharacteristic, complemented
21#11762965—21no—
22#11762940—22no—
23#11762940C2212yescomplemented
24Q8×#3622C2212yescomplemented
25C4×#63231—21no—
26C4×#63231—21no—
27#11762862C2211yescharacteristic
28#11762833—21no—
29#11762833C2211yescharacteristic
30#11762814—21no—
31#11762814—21no—
32C4×#63223—21no—
33C4×#63223C2211yescharacteristic
34#11762561C2212yescomplemented
35#11762559C2212yescomplemented
36#11762557C2212yescomplemented
37#11762556C2212yescomplemented
38#11762553C2212yes—
39#11762553C2212yes—
40C22×SD16×C17—21no—
41C2×C17×#127C2211yescharacteristic
42C2×C4×Q8×C17C2211yescharacteristic
43C17×#460—21no—
44C17×#460—21no—
45C17×#460C2212yes—
46C17×#460C2212yes—
47C2×C17×C4⋊C8C2211yescharacteristic
48C2×C17×#102—21no—
49C22×Q8×C17D411yescharacteristic
50C22×D4×C17—21no—
51C22×D4×C17—21no—
52C22×D4×C17—21no—
53C2×SD16×C17—22no—
54C2×SD16×C17—22no—
55C2×SD16×C17—44no—
56C22×C8×C17—21no—
57C17×#127—21no—
58C17×#127—21no—
59C17×#127—21no—
60C17×#127C2312yes—
61C4×Q8×C17—21no—
62C4×Q8×C17—21no—
63C4×Q8×C17D411yescharacteristic
64C4×Q8×C17D411yescharacteristic
65C4×Q8×C17C2311yescharacteristic
66C4×Q8×C17C2311yescharacteristic
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17D411yescharacteristic
69C2×C42×C17—21no—
70C2×C42×C17C2311yescharacteristic
71C2×#63231—21no—
72C2×#63231—21no—
73#10587612—21no—
74#10587610—21no—
75#10587610—21no—
76C2×#63223—21no—
77C2×#63223D411yescharacteristic
78C2×#63223D411yescharacteristic
79#10587556—21no—
80#10587555C2311yescharacteristic
81#10587555C2311yescharacteristic
82C4×Dic34—21no—
83C4×Dic34—21no—
84C17×C4⋊C8—21no—
85C17×C4⋊C8C2312yes—
86C17×#102—22no—
87C17×#102—22no—
88#10587506—21no—
89#10587506—21no—
90#10587503—22no—
91#10587503—22no—
92#10587503—22no—
93#10587503—22no—
94C22⋊#3622—21no—
95#10587481—21no—
96#10587481—21no—
97#10587478—21no—
98#10587478—21no—
99C4⋊#3622—22no—
100C4⋊#3622—22no—
101C4⋊#3622—21no—
102C4⋊#3622C2311yescharacteristic
103C4⋊#3622C2311yescharacteristic
104C4⋊#3622C2312yes—
105#10587474—22no—
106#10587474D412yes—
107C4×#3622—22no—
108C4×#3622—21no—
109C4×#3622C2312yes—
110C24×C17—41no—
111C2×Q8×C17—21no—
112C2×Q8×C17—22no—
113C2×Q8×C17—41no—
114C2×Q8×C17C2×D411yescharacteristic
115C2×Q8×C17C2×D411yescharacteristic
116C2×D4×C17—22no—
117C2×D4×C17—22no—
118C2×D4×C17—22no—
119C2×D4×C17—42no—
120C2×D4×C17—42no—
121C2×D4×C17—41no—
122C2×D4×C17—41no—
123C2×D4×C17—41no—
124C2×D4×C17—42no—
125C2×D4×C17—42no—
126C2×D4×C17—82no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17#4111yescharacteristic
132C2×Dic34—21no—
133C2×Dic34—21no—
134SD16×C17—44no—
135SD16×C17—44no—
136C2×C8×C17—22no—
137C2×C8×C17—42no—
138C4⋊C4×C17—21no—
139C4⋊C4×C17—21no—
140C4⋊C4×C17—21no—
141C4⋊C4×C17—21no—
142C4⋊C4×C17—21no—
143C4⋊C4×C17C2×D411yescharacteristic
144C4⋊C4×C17C2×D411yescharacteristic
145C42×C17—21no—
146C42×C17—21no—
147C42×C17—21no—
148C42×C17—21no—
149C42×C17C2×D411yescharacteristic
150C42×C17C2×D411yescharacteristic
151C42×C17#4111yescharacteristic
152C42×C17C2411yescharacteristic
153C22⋊Dic17—41no—
154#63231—42no—
155#63231—42no—
156C4⋊Dic17—21no—
157C4⋊Dic17—21no—
158C4⋊Dic17—21no—
159C4⋊Dic17—21no—
160#63225—41no—
161#63225—41no—
162C4×Dic17—41no—
163#63223—22no—
164#63223—22no—
165#63223—42no—
166C2×#3622—22no—
167C2×#3622—21no—
168C2×#3622—21no—
169C2×#3622C2×D412yes—
170C2×#3622C2×D412yes—
171#19174—171nomaximal, Sylow
172C23×C17—42no—
173C23×C17—82no—
174C23×C17—82no—
175C23×C17—82no—
176C23×C17#12111yescharacteristic
177Q8×C17—21no—
178Q8×C17—21no—
179Q8×C17—21no—
180Q8×C17—42no—
181Q8×C17AGL(1,ℤ/8)12yes—
182D4×C17—42no—
183D4×C17—42no—
184D4×C17—42no—
185D4×C17—42no—
186D4×C17—82no—
187D4×C17—82no—
188D4×C17—82no—
189D4×C17—82no—
190C2×C4×C17—21no—
191C2×C4×C17—21no—
192C2×C4×C17—21no—
193C2×C4×C17—21no—
194C2×C4×C17—21no—
195C2×C4×C17—21no—
196C2×C4×C17—41no—
197C2×C4×C17—42no—
198C2×C4×C17—81no—
199C2×C4×C17#12111yescharacteristic
200C2×C4×C17#12111yescharacteristic
201C2×C4×C17#12111yescharacteristic
202C2×C4×C17#13512yes—
203C2×C4×C17#13712yes—
204C2×C4×C17C22×D411yescharacteristic
205C2×C4×C17C22×D411yescharacteristic
206C2×C4×C17C2×#4111yescharacteristic
207C2×Dic17—41no—
208C2×Dic17—41no—
209Dic34—41no—
210Dic34—41no—
211Dic34—81no—
212C8×C17—42no—
213#3622—22no—
214#3622—22no—
215#3622—42no—
216#3622—41no—
217#3622—41no—
218#3138—171no—
219#3104—171no—
220C2×#460D1711yescharacteristic
221Q8×M16—171no—
222#1702—172no—
223#1698—172no—
224#1682—172no—
225#1678—172no—
226#1565—171no—
227#1550—171no—
228#1545—171no—
229C22×C17—21no—
230C22×C17—21no—
231C22×C17—82no—
232C22×C17—82no—
233C22×C17—82no—
234C22×C17—164no—
235C22×C17#46412yes—
236C22×C17C2×#12111yescharacteristic
237C4×C17—21no—
238C4×C17—22no—
239C4×C17—21no—
240C4×C17—21no—
241C4×C17—41no—
242C4×C17—41no—
243C4×C17—41no—
244C4×C17#46412yes—
245C4×C17C2×#13511yescharacteristic
246C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
247C4×C17C2×#13711yescharacteristic
248C4×C17#57511yescharacteristic
249Dic17—81no—
250Dic17—81no—
251C22×SD16—21no—
252#533—171no—
253C2×#127C2×D1711yescharacteristic
254C2×C4×Q8C2×D1711yescharacteristic
255#490—342no—
256#486—172no—
257#486—172no—
258#484—341no—
259#477—171no—
260#477—171no—
261#462—171no—
262#462—171no—
263#460—21no—
264#460—21no—
265#460C2×D1712yes—
266#460C2×D1712yes—
267#451—341no—
268#446—172no—
269#446—342no—
270C8×Q8—172no—
271C4×Q16—341no—
272C4×Q16—341no—
273#431—171no—
274#423—171no—
275#423—341no—
276C2×C4⋊C8C2×D1711yescharacteristic
277#418—341no—
278#418—341no—
279C2×#102—21no—
280C4×M16—171no—
281C4×M16—341no—
282#335—172no—
283#334—172no—
284#333—172no—
285#332—172no—
286#331—172no—
287#331—172no—
288C2×C17—22no—
289C2×C17—162no—
290C2×C17C2×#46411yescharacteristic
291C2×C17#305511yescharacteristic
292C2×C17#306811yescharacteristic
293C22×Q8C22⋊D1711yescharacteristic
294C22×D4—21no—
295C22×D4—21no—
296C22×D4—21no—
297C2×Q16—341no—
298C2×Q16—341no—
299C2×SD16—22no—
300C2×SD16—22no—
301C2×SD16—44no—
302C2×M16—171no—
303C2×M16—171no—
304C2×M16—341no—
305C22×C8—21no—
306C4⋊Q8—171no—
307C4⋊Q8—171no—
308#127—21no—
309#127—21no—
310#127—21no—
311#127C22×D1712yes—
312#125—341no—
313C22⋊Q8—341no—
314C22⋊Q8—341no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—341no—
318C4×Q8—341no—
319C4×Q8C22⋊D1711yescharacteristic
320C4×Q8C22⋊D1711yescharacteristic
321C4×Q8C22×D1711yescharacteristic
322C4×Q8C22×D1711yescharacteristic
323#117—341no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C22⋊D1711yescharacteristic
326C2×C42—21no—
327C2×C42C22×D1711yescharacteristic
328#107—341no—
329#107—341no—
330C4⋊C8—21no—
331C4⋊C8—172no—
332C4⋊C8—171no—
333C4⋊C8—171no—
334C4⋊C8—342no—
335C4⋊C8—342no—
336C4⋊C8—341no—
337C4⋊C8C22×D1712yes—
338#103—341no—
339#103—341no—
340#103—341no—
341#103—341no—
342#102—22no—
343#102—22no—
344#102—342no—
345#102—342no—
346#102—342no—
347#102—342no—
348C22⋊C8—341no—
349#97—172no—
350#97—342no—
351C4×C8—172no—
352C4×C8—342no—
353C4×C8—341no—
354C17#1917411yescharacteristic, complemented, Sylow
355C24—41no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—341no—
360C2×Q8—341no—
361C2×Q8C2×C22⋊D1711yescharacteristic
362C2×Q8C2×C22⋊D1711yescharacteristic
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—82no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4D4×D1711yescharacteristic
377C22×C4#6325411yescharacteristic
378C22×C4C2×C22⋊D1711yescharacteristic
379Q16—682no—
380Q16—682no—
381SD16—44no—
382SD16—44no—
383M16—342no—
384M16—342no—
385M16—682no—
386C2×C8—22no—
387C2×C8—42no—
388C2×C8—172no—
389C2×C8—172no—
390C2×C8—342no—
391C2×C8—341no—
392C2×C8—341no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—341no—
399C4⋊C4—341no—
400C4⋊C4—341no—
401C4⋊C4—341no—
402C4⋊C4—681no—
403C4⋊C4—681no—
404C4⋊C4C2×C22⋊D1711yescharacteristic
405C4⋊C4C2×C22⋊D1711yescharacteristic
406C22⋊C4—681no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—681no—
412C42D4×D1711yescharacteristic
413C42#6325411yescharacteristic
414C42C2×C22⋊D1711yescharacteristic
415C42C23×D1711yescharacteristic
416C23—42no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23#1058762511yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—681no—
426Q8—681no—
427Q8—1361no—
428Q8#1058761812yes—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—681no—
447C2×C4—681no—
448C2×C4#1058760112yes—
449C2×C4#1058760212yes—
450C2×C4#1058762511yescharacteristic
451C2×C4#1058762511yescharacteristic
452C2×C4#1058762511yescharacteristic
453C2×C4C2×D4×D1711yescharacteristic
454C2×C4C2×#6325411yescharacteristic
455C2×C4C22×C22⋊D1711yescharacteristic
456C8—42no—
457C8—342no—
458C8—342no—
459C8—682no—
460C8—681no—
461C8—681no—
462C22—21no—
463C22—21no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22#1176302912yes—
469C22C2×#1058762511yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—1361no—
478C4—1361no—
479C4#1176302912yes—
480C4C2×#1058760211yescharacteristic
481C4C2×#1058761811yescharacteristic
482C4C2×#1058760111yescharacteristic
483C4#1176360711yescharacteristic
484C2—22no—
485C2—162no—
486C2C2×#1176302911yescharacteristic
487C2#1453839911yescharacteristic
488C2#1453850911yescharacteristic
subgroupG/H
centerC22C2×#10587625
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×C17×#460C2
socleC22×C17C2×#121
prime radicalC2×C17×#127C22
hypercenterC2×#460D17
hypocenterC17#19174
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #14538823C2semidirect
3. Q8×#63223C2semidirect
4. #14528350C2semidirect
5. #14528322C2semidirect
6. #14528226C2semidirect
7. #14528198C2semidirect
8. #14527534C2semidirect
9. #14527472C2semidirect
17. #11763104C22semidirect
18. #11763104C22semidirect
19. #11763019C22semidirect
20. #11763019C22semidirect
23. #11762940C22semidirect
24. Q8×#3622C22semidirect
34. #11762561C22semidirect
35. #11762559C22semidirect
36. #11762557C22semidirect
37. #11762556C22semidirect
354. C17#19174semidirect

Maximal subgroups

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

subgroupindex
#145391582
#145388232
Q8×#632232
#145283502
#145283222
#145282262
#145281982
#145275342
#145274722
#145274522
C2×C17×#4602
#1917417