Finite Group Databasebeta

← #17017224

Subgroups — #17017224

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#14539159C211yesmaximal, characteristic
2#14538823C211yesmaximal, characteristic, complemented
3C2×#11763009C211yesmaximal, characteristic
4#14528352C212yesmaximal, complemented
5#14528324C212yesmaximal, complemented
6#14528228C212yesmaximal, complemented
7#14528200C212yesmaximal, complemented
8#14527536C211yesmaximal, characteristic, complemented
9#14527475C211yesmaximal, characteristic, complemented
10#14527453C211yesmaximal, characteristic
11Q8×M16×C17C211yesmaximal, characteristic, complemented
12C22×#63230—21no—
13#11763366C2211yescharacteristic
14C2×#10587560C2211yescharacteristic
15#11763174—22no—
16#11763156C2212yes—
17#11763156C2212yes—
18#11763144—21no—
19#11763104C2211yescharacteristic, complemented
20#11763104C2211yescharacteristic, complemented
21#11763019C2211yescharacteristic, complemented
22#11763019C2211yescharacteristic, complemented
23#11763009—21no—
24#11763009—21no—
25#11763009C2212yes—
26#11763009C2212yes—
27#11762965—21no—
28C4×#63231—21no—
29C4×#63231—21no—
30#11762862C2211yescharacteristic
31C2×C4⋊#3622C2211yescharacteristic
32#11762815—21no—
33#11762815—21no—
34C2×#10587480—21no—
35#11762562C2212yescomplemented
36#11762560C2212yescomplemented
37#11762557C2212yescomplemented
38#11762554C2212yescomplemented
39#11762552C2212yes—
40#11762552C2212yes—
41C2×C4×Q8×C17C2211yescharacteristic
42C17×#446—22no—
43C17×#446C2212yescomplemented
44C8×Q8×C17C2212yescomplemented
45C17×#423—21no—
46C17×#423C2211yescharacteristic
47C4×M16×C17—21no—
48C4×M16×C17C2211yescharacteristic
49C22×Q8×C17D411yescharacteristic
50C22×D68—21no—
51C22×D68—21no—
52C22×D68—21no—
53C2×M16×C17—21no—
54C2×M16×C17D411yescharacteristic
55C2×M16×C17D411yescharacteristic
56C4×Q8×C17—21no—
57C4×Q8×C17—21no—
58C4×Q8×C17D411yescharacteristic
59C4×Q8×C17D411yescharacteristic
60C4×Q8×C17C2311yescharacteristic
61C4×Q8×C17C2311yescharacteristic
62C2×C4⋊C4×C17—21no—
63C2×C4⋊C4×C17D411yescharacteristic
64C2×C42×C17—21no—
65C2×C42×C17C2311yescharacteristic
66C2×#63231—21no—
67C2×#63231—21no—
68C2×#63230—22no—
69C2×#63230—22no—
70C2×#63230—44no—
71#10587612—21no—
72#10587610—21no—
73#10587610—21no—
74C22×#3622—21no—
75#10587560—21no—
76#10587560—21no—
77#10587560—21no—
78#10587560C2312yes—
79#10587556—21no—
80#10587555C2311yescharacteristic
81#10587555C2311yescharacteristic
82C4×Dic34—21no—
83C4×Dic34—21no—
84C17×C4⋊C8—22no—
85C17×C4⋊C8—22no—
86C17×C4⋊C8—21no—
87C17×C4⋊C8C2312yes—
88C17×#97—22no—
89C17×#97D412yes—
90C4×C8×C17—22no—
91C4×C8×C17C2312yes—
92#10587506—21no—
93#10587506—21no—
94C22⋊#3622—21no—
95#10587492—22no—
96#10587492—22no—
97#10587492—22no—
98#10587492—22no—
99#10587481—21no—
100#10587481—21no—
101#10587480—22no—
102#10587480—22no—
103#10587478—21no—
104#10587478—21no—
105C4⋊#3622—21no—
106C4⋊#3622C2312yes—
107C4⋊#3622C2311yescharacteristic
108C4⋊#3622C2311yescharacteristic
109C4×#3622—21no—
110C23×D17—41no—
111C2×Q8×C17—21no—
112C2×Q8×C17—22no—
113C2×Q8×C17—41no—
114C2×Q8×C17C2×D411yescharacteristic
115C2×Q8×C17C2×D411yescharacteristic
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17C2×D411yescharacteristic
119C22×C4×C17C2×D411yescharacteristic
120C22×C4×C17#4111yescharacteristic
121C2×D68—22no—
122C2×D68—22no—
123C2×D68—22no—
124C2×D68—42no—
125C2×D68—42no—
126C2×D68—42no—
127C2×D68—42no—
128C2×D68—41no—
129C2×D68—41no—
130C2×D68—41no—
131C2×D68—82no—
132C2×Dic34—21no—
133C2×Dic34—21no—
134M16×C17—22no—
135M16×C17—22no—
136M16×C17—42no—
137C2×C8×C17—22no—
138C2×C8×C17C2×D412yes—
139C2×C8×C17C2×D412yes—
140C4⋊C4×C17—21no—
141C4⋊C4×C17—21no—
142C4⋊C4×C17—21no—
143C4⋊C4×C17—21no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17C2×D411yescharacteristic
146C4⋊C4×C17C2×D411yescharacteristic
147C42×C17—21no—
148C42×C17—21no—
149C42×C17—21no—
150C42×C17—21no—
151C42×C17C2×D411yescharacteristic
152C42×C17C2×D411yescharacteristic
153C42×C17#4111yescharacteristic
154C42×C17C2411yescharacteristic
155C22⋊Dic17—41no—
156#63231—42no—
157#63231—42no—
158#63230—44no—
159#63230—44no—
160C4⋊Dic17—21no—
161C4⋊Dic17—21no—
162C4⋊Dic17—21no—
163C4⋊Dic17—21no—
164#63225—41no—
165#63225—41no—
166C4×Dic17—41no—
167C2×#3622—21no—
168C2×#3622—21no—
169C2×#3622—22no—
170C2×#3622—42no—
171#19174—171nomaximal, Sylow
172C23×C17#12111yescharacteristic
173C22×D17—42no—
174C22×D17—82no—
175C22×D17—82no—
176C22×D17—82no—
177Q8×C17—21no—
178Q8×C17—21no—
179Q8×C17—21no—
180Q8×C17—42no—
181Q8×C17AGL(1,ℤ/8)12yes—
182C2×C4×C17—21no—
183C2×C4×C17—21no—
184C2×C4×C17—21no—
185C2×C4×C17—21no—
186C2×C4×C17—21no—
187C2×C4×C17—21no—
188C2×C4×C17—41no—
189C2×C4×C17—42no—
190C2×C4×C17—81no—
191C2×C4×C17#12111yescharacteristic
192C2×C4×C17#12111yescharacteristic
193C2×C4×C17#12111yescharacteristic
194C2×C4×C17#13512yes—
195C2×C4×C17#13712yes—
196C2×C4×C17C22×D411yescharacteristic
197C2×C4×C17C22×D411yescharacteristic
198C2×C4×C17C2×#4111yescharacteristic
199C2×Dic17—41no—
200C2×Dic17—41no—
201D68—42no—
202D68—42no—
203D68—42no—
204D68—42no—
205D68—82no—
206D68—82no—
207D68—82no—
208D68—82no—
209Dic34—41no—
210Dic34—41no—
211Dic34—81no—
212C8×C17—22no—
213C8×C17—22no—
214C8×C17—42no—
215#3622—42no—
216#3622—41no—
217#3622—41no—
218#3138—171no—
219#3104—171no—
220C2×#460—171no—
221Q8×M16D1711yescharacteristic, complemented
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#46412yes—
232C22×C17C2×#12111yescharacteristic
233C2×D17—82no—
234C2×D17—82no—
235C2×D17—82no—
236C2×D17—164no—
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—341no—
252#533—171no—
253C2×#127—171no—
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—172no—
264#460—172no—
265#460—341no—
266#460—341no—
267#451—341no—
268#446—22no—
269#446C2×D1712yescomplemented
270C8×Q8C2×D1712yescomplemented
271C4×Q16—341no—
272C4×Q16—341no—
273#431—171no—
274#423—21no—
275#423C2×D1711yescharacteristic
276C2×C4⋊C8—171no—
277#418—341no—
278#418—341no—
279C2×#102—341no—
280C4×M16—21no—
281C4×M16C2×D1711yescharacteristic
282#335—172no—
283#334—172no—
284#333—172no—
285#332—172no—
286#331—172no—
287#331—172no—
288C2×C17—22no—
289C2×C17C2×#46411yescharacteristic
290C2×C17#305511yescharacteristic
291C2×C17#306811yescharacteristic
292D17—162no—
293C22×Q8D6811yescharacteristic
294C22×D4—341no—
295C22×D4—341no—
296C22×D4—341no—
297C2×Q16—341no—
298C2×Q16—341no—
299C2×SD16—342no—
300C2×SD16—342no—
301C2×SD16—684no—
302C2×M16—21no—
303C2×M16C22⋊D1711yescharacteristic
304C2×M16C22⋊D1711yescharacteristic
305C22×C8—341no—
306C4⋊Q8—171no—
307C4⋊Q8—171no—
308#127—172no—
309#127—341no—
310#127—341no—
311#127—341no—
312#125—341no—
313C22⋊Q8—341no—
314C22⋊Q8—341no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—341no—
318C4×Q8—341no—
319C4×Q8D6811yescharacteristic
320C4×Q8D6811yescharacteristic
321C4×Q8C22×D1711yescharacteristic
322C4×Q8C22×D1711yescharacteristic
323#117—341no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4D6811yescharacteristic
326C2×C42—21no—
327C2×C42C22×D1711yescharacteristic
328#107—341no—
329#107—341no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—21no—
333C4⋊C8—171no—
334C4⋊C8—171no—
335C4⋊C8—172no—
336C4⋊C8—341no—
337C4⋊C8C22×D1712yes—
338#103—341no—
339#103—341no—
340#103—341no—
341#103—341no—
342#102—342no—
343#102—342no—
344#102—342no—
345#102—342no—
346#102—342no—
347#102—342no—
348C22⋊C8—341no—
349#97—22no—
350#97C22⋊D1712yes—
351C4×C8—22no—
352C4×C8—341no—
353C4×C8C22×D1712yes—
354C17#1917411yescharacteristic, complemented, Sylow
355C24—681no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—341no—
360C2×Q8—341no—
361C2×Q8C2×D6811yescharacteristic
362C2×Q8C2×D6811yescharacteristic
363C2×D4—342no—
364C2×D4—342no—
365C2×D4—342no—
366C2×D4—682no—
367C2×D4—682no—
368C2×D4—681no—
369C2×D4—681no—
370C2×D4—681no—
371C2×D4—682no—
372C2×D4—682no—
373C2×D4—1362no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4C2×D6811yescharacteristic
377C22×C4#6325211yescharacteristic
378C22×C4C2×C22⋊D1711yescharacteristic
379Q16—682no—
380Q16—682no—
381SD16—684no—
382SD16—684no—
383M16—22no—
384M16—22no—
385M16—42no—
386C2×C8—22no—
387C2×C8—341no—
388C2×C8—341no—
389C2×C8—342no—
390C2×C8—682no—
391C2×C8C2×C22⋊D1712yes—
392C2×C8C2×C22⋊D1712yes—
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×D6811yescharacteristic
405C4⋊C4C2×D6811yescharacteristic
406C22⋊C4—681no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—681no—
412C42C2×D6811yescharacteristic
413C42#6325211yescharacteristic
414C42C2×C22⋊D1711yescharacteristic
415C42C23×D1711yescharacteristic
416C23—682no—
417C23—1362no—
418C23—1362no—
419C23—1362no—
420C23#1058761611yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—681no—
426Q8—681no—
427Q8—1361no—
428Q8#1058759412yes—
429D4—682no—
430D4—682no—
431D4—682no—
432D4—682no—
433D4—1362no—
434D4—1362no—
435D4—1362no—
436D4—1362no—
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#1058761611yescharacteristic
449C2×C4#1058761611yescharacteristic
450C2×C4#1058761611yescharacteristic
451C2×C4#1058762812yes—
452C2×C4#1058763612yes—
453C2×C4C22×D6811yescharacteristic
454C2×C4C2×#6325211yescharacteristic
455C2×C4C22×C22⋊D1711yescharacteristic
456C8—22no—
457C8—22no—
458C8—42no—
459C8—682no—
460C8—681no—
461C8—681no—
462C22—21no—
463C22—21no—
464C22—2724no—
465C22—1362no—
466C22—1362no—
467C22—1362no—
468C22#1176303112yes—
469C22C2×#1058761611yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—1361no—
478C4—1361no—
479C4#1176303112yes—
480C4C2×#1058763611yescharacteristic
481C4C2×#1058759411yescharacteristic
482C4C2×#1058762811yescharacteristic
483C4#1176360911yescharacteristic
484C2—22no—
485C2—2722no—
486C2C2×#1176303111yescharacteristic
487C2#1453840311yescharacteristic
488C2#1453851211yescharacteristic
subgroupG/H
centerC22C2×#10587616
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingQ8×M16×C17C2
socleC22×C17C2×#121
prime radicalC2×#10587560C22
hypercenterQ8×M16D17
hypocenterC17#19174
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #14538823C2semidirect
4. #14528352C2semidirect
5. #14528324C2semidirect
6. #14528228C2semidirect
7. #14528200C2semidirect
8. #14527536C2semidirect
9. #14527475C2semidirect
11. Q8×M16×C17C2semidirect
19. #11763104C22semidirect
20. #11763104C22semidirect
21. #11763019C22semidirect
22. #11763019C22semidirect
35. #11762562C22semidirect
36. #11762560C22semidirect
37. #11762557C22semidirect
38. #11762554C22semidirect
43. C17×#446C22semidirect
44. C8×Q8×C17C22semidirect
221. Q8×M16D17semidirect
269. #446C2×D17semidirect
270. C8×Q8C2×D17semidirect
354. C17#19174semidirect

Maximal subgroups

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

subgroupindex
#145391592
#145388232
C2×#117630092
#145283522
#145283242
#145282282
#145282002
#145275362
#145274752
#145274532
Q8×M16×C172
#1917417