Finite Group Databasebeta

← #17017226

Subgroups — #17017226

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#14539160C211yesmaximal, characteristic
2C2×#11763009C211yesmaximal, characteristic
3Q8×#63223C211yesmaximal, characteristic, complemented
4#14528354C212yesmaximal, complemented
5#14528326C212yesmaximal, complemented
6#14528230C212yesmaximal, complemented
7#14528202C212yesmaximal, complemented
8#14527537C211yesmaximal, characteristic, complemented
9#14527474C211yesmaximal, characteristic, complemented
10#14527451C211yesmaximal, characteristic
11C17×#3104C211yesmaximal, characteristic, complemented
12C22×#63230—21no—
13C2×#10587560C2211yescharacteristic
14#11763175—22no—
15#11763157C2212yes—
16#11763157C2212yes—
17#11763009—21no—
18#11763009—21no—
19#11763009C2212yes—
20#11763009C2212yes—
21#11762940—22no—
22#11762940C2212yescomplemented
23Q8×#3622C2212yescomplemented
24#11762833—21no—
25#11762833C2211yescharacteristic
26C2×C4⋊#3622C2211yescharacteristic
27#11762813—21no—
28#11762813—21no—
29C2×#10587480—21no—
30C4×#63223—21no—
31C4×#63223C2211yescharacteristic
32#11762561C2212yescomplemented
33#11762559C2212yescomplemented
34#11762558C2212yescomplemented
35#11762555C2212yescomplemented
36#11762552C2212yes—
37#11762552C2212yes—
38C17×#533C2211yescharacteristic
39C2×C4×Q8×C17C2211yescharacteristic
40C17×#484—21no—
41C17×#477C2211yescharacteristic, complemented
42C17×#477C2211yescharacteristic, complemented
43C17×#462C2211yescharacteristic, complemented
44C17×#462C2211yescharacteristic, complemented
45C17×#451—21no—
46C4×Q16×C17—21no—
47C4×Q16×C17—21no—
48C17×#431C2211yescharacteristic
49C22×Q8×C17D411yescharacteristic
50C22×D68—21no—
51C22×D68—21no—
52C22×D68—21no—
53C2×Q16×C17—21no—
54C2×Q16×C17—21no—
55C17×C4⋊Q8C2311yescharacteristic
56C17×C4⋊Q8C2311yescharacteristic
57C17×#125—21no—
58C17×C22⋊Q8—21no—
59C17×C22⋊Q8—21no—
60C4×Q8×C17—21no—
61C4×Q8×C17—21no—
62C4×Q8×C17—21no—
63C4×Q8×C17—21no—
64C4×Q8×C17D411yescharacteristic
65C4×Q8×C17D411yescharacteristic
66C4×Q8×C17C2311yescharacteristic
67C4×Q8×C17C2311yescharacteristic
68C17×#117—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17D411yescharacteristic
71C2×C42×C17—21no—
72C2×C42×C17C2311yescharacteristic
73C2×#63230—22no—
74C2×#63230—22no—
75C2×#63230—44no—
76C2×#63223—21no—
77C2×#63223D411yescharacteristic
78C2×#63223D411yescharacteristic
79C22×#3622—21no—
80#10587560—21no—
81#10587560—21no—
82#10587560—21no—
83#10587560C2312yes—
84C17×#107—21no—
85C17×#107—21no—
86C17×C4⋊C8C2311yescharacteristic
87C17×C4⋊C8C2311yescharacteristic
88C17×#103—21no—
89C17×#103—21no—
90C17×#103—21no—
91C17×#103—21no—
92C17×C22⋊C8—21no—
93C4×C8×C17—21no—
94#10587480—22no—
95#10587480—22no—
96#10587480—22no—
97#10587480—22no—
98#10587480—22no—
99#10587480—22no—
100C4⋊#3622—22no—
101C4⋊#3622—22no—
102C4⋊#3622—21no—
103C4⋊#3622—21no—
104C4⋊#3622C2312yes—
105C4⋊#3622C2312yes—
106#10587474—22no—
107#10587474D412yes—
108C4×#3622—22no—
109C4×#3622C2312yes—
110C23×D17—41no—
111C2×Q8×C17—21no—
112C2×Q8×C17—22no—
113C2×Q8×C17—21no—
114C2×Q8×C17—21no—
115C2×Q8×C17—41no—
116C2×Q8×C17C2×D411yescharacteristic
117C2×Q8×C17C2×D411yescharacteristic
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17C2×D411yescharacteristic
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17#4111yescharacteristic
123C2×D68—22no—
124C2×D68—22no—
125C2×D68—22no—
126C2×D68—42no—
127C2×D68—42no—
128C2×D68—42no—
129C2×D68—42no—
130C2×D68—41no—
131C2×D68—41no—
132C2×D68—41no—
133C2×D68—82no—
134Q16×C17—42no—
135Q16×C17—42no—
136C2×C8×C17—21no—
137C2×C8×C17—21no—
138C4⋊C4×C17—21no—
139C4⋊C4×C17—21no—
140C4⋊C4×C17—21no—
141C4⋊C4×C17—21no—
142C4⋊C4×C17—21no—
143C4⋊C4×C17—21no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—21no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—41no—
148C4⋊C4×C17—41no—
149C4⋊C4×C17C2×D411yescharacteristic
150C4⋊C4×C17C2×D411yescharacteristic
151C22⋊C4×C17—41no—
152C42×C17—21no—
153C42×C17—21no—
154C42×C17—21no—
155C42×C17—21no—
156C42×C17—41no—
157C42×C17C2×D411yescharacteristic
158C42×C17C2×D411yescharacteristic
159C42×C17#4111yescharacteristic
160C42×C17C2411yescharacteristic
161#63230—44no—
162#63230—44no—
163#63223—22no—
164#63223—22no—
165#63223—42no—
166C2×#3622—22no—
167C2×#3622—22no—
168C2×#3622—42no—
169C2×#3622C2×D412yes—
170C2×#3622C2×D412yes—
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×C17—41no—
182Q8×C17—41no—
183Q8×C17—81no—
184Q8×C17AGL(1,ℤ/8)12yes—
185C2×C4×C17—21no—
186C2×C4×C17—21no—
187C2×C4×C17—21no—
188C2×C4×C17—21no—
189C2×C4×C17—21no—
190C2×C4×C17—21no—
191C2×C4×C17—41no—
192C2×C4×C17—42no—
193C2×C4×C17—41no—
194C2×C4×C17—41no—
195C2×C4×C17—81no—
196C2×C4×C17#12111yescharacteristic
197C2×C4×C17#12111yescharacteristic
198C2×C4×C17#12111yescharacteristic
199C2×C4×C17#13512yes—
200C2×C4×C17#13712yes—
201C2×C4×C17C22×D411yescharacteristic
202C2×C4×C17C22×D411yescharacteristic
203C2×C4×C17C2×#4111yescharacteristic
204D68—42no—
205D68—42no—
206D68—42no—
207D68—42no—
208D68—82no—
209D68—82no—
210D68—82no—
211D68—82no—
212C8×C17—41no—
213C8×C17—41no—
214#3622—22no—
215#3622—22no—
216#3622—42no—
217#3622—42no—
218#3138—171no—
219#3104D1711yescharacteristic, complemented
220C2×#460—171no—
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#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—81no—
245C4×C17—81no—
246C4×C17#46412yes—
247C4×C17C2×#13511yescharacteristic
248C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
249C4×C17C2×#13711yescharacteristic
250C4×C17#57511yescharacteristic
251C22×SD16—341no—
252#533C2×D1711yescharacteristic
253C2×#127—171no—
254C2×C4×Q8C2×D1711yescharacteristic
255#490—342no—
256#486—172no—
257#486—172no—
258#484—21no—
259#477C2×D1711yescharacteristic, complemented
260#477C2×D1711yescharacteristic, complemented
261#462C2×D1711yescharacteristic, complemented
262#462C2×D1711yescharacteristic, complemented
263#460—172no—
264#460—172no—
265#460—341no—
266#460—341no—
267#451—21no—
268#446—172no—
269#446—342no—
270C8×Q8—172no—
271C4×Q16—21no—
272C4×Q16—21no—
273#431C2×D1711yescharacteristic
274#423—171no—
275#423—341no—
276C2×C4⋊C8—171no—
277#418—341no—
278#418—341no—
279C2×#102—341no—
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×C17C2×#46411yescharacteristic
290C2×C17#305511yescharacteristic
291C2×C17#306811yescharacteristic
292D17—162no—
293C22×Q8C22⋊D1711yescharacteristic
294C22×D4—341no—
295C22×D4—341no—
296C22×D4—341no—
297C2×Q16—21no—
298C2×Q16—21no—
299C2×SD16—342no—
300C2×SD16—342no—
301C2×SD16—684no—
302C2×M16—171no—
303C2×M16—171no—
304C2×M16—341no—
305C22×C8—341no—
306C4⋊Q8C22×D1711yescharacteristic
307C4⋊Q8C22×D1711yescharacteristic
308#127—172no—
309#127—341no—
310#127—341no—
311#127—341no—
312#125—21no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×Q8C22⋊D1711yescharacteristic
320C4×Q8C22⋊D1711yescharacteristic
321C4×Q8C22×D1711yescharacteristic
322C4×Q8C22×D1711yescharacteristic
323#117—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C22⋊D1711yescharacteristic
326C2×C42—21no—
327C2×C42C22×D1711yescharacteristic
328#107—21no—
329#107—21no—
330C4⋊C8—172no—
331C4⋊C8—172no—
332C4⋊C8—342no—
333C4⋊C8—342no—
334C4⋊C8—341no—
335C4⋊C8—341no—
336C4⋊C8C22×D1711yescharacteristic
337C4⋊C8C22×D1711yescharacteristic
338#103—21no—
339#103—21no—
340#103—21no—
341#103—21no—
342#102—342no—
343#102—342no—
344#102—342no—
345#102—342no—
346#102—342no—
347#102—342no—
348C22⋊C8—21no—
349#97—172no—
350#97—342no—
351C4×C8—21no—
352C4×C8—172no—
353C4×C8—342no—
354C17#1917411yescharacteristic, complemented, Sylow
355C24—681no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—41no—
361C2×Q8C2×C22⋊D1711yescharacteristic
362C2×Q8C2×C22⋊D1711yescharacteristic
363C2×D4—342no—
364C2×D4—342no—
365C2×D4—342no—
366C2×D4—682no—
367C2×D4—682no—
368C2×D4—682no—
369C2×D4—682no—
370C2×D4—681no—
371C2×D4—681no—
372C2×D4—681no—
373C2×D4—1362no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4D4×D1711yescharacteristic
377C22×C4#6325411yescharacteristic
378C22×C4C2×C22⋊D1711yescharacteristic
379Q16—42no—
380Q16—42no—
381SD16—684no—
382SD16—684no—
383M16—342no—
384M16—342no—
385M16—682no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—172no—
389C2×C8—172no—
390C2×C8—342no—
391C2×C8—342no—
392C2×C8—682no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4C2×C22⋊D1711yescharacteristic
405C4⋊C4C2×C22⋊D1711yescharacteristic
406C22⋊C4—41no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—41no—
412C42D4×D1711yescharacteristic
413C42#6325411yescharacteristic
414C42C2×C22⋊D1711yescharacteristic
415C42C23×D1711yescharacteristic
416C23—682no—
417C23—1362no—
418C23—1362no—
419C23—1362no—
420C23#1058762511yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—41no—
426Q8—41no—
427Q8—81no—
428Q8#1058763512yes—
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—41no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4#1058760412yes—
449C2×C4#1058760512yes—
450C2×C4#1058762511yescharacteristic
451C2×C4#1058762511yescharacteristic
452C2×C4#1058762511yescharacteristic
453C2×C4C2×D4×D1711yescharacteristic
454C2×C4C2×#6325411yescharacteristic
455C2×C4C22×C22⋊D1711yescharacteristic
456C8—41no—
457C8—41no—
458C8—342no—
459C8—342no—
460C8—682no—
461C8—682no—
462C22—21no—
463C22—21no—
464C22—2724no—
465C22—1362no—
466C22—1362no—
467C22—1362no—
468C22#1176303312yes—
469C22C2×#1058762511yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—81no—
478C4—81no—
479C4#1176303312yes—
480C4C2×#1058760511yescharacteristic
481C4C2×#1058763511yescharacteristic
482C4C2×#1058760411yescharacteristic
483C4#1176360711yescharacteristic
484C2—22no—
485C2—2722no—
486C2C2×#1176303311yescharacteristic
487C2#1453840511yescharacteristic
488C2#1453851411yescharacteristic
subgroupG/H
centerC22C2×#10587625
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3104C2
socleC22×C17C2×#121
prime radicalC2×#10587560C22
hypercenter#3104D17
hypocenterC17#19174
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. Q8×#63223C2semidirect
4. #14528354C2semidirect
5. #14528326C2semidirect
6. #14528230C2semidirect
7. #14528202C2semidirect
8. #14527537C2semidirect
9. #14527474C2semidirect
11. C17×#3104C2semidirect
22. #11762940C22semidirect
23. Q8×#3622C22semidirect
32. #11762561C22semidirect
33. #11762559C22semidirect
34. #11762558C22semidirect
35. #11762555C22semidirect
41. C17×#477C22semidirect
42. C17×#477C22semidirect
43. C17×#462C22semidirect
44. C17×#462C22semidirect
219. #3104D17semidirect
259. #477C2×D17semidirect
260. #477C2×D17semidirect
261. #462C2×D17semidirect
262. #462C2×D17semidirect
354. C17#19174semidirect

Maximal subgroups

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

subgroupindex
#145391602
C2×#117630092
Q8×#632232
#145283542
#145283262
#145282302
#145282022
#145275372
#145274742
#145274512
C17×#31042
#1917417