Finite Group Databasebeta

← #17012904

Subgroups — #17012904

Order 4352.

1170 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539928C211yesmaximal, characteristic, complemented
2#14538792C211yesmaximal, characteristic, complemented
3#14538581C211yesmaximal, characteristic, complemented
4#14528068C214yesmaximal, complemented
5#14527935C214yesmaximal, complemented
6#14527456C211yesmaximal, characteristic, complemented
7#14527209C211yesmaximal, characteristic, complemented
8#14527175C211yesmaximal, characteristic, complemented
9C2×C17×#547C211yesmaximal, characteristic
10#11763366C2211yescharacteristic
11#11763205C2212yescomplemented
12#11763161C2212yescomplemented
13#11763139—21no—
14#11763139—22no—
15#11763109—22no—
16#11763099C2212yescomplemented
17#11763099C2212yescomplemented
18#11763091—22no—
19#11763089C2212yescomplemented
20#11763014C2212yescomplemented
21#11762960—21no—
22C4×#63229—22no—
23#11762862C2211yescharacteristic
24#11762862C2211yescharacteristic
25#11762849—21no—
26#11762833C2211yescharacteristic
27#11762814—21no—
28#11762555C2214yescomplemented
29D4⋊#3622C2214yescomplemented
30D4⋊#3622C2214yescomplemented
31C22×#41×C17—21no—
32C17×#547—22no—
33C17×#547—22no—
34C17×#547—22no—
35C17×#547C2214yes—
36C2×C17×C4⋊Q8C2211yescharacteristic
37C2×C17×#123—22no—
38C2×C17×C22⋊Q8—21no—
39C2×C17×#121—21no—
40C2×C4×D4×C17C2211yescharacteristic
41C2×C4×D4×C17C2211yescharacteristic
42C22×C4⋊C4×C17—21no—
43C2×#41×C17—24no—
44C2×#41×C17—24no—
45C2×#41×C17—44no—
46C2×#41×C17—44no—
47C22×Q8×C17—21no—
48C22×D4×C17—22no—
49C22×D4×C17D411yescharacteristic
50C22×D4×C17D411yescharacteristic
51C23×C4×C17—22no—
52C23×C4×C17—21no—
53C23×C4×C17—21no—
54C17×C4⋊Q8—21no—
55C17×C4⋊Q8—21no—
56C17×C4⋊Q8D412yes—
57C17×C4⋊Q8C2312yes—
58C17×#123—28no—
59C17×#123—28no—
60C17×C22⋊Q8—24no—
61C17×C22⋊Q8—24no—
62C17×#121—24no—
63C17×#121—24no—
64C4×D4×C17—21no—
65C4×D4×C17—21no—
66C4×D4×C17—21no—
67C4×D4×C17—21no—
68C4×D4×C17D412yes—
69C4×D4×C17D412yes—
70C4×D4×C17C2312yes—
71C4×D4×C17C2312yes—
72C2×C4⋊C4×C17—22no—
73C2×C4⋊C4×C17—24no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—24no—
76C2×C4⋊C4×C17D411yescharacteristic
77C2×C4⋊C4×C17D411yescharacteristic
78C2×C4⋊C4×C17D412yes—
79C2×C22⋊C4×C17—22no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C42×C17C2311yescharacteristic
83C2×#63229—22no—
84#10587612—21no—
85#10587610—21no—
86#10587610—21no—
87C2×#63223—21no—
88#10587556—21no—
89#10587555C2311yescharacteristic
90#10587555C2311yescharacteristic
91C4×Dic34—21no—
92C4×Dic34—21no—
93#10587503—22no—
94#10587503—22no—
95#10587503—22no—
96C22⋊#3622—21no—
97C22⋊#3622—21no—
98#10587481—22no—
99#10587481—24no—
100#10587479—24no—
101#10587479—22no—
102#10587479—22no—
103#10587478—22no—
104C4⋊#3622—21no—
105C4⋊#3622—21no—
106C4⋊#3622C2312yes—
107C4⋊#3622C2312yes—
108C4×#3622—21no—
109C4×#3622C2312yes—
110C24×C17—21no—
111C24×C17—21no—
112#41×C17—48no—
113#41×C17—416no—
114#41×C17—48no—
115C2×Q8×C17—22no—
116C2×Q8×C17—22no—
117C2×Q8×C17—42no—
118C2×Q8×C17—42no—
119C2×D4×C17—28no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—22no—
123C2×D4×C17—42no—
124C2×D4×C17—48no—
125C2×D4×C17SD1614yescomplemented
126C2×D4×C17C2×D412yes—
127C2×D4×C17C2×D412yes—
128C22×C4×C17—28no—
129C22×C4×C17—21no—
130C22×C4×C17—22no—
131C22×C4×C17—21no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—44no—
137C22×C4×C17—48no—
138C22×C4×C17—42no—
139C22×C4×C17—42no—
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C2×Dic34—21no—
144C2×Dic34—21no—
145C4⋊C4×C17—28no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—22no—
148C4⋊C4×C17—22no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17C2×D412yes—
152C4⋊C4×C17C2×D414yes—
153C4⋊C4×C17C2×D412yes—
154C22⋊C4×C17—28no—
155C22⋊C4×C17—22no—
156C22⋊C4×C17—22no—
157C22⋊C4×C17—22no—
158C22⋊C4×C17—22no—
159C42×C17C2×D411yescharacteristic
160C42×C17C2×D411yescharacteristic
161C42×C17C2×D411yescharacteristic
162C42×C17C2411yescharacteristic
163C22⋊Dic17—41no—
164#63229—44no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169#63225—41no—
170#63225—41no—
171C4×Dic17—41no—
172#63223—42no—
173C2×#3622—22no—
174C2×#3622—22no—
175C2×#3622—22no—
176#18757—171nomaximal, Sylow
177C23×C17—22no—
178C23×C17—24no—
179C23×C17—22no—
180C23×C17—42no—
181C23×C17—42no—
182C23×C17—42no—
183C23×C17—42no—
184C23×C17—42no—
185C23×C17#12011yescharacteristic
186Q8×C17—42no—
187Q8×C17—42no—
188Q8×C17—44no—
189D4×C17—22no—
190D4×C17—22no—
191D4×C17—22no—
192D4×C17—24no—
193D4×C17—42no—
194D4×C17—416no—
195D4×C17C2×SD1614yes—
196C2×C4×C17—22no—
197C2×C4×C17—22no—
198C2×C4×C17—22no—
199C2×C4×C17—24no—
200C2×C4×C17—41no—
201C2×C4×C17—44no—
202C2×C4×C17—42no—
203C2×C4×C17—416no—
204C2×C4×C17—42no—
205C2×C4×C17—44no—
206C2×C4×C17—42no—
207C2×C4×C17—42no—
208C2×C4×C17—44no—
209C2×C4×C17—44no—
210C2×C4×C17#12011yescharacteristic
211C2×C4×C17#12011yescharacteristic
212C2×C4×C17#12011yescharacteristic
213C2×C4×C17C2×SD1612yes—
214C2×C4×C17AGL(1,ℤ/8)12yes—
215C2×C4×C17C22×D411yescharacteristic
216C2×C4×C17C22×D411yescharacteristic
217C2×C4×C17C22×D411yescharacteristic
218C2×Dic17—41no—
219C2×Dic17—41no—
220Dic34—42no—
221Dic34—42no—
222#3622—42no—
223#3622—42no—
224#3622—42no—
225C2×#547D1711yescharacteristic
226#3229—171no—
227#3101—171no—
228#3077—171no—
229#1654—174no—
230#1635—174no—
231#1546—171no—
232#1486—171no—
233#1478—171no—
234C22×C17—21no—
235C22×C17—21no—
236C22×C17—24no—
237C22×C17—48no—
238C22×C17—42no—
239C22×C17—42no—
240C22×C17—42no—
241C22×C17—42no—
242C22×C17—42no—
243C22×C17—84no—
244C22×C17#45012yes—
245C22×C17C2×#12011yescharacteristic
246C4×C17—21no—
247C4×C17—21no—
248C4×C17—21no—
249C4×C17—42no—
250C4×C17—44no—
251C4×C17—42no—
252C4×C17#45012yes—
253C4×C17C22×SD1611yescharacteristic
254C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C17#57512yes—
256Dic17—81no—
257Dic17—81no—
258C22×#41—21no—
259#547—22no—
260#547—22no—
261#547—22no—
262#547C2×D1714yes—
263#533—171no—
264C2×C4⋊Q8C2×D1711yescharacteristic
265C2×#123—22no—
266C2×C22⋊Q8—21no—
267C2×#121—21no—
268C2×C4×D4C2×D1711yescharacteristic
269C2×C4×D4C2×D1711yescharacteristic
270C22×C4⋊C4—21no—
271#498—172no—
272#487—172no—
273#483—341no—
274#483—342no—
275#478—342no—
276#476—172no—
277#476—172no—
278#475—342no—
279#474—172no—
280#461—172no—
281#450—341no—
282C4×SD16—342no—
283#431—171no—
284#431—171no—
285#428—341no—
286#423—171no—
287#418—341no—
288#332—174no—
289D4⋊C8—174no—
290D4⋊C8—174no—
291C2×C17—22no—
292C2×C17—44no—
293C2×C17—82no—
294C2×C17C2×#45011yescharacteristic
295C2×C17#300511yescharacteristic
296C2×C17#302011yescharacteristic
297C2×#41—24no—
298C2×#41—24no—
299C2×#41—44no—
300C2×#41—44no—
301C22×Q8—21no—
302C22×D4—22no—
303C22×D4C22⋊D1711yescharacteristic
304C22×D4C22⋊D1711yescharacteristic
305C23×C4—22no—
306C23×C4—21no—
307C23×C4—21no—
308C2×SD16—342no—
309C2×M16—341no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—171no—
313C4⋊Q8—171no—
314C4⋊Q8C22⋊D1712yes—
315C4⋊Q8C22×D1712yes—
316#125—341no—
317#123—28no—
318#123—28no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C22⋊Q8—341no—
322C22⋊Q8—341no—
323#121—24no—
324#121—24no—
325C4×Q8—341no—
326C4×Q8—341no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4C22⋊D1712yes—
332C4×D4C22⋊D1712yes—
333C4×D4C22×D1712yes—
334C4×D4C22×D1712yes—
335#117—341no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4C22⋊D1711yescharacteristic
341C2×C4⋊C4C22⋊D1711yescharacteristic
342C2×C4⋊C4C22⋊D1712yes—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C42C22×D1711yescharacteristic
347#107—342no—
348#106—342no—
349#106—342no—
350#106—344no—
351C4⋊C8—172no—
352C4⋊C8—172no—
353C4⋊C8—341no—
354C4⋊C8—341no—
355#103—342no—
356#103—344no—
357#102—342no—
358#102—342no—
359#102—342no—
360C22⋊C8—341no—
361C22⋊C8—341no—
362C4×C8—172no—
363C4×C8—341no—
364C17#1875711yescharacteristic, complemented, Sylow
365C24—21no—
366C24—21no—
367#41—416no—
368#41—48no—
369#41—48no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—341no—
375C2×Q8—341no—
376C2×D4—28no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—42no—
381C2×D4—48no—
382C2×D4#6322914yescomplemented
383C2×D4C2×C22⋊D1712yes—
384C2×D4C2×C22⋊D1712yes—
385C22×C4—28no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4C2×C22⋊D1711yescharacteristic
398C22×C4C2×C22⋊D1711yescharacteristic
399C22×C4C2×C22⋊D1711yescharacteristic
400SD16—684no—
401M16—682no—
402C2×C8—342no—
403C2×C8—342no—
404C2×C8—342no—
405C4⋊C4—28no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—341no—
412C4⋊C4—341no—
413C4⋊C4—341no—
414C4⋊C4—341no—
415C4⋊C4—681no—
416C4⋊C4—681no—
417C4⋊C4C2×C22⋊D1712yes—
418C4⋊C4C2×C22⋊D1714yes—
419C4⋊C4C2×C22⋊D1712yes—
420C22⋊C4—28no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—681no—
426C42—681no—
427C42C2×C22⋊D1711yescharacteristic
428C42C2×C22⋊D1711yescharacteristic
429C42C2×C22⋊D1711yescharacteristic
430C42C23×D1711yescharacteristic
431C23—24no—
432C23—22no—
433C23—22no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23#1058763911yescharacteristic
440Q8—44no—
441Q8—42no—
442Q8—42no—
443Q8—682no—
444Q8—682no—
445D4—24no—
446D4—22no—
447D4—22no—
448D4—22no—
449D4—42no—
450D4—416no—
451D4C2×#6322914yes—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—416no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—681no—
467C2×C4—681no—
468C2×C4#1058761812yes—
469C2×C4C2×#6322912yes—
470C2×C4#1058763911yescharacteristic
471C2×C4#1058763911yescharacteristic
472C2×C4#1058763911yescharacteristic
473C2×C4C22×C22⋊D1711yescharacteristic
474C2×C4C22×C22⋊D1711yescharacteristic
475C2×C4C22×C22⋊D1711yescharacteristic
476C8—682no—
477C8—682no—
478C8—682no—
479C22—21no—
480C22—21no—
481C22—24no—
482C22—48no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22#1176296012yes—
490C22C2×#1058763911yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—44no—
497C4—1361no—
498C4—1361no—
499C4#1176296012yes—
500C4C22×#6322911yescharacteristic
501C4C2×#1058761811yescharacteristic
502C4#1176360712yes—
503C2—22no—
504C2—44no—
505C2—82no—
506C2C2×#1176296011yescharacteristic
507C2#1453802611yescharacteristic
508C2#1453814811yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
Fitting, prime radicalC2×C17×#547C2
socleC22×C17C2×#120
hypercenterC2×#547D17
hypocenterC17#18757
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #14539928C2semidirect
2. #14538792C2semidirect
3. #14538581C2semidirect
4. #14528068C2semidirect
5. #14527935C2semidirect
6. #14527456C2semidirect
7. #14527209C2semidirect
8. #14527175C2semidirect
11. #11763205C22semidirect
12. #11763161C22semidirect
16. #11763099C22semidirect
17. #11763099C22semidirect
19. #11763089C22semidirect
20. #11763014C22semidirect
28. #11762555C22semidirect
29. D4⋊#3622C22semidirect
30. D4⋊#3622C22semidirect
125. C2×D4×C17SD16semidirect
364. C17#18757semidirect
382. C2×D4#63229semidirect

Maximal subgroups

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

subgroupindex
#145399282
#145387922
#145385812
#145280682
#145279352
#145274562
#145272092
#145271752
C2×C17×#5472
#1875717