Finite Group Databasebeta

← #17191297

Subgroups — #17191297

Order 4352.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14541132C211yesmaximal, characteristic, complemented
2#14540471C211yesmaximal, characteristic, complemented
3#14531021C211yesmaximal, characteristic, complemented
4#14530982C211yesmaximal, characteristic, complemented
5#14530970C211yesmaximal, characteristic, complemented
6#14530932C211yesmaximal, characteristic, complemented
7#14530887C211yesmaximal, characteristic, complemented
8#14530880C211yesmaximal, characteristic, complemented
9#14530840C211yesmaximal, characteristic, complemented
10#14530709C211yesmaximal, characteristic, complemented
11#14530698C211yesmaximal, characteristic, complemented
12#14530661C211yesmaximal, characteristic, complemented
13#14530497C211yesmaximal, characteristic, complemented
14C2×#11762611C211yesmaximal, characteristic, complemented
15C17×#2168C211yesmaximal, characteristic, complemented
16C2×#10587590C2211yescharacteristic
17C22×D136C2211yescharacteristic
18#11763463—21no—
19#11763455—21no—
20C2×#10587471C2211yescharacteristic, complemented
21C2×D272C2211yescharacteristic
22C2×D272C2211yescharacteristic
23#11763214C2211yescharacteristic, complemented
24#11763193C2211yescharacteristic
25#11763123—21no—
26#11763123—21no—
27#11763090—21no—
28#11763050C2211yescharacteristic
29#11763045C2211yescharacteristic
30#11763045C2211yescharacteristic
31#11763045C2211yescharacteristic, complemented
32#11763025—21no—
33#11762962—21no—
34#11762956—21no—
35C4×D136C2211yescharacteristic
36C4×D136C2211yescharacteristic, complemented
37C2×#10587489C2211yescharacteristic
38C2×#10587489C2211yescharacteristic, complemented
39#11762637C2211yescharacteristic, complemented
40#11762637C2211yescharacteristic, complemented
41#11762637C2211yescharacteristic, complemented
42#11762614C2211yescharacteristic, complemented
43#11762614C2211yescharacteristic, complemented
44#11762614C2211yescharacteristic, complemented
45#11762611C2211yescharacteristic
46#11762611C2211yescharacteristic
47#11762611C2211yescharacteristic
48#11762611C2211yescharacteristic
49#11762611C2211yescharacteristic
50#11762611C2211yescharacteristic
51#11762611C2211yescharacteristic, complemented
52C2×C17×M32C2211yescharacteristic, complemented
53C22×C16×C17C2211yescharacteristic
54C8×D4×C17C2211yescharacteristic
55C17×C4⋊C16C2211yescharacteristic, complemented
56C17×C22⋊C16C2211yescharacteristic
57C17×C22⋊C16C2211yescharacteristic, complemented
58C17×#346C2211yescharacteristic, complemented
59C2×#63254—41no—
60C2×#63252—21no—
61C22×D68—21no—
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C4×D4×C17D411yescharacteristic
65#10587623—41no—
66#10587616—21no—
67#10587616—21no—
68#10587616—21no—
69C2×C4⋊Dic17—21no—
70C2×C4⋊Dic17—21no—
71#10587610—21no—
72C2×Dic68C2311yescharacteristic
73#10587590—22no—
74#10587590—41no—
75C2×D136—22no—
76C2×D136—21no—
77C2×D136—41no—
78C2×D136C2311yescharacteristic
79C2×D136C2311yescharacteristic
80C2×D136C2311yescharacteristic
81C2×#63219—21no—
82C2×#63219—21no—
83#10587583—41no—
84#10587580—41no—
85#10587572—41no—
86#10587572—41no—
87#10587570—41no—
88C22⋊C4×D17—41no—
89#10587561—21no—
90C4×D68—21no—
91C4×D68—21no—
92#10587555—21no—
93C17×M32—41no—
94C2×C16×C17—41no—
95C2×C16×C17C2311yescharacteristic
96C2×C16×C17C2311yescharacteristic
97C2×C16×C17C2311yescharacteristic
98C2×C16×C17C2311yescharacteristic
99C17×C4⋊C8D411yescharacteristic
100C17×C22⋊C8D411yescharacteristic
101C17×C22⋊C8D411yescharacteristic
102C4×C8×C17C2311yescharacteristic
103#10587492—21no—
104#10587492—21no—
105#10587492—21no—
106#10587492—21no—
107#10587492—21no—
108#10587489—21no—
109#10587489—21no—
110#10587489—21no—
111#10587489C2311yescharacteristic
112#10587489C2311yescharacteristic
113#10587489C2311yescharacteristic
114#10587489C2311yescharacteristic
115#10587487—21no—
116#10587471—41no—
117D272—22no—
118D272—41no—
119C23×D17—41no—
120C2×D4×C17D811yescharacteristic
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17C2×D411yescharacteristic
123C2×C22⋊D17—41no—
124C2×C22⋊D17—41no—
125C2×C22⋊D17—41no—
126C2×C22⋊D17—41no—
127C22×Dic17—41no—
128C22×Dic17—41no—
129#63254—84no—
130#63252—42no—
131#63252—81no—
132C2×D68—21no—
133C2×D68—21no—
134C2×D68—21no—
135C2×D68—42no—
136C2×D68—41no—
137C2×D68—81no—
138C2×C4×D17—41no—
139C2×C4×D17—41no—
140C2×C4×D17—41no—
141C2×Dic34—21no—
142C2×Dic34—41no—
143C2×C8×C17—21no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17C2×D411yescharacteristic
148C2×C8×C17C2×D411yescharacteristic
149C2×C8×C17C2×D411yescharacteristic
150C2×C8×C17C2411yescharacteristic
151C4⋊C4×C17D811yescharacteristic
152C22⋊C4×C17D811yescharacteristic
153C22⋊C4×C17D811yescharacteristic
154C42×C17C2×D411yescharacteristic
155C22⋊Dic17—41no—
156#63227—41no—
157#63227—41no—
158#63227—41no—
159#63227—41no—
160#63227—41no—
161#63227—41no—
162#63227—41no—
163C4⋊Dic17—21no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—41no—
168C4⋊Dic17—41no—
169C4⋊Dic17—41no—
170#63225—41no—
171Dic68—21no—
172Dic68—21no—
173D136—21no—
174D136—21no—
175D136—21no—
176D136—21no—
177D136—21no—
178D136—42no—
179D136#4112yescomplemented
180#63219—42no—
181#63219—81no—
182C16×C17—21no—
183C16×C17—21no—
184C16×C17—21no—
185C16×C17—21no—
186#33873—171nomaximal, Sylow
187C23×C17C2×D811yescharacteristic
188C23×C17C2×D811yescharacteristic
189C22×D17—41no—
190C22×D17—41no—
191C22×D17—41no—
192C22×D17—82no—
193C22×D17—82no—
194C22×D17—161no—
195D4×C17—22no—
196C2×C4×C17—21no—
197C2×C4×C17—21no—
198C2×C4×C17—21no—
199C2×C4×C17—21no—
200C2×C4×C17C2×D811yescharacteristic
201C2×C4×C17C2×D811yescharacteristic
202C2×C4×C17C2×D811yescharacteristic
203C2×C4×C17C2×D811yescharacteristic
204C2×C4×C17C22×D411yescharacteristic
205C22⋊D17—82no—
206C22⋊D17—82no—
207C22⋊D17—82no—
208C22⋊D17—161no—
209C2×Dic17—41no—
210C2×Dic17—41no—
211C2×Dic17—41no—
212C2×Dic17—41no—
213C2×Dic17—41no—
214C2×Dic17—82no—
215C2×Dic17—82no—
216D68—22no—
217D68—41no—
218D68—41no—
219D68—41no—
220D68—41no—
221D68—41no—
222D68—82no—
223C4×D17—82no—
224C4×D17—82no—
225C4×D17—161no—
226Dic34—41no—
227Dic34—41no—
228Dic34—82no—
229C8×C17—21no—
230C8×C17—21no—
231C8×C17—21no—
232C8×C17C2×#4111yescharacteristic
233C8×C17H(5,𝔽2)11yescharacteristic
234#3334—171no—
235#3280—171no—
236#2243—171no—
237#2235—171no—
238#2233—171no—
239#2226—171no—
240#2218—171no—
241#2217—171no—
242#2210—171no—
243#2186—171no—
244#2184—171no—
245#2177—171no—
246#2168D1711yescharacteristic, complemented
247#2139—171no—
248C2×#357—171no—
249C22×C17—21no—
250C22×C17—21no—
251C22×C17—21no—
252C22×C17—21no—
253C22×C17—21no—
254C22×C17#50912yes—
255C22×C17C22×D811yescharacteristic
256C2×D17—42no—
257C2×D17—81no—
258C2×D17—81no—
259C2×D17—81no—
260C2×D17—81no—
261C2×D17—81no—
262C2×D17—81no—
263C2×D17—81no—
264C2×D17—81no—
265C2×D17—164no—
266C4×C17—21no—
267C4×C17—21no—
268C4×C17—21no—
269C4×C17—21no—
270C4×C17#54611yescharacteristic
271C4×C17#57711yescharacteristic
272Dic17—81no—
273Dic17—81no—
274Dic17—81no—
275Dic17—81no—
276Dic17—81no—
277C2×#135—171no—
278C22×D8—171no—
279#547—341no—
280#546—341no—
281C2×SD32—171no—
282C2×D16—171no—
283C2×D16—171no—
284C2×M32C2×D1711yescharacteristic, complemented
285C22×C16C2×D1711yescharacteristic
286#500—171no—
287#495—171no—
288#480—341no—
289#480—341no—
290#474—341no—
291#467—171no—
292#466—171no—
293#466—171no—
294#466—171no—
295#463—341no—
296#450—341no—
297#449—341no—
298C4×D8—171no—
299C4×D8—171no—
300C8×D4C2×D1711yescharacteristic
301C2×#107—171no—
302C2×#107—171no—
303#367—171no—
304#367—171no—
305#367—171no—
306C4⋊C16C2×D1711yescharacteristic, complemented
307#358—171no—
308#358—171no—
309#358—171no—
310#357—171no—
311#357—171no—
312#357—171no—
313#357—171no—
314#357—171no—
315#357—171no—
316#357—171no—
317C22⋊C16C2×D1711yescharacteristic
318C22⋊C16C2×D1711yescharacteristic, complemented
319#346C2×D1711yescharacteristic, complemented
320C2×C17—22no—
321C2×C17—41no—
322C2×C17#329411yescharacteristic
323C2×C17C2×#50911yescharacteristic
324C2×C17#341611yescharacteristic
325D17—82no—
326D17—161no—
327D17—161no—
328C2×#41—341no—
329C2×#41—681no—
330C22×D4—341no—
331#135—342no—
332#135—681no—
333C2×Q16—171no—
334C2×SD16—341no—
335C2×SD16—341no—
336C2×D8—171no—
337C2×D8—171no—
338C2×D8—171no—
339C2×D8—342no—
340C2×D8—341no—
341C2×D8—681no—
342C22×C8C22×D1711yescharacteristic
343C22×C8C22×D1711yescharacteristic
344C4⋊Q8—341no—
345#124—341no—
346#123—681no—
347#123—681no—
348#123—681no—
349C22⋊Q8—341no—
350C22⋊Q8—681no—
351#121—341no—
352#121—341no—
353#121—341no—
354#121—681no—
355#120—681no—
356C4×D4—341no—
357C4×D4—341no—
358C4×D4D6811yescharacteristic
359C2×C4⋊C4—341no—
360C2×C4⋊C4—341no—
361C2×C22⋊C4—681no—
362SD32—681no—
363D16—342no—
364D16—681no—
365M32—41no—
366C2×C16—41no—
367C2×C16C22×D1711yescharacteristic
368C2×C16C22×D1711yescharacteristic
369C2×C16C22×D1711yescharacteristic
370C2×C16C22×D1711yescharacteristic
371#107—171no—
372#107—171no—
373#107—171no—
374#107—171no—
375#107—341no—
376#107—341no—
377#107—341no—
378C4⋊C8D6811yescharacteristic
379#103—341no—
380#102—341no—
381#102—341no—
382#102—341no—
383#102—341no—
384#102—341no—
385C22⋊C8D6811yescharacteristic
386C22⋊C8D6811yescharacteristic
387C4×C8C22×D1711yescharacteristic
388C17#3387311yescharacteristic, complemented, Sylow
389C24—681no—
390#41—682no—
391#41—1361no—
392#41—1364no—
393C2×Q8—341no—
394C2×Q8—681no—
395C2×D4—341no—
396C2×D4—341no—
397C2×D4—341no—
398C2×D4—682no—
399C2×D4—681no—
400C2×D4—681no—
401C2×D4—681no—
402C2×D4—681no—
403C2×D4—681no—
404C2×D4—1361no—
405C2×D4D13611yescharacteristic
406C22×C4—681no—
407C22×C4—681no—
408C22×C4—681no—
409C22×C4—681no—
410C22×C4—681no—
411C22×C4C2×D6811yescharacteristic
412C22×C4C2×D6811yescharacteristic
413Q16—341no—
414Q16—341no—
415SD16—682no—
416SD16—1361no—
417D8—172no—
418D8—341no—
419D8—341no—
420D8—341no—
421D8—341no—
422D8—341no—
423D8—682no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×D6811yescharacteristic
429C2×C8C2×D6811yescharacteristic
430C2×C8C2×D6811yescharacteristic
431C2×C8C23×D1711yescharacteristic
432C4⋊C4—341no—
433C4⋊C4—341no—
434C4⋊C4—341no—
435C4⋊C4—341no—
436C4⋊C4—681no—
437C4⋊C4—681no—
438C4⋊C4—681no—
439C4⋊C4—681no—
440C4⋊C4D13611yescharacteristic
441C22⋊C4—681no—
442C22⋊C4—681no—
443C22⋊C4—681no—
444C22⋊C4—681no—
445C22⋊C4—681no—
446C22⋊C4—681no—
447C22⋊C4—681no—
448C22⋊C4—681no—
449C22⋊C4D13611yescharacteristic
450C22⋊C4D13611yescharacteristic
451C42C2×D6811yescharacteristic
452C16—21no—
453C16—21no—
454C16—21no—
455C16—21no—
456C23—2721no—
457C23—681no—
458C23—681no—
459C23—681no—
460C23—1362no—
461C23—1362no—
462C23C2×D13611yescharacteristic
463C23C2×D13611yescharacteristic
464Q8—681no—
465Q8—681no—
466Q8—1362no—
467D4—22no—
468D4—2721no—
469D4—342no—
470D4—681no—
471D4—681no—
472D4—681no—
473D4—681no—
474D4—681no—
475D4—1362no—
476D4—1362no—
477D4—1362no—
478D4—1362no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—2721no—
484C2×C4—681no—
485C2×C4—681no—
486C2×C4—681no—
487C2×C4—681no—
488C2×C4—681no—
489C2×C4—1362no—
490C2×C4—1362no—
491C2×C4—1362no—
492C2×C4—1362no—
493C2×C4C2×D13611yescharacteristic
494C2×C4C2×D13611yescharacteristic
495C2×C4C2×D13611yescharacteristic
496C2×C4C2×D13611yescharacteristic
497C2×C4C22×D6811yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8#1058769111yescharacteristic
502C8#41×D1711yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—2724no—
509C22—682no—
510C22—1361no—
511C22—1361no—
512C22—1361no—
513C22—1361no—
514C22—1361no—
515C22—1361no—
516C22—1361no—
517C22—1361no—
518C22#1176324712yes—
519C22C22×D13611yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—1361no—
525C4—1361no—
526C4—1361no—
527C4—1361no—
528C4—1361no—
529C4#1176345411yescharacteristic
530C4#1176362111yescharacteristic
531C2—22no—
532C2—41no—
533C2—2721no—
534C2—2721no—
535C2—1362no—
536C2#1454059811yescharacteristic
537C2C2×#1176324711yescharacteristic
538C2#1454193211yescharacteristic
subgroupG/H
centerC22C22×D136
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#2168C2
socleC22×C17C22×D8
hypercenter#2168D17
hypocenterC17#33873
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. #14541132C2semidirect
2. #14540471C2semidirect
3. #14531021C2semidirect
4. #14530982C2semidirect
5. #14530970C2semidirect
6. #14530932C2semidirect
7. #14530887C2semidirect
8. #14530880C2semidirect
9. #14530840C2semidirect
10. #14530709C2semidirect
11. #14530698C2semidirect
12. #14530661C2semidirect
13. #14530497C2semidirect
14. C2×#11762611C2semidirect
15. C17×#2168C2semidirect
20. C2×#10587471C22semidirect
23. #11763214C22semidirect
31. #11763045C22semidirect
36. C4×D136C22semidirect
38. C2×#10587489C22semidirect
39. #11762637C22semidirect
40. #11762637C22semidirect
41. #11762637C22semidirect
42. #11762614C22semidirect
43. #11762614C22semidirect
44. #11762614C22semidirect
51. #11762611C22semidirect
52. C2×C17×M32C22semidirect
55. C17×C4⋊C16C22semidirect
57. C17×C22⋊C16C22semidirect
58. C17×#346C22semidirect
179. D136#41semidirect
246. #2168D17semidirect
284. C2×M32C2×D17semidirect
306. C4⋊C16C2×D17semidirect
318. C22⋊C16C2×D17semidirect
319. #346C2×D17semidirect
388. C17#33873semidirect

Maximal subgroups

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

subgroupindex
#145411322
#145404712
#145310212
#145309822
#145309702
#145309322
#145308872
#145308802
#145308402
#145307092
#145306982
#145306612
#145304972
C2×#117626112
C17×#21682
#3387317