Finite Group Databasebeta

← #28743790

Subgroups — #28743790

Order 15104.

568 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434728C211yesmaximal, characteristic, complemented
2#21432948C211yesmaximal, characteristic, complemented
3#21426305C211yesmaximal, characteristic, complemented
4#21426284C211yesmaximal, characteristic, complemented
5#21426241C211yesmaximal, characteristic, complemented
6#21426231C211yesmaximal, characteristic, complemented
7#21426149C211yesmaximal, characteristic, complemented
8#21426149C211yesmaximal, characteristic, complemented
9#21426025C211yesmaximal, characteristic, complemented
10#21425972C211yesmaximal, characteristic, complemented
11#21425957C211yesmaximal, characteristic, complemented
12#21425951C211yesmaximal, characteristic, complemented
13D16×Dic59C211yesmaximal, characteristic, complemented
14#21425900C211yesmaximal, characteristic, complemented
15C59×#2184C211yesmaximal, characteristic
16#16838019—21no—
17#16837877—21no—
18C2×#14273671C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20#16837806C2211yescharacteristic, complemented
21#16837785C2211yescharacteristic, complemented
22#16837760—21no—
23#16837646—21no—
24#16837643C2211yescharacteristic
25#16837514—21no—
26#16837514—21no—
27#16837499—21no—
28#16837496C2211yescharacteristic, complemented
29D8×Dic59C2211yescharacteristic
30D8×Dic59C2211yescharacteristic
31D8×Dic59C2211yescharacteristic
32D8×Dic59C2211yescharacteristic
33D8×Dic59C2211yescharacteristic, complemented
34#16837440C2211yescharacteristic
35#16837407—21no—
36#16837357C2211yescharacteristic
37#16837230C2211yescharacteristic, complemented
38#16837229C2211yescharacteristic, complemented
39#16837228C2211yescharacteristic, complemented
40#16837227C2211yescharacteristic, complemented
41Dic59⋊C16C2211yescharacteristic, complemented
42Dic59⋊C16C2211yescharacteristic, complemented
43#16837207C2211yescharacteristic, complemented
44#16837207C2211yescharacteristic, complemented
45#16837203C2211yescharacteristic
46#16837203C2211yescharacteristic
47#16837203C2211yescharacteristic, complemented
48#16837203C2211yescharacteristic, complemented
49C22⋊(C59⋊C16)C2211yescharacteristic
50#16837177C2211yescharacteristic, complemented
51C16×Dic59C2211yescharacteristic, complemented
52C22×D8×C59C2211yescharacteristic
53C2×D16×C59C2211yescharacteristic
54C2×D16×C59C2211yescharacteristic
55C59×#466C2211yescharacteristic
56C59×#357C2211yescharacteristic
57C59×#357C2211yescharacteristic
58C59×C22⋊C16C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60C2×#11745670—21no—
61C2×#11745664D411yescharacteristic
62#14273726—21no—
63#14273720—41no—
64#14273714—21no—
65#14273714—41no—
66#14273708—41no—
67#14273706—21no—
68#14273704—41no—
69#14273692—21no—
70D4×Dic59—21no—
71D4×Dic59—21no—
72D4×Dic59—21no—
73D4×Dic59—21no—
74D4×Dic59—21no—
75D4×Dic59—41no—
76D4×Dic59—41no—
77#14273686—21no—
78#14273686D411yescharacteristic
79C2×C22⋊Dic59—41no—
80C2×C22⋊Dic59—41no—
81#14273671—22no—
82#14273671—22no—
83C2×C59⋊C16C2311yescharacteristic
84C2×C59⋊C16C2311yescharacteristic
85#14273661C2311yescharacteristic
86#14273661C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273660—21no—
89#14273659—21no—
90#14273652—21no—
91#14273650—21no—
92#14273648—21no—
93#14273648—21no—
94#14273648—21no—
95#14273648—21no—
96#14273648—21no—
97#14273637D411yescharacteristic
98#14273637D411yescharacteristic
99C8×Dic59C2311yescharacteristic
100C8×Dic59C2311yescharacteristic
101C22×D4×C59—21no—
102C2×D8×C59—22no—
103C2×D8×C59—22no—
104C2×D8×C59—21no—
105C2×D8×C59C2311yescharacteristic
106C2×D8×C59C2311yescharacteristic
107C2×D8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109#121×C59—21no—
110D16×C59—22no—
111D16×C59—22no—
112C2×C16×C59C2311yescharacteristic
113C2×C16×C59C2311yescharacteristic
114#107×C59C2311yescharacteristic
115#102×C59—21no—
116C24×C59—41no—
117C2×D4×C59—24no—
118C2×D4×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—42no—
122C2×D4×C59—41no—
123C2×D4×C59—41no—
124C22×C4×C59C2×D411yescharacteristic
125C22×Dic59—41no—
126C22×Dic59—41no—
127C22×Dic59—41no—
128C22×Dic59—41no—
129C22×Dic59—41no—
130C2×Dic118—21no—
131C2×Dic118—41no—
132D8×C59—24no—
133D8×C59—21no—
134D8×C59—21no—
135D8×C59—21no—
136D8×C59—21no—
137D8×C59#4112yes—
138D8×C59#4112yes—
139C2×C8×C59—21no—
140C2×C8×C59—21no—
141C2×C8×C59C2×D411yescharacteristic
142C2×C8×C59C2411yescharacteristic
143C4⋊C4×C59—21no—
144C22⋊C4×C59—41no—
145C22⋊Dic59—41no—
146C22⋊Dic59—41no—
147C22⋊Dic59—41no—
148C22⋊Dic59—41no—
149C22⋊Dic59—41no—
150C22⋊Dic59—41no—
151C22⋊Dic59—41no—
152C22⋊Dic59—41no—
153C22⋊Dic59—41no—
154C22⋊Dic59D811yescharacteristic
155C22⋊Dic59D811yescharacteristic
156#11745670—42no—
157C4⋊Dic59—21no—
158C4⋊Dic59—21no—
159C4⋊Dic59—21no—
160C4⋊Dic59D811yescharacteristic
161C4⋊Dic59D811yescharacteristic
162Dic59⋊C4—41no—
163Dic59⋊C4—41no—
164Dic59⋊C4—41no—
165C4×Dic59—41no—
166C4×Dic59—41no—
167C4×Dic59C2×D411yescharacteristic
168C4×Dic59C2×D411yescharacteristic
169#11745664—41no—
170C2×C59⋊C8C2×D411yescharacteristic
171C2×C59⋊C8C2×D411yescharacteristic
172Dic236—21no—
173Dic236—21no—
174C16×C59—21no—
175C16×C59—21no—
176C59⋊C16—21no—
177C59⋊C16—21no—
178C23×C59—44no—
179C23×C59—41no—
180C23×C59—41no—
181C23×C59—41no—
182C23×C59—161no—
183C23×C59—161no—
184C23×C59C2×D811yescharacteristic
185D4×C59—22no—
186D4×C59—22no—
187D4×C59—44no—
188D4×C59—41no—
189D4×C59—41no—
190D4×C59—41no—
191D4×C59—41no—
192D4×C59—161no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—41no—
196C2×C4×C59C2×D811yescharacteristic
197C2×C4×C59C22×D411yescharacteristic
198C2×Dic59—41no—
199C2×Dic59—41no—
200C2×Dic59—41no—
201C2×Dic59—41no—
202C2×Dic59—161no—
203C2×Dic59—161no—
204C2×Dic59—161no—
205C2×Dic59—161no—
206C2×Dic59—161no—
207C2×Dic59C2×D811yescharacteristic
208C2×Dic59C2×D811yescharacteristic
209C2×Dic59C2×D811yescharacteristic
210C2×Dic59C2×D811yescharacteristic
211Dic118—41no—
212Dic118—41no—
213Dic118—161no—
214C8×C59—21no—
215C8×C59C2×#4111yescharacteristic
216C8×C59C2×#4111yescharacteristic
217C59⋊C8—21no—
218C59⋊C8—21no—
219#33678—591nomaximal, Sylow
220C22×C59—21no—
221C22×C59—21no—
222C22×C59—21no—
223C22×C59—42no—
224C22×C59—42no—
225C22×C59—81no—
226C22×C59—81no—
227C22×C59—81no—
228C22×C59—81no—
229C22×C59—81no—
230C22×C59—81no—
231C22×C59—81no—
232C22×C59—164no—
233C22×C59C22×D811yescharacteristic
234C4×C59—21no—
235C4×C59—81no—
236C4×C59#52511yescharacteristic
237C4×C59#57511yescharacteristic
238Dic59—21no—
239Dic59—21no—
240Dic59—21no—
241Dic59—21no—
242Dic59—81no—
243Dic59—81no—
244Dic59—81no—
245Dic59—81no—
246#3175—591no—
247#2948—591no—
248#2237—591no—
249#2234—591no—
250#2226—591no—
251#2224—591no—
252#2210—591no—
253#2210—591no—
254#2187—591no—
255#2184D5911yescharacteristic
256#2177—591no—
257#2175—591no—
258C4×SD32—591no—
259C4×D16—591no—
260#2164—591no—
261C2×C59—41no—
262C2×C59—82no—
263C2×C59—82no—
264C2×C59—161no—
265C2×C59#310011yescharacteristic
266C2×C59#341511yescharacteristic
267C2×C59#341611yescharacteristic
268C22×D8C2×D5911yescharacteristic
269#542—1181no—
270#518—1181no—
271C2×SD32—591no—
272C2×SD32—591no—
273C2×D16C2×D5911yescharacteristic
274C2×D16C2×D5911yescharacteristic
275#500—591no—
276#495—591no—
277#489—1181no—
278#468—1181no—
279#467—591no—
280#466C2×D5911yescharacteristic
281#442—1181no—
282#442—1181no—
283#440—1181no—
284C4×Q16—591no—
285C4×D8—591no—
286C4×D8—591no—
287C4×D8—591no—
288C4×D8—591no—
289C4×D8—591no—
290#427—591no—
291#418—1181no—
292#405—591no—
293#367—591no—
294#367—591no—
295#366—591no—
296#366—591no—
297C4⋊C16—591no—
298C4⋊C16—591no—
299#358—591no—
300#358—591no—
301#357—591no—
302#357—591no—
303#357—591no—
304#357—591no—
305#357C2×D5911yescharacteristic
306#357C2×D5911yescharacteristic
307C22⋊C16—591no—
308C22⋊C16C2×D5911yescharacteristic
309#346—591no—
310C4×C16—591no—
311C59#3367811yescharacteristic, complemented, Sylow
312C22×D4—21no—
313C2×Q16—591no—
314C2×SD16—1181no—
315C2×D8—22no—
316C2×D8—22no—
317C2×D8—21no—
318C2×D8C22×D5911yescharacteristic
319C2×D8C22×D5911yescharacteristic
320C2×D8C22×D5911yescharacteristic
321C2×M16—591no—
322C22×C8C22×D5911yescharacteristic
323C4⋊Q8—1181no—
324#126—2361no—
325#124—1181no—
326#124—2361no—
327#123—2361no—
328C22⋊Q8—1181no—
329C22⋊Q8—2361no—
330#121—21no—
331C4×Q8—1181no—
332C4×D4—1181no—
333C4×D4—1181no—
334C4×D4—1181no—
335C4×D4—1181no—
336C4×D4—1181no—
337C4×D4—2361no—
338C4×D4—2361no—
339#117—591no—
340#117—1181no—
341C2×C22⋊C4—2361no—
342C2×C22⋊C4—2361no—
343SD32—1182no—
344SD32—1182no—
345D16—22no—
346D16—22no—
347C2×C16—591no—
348C2×C16—591no—
349C2×C16C22×D5911yescharacteristic
350C2×C16C22×D5911yescharacteristic
351#107—591no—
352#107—591no—
353#107—591no—
354#107—1181no—
355#107C22×D5911yescharacteristic
356#106—1181no—
357#103—1181no—
358#103—1181no—
359#102—21no—
360#102—1181no—
361#102—1181no—
362#102—1181no—
363#102—1181no—
364#102—1181no—
365#97—591no—
366#97—591no—
367C4×C8—591no—
368C4×C8—591no—
369C24—41no—
370C2×Q8—1181no—
371C2×Q8—2361no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—42no—
377C2×D4—41no—
378C2×D4—41no—
379C22×C4—2361no—
380C22×C4—2361no—
381C22×C4—2361no—
382C22×C4—2361no—
383C22×C4—2361no—
384C22×C4D4×D5911yescharacteristic
385Q16—1181no—
386Q16—1181no—
387SD16—2362no—
388D8—24no—
389D8—21no—
390D8—21no—
391D8—21no—
392D8—21no—
393D8#1174568712yes—
394D8#1174568712yes—
395M16—2361no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—591no—
399C2×C8—591no—
400C2×C8D4×D5911yescharacteristic
401C2×C8C23×D5911yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—591no—
404C4⋊C4—591no—
405C4⋊C4—1181no—
406C4⋊C4—1181no—
407C4⋊C4—1181no—
408C4⋊C4—2361no—
409C4⋊C4—2361no—
410C4⋊C4—2361no—
411C22⋊C4—41no—
412C22⋊C4—591no—
413C22⋊C4—591no—
414C22⋊C4—2361no—
415C22⋊C4—2361no—
416C22⋊C4—2361no—
417C22⋊C4—2361no—
418C22⋊C4—2361no—
419C22⋊C4—2361no—
420C22⋊C4—2361no—
421C22⋊C4—2361no—
422C22⋊C4—2361no—
423C42—591no—
424C42—591no—
425C42—2361no—
426C42—2361no—
427C16—21no—
428C16—21no—
429C16—1181no—
430C16—1181no—
431C23—44no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—161no—
436C23—161no—
437C23D8×D5911yescharacteristic
438Q8—9441no—
439Q8—2361no—
440Q8—2361no—
441D4—22no—
442D4—22no—
443D4—44no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—41no—
448D4—161no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—591no—
453C2×C4—591no—
454C2×C4—591no—
455C2×C4—591no—
456C2×C4—9441no—
457C2×C4—9441no—
458C2×C4—9441no—
459C2×C4—9441no—
460C2×C4—9441no—
461C2×C4—2361no—
462C2×C4—2361no—
463C2×C4—2361no—
464C2×C4—2361no—
465C2×C4D8×D5911yescharacteristic
466C2×C4C2×D4×D5911yescharacteristic
467C8—21no—
468C8—1181no—
469C8—1181no—
470C8C2×#1174568711yescharacteristic
471C8C2×#1174568711yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—42no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—164no—
485C22C2×D8×D5911yescharacteristic
486C4—21no—
487C4—81no—
488C4—1181no—
489C4—1181no—
490C4—1181no—
491C4—1181no—
492C4—4721no—
493C4—4721no—
494C4—4721no—
495C4—4721no—
496C4#1683790811yescharacteristic
497C4#1683819811yescharacteristic
498C2—41no—
499C2—82no—
500C2—82no—
501C2—161no—
502C2#2143409111yescharacteristic
503C2#2143723311yescharacteristic
504C2#2143724211yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
Fitting, prime radicalC59×#2184C2
socleC22×C59C22×D8
hypercenter#2184D59
hypocenterC59#33678
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2184

Complements

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

NH ≅ G/Ntype
1. #21434728C2semidirect
2. #21432948C2semidirect
3. #21426305C2semidirect
4. #21426284C2semidirect
5. #21426241C2semidirect
6. #21426231C2semidirect
7. #21426149C2semidirect
8. #21426149C2semidirect
9. #21426025C2semidirect
10. #21425972C2semidirect
11. #21425957C2semidirect
12. #21425951C2semidirect
13. D16×Dic59C2semidirect
14. #21425900C2semidirect
20. #16837806C22semidirect
21. #16837785C22semidirect
28. #16837496C22semidirect
33. D8×Dic59C22semidirect
37. #16837230C22semidirect
38. #16837229C22semidirect
39. #16837228C22semidirect
40. #16837227C22semidirect
41. Dic59⋊C16C22semidirect
42. Dic59⋊C16C22semidirect
43. #16837207C22semidirect
44. #16837207C22semidirect
47. #16837203C22semidirect
48. #16837203C22semidirect
50. #16837177C22semidirect
51. C16×Dic59C22semidirect
311. C59#33678semidirect

Maximal subgroups

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

subgroupindex
#214347282
#214329482
#214263052
#214262842
#214262412
#214262312
#214261492
#214260252
#214259722
#214259572
#214259512
D16×Dic592
#214259002
C59×#21842
#3367859