Finite Group Databasebeta

← #28745519

Subgroups — #28745519

Order 15104.

622 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436486C211yesmaximal, characteristic, complemented
2D8×C22⋊D59C211yesmaximal, characteristic
3#21426318C211yesmaximal, characteristic, complemented
4#21426281C211yesmaximal, characteristic, complemented
5#21426280C211yesmaximal, characteristic, complemented
6#21426241C211yesmaximal, characteristic, complemented
7#21426192C211yesmaximal, characteristic, complemented
8#21426191C211yesmaximal, characteristic, complemented
9#21426135C211yesmaximal, characteristic, complemented
10#21426026C211yesmaximal, characteristic, complemented
11#21426025C211yesmaximal, characteristic, complemented
12#21425951C211yesmaximal, characteristic, complemented
13C16×C22⋊D59C211yesmaximal, characteristic, complemented
14#357×D59C211yesmaximal, characteristic, complemented
15C2×C59×#357C211yesmaximal, characteristic, complemented
16C2×D8×D59C2211yescharacteristic
17#16838107—21no—
18D4×C22⋊D59—21no—
19C2×#14273672C2211yescharacteristic, complemented
20C2×#14273671C2211yescharacteristic
21C2×#14273671C2211yescharacteristic, complemented
22C2×C16×D59C2211yescharacteristic, complemented
23#16837806C2211yescharacteristic, complemented
24#16837777C2211yescharacteristic
25#16837739—21no—
26#16837737—21no—
27#16837696—21no—
28#16837644C2211yescharacteristic, complemented
29#16837643C2211yescharacteristic, complemented
30#16837638C2211yescharacteristic
31#16837635C2211yescharacteristic
32#16837595—21no—
33#16837536—21no—
34#16837534—21no—
35#16837496C2211yescharacteristic, complemented
36D8×Dic59C2211yescharacteristic
37C8×C22⋊D59C2211yescharacteristic
38#107×D59C2211yescharacteristic, complemented
39#16837230C2211yescharacteristic, complemented
40#16837229C2211yescharacteristic, complemented
41#16837229C2211yescharacteristic, complemented
42Dic59⋊C16C2211yescharacteristic, complemented
43#16837207C2211yescharacteristic, complemented
44#16837204C2211yescharacteristic, complemented
45#16837203C2211yescharacteristic
46#16837203C2211yescharacteristic, complemented
47#16837202C2211yescharacteristic, complemented
48#16837202C2211yescharacteristic, complemented
49C22⋊(C59⋊C16)C2211yescharacteristic, complemented
50(C2×D59)⋊C16C2211yescharacteristic, complemented
51C16×Dic59C2211yescharacteristic, complemented
52C22×D8×C59C2211yescharacteristic
53C22×C16×C59C2211yescharacteristic, complemented
54C2×#107×C59C2211yescharacteristic, complemented
55C59×#357C2211yescharacteristic
56C59×#357C2211yescharacteristic
57C59×#357C2211yescharacteristic, complemented
58C59×#357C2211yescharacteristic, complemented
59C22×C22⋊D59—41no—
60C2×D4×D59—21no—
61C2×Dic236C2311yescharacteristic
62C2×D472C2311yescharacteristic
63C2×#11745669—21no—
64C2×#11745669—21no—
65D8×D59—22no—
66D8×D59—41no—
67C2×C8×D59C2311yescharacteristic
68#14273726—21no—
69#14273723—21no—
70#14273717—41no—
71#14273707—21no—
72#14273706—21no—
73#14273706—41no—
74#14273705—41no—
75#14273701—21no—
76#14273699—41no—
77#14273698—21no—
78#14273696—41no—
79#14273695—41no—
80#14273692—21no—
81D4×Dic59—21no—
82C4×C22⋊D59—41no—
83C4×C22⋊D59D411yescharacteristic
84C4⋊C4×D59—21no—
85#14273672—41no—
86#14273671—22no—
87#14273671—41no—
88C2×C59⋊C16C2311yescharacteristic
89C2×C59⋊C16C2311yescharacteristic
90C16×D59—41no—
91#14273661C2311yescharacteristic
92#14273661C2311yescharacteristic
93#14273660—21no—
94#14273660—21no—
95Dic59⋊C8D411yescharacteristic
96#14273652—21no—
97#14273650—21no—
98#14273650—21no—
99#14273649—21no—
100#14273648—21no—
101#14273648—21no—
102C22⋊(C59⋊C8)D411yescharacteristic
103(C2×D59)⋊C8D411yescharacteristic
104C8×Dic59C2311yescharacteristic
105C22×D4×C59—21no—
106C2×D8×C59—22no—
107C2×D8×C59—21no—
108C2×D8×C59—41no—
109C2×D8×C59C2311yescharacteristic
110C2×D8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C2×C4⋊C4×C59—21no—
113C2×C16×C59—41no—
114C2×C16×C59C2311yescharacteristic
115C2×C16×C59C2311yescharacteristic
116#107×C59—21no—
117#107×C59C2311yescharacteristic
118#107×C59C2311yescharacteristic
119C24×C59—41no—
120C23×D59—41no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—42no—
124C2×D4×C59—41no—
125C2×D4×C59—81no—
126C22×C4×C59—41no—
127C22×C4×C59C2×D411yescharacteristic
128C2×C22⋊D59—41no—
129C2×C22⋊D59—41no—
130C2×C22⋊D59—41no—
131C2×C22⋊D59—41no—
132C2×C22⋊D59—41no—
133C2×C22⋊D59—84no—
134C2×C22⋊D59D811yescharacteristic
135C22×Dic59—41no—
136D4×D59—42no—
137D4×D59—81no—
138C2×D236—21no—
139C2×C4×D59—41no—
140C2×C4×D59C2×D411yescharacteristic
141C2×Dic118—21no—
142C2×Dic118—41no—
143D8×C59—21no—
144D8×C59—21no—
145D8×C59—21no—
146D8×C59—42no—
147D8×C59#4112yes—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59C2×D411yescharacteristic
151C2×C8×C59C2411yescharacteristic
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—41no—
155C22⋊Dic59—41no—
156C22⋊Dic59—41no—
157C22⋊Dic59—41no—
158C22⋊Dic59D811yescharacteristic
159#11745669—42no—
160#11745669—81no—
161(C2×D59)⋊C4—41no—
162(C2×D59)⋊C4—41no—
163(C2×D59)⋊C4D811yescharacteristic
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—41no—
167Dic59⋊C4—41no—
168Dic59⋊C4—41no—
169Dic59⋊C4—41no—
170Dic59⋊C4—41no—
171Dic59⋊C4D811yescharacteristic
172C4×Dic59—41no—
173C4×Dic59C2×D411yescharacteristic
174C2×C59⋊C8C2×D411yescharacteristic
175C2×C59⋊C8C2×D411yescharacteristic
176Dic236—21no—
177Dic236—21no—
178D472—21no—
179D472—21no—
180C8×D59—21no—
181C8×D59—21no—
182C16×C59—21no—
183C16×C59—21no—
184C59⋊C16—21no—
185C59⋊C16—21no—
186C23×C59—41no—
187C23×C59—41no—
188C23×C59—82no—
189C23×C59—82no—
190C23×C59—161no—
191C23×C59C2×D811yescharacteristic
192C22×D59—41no—
193C22×D59—82no—
194C22×D59—82no—
195C22×D59—161no—
196C22×D59C2×D811yescharacteristic
197D4×C59—22no—
198D4×C59—41no—
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—82no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—161no—
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C22×D411yescharacteristic
209C22⋊D59—82no—
210C22⋊D59—82no—
211C22⋊D59—82no—
212C22⋊D59—82no—
213C22⋊D59—161no—
214C22⋊D59SD3214yescomplemented
215C2×Dic59—41no—
216C2×Dic59—41no—
217C2×Dic59—41no—
218C2×Dic59—82no—
219C2×Dic59C2×D811yescharacteristic
220C2×Dic59C2×D811yescharacteristic
221C2×Dic59C2×D811yescharacteristic
222D236—41no—
223D236—41no—
224C4×D59—21no—
225C4×D59—21no—
226C4×D59—161no—
227Dic118—41no—
228Dic118—41no—
229Dic118—161no—
230C8×C59—21no—
231C8×C59C2×#4111yescharacteristic
232C8×C59H(5,𝔽2)11yescharacteristic
233C59⋊C8—21no—
234C59⋊C8—21no—
235#33862—591nomaximal, Sylow
236C22×C59—21no—
237C22×C59—21no—
238C22×C59—42no—
239C22×C59—81no—
240C22×C59—81no—
241C22×C59—81no—
242C22×C59—81no—
243C22×C59—81no—
244C22×C59—164no—
245C22×C59C2×SD3212yes—
246C22×C59C22×D811yescharacteristic
247C2×D59—21no—
248C2×D59—21no—
249C2×D59—81no—
250C2×D59—81no—
251C2×D59—81no—
252C2×D59—164no—
253C2×D59C2×SD3212yes—
254C4×C59—21no—
255C4×C59—81no—
256C4×C59—81no—
257C4×C59#54611yescharacteristic
258C4×C59#57711yescharacteristic
259Dic59—21no—
260Dic59—21no—
261Dic59—81no—
262Dic59—81no—
263Dic59—81no—
264Dic59C2×SD3212yes—
265#3338—591no—
266D4×D8—591no—
267#2240—591no—
268#2233—591no—
269#2233—591no—
270#2226—591no—
271#2217—591no—
272#2217—591no—
273#2207—591no—
274#2187—591no—
275#2187—591no—
276C4×SD32—591no—
277D4×C16—591no—
278C2×#357—591no—
279C2×#357D5911yescharacteristic, complemented
280C2×C59—22no—
281C2×C59—82no—
282C2×C59—161no—
283C2×C59#329411yescharacteristic
284C2×C59C22×SD3211yescharacteristic
285C2×C59#341511yescharacteristic
286D59—22no—
287D59—161no—
288C22×D8—591no—
289C22×D8C2×D5911yescharacteristic
290#554—1181no—
291D42—1181no—
292C2×SD32—591no—
293C2×SD32—591no—
294C2×SD32—591no—
295C22×C16—591no—
296C22×C16C2×D5911yescharacteristic, complemented
297#500—591no—
298#493—591no—
299#484—1181no—
300#484—1181no—
301#477—1181no—
302#467—591no—
303#467—591no—
304#466—591no—
305#466—591no—
306#459—1181no—
307#447—1181no—
308#447—1181no—
309C4×Q16—591no—
310C4×D8—591no—
311C8×D4—591no—
312C2×#107—591no—
313C2×#107C2×D5911yescharacteristic, complemented
314#367—591no—
315#367—591no—
316#367—591no—
317C4⋊C16—591no—
318#358—591no—
319#357—591no—
320#357—591no—
321#357—591no—
322#357—591no—
323#357—591no—
324#357C2×D5911yescharacteristic
325#357C2×D5911yescharacteristic
326#357C2×D5911yescharacteristic, complemented
327#357C2×D5911yescharacteristic, complemented
328C22⋊C16—591no—
329C22⋊C16—591no—
330C4×C16—591no—
331C59#3386211yescharacteristic, complemented, Sylow
332C22×D4—21no—
333C22×D4—1181no—
334C22×D4—2361no—
335C2×Q16—591no—
336C2×D8—22no—
337C2×D8—21no—
338C2×D8—41no—
339C2×D8—591no—
340C2×D8—1181no—
341C2×D8—1181no—
342C2×D8—1182no—
343C2×D8—2361no—
344C2×D8C22×D5911yescharacteristic
345C2×D8C22×D5911yescharacteristic
346C22×C8—591no—
347C22×C8C22×D5911yescharacteristic
348C4⋊Q8—1181no—
349#127—1181no—
350#125—2361no—
351C22⋊Q8—1181no—
352C22⋊Q8—1181no—
353C22⋊Q8—2361no—
354C22⋊Q8—2361no—
355#121—1181no—
356#121—1181no—
357#121—2361no—
358#120—2361no—
359#120—2361no—
360C4×Q8—1181no—
361C4×D4—591no—
362C4×D4—1181no—
363C4×D4—2361no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—1181no—
366SD32—1182no—
367SD32—2361no—
368SD32—2361no—
369C2×C16—41no—
370C2×C16—591no—
371C2×C16—591no—
372C2×C16—2361no—
373C2×C16C22×D5911yescharacteristic
374C2×C16C22×D5911yescharacteristic
375#107—21no—
376#107—591no—
377#107—591no—
378#107—1181no—
379#107—1181no—
380#107C22×D5911yescharacteristic
381#107C22×D5911yescharacteristic
382C4⋊C8—591no—
383#103—1181no—
384#103—1181no—
385#103—1181no—
386#102—1181no—
387#102—1181no—
388#102—1181no—
389C22⋊C8—591no—
390C22⋊C8—591no—
391C4×C8—591no—
392C24—41no—
393C24—2361no—
394C2×Q8—1181no—
395C2×Q8—2361no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—81no—
401C2×D4—591no—
402C2×D4—1181no—
403C2×D4—4721no—
404C2×D4—4724no—
405C2×D4—2361no—
406C2×D4—2361no—
407C2×D4—2361no—
408C2×D4—2362no—
409C2×D4—2361no—
410C2×D4—2361no—
411C22×C4—41no—
412C22×C4—591no—
413C22×C4—2361no—
414C22×C4—2361no—
415C22×C4D4×D5911yescharacteristic
416Q16—1181no—
417Q16—1181no—
418D8—21no—
419D8—21no—
420D8—21no—
421D8—42no—
422D8—1181no—
423D8—1181no—
424D8—4721no—
425D8—2362no—
426D8#1174568512yes—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—591no—
430C2×C8—591no—
431C2×C8—1181no—
432C2×C8—1181no—
433C2×C8D4×D5911yescharacteristic
434C2×C8C23×D5911yescharacteristic
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—41no—
438C4⋊C4—591no—
439C4⋊C4—1181no—
440C4⋊C4—1181no—
441C4⋊C4—2361no—
442C4⋊C4—2361no—
443C4⋊C4—2361no—
444C4⋊C4—2361no—
445C4⋊C4—2361no—
446C22⋊C4—591no—
447C22⋊C4—591no—
448C22⋊C4—2361no—
449C22⋊C4—2361no—
450C22⋊C4—2361no—
451C22⋊C4—2361no—
452C22⋊C4—2361no—
453C42—591no—
454C42—2361no—
455C16—21no—
456C16—21no—
457C16—1181no—
458C16—1181no—
459C23—41no—
460C23—41no—
461C23—82no—
462C23—82no—
463C23—161no—
464C23—591no—
465C23—9441no—
466C23—4722no—
467C23—4722no—
468C23—2361no—
469C23D8×D5911yescharacteristic
470Q8—9441no—
471Q8—2361no—
472Q8—2361no—
473D4—22no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—82no—
478D4—594no—
479D4—9441no—
480D4—4722no—
481D4—4722no—
482D4—4722no—
483D4—4722no—
484D4—2361no—
485D4—2361no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—161no—
491C2×C4—591no—
492C2×C4—591no—
493C2×C4—591no—
494C2×C4—1181no—
495C2×C4—1181no—
496C2×C4—9441no—
497C2×C4—4722no—
498C2×C4—2361no—
499C2×C4—2361no—
500C2×C4—2361no—
501C2×C4D8×D5911yescharacteristic
502C2×C4C2×D4×D5911yescharacteristic
503C8—21no—
504C8—1181no—
505C8—1181no—
506C8C2×#1174568511yescharacteristic
507C8#1427377211yescharacteristic
508C22—21no—
509C22—21no—
510C22—42no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—164no—
517C22—592no—
518C22—1181no—
519C22—1181no—
520C22—9444no—
521C22—4721no—
522C22—4721no—
523C22—4721no—
524C22SD32×D5912yes—
525C22C2×D8×D5911yescharacteristic
526C4—21no—
527C4—81no—
528C4—81no—
529C4—592no—
530C4—1181no—
531C4—1181no—
532C4—4721no—
533C4—4721no—
534C4—4721no—
535C4#1683804211yescharacteristic
536C4#1683820811yescharacteristic
537C2—22no—
538C2—82no—
539C2—161no—
540C2—1182no—
541C2—9441no—
542C2#2143590511yescharacteristic
543C2C2×SD32×D5911yescharacteristic
544C2#2143723311yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC2×C59×#357C2
socleC22×C59C22×D8
prime radicalD8×C22⋊D59C2
hypercenterC2×#357D59
hypocenterC59#33862
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. #21436486C2semidirect
3. #21426318C2semidirect
4. #21426281C2semidirect
5. #21426280C2semidirect
6. #21426241C2semidirect
7. #21426192C2semidirect
8. #21426191C2semidirect
9. #21426135C2semidirect
10. #21426026C2semidirect
11. #21426025C2semidirect
12. #21425951C2semidirect
13. C16×C22⋊D59C2semidirect
14. #357×D59C2semidirect
15. C2×C59×#357C2semidirect
19. C2×#14273672C22semidirect
21. C2×#14273671C22semidirect
22. C2×C16×D59C22semidirect
23. #16837806C22semidirect
28. #16837644C22semidirect
29. #16837643C22semidirect
35. #16837496C22semidirect
38. #107×D59C22semidirect
39. #16837230C22semidirect
40. #16837229C22semidirect
41. #16837229C22semidirect
42. Dic59⋊C16C22semidirect
43. #16837207C22semidirect
44. #16837204C22semidirect
46. #16837203C22semidirect
47. #16837202C22semidirect
48. #16837202C22semidirect
49. C22⋊(C59⋊C16)C22semidirect
50. (C2×D59)⋊C16C22semidirect
51. C16×Dic59C22semidirect
53. C22×C16×C59C22semidirect
54. C2×#107×C59C22semidirect
57. C59×#357C22semidirect
58. C59×#357C22semidirect
214. C22⋊D59SD32semidirect
279. C2×#357D59semidirect
296. C22×C16C2×D59semidirect
313. C2×#107C2×D59semidirect
326. #357C2×D59semidirect
327. #357C2×D59semidirect
331. C59#33862semidirect

Maximal subgroups

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

subgroupindex
#214364862
D8×C22⋊D592
#214263182
#214262812
#214262802
#214262412
#214261922
#214261912
#214261352
#214260262
#214260252
#214259512
C16×C22⋊D592
#357×D592
C2×C59×#3572
#3386259