Finite Group Databasebeta

← #28743514

Subgroups — #28743514

Order 15104.

632 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#466×D59C211yesmaximal, characteristic
2#427×D59C211yesmaximal, characteristic, complemented
3#21426288C211yesmaximal, characteristic, complemented
4#21426287C211yesmaximal, characteristic, complemented
5#21426284C211yesmaximal, characteristic, complemented
6#21426283C211yesmaximal, characteristic, complemented
7#21426280C211yesmaximal, characteristic, complemented
8#21426278C211yesmaximal, characteristic, complemented
9#21426275C211yesmaximal, characteristic, complemented
10#21425883C211yesmaximal, characteristic, complemented
11#21425881C211yesmaximal, characteristic, complemented
12#21425810C211yesmaximal, characteristic, complemented
13#21425810C211yesmaximal, characteristic, complemented
14#21425705C211yesmaximal, characteristic, complemented
15C59×#2234C211yesmaximal, characteristic, complemented
16C2×D8×D59C2211yescharacteristic
17C22×C8×D59C2211yescharacteristic
18#121×D59—21no—
19#117×D59—21no—
20C2×#14273666C2211yescharacteristic
21C2×#14273666C2211yescharacteristic
22#16837639C2211yescharacteristic
23#16837638C2211yescharacteristic
24#16837637—21no—
25#16837636—21no—
26#16837635C2211yescharacteristic
27#16837440C2211yescharacteristic, complemented
28#16837439C2211yescharacteristic
29#16837439C2211yescharacteristic
30#16837438—21no—
31#16837437—21no—
32#107×D59C2211yescharacteristic
33#107×D59C2211yescharacteristic, complemented
34#107×D59C2211yescharacteristic, complemented
35#106×D59—21no—
36#102×D59—21no—
37#16837230C2211yescharacteristic, complemented
38#16837229C2211yescharacteristic, complemented
39#16837229C2211yescharacteristic, complemented
40#16837228C2211yescharacteristic, complemented
41#16837227C2211yescharacteristic, complemented
42#16837227C2211yescharacteristic, complemented
43#16837204C2211yescharacteristic, complemented
44#16837204C2211yescharacteristic, complemented
45#16837203C2211yescharacteristic, complemented
46#16837203C2211yescharacteristic, complemented
47#16837202C2211yescharacteristic, complemented
48#16837202C2211yescharacteristic, complemented
49C22⋊(C59⋊C16)C2211yescharacteristic, complemented
50(C2×D59)⋊C16C2211yescharacteristic, complemented
51(C2×D59)⋊C16C2211yescharacteristic, complemented
52C59×#466C2211yescharacteristic
53C59×#427C2211yescharacteristic, complemented
54C59×#367C2211yescharacteristic, complemented
55C59×#366C2211yescharacteristic, complemented
56C59×#357C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic, complemented
58C59×C22⋊C16C2211yescharacteristic, complemented
59C2×D4×D59—21no—
60C2×D4×D59—41no—
61C22×C4×D59D411yescharacteristic
62C2×D472C2311yescharacteristic
63C2×#11745669—21no—
64D8×D59—22no—
65D8×D59—22no—
66C22×C59⋊C8D411yescharacteristic
67C2×C8×D59—22no—
68C2×C8×D59—22no—
69C2×C8×D59D411yescharacteristic
70C2×C8×D59D411yescharacteristic
71C2×C8×D59C2311yescharacteristic
72C2×C8×D59C2311yescharacteristic
73#14273702—21no—
74#14273701—21no—
75#14273700—41no—
76#14273699—41no—
77#14273698—21no—
78#14273686—21no—
79#14273685—21no—
80#14273685—21no—
81#14273684—41no—
82#14273683—41no—
83C4⋊C4×D59—21no—
84C4⋊C4×D59—21no—
85C4⋊C4×D59—21no—
86C22⋊C4×D59—41no—
87C22⋊C4×D59—41no—
88C42×D59—41no—
89#14273666—22no—
90#14273666—22no—
91C2×C59⋊C16C2311yescharacteristic
92C2×C59⋊C16C2311yescharacteristic
93#14273661C2311yescharacteristic
94#14273661C2311yescharacteristic
95#14273661C2311yescharacteristic
96#14273660—21no—
97#14273660—21no—
98#14273660—21no—
99#14273659—21no—
100#14273658—21no—
101#14273658—21no—
102#14273649—21no—
103#14273648—21no—
104#14273647—21no—
105C2×D8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107#121×C59—21no—
108#117×C59—21no—
109C2×C16×C59C2311yescharacteristic
110C2×C16×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112#107×C59C2311yescharacteristic
113#107×C59C2311yescharacteristic
114#106×C59—21no—
115#102×C59—21no—
116C23×D59—41no—
117C23×D59D811yescharacteristic
118C2×D4×C59—21no—
119C2×D4×C59—41no—
120C22×C4×C59C2×D411yescharacteristic
121C2×C22⋊D59—41no—
122C2×C22⋊D59—41no—
123C2×C22⋊D59—41no—
124C22×Dic59D811yescharacteristic
125D4×D59—42no—
126D4×D59—42no—
127D4×D59—84no—
128C2×D236—21no—
129C2×D236—41no—
130C2×C4×D59—22no—
131C2×C4×D59—22no—
132C2×C4×D59—41no—
133C2×C4×D59—41no—
134C2×C4×D59—41no—
135C2×C4×D59D811yescharacteristic
136C2×C4×D59D811yescharacteristic
137C2×C4×D59C2×D411yescharacteristic
138C2×C4×D59C2×D411yescharacteristic
139D8×C59—21no—
140D8×C59—21no—
141C2×C8×C59—21no—
142C2×C8×C59—21no—
143C2×C8×C59C2×D411yescharacteristic
144C2×C8×C59C2411yescharacteristic
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C22⋊C4×C59—41no—
149C22⋊C4×C59—41no—
150C42×C59—41no—
151C22⋊Dic59—41no—
152C22⋊Dic59—41no—
153#11745669—42no—
154(C2×D59)⋊C4—41no—
155(C2×D59)⋊C4—41no—
156(C2×D59)⋊C4—41no—
157(C2×D59)⋊C4—41no—
158C4⋊Dic59—21no—
159C4⋊Dic59—21no—
160C4⋊Dic59—21no—
161Dic59⋊C4—41no—
162Dic59⋊C4—41no—
163Dic59⋊C4—41no—
164C4×Dic59—41no—
165C4×Dic59—41no—
166C4×Dic59—41no—
167C2×C59⋊C8—22no—
168C2×C59⋊C8C2×D411yescharacteristic
169C2×C59⋊C8C2×D411yescharacteristic
170D472—21no—
171D472—21no—
172C8×D59—22no—
173C8×D59—21no—
174C8×D59—21no—
175C8×D59—22no—
176C8×D59#4112yes—
177C8×D59#4112yes—
178C16×C59—21no—
179C16×C59—21no—
180C59⋊C16—21no—
181C59⋊C16—21no—
182C23×C59—41no—
183C23×C59C2×D811yescharacteristic
184C22×D59—22no—
185C22×D59—22no—
186C22×D59—41no—
187C22×D59—41no—
188C22×D59—82no—
189C22×D59—82no—
190C22×D59—82no—
191C22×D59C2×D811yescharacteristic
192C22×D59C2×D811yescharacteristic
193D4×C59—41no—
194D4×C59—41no—
195D4×C59—82no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59—41no—
199C2×C4×C59—41no—
200C2×C4×C59—41no—
201C2×C4×C59C2×D811yescharacteristic
202C2×C4×C59C22×D411yescharacteristic
203C22⋊D59—82no—
204C22⋊D59—82no—
205C22⋊D59—82no—
206C2×Dic59—22no—
207C2×Dic59—41no—
208C2×Dic59—41no—
209C2×Dic59—41no—
210C2×Dic59C2×D811yescharacteristic
211C2×Dic59C2×D811yescharacteristic
212D236—41no—
213D236—41no—
214D236—82no—
215C4×D59—22no—
216C4×D59—21no—
217C4×D59—21no—
218C4×D59—22no—
219C4×D59—82no—
220C4×D59—82no—
221C4×D59—82no—
222C4×D59#12312yes—
223C4×D59#13712yes—
224C8×C59—21no—
225C8×C59C2×#4111yescharacteristic
226C8×C59C2×#4111yescharacteristic
227C59⋊C8—21no—
228C59⋊C8#12312yes—
229#33648—591nomaximal, Sylow
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—81no—
234C22×C59—81no—
235C22×C59—81no—
236C22×C59C22×D811yescharacteristic
237C2×D59—21no—
238C2×D59—21no—
239C2×D59—21no—
240C2×D59—21no—
241C2×D59—44no—
242C2×D59—81no—
243C2×D59—81no—
244C2×D59—81no—
245C2×D59—164no—
246C2×D59#48012yes—
247C2×D59#50912yes—
248C4×C59—21no—
249C4×C59—81no—
250C4×C59—81no—
251C4×C59—81no—
252C4×C59C2×#12311yescharacteristic
253C4×C59C2×#13711yescharacteristic
254Dic59—21no—
255Dic59—81no—
256Dic59—81no—
257Dic59—81no—
258Dic59#48012yes—
259C2×#466—591no—
260C2×#427—591no—
261#2234—591no—
262#2234—591no—
263#2234—591no—
264#2234D5911yescharacteristic, complemented
265#2233—591no—
266#2233—591no—
267#2232—591no—
268#2232—591no—
269#2159—591no—
270#2159—591no—
271#2140—591no—
272#2140—591no—
273#2112—591no—
274C2×C59—41no—
275C2×C59—161no—
276C2×C59C2×#48011yescharacteristic
277C2×C59C2×#50911yescharacteristic
278C2×C59#341411yescharacteristic
279D59—22no—
280D59—22no—
281D59—161no—
282C22×D8—591no—
283C23×C8—591no—
284C2×#121—1181no—
285C2×#117—1181no—
286C2×M32—591no—
287C2×M32—591no—
288#466—591no—
289#466—591no—
290#466—591no—
291#466—1181no—
292#466—1181no—
293#466C2×D5911yescharacteristic
294#427—591no—
295#427—591no—
296#427—591no—
297#427—1181no—
298#427—1181no—
299#427C2×D5911yescharacteristic, complemented
300C2×#107—591no—
301C2×#107—591no—
302C2×#107—591no—
303C2×#106—1181no—
304C2×#102—1181no—
305#367—591no—
306#367—591no—
307#367—591no—
308#367C2×D5911yescharacteristic, complemented
309#366—591no—
310#366—591no—
311#366—591no—
312#366C2×D5911yescharacteristic, complemented
313#357—591no—
314#357—591no—
315#357—591no—
316#357—591no—
317#357—591no—
318#357—591no—
319#357C2×D5911yescharacteristic, complemented
320#357C2×D5911yescharacteristic, complemented
321C22⋊C16—591no—
322C22⋊C16—591no—
323C22⋊C16—591no—
324C22⋊C16C2×D5911yescharacteristic, complemented
325C59#3364811yescharacteristic, complemented, Sylow
326C22×D4—1181no—
327C22×D4—2361no—
328C23×C4—591no—
329C2×D8—591no—
330C2×D8—1182no—
331C2×D8—1182no—
332C2×D8—1181no—
333C2×D8C22×D5911yescharacteristic
334C22×C8—591no—
335C22×C8—591no—
336C22×C8—591no—
337C22×C8—591no—
338C22×C8—591no—
339C22×C8—1182no—
340C22×C8—1182no—
341C22×C8C22×D5911yescharacteristic
342#121—21no—
343#121—1181no—
344#121—1181no—
345#121—1181no—
346#121—2361no—
347#121—2361no—
348#117—21no—
349#117—1181no—
350#117—1181no—
351#117—1181no—
352#117—2361no—
353#117—2361no—
354C2×C4⋊C4—1181no—
355C2×C4⋊C4—1181no—
356C2×C4⋊C4—1181no—
357C2×C22⋊C4—2361no—
358C2×C22⋊C4—2361no—
359C2×C42—2361no—
360M32—1182no—
361M32—1182no—
362C2×C16—591no—
363C2×C16—591no—
364C2×C16C22×D5911yescharacteristic
365C2×C16C22×D5911yescharacteristic
366#107—591no—
367#107—591no—
368#107—591no—
369#107—1181no—
370#107—1181no—
371#107—1181no—
372#107C22×D5911yescharacteristic
373#107C22×D5911yescharacteristic
374#107C22×D5911yescharacteristic
375#106—21no—
376#106—1181no—
377#106—1181no—
378#106—1181no—
379#102—21no—
380#102—1181no—
381#102—1181no—
382#102—1181no—
383C24—591no—
384C24—2361no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—1181no—
388C2×D4—4724no—
389C2×D4—2361no—
390C2×D4—2362no—
391C2×D4—2362no—
392C2×D4—2361no—
393C2×D4—2361no—
394C2×D4—2361no—
395C22×C4—591no—
396C22×C4—591no—
397C22×C4—591no—
398C22×C4—591no—
399C22×C4—591no—
400C22×C4—1182no—
401C22×C4—1182no—
402C22×C4—2361no—
403C22×C4—2361no—
404C22×C4—2361no—
405C22×C4D4×D5911yescharacteristic
406D8—21no—
407D8—21no—
408D8—1181no—
409D8—1181no—
410D8—2362no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—591no—
414C2×C8—592no—
415C2×C8—592no—
416C2×C8—591no—
417C2×C8—1182no—
418C2×C8—1182no—
419C2×C8—1182no—
420C2×C8—1181no—
421C2×C8—1181no—
422C2×C8D4×D5911yescharacteristic
423C2×C8C23×D5911yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—1181no—
428C4⋊C4—1181no—
429C4⋊C4—1181no—
430C4⋊C4—2361no—
431C4⋊C4—2361no—
432C4⋊C4—2361no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—2361no—
436C22⋊C4—2361no—
437C22⋊C4—2361no—
438C22⋊C4—2361no—
439C22⋊C4—2361no—
440C22⋊C4—2361no—
441C42—41no—
442C42—2361no—
443C42—2361no—
444C42—2361no—
445C16—21no—
446C16—21no—
447C16—1181no—
448C16—1181no—
449C23—41no—
450C23—591no—
451C23—591no—
452C23—1182no—
453C23—1182no—
454C23—4722no—
455C23—4722no—
456C23—4722no—
457C23—2361no—
458C23—2361no—
459C23D8×D5911yescharacteristic
460D4—41no—
461D4—41no—
462D4—82no—
463D4—4722no—
464D4—4722no—
465D4—4722no—
466D4—4722no—
467D4—2361no—
468D4—2361no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—591no—
475C2×C4—592no—
476C2×C4—592no—
477C2×C4—591no—
478C2×C4—1182no—
479C2×C4—1182no—
480C2×C4—1182no—
481C2×C4—1181no—
482C2×C4—1181no—
483C2×C4—4722no—
484C2×C4—4722no—
485C2×C4—4722no—
486C2×C4—2361no—
487C2×C4—2361no—
488C2×C4—2361no—
489C2×C4D8×D5911yescharacteristic
490C2×C4C2×D4×D5911yescharacteristic
491C8—21no—
492C8—592no—
493C8—1181no—
494C8#41×D5911yescharacteristic
495C8#41×D5911yescharacteristic
496C22—21no—
497C22—21no—
498C22—21no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—592no—
503C22—592no—
504C22—1181no—
505C22—1181no—
506C22—1181no—
507C22—1181no—
508C22—9444no—
509C22—4721no—
510C22—4721no—
511C22—4721no—
512C22—2364no—
513C22C2×D8×D5911yescharacteristic
514C4—21no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4—592no—
519C4—1181no—
520C4—4721no—
521C4—4721no—
522C4—4721no—
523C4#123×D5911yescharacteristic
524C4#137×D5911yescharacteristic
525C2—41no—
526C2—161no—
527C2—1182no—
528C2—1182no—
529C2—9441no—
530C2#480×D5911yescharacteristic
531C2#509×D5911yescharacteristic
532C2#2143722911yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2234C2
socleC22×C59C22×D8
prime radical#466×D59C2
hypercenter#2234D59
hypocenterC59#33648
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2234

Complements

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

NH ≅ G/Ntype
2. #427×D59C2semidirect
3. #21426288C2semidirect
4. #21426287C2semidirect
5. #21426284C2semidirect
6. #21426283C2semidirect
7. #21426280C2semidirect
8. #21426278C2semidirect
9. #21426275C2semidirect
10. #21425883C2semidirect
11. #21425881C2semidirect
12. #21425810C2semidirect
13. #21425810C2semidirect
14. #21425705C2semidirect
15. C59×#2234C2semidirect
27. #16837440C22semidirect
33. #107×D59C22semidirect
34. #107×D59C22semidirect
37. #16837230C22semidirect
38. #16837229C22semidirect
39. #16837229C22semidirect
40. #16837228C22semidirect
41. #16837227C22semidirect
42. #16837227C22semidirect
43. #16837204C22semidirect
44. #16837204C22semidirect
45. #16837203C22semidirect
46. #16837203C22semidirect
47. #16837202C22semidirect
48. #16837202C22semidirect
49. C22⋊(C59⋊C16)C22semidirect
50. (C2×D59)⋊C16C22semidirect
51. (C2×D59)⋊C16C22semidirect
53. C59×#427C22semidirect
54. C59×#367C22semidirect
55. C59×#366C22semidirect
56. C59×#357C22semidirect
57. C59×#357C22semidirect
58. C59×C22⋊C16C22semidirect
264. #2234D59semidirect
299. #427C2×D59semidirect
308. #367C2×D59semidirect
312. #366C2×D59semidirect
319. #357C2×D59semidirect
320. #357C2×D59semidirect
324. C22⋊C16C2×D59semidirect
325. C59#33648semidirect

Maximal subgroups

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

subgroupindex
#466×D592
#427×D592
#214262882
#214262872
#214262842
#214262832
#214262802
#214262782
#214262752
#214258832
#214258812
#214258102
#214257052
C59×#22342
#3364859