Finite Group Databasebeta

← #28745654

Subgroups — #28745654

Order 15104.

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#21435783C211yesmaximal, characteristic, complemented
2#21426335C211yesmaximal, characteristic, complemented
3#21426293C211yesmaximal, characteristic, complemented
4#21426281C211yesmaximal, characteristic, complemented
5#21426240C211yesmaximal, characteristic, complemented
6#21426197C211yesmaximal, characteristic, complemented
7#21426190C211yesmaximal, characteristic, complemented
8#21426151C211yesmaximal, characteristic, complemented
9#21426017C211yesmaximal, characteristic, complemented
10#21426007C211yesmaximal, characteristic, complemented
11#21425970C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425806C211yesmaximal, characteristic, complemented
14C2×#16837202C211yesmaximal, characteristic, complemented
15C59×#3334C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×D472C2211yescharacteristic
18#16838048—21no—
19C2×#14273671C2211yescharacteristic, complemented
20C2×#14273668C2211yescharacteristic
21C2×#14273668C2211yescharacteristic
22C2×#14273667C2211yescharacteristic, complemented
23C22×C59⋊C16C2211yescharacteristic
24#16837786C2211yescharacteristic
25#16837643C2211yescharacteristic
26#16837638C2211yescharacteristic
27#16837618—21no—
28#16837555—21no—
29#16837549—21no—
30C4×D472C2211yescharacteristic
31#16837229C2211yescharacteristic, complemented
32#16837229C2211yescharacteristic, complemented
33#16837229C2211yescharacteristic, complemented
34C4⋊(C59⋊C16)C2211yescharacteristic, complemented
35#16837205C2211yescharacteristic, complemented
36#16837205C2211yescharacteristic, complemented
37#16837205C2211yescharacteristic, complemented
38#16837203C2211yescharacteristic, complemented
39#16837202C2211yescharacteristic
40#16837202C2211yescharacteristic
41#16837202C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic, complemented
47#16837178C2211yescharacteristic, complemented
48C59×#547—21no—
49C59×#500C2211yescharacteristic, complemented
50C59×#480—21no—
51C59×#480—21no—
52C59×#474—21no—
53C59×#466C2211yescharacteristic
54C59×#466C2211yescharacteristic, complemented
55C4×D8×C59C2211yescharacteristic, complemented
56C8×D4×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic, complemented
59C2×#11745685—21no—
60C22×D236—21no—
61#14273749—22no—
62#14273749—41no—
63C2×Dic236C2311yescharacteristic
64C2×#11745660—21no—
65C2×#11745660—21no—
66C2×D472—22no—
67C2×D472—21no—
68C2×D472—41no—
69C2×D472C2311yescharacteristic
70C2×D472C2311yescharacteristic
71#14273715—21no—
72#14273712—41no—
73#14273706—21no—
74#14273701—21no—
75#14273700—41no—
76#14273695—41no—
77C4×D236—21no—
78C22⋊C4×D59—41no—
79#14273671—41no—
80#14273668—22no—
81#14273668—41no—
82#14273667—41no—
83C2×C59⋊C16—41no—
84C2×C59⋊C16C2311yescharacteristic
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273652—21no—
90#14273649—21no—
91#14273649—21no—
92C2×#41×C59—41no—
93C2×D8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95C22×C8×C59C2311yescharacteristic
96C4⋊Q8×C59—21no—
97#123×C59—41no—
98#123×C59—41no—
99C22⋊Q8×C59—41no—
100#121×C59—21no—
101#121×C59—21no—
102C4×D4×C59—21no—
103C4×D4×C59D411yescharacteristic
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106#107×C59—21no—
107#107×C59—21no—
108#107×C59—21no—
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#102×C59—21no—
114#102×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C23×D59—41no—
120#41×C59—84no—
121C2×Q8×C59—41no—
122C2×D4×C59—21no—
123C2×D4×C59—41no—
124C2×D4×C59—41no—
125C2×D4×C59D811yescharacteristic
126C22×C4×C59—41no—
127C22×C4×C59—41no—
128C22×C4×C59—41no—
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C2×C22⋊D59—41no—
132C2×C22⋊D59—41no—
133#11745685—42no—
134#11745685—81no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—42no—
138C2×D236—41no—
139C2×D236—81no—
140C2×C4×D59—41no—
141C2×C4×D59—41no—
142C2×Dic118—21no—
143D8×C59—21no—
144D8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59C2×D411yescharacteristic
150C2×C8×C59C2×D411yescharacteristic
151C2×C8×C59C2×D411yescharacteristic
152C2×C8×C59C2411yescharacteristic
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—41no—
157C4⋊C4×C59—41no—
158C4⋊C4×C59—41no—
159C4⋊C4×C59D811yescharacteristic
160C22⋊C4×C59—41no—
161C22⋊C4×C59—41no—
162C22⋊C4×C59—41no—
163C22⋊C4×C59D811yescharacteristic
164C22⋊C4×C59D811yescharacteristic
165C42×C59C2×D411yescharacteristic
166C22⋊Dic59—41no—
167(C2×D59)⋊C4—41no—
168(C2×D59)⋊C4—41no—
169(C2×D59)⋊C4—41no—
170(C2×D59)⋊C4—41no—
171C4⋊Dic59—21no—
172Dic59⋊C4—41no—
173Dic236—21no—
174Dic236—21no—
175D472—21no—
176D472—21no—
177D472—21no—
178D472—42no—
179D472#4112yescomplemented
180#11745660—42no—
181#11745660—81no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C59⋊C16—21no—
186C23×C59—41no—
187C23×C59C2×D811yescharacteristic
188C23×C59C2×D811yescharacteristic
189C22×D59—41no—
190C22×D59—41no—
191C22×D59—82no—
192C22×D59—82no—
193C22×D59—161no—
194Q8×C59—82no—
195D4×C59—22no—
196D4×C59—41no—
197D4×C59—41no—
198D4×C59—82no—
199D4×C59—82no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—41no—
207C2×C4×C59—82no—
208C2×C4×C59—82no—
209C2×C4×C59—82no—
210C2×C4×C59C2×D811yescharacteristic
211C2×C4×C59C2×D811yescharacteristic
212C2×C4×C59C2×D811yescharacteristic
213C2×C4×C59C2×D811yescharacteristic
214C2×C4×C59C22×D411yescharacteristic
215C22⋊D59—82no—
216C22⋊D59—161no—
217C2×Dic59—41no—
218C2×Dic59—41no—
219D236—22no—
220D236—41no—
221D236—41no—
222D236—41no—
223D236—82no—
224C4×D59—82no—
225C4×D59—161no—
226Dic118—41no—
227Dic118—41no—
228C8×C59—21no—
229C8×C59—21no—
230C8×C59—21no—
231C8×C59C2×#4111yescharacteristic
232C8×C59H(5,𝔽2)11yescharacteristic
233#33873—591nomaximal, Sylow
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—21no—
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—81no—
240C22×C59—81no—
241C22×C59—81no—
242C22×C59#50912yes—
243C22×C59C22×D811yescharacteristic
244C2×D59—42no—
245C2×D59—81no—
246C2×D59—81no—
247C2×D59—81no—
248C2×D59—81no—
249C2×D59—81no—
250C2×D59—164no—
251C4×C59—21no—
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—21no—
255C4×C59—81no—
256C4×C59—81no—
257C4×C59—81no—
258C4×C59#54611yescharacteristic
259C4×C59#57711yescharacteristic
260Dic59—81no—
261Dic59—81no—
262#3334D5911yescharacteristic, complemented
263#3280—591no—
264#2243—591no—
265#2235—591no—
266#2233—591no—
267#2226—591no—
268#2218—591no—
269#2217—591no—
270#2210—591no—
271#2186—591no—
272#2184—591no—
273#2177—591no—
274#2168—591no—
275#2139—591no—
276C2×#357—591no—
277C2×C59—22no—
278C2×C59—41no—
279C2×C59—161no—
280C2×C59#329411yescharacteristic
281C2×C59C2×#50911yescharacteristic
282C2×C59#341611yescharacteristic
283D59—82no—
284D59—161no—
285C2×#135—591no—
286C22×D8—591no—
287#547—21no—
288#546—1181no—
289C2×SD32—591no—
290C2×D16—591no—
291C2×D16—591no—
292C2×M32—591no—
293C22×C16—591no—
294#500C2×D5911yescharacteristic, complemented
295#495—591no—
296#480—21no—
297#480—21no—
298#474—21no—
299#467—591no—
300#466—591no—
301#466C2×D5911yescharacteristic
302#466C2×D5911yescharacteristic, complemented
303#463—1181no—
304#450—1181no—
305#449—1181no—
306C4×D8—591no—
307C4×D8C2×D5911yescharacteristic, complemented
308C8×D4C2×D5911yescharacteristic
309C2×#107C2×D5911yescharacteristic
310C2×#107C2×D5911yescharacteristic, complemented
311#367—591no—
312#367—591no—
313#367—591no—
314C4⋊C16—591no—
315#358—591no—
316#358—591no—
317#358—591no—
318#357—591no—
319#357—591no—
320#357—591no—
321#357—591no—
322#357—591no—
323#357—591no—
324#357—591no—
325C22⋊C16—591no—
326C22⋊C16—591no—
327#346—591no—
328C59#3387311yescharacteristic, complemented, Sylow
329C2×#41—41no—
330C2×#41—1181no—
331C22×D4—1181no—
332#135—1182no—
333#135—2361no—
334C2×Q16—591no—
335C2×SD16—1181no—
336C2×SD16—1181no—
337C2×D8—591no—
338C2×D8—591no—
339C2×D8—1182no—
340C2×D8—1181no—
341C2×D8—2361no—
342C2×D8C22×D5911yescharacteristic
343C22×C8C22×D5911yescharacteristic
344C22×C8C22×D5911yescharacteristic
345C4⋊Q8—21no—
346#124—1181no—
347#123—41no—
348#123—41no—
349#123—2361no—
350C22⋊Q8—41no—
351C22⋊Q8—1181no—
352#121—21no—
353#121—21no—
354#121—1181no—
355#121—2361no—
356#120—2361no—
357C4×D4—21no—
358C4×D4—1181no—
359C4×D4C22⋊D5911yescharacteristic
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C22⋊C4—2361no—
363SD32—2361no—
364D16—1182no—
365D16—2361no—
366M32—2361no—
367C2×C16—591no—
368C2×C16—591no—
369C2×C16—591no—
370C2×C16—591no—
371C2×C16—2361no—
372#107—21no—
373#107—21no—
374#107—21no—
375#107—591no—
376#107C22×D5911yescharacteristic
377#107C22×D5911yescharacteristic
378#107C22×D5911yescharacteristic
379C4⋊C8C22⋊D5911yescharacteristic
380#103—1181no—
381#102—21no—
382#102—21no—
383#102—21no—
384#102—1181no—
385#102—1181no—
386C22⋊C8C22⋊D5911yescharacteristic
387C22⋊C8C22⋊D5911yescharacteristic
388C4×C8C22×D5911yescharacteristic
389C24—2361no—
390#41—84no—
391#41—4721no—
392#41—2362no—
393C2×Q8—41no—
394C2×Q8—1181no—
395C2×D4—21no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—1181no—
399C2×D4—1181no—
400C2×D4—4721no—
401C2×D4—2362no—
402C2×D4—2361no—
403C2×D4—2361no—
404C2×D4—2361no—
405C2×D4#1174566911yescharacteristic
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—2361no—
410C22×C4—2361no—
411C22×C4C2×C22⋊D5911yescharacteristic
412C22×C4C2×C22⋊D5911yescharacteristic
413Q16—1181no—
414Q16—1181no—
415SD16—4721no—
416SD16—2362no—
417D8—21no—
418D8—21no—
419D8—592no—
420D8—1181no—
421D8—1181no—
422D8—1181no—
423D8—2362no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×C22⋊D5911yescharacteristic
429C2×C8C2×C22⋊D5911yescharacteristic
430C2×C8C2×C22⋊D5911yescharacteristic
431C2×C8C23×D5911yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—1181no—
439C4⋊C4—2361no—
440C4⋊C4#1174566911yescharacteristic
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—2361no—
445C22⋊C4—2361no—
446C22⋊C4—2361no—
447C22⋊C4—2361no—
448C22⋊C4—2361no—
449C22⋊C4#1174566911yescharacteristic
450C22⋊C4#1174566911yescharacteristic
451C42C2×C22⋊D5911yescharacteristic
452C16—1181no—
453C16—1181no—
454C16—1181no—
455C16—1181no—
456C23—41no—
457C23—9441no—
458C23—4722no—
459C23—4722no—
460C23—2361no—
461C23—2361no—
462C23C2×#1174566911yescharacteristic
463C23C2×#1174566911yescharacteristic
464Q8—82no—
465Q8—2361no—
466Q8—2361no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—82no—
471D4—82no—
472D4—1182no—
473D4—9441no—
474D4—4722no—
475D4—4722no—
476D4—2361no—
477D4—2361no—
478D4—2361no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—9441no—
490C2×C4—4722no—
491C2×C4—2361no—
492C2×C4—2361no—
493C2×C4C2×#1174566911yescharacteristic
494C2×C4C2×#1174566911yescharacteristic
495C2×C4C2×#1174566911yescharacteristic
496C2×C4C2×#1174566911yescharacteristic
497C2×C4C22×C22⋊D5911yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8#41×D5911yescharacteristic
502C8#1427377311yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—9444no—
512C22—4721no—
513C22—4721no—
514C22—4721no—
515C22—4721no—
516C22—4721no—
517C22—2362no—
518C22#1683783812yes—
519C22C22×#1174566911yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—4721no—
528C4—4721no—
529C4#1683804011yescharacteristic
530C4#1683821011yescharacteristic
531C2—22no—
532C2—41no—
533C2—161no—
534C2—9441no—
535C2—4722no—
536C2#2143590311yescharacteristic
537C2C2×#1683783811yescharacteristic
538C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3334C2
socleC22×C59C22×D8
hypercenter#3334D59
hypocenterC59#33873
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #21435783C2semidirect
2. #21426335C2semidirect
3. #21426293C2semidirect
4. #21426281C2semidirect
5. #21426240C2semidirect
6. #21426197C2semidirect
7. #21426190C2semidirect
8. #21426151C2semidirect
9. #21426017C2semidirect
10. #21426007C2semidirect
11. #21425970C2semidirect
12. #21425922C2semidirect
13. #21425806C2semidirect
14. C2×#16837202C2semidirect
15. C59×#3334C2semidirect
19. C2×#14273671C22semidirect
22. C2×#14273667C22semidirect
31. #16837229C22semidirect
32. #16837229C22semidirect
33. #16837229C22semidirect
34. C4⋊(C59⋊C16)C22semidirect
35. #16837205C22semidirect
36. #16837205C22semidirect
37. #16837205C22semidirect
38. #16837203C22semidirect
46. C22⋊(C59⋊C16)C22semidirect
47. #16837178C22semidirect
49. C59×#500C22semidirect
54. C59×#466C22semidirect
55. C4×D8×C59C22semidirect
58. C2×#107×C59C22semidirect
179. D472#41semidirect
262. #3334D59semidirect
294. #500C2×D59semidirect
302. #466C2×D59semidirect
307. C4×D8C2×D59semidirect
310. C2×#107C2×D59semidirect
328. C59#33873semidirect

Maximal subgroups

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

subgroupindex
#214357832
#214263352
#214262932
#214262812
#214262402
#214261972
#214261902
#214261512
#214260172
#214260072
#214259702
#214259222
#214258062
C2×#168372022
C59×#33342
#3387359