Finite Group Databasebeta

← #28745844

Subgroups — #28745844

Order 15104.

622 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436501C211yesmaximal, characteristic, complemented
2D8×C22⋊D59C211yesmaximal, characteristic, complemented
3#21426336C211yesmaximal, characteristic, complemented
4#21426286C211yesmaximal, characteristic, complemented
5#21426282C211yesmaximal, characteristic, complemented
6#21426255C211yesmaximal, characteristic, complemented
7#21426196C211yesmaximal, characteristic, complemented
8#21426180C211yesmaximal, characteristic, complemented
9#21426129C211yesmaximal, characteristic, complemented
10#21426006C211yesmaximal, characteristic, complemented
11#21425972C211yesmaximal, characteristic, complemented
12#21425918C211yesmaximal, characteristic, complemented
13#21425810C211yesmaximal, characteristic, complemented
14C2×#16837203C211yesmaximal, characteristic, complemented
15C59×#2184C211yesmaximal, characteristic, complemented
16C2×D8×D59C2211yescharacteristic
17#16838104—21no—
18D4×C22⋊D59—21no—
19C2×#14273668C2211yescharacteristic, complemented
20C2×#14273666C2211yescharacteristic, complemented
21C22×C59⋊C16C2211yescharacteristic, complemented
22#16837803C2211yescharacteristic, complemented
23#16837777C2211yescharacteristic
24#16837724—21no—
25#16837718—21no—
26#16837700—21no—
27#16837638C2211yescharacteristic
28#16837635C2211yescharacteristic
29#16837635C2211yescharacteristic, complemented
30#16837595—21no—
31#16837536—21no—
32#16837534—21no—
33D8×Dic59C2211yescharacteristic
34D8×Dic59C2211yescharacteristic, complemented
35C8×C22⋊D59C2211yescharacteristic
36#16837439C2211yescharacteristic, complemented
37C2×#14273661C2211yescharacteristic, complemented
38#16837230C2211yescharacteristic, complemented
39#16837230C2211yescharacteristic, complemented
40#16837227C2211yescharacteristic, complemented
41Dic59⋊C16C2211yescharacteristic, complemented
42#16837204C2211yescharacteristic, complemented
43#16837204C2211yescharacteristic, complemented
44#16837203C2211yescharacteristic
45#16837203C2211yescharacteristic
46#16837203C2211yescharacteristic, complemented
47#16837203C2211yescharacteristic, complemented
48#16837203C2211yescharacteristic, complemented
49#16837202C2211yescharacteristic, complemented
50(C2×D59)⋊C16C2211yescharacteristic, complemented
51#16837177C2211yescharacteristic, complemented
52C22×D8×C59C2211yescharacteristic
53C2×D16×C59C2211yescharacteristic
54C2×D16×C59C2211yescharacteristic, complemented
55C59×#466C2211yescharacteristic, complemented
56C59×#357C2211yescharacteristic
57C59×#357C2211yescharacteristic, complemented
58C59×C22⋊C16C2211yescharacteristic, complemented
59C22×C22⋊D59—41no—
60C2×D4×D59—21no—
61C2×D472C2311yescharacteristic
62C2×#11745669—21no—
63C2×#11745669—21no—
64D8×D59—22no—
65D8×D59—41no—
66C2×C8×D59C2311yescharacteristic
67#14273723—21no—
68#14273720—41no—
69#14273718—21no—
70#14273711—41no—
71#14273701—21no—
72#14273699—41no—
73#14273699—41no—
74#14273698—21no—
75#14273698—21no—
76#14273696—41no—
77#14273695—41no—
78D4×Dic59—21no—
79D4×Dic59—21no—
80D4×Dic59—41no—
81C4×C22⋊D59D411yescharacteristic
82#14273685—21no—
83C2×C4⋊Dic59—21no—
84#14273668—41no—
85#14273666—41no—
86C2×C59⋊C16—41no—
87C2×C59⋊C16C2311yescharacteristic
88C2×C59⋊C16C2311yescharacteristic
89#14273661—21no—
90#14273661C2311yescharacteristic
91#14273661C2311yescharacteristic
92#14273661C2311yescharacteristic
93#14273658—21no—
94#14273658—21no—
95Dic59⋊C8D411yescharacteristic
96#14273649—21no—
97#14273648—21no—
98#14273648—21no—
99#14273648—21no—
100#14273648—21no—
101C22⋊(C59⋊C8)D411yescharacteristic
102(C2×D59)⋊C8D411yescharacteristic
103C8×Dic59C2311yescharacteristic
104C22×D4×C59—21no—
105C2×D8×C59—22no—
106C2×D8×C59—21no—
107C2×D8×C59—41no—
108C2×D8×C59C2311yescharacteristic
109C2×D8×C59C2311yescharacteristic
110C2×D8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112#121×C59—21no—
113D16×C59—22no—
114D16×C59—41no—
115C2×C16×C59C2311yescharacteristic
116C2×C16×C59C2311yescharacteristic
117#107×C59C2311yescharacteristic
118#102×C59—21no—
119C24×C59—41no—
120C23×D59—41no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—42no—
125C2×D4×C59—41no—
126C2×D4×C59—41no—
127C2×D4×C59—81no—
128C22×C4×C59C2×D411yescharacteristic
129C2×C22⋊D59—41no—
130C2×C22⋊D59—41no—
131C2×C22⋊D59—41no—
132C2×C22⋊D59—41no—
133C2×C22⋊D59—41no—
134C2×C22⋊D59—41no—
135C2×C22⋊D59—84no—
136C2×C22⋊D59D811yescharacteristic
137C22×Dic59—41no—
138C22×Dic59—41no—
139C22×Dic59—41no—
140D4×D59—42no—
141D4×D59—81no—
142C2×D236—21no—
143C2×C4×D59C2×D411yescharacteristic
144D8×C59—21no—
145D8×C59—21no—
146D8×C59—21no—
147D8×C59—21no—
148D8×C59—21no—
149D8×C59—42no—
150D8×C59#4112yes—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59C2×D411yescharacteristic
154C2×C8×C59C2411yescharacteristic
155C4⋊C4×C59—21no—
156C22⋊C4×C59—41no—
157C22⋊Dic59—41no—
158C22⋊Dic59—41no—
159C22⋊Dic59—41no—
160C22⋊Dic59—41no—
161C22⋊Dic59D811yescharacteristic
162#11745669—42no—
163#11745669—81no—
164(C2×D59)⋊C4—41no—
165(C2×D59)⋊C4—41no—
166(C2×D59)⋊C4D811yescharacteristic
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—41no—
171C4⋊Dic59—41no—
172Dic59⋊C4—41no—
173Dic59⋊C4D811yescharacteristic
174C4×Dic59—41no—
175C4×Dic59C2×D411yescharacteristic
176C2×C59⋊C8C2×D411yescharacteristic
177C2×C59⋊C8C2×D411yescharacteristic
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—41no—
189C23×C59—82no—
190C23×C59—82no—
191C23×C59—161no—
192C23×C59C2×D811yescharacteristic
193C22×D59—41no—
194C22×D59—82no—
195C22×D59—82no—
196C22×D59—161no—
197C22×D59C2×D811yescharacteristic
198D4×C59—22no—
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—41no—
202D4×C59—41no—
203D4×C59—41no—
204D4×C59—82no—
205D4×C59—82no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—41no—
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C22×D411yescharacteristic
211C22⋊D59—22no—
212C22⋊D59—82no—
213C22⋊D59—82no—
214C22⋊D59—82no—
215C22⋊D59—82no—
216C22⋊D59—161no—
217C22⋊D59—161no—
218C2×Dic59—41no—
219C2×Dic59—41no—
220C2×Dic59—41no—
221C2×Dic59—82no—
222C2×Dic59—161no—
223C2×Dic59—161no—
224C2×Dic59C2×D811yescharacteristic
225C2×Dic59C2×D811yescharacteristic
226C2×Dic59C2×D811yescharacteristic
227D236—41no—
228D236—41no—
229C4×D59—21no—
230C4×D59—21no—
231C8×C59—21no—
232C8×C59C2×#4111yescharacteristic
233C8×C59H(5,𝔽2)11yescharacteristic
234C59⋊C8—21no—
235C59⋊C8—21no—
236#33885—591nomaximal, Sylow
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—42no—
240C22×C59—81no—
241C22×C59—81no—
242C22×C59—81no—
243C22×C59—81no—
244C22×C59—81no—
245C22×C59—81no—
246C22×C59—81no—
247C22×C59—81no—
248C22×C59—164no—
249C22×C59#50912yes—
250C22×C59C22×D811yescharacteristic
251C2×D59—21no—
252C2×D59—21no—
253C2×D59—21no—
254C2×D59—81no—
255C2×D59—81no—
256C2×D59—81no—
257C2×D59—164no—
258C4×C59—21no—
259C4×C59—81no—
260C4×C59#54611yescharacteristic
261C4×C59#57711yescharacteristic
262Dic59—21no—
263Dic59—21no—
264Dic59—21no—
265Dic59—81no—
266Dic59—81no—
267Dic59—81no—
268#3339—591no—
269D4×D8—591no—
270#2244—591no—
271#2234—591no—
272#2233—591no—
273#2228—591no—
274#2218—591no—
275#2215—591no—
276#2206—591no—
277#2184—591no—
278#2184D5911yescharacteristic, complemented
279#2177—591no—
280#2168—591no—
281#2140—591no—
282C2×#357—591no—
283C2×C59—22no—
284C2×C59—82no—
285C2×C59—161no—
286C2×C59—161no—
287C2×C59#329411yescharacteristic
288C2×C59C2×#50911yescharacteristic
289C2×C59#341511yescharacteristic
290D59—41no—
291D59—161no—
292C22×D8—591no—
293C22×D8C2×D5911yescharacteristic
294#553—1181no—
295D42—1181no—
296C2×D16—591no—
297C2×D16C2×D5911yescharacteristic
298C2×D16C2×D5911yescharacteristic, complemented
299C2×M32—591no—
300C22×C16—591no—
301#499—591no—
302#493—591no—
303#482—1181no—
304#481—1181no—
305#478—1181no—
306#466—591no—
307#466—591no—
308#466—591no—
309#466C2×D5911yescharacteristic, complemented
310#459—1181no—
311#447—1181no—
312#447—1181no—
313C4×D8—591no—
314C4×D8—591no—
315C8×D4—591no—
316#427—591no—
317C2×#107—591no—
318#367—591no—
319#367—591no—
320#366—591no—
321C4⋊C16—591no—
322#357—591no—
323#357—591no—
324#357—591no—
325#357—591no—
326#357—591no—
327#357—591no—
328#357—591no—
329#357—591no—
330#357C2×D5911yescharacteristic
331#357C2×D5911yescharacteristic, complemented
332C22⋊C16—591no—
333C22⋊C16C2×D5911yescharacteristic, complemented
334#346—591no—
335C59#3388511yescharacteristic, complemented, Sylow
336C22×D4—21no—
337C22×D4—1181no—
338C22×D4—2361no—
339C2×D8—22no—
340C2×D8—21no—
341C2×D8—41no—
342C2×D8—591no—
343C2×D8—1181no—
344C2×D8—1181no—
345C2×D8—1182no—
346C2×D8—2361no—
347C2×D8C22×D5911yescharacteristic
348C2×D8C22×D5911yescharacteristic
349C2×D8C22×D5911yescharacteristic
350C22×C8—591no—
351C22×C8C22×D5911yescharacteristic
352#127—1181no—
353#126—2361no—
354#125—1181no—
355#123—2361no—
356#121—21no—
357#121—1181no—
358#121—1181no—
359#121—1181no—
360#121—2361no—
361#121—2361no—
362#120—2361no—
363#120—2361no—
364C4×D4—591no—
365C4×D4—1181no—
366C4×D4—1181no—
367C4×D4—2361no—
368#117—1181no—
369C2×C4⋊C4—1181no—
370D16—22no—
371D16—41no—
372D16—2361no—
373M32—2361no—
374C2×C16—591no—
375C2×C16—591no—
376C2×C16—2361no—
377C2×C16C22×D5911yescharacteristic
378C2×C16C22×D5911yescharacteristic
379#107—591no—
380#107—591no—
381#107—591no—
382#107—1181no—
383#107C22×D5911yescharacteristic
384#106—1181no—
385#106—1181no—
386C4⋊C8—591no—
387#102—21no—
388#102—1181no—
389#102—1181no—
390#102—1181no—
391#102—1181no—
392#102—1181no—
393C22⋊C8—591no—
394C22⋊C8—591no—
395C4×C8—591no—
396C24—41no—
397C24—2361no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—81no—
405C2×D4—591no—
406C2×D4—1181no—
407C2×D4—4721no—
408C2×D4—4724no—
409C2×D4—2361no—
410C2×D4—2361no—
411C2×D4—2361no—
412C2×D4—2362no—
413C2×D4—2361no—
414C2×D4—2361no—
415C2×D4—2361no—
416C22×C4—591no—
417C22×C4—2361no—
418C22×C4—2361no—
419C22×C4—2361no—
420C22×C4D4×D5911yescharacteristic
421D8—21no—
422D8—21no—
423D8—21no—
424D8—21no—
425D8—21no—
426D8—42no—
427D8—1181no—
428D8—1181no—
429D8—4721no—
430D8—2362no—
431D8#1174568712yes—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—591no—
435C2×C8—591no—
436C2×C8—1181no—
437C2×C8—1181no—
438C2×C8D4×D5911yescharacteristic
439C2×C8C23×D5911yescharacteristic
440C4⋊C4—21no—
441C4⋊C4—591no—
442C4⋊C4—1181no—
443C4⋊C4—1181no—
444C4⋊C4—1181no—
445C4⋊C4—2361no—
446C4⋊C4—2361no—
447C4⋊C4—2361no—
448C22⋊C4—41no—
449C22⋊C4—591no—
450C22⋊C4—591no—
451C22⋊C4—2361no—
452C22⋊C4—2361no—
453C22⋊C4—2361no—
454C22⋊C4—2361no—
455C22⋊C4—2361no—
456C22⋊C4—2361no—
457C42—591no—
458C42—2361no—
459C16—21no—
460C16—21no—
461C16—1181no—
462C16—1181no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—82no—
467C23—82no—
468C23—161no—
469C23—591no—
470C23—9441no—
471C23—4722no—
472C23—4722no—
473C23—2361no—
474C23D8×D5911yescharacteristic
475D4—22no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—82no—
482D4—82no—
483D4—1182no—
484D4—9441no—
485D4—9441no—
486D4—4722no—
487D4—4722no—
488D4—4722no—
489D4—4722no—
490D4—2361no—
491D4—2361no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—41no—
495C2×C4—591no—
496C2×C4—591no—
497C2×C4—591no—
498C2×C4—1181no—
499C2×C4—1181no—
500C2×C4—9441no—
501C2×C4—9441no—
502C2×C4—4722no—
503C2×C4—2361no—
504C2×C4—2361no—
505C2×C4—2361no—
506C2×C4D8×D5911yescharacteristic
507C2×C4C2×D4×D5911yescharacteristic
508C8—21no—
509C8—1181no—
510C8—1181no—
511C8C2×#1174568711yescharacteristic
512C8#1427377211yescharacteristic
513C22—21no—
514C22—21no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—164no—
525C22—1181no—
526C22—1181no—
527C22—1181no—
528C22—9444no—
529C22—4721no—
530C22—4721no—
531C22—4721no—
532C22#1683783612yes—
533C22C2×D8×D5911yescharacteristic
534C4—21no—
535C4—81no—
536C4—1181no—
537C4—1181no—
538C4—1181no—
539C4—4721no—
540C4—4721no—
541C4—4721no—
542C4#1683803911yescharacteristic
543C4#1683821211yescharacteristic
544C2—22no—
545C2—82no—
546C2—161no—
547C2—161no—
548C2—9441no—
549C2—2361no—
550C2#2143590211yescharacteristic
551C2C2×#1683783611yescharacteristic
552C2#2143723611yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2184C2
socleC22×C59C22×D8
hypercenter#2184D59
hypocenterC59#33885
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. #21436501C2semidirect
2. D8×C22⋊D59C2semidirect
3. #21426336C2semidirect
4. #21426286C2semidirect
5. #21426282C2semidirect
6. #21426255C2semidirect
7. #21426196C2semidirect
8. #21426180C2semidirect
9. #21426129C2semidirect
10. #21426006C2semidirect
11. #21425972C2semidirect
12. #21425918C2semidirect
13. #21425810C2semidirect
14. C2×#16837203C2semidirect
15. C59×#2184C2semidirect
19. C2×#14273668C22semidirect
20. C2×#14273666C22semidirect
21. C22×C59⋊C16C22semidirect
22. #16837803C22semidirect
29. #16837635C22semidirect
34. D8×Dic59C22semidirect
36. #16837439C22semidirect
37. C2×#14273661C22semidirect
38. #16837230C22semidirect
39. #16837230C22semidirect
40. #16837227C22semidirect
41. Dic59⋊C16C22semidirect
42. #16837204C22semidirect
43. #16837204C22semidirect
46. #16837203C22semidirect
47. #16837203C22semidirect
48. #16837203C22semidirect
49. #16837202C22semidirect
50. (C2×D59)⋊C16C22semidirect
51. #16837177C22semidirect
54. C2×D16×C59C22semidirect
55. C59×#466C22semidirect
57. C59×#357C22semidirect
58. C59×C22⋊C16C22semidirect
278. #2184D59semidirect
298. C2×D16C2×D59semidirect
309. #466C2×D59semidirect
331. #357C2×D59semidirect
333. C22⋊C16C2×D59semidirect
335. C59#33885semidirect

Maximal subgroups

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

subgroupindex
#214365012
D8×C22⋊D592
#214263362
#214262862
#214262822
#214262552
#214261962
#214261802
#214261292
#214260062
#214259722
#214259182
#214258102
C2×#168372032
C59×#21842
#3388559