Finite Group Databasebeta

← #28745784

Subgroups — #28745784

Order 15104.

570 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436485C211yesmaximal, characteristic, complemented
2#21435781C211yesmaximal, characteristic
3#21426331C211yesmaximal, characteristic, complemented
4#21426299C211yesmaximal, characteristic, complemented
5#21426275C211yesmaximal, characteristic, complemented
6#21426231C211yesmaximal, characteristic, complemented
7#21426206C211yesmaximal, characteristic, complemented
8#21426185C211yesmaximal, characteristic, complemented
9#21426156C211yesmaximal, characteristic, complemented
10#21426033C211yesmaximal, characteristic, complemented
11#21425961C211yesmaximal, characteristic, complemented
12#21425918C211yesmaximal, characteristic, complemented
13#21425805C211yesmaximal, characteristic, complemented
14C2×#16837203C211yesmaximal, characteristic, complemented
15C59×#2187C211yesmaximal, characteristic, complemented
16C2×#14273746C2211yescharacteristic
17#16838110—21no—
18#16838046—21no—
19C2×#14273673C2211yescharacteristic, complemented
20C2×#14273666C2211yescharacteristic, complemented
21C22×C59⋊C16C2211yescharacteristic
22#16837806C2211yescharacteristic, complemented
23#16837785C2211yescharacteristic
24#16837736—21no—
25#16837735—21no—
26#16837695—21no—
27#16837643C2211yescharacteristic
28#16837640C2211yescharacteristic, complemented
29#16837635C2211yescharacteristic
30#16837615—21no—
31#16837553—21no—
32#16837544—21no—
33Q16×Dic59C2211yescharacteristic, complemented
34D8×Dic59C2211yescharacteristic
35C8×C22⋊D59C2211yescharacteristic
36C2×#14273661C2211yescharacteristic
37#107×D59C2211yescharacteristic, complemented
38#16837228C2211yescharacteristic, complemented
39#16837228C2211yescharacteristic, complemented
40#16837227C2211yescharacteristic, complemented
41Dic59⋊C16C2211yescharacteristic, complemented
42#16837207C2211yescharacteristic, complemented
43#16837207C2211yescharacteristic, complemented
44#16837206C2211yescharacteristic, complemented
45#16837205C2211yescharacteristic, complemented
46#16837203C2211yescharacteristic
47#16837203C2211yescharacteristic
48#16837203C2211yescharacteristic
49#16837203C2211yescharacteristic
50(C2×D59)⋊C16C2211yescharacteristic, complemented
51#16837177C2211yescharacteristic, complemented
52C22×D8×C59C2211yescharacteristic
53C2×SD32×C59C2211yescharacteristic
54C2×SD32×C59C2211yescharacteristic
55C59×#467C2211yescharacteristic
56C59×#357C2211yescharacteristic
57C59×#357C2211yescharacteristic
58C59×C22⋊C16C2211yescharacteristic
59C2×#11745687—21no—
60#14273746—22no—
61#14273746—41no—
62C2×Dic236C2311yescharacteristic
63C2×#11745670—21no—
64C2×#11745670—21no—
65C2×C8×D59C2311yescharacteristic
66#14273726—21no—
67#14273717—41no—
68#14273714—21no—
69#14273708—41no—
70#14273707—41no—
71#14273706—21no—
72#14273704—41no—
73#14273703—21no—
74#14273699—41no—
75#14273698—21no—
76#14273696—41no—
77Q8×Dic59—21no—
78D4×Dic59—21no—
79C4×C22⋊D59D411yescharacteristic
80(C22⋊D59)⋊C4—41no—
81C2×C4⋊Dic59—21no—
82C4⋊C4×D59—21no—
83C2×C22⋊Dic59—41no—
84#14273673—41no—
85#14273666—41no—
86C2×C59⋊C16—41no—
87C2×C59⋊C16C2311yescharacteristic
88C2×C59⋊C16C2311yescharacteristic
89#14273661—21no—
90#14273661C2311yescharacteristic
91#14273661C2311yescharacteristic
92#14273661C2311yescharacteristic
93#14273660—21no—
94#14273660—21no—
95Dic59⋊C8D411yescharacteristic
96#14273652—21no—
97#14273651—21no—
98#14273651—21no—
99#14273648—21no—
100#14273648—21no—
101C22⋊(C59⋊C8)D411yescharacteristic
102(C2×D59)⋊C8D411yescharacteristic
103C8×Dic59C2311yescharacteristic
104C22×D4×C59—21no—
105C2×Q16×C59C2311yescharacteristic
106C2×D8×C59—22no—
107C2×D8×C59—21no—
108C2×D8×C59—41no—
109C2×D8×C59C2311yescharacteristic
110C2×D8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C22⋊Q8×C59—21no—
113SD32×C59—22no—
114SD32×C59—41no—
115C2×C16×C59C2311yescharacteristic
116C2×C16×C59C2311yescharacteristic
117#107×C59C2311yescharacteristic
118#103×C59—21no—
119C24×C59—41no—
120C2×Q8×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—42no—
124C2×D4×C59—41no—
125C2×D4×C59—81no—
126C22×C4×C59C2×D411yescharacteristic
127C2×C22⋊D59—41no—
128C2×C22⋊D59—41no—
129C2×C22⋊D59D811yescharacteristic
130C22×Dic59—41no—
131C22×Dic59—41no—
132C22×Dic59—41no—
133#11745687—42no—
134#11745687—81no—
135C2×C4×D59—41no—
136C2×C4×D59C2×D411yescharacteristic
137C2×Dic118—21no—
138C2×Dic118—41no—
139Q16×C59—21no—
140Q16×C59—21no—
141D8×C59—21no—
142D8×C59—21no—
143D8×C59—21no—
144D8×C59—42no—
145D8×C59#4112yes—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59C2×D411yescharacteristic
149C2×C8×C59C2411yescharacteristic
150C4⋊C4×C59—21no—
151C4⋊C4×C59—41no—
152C22⋊C4×C59—41no—
153C22⋊Dic59—41no—
154C22⋊Dic59—41no—
155C22⋊Dic59—41no—
156C22⋊Dic59—41no—
157C22⋊Dic59—41no—
158C22⋊Dic59D811yescharacteristic
159#11745670—42no—
160#11745670—81no—
161(C2×D59)⋊C4—41no—
162(C2×D59)⋊C4D811yescharacteristic
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—41no—
167C4⋊Dic59—41no—
168Dic59⋊C4—41no—
169Dic59⋊C4—41no—
170Dic59⋊C4—41no—
171Dic59⋊C4—41no—
172Dic59⋊C4D811yescharacteristic
173C4×Dic59—41no—
174C4×Dic59C2×D411yescharacteristic
175C2×C59⋊C8C2×D411yescharacteristic
176C2×C59⋊C8C2×D411yescharacteristic
177Dic236—21no—
178Dic236—21no—
179C8×D59—21no—
180C8×D59—21no—
181C16×C59—21no—
182C16×C59—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59—41no—
186C23×C59—41no—
187C23×C59—82no—
188C23×C59—82no—
189C23×C59—161no—
190C23×C59C2×D811yescharacteristic
191C22×D59C2×D811yescharacteristic
192Q8×C59—41no—
193Q8×C59—41no—
194D4×C59—22no—
195D4×C59—41no—
196D4×C59—41no—
197D4×C59—41no—
198D4×C59—82no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—41no—
202C2×C4×C59—41no—
203C2×C4×C59C2×D811yescharacteristic
204C2×C4×C59C22×D411yescharacteristic
205C22⋊D59—22no—
206C22⋊D59—82no—
207C22⋊D59—161no—
208C2×Dic59—41no—
209C2×Dic59—41no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—82no—
213C2×Dic59—161no—
214C2×Dic59—161no—
215C2×Dic59C2×D811yescharacteristic
216C2×Dic59C2×D811yescharacteristic
217C2×Dic59C2×D811yescharacteristic
218C4×D59—21no—
219C4×D59—21no—
220C4×D59—161no—
221Dic118—41no—
222Dic118—41no—
223Dic118—161no—
224C8×C59—21no—
225C8×C59C2×#4111yescharacteristic
226C8×C59H(5,𝔽2)11yescharacteristic
227C59⋊C8—21no—
228C59⋊C8—21no—
229#33881—591nomaximal, Sylow
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—42no—
233C22×C59—81no—
234C22×C59—81no—
235C22×C59—81no—
236C22×C59—81no—
237C22×C59—81no—
238C22×C59—164no—
239C22×C59#51012yes—
240C22×C59C22×D811yescharacteristic
241C2×D59—21no—
242C2×D59—21no—
243C2×D59—21no—
244C4×C59—21no—
245C4×C59—81no—
246C4×C59—81no—
247C4×C59#54611yescharacteristic
248C4×C59#57711yescharacteristic
249Dic59—21no—
250Dic59—21no—
251Dic59—21no—
252Dic59—81no—
253Dic59—81no—
254Dic59—81no—
255Dic59—81no—
256#3338—591no—
257#3280—591no—
258#2243—591no—
259#2236—591no—
260#2232—591no—
261#2224—591no—
262#2220—591no—
263#2216—591no—
264#2211—591no—
265#2189—591no—
266#2187D5911yescharacteristic, complemented
267#2175—591no—
268#2168—591no—
269#2139—591no—
270C2×#357—591no—
271C2×C59—22no—
272C2×C59—82no—
273C2×C59—161no—
274C2×C59#329411yescharacteristic
275C2×C59C2×#51011yescharacteristic
276C2×C59#341511yescharacteristic
277D59—41no—
278C2×#135—591no—
279C22×D8C2×D5911yescharacteristic
280#554—1181no—
281#546—1181no—
282C2×Q32—591no—
283C2×SD32C2×D5911yescharacteristic
284C2×SD32C2×D5911yescharacteristic
285C2×M32—591no—
286C22×C16—591no—
287#500—591no—
288#495—591no—
289#484—1181no—
290#484—1181no—
291#477—1181no—
292#467—591no—
293#467—591no—
294#467C2×D5911yescharacteristic
295#466—591no—
296#463—1181no—
297#450—1181no—
298#449—1181no—
299C4×Q16—591no—
300C4×D8—591no—
301C8×D4—591no—
302C2×#107—591no—
303C2×#107—591no—
304#366—591no—
305#366—591no—
306#366—591no—
307C4⋊C16—591no—
308#358—591no—
309#358—591no—
310#358—591no—
311#358—591no—
312#357—591no—
313#357—591no—
314#357—591no—
315#357—591no—
316#357C2×D5911yescharacteristic
317#357C2×D5911yescharacteristic
318C22⋊C16—591no—
319C22⋊C16C2×D5911yescharacteristic
320#346—591no—
321C59#3388111yescharacteristic, complemented, Sylow
322C2×#41—1181no—
323C22×D4—21no—
324#135—1182no—
325#135—2361no—
326C2×Q16—591no—
327C2×Q16C22×D5911yescharacteristic
328C2×SD16—1181no—
329C2×SD16—1181no—
330C2×D8—22no—
331C2×D8—21no—
332C2×D8—41no—
333C2×D8C22×D5911yescharacteristic
334C2×D8C22×D5911yescharacteristic
335C22×C8—591no—
336C22×C8C22×D5911yescharacteristic
337C4⋊Q8—1181no—
338#125—2361no—
339#124—1181no—
340#123—2361no—
341C22⋊Q8—21no—
342C22⋊Q8—1181no—
343C22⋊Q8—1181no—
344C22⋊Q8—2361no—
345C22⋊Q8—2361no—
346#121—1181no—
347#121—2361no—
348#120—2361no—
349C4×Q8—1181no—
350C4×D4—591no—
351C4×D4—1181no—
352C4×D4—2361no—
353C2×C4⋊C4—1181no—
354C2×C4⋊C4—1181no—
355C2×C22⋊C4—2361no—
356Q32—2361no—
357SD32—22no—
358SD32—41no—
359M32—2361no—
360C2×C16—591no—
361C2×C16—591no—
362C2×C16—2361no—
363C2×C16C22×D5911yescharacteristic
364C2×C16C22×D5911yescharacteristic
365#107—591no—
366#107—591no—
367#107—591no—
368#107—1181no—
369#107—1181no—
370#107—1181no—
371#107C22×D5911yescharacteristic
372C4⋊C8—591no—
373#103—21no—
374#103—1181no—
375#103—1181no—
376#103—1181no—
377#102—1181no—
378#102—1181no—
379C22⋊C8—591no—
380C22⋊C8—591no—
381C4×C8—591no—
382C24—41no—
383#41—4721no—
384#41—2362no—
385C2×Q8—21no—
386C2×Q8—1181no—
387C2×Q8—2361no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—41no—
392C2×D4—81no—
393C2×D4—591no—
394C2×D4—2361no—
395C2×D4—2361no—
396C22×C4—591no—
397C22×C4—2361no—
398C22×C4—2361no—
399C22×C4—2361no—
400C22×C4—2361no—
401C22×C4D4×D5911yescharacteristic
402Q16—21no—
403Q16—21no—
404Q16—1181no—
405Q16—1181no—
406SD16—4721no—
407SD16—2362no—
408D8—21no—
409D8—21no—
410D8—21no—
411D8—42no—
412D8#1174568712yes—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—591no—
416C2×C8—591no—
417C2×C8—1181no—
418C2×C8—1181no—
419C2×C8D4×D5911yescharacteristic
420C2×C8C23×D5911yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—591no—
424C4⋊C4—1181no—
425C4⋊C4—1181no—
426C4⋊C4—1181no—
427C4⋊C4—2361no—
428C4⋊C4—2361no—
429C4⋊C4—2361no—
430C4⋊C4—2361no—
431C4⋊C4—2361no—
432C4⋊C4—2361no—
433C22⋊C4—41no—
434C22⋊C4—591no—
435C22⋊C4—591no—
436C22⋊C4—2361no—
437C22⋊C4—2361no—
438C22⋊C4—2361no—
439C22⋊C4—2361no—
440C22⋊C4—2361no—
441C22⋊C4—2361no—
442C42—591no—
443C42—2361no—
444C16—21no—
445C16—21no—
446C16—1181no—
447C16—1181no—
448C23—41no—
449C23—41no—
450C23—82no—
451C23—82no—
452C23—161no—
453C23—591no—
454C23D8×D5911yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—9441no—
458Q8—2361no—
459Q8—2361no—
460D4—22no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—82no—
465D4—1182no—
466D4—9441no—
467D4—4722no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—591no—
473C2×C4—591no—
474C2×C4—591no—
475C2×C4—1181no—
476C2×C4—1181no—
477C2×C4—9441no—
478C2×C4—9441no—
479C2×C4—9441no—
480C2×C4—4722no—
481C2×C4—2361no—
482C2×C4—2361no—
483C2×C4—2361no—
484C2×C4—2361no—
485C2×C4D8×D5911yescharacteristic
486C2×C4C2×D4×D5911yescharacteristic
487C8—21no—
488C8—1181no—
489C8—1181no—
490C8C2×#1174568711yescharacteristic
491C8#1427377211yescharacteristic
492C22—21no—
493C22—21no—
494C22—42no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—164no—
501C22—1181no—
502C22—1181no—
503C22—1181no—
504C22#1683784112yes—
505C22C2×D8×D5911yescharacteristic
506C4—21no—
507C4—81no—
508C4—81no—
509C4—1181no—
510C4—1181no—
511C4—1181no—
512C4—4721no—
513C4—4721no—
514C4—4721no—
515C4—4721no—
516C4#1683803911yescharacteristic
517C4#1683821211yescharacteristic
518C2—22no—
519C2—82no—
520C2—161no—
521C2—2361no—
522C2#2143590211yescharacteristic
523C2C2×#1683784111yescharacteristic
524C2#2143723611yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2187C2
socleC22×C59C22×D8
prime radical#21435781C2
hypercenter#2187D59
hypocenterC59#33881
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #21436485C2semidirect
3. #21426331C2semidirect
4. #21426299C2semidirect
5. #21426275C2semidirect
6. #21426231C2semidirect
7. #21426206C2semidirect
8. #21426185C2semidirect
9. #21426156C2semidirect
10. #21426033C2semidirect
11. #21425961C2semidirect
12. #21425918C2semidirect
13. #21425805C2semidirect
14. C2×#16837203C2semidirect
15. C59×#2187C2semidirect
19. C2×#14273673C22semidirect
20. C2×#14273666C22semidirect
22. #16837806C22semidirect
28. #16837640C22semidirect
33. Q16×Dic59C22semidirect
37. #107×D59C22semidirect
38. #16837228C22semidirect
39. #16837228C22semidirect
40. #16837227C22semidirect
41. Dic59⋊C16C22semidirect
42. #16837207C22semidirect
43. #16837207C22semidirect
44. #16837206C22semidirect
45. #16837205C22semidirect
50. (C2×D59)⋊C16C22semidirect
51. #16837177C22semidirect
266. #2187D59semidirect
321. C59#33881semidirect

Maximal subgroups

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

subgroupindex
#214364852
#214357812
#214263312
#214262992
#214262752
#214262312
#214262062
#214261852
#214261562
#214260332
#214259612
#214259182
#214258052
C2×#168372032
C59×#21872
#3388159