Finite Group Databasebeta

← #28742028

Subgroups — #28742028

Order 15104.

696 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837836C211yesmaximal, characteristic
2#21432966C211yesmaximal, characteristic, complemented
3#21432943C211yesmaximal, characteristic, complemented
4#21425969C211yesmaximal, characteristic, complemented
5#21425962C211yesmaximal, characteristic, complemented
6#21425957C211yesmaximal, characteristic, complemented
7#21425947C211yesmaximal, characteristic, complemented
8C4×#14273671C211yesmaximal, characteristic, complemented
9D16×Dic59C211yesmaximal, characteristic, complemented
10C4×#14273668C211yesmaximal, characteristic, complemented
11#21425883C211yesmaximal, characteristic, complemented
12#21425800C211yesmaximal, characteristic, complemented
13#357×D59C211yesmaximal, characteristic, complemented
14#21425694C211yesmaximal, characteristic, complemented
15C59×#2177C211yesmaximal, characteristic, complemented
16C2×#14273746C2211yescharacteristic
17C2×D8×D59C2211yescharacteristic
18#16837881—21no—
19#16837874—21no—
20#16837836C418yescomplemented
21C2×#14273672C2211yescharacteristic
22C2×#14273671C2211yescharacteristic
23C2×#14273668C2211yescharacteristic
24C2×#14273666C2211yescharacteristic
25C2×C16×D59C2211yescharacteristic
26#16837512—21no—
27#16837511—21no—
28#16837502—21no—
29#16837499—21no—
30C4×Dic236C2211yescharacteristic, complemented
31#16837496C2211yescharacteristic, complemented
32C4×D472C2211yescharacteristic
33D8×Dic59C2211yescharacteristic
34D8×Dic59C2211yescharacteristic, complemented
35#16837484C2211yescharacteristic
36#16837439C2211yescharacteristic
37#107×D59C2211yescharacteristic
38#16837405—21no—
39#16837400—21no—
40#16837356C2211yescharacteristic
41#16837229C2211yescharacteristic, complemented
42#16837228C2211yescharacteristic, complemented
43#16837227C2211yescharacteristic, complemented
44#16837207C2211yescharacteristic, complemented
45#16837205C2211yescharacteristic, complemented
46#16837204C2211yescharacteristic
47#16837203C2211yescharacteristic
48#16837203C2211yescharacteristic, complemented
49#16837202C2211yescharacteristic
50#16837177C2211yescharacteristic, complemented
51C4×C59⋊C16C2211yescharacteristic, complemented
52C16×Dic59C2211yescharacteristic, complemented
53C2×D16×C59C2211yescharacteristic
54C4×D8×C59C2211yescharacteristic
55C4×D8×C59C2211yescharacteristic, complemented
56C59×#367C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic
58C59×#357C2211yescharacteristic, complemented
59C59×#346C2211yescharacteristic, complemented
60C2×#11745687—21no—
61C2×D4×D59—21no—
62#14273746—22no—
63#14273746C2×C414yescomplemented
64C2×Dic236C2311yescharacteristic
65C2×#11745670—21no—
66C2×D472C2311yescharacteristic
67C2×#11745669—21no—
68D8×D59—22no—
69D8×D59C2×C414yescomplemented
70C2×#11745659D411yescharacteristic
71C2×C8×D59C2311yescharacteristic
72C4×Dic118—21no—
73#14273692—21no—
74C4×D236—21no—
75D4×Dic59—21no—
76D4×Dic59—21no—
77#14273689—21no—
78C4×C22⋊D59—41no—
79C4×C22⋊D59—41no—
80(C22⋊D59)⋊C4—41no—
81(C22⋊D59)⋊C4—41no—
82#14273685—21no—
83#14273684D411yescharacteristic
84#14273683—41no—
85C2×Dic59⋊C4—41no—
86C4⋊C4×D59—21no—
87C2×(C2×D59)⋊C4—41no—
88C22⋊C4×D59—41no—
89#14273672C2×C414yescomplemented
90#14273671C2×C414yescomplemented
91#14273668C2×C414yescomplemented
92#14273666C2×C414yescomplemented
93C2×C59⋊C16C2311yescharacteristic
94C2×C59⋊C16C2311yescharacteristic
95C16×D59—22no—
96#14273661C2311yescharacteristic
97#14273661C2311yescharacteristic
98#14273660—21no—
99#14273658—21no—
100#14273652—21no—
101#14273650—21no—
102#14273649—21no—
103#14273648—21no—
104#14273648—21no—
105#14273647—21no—
106#14273638D411yescharacteristic
107#14273637D411yescharacteristic
108C8×Dic59C2311yescharacteristic
109C2×D8×C59C2311yescharacteristic
110C2×D8×C59C2311yescharacteristic
111C4×D4×C59—21no—
112C4×D4×C59—21no—
113D16×C59C2×C414yescomplemented
114C2×C16×C59C2311yescharacteristic
115C2×C16×C59C2311yescharacteristic
116#107×C59C2311yescharacteristic
117#107×C59C2311yescharacteristic
118#102×C59—21no—
119#102×C59—21no—
120C4×C8×C59C2311yescharacteristic
121C23×D59—41no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C22×C4×C59—41no—
125C22×C4×C59—41no—
126C2×C22⋊D59—41no—
127C2×C22⋊D59—41no—
128C22×Dic59—41no—
129C22×Dic59—41no—
130C22×Dic59—41no—
131#11745687—24no—
132#11745687—42no—
133D4×D59—24no—
134D4×D59—42no—
135C2×D236—21no—
136C2×C4×D59—41no—
137C2×C4×D59—41no—
138C2×C4×D59—41no—
139C2×C4×D59C2×D411yescharacteristic
140C2×Dic118—21no—
141D8×C59—21no—
142D8×C59—21no—
143D8×C59C22×C412yes—
144D8×C59C22×C412yes—
145C2×C8×C59C2×D411yescharacteristic
146C2×C8×C59C2411yescharacteristic
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C22⋊C4×C59—41no—
150C22⋊C4×C59—41no—
151C42×C59C2×D411yescharacteristic
152C22⋊Dic59—41no—
153C22⋊Dic59—41no—
154#11745670—24no—
155#11745669—24no—
156(C2×D59)⋊C4—41no—
157(C2×D59)⋊C4—41no—
158(C2×D59)⋊C4—41no—
159(C2×D59)⋊C4D811yescharacteristic
160(C2×D59)⋊C4D811yescharacteristic
161C4⋊Dic59—21no—
162C4⋊Dic59—21no—
163Dic59⋊C4—41no—
164Dic59⋊C4—41no—
165Dic59⋊C4—41no—
166Dic59⋊C4D811yescharacteristic
167Dic59⋊C4D811yescharacteristic
168C4×Dic59—41no—
169C4×Dic59—41no—
170C4×Dic59—41no—
171C4×Dic59C2×D411yescharacteristic
172C2×C59⋊C8C2×D411yescharacteristic
173C2×C59⋊C8C2×D411yescharacteristic
174Dic236—21no—
175Dic236C22×C412yes—
176D472—21no—
177D472C22×C412yes—
178#11745659—22no—
179C8×D59C22×C412yes—
180C8×D59#4112yes—
181C16×C59—21no—
182C16×C59C22×C412yes—
183C59⋊C16—21no—
184C59⋊C16C22×C412yes—
185C23×C59—41no—
186C23×C59—41no—
187C22×D59—41no—
188C22×D59—44no—
189C22×D59—161no—
190C22×D59—161no—
191C22×D59C2×D811yescharacteristic
192D4×C59—22no—
193D4×C59—22no—
194D4×C59—41no—
195D4×C59—41no—
196C2×C4×C59—41no—
197C2×C4×C59—41no—
198C2×C4×C59—161no—
199C2×C4×C59—161no—
200C2×C4×C59C2×D811yescharacteristic
201C2×C4×C59C2×D811yescharacteristic
202C2×C4×C59C22×D411yescharacteristic
203C22⋊D59—44no—
204C22⋊D59—44no—
205C2×Dic59—41no—
206C2×Dic59—44no—
207C2×Dic59—41no—
208C2×Dic59—41no—
209C2×Dic59—161no—
210C2×Dic59—161no—
211C2×Dic59C2×D811yescharacteristic
212C2×Dic59C2×D811yescharacteristic
213C2×Dic59C2×D811yescharacteristic
214D236—22no—
215D236—41no—
216C4×D59—161no—
217C4×D59—161no—
218C4×D59—161no—
219C4×D59C4×D412yes—
220C4×D59#13512yes—
221Dic118—22no—
222Dic118—41no—
223C8×C59—21no—
224C8×C59C23×C411yescharacteristic
225C8×C59C2×#4111yescharacteristic
226C59⋊C8—21no—
227C59⋊C8C4×D412yes—
228#33454—591nomaximal, Sylow
229C22×C59—42no—
230C22×C59—42no—
231C22×C59—81no—
232C22×C59—81no—
233C22×C59—81no—
234C22×C59—81no—
235C22×C59C22×D811yescharacteristic
236C2×D59—21no—
237C2×D59—21no—
238C2×D59—42no—
239C2×D59—81no—
240C2×D59—81no—
241C2×D59—164no—
242C2×D59C4×D812yes—
243C4×C59—21no—
244C4×C59—21no—
245C4×C59—81no—
246C4×C59—81no—
247C4×C59C2×C4×D411yescharacteristic
248C4×C59C2×#13511yescharacteristic
249Dic59—21no—
250Dic59—21no—
251Dic59—42no—
252Dic59—81no—
253Dic59—81no—
254Dic59C4×D812yes—
255C2×#509—591no—
256#2951—591no—
257#2948—591no—
258#2177—591no—
259#2177D5911yescharacteristic, complemented
260#2175—591no—
261#2175—591no—
262C4×SD32—591no—
263C4×SD32—591no—
264C4×D16—591no—
265C4×D16—591no—
266#2159—591no—
267#2138—591no—
268C2×#357—591no—
269#2108—591no—
270C2×C59—82no—
271C2×C59—82no—
272C2×C59C2×C4×D811yescharacteristic
273C2×C59#341511yescharacteristic
274C2×C59#341611yescharacteristic
275D59—22no—
276D59—82no—
277C2×#135—591no—
278C22×D8—591no—
279#520—1181no—
280#518—1181no—
281#509—598no—
282C2×SD32—591no—
283C2×SD32—591no—
284C2×D16—591no—
285C2×D16C2×D5911yescharacteristic
286C2×M32—591no—
287C22×C16—591no—
288#442—1181no—
289#442—1181no—
290#440—1181no—
291#440—1181no—
292C4×Q16—591no—
293C4×Q16—591no—
294C4×D8—591no—
295C4×D8—591no—
296C4×D8—591no—
297C4×D8—591no—
298C4×D8C2×D5911yescharacteristic
299C4×D8C2×D5911yescharacteristic, complemented
300#427—591no—
301C2×#107—591no—
302#418—1181no—
303#417—1181no—
304#405—591no—
305#367—591no—
306#367C2×D5911yescharacteristic, complemented
307#366—591no—
308#366—591no—
309#358—591no—
310#358—591no—
311#357—591no—
312#357—591no—
313#357—591no—
314#357—591no—
315#357C2×D5911yescharacteristic
316#357C2×D5911yescharacteristic, complemented
317#346—591no—
318#346C2×D5911yescharacteristic, complemented
319C4×C16—591no—
320C4×C16—591no—
321C59#3345411yescharacteristic, complemented, Sylow
322C2×#41—1181no—
323C22×D4—1181no—
324#135—594no—
325#135—1182no—
326C2×Q16—591no—
327C2×SD16—1181no—
328C2×D8—594no—
329C2×D8—591no—
330C2×D8—1182no—
331C2×D8—1181no—
332C2×D8C22×D5911yescharacteristic
333C2×D8C22×D5911yescharacteristic
334C2×M16—591no—
335C22×C8—591no—
336C4×Q8—1181no—
337C4×Q8—1181no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4—1181no—
341C4×D4—1181no—
342C4×D4—1181no—
343C4×D4—1181no—
344C4×D4—2361no—
345C4×D4—2361no—
346C4×D4—2361no—
347C4×D4—2361no—
348#117—591no—
349#117—1181no—
350#117—2361no—
351C2×C4⋊C4—1181no—
352C2×C4⋊C4—2361no—
353C2×C22⋊C4—2361no—
354C2×C22⋊C4—2361no—
355SD32—594no—
356SD32—594no—
357D16—594no—
358D16C4×D5914yescomplemented
359M32—594no—
360C2×C16—591no—
361C2×C16—591no—
362C2×C16—1182no—
363C2×C16C22×D5911yescharacteristic
364C2×C16C22×D5911yescharacteristic
365#107—591no—
366#107—591no—
367#107—1181no—
368#107C22×D5911yescharacteristic
369#107C22×D5911yescharacteristic
370#106—1181no—
371#103—1181no—
372#103—1181no—
373#102—21no—
374#102—21no—
375#102—1181no—
376#102—1181no—
377#102—1181no—
378#102—1181no—
379#97—591no—
380#97—591no—
381C4×C8—591no—
382C4×C8C22×D5911yescharacteristic
383C24—2361no—
384#41—1184no—
385#41—2362no—
386C2×Q8—1181no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—1184no—
390C2×D4—1181no—
391C2×D4—2362no—
392C2×D4—2361no—
393C2×D4—2361no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—591no—
397C22×C4—2361no—
398C22×C4—2361no—
399C22×C4—2361no—
400C22×C4—2361no—
401C22×C4—2361no—
402C22×C4—2361no—
403Q16—592no—
404Q16—1181no—
405SD16—1184no—
406D8—21no—
407D8—21no—
408D8—592no—
409D8—1184no—
410D8—1181no—
411D8C2×C4×D5912yes—
412D8C2×C4×D5912yes—
413M16—1182no—
414C2×C8—591no—
415C2×C8—592no—
416C2×C8—592no—
417C2×C8—591no—
418C2×C8D4×D5911yescharacteristic
419C2×C8C23×D5911yescharacteristic
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—591no—
423C4⋊C4—591no—
424C4⋊C4—1181no—
425C4⋊C4—1181no—
426C4⋊C4—2361no—
427C4⋊C4—2361no—
428C4⋊C4—2361no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—591no—
432C22⋊C4—591no—
433C22⋊C4—2361no—
434C22⋊C4—2361no—
435C22⋊C4—2361no—
436C22⋊C4—2361no—
437C22⋊C4—2361no—
438C42—591no—
439C42—2361no—
440C42—2361no—
441C42—2361no—
442C42D4×D5911yescharacteristic
443C16—21no—
444C16—592no—
445C16—1181no—
446C16C2×C4×D5912yes—
447C23—41no—
448C23—41no—
449C23—591no—
450C23—9441no—
451C23—9441no—
452C23—2364no—
453C23—2361no—
454Q8—1182no—
455Q8—2361no—
456D4—22no—
457D4—22no—
458D4—41no—
459D4—41no—
460D4—1182no—
461D4—2364no—
462D4—2364no—
463D4—2361no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—161no—
467C2×C4—161no—
468C2×C4—591no—
469C2×C4—592no—
470C2×C4—592no—
471C2×C4—591no—
472C2×C4—591no—
473C2×C4—9441no—
474C2×C4—9441no—
475C2×C4—9441no—
476C2×C4—9441no—
477C2×C4—9441no—
478C2×C4—2364no—
479C2×C4—2361no—
480C2×C4—2361no—
481C2×C4—2361no—
482C2×C4D8×D5911yescharacteristic
483C2×C4D8×D5911yescharacteristic
484C2×C4C2×D4×D5911yescharacteristic
485C8—21no—
486C8—592no—
487C8—1181no—
488C8C22×C4×D5911yescharacteristic
489C8#41×D5911yescharacteristic
490C22—42no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—592no—
497C22—1181no—
498C22—1181no—
499C22—9444no—
500C22—4721no—
501C22—4721no—
502C22—2362no—
503C22C2×D8×D5911yescharacteristic
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—592no—
509C4—1181no—
510C4—1181no—
511C4—4721no—
512C4—4721no—
513C4—2362no—
514C4C4×D4×D5911yescharacteristic
515C4#135×D5911yescharacteristic
516C2—82no—
517C2—82no—
518C2—1182no—
519C2—4722no—
520C2C4×D8×D5911yescharacteristic
521C2#2143723611yescharacteristic
522C2#2143724211yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutatorC8×C59C23×C4
FrattiniC2×C8C23×D59
FittingC59×#2177C2
socleC22×C59C22×D8
prime radicalC2×#16837836C2
cosocleC2×C8×C59C24
hypercenter#2177D59
hypocenterC59#33454
LCSG ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2177

Complements

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

NH ≅ G/Ntype
2. #21432966C2semidirect
3. #21432943C2semidirect
4. #21425969C2semidirect
5. #21425962C2semidirect
6. #21425957C2semidirect
7. #21425947C2semidirect
8. C4×#14273671C2semidirect
9. D16×Dic59C2semidirect
10. C4×#14273668C2semidirect
11. #21425883C2semidirect
12. #21425800C2semidirect
13. #357×D59C2semidirect
14. #21425694C2semidirect
15. C59×#2177C2semidirect
20. #16837836C4semidirect
30. C4×Dic236C22semidirect
31. #16837496C22semidirect
34. D8×Dic59C22semidirect
41. #16837229C22semidirect
42. #16837228C22semidirect
43. #16837227C22semidirect
44. #16837207C22semidirect
45. #16837205C22semidirect
48. #16837203C22semidirect
50. #16837177C22semidirect
51. C4×C59⋊C16C22semidirect
52. C16×Dic59C22semidirect
55. C4×D8×C59C22semidirect
56. C59×#367C22semidirect
58. C59×#357C22semidirect
59. C59×#346C22semidirect
63. #14273746C2×C4semidirect
69. D8×D59C2×C4semidirect
89. #14273672C2×C4semidirect
90. #14273671C2×C4semidirect
91. #14273668C2×C4semidirect
92. #14273666C2×C4semidirect
113. D16×C59C2×C4semidirect
259. #2177D59semidirect
299. C4×D8C2×D59semidirect
306. #367C2×D59semidirect
316. #357C2×D59semidirect
318. #346C2×D59semidirect
321. C59#33454semidirect
358. D16C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378362
#214329662
#214329432
#214259692
#214259622
#214259572
#214259472
C4×#142736712
D16×Dic592
C4×#142736682
#214258832
#214258002
#357×D592
#214256942
C59×#21772
#3345459