Finite Group Databasebeta

← #28743799

Subgroups — #28743799

Order 15104.

568 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434735C211yesmaximal, characteristic, complemented
2#21432949C211yesmaximal, characteristic, complemented
3#21426305C211yesmaximal, characteristic, complemented
4#21426288C211yesmaximal, characteristic, complemented
5#21426244C211yesmaximal, characteristic, complemented
6#21426234C211yesmaximal, characteristic, complemented
7#21426152C211yesmaximal, characteristic, complemented
8#21426152C211yesmaximal, characteristic, complemented
9#21426027C211yesmaximal, characteristic, complemented
10#21426010C211yesmaximal, characteristic
11#21425973C211yesmaximal, characteristic, complemented
12#21425962C211yesmaximal, characteristic, complemented
13C4×#14273672C211yesmaximal, characteristic, complemented
14C4×D944C211yesmaximal, characteristic, complemented
15C59×#2164C211yesmaximal, characteristic, complemented
16C22×D472C2211yescharacteristic
17#16838023—21no—
18#16837876—21no—
19C2×#14273672C2211yescharacteristic
20C2×#14273672C2211yescharacteristic
21C2×D944C2211yescharacteristic
22C2×D944C2211yescharacteristic
23#16837807C2211yescharacteristic, complemented
24#16837786C2211yescharacteristic, complemented
25#16837762—21no—
26#16837648—21no—
27#16837643C2211yescharacteristic
28#16837638C2211yescharacteristic
29#16837515—21no—
30#16837515—21no—
31#16837504—21no—
32C4×Dic236C2211yescharacteristic, complemented
33C4×D472C2211yescharacteristic
34C4×D472C2211yescharacteristic
35C4×D472C2211yescharacteristic
36C4×D472C2211yescharacteristic
37C4×D472C2211yescharacteristic, complemented
38#16837440C2211yescharacteristic
39#16837408—21no—
40#16837230C2211yescharacteristic, complemented
41#16837230C2211yescharacteristic, complemented
42#16837228C2211yescharacteristic, complemented
43#16837228C2211yescharacteristic, complemented
44#16837207C2211yescharacteristic, complemented
45#16837207C2211yescharacteristic, complemented
46#16837204C2211yescharacteristic
47#16837204C2211yescharacteristic
48#16837204C2211yescharacteristic
49#16837204C2211yescharacteristic
50#16837204C2211yescharacteristic, complemented
51#16837204C2211yescharacteristic, complemented
52C59×#405C2211yescharacteristic
53C59×C4⋊C16C2211yescharacteristic, complemented
54C59×C4⋊C16C2211yescharacteristic, complemented
55C59×C22⋊C16C2211yescharacteristic
56C59×C22⋊C16C2211yescharacteristic
57C59×#346C2211yescharacteristic, complemented
58C4×C16×C59C2211yescharacteristic, complemented
59C22×D236—21no—
60C2×Dic236C2311yescharacteristic
61C2×#11745660—21no—
62C2×D472—21no—
63C2×D472—22no—
64C2×D472—22no—
65C2×D472C2311yescharacteristic
66C2×D472C2311yescharacteristic
67C2×D472C2311yescharacteristic
68#14273727—21no—
69#14273722—41no—
70#14273715—21no—
71#14273713—41no—
72#14273709—41no—
73#14273706—21no—
74#14273705—41no—
75#14273701—21no—
76C4×Dic118—21no—
77C4×D236—21no—
78C4×D236—21no—
79C4×D236—21no—
80C4×D236—21no—
81C4×D236—21no—
82#14273689—41no—
83#14273689—41no—
84#14273686—21no—
85C22⋊C4×D59—41no—
86C22⋊C4×D59—41no—
87#14273672—22no—
88#14273672—22no—
89D944—22no—
90D944—22no—
91#14273661—21no—
92#14273661C2311yescharacteristic
93#14273661C2311yescharacteristic
94#14273661C2311yescharacteristic
95#14273661C2311yescharacteristic
96#14273659—21no—
97#14273652—21no—
98#14273652—21no—
99#14273649—21no—
100#14273649—21no—
101#14273649—21no—
102#14273649—21no—
103#14273649—21no—
104#14273649—21no—
105C2×M16×C59D411yescharacteristic
106C22×C8×C59C2311yescharacteristic
107#117×C59D411yescharacteristic
108C2×C16×C59C2311yescharacteristic
109C2×C16×C59C2311yescharacteristic
110C2×C16×C59C2311yescharacteristic
111C2×C16×C59C2311yescharacteristic
112#97×C59D411yescharacteristic
113#97×C59D411yescharacteristic
114C4×C8×C59C2311yescharacteristic
115C4×C8×C59C2311yescharacteristic
116C23×D59—41no—
117C22×C4×C59C2×D411yescharacteristic
118C2×C22⋊D59—41no—
119C2×D236—24no—
120C2×D236—21no—
121C2×D236—21no—
122C2×D236—21no—
123C2×D236—42no—
124C2×D236—41no—
125C2×C4×D59—41no—
126C2×C4×D59—41no—
127C2×C4×D59—41no—
128C2×C4×D59—41no—
129C2×C4×D59—41no—
130C2×Dic118—21no—
131C2×Dic118—41no—
132M16×C59—41no—
133C2×C8×C59—21no—
134C2×C8×C59—21no—
135C2×C8×C59C2×D411yescharacteristic
136C2×C8×C59C2×D411yescharacteristic
137C2×C8×C59C2×D411yescharacteristic
138C2×C8×C59C2411yescharacteristic
139C4⋊C4×C59D811yescharacteristic
140C4⋊C4×C59D811yescharacteristic
141C22⋊C4×C59D811yescharacteristic
142C22⋊C4×C59D811yescharacteristic
143C42×C59C2×D411yescharacteristic
144C42×C59C2×D411yescharacteristic
145C22⋊Dic59—41no—
146C22⋊Dic59—41no—
147(C2×D59)⋊C4—41no—
148(C2×D59)⋊C4—41no—
149(C2×D59)⋊C4—41no—
150(C2×D59)⋊C4—41no—
151(C2×D59)⋊C4—41no—
152(C2×D59)⋊C4—41no—
153(C2×D59)⋊C4—41no—
154(C2×D59)⋊C4—41no—
155C4⋊Dic59—21no—
156C4⋊Dic59—21no—
157C4⋊Dic59—21no—
158C4⋊Dic59—21no—
159C4⋊Dic59—41no—
160Dic59⋊C4—41no—
161Dic59⋊C4—41no—
162C4×Dic59—41no—
163C4×Dic59—41no—
164Dic236—21no—
165Dic236—21no—
166D472—24no—
167D472—21no—
168D472—21no—
169D472—21no—
170D472—21no—
171D472#4112yes—
172D472#4112yes—
173#11745660—42no—
174C16×C59—21no—
175C16×C59—21no—
176C16×C59—21no—
177C16×C59—21no—
178C23×C59C2×D811yescharacteristic
179C22×D59—41no—
180C22×D59—41no—
181C22×D59—44no—
182C22×D59—41no—
183C22×D59—161no—
184C22×D59—161no—
185C2×C4×C59—21no—
186C2×C4×C59—21no—
187C2×C4×C59C2×D811yescharacteristic
188C2×C4×C59C2×D811yescharacteristic
189C2×C4×C59C2×D811yescharacteristic
190C2×C4×C59C2×D811yescharacteristic
191C2×C4×C59C2×D811yescharacteristic
192C2×C4×C59C22×D411yescharacteristic
193C22⋊D59—161no—
194C2×Dic59—41no—
195C2×Dic59—41no—
196C2×Dic59—41no—
197C2×Dic59—41no—
198C2×Dic59—41no—
199D236—22no—
200D236—22no—
201D236—44no—
202D236—41no—
203D236—41no—
204D236—41no—
205D236—41no—
206C4×D59—161no—
207C4×D59—161no—
208C4×D59—161no—
209C4×D59—161no—
210C4×D59—161no—
211Dic118—41no—
212Dic118—41no—
213Dic118—161no—
214C8×C59—21no—
215C8×C59—21no—
216C8×C59—21no—
217C8×C59C2×#4111yescharacteristic
218C8×C59C2×#4111yescharacteristic
219#33678—591nomaximal, Sylow
220C22×C59—21no—
221C22×C59—21no—
222C22×C59—21no—
223C22×C59C22×D811yescharacteristic
224C2×D59—42no—
225C2×D59—42no—
226C2×D59—81no—
227C2×D59—81no—
228C2×D59—81no—
229C2×D59—81no—
230C2×D59—81no—
231C2×D59—81no—
232C2×D59—81no—
233C2×D59—164no—
234C4×C59—21no—
235C4×C59—21no—
236C4×C59—21no—
237C4×C59—21no—
238C4×C59—21no—
239C4×C59#52511yescharacteristic
240C4×C59#57511yescharacteristic
241Dic59—81no—
242Dic59—81no—
243Dic59—81no—
244Dic59—81no—
245Dic59—81no—
246#3175—591no—
247#2948—591no—
248#2237—591no—
249#2234—591no—
250#2226—591no—
251#2224—591no—
252#2210—591no—
253#2210—591no—
254#2187—591no—
255#2184—591no—
256#2177—591no—
257#2175—591no—
258C4×SD32—591no—
259C4×D16—591no—
260#2164D5911yescharacteristic, complemented
261C2×C59—41no—
262C2×C59#310011yescharacteristic
263C2×C59#341511yescharacteristic
264C2×C59#341611yescharacteristic
265D59—82no—
266D59—82no—
267D59—161no—
268C22×D8—591no—
269#542—1181no—
270#518—1181no—
271C2×SD32—591no—
272C2×SD32—591no—
273C2×D16—591no—
274C2×D16—591no—
275#500—591no—
276#495—591no—
277#489—1181no—
278#468—1181no—
279#467—591no—
280#466—591no—
281#442—1181no—
282#442—1181no—
283#440—1181no—
284C4×Q16—591no—
285C4×D8—591no—
286C4×D8—591no—
287C4×D8—591no—
288C4×D8—591no—
289C4×D8—591no—
290#427—591no—
291#418—1181no—
292#405C2×D5911yescharacteristic
293#367—591no—
294#367—591no—
295#366—591no—
296#366—591no—
297C4⋊C16C2×D5911yescharacteristic, complemented
298C4⋊C16C2×D5911yescharacteristic, complemented
299#358—591no—
300#358—591no—
301#357—591no—
302#357—591no—
303#357—591no—
304#357—591no—
305#357—591no—
306#357—591no—
307C22⋊C16C2×D5911yescharacteristic
308C22⋊C16C2×D5911yescharacteristic
309#346C2×D5911yescharacteristic, complemented
310C4×C16C2×D5911yescharacteristic, complemented
311C59#3367811yescharacteristic, complemented, Sylow
312C22×D4—1181no—
313C2×Q16—591no—
314C2×SD16—1181no—
315C2×D8—591no—
316C2×D8—591no—
317C2×D8—591no—
318C2×D8—1182no—
319C2×D8—1182no—
320C2×D8—1181no—
321C2×M16D23611yescharacteristic
322C22×C8C22×D5911yescharacteristic
323C4⋊Q8—1181no—
324#126—2361no—
325#124—1181no—
326#124—2361no—
327#123—2361no—
328C22⋊Q8—1181no—
329C22⋊Q8—2361no—
330#121—1181no—
331C4×Q8—1181no—
332C4×D4—1181no—
333C4×D4—1181no—
334C4×D4—1181no—
335C4×D4—1181no—
336C4×D4—1181no—
337C4×D4—2361no—
338C4×D4—2361no—
339#117—1181no—
340#117D23611yescharacteristic
341C2×C22⋊C4—2361no—
342C2×C22⋊C4—2361no—
343SD32—1182no—
344SD32—1182no—
345D16—1182no—
346D16—1182no—
347C2×C16C22×D5911yescharacteristic
348C2×C16C22×D5911yescharacteristic
349C2×C16C22×D5911yescharacteristic
350C2×C16C22×D5911yescharacteristic
351#107—591no—
352#107—591no—
353#107—591no—
354#107—591no—
355#107—1181no—
356#106—1181no—
357#103—1181no—
358#103—1181no—
359#102—1181no—
360#102—1181no—
361#102—1181no—
362#102—1181no—
363#102—1181no—
364#102—1181no—
365#97D23611yescharacteristic
366#97D23611yescharacteristic
367C4×C8C22×D5911yescharacteristic
368C4×C8C22×D5911yescharacteristic
369C24—2361no—
370C2×Q8—1181no—
371C2×Q8—2361no—
372C2×D4—1184no—
373C2×D4—1181no—
374C2×D4—1181no—
375C2×D4—1181no—
376C2×D4—2362no—
377C2×D4—2361no—
378C2×D4—2361no—
379C22×C4—2361no—
380C22×C4—2361no—
381C22×C4—2361no—
382C22×C4—2361no—
383C22×C4—2361no—
384C22×C4C2×D23611yescharacteristic
385Q16—1181no—
386Q16—1181no—
387SD16—2362no—
388D8—592no—
389D8—592no—
390D8—1184no—
391D8—1181no—
392D8—1181no—
393D8—1181no—
394D8—1181no—
395M16—41no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8C2×D23611yescharacteristic
399C2×C8C2×D23611yescharacteristic
400C2×C8C2×D23611yescharacteristic
401C2×C8C23×D5911yescharacteristic
402C4⋊C4—1181no—
403C4⋊C4—1181no—
404C4⋊C4—1181no—
405C4⋊C4—1181no—
406C4⋊C4—2361no—
407C4⋊C4—2361no—
408C4⋊C4—2361no—
409C4⋊C4D47211yescharacteristic
410C4⋊C4D47211yescharacteristic
411C22⋊C4—2361no—
412C22⋊C4—2361no—
413C22⋊C4—2361no—
414C22⋊C4—2361no—
415C22⋊C4—2361no—
416C22⋊C4—2361no—
417C22⋊C4—2361no—
418C22⋊C4—2361no—
419C22⋊C4—2361no—
420C22⋊C4—2361no—
421C22⋊C4D47211yescharacteristic
422C22⋊C4D47211yescharacteristic
423C42—2361no—
424C42—2361no—
425C42C2×D23611yescharacteristic
426C42C2×D23611yescharacteristic
427C16—21no—
428C16—21no—
429C16—21no—
430C16—21no—
431C23—9441no—
432C23—9441no—
433C23—2364no—
434C23—2361no—
435C23—2361no—
436C23—2361no—
437C23C2×D47211yescharacteristic
438Q8—9441no—
439Q8—2361no—
440Q8—2361no—
441D4—1182no—
442D4—1182no—
443D4—9441no—
444D4—2364no—
445D4—2361no—
446D4—2361no—
447D4—2361no—
448D4—2361no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—9441no—
452C2×C4—9441no—
453C2×C4—9441no—
454C2×C4—9441no—
455C2×C4—9441no—
456C2×C4—2361no—
457C2×C4—2361no—
458C2×C4—2361no—
459C2×C4—2361no—
460C2×C4—2361no—
461C2×C4C2×D47211yescharacteristic
462C2×C4C2×D47211yescharacteristic
463C2×C4C2×D47211yescharacteristic
464C2×C4C2×D47211yescharacteristic
465C2×C4C2×D47211yescharacteristic
466C2×C4C22×D23611yescharacteristic
467C8—21no—
468C8—21no—
469C8—21no—
470C8#41×D5911yescharacteristic
471C8#41×D5911yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—9444no—
476C22—4721no—
477C22—4721no—
478C22—4721no—
479C22—4721no—
480C22—4721no—
481C22—4721no—
482C22—4721no—
483C22—2362no—
484C22—2362no—
485C22C22×D47211yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—4721no—
492C4—4721no—
493C4—4721no—
494C4—4721no—
495C4—4721no—
496C4#1683791411yescharacteristic
497C4#1683820211yescharacteristic
498C2—41no—
499C2—9441no—
500C2—4722no—
501C2—4722no—
502C2#2143409711yescharacteristic
503C2#2143723711yescharacteristic
504C2#2143724411yescharacteristic
subgroupG/H
centerC22C22×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2164C2
socleC22×C59C22×D8
prime radical#21426010C2
hypercenter#2164D59
hypocenterC59#33678
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2164

Complements

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

NH ≅ G/Ntype
1. #21434735C2semidirect
2. #21432949C2semidirect
3. #21426305C2semidirect
4. #21426288C2semidirect
5. #21426244C2semidirect
6. #21426234C2semidirect
7. #21426152C2semidirect
8. #21426152C2semidirect
9. #21426027C2semidirect
11. #21425973C2semidirect
12. #21425962C2semidirect
13. C4×#14273672C2semidirect
14. C4×D944C2semidirect
15. C59×#2164C2semidirect
23. #16837807C22semidirect
24. #16837786C22semidirect
32. C4×Dic236C22semidirect
37. C4×D472C22semidirect
40. #16837230C22semidirect
41. #16837230C22semidirect
42. #16837228C22semidirect
43. #16837228C22semidirect
44. #16837207C22semidirect
45. #16837207C22semidirect
50. #16837204C22semidirect
51. #16837204C22semidirect
53. C59×C4⋊C16C22semidirect
54. C59×C4⋊C16C22semidirect
57. C59×#346C22semidirect
58. C4×C16×C59C22semidirect
260. #2164D59semidirect
297. C4⋊C16C2×D59semidirect
298. C4⋊C16C2×D59semidirect
309. #346C2×D59semidirect
310. C4×C16C2×D59semidirect
311. C59#33678semidirect

Maximal subgroups

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

subgroupindex
#214347352
#214329492
#214263052
#214262882
#214262442
#214262342
#214261522
#214260272
#214260102
#214259732
#214259622
C4×#142736722
C4×D9442
C59×#21642
#3367859