Finite Group Databasebeta

← #28743791

Subgroups — #28743791

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#21432949C211yesmaximal, characteristic, complemented
2#21426301C211yesmaximal, characteristic, complemented
3#21426284C211yesmaximal, characteristic, complemented
4#21426240C211yesmaximal, characteristic, complemented
5#21426230C211yesmaximal, characteristic, complemented
6#21426148C211yesmaximal, characteristic, complemented
7#21426148C211yesmaximal, characteristic, complemented
8#21426024C211yesmaximal, characteristic, complemented
9#21426007C211yesmaximal, characteristic
10#21425970C211yesmaximal, characteristic, complemented
11#21425958C211yesmaximal, characteristic, complemented
12C4×#14273670C211yesmaximal, characteristic, complemented
13C4×#14273668C211yesmaximal, characteristic, complemented
14#21425903C211yesmaximal, characteristic, complemented
15C59×#3175C211yesmaximal, characteristic, complemented
16C22×D472C2211yescharacteristic
17#16837876—21no—
18C2×#14273670C2211yescharacteristic
19C2×#14273670C2211yescharacteristic
20C2×#14273668C2211yescharacteristic
21C2×#14273668C2211yescharacteristic
22#16837515—21no—
23#16837515—21no—
24C4×D472C2211yescharacteristic
25C4×D472C2211yescharacteristic
26C4×D472C2211yescharacteristic
27C4×D472C2211yescharacteristic
28#16837440C2211yescharacteristic
29#16837408—21no—
30#16837229C2211yescharacteristic, complemented
31#16837229C2211yescharacteristic, complemented
32#16837227C2211yescharacteristic, complemented
33#16837227C2211yescharacteristic, complemented
34C4⋊(C59⋊C16)C2211yescharacteristic, complemented
35C4⋊(C59⋊C16)C2211yescharacteristic, complemented
36#16837206C2211yescharacteristic, complemented
37#16837206C2211yescharacteristic, complemented
38#16837203C2211yescharacteristic, complemented
39#16837203C2211yescharacteristic, complemented
40#16837202C2211yescharacteristic
41#16837202C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44C22⋊(C59⋊C16)C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46#16837178C2211yescharacteristic, complemented
47C4×C59⋊C16C2211yescharacteristic, complemented
48C59×#542—21no—
49C59×#500C2211yescharacteristic, complemented
50C59×#495C2211yescharacteristic, complemented
51C59×#489—21no—
52C59×#468—21no—
53C59×#467C2211yescharacteristic
54C59×#466C2211yescharacteristic
55C59×#440—21no—
56C4×Q16×C59C2211yescharacteristic, complemented
57C4×D8×C59C2211yescharacteristic, complemented
58C59×#405C2211yescharacteristic
59C22×D236—21no—
60C2×D472—21no—
61C2×D472—22no—
62C2×D472—22no—
63C2×D472C2311yescharacteristic
64C2×D472C2311yescharacteristic
65C4×D236—21no—
66C4×D236—21no—
67C4×D236—21no—
68C4×D236—21no—
69#14273689—41no—
70#14273689—41no—
71#14273686—21no—
72C22⋊C4×D59—41no—
73C22⋊C4×D59—41no—
74#14273670—22no—
75#14273670—22no—
76#14273668—22no—
77#14273668—22no—
78C2×C59⋊C16C2311yescharacteristic
79C2×C59⋊C16C2311yescharacteristic
80C2×C59⋊C16C2311yescharacteristic
81C2×C59⋊C16C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273661C2311yescharacteristic
84#14273659—21no—
85#14273649—21no—
86#14273649—21no—
87#14273649—21no—
88#14273649—21no—
89C2×Q16×C59C2311yescharacteristic
90C2×SD16×C59—21no—
91C2×D8×C59C2311yescharacteristic
92C2×M16×C59D411yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C4⋊Q8×C59—21no—
95#126×C59—41no—
96#124×C59—21no—
97#124×C59—41no—
98#123×C59—41no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59—41no—
101#121×C59—21no—
102C4×Q8×C59—21no—
103C4×D4×C59—21no—
104#117×C59D411yescharacteristic
105#107×C59—21no—
106#107×C59C2311yescharacteristic
107#107×C59C2311yescharacteristic
108#103×C59—21no—
109#103×C59—21no—
110#102×C59—21no—
111#102×C59—21no—
112#97×C59D411yescharacteristic
113#97×C59D411yescharacteristic
114C4×C8×C59C2311yescharacteristic
115C4×C8×C59C2311yescharacteristic
116C23×D59—41no—
117C2×Q8×C59—21no—
118C2×Q8×C59—41no—
119C2×D4×C59—21no—
120C2×D4×C59—41no—
121C22×C4×C59—41no—
122C22×C4×C59C2×D411yescharacteristic
123C2×D236—24no—
124C2×D236—21no—
125C2×D236—21no—
126C2×D236—42no—
127C2×D236—41no—
128C2×C4×D59—41no—
129C2×C4×D59—41no—
130C2×C4×D59—41no—
131C2×C4×D59—41no—
132Q16×C59—21no—
133Q16×C59—21no—
134SD16×C59—42no—
135D8×C59—21no—
136D8×C59—21no—
137M16×C59—41no—
138C2×C8×C59—21no—
139C2×C8×C59—21no—
140C2×C8×C59C2×D411yescharacteristic
141C2×C8×C59C2×D411yescharacteristic
142C2×C8×C59C2×D411yescharacteristic
143C2×C8×C59C2411yescharacteristic
144C4⋊C4×C59—21no—
145C4⋊C4×C59—21no—
146C4⋊C4×C59—41no—
147C4⋊C4×C59—41no—
148C4⋊C4×C59—41no—
149C4⋊C4×C59D811yescharacteristic
150C4⋊C4×C59D811yescharacteristic
151C22⋊C4×C59—41no—
152C22⋊C4×C59—41no—
153C22⋊C4×C59—41no—
154C22⋊C4×C59—41no—
155C22⋊C4×C59—41no—
156C22⋊C4×C59D811yescharacteristic
157C22⋊C4×C59D811yescharacteristic
158C42×C59—41no—
159C42×C59C2×D411yescharacteristic
160C42×C59C2×D411yescharacteristic
161C22⋊Dic59—41no—
162(C2×D59)⋊C4—41no—
163(C2×D59)⋊C4—41no—
164(C2×D59)⋊C4—41no—
165(C2×D59)⋊C4—41no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4×Dic59—41no—
169D472—24no—
170D472—21no—
171D472—21no—
172D472#4112yes—
173D472#4112yes—
174C59⋊C16—21no—
175C59⋊C16—21no—
176C59⋊C16—21no—
177C59⋊C16—21no—
178C23×C59—41no—
179C23×C59C2×D811yescharacteristic
180C22×D59—41no—
181C22×D59—41no—
182C22×D59—44no—
183C22×D59—161no—
184C22×D59—161no—
185Q8×C59—41no—
186Q8×C59—41no—
187Q8×C59—161no—
188D4×C59—41no—
189D4×C59—41no—
190D4×C59—161no—
191C2×C4×C59—21no—
192C2×C4×C59—21no—
193C2×C4×C59—41no—
194C2×C4×C59—41no—
195C2×C4×C59—41no—
196C2×C4×C59—161no—
197C2×C4×C59C2×D811yescharacteristic
198C2×C4×C59C2×D811yescharacteristic
199C2×C4×C59C2×D811yescharacteristic
200C2×C4×C59C2×D811yescharacteristic
201C2×C4×C59C2×D811yescharacteristic
202C2×C4×C59C22×D411yescharacteristic
203C2×Dic59—41no—
204C2×Dic59—41no—
205D236—22no—
206D236—22no—
207D236—44no—
208D236—41no—
209D236—41no—
210C4×D59—161no—
211C4×D59—161no—
212C4×D59—161no—
213C4×D59—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×C59—81no—
224C22×C59—81no—
225C22×C59—81no—
226C22×C59C22×D811yescharacteristic
227C2×D59—42no—
228C2×D59—42no—
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—81no—
240C4×C59—81no—
241C4×C59—81no—
242C4×C59#52511yescharacteristic
243C4×C59#57511yescharacteristic
244Dic59—81no—
245Dic59—81no—
246#3175D5911yescharacteristic, complemented
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#2164—591no—
261C2×C59—41no—
262C2×C59—161no—
263C2×C59#310011yescharacteristic
264C2×C59#341511yescharacteristic
265C2×C59#341611yescharacteristic
266D59—82no—
267D59—82no—
268C22×D8—591no—
269#542—21no—
270#518—1181no—
271C2×SD32—591no—
272C2×SD32—591no—
273C2×D16—591no—
274C2×D16—591no—
275#500C2×D5911yescharacteristic, complemented
276#495C2×D5911yescharacteristic, complemented
277#489—21no—
278#468—21no—
279#467C2×D5911yescharacteristic
280#466C2×D5911yescharacteristic
281#442—1181no—
282#442—1181no—
283#440—21no—
284C4×Q16C2×D5911yescharacteristic, complemented
285C4×D8—591no—
286C4×D8—591no—
287C4×D8—591no—
288C4×D8—591no—
289C4×D8C2×D5911yescharacteristic, complemented
290#427—591no—
291#418—1181no—
292#405C2×D5911yescharacteristic
293#367—591no—
294#367—591no—
295#366—591no—
296#366—591no—
297C4⋊C16—591no—
298C4⋊C16—591no—
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⋊C16—591no—
308C22⋊C16—591no—
309#346—591no—
310C4×C16—591no—
311C59#3367811yescharacteristic, complemented, Sylow
312C22×D4—1181no—
313C2×Q16C22×D5911yescharacteristic
314C2×SD16—21no—
315C2×D8—591no—
316C2×D8—591no—
317C2×D8—1182no—
318C2×D8—1182no—
319C2×D8—1181no—
320C2×D8C22×D5911yescharacteristic
321C2×M16C22⋊D5911yescharacteristic
322C22×C8C22×D5911yescharacteristic
323C4⋊Q8—21no—
324#126—41no—
325#124—21no—
326#124—41no—
327#123—41no—
328C22⋊Q8—21no—
329C22⋊Q8—41no—
330#121—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—1181no—
334C4×D4—1181no—
335C4×D4—1181no—
336C4×D4—1181no—
337C4×D4—2361no—
338C4×D4—2361no—
339#117—1181no—
340#117C22⋊D5911yescharacteristic
341C2×C22⋊C4—2361no—
342C2×C22⋊C4—2361no—
343SD32—1182no—
344SD32—1182no—
345D16—1182no—
346D16—1182no—
347C2×C16—591no—
348C2×C16—591no—
349C2×C16—591no—
350C2×C16—591no—
351#107—21no—
352#107—591no—
353#107—591no—
354#107C22×D5911yescharacteristic
355#107C22×D5911yescharacteristic
356#106—1181no—
357#103—21no—
358#103—21no—
359#102—21no—
360#102—21no—
361#102—1181no—
362#102—1181no—
363#102—1181no—
364#102—1181no—
365#97C22⋊D5911yescharacteristic
366#97C22⋊D5911yescharacteristic
367C4×C8C22×D5911yescharacteristic
368C4×C8C22×D5911yescharacteristic
369C24—2361no—
370C2×Q8—21no—
371C2×Q8—41no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—1184no—
375C2×D4—1181no—
376C2×D4—1181no—
377C2×D4—2362no—
378C2×D4—2361no—
379C22×C4—41no—
380C22×C4—2361no—
381C22×C4—2361no—
382C22×C4—2361no—
383C22×C4—2361no—
384C22×C4C2×C22⋊D5911yescharacteristic
385Q16—21no—
386Q16—21no—
387SD16—42no—
388D8—21no—
389D8—21no—
390D8—592no—
391D8—592no—
392D8—1184no—
393D8—1181no—
394D8—1181no—
395M16—41no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8C2×C22⋊D5911yescharacteristic
399C2×C8C2×C22⋊D5911yescharacteristic
400C2×C8C2×C22⋊D5911yescharacteristic
401C2×C8C23×D5911yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—1181no—
408C4⋊C4—1181no—
409C4⋊C4#1174566911yescharacteristic
410C4⋊C4#1174566911yescharacteristic
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—2361no—
417C22⋊C4—2361no—
418C22⋊C4—2361no—
419C22⋊C4—2361no—
420C22⋊C4—2361no—
421C22⋊C4#1174566911yescharacteristic
422C22⋊C4#1174566911yescharacteristic
423C42—41no—
424C42—2361no—
425C42C2×C22⋊D5911yescharacteristic
426C42C2×C22⋊D5911yescharacteristic
427C16—1181no—
428C16—1181no—
429C16—1181no—
430C16—1181no—
431C23—41no—
432C23—9441no—
433C23—9441no—
434C23—2364no—
435C23—2361no—
436C23—2361no—
437C23C2×#1174566911yescharacteristic
438Q8—41no—
439Q8—41no—
440Q8—161no—
441D4—41no—
442D4—41no—
443D4—161no—
444D4—1182no—
445D4—1182no—
446D4—2364no—
447D4—2361no—
448D4—2361no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—161no—
455C2×C4—9441no—
456C2×C4—9441no—
457C2×C4—9441no—
458C2×C4—9441no—
459C2×C4—2361no—
460C2×C4—2361no—
461C2×C4C2×#1174566911yescharacteristic
462C2×C4C2×#1174566911yescharacteristic
463C2×C4C2×#1174566911yescharacteristic
464C2×C4C2×#1174566911yescharacteristic
465C2×C4C2×#1174566911yescharacteristic
466C2×C4C22×C22⋊D5911yescharacteristic
467C8—21no—
468C8—21no—
469C8—21no—
470C8#41×D5911yescharacteristic
471C8#41×D5911yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—9444no—
479C22—4721no—
480C22—4721no—
481C22—4721no—
482C22—4721no—
483C22—2362no—
484C22—2362no—
485C22C22×#1174566911yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—4721no—
495C4—4721no—
496C4#1683790911yescharacteristic
497C4#1683820011yescharacteristic
498C2—41no—
499C2—161no—
500C2—4722no—
501C2—4722no—
502C2#2143409211yescharacteristic
503C2#2143723511yescharacteristic
504C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3175C2
socleC22×C59C22×D8
prime radical#21426007C2
hypercenter#3175D59
hypocenterC59#33678
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #21432949C2semidirect
2. #21426301C2semidirect
3. #21426284C2semidirect
4. #21426240C2semidirect
5. #21426230C2semidirect
6. #21426148C2semidirect
7. #21426148C2semidirect
8. #21426024C2semidirect
10. #21425970C2semidirect
11. #21425958C2semidirect
12. C4×#14273670C2semidirect
13. C4×#14273668C2semidirect
14. #21425903C2semidirect
15. C59×#3175C2semidirect
30. #16837229C22semidirect
31. #16837229C22semidirect
32. #16837227C22semidirect
33. #16837227C22semidirect
34. C4⋊(C59⋊C16)C22semidirect
35. C4⋊(C59⋊C16)C22semidirect
36. #16837206C22semidirect
37. #16837206C22semidirect
38. #16837203C22semidirect
39. #16837203C22semidirect
46. #16837178C22semidirect
47. C4×C59⋊C16C22semidirect
49. C59×#500C22semidirect
50. C59×#495C22semidirect
56. C4×Q16×C59C22semidirect
57. C4×D8×C59C22semidirect
246. #3175D59semidirect
275. #500C2×D59semidirect
276. #495C2×D59semidirect
284. C4×Q16C2×D59semidirect
289. C4×D8C2×D59semidirect
311. C59#33678semidirect

Maximal subgroups

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

subgroupindex
#214329492
#214263012
#214262842
#214262402
#214262302
#214261482
#214260242
#214260072
#214259702
#214259582
C4×#142736702
C4×#142736682
#214259032
C59×#31752
#3367859