Finite Group Databasebeta

← #28574092

Subgroups — #28574092

Order 15104.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435039C211yesmaximal, characteristic, complemented
2#21433884C211yesmaximal, characteristic, complemented
3#21433576C211yesmaximal, characteristic
4#21423512C212yesmaximal, complemented
5#21423505C212yesmaximal, complemented
6#21423444C212yesmaximal, complemented
7#21423437C212yesmaximal, complemented
8#21422867C211yesmaximal, characteristic, complemented
9#21422784C211yesmaximal, characteristic, complemented
10C2×#16837145C211yesmaximal, characteristic, complemented
11C59×#2988C211yesmaximal, characteristic, complemented
12C2×#14273752—21no—
13#16837959C2211yescharacteristic
14C2×#14273724C2211yescharacteristic
15#16837775C2212yes—
16#16837749C2212yes—
17#16837709—21no—
18#16837702—21no—
19#16837697C2211yescharacteristic, complemented
20#16837692C2211yescharacteristic, complemented
21#16837687C2211yescharacteristic, complemented
22#16837682C2211yescharacteristic, complemented
23#16837603—21no—
24#16837603—21no—
25#16837602—21no—
26#16837602C2212yes—
27#16837597—21no—
28#16837597C2212yes—
29#16837555—22no—
30#16837438—21no—
31#16837426C2211yescharacteristic
32C2×C4⋊(C59⋊C8)C2211yescharacteristic
33#16837406—21no—
34#16837398—21no—
35#16837153C2212yescomplemented
36#16837153C2212yescomplemented
37#16837150C2212yescomplemented
38#16837149C2212yescomplemented
39#16837145C2212yes—
40#16837145C2212yes—
41C4×#41×C59C2211yescharacteristic
42C8×Q8×C59C2212yescomplemented
43C8×D4×C59—22no—
44C8×D4×C59C2212yescomplemented
45C59×#431—21no—
46C59×#431C2211yescharacteristic
47C2×C4⋊C8×C59—21no—
48C2×C4⋊C8×C59C2211yescharacteristic
49C22×D236—21no—
50C22×D236—21no—
51C22×D236—21no—
52#14273752—44no—
53C2×#11745660—22no—
54C2×#11745660—22no—
55C2×#11745671—22no—
56C2×#11745669—22no—
57C2×#11745664—21no—
58C22×C59⋊C8—21no—
59#14273727C2311yescharacteristic
60#14273727C2311yescharacteristic
61#14273724—21no—
62#14273724—21no—
63#14273724—21no—
64#14273724C2312yes—
65#14273719—21no—
66#14273706—21no—
67#14273706—21no—
68C4×Dic118—21no—
69C4×Dic118—21no—
70#14273686—21no—
71#14273660—21no—
72#14273660—21no—
73#14273658—21no—
74#14273658—21no—
75C4⋊(C59⋊C8)—21no—
76C4⋊(C59⋊C8)—21no—
77C4⋊(C59⋊C8)C2311yescharacteristic
78C4⋊(C59⋊C8)C2311yescharacteristic
79C4⋊(C59⋊C8)C2312yes—
80#14273651—21no—
81#14273651—21no—
82#14273649—22no—
83#14273649—22no—
84#14273648—21no—
85#14273648—21no—
86#14273647—22no—
87#14273647—22no—
88C2×#41×C59D411yescharacteristic
89C22×C8×C59—21no—
90C22×C8×C59—22no—
91C4×Q8×C59D411yescharacteristic
92C4×Q8×C59C2311yescharacteristic
93C4×D4×C59—21no—
94C4×D4×C59—21no—
95C4×D4×C59D411yescharacteristic
96C4×D4×C59C2311yescharacteristic
97#117×C59—21no—
98#117×C59D411yescharacteristic
99C2×C42×C59—21no—
100C2×C42×C59C2311yescharacteristic
101C4⋊C8×C59—22no—
102C4⋊C8×C59—21no—
103C4⋊C8×C59D411yescharacteristic
104C4⋊C8×C59D411yescharacteristic
105C4⋊C8×C59C2312yes—
106C22⋊C8×C59—22no—
107C22⋊C8×C59D412yes—
108C4×C8×C59—22no—
109C4×C8×C59C2312yes—
110C23×D59—41no—
111#41×C59—22no—
112#41×C59SD1614yescomplemented
113C2×Q8×C59C2×D411yescharacteristic
114C2×D4×C59—21no—
115C2×D4×C59C2×D411yescharacteristic
116C22×C4×C59—21no—
117C22×C4×C59—21no—
118C22×C4×C59—21no—
119C22×C4×C59C2×D411yescharacteristic
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59#4111yescharacteristic
122C2×D236—22no—
123C2×D236—22no—
124C2×D236—22no—
125C2×D236—42no—
126C2×D236—42no—
127C2×D236—41no—
128C2×D236—41no—
129C2×D236—41no—
130C2×D236—81no—
131C2×D236—81no—
132C2×D236—82no—
133C2×Dic118—21no—
134C2×Dic118—21no—
135C2×C8×C59—22no—
136C2×C8×C59—82no—
137C2×C8×C59—81no—
138C2×C8×C59C2×D412yes—
139C2×C8×C59C2×D412yes—
140C4⋊C4×C59—21no—
141C4⋊C4×C59—21no—
142C4⋊C4×C59C2×D411yescharacteristic
143C4⋊C4×C59C2×D411yescharacteristic
144C22⋊C4×C59—21no—
145C22⋊C4×C59—21no—
146C22⋊C4×C59—21no—
147C42×C59—21no—
148C42×C59—21no—
149C42×C59—21no—
150C42×C59C2×D411yescharacteristic
151C42×C59C2×D411yescharacteristic
152C42×C59#4111yescharacteristic
153C42×C59C2411yescharacteristic
154C22⋊Dic59—41no—
155#11745671—44no—
156#11745669—44no—
157C4⋊Dic59—21no—
158C4⋊Dic59—21no—
159C4⋊Dic59—21no—
160C4⋊Dic59—21no—
161Dic59⋊C4—41no—
162Dic59⋊C4—41no—
163C4×Dic59—41no—
164#11745664—42no—
165C2×C59⋊C8—22no—
166C2×C59⋊C8—21no—
167C2×C59⋊C8—21no—
168C2×C59⋊C8—42no—
169#11745660—82no—
170#11745660—82no—
171C23×C59—21no—
172C23×C59#12111yescharacteristic
173C22×D59—42no—
174C22×D59—82no—
175C22×D59—82no—
176C22×D59—82no—
177Q8×C59—21no—
178Q8×C59C2×SD1612yes—
179D4×C59—24no—
180D4×C59—21no—
181D4×C59C2×SD1612yes—
182C2×C4×C59—21no—
183C2×C4×C59—21no—
184C2×C4×C59—21no—
185C2×C4×C59—24no—
186C2×C4×C59—21no—
187C2×C4×C59—21no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—81no—
192C2×C4×C59#12111yescharacteristic
193C2×C4×C59#12111yescharacteristic
194C2×C4×C59#12111yescharacteristic
195C2×C4×C59C2×SD1612yes—
196C2×C4×C59#13512yes—
197C2×C4×C59C22×D411yescharacteristic
198C2×C4×C59C22×D411yescharacteristic
199C2×C4×C59C2×#4111yescharacteristic
200C2×Dic59—41no—
201C2×Dic59—41no—
202D236—42no—
203D236—42no—
204D236—42no—
205D236—42no—
206D236—82no—
207D236—82no—
208D236—82no—
209D236—82no—
210Dic118—41no—
211Dic118—41no—
212Dic118—81no—
213C8×C59—22no—
214C8×C59—22no—
215C8×C59—42no—
216C59⋊C8—42no—
217C59⋊C8—41no—
218C59⋊C8—41no—
219#19409—591nomaximal, Sylow
220C22×C59—21no—
221C22×C59—21no—
222C22×C59—22no—
223C22×C59—41no—
224C22×C59—41no—
225C22×C59#46512yes—
226C22×C59C2×#12111yescharacteristic
227C2×D59—82no—
228C2×D59—82no—
229C2×D59—82no—
230C2×D59—164no—
231C4×C59—21no—
232C4×C59—22no—
233C4×C59—21no—
234C4×C59—21no—
235C4×C59—41no—
236C4×C59—41no—
237C4×C59#46512yes—
238C4×C59C22×SD1611yescharacteristic
239C4×C59C2×#13511yescharacteristic
240C4×C59#57611yescharacteristic
241C4×C59#57711yescharacteristic
242Dic59—81no—
243Dic59—81no—
244#3208—591no—
245#3076—591no—
246#3038—591no—
247#2988D5911yescharacteristic, complemented
248#1678—592no—
249#1677—592no—
250#1666—592no—
251#1665—592no—
252#1570—591no—
253#1550—591no—
254C2×#331—591no—
255C2×C59—22no—
256C2×C59—42no—
257C2×C59#304511yescharacteristic
258C2×C59C2×#46511yescharacteristic
259C2×C59#306811yescharacteristic
260D59—162no—
261C2×AGL(1,ℤ/8)—1181no—
262#533—591no—
263C2×#127—591no—
264C4×#41C2×D5911yescharacteristic
265#492—592no—
266#486—592no—
267#479—1181no—
268#478—1181no—
269#477—591no—
270#476—591no—
271#475—591no—
272#474—591no—
273#460—592no—
274#460—1181no—
275#460—1181no—
276#460—1181no—
277#459—592no—
278#459—1181no—
279#450—1182no—
280C8×Q8C2×D5912yescomplemented
281C8×D4—22no—
282C8×D4C2×D5912yescomplemented
283#431—21no—
284#431C2×D5911yescharacteristic
285#427—1181no—
286#423—591no—
287C2×C4⋊C8—21no—
288C2×C4⋊C8—591no—
289C2×C4⋊C8C2×D5911yescharacteristic
290#418—1181no—
291#416—1181no—
292#334—592no—
293#334—592no—
294#333—592no—
295#332—592no—
296#331—592no—
297#331—592no—
298C59#1940911yescharacteristic, complemented, Sylow
299C2×#41D23611yescharacteristic
300C22×D4—1181no—
301C22×D4—1181no—
302C22×D4—1181no—
303AGL(1,ℤ/8)—2364no—
304C2×SD16—1182no—
305C2×SD16—1182no—
306C2×SD16—1182no—
307C2×D8—1182no—
308C2×M16—1181no—
309C22×C8—21no—
310C22×C8—22no—
311C22×C8—1181no—
312C4⋊Q8—591no—
313C4⋊Q8—591no—
314#127—592no—
315#127—1181no—
316#127—1181no—
317#127—1181no—
318#125—1181no—
319C22⋊Q8—1181no—
320C22⋊Q8—1181no—
321C4×Q8—1181no—
322C4×Q8—1181no—
323C4×Q8D23611yescharacteristic
324C4×Q8C22×D5911yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4D23611yescharacteristic
328C4×D4C22×D5911yescharacteristic
329#117—21no—
330#117—1181no—
331#117D23611yescharacteristic
332C2×C42—21no—
333C2×C42C22×D5911yescharacteristic
334#107—1181no—
335#107—1181no—
336#106—1181no—
337#106—1181no—
338C4⋊C8—22no—
339C4⋊C8—21no—
340C4⋊C8—591no—
341C4⋊C8—591no—
342C4⋊C8—592no—
343C4⋊C8—1181no—
344C4⋊C8—1181no—
345C4⋊C8C22⋊D5911yescharacteristic
346C4⋊C8C22⋊D5911yescharacteristic
347C4⋊C8C22×D5912yes—
348#103—1181no—
349#103—1181no—
350#102—1181no—
351#102—1181no—
352#102—1182no—
353#102—1182no—
354#102—1182no—
355#102—1182no—
356C22⋊C8—22no—
357C22⋊C8C22⋊D5912yes—
358C4×C8—22no—
359C4×C8C22×D5912yes—
360C24—2361no—
361#41—22no—
362#41#1174566014yescomplemented
363C2×Q8—1181no—
364C2×Q8—1181no—
365C2×Q8C2×D23611yescharacteristic
366C2×D4—21no—
367C2×D4—1182no—
368C2×D4—1182no—
369C2×D4—1182no—
370C2×D4—4721no—
371C2×D4—4721no—
372C2×D4—4722no—
373C2×D4—2361no—
374C2×D4—2361no—
375C2×D4—2361no—
376C2×D4—2362no—
377C2×D4—2362no—
378C2×D4C2×D23611yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4C2×D23611yescharacteristic
383C22×C4#1174568511yescharacteristic
384C22×C4C2×C22⋊D5911yescharacteristic
385SD16—4722no—
386SD16—4722no—
387SD16—2364no—
388D8—2364no—
389M16—2362no—
390C2×C8—22no—
391C2×C8—82no—
392C2×C8—81no—
393C2×C8—1182no—
394C2×C8—1181no—
395C2×C8—1181no—
396C2×C8—2362no—
397C2×C8C2×C22⋊D5912yes—
398C2×C8C2×C22⋊D5912yes—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—1181no—
402C4⋊C4—1181no—
403C4⋊C4—1181no—
404C4⋊C4—1181no—
405C4⋊C4—2361no—
406C4⋊C4—2361no—
407C4⋊C4C2×D23611yescharacteristic
408C4⋊C4C2×D23611yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—2361no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—2361no—
417C42C2×D23611yescharacteristic
418C42#1174568511yescharacteristic
419C42C2×C22⋊D5911yescharacteristic
420C42C23×D5911yescharacteristic
421C23—21no—
422C23—4722no—
423C23—4722no—
424C23—4722no—
425C23—2362no—
426C23#1427370111yescharacteristic
427Q8—21no—
428Q8—4721no—
429Q8—2361no—
430Q8—2361no—
431Q8C2×#1174566012yes—
432D4—24no—
433D4—21no—
434D4—4722no—
435D4—4722no—
436D4—4722no—
437D4—4722no—
438D4—2362no—
439D4—2362no—
440D4—2362no—
441D4—2362no—
442D4C2×#1174566012yes—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—2361no—
454C2×C4—2361no—
455C2×C4#1427370111yescharacteristic
456C2×C4#1427370111yescharacteristic
457C2×C4#1427370111yescharacteristic
458C2×C4C2×#1174566012yes—
459C2×C4#1427374912yes—
460C2×C4C22×C22⋊D5911yescharacteristic
461C2×C4C22×D23611yescharacteristic
462C2×C4C2×#1174568511yescharacteristic
463C8—22no—
464C8—22no—
465C8—42no—
466C8—2362no—
467C8—2361no—
468C8—2361no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—9444no—
475C22—4722no—
476C22—4722no—
477C22—4722no—
478C22#1683763112yes—
479C22C2×#1427370111yescharacteristic
480C4—21no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—4721no—
487C4—4721no—
488C4#1683763112yes—
489C4C22×#1174566011yescharacteristic
490C4C2×#1427374911yescharacteristic
491C4#1683820511yescharacteristic
492C4#1683821011yescharacteristic
493C2—22no—
494C2—42no—
495C2—9442no—
496C2#2143363511yescharacteristic
497C2C2×#1683763111yescharacteristic
498C2#2143382411yescharacteristic
subgroupG/H
centerC22C2×#14273701
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC59×#2988C2
socleC22×C59C2×#121
prime radical#21433576C2
hypercenter#2988D59
hypocenterC59#19409
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #21435039C2semidirect
2. #21433884C2semidirect
4. #21423512C2semidirect
5. #21423505C2semidirect
6. #21423444C2semidirect
7. #21423437C2semidirect
8. #21422867C2semidirect
9. #21422784C2semidirect
10. C2×#16837145C2semidirect
11. C59×#2988C2semidirect
19. #16837697C22semidirect
20. #16837692C22semidirect
21. #16837687C22semidirect
22. #16837682C22semidirect
35. #16837153C22semidirect
36. #16837153C22semidirect
37. #16837150C22semidirect
38. #16837149C22semidirect
42. C8×Q8×C59C22semidirect
44. C8×D4×C59C22semidirect
112. #41×C59SD16semidirect
247. #2988D59semidirect
280. C8×Q8C2×D59semidirect
282. C8×D4C2×D59semidirect
298. C59#19409semidirect
362. #41#11745660semidirect

Maximal subgroups

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

subgroupindex
#214350392
#214338842
#214335762
#214235122
#214235052
#214234442
#214234372
#214228672
#214227842
C2×#168371452
C59×#29882
#1940959