Finite Group Databasebeta

← #28744246

Subgroups — #28744246

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434677C211yesmaximal, characteristic, complemented
2#21434670C211yesmaximal, characteristic
3#21426377C211yesmaximal, characteristic, complemented
4#21426350C211yesmaximal, characteristic, complemented
5#21426322C211yesmaximal, characteristic, complemented
6#21426318C211yesmaximal, characteristic, complemented
7#21426194C211yesmaximal, characteristic, complemented
8#21426192C211yesmaximal, characteristic, complemented
9#21426190C211yesmaximal, characteristic, complemented
10#21425951C211yesmaximal, characteristic, complemented
11SD32×Dic59C211yesmaximal, characteristic, complemented
12#21425947C211yesmaximal, characteristic, complemented
13#21425945C211yesmaximal, characteristic, complemented
14C2×C16×Dic59C211yesmaximal, characteristic, complemented
15C59×#2217C211yesmaximal, characteristic, complemented
16#16837954—21no—
17#16837947—21no—
18C2×#14273672C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20C2×#14273670C2211yescharacteristic
21C22×C59⋊C16C2211yescharacteristic
22#16837806C2211yescharacteristic
23#16837806C2211yescharacteristic
24#16837780C2211yescharacteristic
25#16837777C2211yescharacteristic
26#16837751—21no—
27#16837747—21no—
28#16837643C2211yescharacteristic
29#16837641—21no—
30#16837638C2211yescharacteristic
31#16837636—21no—
32Q16×Dic59C2211yescharacteristic
33#16837496C2211yescharacteristic
34#16837494—21no—
35D8×Dic59C2211yescharacteristic
36#16837484C2211yescharacteristic
37#16837482—21no—
38C2×C8×Dic59C2211yescharacteristic
39#16837230C2211yescharacteristic
40#16837229C2211yescharacteristic
41#16837229C2211yescharacteristic
42#16837207C2211yescharacteristic
43#16837206C2211yescharacteristic
44#16837205C2211yescharacteristic
45#16837204C2211yescharacteristic
46#16837203C2211yescharacteristic
47#16837202C2211yescharacteristic
48C16×Dic59C2211yescharacteristic
49C16×Dic59C2211yescharacteristic
50C16×Dic59C2211yescharacteristic
51C16×Dic59C2211yescharacteristic
52C2×SD32×C59C2211yescharacteristic
53C22×C16×C59C2211yescharacteristic
54C59×#467C2211yescharacteristic
55C59×#466C2211yescharacteristic
56C59×#367C2211yescharacteristic
57C59×#358C2211yescharacteristic
58C59×#357C2211yescharacteristic
59C2×#11745687—41no—
60C2×Dic236C2311yescharacteristic
61C2×#11745672—21no—
62C2×D472C2311yescharacteristic
63C2×#11745669—21no—
64C22×C59⋊C8D411yescharacteristic
65#14273726—21no—
66#14273726—21no—
67#14273725—21no—
68#14273723—21no—
69#14273717—41no—
70#14273713—41no—
71#14273706—21no—
72#14273704—41no—
73#14273701—21no—
74#14273699—41no—
75Q8×Dic59—21no—
76#14273692—21no—
77#14273692—41no—
78D4×Dic59—21no—
79#14273689—21no—
80(C22⋊D59)⋊C4—41no—
81#14273683—41no—
82C2×C4×Dic59D411yescharacteristic
83#14273672—22no—
84#14273671—22no—
85#14273670—22no—
86C2×C59⋊C16—22no—
87C2×C59⋊C16C2311yescharacteristic
88C2×C59⋊C16C2311yescharacteristic
89#14273661C2311yescharacteristic
90#14273661C2311yescharacteristic
91#14273660—21no—
92#14273660—21no—
93#14273652—21no—
94#14273651—21no—
95#14273650—21no—
96#14273649—21no—
97#14273648—21no—
98#14273647—21no—
99C8×Dic59D411yescharacteristic
100C8×Dic59D411yescharacteristic
101C8×Dic59C2311yescharacteristic
102C8×Dic59C2311yescharacteristic
103C2×Q16×C59C2311yescharacteristic
104C2×D8×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22⋊Q8×C59—21no—
107#121×C59—21no—
108SD32×C59—22no—
109C2×C16×C59—22no—
110C2×C16×C59C2311yescharacteristic
111C2×C16×C59C2311yescharacteristic
112#107×C59C2311yescharacteristic
113#107×C59C2311yescharacteristic
114#103×C59—21no—
115#102×C59—21no—
116C2×Q8×C59—21no—
117C2×D4×C59—21no—
118C2×D4×C59—41no—
119C22×C4×C59C2×D411yescharacteristic
120C2×C22⋊D59—41no—
121C2×C22⋊D59—41no—
122C2×C22⋊D59—41no—
123C22×Dic59—41no—
124C22×Dic59D811yescharacteristic
125C22×Dic59D811yescharacteristic
126#11745687—84no—
127C2×D236—21no—
128C2×C4×D59—41no—
129C2×Dic118—21no—
130C2×Dic118—41no—
131C2×Dic118—41no—
132Q16×C59—21no—
133Q16×C59—21no—
134D8×C59—21no—
135D8×C59—21no—
136C2×C8×C59—21no—
137C2×C8×C59—21no—
138C2×C8×C59C2×D411yescharacteristic
139C2×C8×C59C2411yescharacteristic
140C4⋊C4×C59—21no—
141C4⋊C4×C59—21no—
142C4⋊C4×C59—41no—
143C22⋊C4×C59—41no—
144C22⋊C4×C59—41no—
145C22⋊Dic59—41no—
146C22⋊Dic59—41no—
147#11745672—42no—
148#11745669—42no—
149(C2×D59)⋊C4—41no—
150(C2×D59)⋊C4—41no—
151C4⋊Dic59—21no—
152C4⋊Dic59—21no—
153C4⋊Dic59—41no—
154Dic59⋊C4—41no—
155Dic59⋊C4—41no—
156Dic59⋊C4—41no—
157Dic59⋊C4—41no—
158Dic59⋊C4—41no—
159Dic59⋊C4—41no—
160C4×Dic59—41no—
161C4×Dic59—41no—
162C4×Dic59D811yescharacteristic
163C4×Dic59D811yescharacteristic
164C4×Dic59C2×D411yescharacteristic
165C4×Dic59C2×D411yescharacteristic
166C2×C59⋊C8—22no—
167C2×C59⋊C8C2×D411yescharacteristic
168C2×C59⋊C8C2×D411yescharacteristic
169Dic236—21no—
170Dic236—21no—
171D472—21no—
172D472—21no—
173C16×C59—21no—
174C16×C59#4112yescomplemented
175C59⋊C16—21no—
176C59⋊C16#4112yescomplemented
177C23×C59—41no—
178C23×C59C2×D811yescharacteristic
179C22×D59—41no—
180Q8×C59—41no—
181Q8×C59—41no—
182D4×C59—41no—
183D4×C59—41no—
184D4×C59—82no—
185C2×C4×C59—21no—
186C2×C4×C59—21no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59C2×D811yescharacteristic
191C2×C4×C59C22×D411yescharacteristic
192C22⋊D59—82no—
193C22⋊D59—82no—
194C22⋊D59—82no—
195C2×Dic59—22no—
196C2×Dic59—41no—
197C2×Dic59—41no—
198C2×Dic59—41no—
199C2×Dic59—82no—
200C2×Dic59SD3214yescomplemented
201C2×Dic59C2×D811yescharacteristic
202C2×Dic59C2×D811yescharacteristic
203C2×Dic59C2×D811yescharacteristic
204C2×Dic59C2×D811yescharacteristic
205D236—41no—
206D236—41no—
207C4×D59—82no—
208Dic118—41no—
209Dic118—41no—
210Dic118—82no—
211Dic118—82no—
212C8×C59—21no—
213C8×C59C2×#4111yescharacteristic
214C8×C59C2×#4111yescharacteristic
215C59⋊C8—21no—
216C59⋊C8#13512yes—
217#33736—591nomaximal, Sylow
218C22×C59—21no—
219C22×C59—21no—
220C22×C59—81no—
221C22×C59—81no—
222C22×C59—81no—
223C22×C59C2×SD3212yes—
224C22×C59C22×D811yescharacteristic
225C2×D59—81no—
226C2×D59—81no—
227C2×D59—81no—
228C4×C59—21no—
229C4×C59—81no—
230C4×C59—81no—
231C4×C59—81no—
232C4×C59#53211yescharacteristic
233C4×C59C2×#13511yescharacteristic
234Dic59—21no—
235Dic59—81no—
236Dic59—81no—
237Dic59—81no—
238Dic59C2×SD3212yes—
239Dic59C2×SD3212yes—
240Dic59#50812yes—
241#3170—591no—
242#3169—591no—
243#2254—591no—
244#2248—591no—
245#2241—591no—
246#2240—591no—
247#2217—591no—
248#2217—591no—
249#2217—591no—
250#2217D5911yescharacteristic, complemented
251C4×SD32—591no—
252C4×SD32—591no—
253C4×SD32—591no—
254C4×SD32—591no—
255C2×C4×C16—591no—
256C2×C59—22no—
257C2×C59—161no—
258C2×C59#316911yescharacteristic
259C2×C59C22×SD3211yescharacteristic
260C2×C59C2×#50811yescharacteristic
261D59—161no—
262#533—1181no—
263#532—1181no—
264C2×SD32—591no—
265C2×SD32—591no—
266C2×SD32—591no—
267C2×SD32C2×D5911yescharacteristic
268C22×C16—591no—
269C22×C16C2×D5911yescharacteristic
270#500—591no—
271#500—591no—
272#494—591no—
273#493—591no—
274#487—1181no—
275#486—1181no—
276#467—591no—
277#467—1181no—
278#467C2×D5911yescharacteristic
279#466—591no—
280#466—1181no—
281#466C2×D5911yescharacteristic
282C4×Q16—591no—
283C4×Q16—591no—
284C4×Q16—1181no—
285C4×D8—591no—
286C4×D8—591no—
287C4×D8—1181no—
288C2×C4×C8—591no—
289#367—591no—
290#367—591no—
291#367—591no—
292#367C2×D5911yescharacteristic
293#358—591no—
294#358—591no—
295#358—591no—
296#358C2×D5911yescharacteristic
297#357—591no—
298#357—591no—
299#357—591no—
300#357C2×D5911yescharacteristic
301C4×C16—591no—
302C4×C16—591no—
303C4×C16—591no—
304C4×C16—591no—
305C59#3373611yescharacteristic, complemented, Sylow
306C2×#41—2361no—
307C2×Q16—591no—
308C2×Q16—1181no—
309C2×Q16C22×D5911yescharacteristic
310C2×D8—591no—
311C2×D8—1181no—
312C2×D8C22×D5911yescharacteristic
313C22×C8—591no—
314C22×C8C22×D5911yescharacteristic
315C4⋊Q8—1181no—
316C4⋊Q8—1181no—
317C4⋊Q8—1181no—
318#127—1181no—
319#125—2361no—
320#124—2361no—
321C22⋊Q8—21no—
322C22⋊Q8—1181no—
323C22⋊Q8—2361no—
324#121—21no—
325#121—1181no—
326#121—2361no—
327C4×Q8—1181no—
328C4×Q8—1181no—
329C4×Q8—2361no—
330C4×D4—1181no—
331C4×D4—1181no—
332C4×D4—2361no—
333#117—2361no—
334C2×C42—591no—
335SD32—22no—
336SD32—1182no—
337SD32—1182no—
338SD32—1182no—
339C2×C16—22no—
340C2×C16—591no—
341C2×C16—591no—
342C2×C16—1182no—
343C2×C16C22×D5911yescharacteristic
344C2×C16C22×D5911yescharacteristic
345#107—591no—
346#107—591no—
347#107—1181no—
348#107—1181no—
349#107C22×D5911yescharacteristic
350#107C22×D5911yescharacteristic
351#103—21no—
352#103—1181no—
353#103—1181no—
354#103—1181no—
355#102—21no—
356#102—1181no—
357#102—1181no—
358#102—1181no—
359C4×C8—591no—
360C4×C8—591no—
361C4×C8—591no—
362C4×C8—591no—
363#41—4724no—
364C2×Q8—21no—
365C2×Q8—1181no—
366C2×Q8—2361no—
367C2×Q8—2361no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—1181no—
371C2×D4—2361no—
372C2×D4—2361no—
373C2×D4—2361no—
374C22×C4—591no—
375C22×C4—591no—
376C22×C4—2361no—
377C22×C4—2361no—
378C22×C4D4×D5911yescharacteristic
379Q16—21no—
380Q16—21no—
381Q16—1181no—
382Q16—1181no—
383Q16—2362no—
384D8—21no—
385D8—21no—
386D8—1181no—
387D8—1181no—
388D8—2362no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—591no—
392C2×C8—591no—
393C2×C8—1182no—
394C2×C8D4×D5911yescharacteristic
395C2×C8C23×D5911yescharacteristic
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—1181no—
400C4⋊C4—1181no—
401C4⋊C4—2361no—
402C4⋊C4—2361no—
403C4⋊C4—2361no—
404C4⋊C4—2361no—
405C4⋊C4—2361no—
406C4⋊C4—2361no—
407C4⋊C4—2361no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—2361no—
411C22⋊C4—2361no—
412C22⋊C4—2361no—
413C22⋊C4—2361no—
414C42—591no—
415C42—591no—
416C42—591no—
417C42—591no—
418C42—2361no—
419C42—2361no—
420C16—21no—
421C16—592no—
422C16—1181no—
423C16#1174568712yescomplemented
424C23—41no—
425C23—2361no—
426C23D8×D5911yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—4722no—
430Q8—4722no—
431Q8—2361no—
432Q8—2361no—
433D4—41no—
434D4—41no—
435D4—82no—
436D4—4722no—
437D4—4722no—
438D4—4722no—
439D4—2361no—
440D4—2361no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—591no—
447C2×C4—591no—
448C2×C4—591no—
449C2×C4—594no—
450C2×C4—591no—
451C2×C4—1182no—
452C2×C4—4722no—
453C2×C4—4722no—
454C2×C4—2361no—
455C2×C4—2361no—
456C2×C4—2361no—
457C2×C4D8×D5911yescharacteristic
458C2×C4C2×D4×D5911yescharacteristic
459C8—21no—
460C8—592no—
461C8—1181no—
462C8#41×D5911yescharacteristic
463C8C2×#1174568711yescharacteristic
464C22—21no—
465C22—21no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—4721no—
470C22—4721no—
471C22—4721no—
472C22SD32×D5912yes—
473C22C2×D8×D5911yescharacteristic
474C4—21no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—592no—
479C4—592no—
480C4—592no—
481C4—1181no—
482C4—4721no—
483C4—4721no—
484C4—4721no—
485C4#1683794711yescharacteristic
486C4#135×D5911yescharacteristic
487C2—22no—
488C2—161no—
489C2—9441no—
490C2#2143467011yescharacteristic
491C2C2×SD32×D5911yescharacteristic
492C2#508×D5911yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2217C2
socleC22×C59C22×D8
prime radical#21434670C2
hypercenter#2217D59
hypocenterC59#33736
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. #21434677C2semidirect
3. #21426377C2semidirect
4. #21426350C2semidirect
5. #21426322C2semidirect
6. #21426318C2semidirect
7. #21426194C2semidirect
8. #21426192C2semidirect
9. #21426190C2semidirect
10. #21425951C2semidirect
11. SD32×Dic59C2semidirect
12. #21425947C2semidirect
13. #21425945C2semidirect
14. C2×C16×Dic59C2semidirect
15. C59×#2217C2semidirect
174. C16×C59#41semidirect
176. C59⋊C16#41semidirect
200. C2×Dic59SD32semidirect
250. #2217D59semidirect
305. C59#33736semidirect
423. C16#11745687semidirect

Maximal subgroups

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

subgroupindex
#214346772
#214346702
#214263772
#214263502
#214263222
#214263182
#214261942
#214261922
#214261902
#214259512
SD32×Dic592
#214259472
#214259452
C2×C16×Dic592
C59×#22172
#3373659