Finite Group Databasebeta

← #28746143

Subgroups — #28746143

Order 15104.

504 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436460C211yesmaximal, characteristic, complemented
2#21436440C211yesmaximal, characteristic
3#21426380C211yesmaximal, characteristic, complemented
4#21426300C211yesmaximal, characteristic, complemented
5#21426294C211yesmaximal, characteristic, complemented
6#21426286C211yesmaximal, characteristic, complemented
7#21426255C211yesmaximal, characteristic, complemented
8#21426247C211yesmaximal, characteristic, complemented
9#21426208C211yesmaximal, characteristic, complemented
10#21426190C211yesmaximal, characteristic, complemented
11#21425957C211yesmaximal, characteristic, complemented
12#21425918C211yesmaximal, characteristic, complemented
13#21425881C211yesmaximal, characteristic, complemented
14C2×#16837227C211yesmaximal, characteristic, complemented
15C59×#2232C211yesmaximal, characteristic, complemented
16#16838087—21no—
17#16838052—21no—
18C2×#14273671C2211yescharacteristic
19C2×#14273666C2211yescharacteristic
20C22×C59⋊C16C2211yescharacteristic
21#16837806C2211yescharacteristic
22#16837803C2211yescharacteristic
23#16837745—21no—
24#16837732—21no—
25#16837714—21no—
26#16837713—21no—
27#16837708—21no—
28#16837680—21no—
29#16837644C2211yescharacteristic
30#16837643C2211yescharacteristic
31#16837635C2211yescharacteristic
32#16837496C2211yescharacteristic
33D8×Dic59C2211yescharacteristic
34C8×C22⋊D59C2211yescharacteristic
35#16837439C2211yescharacteristic
36C2×#14273661C2211yescharacteristic
37#107×D59C2211yescharacteristic
38#16837230C2211yescharacteristic
39#16837230C2211yescharacteristic
40#16837229C2211yescharacteristic
41#16837227C2211yescharacteristic
42#16837227C2211yescharacteristic
43#16837227C2211yescharacteristic
44#16837227C2211yescharacteristic
45Dic59⋊C16C2211yescharacteristic
46#16837207C2211yescharacteristic
47#16837207C2211yescharacteristic
48#16837205C2211yescharacteristic
49#16837203C2211yescharacteristic
50(C2×D59)⋊C16C2211yescharacteristic
51#16837177C2211yescharacteristic
52C59×#466C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C59×#366C2211yescharacteristic
55C59×#366C2211yescharacteristic
56C59×#357C2211yescharacteristic
57C59×#357C2211yescharacteristic
58C59×C22⋊C16C2211yescharacteristic
59C2×#11745687—41no—
60C2×Dic236C2311yescharacteristic
61C2×C8×D59C2311yescharacteristic
62#14273726—21no—
63#14273722—41no—
64#14273718—21no—
65#14273717—41no—
66#14273710—41no—
67#14273709—41no—
68#14273708—41no—
69#14273707—21no—
70#14273706—21no—
71#14273704—41no—
72#14273698—21no—
73#14273692—21no—
74D4×Dic59—21no—
75C4×C22⋊D59D411yescharacteristic
76#14273686—41no—
77#14273685—21no—
78C2×C4⋊Dic59—21no—
79C4⋊C4×D59—21no—
80#14273671—22no—
81#14273666—22no—
82C2×C59⋊C16—22no—
83C2×C59⋊C16C2311yescharacteristic
84C2×C59⋊C16C2311yescharacteristic
85#14273661—21no—
86#14273661C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273660—21no—
90#14273660—21no—
91#14273658—21no—
92#14273658—21no—
93Dic59⋊C8D411yescharacteristic
94#14273652—21no—
95#14273650—21no—
96#14273650—21no—
97#14273648—21no—
98#14273648—21no—
99C22⋊(C59⋊C8)D411yescharacteristic
100(C2×D59)⋊C8D411yescharacteristic
101C8×Dic59C2311yescharacteristic
102C2×D8×C59C2311yescharacteristic
103C22×C8×C59C2311yescharacteristic
104#121×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C16×C59C2311yescharacteristic
107C2×C16×C59C2311yescharacteristic
108#107×C59—21no—
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112#102×C59—21no—
113C2×D4×C59—21no—
114C2×D4×C59—41no—
115C22×C4×C59—41no—
116C22×C4×C59C2×D411yescharacteristic
117C2×C22⋊D59—41no—
118C2×C22⋊D59D811yescharacteristic
119C22×Dic59—41no—
120C22×Dic59—41no—
121#11745687—84no—
122C2×C4×D59—41no—
123C2×C4×D59C2×D411yescharacteristic
124C2×Dic118—21no—
125C2×Dic118—41no—
126D8×C59—21no—
127D8×C59—21no—
128C2×C8×C59—21no—
129C2×C8×C59—21no—
130C2×C8×C59C2×D411yescharacteristic
131C2×C8×C59C2411yescharacteristic
132C4⋊C4×C59—21no—
133C4⋊C4×C59—21no—
134C4⋊C4×C59—21no—
135C4⋊C4×C59—41no—
136C22⋊C4×C59—41no—
137C22⋊Dic59—41no—
138C22⋊Dic59—41no—
139C22⋊Dic59—41no—
140C22⋊Dic59D811yescharacteristic
141(C2×D59)⋊C4—41no—
142(C2×D59)⋊C4—41no—
143(C2×D59)⋊C4D811yescharacteristic
144C4⋊Dic59—21no—
145C4⋊Dic59—21no—
146C4⋊Dic59—21no—
147C4⋊Dic59—41no—
148C4⋊Dic59—41no—
149C4⋊Dic59—41no—
150Dic59⋊C4—41no—
151Dic59⋊C4—41no—
152Dic59⋊C4—41no—
153Dic59⋊C4—41no—
154Dic59⋊C4—41no—
155Dic59⋊C4D811yescharacteristic
156C4×Dic59—41no—
157C4×Dic59—41no—
158C4×Dic59C2×D411yescharacteristic
159C2×C59⋊C8C2×D411yescharacteristic
160C2×C59⋊C8C2×D411yescharacteristic
161Dic236—21no—
162Dic236—21no—
163C8×D59—21no—
164C8×D59—21no—
165C16×C59—21no—
166C16×C59—21no—
167C59⋊C16—21no—
168C59⋊C16#4112yescomplemented
169C23×C59—41no—
170C23×C59C2×D811yescharacteristic
171C22×D59C2×D811yescharacteristic
172D4×C59—41no—
173D4×C59—41no—
174D4×C59—82no—
175C2×C4×C59—21no—
176C2×C4×C59—21no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—82no—
181C2×C4×C59C2×D811yescharacteristic
182C2×C4×C59C22×D411yescharacteristic
183C22⋊D59—22no—
184C22⋊D59—82no—
185C2×Dic59—41no—
186C2×Dic59—41no—
187C2×Dic59—41no—
188C2×Dic59—41no—
189C2×Dic59—82no—
190C2×Dic59—82no—
191C2×Dic59C2×D811yescharacteristic
192C2×Dic59C2×D811yescharacteristic
193C2×Dic59C2×D811yescharacteristic
194C4×D59—21no—
195C4×D59—21no—
196C4×D59—82no—
197Dic118—41no—
198Dic118—41no—
199Dic118—82no—
200C8×C59—21no—
201C8×C59C2×#4111yescharacteristic
202C8×C592−1+411yescharacteristic
203C59⋊C8—21no—
204C59⋊C8—21no—
205#33910—591nomaximal, Sylow
206C22×C59—21no—
207C22×C59—21no—
208C22×C59—81no—
209C22×C59—81no—
210C22×C59—81no—
211C22×C59#50912yes—
212C22×C59C22×D811yescharacteristic
213C2×D59—21no—
214C2×D59—21no—
215C2×D59—21no—
216C4×C59—21no—
217C4×C59—81no—
218C4×C59—81no—
219C4×C59—81no—
220C4×C59#54711yescharacteristic
221C4×C59#57811yescharacteristic
222Dic59—21no—
223Dic59—21no—
224Dic59—21no—
225Dic59—81no—
226Dic59—81no—
227Dic59—81no—
228Dic59—81no—
229#3336—591no—
230#3334—591no—
231#2255—591no—
232#2236—591no—
233#2235—591no—
234#2234—591no—
235#2232D5911yescharacteristic, complemented
236#2228—591no—
237#2227—591no—
238#2220—591no—
239#2217—591no—
240#2175—591no—
241#2168—591no—
242#2159—591no—
243C2×#366—591no—
244C2×C59—22no—
245C2×C59—161no—
246C2×C59#333411yescharacteristic
247C2×C59C2×#50911yescharacteristic
248C2×C59#341711yescharacteristic
249D59—41no—
250#552—1181no—
251#547—1181no—
252C2×SD32—591no—
253C2×M32—591no—
254C22×C16—591no—
255#500—591no—
256#499—591no—
257#485—1181no—
258#483—1181no—
259#480—1181no—
260#480—1181no—
261#479—1181no—
262#474—1181no—
263#467—591no—
264#467—591no—
265#466—591no—
266#466C2×D5911yescharacteristic
267C4×Q16—591no—
268C4×D8—591no—
269C8×D4—591no—
270#427—591no—
271C2×#107—591no—
272C2×#107—591no—
273C2×#107C2×D5911yescharacteristic
274#367—591no—
275#367—591no—
276#367—591no—
277#366—591no—
278#366—591no—
279#366—591no—
280#366—591no—
281#366C2×D5911yescharacteristic
282#366C2×D5911yescharacteristic
283C4⋊C16—591no—
284#358—591no—
285#358—591no—
286#358—591no—
287#357—591no—
288#357C2×D5911yescharacteristic
289#357C2×D5911yescharacteristic
290C22⋊C16—591no—
291C22⋊C16C2×D5911yescharacteristic
292#346—591no—
293C59#3391011yescharacteristic, complemented, Sylow
294C2×#41—2361no—
295C2×Q16—591no—
296C2×D8C22×D5911yescharacteristic
297C22×C8—591no—
298C22×C8C22×D5911yescharacteristic
299C4⋊Q8—1181no—
300#126—2361no—
301#125—1181no—
302#125—2361no—
303#123—2361no—
304#123—2361no—
305#123—2361no—
306C22⋊Q8—1181no—
307C22⋊Q8—1181no—
308C22⋊Q8—2361no—
309#121—21no—
310#121—1181no—
311C4×Q8—1181no—
312C4×D4—591no—
313C4×D4—1181no—
314#117—1181no—
315#117—2361no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—1181no—
318C2×C4⋊C4—1181no—
319SD32—1182no—
320M32—1182no—
321C2×C16—591no—
322C2×C16—591no—
323C2×C16—1182no—
324C2×C16C22×D5911yescharacteristic
325C2×C16C22×D5911yescharacteristic
326#107—21no—
327#107—591no—
328#107—591no—
329#107—591no—
330#107—1181no—
331#107—1181no—
332#107—1181no—
333#107C22×D5911yescharacteristic
334#107C22×D5911yescharacteristic
335#107C22×D5911yescharacteristic
336#106—1181no—
337#106—1181no—
338C4⋊C8—591no—
339#103—1181no—
340#103—1181no—
341#103—1181no—
342#102—21no—
343#102—1181no—
344#102—1181no—
345C22⋊C8—591no—
346C22⋊C8—591no—
347C4×C8—591no—
348#41—4724no—
349C2×Q8—1181no—
350C2×Q8—2361no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—591no—
354C2×D4—2361no—
355C22×C4—41no—
356C22×C4—591no—
357C22×C4—2361no—
358C22×C4—2361no—
359C22×C4—2361no—
360C22×C4D4×D5911yescharacteristic
361Q16—1181no—
362Q16—1181no—
363D8—21no—
364D8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—591no—
368C2×C8—591no—
369C2×C8—1181no—
370C2×C8—1181no—
371C2×C8D4×D5911yescharacteristic
372C2×C8C23×D5911yescharacteristic
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—591no—
378C4⋊C4—1181no—
379C4⋊C4—1181no—
380C4⋊C4—1181no—
381C4⋊C4—2361no—
382C4⋊C4—2361no—
383C4⋊C4—2361no—
384C4⋊C4—2361no—
385C4⋊C4—2361no—
386C4⋊C4—2361no—
387C4⋊C4—2361no—
388C4⋊C4—2361no—
389C22⋊C4—41no—
390C22⋊C4—591no—
391C22⋊C4—591no—
392C22⋊C4—2361no—
393C22⋊C4—2361no—
394C22⋊C4—2361no—
395C22⋊C4—2361no—
396C22⋊C4—2361no—
397C42—591no—
398C42—2361no—
399C42—2361no—
400C16—21no—
401C16—21no—
402C16—592no—
403C16—1181no—
404C23—41no—
405C23—591no—
406C23D8×D5911yescharacteristic
407Q8—4722no—
408Q8—2361no—
409Q8—2361no—
410D4—41no—
411D4—41no—
412D4—82no—
413D4—1182no—
414D4—4722no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—82no—
421C2×C4—591no—
422C2×C4—591no—
423C2×C4—591no—
424C2×C4—1181no—
425C2×C4—1181no—
426C2×C4—4722no—
427C2×C4—4722no—
428C2×C4—4722no—
429C2×C4—2361no—
430C2×C4—2361no—
431C2×C4—2361no—
432C2×C4—2361no—
433C2×C4D8×D5911yescharacteristic
434C2×C4C2×D4×D5911yescharacteristic
435C8—21no—
436C8—1181no—
437C8—1181no—
438C8#41×D5911yescharacteristic
439C8#1427377411yescharacteristic
440C22—21no—
441C22—21no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—1181no—
446C22—1181no—
447C22—1181no—
448C22#1683783612yes—
449C22C2×D8×D5911yescharacteristic
450C4—21no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—1181no—
455C4—1181no—
456C4—1181no—
457C4—4721no—
458C4—4721no—
459C4—4721no—
460C4—4721no—
461C4#1683805011yescharacteristic
462C4#1683821811yescharacteristic
463C2—22no—
464C2—161no—
465C2—2361no—
466C2#2143643811yescharacteristic
467C2C2×#1683783611yescharacteristic
468C2#2143724811yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2232C2
socleC22×C59C22×D8
prime radical#21436440C2
hypercenter#2232D59
hypocenterC59#33910
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2232

Complements

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

NH ≅ G/Ntype
1. #21436460C2semidirect
3. #21426380C2semidirect
4. #21426300C2semidirect
5. #21426294C2semidirect
6. #21426286C2semidirect
7. #21426255C2semidirect
8. #21426247C2semidirect
9. #21426208C2semidirect
10. #21426190C2semidirect
11. #21425957C2semidirect
12. #21425918C2semidirect
13. #21425881C2semidirect
14. C2×#16837227C2semidirect
15. C59×#2232C2semidirect
168. C59⋊C16#41semidirect
235. #2232D59semidirect
293. C59#33910semidirect

Maximal subgroups

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

subgroupindex
#214364602
#214364402
#214263802
#214263002
#214262942
#214262862
#214262552
#214262472
#214262082
#214261902
#214259572
#214259182
#214258812
C2×#168372272
C59×#22322
#3391059