Finite Group Databasebeta

← #28746133

Subgroups — #28746133

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#21436465C211yesmaximal, characteristic, complemented
2#21426379C211yesmaximal, characteristic, complemented
3#21426298C211yesmaximal, characteristic, complemented
4#21426293C211yesmaximal, characteristic, complemented
5#21426284C211yesmaximal, characteristic, complemented
6#21426274C211yesmaximal, characteristic, complemented
7#21426253C211yesmaximal, characteristic, complemented
8#21426245C211yesmaximal, characteristic, complemented
9#21426207C211yesmaximal, characteristic, complemented
10#21426190C211yesmaximal, characteristic, complemented
11#21425960C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425882C211yesmaximal, characteristic, complemented
14C2×#16837227C211yesmaximal, characteristic, complemented
15C59×#3334C211yesmaximal, characteristic
16#16838092—21no—
17C2×#14273671C2211yescharacteristic
18C2×#14273667C2211yescharacteristic
19C22×C59⋊C16C2211yescharacteristic
20#16837804C2211yescharacteristic
21#16837744—21no—
22#16837733—21no—
23#16837710—21no—
24#16837643C2211yescharacteristic
25#16837643C2211yescharacteristic
26C4×Dic236C2211yescharacteristic
27#16837440C2211yescharacteristic
28C2×#14273661C2211yescharacteristic
29#16837229C2211yescharacteristic
30#16837229C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837227C2211yescharacteristic
33#16837227C2211yescharacteristic
34#16837227C2211yescharacteristic
35#16837227C2211yescharacteristic
36#16837227C2211yescharacteristic
37#16837227C2211yescharacteristic
38C4⋊(C59⋊C16)C2211yescharacteristic
39#16837205C2211yescharacteristic
40#16837205C2211yescharacteristic
41#16837205C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44#16837203C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C59×#547—21no—
49C59×#500C2211yescharacteristic
50C59×#480—21no—
51C59×#480—21no—
52C59×#474—21no—
53C59×#466C2211yescharacteristic
54C59×#466C2211yescharacteristic
55C4×D8×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60#14273720—41no—
61#14273719—21no—
62#14273717—41no—
63#14273708—41no—
64#14273706—21no—
65#14273706—21no—
66C4×Dic118—21no—
67#14273686—21no—
68#14273683—41no—
69C2×C4⋊Dic59—21no—
70#14273671—22no—
71#14273667—22no—
72C2×C59⋊C16—22no—
73C2×C59⋊C16C2311yescharacteristic
74C2×C59⋊C16C2311yescharacteristic
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77#14273661—21no—
78#14273661C2311yescharacteristic
79#14273661C2311yescharacteristic
80#14273661C2311yescharacteristic
81#14273659—21no—
82#14273659—21no—
83#14273652—21no—
84#14273652—21no—
85#14273652—21no—
86C2×#41×C59—41no—
87C2×D8×C59C2311yescharacteristic
88C22×C8×C59C2311yescharacteristic
89C22×C8×C59C2311yescharacteristic
90C4⋊Q8×C59—21no—
91#123×C59—41no—
92#123×C59—41no—
93C22⋊Q8×C59—41no—
94#121×C59—21no—
95#121×C59—21no—
96C4×D4×C59—21no—
97C4×D4×C59D411yescharacteristic
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100#107×C59—21no—
101#107×C59—21no—
102#107×C59—21no—
103#107×C59C2311yescharacteristic
104#107×C59C2311yescharacteristic
105#107×C59C2311yescharacteristic
106C4⋊C8×C59D411yescharacteristic
107#102×C59—21no—
108#102×C59—21no—
109#102×C59—21no—
110C22⋊C8×C59D411yescharacteristic
111C22⋊C8×C59D411yescharacteristic
112C4×C8×C59C2311yescharacteristic
113#41×C59—84no—
114C2×Q8×C59—41no—
115C2×D4×C59—21no—
116C2×D4×C59—41no—
117C2×D4×C59—41no—
118C2×D4×C59D811yescharacteristic
119C22×C4×C59—41no—
120C22×C4×C59—41no—
121C22×C4×C59—41no—
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×Dic59—41no—
125C2×Dic118—21no—
126D8×C59—21no—
127D8×C59—21no—
128C2×C8×C59—21no—
129C2×C8×C59—21no—
130C2×C8×C59—21no—
131C2×C8×C59—21no—
132C2×C8×C59C2×D411yescharacteristic
133C2×C8×C59C2×D411yescharacteristic
134C2×C8×C59C2×D411yescharacteristic
135C2×C8×C59C2411yescharacteristic
136C4⋊C4×C59—21no—
137C4⋊C4×C59—21no—
138C4⋊C4×C59—21no—
139C4⋊C4×C59—41no—
140C4⋊C4×C59—41no—
141C4⋊C4×C59—41no—
142C4⋊C4×C59D811yescharacteristic
143C22⋊C4×C59—41no—
144C22⋊C4×C59—41no—
145C22⋊C4×C59—41no—
146C22⋊C4×C59D811yescharacteristic
147C22⋊C4×C59D811yescharacteristic
148C42×C59C2×D411yescharacteristic
149C22⋊Dic59—41no—
150C22⋊Dic59—41no—
151C22⋊Dic59—41no—
152C4⋊Dic59—21no—
153C4⋊Dic59—21no—
154C4⋊Dic59—21no—
155C4⋊Dic59—41no—
156Dic59⋊C4—41no—
157Dic59⋊C4—41no—
158Dic59⋊C4—41no—
159Dic59⋊C4—41no—
160Dic59⋊C4—41no—
161C4×Dic59—41no—
162C4×Dic59—41no—
163Dic236—21no—
164Dic236—21no—
165C59⋊C16—21no—
166C59⋊C16—21no—
167C59⋊C16—21no—
168C59⋊C16#4112yescomplemented
169C23×C59—41no—
170C23×C59C2×D811yescharacteristic
171C23×C59C2×D811yescharacteristic
172Q8×C59—82no—
173D4×C59—22no—
174D4×C59—41no—
175D4×C59—41no—
176D4×C59—82no—
177D4×C59—82no—
178C2×C4×C59—21no—
179C2×C4×C59—21no—
180C2×C4×C59—21no—
181C2×C4×C59—21no—
182C2×C4×C59—41no—
183C2×C4×C59—41no—
184C2×C4×C59—41no—
185C2×C4×C59—82no—
186C2×C4×C59—82no—
187C2×C4×C59—82no—
188C2×C4×C59C2×D811yescharacteristic
189C2×C4×C59C2×D811yescharacteristic
190C2×C4×C59C2×D811yescharacteristic
191C2×C4×C59C2×D811yescharacteristic
192C2×C4×C59C22×D411yescharacteristic
193C2×Dic59—41no—
194C2×Dic59—41no—
195C2×Dic59—41no—
196C2×Dic59—41no—
197C2×Dic59—82no—
198Dic118—41no—
199Dic118—41no—
200C8×C59—21no—
201C8×C59—21no—
202C8×C59—21no—
203C8×C59C2×#4111yescharacteristic
204C8×C592−1+411yescharacteristic
205#33910—591nomaximal, Sylow
206C22×C59—21no—
207C22×C59—21no—
208C22×C59—21no—
209C22×C59—21no—
210C22×C59—21no—
211C22×C59—81no—
212C22×C59—81no—
213C22×C59—81no—
214C22×C59#50912yes—
215C22×C59C22×D811yescharacteristic
216C4×C59—21no—
217C4×C59—21no—
218C4×C59—21no—
219C4×C59—21no—
220C4×C59—81no—
221C4×C59—81no—
222C4×C59—81no—
223C4×C59#54711yescharacteristic
224C4×C59#57811yescharacteristic
225Dic59—81no—
226Dic59—81no—
227Dic59—81no—
228Dic59—81no—
229#3336—591no—
230#3334D5911yescharacteristic
231#2255—591no—
232#2236—591no—
233#2235—591no—
234#2234—591no—
235#2232—591no—
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—41no—
246C2×C59—161no—
247C2×C59#333411yescharacteristic
248C2×C59C2×#50911yescharacteristic
249C2×C59#341711yescharacteristic
250#552—1181no—
251#547—21no—
252C2×SD32—591no—
253C2×M32—591no—
254C22×C16—591no—
255#500C2×D5911yescharacteristic
256#499—591no—
257#485—1181no—
258#483—1181no—
259#480—21no—
260#480—21no—
261#479—1181no—
262#474—21no—
263#467—591no—
264#467—591no—
265#466C2×D5911yescharacteristic
266#466C2×D5911yescharacteristic
267C4×Q16—591no—
268C4×D8C2×D5911yescharacteristic
269C8×D4C2×D5911yescharacteristic
270#427—591no—
271C2×#107—591no—
272C2×#107C2×D5911yescharacteristic
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#366—591no—
282#366—591no—
283C4⋊C16—591no—
284#358—591no—
285#358—591no—
286#358—591no—
287#357—591no—
288#357—591no—
289#357—591no—
290C22⋊C16—591no—
291C22⋊C16—591no—
292#346—591no—
293C59#3391011yescharacteristic, complemented, Sylow
294C2×#41—41no—
295C2×Q16—591no—
296C2×D8C22×D5911yescharacteristic
297C22×C8C22×D5911yescharacteristic
298C22×C8C22×D5911yescharacteristic
299C4⋊Q8—21no—
300#126—2361no—
301#125—1181no—
302#125—2361no—
303#123—41no—
304#123—41no—
305#123—2361no—
306C22⋊Q8—41no—
307C22⋊Q8—1181no—
308C22⋊Q8—1181no—
309#121—21no—
310#121—21no—
311C4×Q8—1181no—
312C4×D4—21no—
313C4×D4C22⋊D5911yescharacteristic
314#117—1181no—
315#117—2361no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—1181no—
319SD32—1182no—
320M32—1182no—
321C2×C16—591no—
322C2×C16—591no—
323C2×C16—591no—
324C2×C16—591no—
325C2×C16—1182no—
326#107—21no—
327#107—21no—
328#107—21no—
329#107—591no—
330#107—591no—
331#107—591no—
332#107—1181no—
333#107C22×D5911yescharacteristic
334#107C22×D5911yescharacteristic
335#107C22×D5911yescharacteristic
336#106—1181no—
337#106—1181no—
338C4⋊C8C22⋊D5911yescharacteristic
339#103—1181no—
340#103—1181no—
341#103—1181no—
342#102—21no—
343#102—21no—
344#102—21no—
345C22⋊C8C22⋊D5911yescharacteristic
346C22⋊C8C22⋊D5911yescharacteristic
347C4×C8C22×D5911yescharacteristic
348#41—84no—
349C2×Q8—41no—
350C2×Q8—1181no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4#1174566911yescharacteristic
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—2361no—
359C22×C4C2×C22⋊D5911yescharacteristic
360C22×C4C2×C22⋊D5911yescharacteristic
361Q16—1181no—
362Q16—1181no—
363D8—21no—
364D8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8C2×C22⋊D5911yescharacteristic
370C2×C8C2×C22⋊D5911yescharacteristic
371C2×C8C2×C22⋊D5911yescharacteristic
372C2×C8C23×D5911yescharacteristic
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—1181no—
380C4⋊C4—1181no—
381C4⋊C4—1181no—
382C4⋊C4—2361no—
383C4⋊C4—2361no—
384C4⋊C4—2361no—
385C4⋊C4—2361no—
386C4⋊C4—2361no—
387C4⋊C4—2361no—
388C4⋊C4#1174566911yescharacteristic
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—2361no—
393C22⋊C4—2361no—
394C22⋊C4—2361no—
395C22⋊C4#1174566911yescharacteristic
396C22⋊C4#1174566911yescharacteristic
397C42—2361no—
398C42—2361no—
399C42C2×C22⋊D5911yescharacteristic
400C16—592no—
401C16—1181no—
402C16—1181no—
403C16—1181no—
404C23—41no—
405C23C2×#1174566911yescharacteristic
406C23C2×#1174566911yescharacteristic
407Q8—82no—
408Q8—2361no—
409Q8—2361no—
410D4—22no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—82no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—4722no—
426C2×C4—2361no—
427C2×C4—2361no—
428C2×C4—2361no—
429C2×C4—2361no—
430C2×C4C2×#1174566911yescharacteristic
431C2×C4C2×#1174566911yescharacteristic
432C2×C4C2×#1174566911yescharacteristic
433C2×C4C2×#1174566911yescharacteristic
434C2×C4C22×C22⋊D5911yescharacteristic
435C8—21no—
436C8—21no—
437C8—21no—
438C8#41×D5911yescharacteristic
439C8#1427377411yescharacteristic
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22#1683783812yes—
449C22C22×#1174566911yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—4721no—
458C4—4721no—
459C4—4721no—
460C4—4721no—
461C4#1683805111yescharacteristic
462C4#1683821711yescharacteristic
463C2—22no—
464C2—41no—
465C2—161no—
466C2#2143643911yescharacteristic
467C2C2×#1683783811yescharacteristic
468C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
Fitting, prime radicalC59×#3334C2
socleC22×C59C22×D8
hypercenter#3334D59
hypocenterC59#33910
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #21436465C2semidirect
2. #21426379C2semidirect
3. #21426298C2semidirect
4. #21426293C2semidirect
5. #21426284C2semidirect
6. #21426274C2semidirect
7. #21426253C2semidirect
8. #21426245C2semidirect
9. #21426207C2semidirect
10. #21426190C2semidirect
11. #21425960C2semidirect
12. #21425922C2semidirect
13. #21425882C2semidirect
14. C2×#16837227C2semidirect
168. C59⋊C16#41semidirect
293. C59#33910semidirect

Maximal subgroups

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

subgroupindex
#214364652
#214263792
#214262982
#214262932
#214262842
#214262742
#214262532
#214262452
#214262072
#214261902
#214259602
#214259222
#214258822
C2×#168372272
C59×#33342
#3391059