Finite Group Databasebeta

← #28746461

Subgroups — #28746461

Order 15104.

480 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436988C211yesmaximal, characteristic, complemented
2#21426337C211yesmaximal, characteristic
3#21426332C211yesmaximal, characteristic, complemented
4#21426327C211yesmaximal, characteristic, complemented
5#21426260C211yesmaximal, characteristic, complemented
6#21426253C211yesmaximal, characteristic, complemented
7#21426243C211yesmaximal, characteristic, complemented
8#21426155C211yesmaximal, characteristic, complemented
9#21426155C211yesmaximal, characteristic, complemented
10#21426133C211yesmaximal, characteristic
11#21426005C211yesmaximal, characteristic, complemented
12#21425965C211yesmaximal, characteristic, complemented
13#21425960C211yesmaximal, characteristic, complemented
14C4×#14273671C211yesmaximal, characteristic, complemented
15C59×#3396C211yesmaximal, characteristic
16#16838130—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273671C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20#16837807C2211yescharacteristic
21#16837804C2211yescharacteristic
22#16837804C2211yescharacteristic
23#16837710—21no—
24#16837710—21no—
25#16837688—21no—
26C4×Dic236C2211yescharacteristic
27C4×Dic236C2211yescharacteristic
28C4×Dic236C2211yescharacteristic
29#16837229C2211yescharacteristic
30#16837229C2211yescharacteristic
31#16837227C2211yescharacteristic
32C4⋊(C59⋊C16)C2211yescharacteristic
33C4⋊(C59⋊C16)C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837203C2211yescharacteristic
40#16837203C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44#16837203C2211yescharacteristic
45#16837178C2211yescharacteristic
46#16837178C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C59×#559—21no—
49C59×#495C2211yescharacteristic
50C59×#495C2211yescharacteristic
51C59×#493C2211yescharacteristic
52C59×#463—21no—
53C59×#463—21no—
54C59×#460—21no—
55C8×Q8×C59C2211yescharacteristic
56C4×Q16×C59C2211yescharacteristic
57C4×D8×C59C2211yescharacteristic
58C4×D8×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60#14273727—21no—
61#14273725—41no—
62#14273719—21no—
63#14273719—21no—
64#14273717—41no—
65#14273717—41no—
66C4×Dic118—21no—
67C4×Dic118—21no—
68C4×Dic118—21no—
69#14273692—41no—
70#14273673—22no—
71#14273671—22no—
72#14273671—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×Q16×C59C2311yescharacteristic
87C2×SD16×C59—21no—
88C2×SD16×C59—21no—
89C2×D8×C59—21no—
90C2×D8×C59C2311yescharacteristic
91C2×D8×C59C2311yescharacteristic
92#127×C59—21no—
93#124×C59—21no—
94#124×C59—21no—
95#124×C59—41no—
96#123×C59—41no—
97#123×C59—41no—
98C4×Q8×C59—21no—
99C4×Q8×C59D411yescharacteristic
100C4×D4×C59—21no—
101C4×D4×C59—21no—
102C4×D4×C59—41no—
103#107×C59C2311yescharacteristic
104C4⋊C8×C59D411yescharacteristic
105C4⋊C8×C59D411yescharacteristic
106C4⋊C8×C59D411yescharacteristic
107#103×C59—21no—
108#102×C59—21no—
109#102×C59—21no—
110C4×C8×C59C2311yescharacteristic
111C4×C8×C59C2311yescharacteristic
112C4×C8×C59C2311yescharacteristic
113C2×Q8×C59—21no—
114C2×Q8×C59D811yescharacteristic
115C2×D4×C59—21no—
116C2×D4×C59—21no—
117C2×D4×C59—41no—
118C2×D4×C59—41no—
119C22×C4×C59—41no—
120C22×C4×C59—41no—
121C2×Dic118—21no—
122C2×Dic118—41no—
123Q16×C59—21no—
124Q16×C59—21no—
125SD16×C59—81no—
126SD16×C59—81no—
127D8×C59—21no—
128D8×C59—21no—
129D8×C59—21no—
130D8×C59—21no—
131D8×C59—81no—
132C2×C8×C59C2×D411yescharacteristic
133C2×C8×C59C2×D411yescharacteristic
134C2×C8×C59C2×D411yescharacteristic
135C2×C8×C59C2411yescharacteristic
136C4⋊C4×C59—21no—
137C4⋊C4×C59—41no—
138C4⋊C4×C59D811yescharacteristic
139C4⋊C4×C59D811yescharacteristic
140C4⋊C4×C59D811yescharacteristic
141C22⋊C4×C59—41no—
142C22⋊C4×C59—41no—
143C22⋊C4×C59—41no—
144C22⋊C4×C59—41no—
145C22⋊C4×C59—41no—
146C22⋊C4×C59—41no—
147C42×C59—41no—
148C42×C59C2×D411yescharacteristic
149C42×C59C2×D411yescharacteristic
150C42×C59C2×D411yescharacteristic
151C4⋊Dic59—21no—
152C4⋊Dic59—21no—
153C4⋊Dic59—21no—
154C4⋊Dic59—41no—
155Dic59⋊C4—41no—
156Dic59⋊C4—41no—
157Dic59⋊C4—41no—
158Dic59⋊C4—41no—
159Dic59⋊C4—41no—
160Dic59⋊C4—41no—
161Dic59⋊C4—41no—
162C4×Dic59—41no—
163C4×Dic59—41no—
164C4×Dic59—41no—
165Dic236—21no—
166Dic236#4112yes—
167C59⋊C16—21no—
168C59⋊C16—21no—
169C59⋊C16—21no—
170C59⋊C16—21no—
171C23×C59—41no—
172C23×C59—41no—
173Q8×C59—41no—
174Q8×C59—41no—
175Q8×C59—41no—
176D4×C59—41no—
177D4×C59—41no—
178D4×C59—41no—
179D4×C59—41no—
180D4×C59—82no—
181D4×C59—161no—
182C2×C4×C59—41no—
183C2×C4×C59—41no—
184C2×C4×C59—161no—
185C2×C4×C59—161no—
186C2×C4×C59C2×D811yescharacteristic
187C2×C4×C59C2×D811yescharacteristic
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—
197Dic118—22no—
198Dic118—41no—
199Dic118—82no—
200C8×C59—21no—
201C8×C59—21no—
202C8×C59—21no—
203C8×C59C2×#4111yescharacteristic
204C8×C592−1+411yescharacteristic
205#33938—591nomaximal, Sylow
206C22×C59—81no—
207C22×C59—81no—
208C22×C59—81no—
209C22×C59—81no—
210C22×C59—81no—
211C22×C59—81no—
212C22×C59C22×D811yescharacteristic
213C4×C59—21no—
214C4×C59—21no—
215C4×C59—21no—
216C4×C59—21no—
217C4×C59—21no—
218C4×C59—21no—
219C4×C59—81no—
220C4×C59—81no—
221C4×C59#54811yescharacteristic
222C4×C59#57711yescharacteristic
223Dic59—42no—
224Dic59—81no—
225Dic59—81no—
226Dic59—81no—
227#3396D5911yescharacteristic
228#3383—591no—
229#2244—591no—
230#2243—591no—
231#2242—591no—
232#2229—591no—
233#2228—591no—
234#2226—591no—
235#2211—591no—
236#2211—591no—
237#2207—591no—
238#2183—591no—
239#2176—591no—
240#2175—591no—
241C4×SD32—591no—
242C2×C59—161no—
243C2×C59—161no—
244C2×C59#335811yescharacteristic
245C2×C59#341411yescharacteristic
246C2×C59#341611yescharacteristic
247#559—21no—
248#557—1181no—
249C2×Q32—591no—
250C2×SD32—591no—
251C2×SD32—591no—
252#500—591no—
253#499—591no—
254#499—591no—
255#495C2×D5911yescharacteristic
256#495C2×D5911yescharacteristic
257#493C2×D5911yescharacteristic
258#479—1181no—
259#479—1181no—
260#475—1181no—
261#463—21no—
262#463—21no—
263#460—21no—
264C8×Q8C2×D5911yescharacteristic
265C4×Q16—591no—
266C4×Q16—591no—
267C4×Q16—591no—
268C4×Q16C2×D5911yescharacteristic
269C4×D8C2×D5911yescharacteristic
270C4×D8C2×D5911yescharacteristic
271#367—591no—
272#367—591no—
273#366—591no—
274C4⋊C16—591no—
275C4⋊C16—591no—
276C4⋊C16—591no—
277#358—591no—
278#358—591no—
279#358—591no—
280#358—591no—
281#357—591no—
282#357—591no—
283#357—591no—
284#357—591no—
285#357—591no—
286#357—591no—
287#346—591no—
288#346—591no—
289C4×C16—591no—
290C59#3393811yescharacteristic, complemented, Sylow
291C2×Q16—591no—
292C2×Q16C22×D5911yescharacteristic
293C2×SD16—21no—
294C2×SD16—21no—
295C2×D8—21no—
296C2×D8C22×D5911yescharacteristic
297C2×D8C22×D5911yescharacteristic
298C4⋊Q8—1181no—
299C4⋊Q8—2361no—
300#127—21no—
301#125—1181no—
302#125—1181no—
303#125—2361no—
304#125—2361no—
305#124—21no—
306#124—21no—
307#124—41no—
308#123—41no—
309#123—41no—
310C4×Q8—21no—
311C4×Q8—1181no—
312C4×Q8—1181no—
313C4×Q8—1181no—
314C4×Q8—2361no—
315C4×Q8C22⋊D5911yescharacteristic
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—41no—
319Q32—1182no—
320SD32—1182no—
321SD32—1182no—
322C2×C16—591no—
323C2×C16—591no—
324C2×C16—591no—
325C2×C16—591no—
326#107—591no—
327#107—591no—
328#107—591no—
329#107—1181no—
330#107C22×D5911yescharacteristic
331#106—1181no—
332#106—1181no—
333C4⋊C8C22⋊D5911yescharacteristic
334C4⋊C8C22⋊D5911yescharacteristic
335C4⋊C8C22⋊D5911yescharacteristic
336#103—21no—
337#103—1181no—
338#103—1181no—
339#103—1181no—
340#102—21no—
341#102—21no—
342C4×C8C22×D5911yescharacteristic
343C4×C8C22×D5911yescharacteristic
344C4×C8C22×D5911yescharacteristic
345C2×Q8—21no—
346C2×Q8—1181no—
347C2×Q8—2361no—
348C2×Q8#1174566911yescharacteristic
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—41no—
352C2×D4—41no—
353C22×C4—41no—
354C22×C4—41no—
355Q16—21no—
356Q16—21no—
357Q16—592no—
358Q16—1181no—
359SD16—81no—
360SD16—81no—
361D8—21no—
362D8—21no—
363D8—21no—
364D8—21no—
365D8—81no—
366C2×C8C2×C22⋊D5911yescharacteristic
367C2×C8C2×C22⋊D5911yescharacteristic
368C2×C8C2×C22⋊D5911yescharacteristic
369C2×C8C23×D5911yescharacteristic
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—1181no—
373C4⋊C4—1181no—
374C4⋊C4—1181no—
375C4⋊C4—2361no—
376C4⋊C4—2361no—
377C4⋊C4—2361no—
378C4⋊C4—2361no—
379C4⋊C4—2361no—
380C4⋊C4—2361no—
381C4⋊C4—2361no—
382C4⋊C4—2361no—
383C4⋊C4#1174566911yescharacteristic
384C4⋊C4#1174566911yescharacteristic
385C4⋊C4#1174566911yescharacteristic
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C42—41no—
393C42—2361no—
394C42—2361no—
395C42—2361no—
396C42C2×C22⋊D5911yescharacteristic
397C42C2×C22⋊D5911yescharacteristic
398C42C2×C22⋊D5911yescharacteristic
399C16—1181no—
400C16—1181no—
401C16—1181no—
402C16—1181no—
403C23—41no—
404C23—41no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408Q8—1182no—
409Q8—4722no—
410Q8—2361no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—82no—
416D4—161no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—161no—
420C2×C4—161no—
421C2×C4—2361no—
422C2×C4—2361no—
423C2×C4—2361no—
424C2×C4—2361no—
425C2×C4C2×#1174566911yescharacteristic
426C2×C4C2×#1174566911yescharacteristic
427C2×C4C2×#1174566911yescharacteristic
428C2×C4C2×#1174566911yescharacteristic
429C2×C4C2×#1174566911yescharacteristic
430C2×C4C2×#1174566911yescharacteristic
431C2×C4C22×C22⋊D5911yescharacteristic
432C8—21no—
433C8—21no—
434C8—21no—
435C8#41×D5911yescharacteristic
436C8#1427377411yescharacteristic
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22C22×#1174566911yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—81no—
451C4—81no—
452C4—4721no—
453C4—4721no—
454C4—4721no—
455C4—2362no—
456C4#1683805911yescharacteristic
457C4#1683821311yescharacteristic
458C2—161no—
459C2—161no—
460C2#2143672711yescharacteristic
461C2#2143723011yescharacteristic
462C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3396C2
socleC22×C59C22×D8
prime radicalC59×#493C22
hypercenter#3396D59
hypocenterC59#33938
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3396

Complements

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

NH ≅ G/Ntype
1. #21436988C2semidirect
3. #21426332C2semidirect
4. #21426327C2semidirect
5. #21426260C2semidirect
6. #21426253C2semidirect
7. #21426243C2semidirect
8. #21426155C2semidirect
9. #21426155C2semidirect
11. #21426005C2semidirect
12. #21425965C2semidirect
13. #21425960C2semidirect
14. C4×#14273671C2semidirect
290. C59#33938semidirect

Maximal subgroups

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

subgroupindex
#214369882
#214263372
#214263322
#214263272
#214262602
#214262532
#214262432
#214261552
#214261332
#214260052
#214259652
#214259602
C4×#142736712
C59×#33962
#3393859