Finite Group Databasebeta

← #28745132

Subgroups — #28745132

Order 15104.

488 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434735C211yesmaximal, characteristic, complemented
2#21426243C211yesmaximal, characteristic, complemented
3#21426240C211yesmaximal, characteristic, complemented
4#21426238C211yesmaximal, characteristic, complemented
5#21426235C211yesmaximal, characteristic, complemented
6#21426202C211yesmaximal, characteristic, complemented
7#21426197C211yesmaximal, characteristic
8#21426194C211yesmaximal, characteristic, complemented
9#21426190C211yesmaximal, characteristic, complemented
10#21426158C211yesmaximal, characteristic, complemented
11#21426155C211yesmaximal, characteristic, complemented
12#21426151C211yesmaximal, characteristic, complemented
13#21426148C211yesmaximal, characteristic, complemented
14#21425855C211yesmaximal, characteristic, complemented
15C59×#3175C211yesmaximal, characteristic, complemented
16#16838023—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273671C2211yescharacteristic
19C2×#14273670C2211yescharacteristic
20C2×#14273668C2211yescharacteristic
21C2×#14273667C2211yescharacteristic
22C22×C59⋊C16C2211yescharacteristic
23#16837807C2211yescharacteristic
24#16837786C2211yescharacteristic
25#16837762—21no—
26#16837648—21no—
27#16837643C2211yescharacteristic
28#16837638C2211yescharacteristic
29#16837504—21no—
30C4×Dic236C2211yescharacteristic
31C4×D472C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837229C2211yescharacteristic
34#16837229C2211yescharacteristic
35#16837229C2211yescharacteristic
36C4⋊(C59⋊C16)C2211yescharacteristic
37C4⋊(C59⋊C16)C2211yescharacteristic
38C4⋊(C59⋊C16)C2211yescharacteristic
39C4⋊(C59⋊C16)C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837206C2211yescharacteristic
42#16837205C2211yescharacteristic
43#16837205C2211yescharacteristic
44#16837203C2211yescharacteristic
45#16837203C2211yescharacteristic
46#16837202C2211yescharacteristic
47#16837202C2211yescharacteristic
48C59×#542—21no—
49C59×#500C2211yescharacteristic
50C59×#495C2211yescharacteristic
51C59×#489—21no—
52C59×#468—21no—
53C59×#467C2211yescharacteristic
54C59×#466C2211yescharacteristic
55C59×#440—21no—
56C4×Q16×C59C2211yescharacteristic
57C4×D8×C59C2211yescharacteristic
58C59×#405C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60C2×#11745660—21no—
61C2×D472C2311yescharacteristic
62#14273727—21no—
63#14273722—41no—
64#14273715—21no—
65#14273713—41no—
66#14273709—41no—
67#14273706—21no—
68#14273705—41no—
69#14273701—21no—
70C4×Dic118—21no—
71C4×D236—21no—
72#14273673—41no—
73#14273671—41no—
74#14273670—41no—
75#14273668—41no—
76#14273667—41no—
77C2×C59⋊C16—41no—
78C2×C59⋊C16C2311yescharacteristic
79C2×C59⋊C16C2311yescharacteristic
80C2×C59⋊C16C2311yescharacteristic
81C2×C59⋊C16C2311yescharacteristic
82#14273661—21no—
83#14273661C2311yescharacteristic
84#14273661C2311yescharacteristic
85#14273652—21no—
86#14273652—21no—
87#14273649—21no—
88#14273649—21no—
89C2×Q16×C59C2311yescharacteristic
90C2×SD16×C59—21no—
91C2×D8×C59C2311yescharacteristic
92C2×M16×C59D411yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C4⋊Q8×C59—21no—
95#126×C59—41no—
96#124×C59—21no—
97#124×C59—41no—
98#123×C59—41no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59—41no—
101#121×C59—21no—
102C4×Q8×C59—21no—
103C4×D4×C59—21no—
104#117×C59D411yescharacteristic
105#107×C59—21no—
106#107×C59C2311yescharacteristic
107#107×C59C2311yescharacteristic
108#103×C59—21no—
109#103×C59—21no—
110#102×C59—21no—
111#102×C59—21no—
112#97×C59D411yescharacteristic
113#97×C59D411yescharacteristic
114C4×C8×C59C2311yescharacteristic
115C4×C8×C59C2311yescharacteristic
116C2×Q8×C59—21no—
117C2×Q8×C59—41no—
118C2×D4×C59—21no—
119C2×D4×C59—41no—
120C22×C4×C59—41no—
121C22×C4×C59C2×D411yescharacteristic
122C2×C22⋊D59—41no—
123C2×D236—21no—
124C2×C4×D59—41no—
125C2×Dic118—21no—
126C2×Dic118—41no—
127Q16×C59—21no—
128Q16×C59—21no—
129SD16×C59—42no—
130D8×C59—21no—
131D8×C59—21no—
132M16×C59—41no—
133C2×C8×C59—21no—
134C2×C8×C59—21no—
135C2×C8×C59C2×D411yescharacteristic
136C2×C8×C59C2×D411yescharacteristic
137C2×C8×C59C2×D411yescharacteristic
138C2×C8×C59C2411yescharacteristic
139C4⋊C4×C59—21no—
140C4⋊C4×C59—21no—
141C4⋊C4×C59—41no—
142C4⋊C4×C59—41no—
143C4⋊C4×C59—41no—
144C4⋊C4×C59D811yescharacteristic
145C4⋊C4×C59D811yescharacteristic
146C22⋊C4×C59—41no—
147C22⋊C4×C59—41no—
148C22⋊C4×C59—41no—
149C22⋊C4×C59—41no—
150C22⋊C4×C59—41no—
151C22⋊C4×C59D811yescharacteristic
152C22⋊C4×C59D811yescharacteristic
153C42×C59—41no—
154C42×C59C2×D411yescharacteristic
155C42×C59C2×D411yescharacteristic
156C22⋊Dic59—41no—
157(C2×D59)⋊C4—41no—
158(C2×D59)⋊C4—41no—
159(C2×D59)⋊C4—41no—
160(C2×D59)⋊C4—41no—
161C4⋊Dic59—21no—
162C4⋊Dic59—21no—
163C4⋊Dic59—41no—
164Dic59⋊C4—41no—
165Dic59⋊C4—41no—
166C4×Dic59—41no—
167Dic236—21no—
168Dic236—21no—
169D472—21no—
170D472—21no—
171#11745660—42no—
172C59⋊C16—21no—
173C59⋊C16—21no—
174C59⋊C16—21no—
175C59⋊C16—21no—
176C23×C59—41no—
177C23×C59C2×D811yescharacteristic
178C22×D59—41no—
179Q8×C59—41no—
180Q8×C59—41no—
181Q8×C59—161no—
182D4×C59—41no—
183D4×C59—41no—
184D4×C59—161no—
185C2×C4×C59—21no—
186C2×C4×C59—21no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—161no—
191C2×C4×C59C2×D811yescharacteristic
192C2×C4×C59C2×D811yescharacteristic
193C2×C4×C59C2×D811yescharacteristic
194C2×C4×C59C2×D811yescharacteristic
195C2×C4×C59C2×D811yescharacteristic
196C2×C4×C59C22×D411yescharacteristic
197C22⋊D59—161no—
198C2×Dic59—41no—
199C2×Dic59—41no—
200C2×Dic59—41no—
201D236—41no—
202D236—41no—
203C4×D59—161no—
204Dic118—41no—
205Dic118—41no—
206Dic118—161no—
207C8×C59—21no—
208C8×C59—21no—
209C8×C59—21no—
210C8×C59H(5,𝔽2)11yescharacteristic
211C8×C592−1+411yescharacteristic
212#33829—591nomaximal, Sylow
213C22×C59—21no—
214C22×C59—21no—
215C22×C59—21no—
216C22×C59—81no—
217C22×C59—81no—
218C22×C59—81no—
219C22×C59C22×D811yescharacteristic
220C2×D59—81no—
221C2×D59—81no—
222C2×D59—81no—
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—21no—
226C4×C59—21no—
227C4×C59—21no—
228C4×C59—81no—
229C4×C59—81no—
230C4×C59—81no—
231C4×C59#53711yescharacteristic
232C4×C59#57511yescharacteristic
233Dic59—81no—
234Dic59—81no—
235Dic59—81no—
236#3175—591no—
237#3175D5911yescharacteristic, complemented
238#2226—591no—
239#2226—591no—
240#2225—591no—
241#2225—591no—
242#2219—591no—
243#2218—591no—
244#2217—591no—
245#2217—591no—
246#2211—591no—
247#2211—591no—
248#2210—591no—
249#2210—591no—
250#2151—591no—
251C2×C59—41no—
252C2×C59—161no—
253C2×C59#324511yescharacteristic
254C2×C59#341611yescharacteristic
255C2×C59#341611yescharacteristic
256D59—161no—
257#542—21no—
258#542—1181no—
259C2×Q32—591no—
260C2×SD32—591no—
261C2×SD32—591no—
262C2×D16—591no—
263C2×M32—591no—
264C22×C16—591no—
265#500—591no—
266#500C2×D5911yescharacteristic
267#495—591no—
268#495C2×D5911yescharacteristic
269#489—21no—
270#489—1181no—
271#468—21no—
272#468—1181no—
273#467—591no—
274#467C2×D5911yescharacteristic
275#466—591no—
276#466C2×D5911yescharacteristic
277#440—21no—
278#440—1181no—
279C4×Q16—591no—
280C4×Q16C2×D5911yescharacteristic
281C4×D8—591no—
282C4×D8C2×D5911yescharacteristic
283#405C2×D5911yescharacteristic
284#367—591no—
285#367—591no—
286#367—591no—
287#367—591no—
288C4⋊C16—591no—
289C4⋊C16—591no—
290C4⋊C16—591no—
291C4⋊C16—591no—
292#358—591no—
293#358—591no—
294#358—591no—
295#358—591no—
296#357—591no—
297#357—591no—
298#357—591no—
299#357—591no—
300C59#3382911yescharacteristic, complemented, Sylow
301C2×Q16—591no—
302C2×Q16C22×D5911yescharacteristic
303C2×SD16—21no—
304C2×SD16—1181no—
305C2×D8—591no—
306C2×D8C22×D5911yescharacteristic
307C2×M16C22⋊D5911yescharacteristic
308C22×C8C22×D5911yescharacteristic
309C4⋊Q8—21no—
310C4⋊Q8—1181no—
311#126—41no—
312#126—2361no—
313#124—21no—
314#124—41no—
315#124—1181no—
316#124—2361no—
317#123—41no—
318#123—2361no—
319C22⋊Q8—21no—
320C22⋊Q8—41no—
321C22⋊Q8—1181no—
322C22⋊Q8—2361no—
323#121—21no—
324#121—1181no—
325C4×Q8—21no—
326C4×Q8—1181no—
327C4×D4—21no—
328C4×D4—1181no—
329#117C22⋊D5911yescharacteristic
330Q32—2361no—
331SD32—2361no—
332SD32—2361no—
333D16—2361no—
334M32—2361no—
335C2×C16—591no—
336C2×C16—591no—
337C2×C16—591no—
338C2×C16—591no—
339C2×C16—2361no—
340#107—21no—
341#107—591no—
342#107—591no—
343#107—1181no—
344#107C22×D5911yescharacteristic
345#107C22×D5911yescharacteristic
346#103—21no—
347#103—21no—
348#103—1181no—
349#103—1181no—
350#102—21no—
351#102—21no—
352#102—1181no—
353#102—1181no—
354#97C22⋊D5911yescharacteristic
355#97C22⋊D5911yescharacteristic
356C4×C8C22×D5911yescharacteristic
357C4×C8C22×D5911yescharacteristic
358C2×Q8—21no—
359C2×Q8—41no—
360C2×Q8—1181no—
361C2×Q8—2361no—
362C2×D4—21no—
363C2×D4—41no—
364C2×D4—1181no—
365C2×D4—2361no—
366C22×C4—41no—
367C22×C4—2361no—
368C22×C4C2×C22⋊D5911yescharacteristic
369Q16—21no—
370Q16—21no—
371Q16—1181no—
372Q16—1181no—
373SD16—42no—
374SD16—2362no—
375D8—21no—
376D8—21no—
377D8—1181no—
378D8—1181no—
379M16—41no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8C2×C22⋊D5911yescharacteristic
383C2×C8C2×C22⋊D5911yescharacteristic
384C2×C8C2×C22⋊D5911yescharacteristic
385C2×C8C23×D5911yescharacteristic
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—1181no—
392C4⋊C4—1181no—
393C4⋊C4—2361no—
394C4⋊C4—2361no—
395C4⋊C4—2361no—
396C4⋊C4#1174566911yescharacteristic
397C4⋊C4#1174566911yescharacteristic
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—2361no—
404C22⋊C4—2361no—
405C22⋊C4—2361no—
406C22⋊C4—2361no—
407C22⋊C4—2361no—
408C22⋊C4#1174566911yescharacteristic
409C22⋊C4#1174566911yescharacteristic
410C42—41no—
411C42—2361no—
412C42C2×C22⋊D5911yescharacteristic
413C42C2×C22⋊D5911yescharacteristic
414C16—1181no—
415C16—1181no—
416C16—1181no—
417C16—1181no—
418C23—41no—
419C23—2361no—
420C23C2×#1174566911yescharacteristic
421Q8—41no—
422Q8—41no—
423Q8—161no—
424Q8—9441no—
425Q8—2361no—
426Q8—2361no—
427D4—41no—
428D4—41no—
429D4—161no—
430D4—9441no—
431D4—2361no—
432D4—2361no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—161no—
439C2×C4—9441no—
440C2×C4—2361no—
441C2×C4—2361no—
442C2×C4—2361no—
443C2×C4C2×#1174566911yescharacteristic
444C2×C4C2×#1174566911yescharacteristic
445C2×C4C2×#1174566911yescharacteristic
446C2×C4C2×#1174566911yescharacteristic
447C2×C4C2×#1174566911yescharacteristic
448C2×C4C22×C22⋊D5911yescharacteristic
449C8—21no—
450C8—21no—
451C8—21no—
452C8#1427377311yescharacteristic
453C8#1427377411yescharacteristic
454C22—21no—
455C22—21no—
456C22—21no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—4721no—
461C22—4721no—
462C22—4721no—
463C22C22×#1174566911yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—4721no—
473C4—4721no—
474C4—4721no—
475C4#1683798011yescharacteristic
476C4#1683820011yescharacteristic
477C2—41no—
478C2—161no—
479C2—9441no—
480C2#2143540311yescharacteristic
481C2#2143724011yescharacteristic
482C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3175C2
socleC22×C59C22×D8
prime radical#21426197C2
hypercenter#3175D59
hypocenterC59#33829
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #21434735C2semidirect
2. #21426243C2semidirect
3. #21426240C2semidirect
4. #21426238C2semidirect
5. #21426235C2semidirect
6. #21426202C2semidirect
8. #21426194C2semidirect
9. #21426190C2semidirect
10. #21426158C2semidirect
11. #21426155C2semidirect
12. #21426151C2semidirect
13. #21426148C2semidirect
14. #21425855C2semidirect
15. C59×#3175C2semidirect
237. #3175D59semidirect
300. C59#33829semidirect

Maximal subgroups

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

subgroupindex
#214347352
#214262432
#214262402
#214262382
#214262352
#214262022
#214261972
#214261942
#214261902
#214261582
#214261552
#214261512
#214261482
#214258552
C59×#31752
#3382959