Finite Group Databasebeta

← #28745087

Subgroups — #28745087

Order 15104.

476 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434735C211yesmaximal, characteristic
2#21426257C211yesmaximal, characteristic, complemented
3#21426250C211yesmaximal, characteristic, complemented
4#21426248C211yesmaximal, characteristic, complemented
5#21426238C211yesmaximal, characteristic, complemented
6#21426211C211yesmaximal, characteristic
7#21426202C211yesmaximal, characteristic, complemented
8#21426194C211yesmaximal, characteristic
9#21426185C211yesmaximal, characteristic, complemented
10#21426158C211yesmaximal, characteristic, complemented
11#21426158C211yesmaximal, characteristic, complemented
12#21426143C211yesmaximal, characteristic, complemented
13#21426138C211yesmaximal, characteristic, complemented
14#21425855C211yesmaximal, characteristic, complemented
15C59×#3177C211yesmaximal, characteristic, complemented
16#16838023—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273673C2211yescharacteristic
19C2×#14273670C2211yescharacteristic
20C2×#14273670C2211yescharacteristic
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#16837227C2211yescharacteristic
35#16837227C2211yescharacteristic
36C4⋊(C59⋊C16)C2211yescharacteristic
37C4⋊(C59⋊C16)C2211yescharacteristic
38C4⋊(C59⋊C16)C2211yescharacteristic
39C4⋊(C59⋊C16)C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837206C2211yescharacteristic
42#16837206C2211yescharacteristic
43#16837206C2211yescharacteristic
44#16837205C2211yescharacteristic
45#16837205C2211yescharacteristic
46#16837202C2211yescharacteristic
47#16837202C2211yescharacteristic
48C59×#541—21no—
49C59×#499C2211yescharacteristic
50C59×#494C2211yescharacteristic
51C59×#491—21no—
52C59×#470—21no—
53C59×#467C2211yescharacteristic
54C59×#467C2211yescharacteristic
55C59×#441—21no—
56C4×Q16×C59C2211yescharacteristic
57C4×Q16×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#14273673—41no—
74#14273670—41no—
75#14273670—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×C59—21no—
90C2×Q16×C59C2311yescharacteristic
91C2×Q16×C59C2311yescharacteristic
92C2×M16×C59D411yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C4⋊Q8×C59—21no—
95#126×C59—41no—
96#126×C59—41no—
97#125×C59—21no—
98#125×C59—41no—
99#125×C59—41no—
100C22⋊Q8×C59—21no—
101C22⋊Q8×C59—21no—
102C4×Q8×C59—21no—
103C4×Q8×C59—21no—
104#117×C59D411yescharacteristic
105#107×C59C2311yescharacteristic
106#107×C59C2311yescharacteristic
107#106×C59—21no—
108#103×C59—21no—
109#103×C59—21no—
110#103×C59—21no—
111#103×C59—21no—
112#97×C59D411yescharacteristic
113#97×C59D411yescharacteristic
114C4×C8×C59C2311yescharacteristic
115C4×C8×C59C2311yescharacteristic
116C2×Q8×C59—21no—
117C2×Q8×C59—21no—
118C22×C4×C59C2×D411yescharacteristic
119C2×C22⋊D59—41no—
120C2×D236—21no—
121C2×C4×D59—41no—
122C2×Dic118—21no—
123C2×Dic118—41no—
124Q16×C59—21no—
125Q16×C59—21no—
126Q16×C59—21no—
127Q16×C59—21no—
128Q16×C59—42no—
129M16×C59—41no—
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—41no—
139C4⋊C4×C59—41no—
140C4⋊C4×C59—41no—
141C4⋊C4×C59—41no—
142C4⋊C4×C59—41no—
143C4⋊C4×C59—41no—
144C4⋊C4×C59—41no—
145C4⋊C4×C59—41no—
146C4⋊C4×C59D811yescharacteristic
147C4⋊C4×C59D811yescharacteristic
148C22⋊C4×C59—41no—
149C22⋊C4×C59—41no—
150C22⋊C4×C59D811yescharacteristic
151C22⋊C4×C59D811yescharacteristic
152C42×C59—41no—
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×C59C2×D811yescharacteristic
177C22×D59—41no—
178Q8×C59—41no—
179Q8×C59—41no—
180Q8×C59—41no—
181Q8×C59—41no—
182C2×C4×C59—21no—
183C2×C4×C59—21no—
184C2×C4×C59—41no—
185C2×C4×C59—41no—
186C2×C4×C59—41no—
187C2×C4×C59—41no—
188C2×C4×C59C2×D811yescharacteristic
189C2×C4×C59C2×D811yescharacteristic
190C2×C4×C59C2×D811yescharacteristic
191C2×C4×C59C2×D811yescharacteristic
192C2×C4×C59C2×D811yescharacteristic
193C2×C4×C59C22×D411yescharacteristic
194C22⋊D59—161no—
195C2×Dic59—41no—
196C2×Dic59—41no—
197C2×Dic59—41no—
198D236—41no—
199D236—41no—
200C4×D59—161no—
201Dic118—41no—
202Dic118—41no—
203Dic118—161no—
204C8×C59—21no—
205C8×C59—21no—
206C8×C59—21no—
207C8×C59H(5,𝔽2)11yescharacteristic
208C8×C592−1+411yescharacteristic
209#33824—591nomaximal, Sylow
210C22×C59—21no—
211C22×C59—21no—
212C22×C59—21no—
213C22×C59C22×D811yescharacteristic
214C2×D59—81no—
215C2×D59—81no—
216C2×D59—81no—
217C4×C59—21no—
218C4×C59—21no—
219C4×C59—21no—
220C4×C59—21no—
221C4×C59—21no—
222C4×C59—81no—
223C4×C59—81no—
224C4×C59—81no—
225C4×C59—81no—
226C4×C59#53711yescharacteristic
227C4×C59#57511yescharacteristic
228Dic59—81no—
229Dic59—81no—
230Dic59—81no—
231#3177D5911yescharacteristic, complemented
232#3175—591no—
233#2229—591no—
234#2228—591no—
235#2227—591no—
236#2225—591no—
237#2220—591no—
238#2219—591no—
239#2217—591no—
240#2216—591no—
241#2211—591no—
242#2211—591no—
243#2209—591no—
244#2208—591no—
245#2151—591no—
246C2×C59—41no—
247C2×C59#324511yescharacteristic
248C2×C59#341611yescharacteristic
249C2×C59#341711yescharacteristic
250D59—161no—
251#542—1181no—
252#541—21no—
253C2×Q32—591no—
254C2×Q32—591no—
255C2×SD32—591no—
256C2×SD32—591no—
257C2×M32—591no—
258C22×C16—591no—
259#500—591no—
260#499C2×D5911yescharacteristic
261#495—591no—
262#494C2×D5911yescharacteristic
263#491—21no—
264#489—1181no—
265#470—21no—
266#468—1181no—
267#467—591no—
268#467C2×D5911yescharacteristic
269#467C2×D5911yescharacteristic
270#466—591no—
271#441—21no—
272#440—1181no—
273C4×Q16—591no—
274C4×Q16C2×D5911yescharacteristic
275C4×Q16C2×D5911yescharacteristic
276C4×D8—591no—
277#405C2×D5911yescharacteristic
278#367—591no—
279#367—591no—
280#366—591no—
281#366—591no—
282C4⋊C16—591no—
283C4⋊C16—591no—
284C4⋊C16—591no—
285C4⋊C16—591no—
286#358—591no—
287#358—591no—
288#358—591no—
289#358—591no—
290#358—591no—
291#358—591no—
292#357—591no—
293#357—591no—
294C59#3382411yescharacteristic, complemented, Sylow
295C2×Q16—21no—
296C2×Q16—591no—
297C2×Q16C22×D5911yescharacteristic
298C2×Q16C22×D5911yescharacteristic
299C2×SD16—1181no—
300C2×D8—591no—
301C2×M16C22⋊D5911yescharacteristic
302C22×C8C22×D5911yescharacteristic
303C4⋊Q8—21no—
304C4⋊Q8—1181no—
305#126—41no—
306#126—41no—
307#126—2361no—
308#125—21no—
309#125—41no—
310#125—41no—
311#124—1181no—
312#124—2361no—
313#123—2361no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C22⋊Q8—1181no—
317C22⋊Q8—2361no—
318#121—1181no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8—1181no—
322C4×D4—1181no—
323#117C22⋊D5911yescharacteristic
324Q32—2361no—
325Q32—2361no—
326SD32—2361no—
327SD32—2361no—
328M32—2361no—
329C2×C16—591no—
330C2×C16—591no—
331C2×C16—591no—
332C2×C16—591no—
333C2×C16—2361no—
334#107—591no—
335#107—591no—
336#107—1181no—
337#107C22×D5911yescharacteristic
338#107C22×D5911yescharacteristic
339#106—21no—
340#103—21no—
341#103—21no—
342#103—21no—
343#103—21no—
344#103—1181no—
345#103—1181no—
346#102—1181no—
347#102—1181no—
348#97C22⋊D5911yescharacteristic
349#97C22⋊D5911yescharacteristic
350C4×C8C22×D5911yescharacteristic
351C4×C8C22×D5911yescharacteristic
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—1181no—
355C2×Q8—2361no—
356C2×D4—1181no—
357C2×D4—2361no—
358C22×C4—2361no—
359C22×C4C2×C22⋊D5911yescharacteristic
360Q16—21no—
361Q16—21no—
362Q16—21no—
363Q16—21no—
364Q16—42no—
365Q16—1181no—
366Q16—1181no—
367SD16—2362no—
368D8—1181no—
369D8—1181no—
370M16—41no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8C2×C22⋊D5911yescharacteristic
374C2×C8C2×C22⋊D5911yescharacteristic
375C2×C8C2×C22⋊D5911yescharacteristic
376C2×C8C23×D5911yescharacteristic
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—1181no—
388C4⋊C4—1181no—
389C4⋊C4—2361no—
390C4⋊C4—2361no—
391C4⋊C4—2361no—
392C4⋊C4#1174566911yescharacteristic
393C4⋊C4#1174566911yescharacteristic
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—2361no—
397C22⋊C4—2361no—
398C22⋊C4—2361no—
399C22⋊C4—2361no—
400C22⋊C4—2361no—
401C22⋊C4#1174566911yescharacteristic
402C22⋊C4#1174566911yescharacteristic
403C42—41no—
404C42—41no—
405C42—2361no—
406C42C2×C22⋊D5911yescharacteristic
407C42C2×C22⋊D5911yescharacteristic
408C16—1181no—
409C16—1181no—
410C16—1181no—
411C16—1181no—
412C23—2361no—
413C23C2×#1174566911yescharacteristic
414Q8—41no—
415Q8—41no—
416Q8—41no—
417Q8—41no—
418Q8—9441no—
419Q8—2361no—
420Q8—2361no—
421D4—9441no—
422D4—2361no—
423D4—2361no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—9441no—
431C2×C4—2361no—
432C2×C4—2361no—
433C2×C4—2361no—
434C2×C4C2×#1174566911yescharacteristic
435C2×C4C2×#1174566911yescharacteristic
436C2×C4C2×#1174566911yescharacteristic
437C2×C4C2×#1174566911yescharacteristic
438C2×C4C2×#1174566911yescharacteristic
439C2×C4C22×C22⋊D5911yescharacteristic
440C8—21no—
441C8—21no—
442C8—21no—
443C8#1427377311yescharacteristic
444C8#1427377411yescharacteristic
445C22—21no—
446C22—21no—
447C22—21no—
448C22—4721no—
449C22—4721no—
450C22—4721no—
451C22C22×#1174566911yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—4721no—
462C4—4721no—
463C4—4721no—
464C4#1683798011yescharacteristic
465C4#1683820011yescharacteristic
466C2—41no—
467C2—9441no—
468C2#2143540311yescharacteristic
469C2#2143724011yescharacteristic
470C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3177C2
socleC22×C59C22×D8
prime radical#16837638C22
hypercenter#3177D59
hypocenterC59#33824
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3177

Complements

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

NH ≅ G/Ntype
2. #21426257C2semidirect
3. #21426250C2semidirect
4. #21426248C2semidirect
5. #21426238C2semidirect
7. #21426202C2semidirect
9. #21426185C2semidirect
10. #21426158C2semidirect
11. #21426158C2semidirect
12. #21426143C2semidirect
13. #21426138C2semidirect
14. #21425855C2semidirect
15. C59×#3177C2semidirect
231. #3177D59semidirect
294. C59#33824semidirect

Maximal subgroups

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

subgroupindex
#214347352
#214262572
#214262502
#214262482
#214262382
#214262112
#214262022
#214261942
#214261852
#214261582
#214261432
#214261382
#214258552
C59×#31772
#3382459