Finite Group Databasebeta

← #28746467

Subgroups — #28746467

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#21437109C211yesmaximal, characteristic
2#21426339C211yesmaximal, characteristic
3#21426335C211yesmaximal, characteristic, complemented
4#21426330C211yesmaximal, characteristic, complemented
5#21426257C211yesmaximal, characteristic, complemented
6#21426250C211yesmaximal, characteristic, complemented
7#21426240C211yesmaximal, characteristic, complemented
8#21426158C211yesmaximal, characteristic, complemented
9#21426158C211yesmaximal, characteristic, complemented
10#21426136C211yesmaximal, characteristic
11#21426005C211yesmaximal, characteristic, complemented
12#21425965C211yesmaximal, characteristic, complemented
13#21425958C211yesmaximal, characteristic, complemented
14C4×#14273670C211yesmaximal, characteristic, complemented
15C59×#3383C211yesmaximal, characteristic, complemented
16#16838137—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273670C2211yescharacteristic
19C2×#14273670C2211yescharacteristic
20#16837786C2211yescharacteristic
21#16837786C2211yescharacteristic
22#16837779C2211yescharacteristic
23#16837618—21no—
24#16837618—21no—
25#16837603—21no—
26C4×Dic236C2211yescharacteristic
27C4×D472C2211yescharacteristic
28C4×D472C2211yescharacteristic
29#16837229C2211yescharacteristic
30#16837229C2211yescharacteristic
31#16837227C2211yescharacteristic
32C4⋊(C59⋊C16)C2211yescharacteristic
33C4⋊(C59⋊C16)C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837205C2211yescharacteristic
37#16837205C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837202C2211yescharacteristic
40#16837202C2211yescharacteristic
41#16837202C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44#16837202C2211yescharacteristic
45#16837178C2211yescharacteristic
46#16837178C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C59×#557—21no—
49C59×#500C2211yescharacteristic
50C59×#499C2211yescharacteristic
51C59×#499C2211yescharacteristic
52C59×#479—21no—
53C59×#479—21no—
54C59×#475—21no—
55C8×Q8×C59C2211yescharacteristic
56C4×Q16×C59C2211yescharacteristic
57C4×Q16×C59C2211yescharacteristic
58C4×Q16×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60C2×#11745660—21no—
61C2×#11745660—21no—
62C2×D472—21no—
63C2×D472C2311yescharacteristic
64C2×D472C2311yescharacteristic
65#14273724—21no—
66#14273716—41no—
67#14273715—21no—
68#14273715—21no—
69#14273712—41no—
70#14273712—41no—
71C4×Dic118—21no—
72C4×D236—21no—
73C4×D236—21no—
74#14273689—41no—
75#14273673—22no—
76#14273670—22no—
77#14273670—22no—
78C2×C59⋊C16C2311yescharacteristic
79C2×C59⋊C16C2311yescharacteristic
80C2×C59⋊C16C2311yescharacteristic
81C2×C59⋊C16C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273652—21no—
84#14273649—21no—
85#14273649—21no—
86C2×Q16×C59C2311yescharacteristic
87C4⋊Q8×C59—21no—
88C4⋊Q8×C59—41no—
89#125×C59—21no—
90#125×C59—21no—
91#125×C59—41no—
92#125×C59—41no—
93C4×Q8×C59—21no—
94C4×Q8×C59—21no—
95C4×Q8×C59—21no—
96C4×Q8×C59—41no—
97C4×Q8×C59D411yescharacteristic
98#107×C59—21no—
99#107×C59C2311yescharacteristic
100#107×C59C2311yescharacteristic
101#107×C59C2311yescharacteristic
102#106×C59—21no—
103#106×C59—21no—
104C4⋊C8×C59D411yescharacteristic
105C4⋊C8×C59D411yescharacteristic
106C4⋊C8×C59D411yescharacteristic
107#103×C59—21no—
108#103×C59—21no—
109#103×C59—21no—
110C4×C8×C59C2311yescharacteristic
111C4×C8×C59C2311yescharacteristic
112C4×C8×C59C2311yescharacteristic
113C2×Q8×C59—21no—
114C2×Q8×C59—41no—
115C2×Q8×C59D811yescharacteristic
116C2×D236—21no—
117C2×D236—21no—
118C2×D236—41no—
119C2×D236—41no—
120C2×C4×D59—41no—
121C2×C4×D59—41no—
122C2×Dic118—21no—
123Q16×C59—21no—
124Q16×C59#4112yes—
125C2×C8×C59C2×D411yescharacteristic
126C2×C8×C59C2×D411yescharacteristic
127C2×C8×C59C2×D411yescharacteristic
128C2×C8×C59C2411yescharacteristic
129C4⋊C4×C59—21no—
130C4⋊C4×C59—21no—
131C4⋊C4×C59—21no—
132C4⋊C4×C59—41no—
133C4⋊C4×C59—41no—
134C4⋊C4×C59—41no—
135C4⋊C4×C59—41no—
136C4⋊C4×C59—41no—
137C4⋊C4×C59—41no—
138C4⋊C4×C59—41no—
139C4⋊C4×C59—41no—
140C4⋊C4×C59D811yescharacteristic
141C4⋊C4×C59D811yescharacteristic
142C4⋊C4×C59D811yescharacteristic
143C42×C59—41no—
144C42×C59—41no—
145C42×C59—41no—
146C42×C59C2×D411yescharacteristic
147C42×C59C2×D411yescharacteristic
148C42×C59C2×D411yescharacteristic
149(C2×D59)⋊C4—41no—
150(C2×D59)⋊C4—41no—
151(C2×D59)⋊C4—41no—
152(C2×D59)⋊C4—41no—
153(C2×D59)⋊C4—41no—
154(C2×D59)⋊C4—41no—
155C4⋊Dic59—21no—
156Dic59⋊C4—41no—
157C4×Dic59—41no—
158Dic236—21no—
159Dic236—21no—
160D472—21no—
161D472—21no—
162D472—21no—
163D472—21no—
164D472—81no—
165#11745660—81no—
166#11745660—81no—
167C59⋊C16—21no—
168C59⋊C16—21no—
169C59⋊C16—21no—
170C59⋊C16—21no—
171C22×D59—41no—
172C22×D59—41no—
173Q8×C59—22no—
174Q8×C59—41no—
175Q8×C59—41no—
176Q8×C59—82no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—41no—
181C2×C4×C59C2×D811yescharacteristic
182C2×C4×C59C2×D811yescharacteristic
183C2×C4×C59C2×D811yescharacteristic
184C2×C4×C59C2×D811yescharacteristic
185C2×C4×C59C2×D811yescharacteristic
186C2×C4×C59C2×D811yescharacteristic
187C2×C4×C59C22×D411yescharacteristic
188C2×Dic59—41no—
189C2×Dic59—41no—
190D236—41no—
191D236—41no—
192D236—41no—
193D236—41no—
194D236—82no—
195D236—161no—
196C4×D59—161no—
197C4×D59—161no—
198Dic118—41no—
199Dic118—41no—
200C8×C59—21no—
201C8×C59—21no—
202C8×C59—21no—
203C8×C59C2×#4111yescharacteristic
204C8×C592−1+411yescharacteristic
205#33938—591nomaximal, Sylow
206C22×C59C22×D811yescharacteristic
207C2×D59—81no—
208C2×D59—81no—
209C2×D59—81no—
210C2×D59—81no—
211C2×D59—81no—
212C2×D59—81no—
213C4×C59—21no—
214C4×C59—21no—
215C4×C59—21no—
216C4×C59—21no—
217C4×C59—21no—
218C4×C59—21no—
219C4×C59—42no—
220C4×C59—81no—
221C4×C59—81no—
222C4×C59—81no—
223C4×C59#54811yescharacteristic
224C4×C59#57711yescharacteristic
225Dic59—81no—
226Dic59—81no—
227#3396—591no—
228#3383D5911yescharacteristic, complemented
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#335811yescharacteristic
243C2×C59#341411yescharacteristic
244C2×C59#341611yescharacteristic
245D59—161no—
246D59—161no—
247#559—1181no—
248#557—21no—
249C2×Q32—591no—
250C2×SD32—591no—
251C2×SD32—591no—
252#500C2×D5911yescharacteristic
253#499C2×D5911yescharacteristic
254#499C2×D5911yescharacteristic
255#495—591no—
256#495—591no—
257#493—591no—
258#479—21no—
259#479—21no—
260#475—21no—
261#463—1181no—
262#463—1181no—
263#460—1181no—
264C8×Q8C2×D5911yescharacteristic
265C4×Q16—591no—
266C4×Q16C2×D5911yescharacteristic
267C4×Q16C2×D5911yescharacteristic
268C4×Q16C2×D5911yescharacteristic
269C4×D8—591no—
270C4×D8—591no—
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—1181no—
294C2×SD16—1181no—
295C2×D8—591no—
296C2×D8—591no—
297C2×D8—1181no—
298C4⋊Q8—21no—
299C4⋊Q8—41no—
300#127—1181no—
301#125—21no—
302#125—21no—
303#125—41no—
304#125—41no—
305#124—1181no—
306#124—1181no—
307#124—2361no—
308#123—2361no—
309#123—2361no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—41no—
314C4×Q8—1181no—
315C4×Q8C22⋊D5911yescharacteristic
316C4×D4—1181no—
317C4×D4—1181no—
318C4×D4—2361no—
319Q32—1182no—
320SD32—1182no—
321SD32—1182no—
322C2×C16—591no—
323C2×C16—591no—
324C2×C16—591no—
325C2×C16—591no—
326#107—21no—
327#107—591no—
328#107C22×D5911yescharacteristic
329#107C22×D5911yescharacteristic
330#107C22×D5911yescharacteristic
331#106—21no—
332#106—21no—
333C4⋊C8C22⋊D5911yescharacteristic
334C4⋊C8C22⋊D5911yescharacteristic
335C4⋊C8C22⋊D5911yescharacteristic
336#103—21no—
337#103—21no—
338#103—21no—
339#103—1181no—
340#102—1181no—
341#102—1181no—
342C4×C8C22×D5911yescharacteristic
343C4×C8C22×D5911yescharacteristic
344C4×C8C22×D5911yescharacteristic
345C2×Q8—21no—
346C2×Q8—41no—
347C2×Q8—1181no—
348C2×Q8#1174566911yescharacteristic
349C2×D4—1181no—
350C2×D4—1181no—
351C2×D4—2361no—
352C2×D4—2361no—
353C22×C4—2361no—
354C22×C4—2361no—
355Q16—21no—
356Q16—1181no—
357Q16—1181no—
358Q16#1174568912yes—
359SD16—4721no—
360SD16—4721no—
361D8—1181no—
362D8—1181no—
363D8—1181no—
364D8—1181no—
365D8—4721no—
366C2×C8C2×C22⋊D5911yescharacteristic
367C2×C8C2×C22⋊D5911yescharacteristic
368C2×C8C2×C22⋊D5911yescharacteristic
369C2×C8C23×D5911yescharacteristic
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—1181no—
382C4⋊C4—2361no—
383C4⋊C4#1174566911yescharacteristic
384C4⋊C4#1174566911yescharacteristic
385C4⋊C4#1174566911yescharacteristic
386C22⋊C4—2361no—
387C22⋊C4—2361no—
388C22⋊C4—2361no—
389C22⋊C4—2361no—
390C22⋊C4—2361no—
391C22⋊C4—2361no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—2361no—
396C42C2×C22⋊D5911yescharacteristic
397C42C2×C22⋊D5911yescharacteristic
398C42C2×C22⋊D5911yescharacteristic
399C16—1181no—
400C16—1181no—
401C16—1181no—
402C16—1181no—
403C23—2361no—
404C23—2361no—
405Q8—22no—
406Q8—41no—
407Q8—41no—
408Q8—82no—
409Q8—2361no—
410Q8—2361no—
411D4—9441no—
412D4—4722no—
413D4—2361no—
414D4—2361no—
415D4—2361no—
416D4—2361no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—9441no—
422C2×C4—9441no—
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—
435C8C2×#1174568911yescharacteristic
436C8#1427377511yescharacteristic
437C22—4721no—
438C22—4721no—
439C22—4721no—
440C22—4721no—
441C22—4721no—
442C22—4721no—
443C22C22×#1174566911yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—42no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—4721no—
455C4—4721no—
456C4#1683806311yescharacteristic
457C4#1683821011yescharacteristic
458C2—9441no—
459C2—9441no—
460C2#2143673111yescharacteristic
461C2#2143723011yescharacteristic
462C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3383C2
socleC22×C59C22×D8
prime radical#16837779C22
hypercenter#3383D59
hypocenterC59#33938
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3383

Complements

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

NH ≅ G/Ntype
3. #21426335C2semidirect
4. #21426330C2semidirect
5. #21426257C2semidirect
6. #21426250C2semidirect
7. #21426240C2semidirect
8. #21426158C2semidirect
9. #21426158C2semidirect
11. #21426005C2semidirect
12. #21425965C2semidirect
13. #21425958C2semidirect
14. C4×#14273670C2semidirect
15. C59×#3383C2semidirect
228. #3383D59semidirect
290. C59#33938semidirect

Maximal subgroups

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

subgroupindex
#214371092
#214263392
#214263352
#214263302
#214262572
#214262502
#214262402
#214261582
#214261362
#214260052
#214259652
#214259582
C4×#142736702
C59×#33832
#3393859