Finite Group Databasebeta

← #28745774

Subgroups — #28745774

Order 15104.

570 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435783C211yesmaximal, characteristic
2#21426335C211yesmaximal, characteristic, complemented
3#21426298C211yesmaximal, characteristic, complemented
4#21426276C211yesmaximal, characteristic, complemented
5#21426230C211yesmaximal, characteristic, complemented
6#21426211C211yesmaximal, characteristic, complemented
7#21426185C211yesmaximal, characteristic, complemented
8#21426158C211yesmaximal, characteristic, complemented
9#21426034C211yesmaximal, characteristic, complemented
10#21426024C211yesmaximal, characteristic, complemented
11#21425958C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425806C211yesmaximal, characteristic, complemented
14C2×#16837202C211yesmaximal, characteristic, complemented
15C59×#3338C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×D472C2211yescharacteristic
18#16838048—21no—
19C2×#14273673C2211yescharacteristic, complemented
20C2×#14273670C2211yescharacteristic
21C2×#14273670C2211yescharacteristic
22C2×#14273667C2211yescharacteristic, complemented
23C22×C59⋊C16C2211yescharacteristic
24#16837786C2211yescharacteristic
25#16837643C2211yescharacteristic
26#16837638C2211yescharacteristic
27#16837618—21no—
28#16837555—21no—
29#16837549—21no—
30C4×D472C2211yescharacteristic
31#16837227C2211yescharacteristic, complemented
32#16837227C2211yescharacteristic, complemented
33#16837227C2211yescharacteristic, complemented
34C4⋊(C59⋊C16)C2211yescharacteristic, complemented
35#16837206C2211yescharacteristic, complemented
36#16837205C2211yescharacteristic, complemented
37#16837205C2211yescharacteristic, complemented
38#16837205C2211yescharacteristic, complemented
39#16837202C2211yescharacteristic
40#16837202C2211yescharacteristic
41#16837202C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic, complemented
47#16837178C2211yescharacteristic, complemented
48C59×#554—21no—
49C59×#500C2211yescharacteristic, complemented
50C59×#484—21no—
51C59×#484—21no—
52C59×#477—21no—
53C59×#467C2211yescharacteristic
54C59×#467C2211yescharacteristic, complemented
55C4×Q16×C59C2211yescharacteristic, complemented
56C8×D4×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic, complemented
59C2×#11745685—21no—
60C22×D236—21no—
61#14273749—22no—
62#14273749—41no—
63C2×Dic236C2311yescharacteristic
64C2×#11745660—21no—
65C2×#11745660—21no—
66C2×D472—22no—
67C2×D472—21no—
68C2×D472—41no—
69C2×D472C2311yescharacteristic
70C2×D472C2311yescharacteristic
71#14273715—21no—
72#14273712—41no—
73#14273706—21no—
74#14273701—21no—
75#14273700—41no—
76#14273695—41no—
77C4×D236—21no—
78C22⋊C4×D59—41no—
79#14273673—41no—
80#14273670—22no—
81#14273670—41no—
82#14273667—41no—
83C2×C59⋊C16—41no—
84C2×C59⋊C16C2311yescharacteristic
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273652—21no—
90#14273649—21no—
91#14273649—21no—
92C2×Q16×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95C4⋊Q8×C59—21no—
96#125×C59—41no—
97C22⋊Q8×C59—21no—
98C22⋊Q8×C59—21no—
99C22⋊Q8×C59—41no—
100C22⋊Q8×C59—41no—
101C4×Q8×C59—21no—
102C4×D4×C59—41no—
103C4×D4×C59D411yescharacteristic
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106#107×C59—21no—
107#107×C59—21no—
108#107×C59—21no—
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#103×C59—21no—
114#103×C59—21no—
115#103×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C23×D59—41no—
120C2×Q8×C59—21no—
121C2×Q8×C59—41no—
122C2×D4×C59D811yescharacteristic
123C22×C4×C59—41no—
124C22×C4×C59—41no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C2×C22⋊D59—41no—
128C2×C22⋊D59—41no—
129#11745685—42no—
130#11745685—81no—
131C2×D236—21no—
132C2×D236—21no—
133C2×D236—42no—
134C2×D236—41no—
135C2×D236—81no—
136C2×C4×D59—41no—
137C2×C4×D59—41no—
138C2×Dic118—21no—
139Q16×C59—21no—
140Q16×C59—21no—
141C2×C8×C59—21no—
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59C2×D411yescharacteristic
146C2×C8×C59C2×D411yescharacteristic
147C2×C8×C59C2×D411yescharacteristic
148C2×C8×C59C2411yescharacteristic
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—41no—
153C4⋊C4×C59—41no—
154C4⋊C4×C59—41no—
155C4⋊C4×C59—41no—
156C4⋊C4×C59—41no—
157C4⋊C4×C59—41no—
158C4⋊C4×C59D811yescharacteristic
159C22⋊C4×C59—41no—
160C22⋊C4×C59—41no—
161C22⋊C4×C59D811yescharacteristic
162C22⋊C4×C59D811yescharacteristic
163C42×C59—41no—
164C42×C59C2×D411yescharacteristic
165C22⋊Dic59—41no—
166(C2×D59)⋊C4—41no—
167(C2×D59)⋊C4—41no—
168(C2×D59)⋊C4—41no—
169(C2×D59)⋊C4—41no—
170C4⋊Dic59—21no—
171Dic59⋊C4—41no—
172Dic236—21no—
173Dic236—21no—
174D472—21no—
175D472—21no—
176D472—21no—
177D472—42no—
178D472#4112yes—
179#11745660—42no—
180#11745660—81no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59C2×D811yescharacteristic
186C23×C59C2×D811yescharacteristic
187C22×D59—41no—
188C22×D59—41no—
189C22×D59—82no—
190C22×D59—82no—
191C22×D59—161no—
192Q8×C59—41no—
193Q8×C59—41no—
194Q8×C59—161no—
195D4×C59—22no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—41no—
201C2×C4×C59—41no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—161no—
205C2×C4×C59—161no—
206C2×C4×C59C2×D811yescharacteristic
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C2×D811yescharacteristic
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C22×D411yescharacteristic
211C22⋊D59—82no—
212C22⋊D59—161no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215D236—22no—
216D236—41no—
217D236—41no—
218D236—41no—
219D236—82no—
220C4×D59—82no—
221C4×D59—161no—
222Dic118—41no—
223Dic118—41no—
224C8×C59—21no—
225C8×C59—21no—
226C8×C59—21no—
227C8×C59C2×#4111yescharacteristic
228C8×C59H(5,𝔽2)11yescharacteristic
229#33881—591nomaximal, Sylow
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—21no—
235C22×C59#51012yes—
236C22×C59C22×D811yescharacteristic
237C2×D59—42no—
238C2×D59—81no—
239C2×D59—81no—
240C2×D59—81no—
241C2×D59—81no—
242C2×D59—81no—
243C2×D59—164no—
244C4×C59—21no—
245C4×C59—21no—
246C4×C59—21no—
247C4×C59—21no—
248C4×C59—81no—
249C4×C59—81no—
250C4×C59—81no—
251C4×C59—81no—
252C4×C59#54611yescharacteristic
253C4×C59#57711yescharacteristic
254Dic59—81no—
255Dic59—81no—
256#3338D5911yescharacteristic, complemented
257#3280—591no—
258#2243—591no—
259#2236—591no—
260#2232—591no—
261#2224—591no—
262#2220—591no—
263#2216—591no—
264#2211—591no—
265#2189—591no—
266#2187—591no—
267#2175—591no—
268#2168—591no—
269#2139—591no—
270C2×#357—591no—
271C2×C59—22no—
272C2×C59—41no—
273C2×C59#329411yescharacteristic
274C2×C59C2×#51011yescharacteristic
275C2×C59#341511yescharacteristic
276D59—82no—
277D59—161no—
278C2×#135—591no—
279C22×D8—591no—
280#554—21no—
281#546—1181no—
282C2×Q32—591no—
283C2×SD32—591no—
284C2×SD32—591no—
285C2×M32—591no—
286C22×C16—591no—
287#500C2×D5911yescharacteristic, complemented
288#495—591no—
289#484—21no—
290#484—21no—
291#477—21no—
292#467—591no—
293#467C2×D5911yescharacteristic
294#467C2×D5911yescharacteristic, complemented
295#466—591no—
296#463—1181no—
297#450—1181no—
298#449—1181no—
299C4×Q16C2×D5911yescharacteristic, complemented
300C4×D8—591no—
301C8×D4C2×D5911yescharacteristic
302C2×#107C2×D5911yescharacteristic
303C2×#107C2×D5911yescharacteristic, complemented
304#366—591no—
305#366—591no—
306#366—591no—
307C4⋊C16—591no—
308#358—591no—
309#358—591no—
310#358—591no—
311#358—591no—
312#357—591no—
313#357—591no—
314#357—591no—
315#357—591no—
316#357—591no—
317#357—591no—
318C22⋊C16—591no—
319C22⋊C16—591no—
320#346—591no—
321C59#3388111yescharacteristic, complemented, Sylow
322C2×#41—1181no—
323C22×D4—1181no—
324#135—1182no—
325#135—2361no—
326C2×Q16—591no—
327C2×Q16C22×D5911yescharacteristic
328C2×SD16—1181no—
329C2×SD16—1181no—
330C2×D8—591no—
331C2×D8—591no—
332C2×D8—1182no—
333C2×D8—1181no—
334C2×D8—2361no—
335C22×C8C22×D5911yescharacteristic
336C22×C8C22×D5911yescharacteristic
337C4⋊Q8—21no—
338#125—41no—
339#124—1181no—
340#123—2361no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—41no—
344C22⋊Q8—41no—
345C22⋊Q8—1181no—
346#121—1181no—
347#121—2361no—
348#120—2361no—
349C4×Q8—21no—
350C4×D4—41no—
351C4×D4—1181no—
352C4×D4C22⋊D5911yescharacteristic
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C22⋊C4—2361no—
356Q32—2361no—
357SD32—1182no—
358SD32—2361no—
359M32—2361no—
360C2×C16—591no—
361C2×C16—591no—
362C2×C16—591no—
363C2×C16—591no—
364C2×C16—2361no—
365#107—21no—
366#107—21no—
367#107—21no—
368#107—591no—
369#107C22×D5911yescharacteristic
370#107C22×D5911yescharacteristic
371#107C22×D5911yescharacteristic
372C4⋊C8C22⋊D5911yescharacteristic
373#103—21no—
374#103—21no—
375#103—21no—
376#103—1181no—
377#102—1181no—
378#102—1181no—
379C22⋊C8C22⋊D5911yescharacteristic
380C22⋊C8C22⋊D5911yescharacteristic
381C4×C8C22×D5911yescharacteristic
382C24—2361no—
383#41—4721no—
384#41—2362no—
385C2×Q8—21no—
386C2×Q8—41no—
387C2×Q8—1181no—
388C2×D4—1181no—
389C2×D4—1181no—
390C2×D4—4721no—
391C2×D4—2362no—
392C2×D4—2361no—
393C2×D4—2361no—
394C2×D4—2361no—
395C2×D4#1174566911yescharacteristic
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—2361no—
399C22×C4—2361no—
400C22×C4C2×C22⋊D5911yescharacteristic
401C22×C4C2×C22⋊D5911yescharacteristic
402Q16—21no—
403Q16—21no—
404Q16—1181no—
405Q16—1181no—
406SD16—4721no—
407SD16—2362no—
408D8—592no—
409D8—1181no—
410D8—1181no—
411D8—1181no—
412D8—2362no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8C2×C22⋊D5911yescharacteristic
418C2×C8C2×C22⋊D5911yescharacteristic
419C2×C8C2×C22⋊D5911yescharacteristic
420C2×C8C23×D5911yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—1181no—
431C4⋊C4—2361no—
432C4⋊C4#1174566911yescharacteristic
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—2361no—
436C22⋊C4—2361no—
437C22⋊C4—2361no—
438C22⋊C4—2361no—
439C22⋊C4—2361no—
440C22⋊C4#1174566911yescharacteristic
441C22⋊C4#1174566911yescharacteristic
442C42—41no—
443C42C2×C22⋊D5911yescharacteristic
444C16—1181no—
445C16—1181no—
446C16—1181no—
447C16—1181no—
448C23—9441no—
449C23—4722no—
450C23—4722no—
451C23—2361no—
452C23—2361no—
453C23C2×#1174566911yescharacteristic
454C23C2×#1174566911yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—161no—
458Q8—2361no—
459Q8—2361no—
460D4—22no—
461D4—1182no—
462D4—9441no—
463D4—4722no—
464D4—4722no—
465D4—2361no—
466D4—2361no—
467D4—2361no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—161no—
477C2×C4—161no—
478C2×C4—9441no—
479C2×C4—4722no—
480C2×C4—2361no—
481C2×C4—2361no—
482C2×C4C2×#1174566911yescharacteristic
483C2×C4C2×#1174566911yescharacteristic
484C2×C4C2×#1174566911yescharacteristic
485C2×C4C2×#1174566911yescharacteristic
486C2×C4C22×C22⋊D5911yescharacteristic
487C8—21no—
488C8—21no—
489C8—21no—
490C8#41×D5911yescharacteristic
491C8#1427377311yescharacteristic
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—9444no—
498C22—4721no—
499C22—4721no—
500C22—4721no—
501C22—4721no—
502C22—4721no—
503C22—2362no—
504C22#1683784312yes—
505C22C22×#1174566911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—4721no—
515C4—4721no—
516C4#1683804011yescharacteristic
517C4#1683821011yescharacteristic
518C2—22no—
519C2—41no—
520C2—9441no—
521C2—4722no—
522C2#2143590311yescharacteristic
523C2C2×#1683784311yescharacteristic
524C2#2143723511yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3338C2
socleC22×C59C22×D8
prime radical#21435783C2
hypercenter#3338D59
hypocenterC59#33881
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3338

Complements

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

NH ≅ G/Ntype
2. #21426335C2semidirect
3. #21426298C2semidirect
4. #21426276C2semidirect
5. #21426230C2semidirect
6. #21426211C2semidirect
7. #21426185C2semidirect
8. #21426158C2semidirect
9. #21426034C2semidirect
10. #21426024C2semidirect
11. #21425958C2semidirect
12. #21425922C2semidirect
13. #21425806C2semidirect
14. C2×#16837202C2semidirect
15. C59×#3338C2semidirect
19. C2×#14273673C22semidirect
22. C2×#14273667C22semidirect
31. #16837227C22semidirect
32. #16837227C22semidirect
33. #16837227C22semidirect
34. C4⋊(C59⋊C16)C22semidirect
35. #16837206C22semidirect
36. #16837205C22semidirect
37. #16837205C22semidirect
38. #16837205C22semidirect
46. C22⋊(C59⋊C16)C22semidirect
47. #16837178C22semidirect
49. C59×#500C22semidirect
54. C59×#467C22semidirect
55. C4×Q16×C59C22semidirect
58. C2×#107×C59C22semidirect
256. #3338D59semidirect
287. #500C2×D59semidirect
294. #467C2×D59semidirect
299. C4×Q16C2×D59semidirect
303. C2×#107C2×D59semidirect
321. C59#33881semidirect

Maximal subgroups

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

subgroupindex
#214357832
#214263352
#214262982
#214262762
#214262302
#214262112
#214261852
#214261582
#214260342
#214260242
#214259582
#214259222
#214258062
C2×#168372022
C59×#33382
#3388159