Finite Group Databasebeta

← #28745782

Subgroups — #28745782

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#21436489C211yesmaximal, characteristic, complemented
2#21435783C211yesmaximal, characteristic
3#21426333C211yesmaximal, characteristic, complemented
4#21426299C211yesmaximal, characteristic, complemented
5#21426277C211yesmaximal, characteristic, complemented
6#21426234C211yesmaximal, characteristic, complemented
7#21426209C211yesmaximal, characteristic, complemented
8#21426187C211yesmaximal, characteristic, complemented
9#21426159C211yesmaximal, characteristic, complemented
10#21426038C211yesmaximal, characteristic, complemented
11#21426027C211yesmaximal, characteristic, complemented
12#21425962C211yesmaximal, characteristic, complemented
13#21425809C211yesmaximal, characteristic, complemented
14C2×#16837204C211yesmaximal, characteristic, complemented
15C59×#2168C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×D472C2211yescharacteristic
18#16838109—21no—
19#16838048—21no—
20C2×Dic472C2211yescharacteristic, complemented
21C2×#14273672C2211yescharacteristic
22C2×#14273672C2211yescharacteristic
23#16837807C2211yescharacteristic, complemented
24#16837786C2211yescharacteristic
25#16837738—21no—
26#16837738—21no—
27#16837698—21no—
28#16837643C2211yescharacteristic
29#16837643C2211yescharacteristic
30#16837643C2211yescharacteristic, complemented
31#16837638C2211yescharacteristic
32#16837618—21no—
33#16837555—21no—
34#16837549—21no—
35C4×Dic236C2211yescharacteristic, complemented
36C4×D472C2211yescharacteristic
37C2×#14273661C2211yescharacteristic
38C2×#14273661C2211yescharacteristic, complemented
39#16837228C2211yescharacteristic, complemented
40#16837228C2211yescharacteristic, complemented
41#16837228C2211yescharacteristic, complemented
42#16837207C2211yescharacteristic, complemented
43#16837207C2211yescharacteristic, complemented
44#16837207C2211yescharacteristic, complemented
45#16837207C2211yescharacteristic, complemented
46#16837204C2211yescharacteristic
47#16837204C2211yescharacteristic
48#16837204C2211yescharacteristic
49#16837204C2211yescharacteristic
50#16837204C2211yescharacteristic
51#16837204C2211yescharacteristic
52C2×M32×C59C2211yescharacteristic, complemented
53C22×C16×C59C2211yescharacteristic
54C8×D4×C59C2211yescharacteristic
55C59×C4⋊C16C2211yescharacteristic, complemented
56C59×C22⋊C16C2211yescharacteristic
57C59×C22⋊C16C2211yescharacteristic, complemented
58C59×#346C2211yescharacteristic, complemented
59C2×#11745685—21no—
60C22×D236—21no—
61#14273749—22no—
62#14273749—41no—
63C2×Dic236C2311yescharacteristic
64C2×Dic236C2311yescharacteristic
65C2×#11745660—21no—
66C2×#11745660—21no—
67C2×D472—22no—
68C2×D472—21no—
69C2×D472—41no—
70C2×D472C2311yescharacteristic
71C2×D472C2311yescharacteristic
72#14273727—21no—
73#14273718—41no—
74#14273715—21no—
75#14273712—41no—
76#14273706—21no—
77#14273706—21no—
78#14273706—21no—
79#14273704—41no—
80#14273704—41no—
81#14273701—21no—
82#14273700—41no—
83#14273695—41no—
84C4×Dic118—21no—
85C4×D236—21no—
86D4×Dic59—41no—
87C2×C4⋊Dic59—21no—
88C2×C4⋊Dic59—21no—
89C22⋊C4×D59—41no—
90Dic472—41no—
91#14273672—22no—
92#14273672—41no—
93#14273661—21no—
94#14273661—21no—
95#14273661—21no—
96#14273661C2311yescharacteristic
97#14273661C2311yescharacteristic
98#14273661C2311yescharacteristic
99#14273661C2311yescharacteristic
100#14273652—21no—
101#14273652—21no—
102#14273652—21no—
103#14273652—21no—
104#14273649—21no—
105#14273649—21no—
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C4×D4×C59D411yescharacteristic
109M32×C59—41no—
110C2×C16×C59—41no—
111C2×C16×C59C2311yescharacteristic
112C2×C16×C59C2311yescharacteristic
113C2×C16×C59C2311yescharacteristic
114C2×C16×C59C2311yescharacteristic
115C4⋊C8×C59D411yescharacteristic
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C23×D59—41no—
120C2×D4×C59D811yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C2×C22⋊D59—41no—
124C2×C22⋊D59—41no—
125C22×Dic59—41no—
126C22×Dic59—41no—
127#11745685—42no—
128#11745685—81no—
129C2×D236—21no—
130C2×D236—21no—
131C2×D236—42no—
132C2×D236—41no—
133C2×D236—81no—
134C2×C4×D59—41no—
135C2×C4×D59—41no—
136C2×Dic118—21no—
137C2×Dic118—21no—
138C2×Dic118—41no—
139C2×C8×C59—21no—
140C2×C8×C59—21no—
141C2×C8×C59—21no—
142C2×C8×C59—21no—
143C2×C8×C59C2×D411yescharacteristic
144C2×C8×C59C2×D411yescharacteristic
145C2×C8×C59C2×D411yescharacteristic
146C2×C8×C59C2411yescharacteristic
147C4⋊C4×C59D811yescharacteristic
148C22⋊C4×C59D811yescharacteristic
149C22⋊C4×C59D811yescharacteristic
150C42×C59C2×D411yescharacteristic
151C22⋊Dic59—41no—
152C22⋊Dic59—41no—
153C22⋊Dic59—41no—
154(C2×D59)⋊C4—41no—
155(C2×D59)⋊C4—41no—
156(C2×D59)⋊C4—41no—
157(C2×D59)⋊C4—41no—
158C4⋊Dic59—21no—
159C4⋊Dic59—21no—
160C4⋊Dic59—21no—
161C4⋊Dic59—21no—
162C4⋊Dic59—41no—
163C4⋊Dic59—41no—
164C4⋊Dic59—41no—
165Dic59⋊C4—41no—
166Dic59⋊C4—41no—
167Dic59⋊C4—41no—
168Dic59⋊C4—41no—
169C4×Dic59—41no—
170Dic236—21no—
171Dic236—21no—
172Dic236—21no—
173Dic236—21no—
174D472—21no—
175D472—21no—
176D472—21no—
177D472—42no—
178D472#4112yes—
179#11745660—42no—
180#11745660—81no—
181C16×C59—21no—
182C16×C59—21no—
183C16×C59—21no—
184C16×C59—21no—
185C23×C59C2×D811yescharacteristic
186C23×C59C2×D811yescharacteristic
187C22×D59—41no—
188C22×D59—41no—
189C22×D59—82no—
190C22×D59—82no—
191C22×D59—161no—
192D4×C59—22no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59C2×D811yescharacteristic
198C2×C4×C59C2×D811yescharacteristic
199C2×C4×C59C2×D811yescharacteristic
200C2×C4×C59C2×D811yescharacteristic
201C2×C4×C59C22×D411yescharacteristic
202C22⋊D59—82no—
203C22⋊D59—161no—
204C2×Dic59—41no—
205C2×Dic59—41no—
206C2×Dic59—41no—
207C2×Dic59—41no—
208C2×Dic59—41no—
209C2×Dic59—41no—
210C2×Dic59—161no—
211C2×Dic59—161no—
212D236—22no—
213D236—41no—
214D236—41no—
215D236—41no—
216D236—82no—
217C4×D59—82no—
218C4×D59—161no—
219Dic118—41no—
220Dic118—41no—
221Dic118—41no—
222Dic118—41no—
223Dic118—161no—
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#54611yescharacteristic
249C4×C59#57711yescharacteristic
250Dic59—81no—
251Dic59—81no—
252Dic59—81no—
253Dic59—81no—
254Dic59—81no—
255Dic59—81no—
256#3338—591no—
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#2168D5911yescharacteristic, complemented
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—1181no—
281#546—1181no—
282C2×Q32—591no—
283C2×SD32—591no—
284C2×SD32—591no—
285C2×M32C2×D5911yescharacteristic, complemented
286C22×C16C2×D5911yescharacteristic
287#500—591no—
288#495—591no—
289#484—1181no—
290#484—1181no—
291#477—1181no—
292#467—591no—
293#467—591no—
294#467—591no—
295#466—591no—
296#463—1181no—
297#450—1181no—
298#449—1181no—
299C4×Q16—591no—
300C4×D8—591no—
301C8×D4C2×D5911yescharacteristic
302C2×#107—591no—
303C2×#107—591no—
304#366—591no—
305#366—591no—
306#366—591no—
307C4⋊C16C2×D5911yescharacteristic, complemented
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⋊C16C2×D5911yescharacteristic
319C22⋊C16C2×D5911yescharacteristic, complemented
320#346C2×D5911yescharacteristic, complemented
321C59#3388111yescharacteristic, complemented, Sylow
322C2×#41—1181no—
323C22×D4—1181no—
324#135—1182no—
325#135—2361no—
326C2×Q16—591no—
327C2×Q16—591no—
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—1181no—
338#125—2361no—
339#124—1181no—
340#123—2361no—
341C22⋊Q8—1181no—
342C22⋊Q8—1181no—
343C22⋊Q8—1181no—
344C22⋊Q8—2361no—
345C22⋊Q8—2361no—
346#121—1181no—
347#121—2361no—
348#120—2361no—
349C4×Q8—1181no—
350C4×D4—1181no—
351C4×D4—2361no—
352C4×D4D23611yescharacteristic
353C2×C4⋊C4—1181no—
354C2×C4⋊C4—1181no—
355C2×C22⋊C4—2361no—
356Q32—2361no—
357SD32—1182no—
358SD32—2361no—
359M32—41no—
360C2×C16—41no—
361C2×C16C22×D5911yescharacteristic
362C2×C16C22×D5911yescharacteristic
363C2×C16C22×D5911yescharacteristic
364C2×C16C22×D5911yescharacteristic
365#107—591no—
366#107—591no—
367#107—591no—
368#107—591no—
369#107—1181no—
370#107—1181no—
371#107—1181no—
372C4⋊C8D23611yescharacteristic
373#103—1181no—
374#103—1181no—
375#103—1181no—
376#103—1181no—
377#102—1181no—
378#102—1181no—
379C22⋊C8D23611yescharacteristic
380C22⋊C8D23611yescharacteristic
381C4×C8C22×D5911yescharacteristic
382C24—2361no—
383#41—4721no—
384#41—2362no—
385C2×Q8—1181no—
386C2×Q8—1181no—
387C2×Q8—2361no—
388C2×D4—1181no—
389C2×D4—1181no—
390C2×D4—4721no—
391C2×D4—2362no—
392C2×D4—2361no—
393C2×D4—2361no—
394C2×D4—2361no—
395C2×D4D47211yescharacteristic
396C22×C4—2361no—
397C22×C4—2361no—
398C22×C4—2361no—
399C22×C4—2361no—
400C22×C4C2×D23611yescharacteristic
401C22×C4C2×D23611yescharacteristic
402Q16—1181no—
403Q16—1181no—
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×D23611yescharacteristic
418C2×C8C2×D23611yescharacteristic
419C2×C8C2×D23611yescharacteristic
420C2×C8C23×D5911yescharacteristic
421C4⋊C4—1181no—
422C4⋊C4—1181no—
423C4⋊C4—1181no—
424C4⋊C4—1181no—
425C4⋊C4—2361no—
426C4⋊C4—2361no—
427C4⋊C4—2361no—
428C4⋊C4—2361no—
429C4⋊C4—2361no—
430C4⋊C4—2361no—
431C4⋊C4—2361no—
432C4⋊C4D47211yescharacteristic
433C22⋊C4—2361no—
434C22⋊C4—2361no—
435C22⋊C4—2361no—
436C22⋊C4—2361no—
437C22⋊C4—2361no—
438C22⋊C4—2361no—
439C22⋊C4—2361no—
440C22⋊C4D47211yescharacteristic
441C22⋊C4D47211yescharacteristic
442C42—2361no—
443C42C2×D23611yescharacteristic
444C16—21no—
445C16—21no—
446C16—21no—
447C16—21no—
448C23—9441no—
449C23—4722no—
450C23—4722no—
451C23—2361no—
452C23—2361no—
453C23C2×D47211yescharacteristic
454C23C2×D47211yescharacteristic
455Q8—9441no—
456Q8—2361no—
457Q8—2361no—
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—9441no—
473C2×C4—9441no—
474C2×C4—9441no—
475C2×C4—4722no—
476C2×C4—2361no—
477C2×C4—2361no—
478C2×C4—2361no—
479C2×C4—2361no—
480C2×C4—2361no—
481C2×C4—2361no—
482C2×C4C2×D47211yescharacteristic
483C2×C4C2×D47211yescharacteristic
484C2×C4C2×D47211yescharacteristic
485C2×C4C2×D47211yescharacteristic
486C2×C4C22×D23611yescharacteristic
487C8—21no—
488C8—21no—
489C8—21no—
490C8#41×D5911yescharacteristic
491C8#1427377211yescharacteristic
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#1683784512yes—
505C22C22×D47211yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—4721no—
511C4—4721no—
512C4—4721no—
513C4—4721no—
514C4—4721no—
515C4—4721no—
516C4#1683804711yescharacteristic
517C4#1683821411yescharacteristic
518C2—22no—
519C2—41no—
520C2—9441no—
521C2—4722no—
522C2#2143591011yescharacteristic
523C2C2×#1683784511yescharacteristic
524C2#2143723711yescharacteristic
subgroupG/H
centerC22C22×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2168C2
socleC22×C59C22×D8
prime radical#21435783C2
hypercenter#2168D59
hypocenterC59#33881
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. #21436489C2semidirect
3. #21426333C2semidirect
4. #21426299C2semidirect
5. #21426277C2semidirect
6. #21426234C2semidirect
7. #21426209C2semidirect
8. #21426187C2semidirect
9. #21426159C2semidirect
10. #21426038C2semidirect
11. #21426027C2semidirect
12. #21425962C2semidirect
13. #21425809C2semidirect
14. C2×#16837204C2semidirect
15. C59×#2168C2semidirect
20. C2×Dic472C22semidirect
23. #16837807C22semidirect
30. #16837643C22semidirect
35. C4×Dic236C22semidirect
38. C2×#14273661C22semidirect
39. #16837228C22semidirect
40. #16837228C22semidirect
41. #16837228C22semidirect
42. #16837207C22semidirect
43. #16837207C22semidirect
44. #16837207C22semidirect
45. #16837207C22semidirect
52. C2×M32×C59C22semidirect
55. C59×C4⋊C16C22semidirect
57. C59×C22⋊C16C22semidirect
58. C59×#346C22semidirect
268. #2168D59semidirect
285. C2×M32C2×D59semidirect
307. C4⋊C16C2×D59semidirect
319. C22⋊C16C2×D59semidirect
320. #346C2×D59semidirect
321. C59#33881semidirect

Maximal subgroups

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

subgroupindex
#214364892
#214357832
#214263332
#214262992
#214262772
#214262342
#214262092
#214261872
#214261592
#214260382
#214260272
#214259622
#214258092
C2×#168372042
C59×#21682
#3388159