Finite Group Databasebeta

← #28745714

Subgroups — #28745714

Order 15104.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D472C211yesmaximal, characteristic
2#21426339C211yesmaximal, characteristic, complemented
3#21426287C211yesmaximal, characteristic, complemented
4#21426276C211yesmaximal, characteristic, complemented
5#21426250C211yesmaximal, characteristic, complemented
6#21426211C211yesmaximal, characteristic, complemented
7#21426194C211yesmaximal, characteristic, complemented
8#21426136C211yesmaximal, characteristic, complemented
9#21426024C211yesmaximal, characteristic, complemented
10#21426024C211yesmaximal, characteristic, complemented
11#21425958C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425811C211yesmaximal, characteristic, complemented
14C2×#16837202C211yesmaximal, characteristic, complemented
15C59×#3336C211yesmaximal, characteristic, complemented
16C22×D472C2211yescharacteristic
17C22×D472C2211yescharacteristic
18D4×D236—21no—
19C2×#14273670C2211yescharacteristic
20C2×#14273670C2211yescharacteristic, complemented
21C2×#14273670C2211yescharacteristic, complemented
22C2×#14273667C2211yescharacteristic, complemented
23C22×C59⋊C16C2211yescharacteristic, complemented
24#16837779C2211yescharacteristic
25#16837638C2211yescharacteristic
26#16837638C2211yescharacteristic
27#16837598—21no—
28#16837538—21no—
29#16837538—21no—
30C4×D472C2211yescharacteristic
31#16837229C2211yescharacteristic, complemented
32#16837227C2211yescharacteristic, complemented
33#16837227C2211yescharacteristic, complemented
34C4⋊(C59⋊C16)C2211yescharacteristic, complemented
35#16837206C2211yescharacteristic, complemented
36#16837202C2211yescharacteristic
37#16837202C2211yescharacteristic
38#16837202C2211yescharacteristic
39#16837202C2211yescharacteristic, complemented
40#16837202C2211yescharacteristic, complemented
41#16837202C2211yescharacteristic, complemented
42#16837202C2211yescharacteristic, complemented
43#16837202C2211yescharacteristic, complemented
44#16837202C2211yescharacteristic, complemented
45C22⋊(C59⋊C16)C2211yescharacteristic, complemented
46C22⋊(C59⋊C16)C2211yescharacteristic, complemented
47#16837178C2211yescharacteristic, complemented
48C59×#552—21no—
49C59×#499C2211yescharacteristic, complemented
50C59×#485—21no—
51C59×#483—21no—
52C59×#479—21no—
53C59×#467C2211yescharacteristic, complemented
54C59×#467C2211yescharacteristic, complemented
55C4×Q16×C59C2211yescharacteristic, complemented
56C8×D4×C59C2211yescharacteristic
57C59×#427C2211yescharacteristic, complemented
58C2×#107×C59C2211yescharacteristic, complemented
59C22×D236—21no—
60C22×D236—21no—
61C2×D4×D59—41no—
62C2×D472—22no—
63C2×D472—22no—
64C2×D472—21no—
65C2×D472—21no—
66C2×D472—21no—
67C2×D472—41no—
68C2×D472—41no—
69C2×D472C2311yescharacteristic
70C2×D472C2311yescharacteristic
71C2×D472C2311yescharacteristic
72#14273724—21no—
73#14273702—41no—
74#14273701—21no—
75#14273701—21no—
76#14273697—41no—
77#14273697—41no—
78C4×D236—21no—
79#14273670—22no—
80#14273670—41no—
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#14273649—21no—
90#14273649—21no—
91#14273649—21no—
92C2×Q16×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95#126×C59—41no—
96#125×C59—21no—
97#125×C59—41no—
98#123×C59—41no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59—21no—
101C4×Q8×C59—21no—
102C4×D4×C59D411yescharacteristic
103#117×C59—21no—
104#117×C59—41no—
105C2×C4⋊C4×C59—21no—
106#107×C59—21no—
107#107×C59C2311yescharacteristic
108#107×C59C2311yescharacteristic
109#107×C59C2311yescharacteristic
110#106×C59—21no—
111#106×C59—21no—
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—
120C23×D59—41no—
121C2×Q8×C59—21no—
122C2×D4×C59D811yescharacteristic
123C22×C4×C59—41no—
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C2×C22⋊D59—41no—
127C2×C22⋊D59—41no—
128D4×D59—84no—
129C2×D236—21no—
130C2×D236—21no—
131C2×D236—21no—
132C2×D236—42no—
133C2×D236—42no—
134C2×D236—41no—
135C2×D236—41no—
136C2×D236—41no—
137C2×D236—41no—
138C2×D236—81no—
139C2×D236—81no—
140C2×C4×D59—41no—
141Q16×C59—21no—
142Q16×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59C2×D411yescharacteristic
148C2×C8×C59C2×D411yescharacteristic
149C2×C8×C59C2×D411yescharacteristic
150C2×C8×C59C2411yescharacteristic
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—41no—
155C4⋊C4×C59—41no—
156C4⋊C4×C59—41no—
157C4⋊C4×C59—41no—
158C4⋊C4×C59—41no—
159C4⋊C4×C59—41no—
160C4⋊C4×C59D811yescharacteristic
161C22⋊C4×C59—41no—
162C22⋊C4×C59—41no—
163C22⋊C4×C59—41no—
164C22⋊C4×C59D811yescharacteristic
165C22⋊C4×C59D811yescharacteristic
166C42×C59—41no—
167C42×C59—41no—
168C42×C59C2×D411yescharacteristic
169(C2×D59)⋊C4—41no—
170(C2×D59)⋊C4—41no—
171(C2×D59)⋊C4—41no—
172C4⋊Dic59—21no—
173D472—21no—
174D472—21no—
175D472—21no—
176D472—21no—
177D472—21no—
178D472—42no—
179D472—42no—
180D472—81no—
181D472#4112yes—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C59⋊C16—21no—
186C23×C59C2×D811yescharacteristic
187C23×C59C2×D811yescharacteristic
188C22×D59—41no—
189C22×D59—41no—
190C22×D59—41no—
191C22×D59—82no—
192C22×D59—82no—
193C22×D59—82no—
194C22×D59—82no—
195C22×D59—161no—
196C22×D59—161no—
197Q8×C59—41no—
198Q8×C59—41no—
199D4×C59—22no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—82no—
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C2×D811yescharacteristic
211C2×C4×C59C2×D811yescharacteristic
212C2×C4×C59C2×D811yescharacteristic
213C2×C4×C59C22×D411yescharacteristic
214C22⋊D59—82no—
215C22⋊D59—82no—
216C2×Dic59—41no—
217D236—22no—
218D236—41no—
219D236—41no—
220D236—41no—
221D236—41no—
222D236—41no—
223D236—82no—
224D236—82no—
225D236—82no—
226D236—161no—
227C4×D59—82no—
228C8×C59—21no—
229C8×C59—21no—
230C8×C59—21no—
231C8×C59C2×#4111yescharacteristic
232C8×C59H(5,𝔽2)11yescharacteristic
233#33877—591nomaximal, Sylow
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—21no—
237C22×C59—21no—
238C22×C59—21no—
239C22×C59#50912yes—
240C22×C59C22×D811yescharacteristic
241C2×D59—42no—
242C2×D59—81no—
243C2×D59—81no—
244C2×D59—81no—
245C2×D59—81no—
246C2×D59—81no—
247C2×D59—81no—
248C2×D59—81no—
249C2×D59—81no—
250C2×D59—164no—
251C2×D59—164no—
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—21no—
255C4×C59—21no—
256C4×C59—81no—
257C4×C59—81no—
258C4×C59—81no—
259C4×C59—81no—
260C4×C59#54611yescharacteristic
261C4×C59#57711yescharacteristic
262Dic59—81no—
263#3336D5911yescharacteristic, complemented
264D4×D8—591no—
265#2244—591no—
266#2234—591no—
267#2232—591no—
268#2228—591no—
269#2220—591no—
270#2217—591no—
271#2207—591no—
272#2187—591no—
273#2187—591no—
274#2175—591no—
275#2168—591no—
276#2140—591no—
277C2×#357—591no—
278C2×C59—22no—
279C2×C59—41no—
280C2×C59#329411yescharacteristic
281C2×C59C2×#50911yescharacteristic
282C2×C59#341611yescharacteristic
283D59—82no—
284D59—161no—
285D59—161no—
286C22×D8—591no—
287C22×D8—591no—
288#552—21no—
289D42—1181no—
290C2×SD32—591no—
291C2×SD32—591no—
292C2×SD32—591no—
293C2×M32—591no—
294C22×C16—591no—
295#499C2×D5911yescharacteristic, complemented
296#493—591no—
297#485—21no—
298#483—21no—
299#479—21no—
300#467C2×D5911yescharacteristic, complemented
301#467C2×D5911yescharacteristic, complemented
302#466—591no—
303#466—591no—
304#459—1181no—
305#447—1181no—
306#447—1181no—
307C4×Q16C2×D5911yescharacteristic, complemented
308C4×D8—591no—
309C8×D4C2×D5911yescharacteristic
310#427C2×D5911yescharacteristic, complemented
311C2×#107C2×D5911yescharacteristic, complemented
312#367—591no—
313#366—591no—
314#366—591no—
315C4⋊C16—591no—
316#358—591no—
317#357—591no—
318#357—591no—
319#357—591no—
320#357—591no—
321#357—591no—
322#357—591no—
323#357—591no—
324#357—591no—
325#357—591no—
326C22⋊C16—591no—
327C22⋊C16—591no—
328#346—591no—
329C59#3387711yescharacteristic, complemented, Sylow
330C22×D4—1181no—
331C22×D4—1181no—
332C22×D4—2361no—
333C2×Q16C22×D5911yescharacteristic
334C2×D8—591no—
335C2×D8—591no—
336C2×D8—591no—
337C2×D8—1182no—
338C2×D8—1182no—
339C2×D8—1181no—
340C2×D8—1181no—
341C2×D8—1181no—
342C2×D8—2361no—
343C2×D8—2361no—
344C22×C8C22×D5911yescharacteristic
345C22×C8C22×D5911yescharacteristic
346#127—1181no—
347#126—41no—
348#125—21no—
349#125—41no—
350#123—41no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353#121—1181no—
354#121—1181no—
355#121—2361no—
356#120—2361no—
357#120—2361no—
358C4×Q8—21no—
359C4×D4—1181no—
360C4×D4C22⋊D5911yescharacteristic
361#117—21no—
362#117—41no—
363C2×C4⋊C4—21no—
364SD32—1182no—
365SD32—2361no—
366SD32—2361no—
367M32—2361no—
368C2×C16—591no—
369C2×C16—591no—
370C2×C16—591no—
371C2×C16—591no—
372C2×C16—2361no—
373#107—21no—
374#107—591no—
375#107C22×D5911yescharacteristic
376#107C22×D5911yescharacteristic
377#107C22×D5911yescharacteristic
378#106—21no—
379#106—21no—
380C4⋊C8C22⋊D5911yescharacteristic
381#103—21no—
382#103—21no—
383#103—21no—
384#102—1181no—
385#102—1181no—
386#102—1181no—
387C22⋊C8C22⋊D5911yescharacteristic
388C22⋊C8C22⋊D5911yescharacteristic
389C4×C8C22×D5911yescharacteristic
390C24—2361no—
391C24—2361no—
392C2×Q8—21no—
393C2×D4—1181no—
394C2×D4—1181no—
395C2×D4—1181no—
396C2×D4—4721no—
397C2×D4—4721no—
398C2×D4—4724no—
399C2×D4—2362no—
400C2×D4—2362no—
401C2×D4—2361no—
402C2×D4—2361no—
403C2×D4—2361no—
404C2×D4—2361no—
405C2×D4—2361no—
406C2×D4—2361no—
407C2×D4#1174566911yescharacteristic
408C22×C4—41no—
409C22×C4—2361no—
410C22×C4C2×C22⋊D5911yescharacteristic
411C22×C4C2×C22⋊D5911yescharacteristic
412Q16—21no—
413Q16—21no—
414D8—592no—
415D8—1181no—
416D8—1181no—
417D8—1181no—
418D8—1181no—
419D8—1181no—
420D8—4721no—
421D8—2362no—
422D8—2362no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×C22⋊D5911yescharacteristic
428C2×C8C2×C22⋊D5911yescharacteristic
429C2×C8C2×C22⋊D5911yescharacteristic
430C2×C8C23×D5911yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—1181no—
441C4⋊C4#1174566911yescharacteristic
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—2361no—
446C22⋊C4—2361no—
447C22⋊C4—2361no—
448C22⋊C4#1174566911yescharacteristic
449C22⋊C4#1174566911yescharacteristic
450C42—41no—
451C42—41no—
452C42C2×C22⋊D5911yescharacteristic
453C16—1181no—
454C16—1181no—
455C16—1181no—
456C16—1181no—
457C23—9441no—
458C23—9441no—
459C23—4722no—
460C23—4722no—
461C23—4722no—
462C23—4722no—
463C23—2361no—
464C23—2361no—
465C23—2361no—
466C23C2×#1174566911yescharacteristic
467C23C2×#1174566911yescharacteristic
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—1182no—
472D4—9441no—
473D4—4722no—
474D4—4722no—
475D4—4722no—
476D4—4722no—
477D4—4722no—
478D4—2361no—
479D4—2361no—
480D4—2361no—
481D4—2361no—
482D4—2361no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4—4722no—
493C2×C4—2361no—
494C2×C4C2×#1174566911yescharacteristic
495C2×C4C2×#1174566911yescharacteristic
496C2×C4C2×#1174566911yescharacteristic
497C2×C4C2×#1174566911yescharacteristic
498C2×C4C22×C22⋊D5911yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8#41×D5911yescharacteristic
503C8#1427377311yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—9444no—
510C22—9444no—
511C22—4721no—
512C22—4721no—
513C22—4721no—
514C22—4721no—
515C22—4721no—
516C22—4721no—
517C22—4721no—
518C22—4721no—
519C22—2362no—
520C22#1683783712yes—
521C22C22×#1174566911yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—4721no—
531C4#1683804011yescharacteristic
532C4#1683821011yescharacteristic
533C2—22no—
534C2—41no—
535C2—9441no—
536C2—9441no—
537C2—4722no—
538C2#2143590311yescharacteristic
539C2C2×#1683783711yescharacteristic
540C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3336C2
socleC22×C59C22×D8
prime radicalD4×D472C2
hypercenter#3336D59
hypocenterC59#33877
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #21426339C2semidirect
3. #21426287C2semidirect
4. #21426276C2semidirect
5. #21426250C2semidirect
6. #21426211C2semidirect
7. #21426194C2semidirect
8. #21426136C2semidirect
9. #21426024C2semidirect
10. #21426024C2semidirect
11. #21425958C2semidirect
12. #21425922C2semidirect
13. #21425811C2semidirect
14. C2×#16837202C2semidirect
15. C59×#3336C2semidirect
20. C2×#14273670C22semidirect
21. C2×#14273670C22semidirect
22. C2×#14273667C22semidirect
23. C22×C59⋊C16C22semidirect
31. #16837229C22semidirect
32. #16837227C22semidirect
33. #16837227C22semidirect
34. C4⋊(C59⋊C16)C22semidirect
35. #16837206C22semidirect
39. #16837202C22semidirect
40. #16837202C22semidirect
41. #16837202C22semidirect
42. #16837202C22semidirect
43. #16837202C22semidirect
44. #16837202C22semidirect
45. C22⋊(C59⋊C16)C22semidirect
46. C22⋊(C59⋊C16)C22semidirect
47. #16837178C22semidirect
49. C59×#499C22semidirect
53. C59×#467C22semidirect
54. C59×#467C22semidirect
55. C4×Q16×C59C22semidirect
57. C59×#427C22semidirect
58. C2×#107×C59C22semidirect
263. #3336D59semidirect
295. #499C2×D59semidirect
300. #467C2×D59semidirect
301. #467C2×D59semidirect
307. C4×Q16C2×D59semidirect
310. #427C2×D59semidirect
311. C2×#107C2×D59semidirect
329. C59#33877semidirect

Maximal subgroups

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

subgroupindex
D4×D4722
#214263392
#214262872
#214262762
#214262502
#214262112
#214261942
#214261362
#214260242
#214259582
#214259222
#214258112
C2×#168372022
C59×#33362
#3387759