Finite Group Databasebeta

← #28745834

Subgroups — #28745834

Order 15104.

622 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D472C211yesmaximal, characteristic, complemented
2#21426339C211yesmaximal, characteristic, complemented
3#21426287C211yesmaximal, characteristic, complemented
4#21426281C211yesmaximal, characteristic, complemented
5#21426250C211yesmaximal, characteristic, complemented
6#21426197C211yesmaximal, characteristic, complemented
7#21426180C211yesmaximal, characteristic, complemented
8#21426131C211yesmaximal, characteristic, complemented
9#21426007C211yesmaximal, characteristic, complemented
10#21426007C211yesmaximal, characteristic, complemented
11#21425970C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425811C211yesmaximal, characteristic, complemented
14C2×#16837202C211yesmaximal, characteristic, complemented
15C59×#3339C211yesmaximal, characteristic, complemented
16C22×D472C2211yescharacteristic
17C22×D472C2211yescharacteristic
18D4×D236—21no—
19C2×#14273668C2211yescharacteristic
20C2×#14273668C2211yescharacteristic, complemented
21C2×#14273668C2211yescharacteristic, 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#16837229C2211yescharacteristic, complemented
33#16837227C2211yescharacteristic, complemented
34C4⋊(C59⋊C16)C2211yescharacteristic, complemented
35#16837203C2211yescharacteristic, 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×#553—21no—
49C59×#499C2211yescharacteristic, complemented
50C59×#482—21no—
51C59×#481—21no—
52C59×#478—21no—
53C59×#466C2211yescharacteristic, complemented
54C59×#466C2211yescharacteristic, complemented
55C4×D8×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#14273668—22no—
80#14273668—41no—
81#14273668—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×D8×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95#126×C59—41no—
96#125×C59—21no—
97#123×C59—41no—
98#121×C59—21no—
99#121×C59—21no—
100#121×C59—41no—
101C4×D4×C59—21no—
102C4×D4×C59—41no—
103C4×D4×C59D411yescharacteristic
104#117×C59—21no—
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#102×C59—21no—
114#102×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C23×D59—41no—
120C23×D59—41no—
121C2×D4×C59—21no—
122C2×D4×C59—41no—
123C2×D4×C59—41no—
124C2×D4×C59D811yescharacteristic
125C22×C4×C59—41no—
126C22×C4×C59—41no—
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C2×C22⋊D59—41no—
130C2×C22⋊D59—41no—
131D4×D59—84no—
132C2×D236—21no—
133C2×D236—21no—
134C2×D236—21no—
135C2×D236—42no—
136C2×D236—42no—
137C2×D236—41no—
138C2×D236—41no—
139C2×D236—41no—
140C2×D236—41no—
141C2×D236—81no—
142C2×D236—81no—
143C2×C4×D59—41no—
144D8×C59—21no—
145D8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59C2×D411yescharacteristic
151C2×C8×C59C2×D411yescharacteristic
152C2×C8×C59C2×D411yescharacteristic
153C2×C8×C59C2411yescharacteristic
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
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×C59—41no—
165C22⋊C4×C59D811yescharacteristic
166C22⋊C4×C59D811yescharacteristic
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×C59—41no—
187C23×C59C2×D811yescharacteristic
188C23×C59C2×D811yescharacteristic
189C22×D59—41no—
190C22×D59—41no—
191C22×D59—41no—
192C22×D59—82no—
193C22×D59—82no—
194C22×D59—82no—
195C22×D59—82no—
196C22×D59—161no—
197C22×D59—161no—
198D4×C59—22no—
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—82no—
202D4×C59—161no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—41no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—161no—
211C2×C4×C59—161no—
212C2×C4×C59C2×D811yescharacteristic
213C2×C4×C59C2×D811yescharacteristic
214C2×C4×C59C2×D811yescharacteristic
215C2×C4×C59C2×D811yescharacteristic
216C2×C4×C59C22×D411yescharacteristic
217C22⋊D59—82no—
218C22⋊D59—82no—
219C2×Dic59—41no—
220D236—22no—
221D236—41no—
222D236—41no—
223D236—41no—
224D236—41no—
225D236—41no—
226D236—82no—
227D236—82no—
228D236—82no—
229D236—161no—
230C4×D59—82no—
231C8×C59—21no—
232C8×C59—21no—
233C8×C59—21no—
234C8×C59C2×#4111yescharacteristic
235C8×C59H(5,𝔽2)11yescharacteristic
236#33885—591nomaximal, Sylow
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—21no—
240C22×C59—21no—
241C22×C59—21no—
242C22×C59—81no—
243C22×C59—81no—
244C22×C59—81no—
245C22×C59#50912yes—
246C22×C59C22×D811yescharacteristic
247C2×D59—42no—
248C2×D59—81no—
249C2×D59—81no—
250C2×D59—81no—
251C2×D59—81no—
252C2×D59—81no—
253C2×D59—81no—
254C2×D59—81no—
255C2×D59—81no—
256C2×D59—164no—
257C2×D59—164no—
258C4×C59—21no—
259C4×C59—21no—
260C4×C59—21no—
261C4×C59—21no—
262C4×C59—81no—
263C4×C59—81no—
264C4×C59—81no—
265C4×C59#54611yescharacteristic
266C4×C59#57711yescharacteristic
267Dic59—81no—
268#3339D5911yescharacteristic, complemented
269D4×D8—591no—
270#2244—591no—
271#2234—591no—
272#2233—591no—
273#2228—591no—
274#2218—591no—
275#2215—591no—
276#2206—591no—
277#2184—591no—
278#2184—591no—
279#2177—591no—
280#2168—591no—
281#2140—591no—
282C2×#357—591no—
283C2×C59—22no—
284C2×C59—41no—
285C2×C59—161no—
286C2×C59#329411yescharacteristic
287C2×C59C2×#50911yescharacteristic
288C2×C59#341511yescharacteristic
289D59—82no—
290D59—161no—
291D59—161no—
292C22×D8—591no—
293C22×D8—591no—
294#553—21no—
295D42—1181no—
296C2×D16—591no—
297C2×D16—591no—
298C2×D16—591no—
299C2×M32—591no—
300C22×C16—591no—
301#499C2×D5911yescharacteristic, complemented
302#493—591no—
303#482—21no—
304#481—21no—
305#478—21no—
306#466—591no—
307#466—591no—
308#466C2×D5911yescharacteristic, complemented
309#466C2×D5911yescharacteristic, complemented
310#459—1181no—
311#447—1181no—
312#447—1181no—
313C4×D8—591no—
314C4×D8C2×D5911yescharacteristic, complemented
315C8×D4C2×D5911yescharacteristic
316#427C2×D5911yescharacteristic, complemented
317C2×#107C2×D5911yescharacteristic, complemented
318#367—591no—
319#367—591no—
320#366—591no—
321C4⋊C16—591no—
322#357—591no—
323#357—591no—
324#357—591no—
325#357—591no—
326#357—591no—
327#357—591no—
328#357—591no—
329#357—591no—
330#357—591no—
331#357—591no—
332C22⋊C16—591no—
333C22⋊C16—591no—
334#346—591no—
335C59#3388511yescharacteristic, complemented, Sylow
336C22×D4—1181no—
337C22×D4—1181no—
338C22×D4—2361no—
339C2×D8—591no—
340C2×D8—591no—
341C2×D8—591no—
342C2×D8—1182no—
343C2×D8—1182no—
344C2×D8—1181no—
345C2×D8—1181no—
346C2×D8—1181no—
347C2×D8—2361no—
348C2×D8—2361no—
349C2×D8C22×D5911yescharacteristic
350C22×C8C22×D5911yescharacteristic
351C22×C8C22×D5911yescharacteristic
352#127—1181no—
353#126—41no—
354#125—21no—
355#123—41no—
356#121—21no—
357#121—21no—
358#121—41no—
359#121—1181no—
360#121—1181no—
361#121—2361no—
362#120—2361no—
363#120—2361no—
364C4×D4—21no—
365C4×D4—41no—
366C4×D4—1181no—
367C4×D4C22⋊D5911yescharacteristic
368#117—21no—
369C2×C4⋊C4—21no—
370D16—1182no—
371D16—2361no—
372D16—2361no—
373M32—2361no—
374C2×C16—591no—
375C2×C16—591no—
376C2×C16—591no—
377C2×C16—591no—
378C2×C16—2361no—
379#107—21no—
380#107—591no—
381#107C22×D5911yescharacteristic
382#107C22×D5911yescharacteristic
383#107C22×D5911yescharacteristic
384#106—21no—
385#106—21no—
386C4⋊C8C22⋊D5911yescharacteristic
387#102—21no—
388#102—21no—
389#102—21no—
390#102—1181no—
391#102—1181no—
392#102—1181no—
393C22⋊C8C22⋊D5911yescharacteristic
394C22⋊C8C22⋊D5911yescharacteristic
395C4×C8C22×D5911yescharacteristic
396C24—2361no—
397C24—2361no—
398C2×D4—21no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—1181no—
402C2×D4—1181no—
403C2×D4—1181no—
404C2×D4—4721no—
405C2×D4—4721no—
406C2×D4—4724no—
407C2×D4—2362no—
408C2×D4—2362no—
409C2×D4—2361no—
410C2×D4—2361no—
411C2×D4—2361no—
412C2×D4—2361no—
413C2×D4—2361no—
414C2×D4—2361no—
415C2×D4#1174566911yescharacteristic
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—2361no—
419C22×C4C2×C22⋊D5911yescharacteristic
420C22×C4C2×C22⋊D5911yescharacteristic
421D8—21no—
422D8—21no—
423D8—592no—
424D8—1181no—
425D8—1181no—
426D8—1181no—
427D8—1181no—
428D8—1181no—
429D8—4721no—
430D8—2362no—
431D8—2362no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8C2×C22⋊D5911yescharacteristic
437C2×C8C2×C22⋊D5911yescharacteristic
438C2×C8C2×C22⋊D5911yescharacteristic
439C2×C8C23×D5911yescharacteristic
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—1181no—
447C4⋊C4#1174566911yescharacteristic
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—2361no—
453C22⋊C4—2361no—
454C22⋊C4—2361no—
455C22⋊C4#1174566911yescharacteristic
456C22⋊C4#1174566911yescharacteristic
457C42—41no—
458C42C2×C22⋊D5911yescharacteristic
459C16—1181no—
460C16—1181no—
461C16—1181no—
462C16—1181no—
463C23—41no—
464C23—9441no—
465C23—9441no—
466C23—4722no—
467C23—4722no—
468C23—4722no—
469C23—4722no—
470C23—2361no—
471C23—2361no—
472C23—2361no—
473C23C2×#1174566911yescharacteristic
474C23C2×#1174566911yescharacteristic
475D4—22no—
476D4—41no—
477D4—41no—
478D4—82no—
479D4—161no—
480D4—1182no—
481D4—9441no—
482D4—4722no—
483D4—4722no—
484D4—4722no—
485D4—4722no—
486D4—4722no—
487D4—2361no—
488D4—2361no—
489D4—2361no—
490D4—2361no—
491D4—2361no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—161no—
500C2×C4—161no—
501C2×C4—4722no—
502C2×C4—2361no—
503C2×C4C2×#1174566911yescharacteristic
504C2×C4C2×#1174566911yescharacteristic
505C2×C4C2×#1174566911yescharacteristic
506C2×C4C2×#1174566911yescharacteristic
507C2×C4C22×C22⋊D5911yescharacteristic
508C8—21no—
509C8—21no—
510C8—21no—
511C8#41×D5911yescharacteristic
512C8#1427377311yescharacteristic
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—9444no—
522C22—9444no—
523C22—4721no—
524C22—4721no—
525C22—4721no—
526C22—4721no—
527C22—4721no—
528C22—4721no—
529C22—4721no—
530C22—4721no—
531C22—2362no—
532C22#1683783712yes—
533C22C22×#1174566911yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—4721no—
542C4#1683804011yescharacteristic
543C4#1683821011yescharacteristic
544C2—22no—
545C2—41no—
546C2—161no—
547C2—9441no—
548C2—9441no—
549C2—4722no—
550C2#2143590311yescharacteristic
551C2C2×#1683783711yescharacteristic
552C2#2143723511yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3339C2
socleC22×C59C22×D8
hypercenter#3339D59
hypocenterC59#33885
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3339

Complements

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

NH ≅ G/Ntype
1. D4×D472C2semidirect
2. #21426339C2semidirect
3. #21426287C2semidirect
4. #21426281C2semidirect
5. #21426250C2semidirect
6. #21426197C2semidirect
7. #21426180C2semidirect
8. #21426131C2semidirect
9. #21426007C2semidirect
10. #21426007C2semidirect
11. #21425970C2semidirect
12. #21425922C2semidirect
13. #21425811C2semidirect
14. C2×#16837202C2semidirect
15. C59×#3339C2semidirect
20. C2×#14273668C22semidirect
21. C2×#14273668C22semidirect
22. C2×#14273667C22semidirect
23. C22×C59⋊C16C22semidirect
31. #16837229C22semidirect
32. #16837229C22semidirect
33. #16837227C22semidirect
34. C4⋊(C59⋊C16)C22semidirect
35. #16837203C22semidirect
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×#466C22semidirect
54. C59×#466C22semidirect
55. C4×D8×C59C22semidirect
57. C59×#427C22semidirect
58. C2×#107×C59C22semidirect
268. #3339D59semidirect
301. #499C2×D59semidirect
308. #466C2×D59semidirect
309. #466C2×D59semidirect
314. C4×D8C2×D59semidirect
316. #427C2×D59semidirect
317. C2×#107C2×D59semidirect
335. C59#33885semidirect

Maximal subgroups

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

subgroupindex
D4×D4722
#214263392
#214262872
#214262812
#214262502
#214261972
#214261802
#214261312
#214260072
#214259702
#214259222
#214258112
C2×#168372022
C59×#33392
#3388559