Finite Group Databasebeta

← #28745551

Subgroups — #28745551

Order 15104.

580 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436499C211yesmaximal, characteristic, complemented
2#21435783C211yesmaximal, characteristic, complemented
3#21426328C211yesmaximal, characteristic, complemented
4#21426299C211yesmaximal, characteristic, complemented
5#21426288C211yesmaximal, characteristic, complemented
6#21426254C211yesmaximal, characteristic, complemented
7#21426192C211yesmaximal, characteristic, complemented
8#21426182C211yesmaximal, characteristic, complemented
9#21426152C211yesmaximal, characteristic, complemented
10#21426021C211yesmaximal, characteristic, complemented
11#21426010C211yesmaximal, characteristic, complemented
12C4×D944C211yesmaximal, characteristic, complemented
13#21425804C211yesmaximal, characteristic, complemented
14C2×#16837204C211yesmaximal, characteristic, complemented
15D4×C16×C59C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×D472C2211yescharacteristic
18#16838098—21no—
19#16838048—21no—
20C2×#14273672C2211yescharacteristic, complemented
21C2×D944C2211yescharacteristic
22C2×D944C2211yescharacteristic
23#16837804C2211yescharacteristic, complemented
24#16837786C2211yescharacteristic
25#16837727—21no—
26#16837721—21no—
27#16837703—21no—
28#16837643C2211yescharacteristic
29#16837638C2211yescharacteristic
30#16837638C2211yescharacteristic
31#16837638C2211yescharacteristic, complemented
32#16837618—21no—
33#16837555—21no—
34#16837549—21no—
35C4×D472C2211yescharacteristic
36C4×D472C2211yescharacteristic, complemented
37#16837440C2211yescharacteristic, complemented
38C2×#14273661C2211yescharacteristic
39#16837230C2211yescharacteristic, complemented
40#16837228C2211yescharacteristic, complemented
41#16837228C2211yescharacteristic, complemented
42#16837207C2211yescharacteristic, complemented
43#16837207C2211yescharacteristic, complemented
44#16837207C2211yescharacteristic, complemented
45#16837204C2211yescharacteristic
46#16837204C2211yescharacteristic
47#16837204C2211yescharacteristic
48#16837204C2211yescharacteristic
49#16837204C2211yescharacteristic
50#16837204C2211yescharacteristic
51#16837204C2211yescharacteristic, complemented
52C22×C16×C59C2211yescharacteristic
53C22×C16×C59C2211yescharacteristic, complemented
54C8×D4×C59C2211yescharacteristic
55C59×C4⋊C16C2211yescharacteristic, complemented
56C59×C22⋊C16C2211yescharacteristic
57C59×C22⋊C16C2211yescharacteristic, complemented
58C4×C16×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
71C2×D472C2311yescharacteristic
72#14273722—41no—
73#14273719—21no—
74#14273715—21no—
75#14273712—41no—
76#14273709—41no—
77#14273706—21no—
78#14273701—21no—
79#14273701—21no—
80#14273701—21no—
81#14273700—41no—
82#14273699—41no—
83#14273695—41no—
84C4×D236—21no—
85C4×D236—21no—
86(C22⋊D59)⋊C4—41no—
87#14273686—21no—
88C2×C4⋊Dic59—21no—
89C22⋊C4×D59—41no—
90#14273672—41no—
91D944—22no—
92D944—41no—
93#14273661—21no—
94#14273661C2311yescharacteristic
95#14273661C2311yescharacteristic
96#14273661C2311yescharacteristic
97#14273661C2311yescharacteristic
98#14273659—21no—
99#14273659—21no—
100#14273652—21no—
101#14273649—21no—
102#14273649—21no—
103#14273649—21no—
104#14273649—21no—
105#14273649—21no—
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C4×D4×C59D411yescharacteristic
109C2×C16×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—
125C2×C22⋊D59—41no—
126C2×C22⋊D59—41no—
127C22×Dic59—41no—
128#11745685—42no—
129#11745685—81no—
130C2×D236—21no—
131C2×D236—21no—
132C2×D236—21no—
133C2×D236—42no—
134C2×D236—41no—
135C2×D236—81no—
136C2×C4×D59—41no—
137C2×C4×D59—41no—
138C2×C4×D59—41no—
139C2×Dic118—21no—
140C2×C8×C59—21no—
141C2×C8×C59—21no—
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59C2×D411yescharacteristic
145C2×C8×C59C2×D411yescharacteristic
146C2×C8×C59C2×D411yescharacteristic
147C2×C8×C59C2411yescharacteristic
148C4⋊C4×C59D811yescharacteristic
149C22⋊C4×C59D811yescharacteristic
150C22⋊C4×C59D811yescharacteristic
151C42×C59C2×D411yescharacteristic
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—
158(C2×D59)⋊C4—41no—
159(C2×D59)⋊C4—41no—
160(C2×D59)⋊C4—41no—
161C4⋊Dic59—21no—
162C4⋊Dic59—21no—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—41no—
166Dic59⋊C4—41no—
167Dic59⋊C4—41no—
168Dic59⋊C4—41no—
169C4×Dic59—41no—
170Dic236—21no—
171Dic236—21no—
172D472—21no—
173D472—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—41no—
190C22×D59—82no—
191C22×D59—82no—
192C22×D59—161no—
193D4×C59—22no—
194C2×C4×C59—21no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59C2×D811yescharacteristic
199C2×C4×C59C2×D811yescharacteristic
200C2×C4×C59C2×D811yescharacteristic
201C2×C4×C59C2×D811yescharacteristic
202C2×C4×C59C22×D411yescharacteristic
203C22⋊D59—82no—
204C22⋊D59—82no—
205C22⋊D59—161no—
206C22⋊D59—161no—
207C2×Dic59—41no—
208C2×Dic59—41no—
209C2×Dic59—41no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—161no—
213D236—22no—
214D236—41no—
215D236—41no—
216D236—41no—
217D236—41no—
218D236—41no—
219D236—82no—
220C4×D59—82no—
221C4×D59—161no—
222C4×D59—161no—
223Dic118—41no—
224Dic118—41no—
225C8×C59—21no—
226C8×C59—21no—
227C8×C59—21no—
228C8×C59C2×#4111yescharacteristic
229C8×C59H(5,𝔽2)11yescharacteristic
230#33864—591nomaximal, Sylow
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—21no—
235C22×C59—21no—
236C22×C59#50812yes—
237C22×C59C22×D811yescharacteristic
238C2×D59—42no—
239C2×D59—81no—
240C2×D59—81no—
241C2×D59—81no—
242C2×D59—81no—
243C2×D59—81no—
244C2×D59—81no—
245C2×D59—81no—
246C2×D59—81no—
247C2×D59—164no—
248C4×C59—21no—
249C4×C59—21no—
250C4×C59—21no—
251C4×C59—21no—
252C4×C59#54611yescharacteristic
253C4×C59#57711yescharacteristic
254Dic59—81no—
255Dic59—81no—
256Dic59—81no—
257Dic59—81no—
258Dic59—81no—
259#3339—591no—
260#3280—591no—
261#2242—591no—
262#2236—591no—
263#2234—591no—
264#2228—591no—
265#2217—591no—
266#2215—591no—
267#2210—591no—
268#2186—591no—
269#2184—591no—
270C4×D16—591no—
271D4×C16D5911yescharacteristic, complemented
272#2138—591no—
273C2×#357—591no—
274C2×C59—22no—
275C2×C59—41no—
276C2×C59#329411yescharacteristic
277C2×C59C2×#50811yescharacteristic
278C2×C59#341511yescharacteristic
279D59—82no—
280D59—161no—
281D59—161no—
282C2×#135—591no—
283C22×D8—591no—
284#553—1181no—
285#546—1181no—
286C2×SD32—591no—
287C2×D16—591no—
288C2×D16—591no—
289C22×C16C2×D5911yescharacteristic
290C22×C16C2×D5911yescharacteristic, complemented
291#499—591no—
292#495—591no—
293#482—1181no—
294#481—1181no—
295#478—1181no—
296#467—591no—
297#466—591no—
298#466—591no—
299#466—591no—
300#463—1181no—
301#450—1181no—
302#449—1181no—
303C4×D8—591no—
304C4×D8—591no—
305C8×D4C2×D5911yescharacteristic
306#427—591no—
307C2×#107—591no—
308#367—591no—
309#366—591no—
310#366—591no—
311C4⋊C16C2×D5911yescharacteristic, complemented
312#358—591no—
313#358—591no—
314#358—591no—
315#357—591no—
316#357—591no—
317#357—591no—
318#357—591no—
319#357—591no—
320#357—591no—
321#357—591no—
322C22⋊C16C2×D5911yescharacteristic
323C22⋊C16C2×D5911yescharacteristic, complemented
324C4×C16C2×D5911yescharacteristic, complemented
325C59#3386411yescharacteristic, complemented, Sylow
326C2×#41—1181no—
327C22×D4—1181no—
328#135—1182no—
329#135—2361no—
330C2×Q16—591no—
331C2×SD16—1181no—
332C2×SD16—1181no—
333C2×D8—591no—
334C2×D8—591no—
335C2×D8—591no—
336C2×D8—1182no—
337C2×D8—1181no—
338C2×D8—2361no—
339C22×C8C22×D5911yescharacteristic
340C22×C8C22×D5911yescharacteristic
341#126—2361no—
342#125—1181no—
343#124—1181no—
344#123—2361no—
345#123—2361no—
346C22⋊Q8—1181no—
347#121—1181no—
348#121—1181no—
349#121—1181no—
350#121—2361no—
351#121—2361no—
352#120—2361no—
353C4×D4—1181no—
354C4×D4—1181no—
355C4×D4—2361no—
356C4×D4D23611yescharacteristic
357#117—1181no—
358C2×C4⋊C4—1181no—
359C2×C22⋊C4—2361no—
360SD32—2361no—
361D16—1182no—
362D16—2361no—
363C2×C16—41no—
364C2×C16—41no—
365C2×C16C22×D5911yescharacteristic
366C2×C16C22×D5911yescharacteristic
367C2×C16C22×D5911yescharacteristic
368C2×C16C22×D5911yescharacteristic
369#107—591no—
370#107—591no—
371#107—591no—
372#107—591no—
373#107—1181no—
374#106—1181no—
375#106—1181no—
376C4⋊C8D23611yescharacteristic
377#103—1181no—
378#102—1181no—
379#102—1181no—
380#102—1181no—
381#102—1181no—
382#102—1181no—
383C22⋊C8D23611yescharacteristic
384C22⋊C8D23611yescharacteristic
385C4×C8C22×D5911yescharacteristic
386C24—2361no—
387#41—4721no—
388#41—2362no—
389C2×Q8—1181no—
390C2×D4—1181no—
391C2×D4—1181no—
392C2×D4—1181no—
393C2×D4—4721no—
394C2×D4—2362no—
395C2×D4—2361no—
396C2×D4—2361no—
397C2×D4—2361no—
398C2×D4—2361no—
399C2×D4—2361no—
400C2×D4D47211yescharacteristic
401C22×C4—2361no—
402C22×C4—2361no—
403C22×C4—2361no—
404C22×C4—2361no—
405C22×C4C2×D23611yescharacteristic
406C22×C4C2×D23611yescharacteristic
407Q16—1181no—
408Q16—1181no—
409SD16—4721no—
410SD16—2362no—
411D8—592no—
412D8—1181no—
413D8—1181no—
414D8—1181no—
415D8—1181no—
416D8—1181no—
417D8—2362no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8C2×D23611yescharacteristic
423C2×C8C2×D23611yescharacteristic
424C2×C8C2×D23611yescharacteristic
425C2×C8C23×D5911yescharacteristic
426C4⋊C4—1181no—
427C4⋊C4—1181no—
428C4⋊C4—1181no—
429C4⋊C4—1181no—
430C4⋊C4—2361no—
431C4⋊C4—2361no—
432C4⋊C4—2361no—
433C4⋊C4—2361no—
434C4⋊C4D47211yescharacteristic
435C22⋊C4—2361no—
436C22⋊C4—2361no—
437C22⋊C4—2361no—
438C22⋊C4—2361no—
439C22⋊C4—2361no—
440C22⋊C4—2361no—
441C22⋊C4—2361no—
442C22⋊C4—2361no—
443C22⋊C4—2361no—
444C22⋊C4D47211yescharacteristic
445C22⋊C4D47211yescharacteristic
446C42—2361no—
447C42C2×D23611yescharacteristic
448C16—21no—
449C16—21no—
450C16—21no—
451C16—21no—
452C23—9441no—
453C23—4722no—
454C23—4722no—
455C23—2361no—
456C23—2361no—
457C23—2361no—
458C23C2×D47211yescharacteristic
459C23C2×D47211yescharacteristic
460Q8—2361no—
461Q8—2361no—
462D4—22no—
463D4—1182no—
464D4—9441no—
465D4—9441no—
466D4—4722no—
467D4—4722no—
468D4—4722no—
469D4—2361no—
470D4—2361no—
471D4—2361no—
472D4—2361no—
473D4—2361no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—9441no—
479C2×C4—9441no—
480C2×C4—9441no—
481C2×C4—4722no—
482C2×C4—2361no—
483C2×C4—2361no—
484C2×C4—2361no—
485C2×C4—2361no—
486C2×C4—2361no—
487C2×C4C2×D47211yescharacteristic
488C2×C4C2×D47211yescharacteristic
489C2×C4C2×D47211yescharacteristic
490C2×C4C2×D47211yescharacteristic
491C2×C4C22×D23611yescharacteristic
492C8—21no—
493C8—21no—
494C8—21no—
495C8#41×D5911yescharacteristic
496C8#1427377211yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—9444no—
503C22—4721no—
504C22—4721no—
505C22—4721no—
506C22—4721no—
507C22—4721no—
508C22—4721no—
509C22—4721no—
510C22—4721no—
511C22—2362no—
512C22#1683783412yes—
513C22C22×D47211yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—4721no—
519C4—4721no—
520C4—4721no—
521C4—4721no—
522C4—4721no—
523C4#1683804711yescharacteristic
524C4#1683821411yescharacteristic
525C2—22no—
526C2—41no—
527C2—9441no—
528C2—9441no—
529C2—4722no—
530C2#2143591011yescharacteristic
531C2C2×#1683783411yescharacteristic
532C2#2143723711yescharacteristic
subgroupG/H
centerC22C22×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingD4×C16×C59C2
socleC22×C59C22×D8
hypercenterD4×C16D59
hypocenterC59#33864
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #21436499C2semidirect
2. #21435783C2semidirect
3. #21426328C2semidirect
4. #21426299C2semidirect
5. #21426288C2semidirect
6. #21426254C2semidirect
7. #21426192C2semidirect
8. #21426182C2semidirect
9. #21426152C2semidirect
10. #21426021C2semidirect
11. #21426010C2semidirect
12. C4×D944C2semidirect
13. #21425804C2semidirect
14. C2×#16837204C2semidirect
15. D4×C16×C59C2semidirect
20. C2×#14273672C22semidirect
23. #16837804C22semidirect
31. #16837638C22semidirect
36. C4×D472C22semidirect
37. #16837440C22semidirect
39. #16837230C22semidirect
40. #16837228C22semidirect
41. #16837228C22semidirect
42. #16837207C22semidirect
43. #16837207C22semidirect
44. #16837207C22semidirect
51. #16837204C22semidirect
53. C22×C16×C59C22semidirect
55. C59×C4⋊C16C22semidirect
57. C59×C22⋊C16C22semidirect
58. C4×C16×C59C22semidirect
271. D4×C16D59semidirect
290. C22×C16C2×D59semidirect
311. C4⋊C16C2×D59semidirect
323. C22⋊C16C2×D59semidirect
324. C4×C16C2×D59semidirect
325. C59#33864semidirect

Maximal subgroups

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

subgroupindex
#214364992
#214357832
#214263282
#214262992
#214262882
#214262542
#214261922
#214261822
#214261522
#214260212
#214260102
C4×D9442
#214258042
C2×#168372042
D4×C16×C592
#3386459