Finite Group Databasebeta

← C59×#62248 #28474357

Subgroups — C59×#62248 #28474357

Order 15104.

1902 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3506C212yesmaximal, complemented
2C59×#3340C214yesmaximal, complemented
3C59×#3305C214yesmaximal, complemented
4C59×#3302C214yesmaximal, complemented
5C59×#3284C214yesmaximal, complemented
6C59×#3249C211yesmaximal, characteristic, complemented
7C59×#3245C211yesmaximal, characteristic, complemented
8C59×#3191C214yesmaximal, complemented
9C59×#3167C212yesmaximal, complemented
10C59×#3050C214yesmaximal, complemented
11C2×C8×D4×C59C211yesmaximal, characteristic, complemented
12C2×#135×C59C2214yescomplemented
13C23×C8×C59C2212yes—
14C59×#559—24no—
15C59×#556C2214yes—
16C59×#555C2214yes—
17C59×#553—24no—
18C59×#550C2214yes—
19C59×#547C2214yes—
20C59×#538—24no—
21C59×#535—22no—
22C59×#533C2212yes—
23C59×#532C2212yes—
24C59×#525C2218yes—
25C59×#520—22no—
26C2×C4×D4×C59C2211yescharacteristic
27C59×#498C2212yescomplemented
28C59×#492C2212yescomplemented
29C59×#487C2212yescomplemented
30C59×#486C2212yescomplemented
31C59×#485C2218yescomplemented
32C59×#482C2218yescomplemented
33C59×#477C2214yescomplemented
34C59×#474C2214yescomplemented
35C59×#467C2214yes—
36C59×#467C2214yescomplemented
37C59×#466C2214yes—
38C59×#466C2214yescomplemented
39C59×#465C2214yes—
40C59×#465C2214yes—
41C59×#465C2218yescomplemented
42C59×#462C2214yescomplemented
43C59×#459C2214yescomplemented
44C59×#452C2218yescomplemented
45C59×#449C2218yescomplemented
46C4×SD16×C59C2214yescomplemented
47C4×SD16×C59C2214yescomplemented
48C8×D4×C59C2214yes—
49C8×D4×C59C2214yes—
50C59×#427C2214yescomplemented
51C2×C4⋊C8×C59C2211yescharacteristic
52C59×#416C2218yescomplemented
53C2×C22⋊C8×C59C2212yes—
54C2×C4×C8×C59C2211yescharacteristic
55C2×#41×C59—22no—
56C2×#41×C59C2318yes—
57C22×D4×C59D411yescharacteristic
58C23×C4×C59C2312yes—
59#135×C59—48no—
60C2×Q16×C59C2314yes—
61C2×SD16×C59C2314yes—
62C2×SD16×C59C2314yes—
63C2×D8×C59C2314yes—
64C22×C8×C59—44no—
65C22×C8×C59C2311yescharacteristic
66C22×C8×C59C2314yes—
67C22×C8×C59C2311yescharacteristic
68C22×C8×C59C2312yes—
69C22×C8×C59C2314yes—
70C4⋊Q8×C59C2312yes—
71C4⋊Q8×C59C2312yes—
72C4⋊Q8×C59C2312yes—
73#127×C59—24no—
74#127×C59C2312yes—
75#126×C59—28no—
76#125×C59—22no—
77#124×C59—24no—
78#124×C59—28no—
79#124×C59—22no—
80#123×C59—28no—
81#123×C59—28no—
82C22⋊Q8×C59—22no—
83C22⋊Q8×C59C2318yes—
84C22⋊Q8×C59C2314yes—
85#121×C59—24no—
86#121×C59—24no—
87#121×C59—22no—
88#121×C59—28no—
89#121×C59C2318yes—
90#121×C59C2314yes—
91#120×C59—28no—
92C4×Q8×C59—24no—
93C4×Q8×C59—22no—
94C4×Q8×C59C2314yes—
95C4×D4×C59—28no—
96C4×D4×C59—28no—
97C4×D4×C59—24no—
98C4×D4×C59—22no—
99C4×D4×C59D414yes—
100C4×D4×C59C2314yes—
101C4×D4×C59C2314yes—
102#117×C59—22no—
103#117×C59—24no—
104#117×C59C2318yes—
105C2×C4⋊C4×C59—24no—
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C22⋊C4×C59D412yes—
108C2×C42×C59C2311yescharacteristic
109#107×C59C2314yes—
110#107×C59C2314yes—
111#106×C59C2314yes—
112#106×C59C2314yes—
113C4⋊C8×C59C2314yes—
114#103×C59C2314yes—
115#103×C59C2318yes—
116#103×C59C2314yes—
117#102×C59C2314yes—
118#102×C59C2318yes—
119#102×C59C2314yes—
120C22⋊C8×C59C2318yes—
121C4×C8×C59C2312yes—
122C4×C8×C59C2312yes—
123C24×C59C2×D412yes—
124#41×C59—48no—
125#41×C59—48no—
126#41×C59—48no—
127C2×Q8×C59—22no—
128C2×Q8×C59—22no—
129C2×Q8×C59C2414yes—
130C2×D4×C59—24no—
131C2×D4×C59—28no—
132C2×D4×C59—28no—
133C2×D4×C59—24no—
134C2×D4×C59—22no—
135C2×D4×C59—22no—
136C2×D4×C59C2×D414yes—
137C2×D4×C59C2414yes—
138C22×C4×C59—24no—
139C22×C4×C59—28no—
140C22×C4×C59—42no—
141C22×C4×C59—42no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D412yes—
144C22×C4×C59C2×D414yes—
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2414yes—
147C22×C4×C59C2411yescharacteristic
148Q16×C59—44no—
149SD16×C59—28no—
150SD16×C59—44no—
151D8×C59—44no—
152C2×C8×C59—22no—
153C2×C8×C59—28no—
154C2×C8×C59—44no—
155C2×C8×C59—42no—
156C2×C8×C59—41no—
157C2×C8×C59C2412yes—
158C2×C8×C59C2411yescharacteristic
159C2×C8×C59C2414yes—
160C2×C8×C59C2411yescharacteristic
161C4⋊C4×C59—22no—
162C4⋊C4×C59—28no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—24no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—24no—
168C4⋊C4×C59C2×D414yes—
169C4⋊C4×C59C2414yes—
170C4⋊C4×C59C2414yes—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—28no—
173C22⋊C4×C59—28no—
174C22⋊C4×C59—28no—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—24no—
177C22⋊C4×C59C2×D418yes—
178C42×C59—28no—
179C42×C59—24no—
180C42×C59C2×D412yes—
181C42×C59C2412yes—
182C23×C59—24no—
183C23×C59—24no—
184C23×C59—42no—
185C23×C59—42no—
186C23×C59C22×D414yes—
187C23×C59C22×D411yescharacteristic
188Q8×C59—24no—
189Q8×C59—24no—
190Q8×C59—44no—
191Q8×C59—82no—
192D4×C59—28no—
193D4×C59—24no—
194D4×C59—24no—
195D4×C59—48no—
196D4×C59—44no—
197D4×C59—82no—
198D4×C59—88no—
199D4×C59—88no—
200C2×C4×C59—21no—
201C2×C4×C59—24no—
202C2×C4×C59—21no—
203C2×C4×C59—24no—
204C2×C4×C59—24no—
205C2×C4×C59—24no—
206C2×C4×C59—24no—
207C2×C4×C59—44no—
208C2×C4×C59—42no—
209C2×C4×C59—41no—
210C2×C4×C59—48no—
211C2×C4×C59—88no—
212C2×C4×C59#13514yescomplemented
213C2×C4×C59C22×D414yes—
214C2×C4×C59C22×D411yescharacteristic
215C2×C4×C59C22×D412yes—
216C2×C4×C59C22×D412yes—
217C2×C4×C59C2511yescharacteristic
218C8×C59—21no—
219C8×C59—24no—
220C8×C59—22no—
221C8×C59H(5,𝔽2)12yes—
222#62248C5911yesmaximal, characteristic, direct product factor, Sylow
223C22×C59—21no—
224C22×C59—21no—
225C22×C59—24no—
226C22×C59—24no—
227C22×C59—24no—
228C22×C59—48no—
229C22×C59—44no—
230C22×C59—44no—
231C22×C59—44no—
232C22×C59C2×#13512yes—
233C22×C59C23×D411yescharacteristic
234C4×C59—22no—
235C4×C59—21no—
236C4×C59—24no—
237C4×C59—44no—
238C4×C59—44no—
239C4×C59—44no—
240C4×C59C2×#13514yes—
241C4×C59C2×H(5,𝔽2)11yescharacteristic
242C4×C59#58511yescharacteristic
243#3506C2×C5912yescomplemented
244#3340C2×C5914yescomplemented
245#3305C2×C5914yescomplemented
246#3302C2×C5914yescomplemented
247#3284C2×C5914yescomplemented
248#3249C2×C5911yescharacteristic, complemented
249#3245C2×C5911yescharacteristic, complemented
250#3191C2×C5914yescomplemented
251#3167C2×C5912yescomplemented
252#3050C2×C5914yescomplemented
253C2×C8×D4C2×C5911yescharacteristic, complemented
254C2×C59—22no—
255C2×C59—44no—
256C2×C59—84no—
257C2×C59#349611yescharacteristic
258C2×C59C22×#13511yescharacteristic
259C2×C59#358411yescharacteristic
260C2×#135C22×C5914yescomplemented
261C23×C8C22×C5912yes—
262#559—24no—
263#556C22×C5914yes—
264#555C22×C5914yes—
265#553—24no—
266#550C22×C5914yes—
267#547C22×C5914yes—
268#538—24no—
269#535—22no—
270#533C22×C5912yes—
271#532C22×C5912yes—
272#525C22×C5918yes—
273#520—22no—
274C2×C4×D4C22×C5911yescharacteristic
275#498C22×C5912yescomplemented
276#492C22×C5912yescomplemented
277#487C22×C5912yescomplemented
278#486C22×C5912yescomplemented
279#485C22×C5918yescomplemented
280#482C22×C5918yescomplemented
281#477C22×C5914yescomplemented
282#474C22×C5914yescomplemented
283#467C22×C5914yes—
284#467C22×C5914yescomplemented
285#466C22×C5914yes—
286#466C22×C5914yescomplemented
287#465C22×C5914yes—
288#465C22×C5914yes—
289#465C22×C5918yescomplemented
290#462C22×C5914yescomplemented
291#459C22×C5914yescomplemented
292#452C22×C5918yescomplemented
293#449C22×C5918yescomplemented
294C4×SD16C22×C5914yescomplemented
295C4×SD16C22×C5914yescomplemented
296C8×D4C22×C5914yes—
297C8×D4C22×C5914yes—
298#427C22×C5914yescomplemented
299C2×C4⋊C8C22×C5911yescharacteristic
300#416C22×C5918yescomplemented
301C2×C22⋊C8C22×C5912yes—
302C2×C4×C8C22×C5911yescharacteristic
303C59#6224811yescharacteristic, direct product factor, Sylow
304C2×#41—22no—
305C2×#41C23×C5918yes—
306C22×D4D4×C5911yescharacteristic
307C23×C4C23×C5912yes—
308#135—48no—
309C2×Q16C23×C5914yes—
310C2×SD16C23×C5914yes—
311C2×SD16C23×C5914yes—
312C2×D8C23×C5914yes—
313C22×C8—44no—
314C22×C8C23×C5911yescharacteristic
315C22×C8C23×C5914yes—
316C22×C8C23×C5912yes—
317C22×C8C23×C5914yes—
318C22×C8C23×C5911yescharacteristic
319C4⋊Q8C23×C5912yes—
320C4⋊Q8C23×C5912yes—
321C4⋊Q8C23×C5912yes—
322#127—24no—
323#127C23×C5912yes—
324#126—28no—
325#125—22no—
326#124—24no—
327#124—28no—
328#124—22no—
329#123—28no—
330#123—28no—
331C22⋊Q8—22no—
332C22⋊Q8C23×C5914yes—
333C22⋊Q8C23×C5918yes—
334#121—24no—
335#121—24no—
336#121—22no—
337#121—28no—
338#121C23×C5914yes—
339#121C23×C5918yes—
340#120—28no—
341C4×Q8—24no—
342C4×Q8—22no—
343C4×Q8C23×C5914yes—
344C4×D4—28no—
345C4×D4—28no—
346C4×D4—24no—
347C4×D4—22no—
348C4×D4D4×C5914yes—
349C4×D4C23×C5914yes—
350C4×D4C23×C5914yes—
351#117—22no—
352#117—24no—
353#117C23×C5918yes—
354C2×C4⋊C4—24no—
355C2×C4⋊C4D4×C5911yescharacteristic
356C2×C22⋊C4D4×C5912yes—
357C2×C42C23×C5911yescharacteristic
358#107C23×C5914yes—
359#107C23×C5914yes—
360#106C23×C5914yes—
361#106C23×C5914yes—
362C4⋊C8C23×C5914yes—
363#103C23×C5914yes—
364#103C23×C5918yes—
365#103C23×C5914yes—
366#102C23×C5914yes—
367#102C23×C5918yes—
368#102C23×C5914yes—
369C22⋊C8C23×C5918yes—
370C4×C8C23×C5912yes—
371C4×C8C23×C5912yes—
372C24C2×D4×C5912yes—
373#41—48no—
374#41—48no—
375#41—48no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8C24×C5914yes—
379C2×D4—24no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4C2×D4×C5914yes—
386C2×D4C24×C5914yes—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4C2×D4×C5911yescharacteristic
392C22×C4C2×D4×C5912yes—
393C22×C4C2×D4×C5914yes—
394C22×C4C2×D4×C5911yescharacteristic
395C22×C4C24×C5914yes—
396C22×C4C24×C5911yescharacteristic
397Q16—44no—
398SD16—28no—
399SD16—44no—
400D8—44no—
401C2×C8—22no—
402C2×C8—28no—
403C2×C8—44no—
404C2×C8—42no—
405C2×C8—41no—
406C2×C8C24×C5912yes—
407C2×C8C24×C5911yescharacteristic
408C2×C8C24×C5914yes—
409C2×C8C24×C5911yescharacteristic
410C4⋊C4—22no—
411C4⋊C4—28no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4C2×D4×C5914yes—
418C4⋊C4C24×C5914yes—
419C4⋊C4C24×C5914yes—
420C22⋊C4—24no—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4C2×D4×C5918yes—
427C42—28no—
428C42—24no—
429C42C2×D4×C5912yes—
430C42C24×C5912yes—
431C23—24no—
432C23—24no—
433C23—42no—
434C23—42no—
435C23C22×D4×C5914yes—
436C23C22×D4×C5911yescharacteristic
437Q8—24no—
438Q8—24no—
439Q8—44no—
440Q8—82no—
441D4—28no—
442D4—24no—
443D4—24no—
444D4—48no—
445D4—44no—
446D4—82no—
447D4—88no—
448D4—88no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—48no—
460C2×C4—88no—
461C2×C4#135×C5914yescomplemented
462C2×C4C22×D4×C5914yes—
463C2×C4C22×D4×C5911yescharacteristic
464C2×C4C22×D4×C5912yes—
465C2×C4C22×D4×C5912yes—
466C2×C4C25×C5911yescharacteristic
467C8—21no—
468C8—24no—
469C8—22no—
470C8H(5,𝔽2)×C5912yes—
471C22—21no—
472C22—21no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—48no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22C2×#135×C5912yes—
481C22C23×D4×C5911yescharacteristic
482C4—22no—
483C4—21no—
484C4—24no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4C2×#135×C5914yes—
489C4C2×H(5,𝔽2)×C5911yescharacteristic
490C4C59×#58511yescharacteristic
491C2—22no—
492C2—44no—
493C2—84no—
494C2C59×#349611yescharacteristic
495C2C22×#135×C5911yescharacteristic
496C2C59×#358411yescharacteristic
subgroupG/H
center, socleC22×C59C23×D4
commutator, Frattini, cosocleC2×C4C25×C59
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C4×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3506C2semidirect
2. C59×#3340C2semidirect
3. C59×#3305C2semidirect
4. C59×#3302C2semidirect
5. C59×#3284C2semidirect
6. C59×#3249C2semidirect
7. C59×#3245C2semidirect
8. C59×#3191C2semidirect
9. C59×#3167C2semidirect
10. C59×#3050C2semidirect
11. C2×C8×D4×C59C2semidirect
12. C2×#135×C59C22semidirect
27. C59×#498C22semidirect
28. C59×#492C22semidirect
29. C59×#487C22semidirect
30. C59×#486C22semidirect
31. C59×#485C22semidirect
32. C59×#482C22semidirect
33. C59×#477C22semidirect
34. C59×#474C22semidirect
36. C59×#467C22semidirect
38. C59×#466C22semidirect
41. C59×#465C22semidirect
42. C59×#462C22semidirect
43. C59×#459C22semidirect
44. C59×#452C22semidirect
45. C59×#449C22semidirect
46. C4×SD16×C59C22semidirect
47. C4×SD16×C59C22semidirect
50. C59×#427C22semidirect
52. C59×#416C22semidirect
212. C2×C4×C59#135semidirect
222. #62248C59direct (N × H)
243. #3506C2×C59semidirect
244. #3340C2×C59semidirect
245. #3305C2×C59semidirect
246. #3302C2×C59semidirect
247. #3284C2×C59semidirect
248. #3249C2×C59semidirect
249. #3245C2×C59semidirect
250. #3191C2×C59semidirect
251. #3167C2×C59semidirect
252. #3050C2×C59semidirect
253. C2×C8×D4C2×C59semidirect
260. C2×#135C22×C59semidirect
275. #498C22×C59semidirect
276. #492C22×C59semidirect
277. #487C22×C59semidirect
278. #486C22×C59semidirect
279. #485C22×C59semidirect
280. #482C22×C59semidirect
281. #477C22×C59semidirect
282. #474C22×C59semidirect
284. #467C22×C59semidirect
286. #466C22×C59semidirect
289. #465C22×C59semidirect
290. #462C22×C59semidirect
291. #459C22×C59semidirect
292. #452C22×C59semidirect
293. #449C22×C59semidirect
294. C4×SD16C22×C59semidirect
295. C4×SD16C22×C59semidirect
298. #427C22×C59semidirect
300. #416C22×C59semidirect
303. C59#62248direct (N × H)
461. C2×C4#135×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#35062
C59×#33402
C59×#33052
C59×#33022
C59×#32842
C59×#32492
C59×#32452
C59×#31912
C59×#31672
C59×#30502
C2×C8×D4×C592
#6224859