Finite Group Databasebeta

← C59×#18352 #28430461

Subgroups — C59×#18352 #28430461

Order 15104.

996 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3501C211yesmaximal, characteristic, complemented
2C59×#3193C212yesmaximal, complemented
3C59×#2635C211yesmaximal, characteristic
4C59×#1617C212yesmaximal, complemented
5C59×#1614C212yesmaximal, complemented
6C59×#1599C212yesmaximal, complemented
7C59×#1596C212yesmaximal, complemented
8C59×#1511C212yesmaximal, complemented
9C59×#1465C211yesmaximal, characteristic, complemented
10C59×#554—24no—
11C59×#554—22no—
12D4×Q8×C59—22no—
13C59×#544C2212yes—
14C59×#543—21no—
15C59×#543C2212yes—
16C59×#533—21no—
17C2×C22⋊Q8×C59—21no—
18C2×C22⋊Q8×C59C2212yes—
19C2×#121×C59C2211yescharacteristic
20C59×#518—21no—
21C2×C4×D4×C59C2211yescharacteristic
22C59×#484C2214yes—
23C59×#480—22no—
24C59×#468—24no—
25C59×#450C2214yes—
26C59×#448—22no—
27C59×#407C2212yes—
28C59×#400C2211yescharacteristic
29C59×#399C2211yescharacteristic
30C59×#394—21no—
31C59×#394C2211yescharacteristic
32C59×#392—21no—
33C59×#392C2211yescharacteristic
34C59×#388—21no—
35C59×#386—21no—
36C59×#381C2211yescharacteristic
37C59×#328C2214yes—
38C59×#328C2214yes—
39C59×#324C2212yes—
40C59×#323C2212yes—
41C22×Q8×C59—21no—
42C22×Q8×C59D412yescomplemented
43C22×D4×C59—21no—
44C22×D4×C59—21no—
45C22×D4×C59D411yescharacteristic
46C22×D4×C59C2311yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59C2311yescharacteristic
49C2×SD16×C59—24no—
50C2×M16×C59—22no—
51C4⋊Q8×C59—21no—
52C4⋊Q8×C59—21no—
53C4⋊Q8×C59—24no—
54#125×C59—22no—
55#125×C59—24no—
56C22⋊Q8×C59—22no—
57C22⋊Q8×C59—22no—
58C22⋊Q8×C59—24no—
59C22⋊Q8×C59—24no—
60C22⋊Q8×C59—24no—
61C22⋊Q8×C59—22no—
62C22⋊Q8×C59—22no—
63C22⋊Q8×C59—22no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59—24no—
66C22⋊Q8×C59D414yescomplemented
67C22⋊Q8×C59C2314yes—
68#121×C59—22no—
69#121×C59—22no—
70C4×Q8×C59—22no—
71C4×Q8×C59—22no—
72C4×D4×C59—24no—
73C4×D4×C59—24no—
74C4×D4×C59—22no—
75C4×D4×C59—22no—
76#117×C59—22no—
77C2×C4⋊C4×C59—21no—
78C2×C4⋊C4×C59—22no—
79C2×C4⋊C4×C59—22no—
80C2×C4⋊C4×C59—21no—
81C2×C4⋊C4×C59D412yescomplemented
82C2×C4⋊C4×C59C2311yescharacteristic
83C2×C4⋊C4×C59C2311yescharacteristic
84C2×C22⋊C4×C59—22no—
85C2×C22⋊C4×C59—22no—
86C2×C22⋊C4×C59—21no—
87C2×C22⋊C4×C59—21no—
88C2×C22⋊C4×C59—21no—
89C2×C22⋊C4×C59—21no—
90C2×C22⋊C4×C59—21no—
91C2×C22⋊C4×C59—21no—
92C2×C22⋊C4×C59D411yescharacteristic
93C2×C22⋊C4×C59C2311yescharacteristic
94C2×C42×C59D411yescharacteristic
95#107×C59—24no—
96#103×C59—24no—
97#102×C59—24no—
98C22⋊C8×C59—22no—
99C22⋊C8×C59C2314yes—
100#95×C59—21no—
101#95×C59—21no—
102#95×C59—21no—
103#95×C59C2311yescharacteristic
104#95×C59C2311yescharacteristic
105C24×C59—21no—
106C24×C59C2×D411yescharacteristic
107C24×C59C2×D411yescharacteristic
108C2×Q8×C59—22no—
109C2×Q8×C59—22no—
110C2×Q8×C59—22no—
111C2×Q8×C59—82no—
112C2×Q8×C59—82no—
113C2×Q8×C59C2×D414yes—
114C2×D4×C59—22no—
115C2×D4×C59—22no—
116C2×D4×C59—22no—
117C2×D4×C59—22no—
118C2×D4×C59—42no—
119C2×D4×C59—42no—
120C2×D4×C59—81no—
121C2×D4×C59—81no—
122C2×D4×C59—81no—
123C2×D4×C59—81no—
124C2×D4×C59—81no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—81no—
135C22×C4×C59—81no—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D412yes—
141C22×C4×C59C2×D412yes—
142C22×C4×C59C2411yescharacteristic
143SD16×C59—84no—
144M16×C59—82no—
145C2×C8×C59—24no—
146C4⋊C4×C59—22no—
147C4⋊C4×C59—24no—
148C4⋊C4×C59—22no—
149C4⋊C4×C59—22no—
150C4⋊C4×C59—24no—
151C4⋊C4×C59—22no—
152C4⋊C4×C59—24no—
153C4⋊C4×C59—22no—
154C4⋊C4×C59—24no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—81no—
158C4⋊C4×C59C2×D414yes—
159C22⋊C4×C59—24no—
160C22⋊C4×C59—24no—
161C22⋊C4×C59—22no—
162C22⋊C4×C59—24no—
163C22⋊C4×C59—22no—
164C22⋊C4×C59—41no—
165C22⋊C4×C59—41no—
166C22⋊C4×C59—42no—
167C22⋊C4×C59—81no—
168C22⋊C4×C59—81no—
169C22⋊C4×C59—81no—
170C22⋊C4×C59—81no—
171C42×C59—24no—
172C42×C59—21no—
173C42×C59—21no—
174C23×C59—21no—
175C23×C59—21no—
176C23×C59—21no—
177C23×C59—21no—
178C23×C59—41no—
179C23×C59—41no—
180C23×C59—41no—
181C23×C59—41no—
182C23×C59—41no—
183C23×C59—41no—
184C23×C59—41no—
185C23×C59—41no—
186C23×C59—41no—
187C23×C59—41no—
188C23×C59—41no—
189C23×C59#12012yes—
190C23×C59#13712yes—
191C23×C59C22×D411yescharacteristic
192Q8×C59—44no—
193Q8×C59—44no—
194Q8×C59—82no—
195Q8×C59—82no—
196D4×C59—28no—
197D4×C59—44no—
198D4×C59—84no—
199D4×C59—84no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—24no—
204C2×C4×C59—24no—
205C2×C4×C59—21no—
206C2×C4×C59—42no—
207C2×C4×C59—41no—
208C2×C4×C59—42no—
209C2×C4×C59—41no—
210C2×C4×C59—42no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—81no—
223C2×C4×C59—82no—
224C2×C4×C59—82no—
225C2×C4×C59—82no—
226C2×C4×C59—82no—
227C2×C4×C59#12011yescharacteristic
228C2×C4×C59#12011yescharacteristic
229C2×C4×C59#13712yes—
230C2×C4×C59C22×D412yes—
231C8×C59—44no—
232#18352C5911yesmaximal, characteristic, direct product factor, Sylow
233C22×C59—21no—
234C22×C59—21no—
235C22×C59—22no—
236C22×C59—22no—
237C22×C59—22no—
238C22×C59—22no—
239C22×C59—42no—
240C22×C59—41no—
241C22×C59—41no—
242C22×C59—81no—
243C22×C59—81no—
244C22×C59—81no—
245C22×C59—81no—
246C22×C59—81no—
247C22×C59—81no—
248C22×C59—81no—
249C22×C59—81no—
250C22×C59—81no—
251C22×C59—81no—
252C22×C59C2×#12011yescharacteristic
253C22×C59C2×#13711yescharacteristic
254C22×C59C2×#13711yescharacteristic
255C4×C59—22no—
256C4×C59—22no—
257C4×C59—21no—
258C4×C59—21no—
259C4×C59—41no—
260C4×C59—44no—
261C4×C59—41no—
262C4×C59—44no—
263C4×C59—81no—
264C4×C59—81no—
265C4×C59—81no—
266#3501C2×C5911yescharacteristic, complemented
267#3193C2×C5912yescomplemented
268#2635C2×C5911yescharacteristic
269#1617C2×C5912yescomplemented
270#1614C2×C5912yescomplemented
271#1599C2×C5912yescomplemented
272#1596C2×C5912yescomplemented
273#1511C2×C5912yescomplemented
274#1465C2×C5911yescharacteristic, complemented
275C2×C59—21no—
276C2×C59—21no—
277C2×C59—42no—
278C2×C59—42no—
279C2×C59—161no—
280C2×C59#300411yescharacteristic
281C2×C59#302111yescharacteristic
282C2×C59#302611yescharacteristic
283#554—24no—
284#554—22no—
285D4×Q8—22no—
286#544C22×C5912yes—
287#543—21no—
288#543C22×C5912yes—
289#533—21no—
290C2×C22⋊Q8—21no—
291C2×C22⋊Q8C22×C5912yes—
292C2×#121C22×C5911yescharacteristic
293#518—21no—
294C2×C4×D4C22×C5911yescharacteristic
295#484C22×C5914yes—
296#480—22no—
297#468—24no—
298#450C22×C5914yes—
299#448—22no—
300#407C22×C5912yes—
301#400C22×C5911yescharacteristic
302#399C22×C5911yescharacteristic
303#394—21no—
304#394C22×C5911yescharacteristic
305#392—21no—
306#392C22×C5911yescharacteristic
307#388—21no—
308#386—21no—
309#381C22×C5911yescharacteristic
310#328C22×C5914yes—
311#328C22×C5914yes—
312#324C22×C5912yes—
313#323C22×C5912yes—
314C59#1835211yescharacteristic, direct product factor, Sylow
315C22×Q8—21no—
316C22×Q8D4×C5912yescomplemented
317C22×D4—21no—
318C22×D4—21no—
319C22×D4D4×C5911yescharacteristic
320C22×D4C23×C5911yescharacteristic
321C23×C4D4×C5911yescharacteristic
322C23×C4C23×C5911yescharacteristic
323C2×SD16—24no—
324C2×M16—22no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—24no—
328#125—22no—
329#125—24no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—24no—
340C22⋊Q8D4×C5914yescomplemented
341C22⋊Q8C23×C5914yes—
342#121—22no—
343#121—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—24no—
347C4×D4—24no—
348C4×D4—22no—
349C4×D4—22no—
350#117—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4D4×C5912yescomplemented
356C2×C4⋊C4C23×C5911yescharacteristic
357C2×C4⋊C4C23×C5911yescharacteristic
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4D4×C5911yescharacteristic
367C2×C22⋊C4C23×C5911yescharacteristic
368C2×C42D4×C5911yescharacteristic
369#107—24no—
370#103—24no—
371#102—24no—
372C22⋊C8—22no—
373C22⋊C8C23×C5914yes—
374#95—21no—
375#95—21no—
376#95—21no—
377#95C23×C5911yescharacteristic
378#95C23×C5911yescharacteristic
379C24—21no—
380C24C2×D4×C5911yescharacteristic
381C24C2×D4×C5911yescharacteristic
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8C2×D4×C5914yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—81no—
395C2×D4—81no—
396C2×D4—81no—
397C2×D4—81no—
398C2×D4—81no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4C2×D4×C5911yescharacteristic
411C22×C4C2×D4×C5911yescharacteristic
412C22×C4C2×D4×C5911yescharacteristic
413C22×C4C2×D4×C5911yescharacteristic
414C22×C4C2×D4×C5912yes—
415C22×C4C2×D4×C5912yes—
416C22×C4C24×C5911yescharacteristic
417SD16—84no—
418M16—82no—
419C2×C8—24no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—81no—
432C4⋊C4C2×D4×C5914yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—22no—
436C22⋊C4—24no—
437C22⋊C4—22no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—42no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C22⋊C4—81no—
444C22⋊C4—81no—
445C42—24no—
446C42—21no—
447C42—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23#120×C5912yes—
464C23#137×C5912yes—
465C23C22×D4×C5911yescharacteristic
466Q8—44no—
467Q8—44no—
468Q8—82no—
469Q8—82no—
470D4—28no—
471D4—44no—
472D4—84no—
473D4—84no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4#120×C5911yescharacteristic
502C2×C4#120×C5911yescharacteristic
503C2×C4#137×C5912yes—
504C2×C4C22×D4×C5912yes—
505C8—44no—
506C22—21no—
507C22—21no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—42no—
513C22—41no—
514C22—41no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22C2×#120×C5911yescharacteristic
526C22C2×#137×C5911yescharacteristic
527C22C2×#137×C5911yescharacteristic
528C4—22no—
529C4—22no—
530C4—21no—
531C4—21no—
532C4—41no—
533C4—44no—
534C4—41no—
535C4—44no—
536C4—81no—
537C4—81no—
538C4—81no—
539C2—21no—
540C2—21no—
541C2—42no—
542C2—42no—
543C2—161no—
544C2C59×#300411yescharacteristic
545C2C59×#302111yescharacteristic
546C2C59×#302611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#2635C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3501C2semidirect
2. C59×#3193C2semidirect
4. C59×#1617C2semidirect
5. C59×#1614C2semidirect
6. C59×#1599C2semidirect
7. C59×#1596C2semidirect
8. C59×#1511C2semidirect
9. C59×#1465C2semidirect
42. C22×Q8×C59D4semidirect
66. C22⋊Q8×C59D4semidirect
81. C2×C4⋊C4×C59D4semidirect
232. #18352C59direct (N × H)
266. #3501C2×C59semidirect
267. #3193C2×C59semidirect
269. #1617C2×C59semidirect
270. #1614C2×C59semidirect
271. #1599C2×C59semidirect
272. #1596C2×C59semidirect
273. #1511C2×C59semidirect
274. #1465C2×C59semidirect
314. C59#18352direct (N × H)
316. C22×Q8D4×C59semidirect
340. C22⋊Q8D4×C59semidirect
355. C2×C4⋊C4D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#35012
C59×#31932
C59×#26352
C59×#16172
C59×#16142
C59×#15992
C59×#15962
C59×#15112
C59×#14652
#1835259