Finite Group Databasebeta

← C59×#23593 #28435702

Subgroups — C59×#23593 #28435702

Order 15104.

1118 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3460C211yesmaximal, characteristic
2C4×#137×C59C212yesmaximal, complemented
3C59×#2326C211yesmaximal, characteristic, complemented
4C59×#2060C211yesmaximal, characteristic, complemented
5C59×#2039C211yesmaximal, characteristic, complemented
6C59×#2037C212yesmaximal, complemented
7C59×#2028C211yesmaximal, characteristic, complemented
8C59×#2007C211yesmaximal, characteristic, complemented
9C59×#2004C212yesmaximal, complemented
10C59×#1759C212yesmaximal, complemented
11C4×#100×C59C211yesmaximal, characteristic, complemented
12C2×#137×C59C2212yes—
13C59×#544C2211yescharacteristic
14C59×#542—22no—
15C59×#540—22no—
16C59×#538—22no—
17C59×#537C2211yescharacteristic
18C59×#535C2212yes—
19C59×#532—22no—
20C59×#529—22no—
21C2×#124×C59C2211yescharacteristic
22C59×#525—22no—
23C4×#41×C59C2212yes—
24C2×C4×Q8×C59C2211yescharacteristic
25C2×C4×D4×C59C2211yescharacteristic
26C59×#441—22no—
27C59×#440—22no—
28C4×Q16×C59—22no—
29C4×SD16×C59—22no—
30C59×#428C2212yes—
31C2×#104×C59C2212yes—
32C2×#104×C59C2212yes—
33C59×#419C2212yes—
34C59×#417C2212yes—
35C2×#100×C59C2212yes—
36C4×M16×C59C2212yes—
37C59×#391C2211yescharacteristic
38C59×#390C2211yescharacteristic
39C59×#388—22no—
40C59×#385—22no—
41C59×#384C2211yescharacteristic
42C59×#376C2211yescharacteristic
43C59×#343C2211yescharacteristic
44C59×#343C2211yescharacteristic
45C59×#339C2212yes—
46C59×#339C2212yes—
47C2×#41×C59—22no—
48C2×#41×C59C2312yes—
49C22×Q8×C59C2311yescharacteristic
50C22×D4×C59C2311yescharacteristic
51C23×C4×C59D412yes—
52#137×C59—24no—
53#137×C59—24no—
54C2×Q16×C59—22no—
55C2×SD16×C59—22no—
56C2×M16×C59C2312yes—
57C2×M16×C59C2312yes—
58C4⋊Q8×C59—22no—
59#127×C59—22no—
60#126×C59—24no—
61#125×C59—22no—
62#124×C59—24no—
63#124×C59—22no—
64#124×C59—22no—
65#124×C59—22no—
66#123×C59—24no—
67#123×C59—24no—
68C22⋊Q8×C59—22no—
69C22⋊Q8×C59—22no—
70C22⋊Q8×C59—22no—
71#121×C59—22no—
72#121×C59—24no—
73#121×C59—22no—
74#121×C59—22no—
75#121×C59—22no—
76#121×C59—22no—
77#120×C59—24no—
78C4×Q8×C59—22no—
79C4×Q8×C59—22no—
80C4×Q8×C59D412yes—
81C4×Q8×C59D412yes—
82C4×D4×C59—24no—
83C4×D4×C59—22no—
84C4×D4×C59—22no—
85C4×D4×C59—22no—
86C4×D4×C59—22no—
87C4×D4×C59D412yes—
88C4×D4×C59D412yes—
89#117×C59—22no—
90#117×C59—22no—
91#117×C59C2312yes—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59—22no—
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C4⋊C4×C59C2311yescharacteristic
96C2×C22⋊C4×C59—22no—
97C2×C22⋊C4×C59—22no—
98C2×C22⋊C4×C59D412yes—
99C2×C42×C59—22no—
100C2×C42×C59—21no—
101C2×C42×C59C2311yescharacteristic
102C2×C42×C59C2311yescharacteristic
103C2×C42×C59C2311yescharacteristic
104#107×C59—22no—
105#106×C59—22no—
106#104×C59—24no—
107#104×C59—24no—
108#103×C59—22no—
109#103×C59—22no—
110#103×C59—22no—
111#102×C59—22no—
112#100×C59—24no—
113#97×C59—22no—
114C4×C8×C59—22no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—22no—
118C24×C59C2×D412yes—
119#41×C59—22no—
120#41×C59—22no—
121#41×C59—22no—
122#41×C59—22no—
123#41×C59—84no—
124C2×Q8×C59—22no—
125C2×Q8×C59—21no—
126C2×Q8×C59—22no—
127C2×Q8×C59—22no—
128C2×Q8×C59C2×D412yes—
129C2×Q8×C59#4112yescomplemented
130C2×D4×C59—22no—
131C2×D4×C59—24no—
132C2×D4×C59—24no—
133C2×D4×C59—22no—
134C2×D4×C59—22no—
135C2×D4×C59—22no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59C2×D412yes—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—82no—
149C22×C4×C59C2×D412yes—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2411yescharacteristic
153Q16×C59—28no—
154SD16×C59—28no—
155M16×C59—22no—
156M16×C59—22no—
157M16×C59—22no—
158M16×C59#4114yescomplemented
159C2×C8×C59—22no—
160C2×C8×C59—22no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—24no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—41no—
169C4⋊C4×C59—41no—
170C4⋊C4×C59—82no—
171C4⋊C4×C59C2×D412yes—
172C4⋊C4×C59C2×D412yes—
173C22⋊C4×C59—28no—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—24no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—24no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—82no—
182C42×C59—22no—
183C42×C59—22no—
184C42×C59—22no—
185C42×C59—21no—
186C42×C59—22no—
187C42×C59—42no—
188C42×C59C2×D412yes—
189C42×C59C2×D412yes—
190C42×C59#4112yescomplemented
191C23×C59—22no—
192C23×C59—22no—
193C23×C59—22no—
194C23×C59—22no—
195C23×C59—42no—
196C23×C59—42no—
197C23×C59C22×D411yescharacteristic
198Q8×C59—24no—
199Q8×C59—22no—
200Q8×C59—22no—
201Q8×C59—42no—
202Q8×C59—82no—
203D4×C59—22no—
204D4×C59—22no—
205D4×C59—42no—
206D4×C59—42no—
207D4×C59—42no—
208D4×C59—42no—
209D4×C59—84no—
210D4×C59—82no—
211D4×C59—84no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—44no—
221C2×C4×C59—44no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—41no—
230C2×C4×C59—42no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—82no—
235C2×C4×C59—82no—
236C2×C4×C59#12011yescharacteristic
237C2×C4×C59#12012yes—
238C2×C4×C59#12011yescharacteristic
239C2×C4×C59C22×D412yes—
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4112yes—
242C2×C4×C59C2×#4111yescharacteristic
243C8×C59—24no—
244C8×C59—42no—
245#23593C5911yesmaximal, characteristic, direct product factor, Sylow
246C22×C59—21no—
247C22×C59—21no—
248C22×C59—44no—
249C22×C59—42no—
250C22×C59—42no—
251C22×C59—42no—
252C22×C59—42no—
253C22×C59—42no—
254C22×C59—42no—
255C22×C59—82no—
256C22×C59C2×#12011yescharacteristic
257C22×C59#52511yescharacteristic
258C22×C59#53211yescharacteristic
259C4×C59—21no—
260C4×C59—21no—
261C4×C59—22no—
262C4×C59—42no—
263C4×C59—42no—
264C4×C59—42no—
265C4×C59—42no—
266C4×C59—42no—
267C4×C59—42no—
268C4×C59—81no—
269C4×C59—81no—
270C4×C59#52512yes—
271C4×C59#53212yes—
272#3460C2×C5911yescharacteristic
273C4×#137C2×C5912yescomplemented
274#2326C2×C5911yescharacteristic, complemented
275#2060C2×C5911yescharacteristic, complemented
276#2039C2×C5911yescharacteristic, complemented
277#2037C2×C5912yescomplemented
278#2028C2×C5911yescharacteristic, complemented
279#2007C2×C5911yescharacteristic, complemented
280#2004C2×C5912yescomplemented
281#1759C2×C5912yescomplemented
282C4×#100C2×C5911yescharacteristic, complemented
283C2×C59—21no—
284C2×C59—21no—
285C2×C59—82no—
286C2×C59—82no—
287C2×C59#238811yescharacteristic
288C2×C59#301911yescharacteristic
289C2×C59#302211yescharacteristic
290C2×#137C22×C5912yes—
291#544C22×C5911yescharacteristic
292#542—22no—
293#540—22no—
294#538—22no—
295#537C22×C5911yescharacteristic
296#535C22×C5912yes—
297#532—22no—
298#529—22no—
299C2×#124C22×C5911yescharacteristic
300#525—22no—
301C4×#41C22×C5912yes—
302C2×C4×Q8C22×C5911yescharacteristic
303C2×C4×D4C22×C5911yescharacteristic
304#441—22no—
305#440—22no—
306C4×Q16—22no—
307C4×SD16—22no—
308#428C22×C5912yes—
309C2×#104C22×C5912yes—
310C2×#104C22×C5912yes—
311#419C22×C5912yes—
312#417C22×C5912yes—
313C2×#100C22×C5912yes—
314C4×M16C22×C5912yes—
315#391C22×C5911yescharacteristic
316#390C22×C5911yescharacteristic
317#388—22no—
318#385—22no—
319#384C22×C5911yescharacteristic
320#376C22×C5911yescharacteristic
321#343C22×C5911yescharacteristic
322#343C22×C5911yescharacteristic
323#339C22×C5912yes—
324#339C22×C5912yes—
325C59#2359311yescharacteristic, direct product factor, Sylow
326C2×#41—22no—
327C2×#41C23×C5912yes—
328C22×Q8C23×C5911yescharacteristic
329C22×D4C23×C5911yescharacteristic
330C23×C4D4×C5912yes—
331#137—24no—
332#137—24no—
333C2×Q16—22no—
334C2×SD16—22no—
335C2×M16C23×C5912yes—
336C2×M16C23×C5912yes—
337C4⋊Q8—22no—
338#127—22no—
339#126—24no—
340#125—22no—
341#124—24no—
342#124—22no—
343#124—22no—
344#124—22no—
345#123—24no—
346#123—24no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350#121—22no—
351#121—24no—
352#121—22no—
353#121—22no—
354#121—22no—
355#121—22no—
356#120—24no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×Q8D4×C5912yes—
360C4×Q8D4×C5912yes—
361C4×D4—24no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4D4×C5912yes—
367C4×D4D4×C5912yes—
368#117—22no—
369#117—22no—
370#117C23×C5912yes—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4C23×C5911yescharacteristic
374C2×C4⋊C4C23×C5911yescharacteristic
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4D4×C5912yes—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42C23×C5911yescharacteristic
381C2×C42C23×C5911yescharacteristic
382C2×C42C23×C5911yescharacteristic
383#107—22no—
384#106—22no—
385#104—24no—
386#104—24no—
387#103—22no—
388#103—22no—
389#103—22no—
390#102—22no—
391#100—24no—
392#97—22no—
393C4×C8—22no—
394#95—21no—
395#95—21no—
396#95—22no—
397C24C2×D4×C5912yes—
398#41—22no—
399#41—22no—
400#41—22no—
401#41—22no—
402#41—84no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8C2×D4×C5912yes—
408C2×Q8#41×C5912yescomplemented
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4C2×D4×C5912yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—82no—
428C22×C4C2×D4×C5912yes—
429C22×C4C2×D4×C5911yescharacteristic
430C22×C4C2×D4×C5911yescharacteristic
431C22×C4C24×C5911yescharacteristic
432Q16—28no—
433SD16—28no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16#41×C5914yescomplemented
438C2×C8—22no—
439C2×C8—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—82no—
450C4⋊C4C2×D4×C5912yes—
451C4⋊C4C2×D4×C5912yes—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—82no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42—22no—
466C42—42no—
467C42C2×D4×C5912yes—
468C42C2×D4×C5912yes—
469C42#41×C5912yescomplemented
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—42no—
475C23—42no—
476C23C22×D4×C5911yescharacteristic
477Q8—24no—
478Q8—22no—
479Q8—22no—
480Q8—42no—
481Q8—82no—
482D4—22no—
483D4—22no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—84no—
489D4—82no—
490D4—84no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4#120×C5911yescharacteristic
516C2×C4#120×C5912yes—
517C2×C4#120×C5911yescharacteristic
518C2×C4C22×D4×C5912yes—
519C2×C4C2×#41×C5911yescharacteristic
520C2×C4C2×#41×C5912yes—
521C2×C4C2×#41×C5911yescharacteristic
522C8—24no—
523C8—42no—
524C22—21no—
525C22—21no—
526C22—44no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—82no—
534C22C2×#120×C5911yescharacteristic
535C22C59×#52511yescharacteristic
536C22C59×#53211yescharacteristic
537C4—21no—
538C4—21no—
539C4—22no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—81no—
547C4—81no—
548C4C59×#52512yes—
549C4C59×#53212yes—
550C2—21no—
551C2—21no—
552C2—82no—
553C2—82no—
554C2C59×#238811yescharacteristic
555C2C59×#301911yescharacteristic
556C2C59×#302211yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3460C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#137×C59C2semidirect
3. C59×#2326C2semidirect
4. C59×#2060C2semidirect
5. C59×#2039C2semidirect
6. C59×#2037C2semidirect
7. C59×#2028C2semidirect
8. C59×#2007C2semidirect
9. C59×#2004C2semidirect
10. C59×#1759C2semidirect
11. C4×#100×C59C2semidirect
129. C2×Q8×C59#41semidirect
158. M16×C59#41semidirect
190. C42×C59#41semidirect
245. #23593C59direct (N × H)
273. C4×#137C2×C59semidirect
274. #2326C2×C59semidirect
275. #2060C2×C59semidirect
276. #2039C2×C59semidirect
277. #2037C2×C59semidirect
278. #2028C2×C59semidirect
279. #2007C2×C59semidirect
280. #2004C2×C59semidirect
281. #1759C2×C59semidirect
282. C4×#100C2×C59semidirect
325. C59#23593direct (N × H)
408. C2×Q8#41×C59semidirect
437. M16#41×C59semidirect
469. C42#41×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34602
C4×#137×C592
C59×#23262
C59×#20602
C59×#20392
C59×#20372
C59×#20282
C59×#20072
C59×#20042
C59×#17592
C4×#100×C592
#2359359