Finite Group Databasebeta

← C59×#18191 #28430300

Subgroups — C59×#18191 #28430300

Order 15104.

990 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3506C211yesmaximal, characteristic
2C59×#3113C211yesmaximal, characteristic, complemented
3C59×#3112C211yesmaximal, characteristic, complemented
4C59×#2398C211yesmaximal, characteristic, complemented
5C59×#1612C214yesmaximal, complemented
6C59×#1606C214yesmaximal, complemented
7C59×#1506C211yesmaximal, characteristic, complemented
8C59×#1505C211yesmaximal, characteristic, complemented
9C59×#1466C211yesmaximal, characteristic, complemented
10C59×#559—22no—
11C59×#556—21no—
12C59×#555—21no—
13C59×#553—22no—
14C59×#550—21no—
15C59×#547—21no—
16C59×#538C2214yes—
17C59×#535—21no—
18C59×#533C2211yescharacteristic
19C59×#532C2211yescharacteristic
20C59×#525—22no—
21C59×#520—21no—
22C2×C4×Q8×C59C2211yescharacteristic
23C2×C4×D4×C59C2211yescharacteristic
24C59×#483C2212yes—
25C59×#480C2212yes—
26C59×#477—21no—
27C59×#475—21no—
28C59×#462—21no—
29C59×#460—21no—
30C59×#451C2212yes—
31C59×#448C2212yes—
32C59×#441—22no—
33C59×#440—22no—
34C59×#432C2211yescharacteristic
35C59×#432C2211yescharacteristic
36C59×#400—21no—
37C59×#397C2212yes—
38C59×#394—21no—
39C59×#393C2212yes—
40C4×C22⋊C4×C59—22no—
41C59×#376C2211yescharacteristic
42C59×#328C2214yes—
43C59×#327C2214yes—
44C59×#323C2214yes—
45C2×#41×C59—22no—
46C2×#41×C59—21no—
47C22×Q8×C59C2311yescharacteristic
48C22×D4×C59D411yescharacteristic
49C23×C4×C59D412yes—
50C2×Q16×C59—22no—
51C2×SD16×C59—22no—
52C4⋊Q8×C59D411yescharacteristic
53C4⋊Q8×C59D411yescharacteristic
54C4⋊Q8×C59D411yescharacteristic
55#127×C59—22no—
56#127×C59D411yescharacteristic
57#126×C59—24no—
58#125×C59—21no—
59#124×C59—22no—
60#124×C59—24no—
61#124×C59—21no—
62#123×C59—24no—
63#123×C59—24no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59—21no—
66C22⋊Q8×C59C2312yes—
67#121×C59—22no—
68#121×C59—22no—
69#121×C59—22no—
70#121×C59—24no—
71#121×C59—21no—
72#121×C59C2312yes—
73#120×C59—24no—
74C4×Q8×C59—22no—
75C4×Q8×C59—21no—
76C4×Q8×C59—21no—
77C4×Q8×C59—21no—
78C4×Q8×C59—21no—
79C4×Q8×C59—21no—
80C4×Q8×C59D412yes—
81C4×D4×C59—22no—
82C4×D4×C59—24no—
83C4×D4×C59—24no—
84C4×D4×C59—21no—
85C4×D4×C59—21no—
86C4×D4×C59—21no—
87C4×D4×C59—21no—
88C4×D4×C59—21no—
89C4×D4×C59D412yes—
90#117×C59—22no—
91#117×C59—22no—
92#117×C59—21no—
93C2×C4⋊C4×C59—22no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59D411yescharacteristic
96C2×C4⋊C4×C59C2311yescharacteristic
97C2×C22⋊C4×C59—22no—
98C2×C22⋊C4×C59—22no—
99C2×C22⋊C4×C59C2312yes—
100C2×C42×C59—21no—
101C2×C42×C59—21no—
102C2×C42×C59C2311yescharacteristic
103#107×C59—22no—
104#106×C59—22no—
105C4⋊C8×C59—21no—
106C4⋊C8×C59—21no—
107#103×C59—22no—
108#103×C59—22no—
109#103×C59—22no—
110#102×C59—22no—
111C22⋊C8×C59C2312yes—
112C22⋊C8×C59C2312yes—
113#97×C59—21no—
114#97×C59—21no—
115#95×C59—21no—
116#95×C59—22no—
117#95×C59—22no—
118#95×C59C2312yes—
119C24×C59C2×D412yes—
120#41×C59—44no—
121#41×C59—84no—
122C2×Q8×C59—21no—
123C2×Q8×C59—21no—
124C2×Q8×C59—21no—
125C2×Q8×C59—21no—
126C2×Q8×C59—42no—
127C2×Q8×C59C2×D412yes—
128C2×D4×C59—22no—
129C2×D4×C59—24no—
130C2×D4×C59—24no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—42no—
136C2×D4×C59C2×D412yes—
137C22×C4×C59—21no—
138C22×C4×C59—24no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—21no—
144C22×C4×C59—44no—
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D412yes—
148C22×C4×C59C2411yescharacteristic
149Q16×C59—44no—
150SD16×C59—44no—
151C2×C8×C59—22no—
152C2×C8×C59—22no—
153C4⋊C4×C59—24no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—41no—
165C4⋊C4×C59—41no—
166C4⋊C4×C59C2×D412yes—
167C4⋊C4×C59C2×D412yes—
168C22⋊C4×C59—24no—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—44no—
177C22⋊C4×C59—44no—
178C42×C59—24no—
179C42×C59—22no—
180C42×C59—42no—
181C42×C59—41no—
182C42×C59—41no—
183C42×C59C2×D411yescharacteristic
184C42×C59C2×D411yescharacteristic
185C42×C59C2×D411yescharacteristic
186C42×C59C2×D411yescharacteristic
187C23×C59—22no—
188C23×C59—22no—
189C23×C59—22no—
190C23×C59—22no—
191C23×C59—42no—
192C23×C59—42no—
193C23×C59C22×D411yescharacteristic
194Q8×C59—22no—
195Q8×C59—22no—
196Q8×C59—42no—
197Q8×C59—41no—
198Q8×C59—41no—
199Q8×C59—42no—
200Q8×C59—42no—
201D4×C59—42no—
202D4×C59—42no—
203D4×C59—42no—
204D4×C59—42no—
205D4×C59—44no—
206D4×C59—42no—
207D4×C59—84no—
208D4×C59—84no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—41no—
217C2×C4×C59—44no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—44no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—41no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—82no—
228C2×C4×C59—84no—
229C2×C4×C59#12011yescharacteristic
230C2×C4×C59#12011yescharacteristic
231C2×C4×C59#12011yescharacteristic
232C2×C4×C59#12011yescharacteristic
233C2×C4×C59C22×D411yescharacteristic
234C2×C4×C59C22×D411yescharacteristic
235C8×C59—42no—
236C8×C59—42no—
237#18191C5911yesmaximal, characteristic, direct product factor, Sylow
238C22×C59—21no—
239C22×C59—21no—
240C22×C59—44no—
241C22×C59—42no—
242C22×C59—42no—
243C22×C59—42no—
244C22×C59—42no—
245C22×C59—42no—
246C22×C59—42no—
247C22×C59—82no—
248C22×C59C2×#12011yescharacteristic
249C22×C59#57512yes—
250C4×C59—21no—
251C4×C59—21no—
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—21no—
255C4×C59—42no—
256C4×C59—42no—
257C4×C59—42no—
258C4×C59—42no—
259C4×C59—42no—
260C4×C59—82no—
261C4×C59—81no—
262C4×C59#45512yes—
263#3506C2×C5911yescharacteristic
264#3113C2×C5911yescharacteristic, complemented
265#3112C2×C5911yescharacteristic, complemented
266#2398C2×C5911yescharacteristic, complemented
267#1612C2×C5914yescomplemented
268#1606C2×C5914yescomplemented
269#1506C2×C5911yescharacteristic, complemented
270#1505C2×C5911yescharacteristic, complemented
271#1466C2×C5911yescharacteristic, complemented
272C2×C59—22no—
273C2×C59—82no—
274C2×C59—82no—
275C2×C59#300511yescharacteristic
276C2×C59C2×#45511yescharacteristic
277C2×C59#302511yescharacteristic
278#559—22no—
279#556—21no—
280#555—21no—
281#553—22no—
282#550—21no—
283#547—21no—
284#538C22×C5914yes—
285#535—21no—
286#533C22×C5911yescharacteristic
287#532C22×C5911yescharacteristic
288#525—22no—
289#520—21no—
290C2×C4×Q8C22×C5911yescharacteristic
291C2×C4×D4C22×C5911yescharacteristic
292#483C22×C5912yes—
293#480C22×C5912yes—
294#477—21no—
295#475—21no—
296#462—21no—
297#460—21no—
298#451C22×C5912yes—
299#448C22×C5912yes—
300#441—22no—
301#440—22no—
302#432C22×C5911yescharacteristic
303#432C22×C5911yescharacteristic
304#400—21no—
305#397C22×C5912yes—
306#394—21no—
307#393C22×C5912yes—
308C4×C22⋊C4—22no—
309#376C22×C5911yescharacteristic
310#328C22×C5914yes—
311#327C22×C5914yes—
312#323C22×C5914yes—
313C59#1819111yescharacteristic, direct product factor, Sylow
314C2×#41—22no—
315C2×#41—21no—
316C22×Q8C23×C5911yescharacteristic
317C22×D4D4×C5911yescharacteristic
318C23×C4D4×C5912yes—
319C2×Q16—22no—
320C2×SD16—22no—
321C4⋊Q8D4×C5911yescharacteristic
322C4⋊Q8D4×C5911yescharacteristic
323C4⋊Q8D4×C5911yescharacteristic
324#127—22no—
325#127D4×C5911yescharacteristic
326#126—24no—
327#125—21no—
328#124—22no—
329#124—24no—
330#124—21no—
331#123—24no—
332#123—24no—
333C22⋊Q8—22no—
334C22⋊Q8—21no—
335C22⋊Q8C23×C5912yes—
336#121—22no—
337#121—22no—
338#121—22no—
339#121—24no—
340#121—21no—
341#121C23×C5912yes—
342#120—24no—
343C4×Q8—22no—
344C4×Q8—21no—
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8—21no—
348C4×Q8—21no—
349C4×Q8D4×C5912yes—
350C4×D4—22no—
351C4×D4—24no—
352C4×D4—24no—
353C4×D4—21no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4D4×C5912yes—
359#117—22no—
360#117—22no—
361#117—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4D4×C5911yescharacteristic
365C2×C4⋊C4C23×C5911yescharacteristic
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4C23×C5912yes—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42C23×C5911yescharacteristic
372#107—22no—
373#106—22no—
374C4⋊C8—21no—
375C4⋊C8—21no—
376#103—22no—
377#103—22no—
378#103—22no—
379#102—22no—
380C22⋊C8C23×C5912yes—
381C22⋊C8C23×C5912yes—
382#97—21no—
383#97—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95C23×C5912yes—
388C24C2×D4×C5912yes—
389#41—44no—
390#41—84no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—42no—
396C2×Q8C2×D4×C5912yes—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—42no—
405C2×D4C2×D4×C5912yes—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—44no—
414C22×C4C2×D4×C5911yescharacteristic
415C22×C4C2×D4×C5911yescharacteristic
416C22×C4C2×D4×C5912yes—
417C22×C4C24×C5911yescharacteristic
418Q16—44no—
419SD16—44no—
420C2×C8—22no—
421C2×C8—22no—
422C4⋊C4—24no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4C2×D4×C5912yes—
436C4⋊C4C2×D4×C5912yes—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C42—24no—
448C42—22no—
449C42—42no—
450C42—41no—
451C42—41no—
452C42C2×D4×C5911yescharacteristic
453C42C2×D4×C5911yescharacteristic
454C42C2×D4×C5911yescharacteristic
455C42C2×D4×C5911yescharacteristic
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—42no—
461C23—42no—
462C23C22×D4×C5911yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—42no—
466Q8—41no—
467Q8—41no—
468Q8—42no—
469Q8—42no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—44no—
475D4—42no—
476D4—84no—
477D4—84no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—82no—
497C2×C4—84no—
498C2×C4#120×C5911yescharacteristic
499C2×C4#120×C5911yescharacteristic
500C2×C4#120×C5911yescharacteristic
501C2×C4#120×C5911yescharacteristic
502C2×C4C22×D4×C5911yescharacteristic
503C2×C4C22×D4×C5911yescharacteristic
504C8—42no—
505C8—42no—
506C22—21no—
507C22—21no—
508C22—44no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—82no—
516C22C2×#120×C5911yescharacteristic
517C22C59×#57512yes—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—82no—
529C4—81no—
530C4C59×#45512yes—
531C2—22no—
532C2—82no—
533C2—82no—
534C2C59×#300511yescharacteristic
535C2C2×C59×#45511yescharacteristic
536C2C59×#302511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3506C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3113C2semidirect
3. C59×#3112C2semidirect
4. C59×#2398C2semidirect
5. C59×#1612C2semidirect
6. C59×#1606C2semidirect
7. C59×#1506C2semidirect
8. C59×#1505C2semidirect
9. C59×#1466C2semidirect
237. #18191C59direct (N × H)
264. #3113C2×C59semidirect
265. #3112C2×C59semidirect
266. #2398C2×C59semidirect
267. #1612C2×C59semidirect
268. #1606C2×C59semidirect
269. #1506C2×C59semidirect
270. #1505C2×C59semidirect
271. #1466C2×C59semidirect
313. C59#18191direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35062
C59×#31132
C59×#31122
C59×#23982
C59×#16122
C59×#16062
C59×#15062
C59×#15052
C59×#14662
#1819159