Finite Group Databasebeta

← C59×#18153 #28430262

Subgroups — C59×#18153 #28430262

Order 15104.

1184 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3449C211yesmaximal, characteristic
2C59×#3112C212yesmaximal, complemented
3C59×#2599C211yesmaximal, characteristic, complemented
4C59×#1610C212yesmaximal, complemented
5C59×#1607C212yesmaximal, complemented
6C59×#1604C212yesmaximal, complemented
7C59×#1601C212yesmaximal, complemented
8C59×#1514C212yesmaximal, complemented
9C59×#1464C211yesmaximal, characteristic, complemented
10C22×#41×C59—21no—
11C59×#537—21no—
12C59×#537C2212yes—
13C59×#536—22no—
14C59×#535—22no—
15C59×#534—21no—
16C59×#534C2212yes—
17C59×#532—21no—
18C59×#532C2212yes—
19C59×#525—24no—
20C2×C4×Q8×C59C2211yescharacteristic, complemented
21C2×#117×C59C2211yescharacteristic
22C59×#480C2214yescomplemented
23C59×#477—22no—
24C59×#460—22no—
25C59×#448C2214yescomplemented
26C59×#440—24no—
27C59×#432C2212yescomplemented
28C59×#398C2211yescharacteristic, complemented
29C59×#397C2211yescharacteristic
30C59×#395C2211yescharacteristic, complemented
31C59×#393C2211yescharacteristic
32C59×#388—21no—
33C59×#387—21no—
34C59×#387—21no—
35C59×#385—21no—
36C59×#383C2211yescharacteristic, complemented
37C59×#327C2214yescomplemented
38C59×#327C2214yescomplemented
39C59×#324C2212yescomplemented
40C59×#323C2212yes—
41C2×#41×C59—22no—
42C2×#41×C59—22no—
43C2×#41×C59—22no—
44C2×#41×C59—22no—
45C2×#41×C59—44no—
46C2×#41×C59—44no—
47C22×Q8×C59—21no—
48C22×Q8×C59C2311yescharacteristic
49C22×D4×C59—21no—
50C22×D4×C59—22no—
51C23×C4×C59—22no—
52C23×C4×C59D411yescharacteristic
53C2×SD16×C59—24no—
54C4⋊Q8×C59—21no—
55C4⋊Q8×C59D412yes—
56#127×C59—21no—
57#127×C59D412yes—
58#124×C59—22no—
59#124×C59—22no—
60#123×C59—24no—
61#123×C59—24no—
62C22⋊Q8×C59—24no—
63C22⋊Q8×C59—24no—
64#121×C59—24no—
65#121×C59—24no—
66#121×C59—22no—
67#121×C59—22no—
68#121×C59—22no—
69#121×C59C2314yes—
70#120×C59—24no—
71#120×C59—24no—
72C4×Q8×C59—22no—
73C4×Q8×C59—22no—
74C4×D4×C59—22no—
75C4×D4×C59—22no—
76C4×D4×C59—22no—
77C4×D4×C59D414yes—
78#117×C59—22no—
79#117×C59—22no—
80C2×C4⋊C4×C59—21no—
81C2×C4⋊C4×C59—21no—
82C2×C4⋊C4×C59—21no—
83C2×C4⋊C4×C59D411yescharacteristic
84C2×C4⋊C4×C59C2311yescharacteristic
85C2×C4⋊C4×C59C2311yescharacteristic
86C2×C22⋊C4×C59—21no—
87C2×C22⋊C4×C59—21no—
88C2×C22⋊C4×C59D411yescharacteristic
89C2×C22⋊C4×C59C2311yescharacteristic
90C2×C42×C59—21no—
91C2×C42×C59D411yescharacteristic
92C2×C42×C59C2311yescharacteristic
93#107×C59—24no—
94C4⋊C8×C59—22no—
95#103×C59—24no—
96#102×C59—24no—
97C22⋊C8×C59C2314yes—
98#97×C59—22no—
99#95×C59—21no—
100#95×C59—21no—
101#95×C59—21no—
102#95×C59—21no—
103#95×C59—21no—
104#95×C59—21no—
105#95×C59C2311yescharacteristic
106#95×C59C2311yescharacteristic
107C24×C59—22no—
108C24×C59C2×D411yescharacteristic
109#41×C59—48no—
110#41×C59—48no—
111#41×C59—44no—
112#41×C59—44no—
113#41×C59—44no—
114#41×C59—44no—
115C2×Q8×C59—22no—
116C2×Q8×C59—22no—
117C2×Q8×C59—22no—
118C2×Q8×C59—42no—
119C2×Q8×C59—44no—
120C2×D4×C59—24no—
121C2×D4×C59—22no—
122C2×D4×C59—24no—
123C2×D4×C59—22no—
124C2×D4×C59—22no—
125C2×D4×C59—22no—
126C2×D4×C59—22no—
127C2×D4×C59—44no—
128C2×D4×C59—44no—
129C2×D4×C59—44no—
130C2×D4×C59C2×D414yes—
131C22×C4×C59—22no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—22no—
135C22×C4×C59—24no—
136C22×C4×C59—22no—
137C22×C4×C59—21no—
138C22×C4×C59—24no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—48no—
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149SD16×C59—48no—
150C2×C8×C59—24no—
151C4⋊C4×C59—22no—
152C4⋊C4×C59—22no—
153C4⋊C4×C59—22no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—42no—
158C4⋊C4×C59—41no—
159C4⋊C4×C59—41no—
160C4⋊C4×C59—42no—
161C4⋊C4×C59C2×D414yes—
162C22⋊C4×C59—24no—
163C22⋊C4×C59—24no—
164C22⋊C4×C59—22no—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—22no—
167C22⋊C4×C59—24no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—42no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—42no—
173C42×C59C2×D412yes—
174C42×C59C2×D412yes—
175C23×C59—21no—
176C23×C59—21no—
177C23×C59—24no—
178C23×C59—41no—
179C23×C59—41no—
180C23×C59—44no—
181C23×C59—82no—
182C23×C59—82no—
183C23×C59AGL(1,ℤ/8)12yes—
184C23×C59C22×D411yescharacteristic
185Q8×C59—24no—
186Q8×C59—42no—
187Q8×C59—44no—
188Q8×C59—44no—
189D4×C59—44no—
190D4×C59—44no—
191D4×C59—44no—
192D4×C59—48no—
193D4×C59—44no—
194D4×C59—44no—
195D4×C59—82no—
196D4×C59—82no—
197D4×C59—82no—
198C2×C4×C59—24no—
199C2×C4×C59—24no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—22no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—24no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—48no—
211C2×C4×C59—48no—
212C2×C4×C59—44no—
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—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59#12011yescharacteristic
227C2×C4×C59#12011yescharacteristic
228C2×C4×C59#12012yes—
229C2×C4×C59AGL(1,ℤ/8)12yes—
230C2×C4×C59C22×D412yes—
231C8×C59—44no—
232#18153C5911yesmaximal, characteristic, direct product factor, Sylow
233C22×C59—21no—
234C22×C59—21no—
235C22×C59—22no—
236C22×C59—42no—
237C22×C59—41no—
238C22×C59—41no—
239C22×C59—44no—
240C22×C59—44no—
241C22×C59—44no—
242C22×C59—81no—
243C22×C59—88no—
244C22×C59C2×#12011yescharacteristic
245C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
247C4×C59—21no—
248C4×C59—22no—
249C4×C59—21no—
250C4×C59—22no—
251C4×C59—22no—
252C4×C59—44no—
253C4×C59—41no—
254C4×C59—42no—
255C4×C59—41no—
256C4×C59—81no—
257C4×C59—81no—
258C4×C59—81no—
259#3449C2×C5911yescharacteristic
260#3112C2×C5912yescomplemented
261#2599C2×C5911yescharacteristic, complemented
262#1610C2×C5912yescomplemented
263#1607C2×C5912yescomplemented
264#1604C2×C5912yescomplemented
265#1601C2×C5912yescomplemented
266#1514C2×C5912yescomplemented
267#1464C2×C5911yescharacteristic, complemented
268C2×C59—21no—
269C2×C59—21no—
270C2×C59—42no—
271C2×C59—84no—
272C2×C59#300311yescharacteristic
273C2×C59#302211yescharacteristic
274C2×C59#302511yescharacteristic
275C22×#41—21no—
276#537—21no—
277#537C22×C5912yes—
278#536—22no—
279#535—22no—
280#534—21no—
281#534C22×C5912yes—
282#532—21no—
283#532C22×C5912yes—
284#525—24no—
285C2×C4×Q8C22×C5911yescharacteristic, complemented
286C2×#117C22×C5911yescharacteristic
287#480C22×C5914yescomplemented
288#477—22no—
289#460—22no—
290#448C22×C5914yescomplemented
291#440—24no—
292#432C22×C5912yescomplemented
293#398C22×C5911yescharacteristic, complemented
294#397C22×C5911yescharacteristic
295#395C22×C5911yescharacteristic, complemented
296#393C22×C5911yescharacteristic
297#388—21no—
298#387—21no—
299#387—21no—
300#385—21no—
301#383C22×C5911yescharacteristic, complemented
302#327C22×C5914yescomplemented
303#327C22×C5914yescomplemented
304#324C22×C5912yescomplemented
305#323C22×C5912yes—
306C59#1815311yescharacteristic, direct product factor, Sylow
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—44no—
312C2×#41—44no—
313C22×Q8—21no—
314C22×Q8C23×C5911yescharacteristic
315C22×D4—21no—
316C22×D4—22no—
317C23×C4—22no—
318C23×C4D4×C5911yescharacteristic
319C2×SD16—24no—
320C4⋊Q8—21no—
321C4⋊Q8D4×C5912yes—
322#127—21no—
323#127D4×C5912yes—
324#124—22no—
325#124—22no—
326#123—24no—
327#123—24no—
328C22⋊Q8—24no—
329C22⋊Q8—24no—
330#121—24no—
331#121—24no—
332#121—22no—
333#121—22no—
334#121—22no—
335#121C23×C5914yes—
336#120—24no—
337#120—24no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×D4—22no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4D4×C5914yes—
344#117—22no—
345#117—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4D4×C5911yescharacteristic
350C2×C4⋊C4C23×C5911yescharacteristic
351C2×C4⋊C4C23×C5911yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4D4×C5911yescharacteristic
355C2×C22⋊C4C23×C5911yescharacteristic
356C2×C42—21no—
357C2×C42D4×C5911yescharacteristic
358C2×C42C23×C5911yescharacteristic
359#107—24no—
360C4⋊C8—22no—
361#103—24no—
362#102—24no—
363C22⋊C8C23×C5914yes—
364#97—22no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95C23×C5911yescharacteristic
372#95C23×C5911yescharacteristic
373C24—22no—
374C24C2×D4×C5911yescharacteristic
375#41—48no—
376#41—48no—
377#41—44no—
378#41—44no—
379#41—44no—
380#41—44no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4C2×D4×C5914yes—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—48no—
409C22×C4C2×D4×C5911yescharacteristic
410C22×C4C2×D4×C5911yescharacteristic
411C22×C4C2×D4×C5911yescharacteristic
412C22×C4C2×D4×C5911yescharacteristic
413C22×C4C2×D4×C5911yescharacteristic
414C22×C4C24×C5911yescharacteristic
415SD16—48no—
416C2×C8—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4C2×D4×C5914yes—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C42—21no—
437C42—21no—
438C42—42no—
439C42C2×D4×C5912yes—
440C42C2×D4×C5912yes—
441C23—21no—
442C23—21no—
443C23—24no—
444C23—41no—
445C23—41no—
446C23—44no—
447C23—82no—
448C23—82no—
449C23AGL(1,ℤ/8)×C5912yes—
450C23C22×D4×C5911yescharacteristic
451Q8—24no—
452Q8—42no—
453Q8—44no—
454Q8—44no—
455D4—44no—
456D4—44no—
457D4—44no—
458D4—48no—
459D4—44no—
460D4—44no—
461D4—82no—
462D4—82no—
463D4—82no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4#120×C5911yescharacteristic
493C2×C4#120×C5911yescharacteristic
494C2×C4#120×C5912yes—
495C2×C4AGL(1,ℤ/8)×C5912yes—
496C2×C4C22×D4×C5912yes—
497C8—44no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—81no—
508C22—88no—
509C22C2×#120×C5911yescharacteristic
510C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
511C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
512C4—21no—
513C4—22no—
514C4—21no—
515C4—22no—
516C4—22no—
517C4—44no—
518C4—41no—
519C4—42no—
520C4—41no—
521C4—81no—
522C4—81no—
523C4—81no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—84no—
528C2C59×#300311yescharacteristic
529C2C59×#302211yescharacteristic
530C2C59×#302511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3449C2
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×#3112C2semidirect
3. C59×#2599C2semidirect
4. C59×#1610C2semidirect
5. C59×#1607C2semidirect
6. C59×#1604C2semidirect
7. C59×#1601C2semidirect
8. C59×#1514C2semidirect
9. C59×#1464C2semidirect
20. C2×C4×Q8×C59C22semidirect
22. C59×#480C22semidirect
25. C59×#448C22semidirect
27. C59×#432C22semidirect
28. C59×#398C22semidirect
30. C59×#395C22semidirect
36. C59×#383C22semidirect
37. C59×#327C22semidirect
38. C59×#327C22semidirect
39. C59×#324C22semidirect
232. #18153C59direct (N × H)
260. #3112C2×C59semidirect
261. #2599C2×C59semidirect
262. #1610C2×C59semidirect
263. #1607C2×C59semidirect
264. #1604C2×C59semidirect
265. #1601C2×C59semidirect
266. #1514C2×C59semidirect
267. #1464C2×C59semidirect
285. C2×C4×Q8C22×C59semidirect
287. #480C22×C59semidirect
290. #448C22×C59semidirect
292. #432C22×C59semidirect
293. #398C22×C59semidirect
295. #395C22×C59semidirect
301. #383C22×C59semidirect
302. #327C22×C59semidirect
303. #327C22×C59semidirect
304. #324C22×C59semidirect
306. C59#18153direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34492
C59×#31122
C59×#25992
C59×#16102
C59×#16072
C59×#16042
C59×#16012
C59×#15142
C59×#14642
#1815359