Finite Group Databasebeta

← C59×#12190 #28424299

Subgroups — C59×#12190 #28424299

Order 15104.

678 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#2670C211yesmaximal, characteristic, complemented
2C59×#2021C211yesmaximal, characteristic, complemented
3C59×#1601C211yesmaximal, characteristic, complemented
4C59×#1595C211yesmaximal, characteristic, complemented
5C59×#1351C211yesmaximal, characteristic, complemented
6C59×#1343C211yesmaximal, characteristic, complemented
7C59×#1316C211yesmaximal, characteristic, complemented
8C59×#534C2211yescharacteristic
9C2×#124×C59—21no—
10C2×#120×C59—21no—
11C2×#120×C59—21no—
12C59×#448—21no—
13C59×#447—21no—
14C2×#99×C59—21no—
15C59×#394—21no—
16C59×#393C2211yescharacteristic
17C59×#392—21no—
18C59×#392C2211yescharacteristic
19C59×#390—21no—
20C59×#388—21no—
21C59×#379C2211yescharacteristic
22C59×#342C2211yescharacteristic
23C59×#327—21no—
24C59×#327—21no—
25C59×#323C2211yescharacteristic
26C59×#323C2211yescharacteristic
27C25×C59D411yescharacteristic
28C22×Q8×C59D411yescharacteristic, complemented
29C22×D4×C59—21no—
30C22×D4×C59—21no—
31C22×D4×C59—21no—
32C22×D4×C59—21no—
33C22×D4×C59—41no—
34C22×D4×C59D411yescharacteristic, complemented
35C2×SD16×C59—41no—
36C2×D8×C59—41no—
37#124×C59—44no—
38#123×C59—41no—
39#121×C59—21no—
40#121×C59—21no—
41#120×C59—44no—
42#120×C59—42no—
43#120×C59—41no—
44#120×C59—42no—
45C2×C4⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—41no—
56C2×C22⋊C4×C59D411yescharacteristic
57C2×C22⋊C4×C59C2311yescharacteristic
58C2×C42×C59—21no—
59#103×C59—41no—
60#102×C59—41no—
61#99×C59—42no—
62C22⋊C8×C59—21no—
63C22⋊C8×C59—21no—
64#95×C59—21no—
65#95×C59D411yescharacteristic, complemented
66#95×C59D411yescharacteristic, complemented
67C24×C59—21no—
68C24×C59—21no—
69C24×C59—21no—
70C24×C59—21no—
71C24×C59—21no—
72C24×C59—41no—
73C24×C59—41no—
74C24×C59—42no—
75C24×C59—81no—
76C24×C59C2×D411yescharacteristic
77C2×Q8×C59—42no—
78C2×Q8×C59—41no—
79C2×D4×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59—41no—
82C2×D4×C59—42no—
83C2×D4×C59—42no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C2×D4×C59—42no—
87C2×D4×C59—41no—
88C2×D4×C59—42no—
89C2×D4×C59—44no—
90C2×D4×C59—41no—
91C2×D4×C59—41no—
92C2×D4×C59—41no—
93C2×D4×C59—41no—
94C2×D4×C59—41no—
95C2×D4×C59—81no—
96C2×D4×C59—81no—
97C2×D4×C59—81no—
98C2×D4×C59—81no—
99C2×D4×C59—81no—
100C2×D4×C59—82no—
101C2×D4×C59—82no—
102C22×C4×C59—21no—
103C22×C4×C59—21no—
104C22×C4×C59—21no—
105C22×C4×C59—21no—
106C22×C4×C59—21no—
107C22×C4×C59—41no—
108C22×C4×C59C2×D411yescharacteristic
109C22×C4×C59C2×D411yescharacteristic
110SD16×C59—82no—
111D8×C59—82no—
112C2×C8×C59—41no—
113C2×C8×C59—41no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59—21no—
116C4⋊C4×C59—81no—
117C22⋊C4×C59—24no—
118C22⋊C4×C59—22no—
119C22⋊C4×C59—41no—
120C22⋊C4×C59—42no—
121C22⋊C4×C59—42no—
122C22⋊C4×C59—42no—
123C22⋊C4×C59—42no—
124C22⋊C4×C59—41no—
125C22⋊C4×C59—41no—
126C22⋊C4×C59—42no—
127C22⋊C4×C59—44no—
128C22⋊C4×C59—41no—
129C22⋊C4×C59—41no—
130C22⋊C4×C59—81no—
131C22⋊C4×C59—81no—
132C42×C59—42no—
133C23×C59—21no—
134C23×C59—21no—
135C23×C59—22no—
136C23×C59—41no—
137C23×C59—41no—
138C23×C59—41no—
139C23×C59—42no—
140C23×C59—41no—
141C23×C59—41no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59—41no—
145C23×C59—41no—
146C23×C59—41no—
147C23×C59—41no—
148C23×C59—42no—
149C23×C59—42no—
150C23×C59—41no—
151C23×C59—41no—
152C23×C59—41no—
153C23×C59—41no—
154C23×C59—41no—
155C23×C59—41no—
156C23×C59—41no—
157C23×C59—81no—
158C23×C59—82no—
159C23×C59—81no—
160C23×C59—82no—
161C23×C59—84no—
162C23×C59—81no—
163C23×C59—81no—
164C23×C59—81no—
165C23×C59#12011yescharacteristic
166C23×C59#13511yescharacteristic
167C23×C59AGL(1,ℤ/8)11yescharacteristic
168Q8×C59—42no—
169Q8×C59—81no—
170D4×C59—42no—
171D4×C59—82no—
172D4×C59—82no—
173D4×C59—84no—
174D4×C59—81no—
175D4×C59—81no—
176D4×C59—81no—
177D4×C59—81no—
178D4×C59—81no—
179D4×C59—84no—
180D4×C59—88no—
181D4×C59—81no—
182D4×C59—81no—
183C2×C4×C59—22no—
184C2×C4×C59—22no—
185C2×C4×C59—21no—
186C2×C4×C59—21no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59—41no—
193C2×C4×C59—41no—
194C2×C4×C59—42no—
195C2×C4×C59—41no—
196C2×C4×C59—41no—
197C2×C4×C59—41no—
198C2×C4×C59—41no—
199C2×C4×C59—41no—
200C2×C4×C59—41no—
201C2×C4×C59—41no—
202C2×C4×C59—41no—
203C2×C4×C59—81no—
204C2×C4×C59—81no—
205C8×C59—81no—
206C8×C59—81no—
207#12190C5911yesmaximal, characteristic, direct product factor, Sylow
208C22×C59—21no—
209C22×C59—21no—
210C22×C59—21no—
211C22×C59—21no—
212C22×C59—41no—
213C22×C59—41no—
214C22×C59—42no—
215C22×C59—42no—
216C22×C59—41no—
217C22×C59—41no—
218C22×C59—44no—
219C22×C59—42no—
220C22×C59—81no—
221C22×C59—81no—
222C22×C59—81no—
223C22×C59—81no—
224C22×C59—81no—
225C22×C59—81no—
226C22×C59—81no—
227C22×C59—82no—
228C22×C59—82no—
229C22×C59—82no—
230C22×C59—82no—
231C22×C59—81no—
232C22×C59—81no—
233C22×C59—81no—
234C22×C59—81no—
235C22×C59—81no—
236C22×C59—81no—
237C22×C59—81no—
238C22×C59—81no—
239C22×C59—161no—
240C22×C59#44911yescharacteristic
241C22×C59#45311yescharacteristic
242C22×C59B(4,𝔽2)11yescharacteristic
243C4×C59—42no—
244C4×C59—41no—
245C4×C59—42no—
246C4×C59—41no—
247C4×C59—81no—
248C4×C59—81no—
249C4×C59—82no—
250C4×C59—81no—
251#2670C2×C5911yescharacteristic, complemented
252#2021C2×C5911yescharacteristic, complemented
253#1601C2×C5911yescharacteristic, complemented
254#1595C2×C5911yescharacteristic, complemented
255#1351C2×C5911yescharacteristic, complemented
256#1343C2×C5911yescharacteristic, complemented
257#1316C2×C5911yescharacteristic, complemented
258C2×C59—21no—
259C2×C59—21no—
260C2×C59—41no—
261C2×C59—41no—
262C2×C59—82no—
263C2×C59—161no—
264C2×C59—161no—
265C2×C59#160011yescharacteristic
266C2×C59#219611yescharacteristic
267C2×C59#219911yescharacteristic
268#534C22×C5911yescharacteristic
269C2×#124—21no—
270C2×#120—21no—
271C2×#120—21no—
272#448—21no—
273#447—21no—
274C2×#99—21no—
275#394—21no—
276#393C22×C5911yescharacteristic
277#392—21no—
278#392C22×C5911yescharacteristic
279#390—21no—
280#388—21no—
281#379C22×C5911yescharacteristic
282#342C22×C5911yescharacteristic
283#327—21no—
284#327—21no—
285#323C22×C5911yescharacteristic
286#323C22×C5911yescharacteristic
287C59#1219011yescharacteristic, direct product factor, Sylow
288C25D4×C5911yescharacteristic
289C22×Q8D4×C5911yescharacteristic, complemented
290C22×D4—21no—
291C22×D4—21no—
292C22×D4—21no—
293C22×D4—21no—
294C22×D4—41no—
295C22×D4D4×C5911yescharacteristic, complemented
296C2×SD16—41no—
297C2×D8—41no—
298#124—44no—
299#123—41no—
300#121—21no—
301#121—21no—
302#120—44no—
303#120—42no—
304#120—41no—
305#120—42no—
306C2×C4⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—41no—
317C2×C22⋊C4D4×C5911yescharacteristic
318C2×C22⋊C4C23×C5911yescharacteristic
319C2×C42—21no—
320#103—41no—
321#102—41no—
322#99—42no—
323C22⋊C8—21no—
324C22⋊C8—21no—
325#95—21no—
326#95D4×C5911yescharacteristic, complemented
327#95D4×C5911yescharacteristic, complemented
328C24—21no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—41no—
334C24—41no—
335C24—42no—
336C24—81no—
337C24C2×D4×C5911yescharacteristic
338C2×Q8—42no—
339C2×Q8—41no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—42no—
348C2×D4—41no—
349C2×D4—42no—
350C2×D4—44no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—81no—
357C2×D4—81no—
358C2×D4—81no—
359C2×D4—81no—
360C2×D4—81no—
361C2×D4—82no—
362C2×D4—82no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—41no—
369C22×C4C2×D4×C5911yescharacteristic
370C22×C4C2×D4×C5911yescharacteristic
371SD16—82no—
372D8—82no—
373C2×C8—41no—
374C2×C8—41no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—81no—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—41no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—42no—
388C22⋊C4—44no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—81no—
392C22⋊C4—81no—
393C42—42no—
394C23—21no—
395C23—21no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—81no—
419C23—82no—
420C23—81no—
421C23—82no—
422C23—84no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23#120×C5911yescharacteristic
427C23#135×C5911yescharacteristic
428C23AGL(1,ℤ/8)×C5911yescharacteristic
429Q8—42no—
430Q8—81no—
431D4—42no—
432D4—82no—
433D4—82no—
434D4—84no—
435D4—81no—
436D4—81no—
437D4—81no—
438D4—81no—
439D4—81no—
440D4—84no—
441D4—88no—
442D4—81no—
443D4—81no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—81no—
465C2×C4—81no—
466C8—81no—
467C8—81no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—44no—
479C22—42no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—161no—
500C22C59×#44911yescharacteristic
501C22C59×#45311yescharacteristic
502C22C59×B(4,𝔽2)11yescharacteristic
503C4—42no—
504C4—41no—
505C4—42no—
506C4—41no—
507C4—81no—
508C4—81no—
509C4—82no—
510C4—81no—
511C2—21no—
512C2—21no—
513C2—41no—
514C2—41no—
515C2—82no—
516C2—161no—
517C2—161no—
518C2C59×#160011yescharacteristic
519C2C59×#219611yescharacteristic
520C2C59×#219911yescharacteristic
subgroupG/H
center, socleC22×C59B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C59
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C59 ◁ C2×C22⋊C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#2670C2semidirect
2. C59×#2021C2semidirect
3. C59×#1601C2semidirect
4. C59×#1595C2semidirect
5. C59×#1351C2semidirect
6. C59×#1343C2semidirect
7. C59×#1316C2semidirect
28. C22×Q8×C59D4semidirect
34. C22×D4×C59D4semidirect
65. #95×C59D4semidirect
66. #95×C59D4semidirect
207. #12190C59direct (N × H)
251. #2670C2×C59semidirect
252. #2021C2×C59semidirect
253. #1601C2×C59semidirect
254. #1595C2×C59semidirect
255. #1351C2×C59semidirect
256. #1343C2×C59semidirect
257. #1316C2×C59semidirect
287. C59#12190direct (N × H)
289. C22×Q8D4×C59semidirect
295. C22×D4D4×C59semidirect
326. #95D4×C59semidirect
327. #95D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#26702
C59×#20212
C59×#16012
C59×#15952
C59×#13512
C59×#13432
C59×#13162
#1219059