Finite Group Databasebeta

← C59×#9931 #28422040

Subgroups — C59×#9931 #28422040

Order 15104.

792 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3193C211yesmaximal, characteristic, complemented
2C59×#3006C211yesmaximal, characteristic
3C59×#2616C211yesmaximal, characteristic, complemented
4C59×#1964C211yesmaximal, characteristic, complemented
5C59×#1789C211yesmaximal, characteristic, complemented
6C59×#1508C211yesmaximal, characteristic, complemented
7C59×#1305C211yesmaximal, characteristic, complemented
8C22×#41×C59C2211yescharacteristic
9C2×AGL(1,ℤ/8)×C59—21no—
10C2×C22⋊Q8×C59—21no—
11C2×C22⋊Q8×C59C2211yescharacteristic
12C2×#121×C59C2211yescharacteristic
13C59×#484—21no—
14C59×#480—21no—
15C59×#468—22no—
16C59×#450—21no—
17C59×#449—22no—
18C59×#448—21no—
19C59×#448—21no—
20C59×#447—21no—
21C2×#104×C59—22no—
22C59×#417—21no—
23C2×#99×C59—22no—
24C59×#407C2211yescharacteristic
25C59×#407C2211yescharacteristic
26C59×#395—21no—
27C59×#394—21no—
28C59×#394—22no—
29C59×#380—21no—
30C59×#380C2211yescharacteristic
31C4×C22⋊C4×C59C2211yescharacteristic
32C59×#328—22no—
33C2×#41×C59—21no—
34C2×#41×C59—21no—
35C2×#41×C59—21no—
36C2×#41×C59—21no—
37C2×#41×C59—44no—
38C22×Q8×C59D411yescharacteristic
39C22×Q8×C59D411yescharacteristic
40C22×D4×C59—21no—
41C22×D4×C59—21no—
42C22×D4×C59D411yescharacteristic
43C22×D4×C59D411yescharacteristic
44C23×C4×C59—21no—
45C23×C4×C59C2311yescharacteristic
46AGL(1,ℤ/8)×C59—44no—
47C2×SD16×C59—22no—
48C2×SD16×C59—22no—
49C2×D8×C59—22no—
50C2×M16×C59—21no—
51C2×M16×C59D412yes—
52C22⋊Q8×C59—21no—
53C22⋊Q8×C59—21no—
54C22⋊Q8×C59—21no—
55C22⋊Q8×C59—21no—
56C22⋊Q8×C59—44no—
57#121×C59—21no—
58#121×C59—21no—
59#121×C59—21no—
60#121×C59—21no—
61C2×C4⋊C4×C59—22no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59D411yescharacteristic
65C2×C4⋊C4×C59D411yescharacteristic
66C2×C22⋊C4×C59—21no—
67C2×C22⋊C4×C59—22no—
68C2×C22⋊C4×C59—21no—
69C2×C22⋊C4×C59—21no—
70C2×C22⋊C4×C59—22no—
71C2×C22⋊C4×C59D412yes—
72C2×C42×C59—21no—
73#107×C59—22no—
74#104×C59—44no—
75#103×C59—22no—
76#103×C59—22no—
77#102×C59—22no—
78#102×C59—22no—
79#99×C59—44no—
80C22⋊C8×C59—21no—
81C22⋊C8×C59—21no—
82C22⋊C8×C59—21no—
83C22⋊C8×C59D412yes—
84#95×C59—22no—
85#95×C59—21no—
86#95×C59—21no—
87C24×C59—21no—
88C24×C59—21no—
89C24×C59C2×D411yescharacteristic
90#41×C59—24no—
91#41×C59—24no—
92#41×C59—42no—
93#41×C59—42no—
94#41×C59—42no—
95#41×C59—42no—
96#41×C59—42no—
97#41×C59—42no—
98C2×Q8×C59—21no—
99C2×Q8×C59—21no—
100C2×Q8×C59—21no—
101C2×Q8×C59—21no—
102C2×Q8×C59—42no—
103C2×Q8×C59—42no—
104C2×D4×C59—21no—
105C2×D4×C59—21no—
106C2×D4×C59—21no—
107C2×D4×C59—21no—
108C2×D4×C59—42no—
109C2×D4×C59—42no—
110C2×D4×C59—42no—
111C2×D4×C59—42no—
112C2×D4×C59—41no—
113C2×D4×C59—41no—
114C2×D4×C59—42no—
115C2×D4×C59—42no—
116C2×D4×C59—41no—
117C2×D4×C59—41no—
118C22×C4×C59—24no—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—22no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—42no—
125C22×C4×C59—42no—
126C22×C4×C59—41no—
127C22×C4×C59—41no—
128C22×C4×C59—41no—
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59#4111yescharacteristic
135SD16×C59—44no—
136SD16×C59—82no—
137D8×C59—44no—
138M16×C59—42no—
139M16×C59—42no—
140C2×C8×C59—22no—
141C2×C8×C59—22no—
142C4⋊C4×C59—21no—
143C4⋊C4×C59—21no—
144C4⋊C4×C59—21no—
145C4⋊C4×C59—21no—
146C4⋊C4×C59—42no—
147C4⋊C4×C59—44no—
148C4⋊C4×C59—41no—
149C4⋊C4×C59—41no—
150C22⋊C4×C59—22no—
151C22⋊C4×C59—41no—
152C22⋊C4×C59—44no—
153C22⋊C4×C59—41no—
154C22⋊C4×C59—41no—
155C22⋊C4×C59—41no—
156C22⋊C4×C59—41no—
157C22⋊C4×C59—42no—
158C22⋊C4×C59—82no—
159C42×C59—42no—
160C23×C59—21no—
161C23×C59—21no—
162C23×C59—22no—
163C23×C59—41no—
164C23×C59—41no—
165C23×C59—41no—
166C23×C59—41no—
167C23×C59—41no—
168C23×C59—42no—
169C23×C59—41no—
170C23×C59—41no—
171C23×C59—41no—
172C23×C59—41no—
173C23×C59—81no—
174C23×C59—81no—
175C23×C59#12011yescharacteristic
176C23×C59#12111yescharacteristic
177C23×C59#12711yescharacteristic
178Q8×C59—22no—
179Q8×C59—22no—
180Q8×C59—41no—
181Q8×C59—41no—
182Q8×C59—42no—
183Q8×C59—81no—
184D4×C59—22no—
185D4×C59—22no—
186D4×C59—41no—
187D4×C59—41no—
188D4×C59—42no—
189D4×C59—44no—
190D4×C59—44no—
191D4×C59—42no—
192D4×C59—82no—
193D4×C59—81no—
194D4×C59—81no—
195C2×C4×C59—22no—
196C2×C4×C59—22no—
197C2×C4×C59—22no—
198C2×C4×C59—22no—
199C2×C4×C59—41no—
200C2×C4×C59—41no—
201C2×C4×C59—41no—
202C2×C4×C59—44no—
203C2×C4×C59—41no—
204C2×C4×C59—41no—
205C2×C4×C59—44no—
206C2×C4×C59—42no—
207C2×C4×C59—42no—
208C2×C4×C59—42no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—42no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—81no—
218C2×C4×C59—81no—
219C2×C4×C59#12011yescharacteristic
220C2×C4×C59#12011yescharacteristic
221C2×C4×C59#12111yescharacteristic
222C2×C4×C59#12111yescharacteristic
223C8×C59—42no—
224C8×C59—42no—
225#9931C5911yesmaximal, characteristic, direct product factor, Sylow
226C22×C59—21no—
227C22×C59—21no—
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—42no—
235C22×C59—42no—
236C22×C59—42no—
237C22×C59—81no—
238C22×C59—81no—
239C22×C59—81no—
240C22×C59—81no—
241C22×C59—81no—
242C22×C59—81no—
243C22×C59—81no—
244C22×C59—81no—
245C22×C59—81no—
246C22×C59—162no—
247C22×C59#39211yescharacteristic
248C22×C59#45311yescharacteristic
249C22×C59#45411yescharacteristic
250C4×C59—21no—
251C4×C59—21no—
252C4×C59—21no—
253C4×C59—42no—
254C4×C59—82no—
255C4×C59—82no—
256C4×C59—81no—
257C4×C59—81no—
258C4×C59—81no—
259C4×C59B(4,𝔽2)12yes—
260#3193C2×C5911yescharacteristic, complemented
261#3006C2×C5911yescharacteristic
262#2616C2×C5911yescharacteristic, complemented
263#1964C2×C5911yescharacteristic, complemented
264#1789C2×C5911yescharacteristic, complemented
265#1508C2×C5911yescharacteristic, complemented
266#1305C2×C5911yescharacteristic, complemented
267C2×C59—21no—
268C2×C59—21no—
269C2×C59—41no—
270C2×C59—41no—
271C2×C59—82no—
272C2×C59—161no—
273C2×C59#200611yescharacteristic
274C2×C59#200911yescharacteristic
275C2×C59#202111yescharacteristic
276C22×#41C22×C5911yescharacteristic
277C2×AGL(1,ℤ/8)—21no—
278C2×C22⋊Q8—21no—
279C2×C22⋊Q8C22×C5911yescharacteristic
280C2×#121C22×C5911yescharacteristic
281#484—21no—
282#480—21no—
283#468—22no—
284#450—21no—
285#449—22no—
286#448—21no—
287#448—21no—
288#447—21no—
289C2×#104—22no—
290#417—21no—
291C2×#99—22no—
292#407C22×C5911yescharacteristic
293#407C22×C5911yescharacteristic
294#395—21no—
295#394—21no—
296#394—22no—
297#380—21no—
298#380C22×C5911yescharacteristic
299C4×C22⋊C4C22×C5911yescharacteristic
300#328—22no—
301C59#993111yescharacteristic, direct product factor, Sylow
302C2×#41—21no—
303C2×#41—21no—
304C2×#41—21no—
305C2×#41—21no—
306C2×#41—44no—
307C22×Q8D4×C5911yescharacteristic
308C22×Q8D4×C5911yescharacteristic
309C22×D4—21no—
310C22×D4—21no—
311C22×D4D4×C5911yescharacteristic
312C22×D4D4×C5911yescharacteristic
313C23×C4—21no—
314C23×C4C23×C5911yescharacteristic
315AGL(1,ℤ/8)—44no—
316C2×SD16—22no—
317C2×SD16—22no—
318C2×D8—22no—
319C2×M16—21no—
320C2×M16D4×C5912yes—
321C22⋊Q8—21no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—44no—
326#121—21no—
327#121—21no—
328#121—21no—
329#121—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4D4×C5911yescharacteristic
334C2×C4⋊C4D4×C5911yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4D4×C5912yes—
341C2×C42—21no—
342#107—22no—
343#104—44no—
344#103—22no—
345#103—22no—
346#102—22no—
347#102—22no—
348#99—44no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8D4×C5912yes—
353#95—22no—
354#95—21no—
355#95—21no—
356C24—21no—
357C24—21no—
358C24C2×D4×C5911yescharacteristic
359#41—24no—
360#41—24no—
361#41—42no—
362#41—42no—
363#41—42no—
364#41—42no—
365#41—42no—
366#41—42no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—41no—
386C2×D4—41no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4C2×D4×C5911yescharacteristic
399C22×C4C2×D4×C5911yescharacteristic
400C22×C4C2×D4×C5911yescharacteristic
401C22×C4C2×D4×C5911yescharacteristic
402C22×C4C2×D4×C5911yescharacteristic
403C22×C4#41×C5911yescharacteristic
404SD16—44no—
405SD16—82no—
406D8—44no—
407M16—42no—
408M16—42no—
409C2×C8—22no—
410C2×C8—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C22⋊C4—22no—
420C22⋊C4—44no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—82no—
428C42—42no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—81no—
443C23—81no—
444C23#120×C5911yescharacteristic
445C23#121×C5911yescharacteristic
446C23#127×C5911yescharacteristic
447Q8—22no—
448Q8—22no—
449Q8—41no—
450Q8—41no—
451Q8—42no—
452Q8—81no—
453D4—22no—
454D4—22no—
455D4—41no—
456D4—41no—
457D4—42no—
458D4—44no—
459D4—44no—
460D4—42no—
461D4—82no—
462D4—81no—
463D4—81no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4#120×C5911yescharacteristic
489C2×C4#120×C5911yescharacteristic
490C2×C4#121×C5911yescharacteristic
491C2×C4#121×C5911yescharacteristic
492C8—42no—
493C8—42no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—162no—
515C22C59×#39211yescharacteristic
516C22C59×#45311yescharacteristic
517C22C59×#45411yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—82no—
523C4—82no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4C59×B(4,𝔽2)12yes—
528C2—21no—
529C2—21no—
530C2—41no—
531C2—41no—
532C2—82no—
533C2—161no—
534C2C59×#200611yescharacteristic
535C2C59×#200911yescharacteristic
536C2C59×#202111yescharacteristic
subgroupG/H
center, socleC22×C59#392
commutator, Frattini, cosocleC23×C4C23×C59
prime radicalC59×#3006C2
LCSG ▷ C23×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×#3193C2semidirect
3. C59×#2616C2semidirect
4. C59×#1964C2semidirect
5. C59×#1789C2semidirect
6. C59×#1508C2semidirect
7. C59×#1305C2semidirect
225. #9931C59direct (N × H)
260. #3193C2×C59semidirect
262. #2616C2×C59semidirect
263. #1964C2×C59semidirect
264. #1789C2×C59semidirect
265. #1508C2×C59semidirect
266. #1305C2×C59semidirect
301. C59#9931direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#31932
C59×#30062
C59×#26162
C59×#19642
C59×#17892
C59×#15082
C59×#13052
#993159