Finite Group Databasebeta

← C59×#9497 #28421606

Subgroups — C59×#9497 #28421606

Order 15104.

1184 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2403C211yesmaximal, characteristic
2C59×#2309C211yesmaximal, characteristic, complemented
3C59×#1781C211yesmaximal, characteristic, complemented
4C59×#1781C212yesmaximal, complemented
5C59×#1304C212yesmaximal, complemented
6C23×D4×C59C2211yescharacteristic
7C23×D4×C59C2211yescharacteristic
8C2×#120×C59—21no—
9C2×#120×C59C412yescomplemented
10C2×#117×C59—21no—
11C2×#99×C59—22no—
12C2×#99×C59—24no—
13C59×#392—22no—
14C59×#390—24no—
15C59×#386—21no—
16C59×#386C412yescomplemented
17C59×#379—21no—
18C59×#379C2212yes—
19C59×#379C2211yescharacteristic
20C59×#379C2212yes—
21C25×C59C2×C411yescharacteristic
22C25×C59D411yescharacteristic
23C25×C59C2311yescharacteristic
24C22×D4×C59—21no—
25C22×D4×C59—22no—
26C22×D4×C59—22no—
27C22×D4×C59—22no—
28C22×D4×C59—21no—
29C22×D4×C59—22no—
30C22×D4×C59—22no—
31C22×D4×C59—44no—
32C22×D4×C59—44no—
33C22×D4×C59C2×C412yes—
34C22×D4×C59D412yes—
35C23×C4×C59C2×C411yescharacteristic
36C23×C4×C59D411yescharacteristic
37#120×C59—44no—
38#120×C59—44no—
39#117×C59—44no—
40C2×C4⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59—22no—
48C2×C22⋊C4×C59—22no—
49C2×C22⋊C4×C59—22no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—22no—
52C2×C22⋊C4×C59C2×C412yes—
53C2×C22⋊C4×C59D414yescomplemented
54C2×C42×C59—22no—
55#99×C59—48no—
56#99×C59—44no—
57#95×C59—22no—
58C24×C59—21no—
59C24×C59—21no—
60C24×C59—22no—
61C24×C59—21no—
62C24×C59—42no—
63C24×C59—42no—
64C24×C59—42no—
65C24×C59—42no—
66C24×C59—42no—
67C24×C59—42no—
68C24×C59—81no—
69C24×C59—81no—
70C24×C59—81no—
71C24×C59C22⋊C412yes—
72C24×C59C22×C411yescharacteristic
73C24×C59C2×D411yescharacteristic
74C24×C59C2×D411yescharacteristic
75C24×C59C2×D412yes—
76C24×C59#4112yes—
77C2×D4×C59—28no—
78C2×D4×C59—24no—
79C2×D4×C59—24no—
80C2×D4×C59—24no—
81C2×D4×C59—24no—
82C2×D4×C59—24no—
83C2×D4×C59—42no—
84C2×D4×C59—42no—
85C2×D4×C59—44no—
86C2×D4×C59—44no—
87C2×D4×C59—48no—
88C2×D4×C59—42no—
89C2×D4×C59—44no—
90C2×D4×C59—42no—
91C2×D4×C59—44no—
92C2×D4×C59—42no—
93C2×D4×C59—44no—
94C2×D4×C59—41no—
95C2×D4×C59—42no—
96C2×D4×C59—41no—
97C2×D4×C59—44no—
98C2×D4×C59—44no—
99C2×D4×C59—82no—
100C22×C4×C59—21no—
101C22×C4×C59—21no—
102C22×C4×C59—24no—
103C22×C4×C59—21no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—24no—
107C22×C4×C59—42no—
108C22×C4×C59—42no—
109C22×C4×C59C22⋊C412yes—
110C4⋊C4×C59—44no—
111C22⋊C4×C59—44no—
112C22⋊C4×C59—44no—
113C22⋊C4×C59—42no—
114C22⋊C4×C59—44no—
115C22⋊C4×C59—44no—
116C22⋊C4×C59—42no—
117C22⋊C4×C59—44no—
118C22⋊C4×C59—44no—
119C22⋊C4×C59—82no—
120C22⋊C4×C59—82no—
121C22⋊C4×C59—81no—
122C22⋊C4×C59—82no—
123C22⋊C4×C59—82no—
124C42×C59—44no—
125C23×C59—22no—
126C23×C59—22no—
127C23×C59—22no—
128C23×C59—22no—
129C23×C59—24no—
130C23×C59—22no—
131C23×C59—24no—
132C23×C59—22no—
133C23×C59—22no—
134C23×C59—22no—
135C23×C59—41no—
136C23×C59—41no—
137C23×C59—41no—
138C23×C59—42no—
139C23×C59—41no—
140C23×C59—42no—
141C23×C59—41no—
142C23×C59—42no—
143C23×C59—42no—
144C23×C59—42no—
145C23×C59—42no—
146C23×C59—42no—
147C23×C59—42no—
148C23×C59—41no—
149C23×C59—42no—
150C23×C59—42no—
151C23×C59—42no—
152C23×C59—42no—
153C23×C59—41no—
154C23×C59—42no—
155C23×C59—42no—
156C23×C59—81no—
157C23×C59—81no—
158C23×C59—82no—
159C23×C59—82no—
160C23×C59—88no—
161C23×C59—88no—
162C23×C59—88no—
163C23×C59C2×C22⋊C411yescharacteristic
164C23×C59C4×D412yes—
165C23×C59#12011yescharacteristic
166C23×C59#12112yes—
167C23×C59#12311yescharacteristic
168D4×C59—48no—
169D4×C59—48no—
170D4×C59—48no—
171D4×C59—44no—
172D4×C59—44no—
173D4×C59—88no—
174C2×C4×C59—22no—
175C2×C4×C59—22no—
176C2×C4×C59—24no—
177C2×C4×C59—22no—
178C2×C4×C59—22no—
179C2×C4×C59—22no—
180C2×C4×C59—24no—
181C2×C4×C59—41no—
182C2×C4×C59—42no—
183C2×C4×C59—41no—
184C2×C4×C59—44no—
185C2×C4×C59—42no—
186C2×C4×C59—44no—
187C2×C4×C59—42no—
188C2×C4×C59—42no—
189C2×C4×C59—44no—
190C2×C4×C59—82no—
191C2×C4×C59—82no—
192C2×C4×C59—82no—
193C2×C4×C59#9914yescomplemented
194#9497C5911yesmaximal, characteristic, direct product factor, Sylow
195C22×C59—21no—
196C22×C59—21no—
197C22×C59—21no—
198C22×C59—21no—
199C22×C59—22no—
200C22×C59—22no—
201C22×C59—21no—
202C22×C59—21no—
203C22×C59—22no—
204C22×C59—22no—
205C22×C59—22no—
206C22×C59—22no—
207C22×C59—21no—
208C22×C59—44no—
209C22×C59—44no—
210C22×C59—44no—
211C22×C59—44no—
212C22×C59—42no—
213C22×C59—42no—
214C22×C59—42no—
215C22×C59—42no—
216C22×C59—84no—
217C22×C59—82no—
218C22×C59—82no—
219C22×C59—84no—
220C22×C59—82no—
221C22×C59—84no—
222C22×C59—84no—
223C22×C59—84no—
224C22×C59—84no—
225C22×C59—81no—
226C22×C59—81no—
227C22×C59#38611yescharacteristic
228C22×C59C2×#9912yes—
229C22×C59B(4,𝔽2)12yes—
230C4×C59—24no—
231C4×C59—42no—
232C4×C59—84no—
233C4×C59—82no—
234C4×C59—162no—
235#2403C2×C5911yescharacteristic
236#2309C2×C5911yescharacteristic, complemented
237#1781C2×C5912yescomplemented
238#1781C2×C5911yescharacteristic, complemented
239#1304C2×C5912yescomplemented
240C2×C59—22no—
241C2×C59—22no—
242C2×C59—42no—
243C2×C59—42no—
244C2×C59—41no—
245C2×C59—82no—
246C2×C59—82no—
247C2×C59#188911yescharacteristic
248C2×C59#189811yescharacteristic
249C2×C59#191311yescharacteristic
250C23×D4C22×C5911yescharacteristic
251C23×D4C22×C5911yescharacteristic
252C2×#120—21no—
253C2×#120C4×C5912yescomplemented
254C2×#117—21no—
255C2×#99—22no—
256C2×#99—24no—
257#392—22no—
258#390—24no—
259#386—21no—
260#386C4×C5912yescomplemented
261#379—21no—
262#379C22×C5912yes—
263#379C22×C5911yescharacteristic
264#379C22×C5912yes—
265C59#949711yescharacteristic, direct product factor, Sylow
266C25C2×C4×C5911yescharacteristic
267C25D4×C5911yescharacteristic
268C25C23×C5911yescharacteristic
269C22×D4—21no—
270C22×D4—22no—
271C22×D4—22no—
272C22×D4—22no—
273C22×D4—21no—
274C22×D4—22no—
275C22×D4—22no—
276C22×D4—44no—
277C22×D4—44no—
278C22×D4C2×C4×C5912yes—
279C22×D4D4×C5912yes—
280C23×C4C2×C4×C5911yescharacteristic
281C23×C4D4×C5911yescharacteristic
282#120—44no—
283#120—44no—
284#117—44no—
285C2×C4⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4C2×C4×C5912yes—
298C2×C22⋊C4D4×C5914yescomplemented
299C2×C42—22no—
300#99—48no—
301#99—44no—
302#95—22no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—21no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—81no—
314C24—81no—
315C24—81no—
316C24C22⋊C4×C5912yes—
317C24C22×C4×C5911yescharacteristic
318C24C2×D4×C5911yescharacteristic
319C24C2×D4×C5911yescharacteristic
320C24C2×D4×C5912yes—
321C24#41×C5912yes—
322C2×D4—28no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—48no—
333C2×D4—42no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—82no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4C22⋊C4×C5912yes—
355C4⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—81no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—44no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—22no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—82no—
404C23—82no—
405C23—88no—
406C23—88no—
407C23—88no—
408C23C2×C22⋊C4×C5911yescharacteristic
409C23C4×D4×C5912yes—
410C23#120×C5911yescharacteristic
411C23#121×C5912yes—
412C23#123×C5911yescharacteristic
413D4—48no—
414D4—48no—
415D4—48no—
416D4—44no—
417D4—44no—
418D4—88no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4#99×C5914yescomplemented
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—21no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—84no—
461C22—82no—
462C22—82no—
463C22—84no—
464C22—82no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—81no—
470C22—81no—
471C22C59×#38611yescharacteristic
472C22C2×#99×C5912yes—
473C22C59×B(4,𝔽2)12yes—
474C4—24no—
475C4—42no—
476C4—82no—
477C4—84no—
478C4—162no—
479C2—22no—
480C2—22no—
481C2—42no—
482C2—42no—
483C2—41no—
484C2—82no—
485C2—82no—
486C2C59×#188911yescharacteristic
487C2C59×#189811yescharacteristic
488C2C59×#191311yescharacteristic
subgroupG/H
center, socleC22×C59#386
commutatorC24C22×C4×C59
Frattini, cosocleC25C23×C59
prime radicalC59×#2403C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C25×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#2309C2semidirect
3. C59×#1781C2semidirect
4. C59×#1781C2semidirect
5. C59×#1304C2semidirect
9. C2×#120×C59C4semidirect
16. C59×#386C4semidirect
53. C2×C22⋊C4×C59D4semidirect
193. C2×C4×C59#99semidirect
194. #9497C59direct (N × H)
236. #2309C2×C59semidirect
237. #1781C2×C59semidirect
238. #1781C2×C59semidirect
239. #1304C2×C59semidirect
253. C2×#120C4×C59semidirect
260. #386C4×C59semidirect
265. C59#9497direct (N × H)
298. C2×C22⋊C4D4×C59semidirect
438. C2×C4#99×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#24032
C59×#23092
C59×#17812
C59×#13042
#949759