Finite Group Databasebeta

← C59×#19200 #28431309

Subgroups — C59×#19200 #28431309

Order 15104.

660 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3246C211yesmaximal, characteristic, complemented
2C59×#3245C211yesmaximal, characteristic
3C59×#3131C211yesmaximal, characteristic, complemented
4C59×#2931C211yesmaximal, characteristic, complemented
5C59×#1703C212yesmaximal, complemented
6C59×#1701C212yesmaximal, complemented
7C59×#1679C212yesmaximal, complemented
8C59×#1677C212yesmaximal, complemented
9C2×C59×#334C211yesmaximal, characteristic, complemented
10C59×#1550C211yesmaximal, characteristic, complemented
11C59×#1547C211yesmaximal, characteristic, complemented
12C2×#131×C59—21no—
13C59×#533C2211yescharacteristic
14C59×#532C2211yescharacteristic
15C59×#532C2211yescharacteristic
16C59×#488—21no—
17C59×#488—21no—
18C59×#487C2211yescharacteristic
19C59×#487C2211yescharacteristic
20C59×#486C2211yescharacteristic
21C59×#486C2211yescharacteristic
22C59×#476C2212yes—
23C59×#475—21no—
24C59×#474—21no—
25C59×#474C2212yes—
26C59×#466—22no—
27C59×#465—22no—
28C59×#461C2212yes—
29C59×#460—21no—
30C59×#459—21no—
31C59×#459C2212yes—
32C59×#436—21no—
33C59×#436C2212yes—
34C8×D4×C59—21no—
35C8×D4×C59C2212yes—
36C59×#423C2211yescharacteristic
37C2×C4⋊C8×C59C2211yescharacteristic
38C2×C4⋊C8×C59C2211yescharacteristic
39C59×#416—21no—
40C59×#416—21no—
41C59×#408—21no—
42C2×C4×C8×C59C2211yescharacteristic
43C59×#334C2212yes—
44C59×#334C2212yes—
45C59×#333C2212yes—
46C59×#332C2212yes—
47C59×#332C2212yes—
48C59×#331C2212yes—
49C2×#41×C59—21no—
50C2×#41×C59—21no—
51C2×SD16×C59—22no—
52C2×D8×C59—22no—
53#131×C59—44no—
54C2×M16×C59—21no—
55C2×M16×C59—22no—
56C22×C8×C59—21no—
57C22×C8×C59—21no—
58C22×C8×C59—22no—
59C22×C8×C59D411yescharacteristic
60C22×C8×C59D411yescharacteristic
61C4⋊Q8×C59D411yescharacteristic
62C4⋊Q8×C59C2311yescharacteristic
63C4⋊Q8×C59C2311yescharacteristic
64C4⋊Q8×C59C2311yescharacteristic
65#127×C59D411yescharacteristic
66#127×C59C2311yescharacteristic
67#125×C59—21no—
68#124×C59—21no—
69#124×C59—21no—
70C22⋊Q8×C59—21no—
71C22⋊Q8×C59—21no—
72#121×C59—21no—
73#121×C59—21no—
74#121×C59—21no—
75#121×C59D412yes—
76C4×Q8×C59—21no—
77C4×Q8×C59—21no—
78C4×D4×C59—21no—
79C4×D4×C59—21no—
80C4×D4×C59—21no—
81C4×D4×C59C2312yes—
82#117×C59—21no—
83C2×C42×C59C2311yescharacteristic
84#107×C59—22no—
85#106×C59—22no—
86C4⋊C8×C59—21no—
87C4⋊C8×C59—21no—
88C4⋊C8×C59—21no—
89C4⋊C8×C59C2312yes—
90C4⋊C8×C59C2312yes—
91C4⋊C8×C59C2312yes—
92#103×C59—21no—
93#103×C59—21no—
94#103×C59—21no—
95#103×C59—21no—
96#103×C59—22no—
97#102×C59—22no—
98#102×C59—22no—
99#102×C59—21no—
100#102×C59—21no—
101#102×C59—21no—
102#102×C59—21no—
103#102×C59—22no—
104C22⋊C8×C59—22no—
105C22⋊C8×C59—22no—
106C4×C8×C59D411yescharacteristic
107C4×C8×C59D411yescharacteristic
108C4×C8×C59C2311yescharacteristic
109C4×C8×C59C2311yescharacteristic
110#41×C59—42no—
111#41×C59—42no—
112#41×C59—42no—
113#41×C59—42no—
114C2×Q8×C59—21no—
115C2×Q8×C59—21no—
116C2×Q8×C59—21no—
117C2×Q8×C59—21no—
118C2×D4×C59—22no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—41no—
125C2×D4×C59C2×D412yes—
126C22×C4×C59—22no—
127C22×C4×C59—41no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59#4111yescharacteristic
131SD16×C59—82no—
132D8×C59—82no—
133M16×C59—44no—
134M16×C59—42no—
135C2×C8×C59—22no—
136C2×C8×C59—22no—
137C2×C8×C59—22no—
138C2×C8×C59—22no—
139C2×C8×C59—22no—
140C2×C8×C59—44no—
141C2×C8×C59—81no—
142C2×C8×C59—81no—
143C2×C8×C59C2×D411yescharacteristic
144C2×C8×C59C2×D411yescharacteristic
145C2×C8×C59C2×D411yescharacteristic
146C2×C8×C59C2×D411yescharacteristic
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—41no—
155C4⋊C4×C59—41no—
156C4⋊C4×C59C2×D412yes—
157C22⋊C4×C59—22no—
158C22⋊C4×C59—22no—
159C22⋊C4×C59—41no—
160C22⋊C4×C59—41no—
161C22⋊C4×C59—41no—
162C42×C59—41no—
163C42×C59C2×D411yescharacteristic
164C42×C59C2×D411yescharacteristic
165C42×C59#4111yescharacteristic
166C42×C59C2411yescharacteristic
167C23×C59—22no—
168C23×C59—41no—
169C23×C59#12111yescharacteristic
170Q8×C59—42no—
171Q8×C59—41no—
172Q8×C59—41no—
173Q8×C59—41no—
174Q8×C59—41no—
175Q8×C59—81no—
176D4×C59—22no—
177D4×C59—22no—
178D4×C59—42no—
179D4×C59—44no—
180D4×C59—41no—
181D4×C59—41no—
182D4×C59—41no—
183D4×C59—41no—
184D4×C59—42no—
185D4×C59—81no—
186D4×C59—82no—
187C2×C4×C59—21no—
188C2×C4×C59—21no—
189C2×C4×C59—22no—
190C2×C4×C59—41no—
191C2×C4×C59—42no—
192C2×C4×C59—42no—
193C2×C4×C59—41no—
194C2×C4×C59—41no—
195C2×C4×C59—41no—
196C2×C4×C59—82no—
197C2×C4×C59#12111yescharacteristic
198C2×C4×C59#12111yescharacteristic
199C2×C4×C59#12111yescharacteristic
200C2×C4×C59#13512yes—
201C2×C4×C59AGL(1,ℤ/8)12yes—
202C2×C4×C59C22×D411yescharacteristic
203C2×C4×C59C22×D411yescharacteristic
204C2×C4×C59C2×#4111yescharacteristic
205C8×C59—21no—
206C8×C59—21no—
207C8×C59—21no—
208C8×C59—21no—
209C8×C59—42no—
210C8×C59—42no—
211C8×C59—42no—
212#19200C5911yesmaximal, characteristic, direct product factor, Sylow
213C22×C59—21no—
214C22×C59—21no—
215C22×C59—42no—
216C22×C59—42no—
217C22×C59—42no—
218C22×C59—81no—
219C22×C59—81no—
220C22×C59—81no—
221C22×C59#46312yes—
222C22×C59C2×#12111yescharacteristic
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—21no—
226C4×C59—42no—
227C4×C59—81no—
228C4×C59—81no—
229C4×C59—81no—
230C4×C59#46312yes—
231C4×C59C2×#13511yescharacteristic
232C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
233C4×C59#57711yescharacteristic
234C4×C59#57711yescharacteristic
235#3246C2×C5911yescharacteristic, complemented
236#3245C2×C5911yescharacteristic
237#3131C2×C5911yescharacteristic, complemented
238#2931C2×C5911yescharacteristic, complemented
239#1703C2×C5912yescomplemented
240#1701C2×C5912yescomplemented
241#1679C2×C5912yescomplemented
242#1677C2×C5912yescomplemented
243C2×#334C2×C5911yescharacteristic, complemented
244#1550C2×C5911yescharacteristic, complemented
245#1547C2×C5911yescharacteristic, complemented
246C2×C59—22no—
247C2×C59—82no—
248C2×C59—161no—
249C2×C59C2×#46311yescharacteristic
250C2×C59#306011yescharacteristic
251C2×C59#306711yescharacteristic
252C2×#131—21no—
253#533C22×C5911yescharacteristic
254#532C22×C5911yescharacteristic
255#532C22×C5911yescharacteristic
256#488—21no—
257#488—21no—
258#487C22×C5911yescharacteristic
259#487C22×C5911yescharacteristic
260#486C22×C5911yescharacteristic
261#486C22×C5911yescharacteristic
262#476C22×C5912yes—
263#475—21no—
264#474—21no—
265#474C22×C5912yes—
266#466—22no—
267#465—22no—
268#461C22×C5912yes—
269#460—21no—
270#459—21no—
271#459C22×C5912yes—
272#436—21no—
273#436C22×C5912yes—
274C8×D4—21no—
275C8×D4C22×C5912yes—
276#423C22×C5911yescharacteristic
277C2×C4⋊C8C22×C5911yescharacteristic
278C2×C4⋊C8C22×C5911yescharacteristic
279#416—21no—
280#416—21no—
281#408—21no—
282C2×C4×C8C22×C5911yescharacteristic
283#334C22×C5912yes—
284#334C22×C5912yes—
285#333C22×C5912yes—
286#332C22×C5912yes—
287#332C22×C5912yes—
288#331C22×C5912yes—
289C59#1920011yescharacteristic, direct product factor, Sylow
290C2×#41—21no—
291C2×#41—21no—
292C2×SD16—22no—
293C2×D8—22no—
294#131—44no—
295C2×M16—21no—
296C2×M16—22no—
297C22×C8—21no—
298C22×C8—21no—
299C22×C8—22no—
300C22×C8D4×C5911yescharacteristic
301C22×C8D4×C5911yescharacteristic
302C4⋊Q8D4×C5911yescharacteristic
303C4⋊Q8C23×C5911yescharacteristic
304C4⋊Q8C23×C5911yescharacteristic
305C4⋊Q8C23×C5911yescharacteristic
306#127D4×C5911yescharacteristic
307#127C23×C5911yescharacteristic
308#125—21no—
309#124—21no—
310#124—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313#121—21no—
314#121—21no—
315#121—21no—
316#121D4×C5912yes—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4C23×C5912yes—
323#117—21no—
324C2×C42C23×C5911yescharacteristic
325#107—22no—
326#106—22no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8C23×C5912yes—
331C4⋊C8C23×C5912yes—
332C4⋊C8C23×C5912yes—
333#103—21no—
334#103—21no—
335#103—21no—
336#103—21no—
337#103—22no—
338#102—22no—
339#102—22no—
340#102—21no—
341#102—21no—
342#102—21no—
343#102—21no—
344#102—22no—
345C22⋊C8—22no—
346C22⋊C8—22no—
347C4×C8D4×C5911yescharacteristic
348C4×C8D4×C5911yescharacteristic
349C4×C8C23×C5911yescharacteristic
350C4×C8C23×C5911yescharacteristic
351#41—42no—
352#41—42no—
353#41—42no—
354#41—42no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4C2×D4×C5912yes—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4C2×D4×C5911yescharacteristic
370C22×C4C2×D4×C5911yescharacteristic
371C22×C4#41×C5911yescharacteristic
372SD16—82no—
373D8—82no—
374M16—44no—
375M16—42no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—44no—
382C2×C8—81no—
383C2×C8—81no—
384C2×C8C2×D4×C5911yescharacteristic
385C2×C8C2×D4×C5911yescharacteristic
386C2×C8C2×D4×C5911yescharacteristic
387C2×C8C2×D4×C5911yescharacteristic
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4C2×D4×C5912yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C42—41no—
404C42C2×D4×C5911yescharacteristic
405C42C2×D4×C5911yescharacteristic
406C42#41×C5911yescharacteristic
407C42C24×C5911yescharacteristic
408C23—22no—
409C23—41no—
410C23#121×C5911yescharacteristic
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—41no—
415Q8—41no—
416Q8—81no—
417D4—22no—
418D4—22no—
419D4—42no—
420D4—44no—
421D4—42no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—81no—
427D4—82no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—82no—
438C2×C4#121×C5911yescharacteristic
439C2×C4#121×C5911yescharacteristic
440C2×C4#121×C5911yescharacteristic
441C2×C4#135×C5912yes—
442C2×C4AGL(1,ℤ/8)×C5912yes—
443C2×C4C22×D4×C5911yescharacteristic
444C2×C4C22×D4×C5911yescharacteristic
445C2×C4C2×#41×C5911yescharacteristic
446C8—21no—
447C8—21no—
448C8—21no—
449C8—21no—
450C8—42no—
451C8—42no—
452C8—42no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22C59×#46312yes—
462C22C2×#121×C5911yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4C59×#46312yes—
471C4C2×#135×C5911yescharacteristic
472C4C2×AGL(1,ℤ/8)×C5911yescharacteristic
473C4C59×#57711yescharacteristic
474C4C59×#57711yescharacteristic
475C2—22no—
476C2—82no—
477C2—161no—
478C2C2×C59×#46311yescharacteristic
479C2C59×#306011yescharacteristic
480C2C59×#306711yescharacteristic
subgroupG/H
center, socleC22×C59C2×#121
commutator, Frattini, cosocleC42C24×C59
prime radicalC59×#3245C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3246C2semidirect
3. C59×#3131C2semidirect
4. C59×#2931C2semidirect
5. C59×#1703C2semidirect
6. C59×#1701C2semidirect
7. C59×#1679C2semidirect
8. C59×#1677C2semidirect
9. C2×C59×#334C2semidirect
10. C59×#1550C2semidirect
11. C59×#1547C2semidirect
212. #19200C59direct (N × H)
235. #3246C2×C59semidirect
237. #3131C2×C59semidirect
238. #2931C2×C59semidirect
239. #1703C2×C59semidirect
240. #1701C2×C59semidirect
241. #1679C2×C59semidirect
242. #1677C2×C59semidirect
243. C2×#334C2×C59semidirect
244. #1550C2×C59semidirect
245. #1547C2×C59semidirect
289. C59#19200direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#32462
C59×#32452
C59×#31312
C59×#29312
C59×#17032
C59×#17012
C59×#16792
C59×#16772
C2×C59×#3342
C59×#15502
C59×#15472
#1920059