Finite Group Databasebeta

← C59×#33571 #28445680

Subgroups — C59×#33571 #28445680

Order 15104.

1084 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3550C211yesmaximal, characteristic, complemented
2C59×#3022C212yesmaximal, complemented
3C59×#3018C211yesmaximal, characteristic
4C59×#2205C214yesmaximal, complemented
5C59×#2203C214yesmaximal, complemented
6C59×#2135C212yesmaximal, complemented
7C59×#2132C211yesmaximal, characteristic, complemented
8C59×#585C2211yescharacteristic
9C59×#575—21no—
10C2×#137×C59—22no—
11C59×#563—21no—
12C59×#563C2212yescomplemented
13C59×#559—21no—
14Q82×C59C2212yescomplemented
15C59×#556—22no—
16D4×Q8×C59—22no—
17C59×#542—24no—
18C59×#536—21no—
19C59×#536C2212yes—
20C59×#533—21no—
21C59×#532C2211yescharacteristic
22C59×#456C2212yes—
23C59×#456C2218yes—
24C59×#455—21no—
25C59×#455—24no—
26C59×#454C2212yes—
27C59×#453—21no—
28C59×#421—22no—
29C2×#104×C59—21no—
30C59×#413C2211yescharacteristic
31C59×#413C2212yes—
32C59×#356C2212yescomplemented
33C59×#356C2218yescomplemented
34C59×#355C2212yescomplemented
352−1+4×C59—41no—
362−1+4×C59C2312yes—
37H(5,𝔽2)×C59—21no—
38H(5,𝔽2)×C59—22no—
39H(5,𝔽2)×C59—41no—
40C2×#41×C59—21no—
41C2×#41×C59—22no—
42C2×#41×C59—42no—
43C2×#41×C59D411yescharacteristic
44C2×#41×C59C2311yescharacteristic
45C22×Q8×C59—21no—
46C22×Q8×C59D412yes—
47#137×C59—28no—
48#137×C59—22no—
49#137×C59—41no—
50#137×C59—44no—
51AGL(1,ℤ/8)×C59—22no—
52AGL(1,ℤ/8)×C59—41no—
53#135×C59—42no—
54C2×Q16×C59—42no—
55C2×SD16×C59—42no—
56C2×M16×C59—21no—
57C2×M16×C59—22no—
58C4⋊Q8×C59—21no—
59C4⋊Q8×C59—21no—
60C4⋊Q8×C59—22no—
61C4⋊Q8×C59—24no—
62C4⋊Q8×C59C2311yescharacteristic
63C4⋊Q8×C59C2314yes—
64#127×C59D411yescharacteristic
65#126×C59—24no—
66#126×C59—24no—
67#125×C59—22no—
68#124×C59—22no—
69#124×C59—24no—
70#124×C59D411yescharacteristic
71#124×C59D414yes—
72#124×C59C2311yescharacteristic
73#123×C59—22no—
74#123×C59—22no—
75#123×C59—22no—
76#123×C59—22no—
77C22⋊Q8×C59—22no—
78C22⋊Q8×C59—22no—
79C22⋊Q8×C59—24no—
80C22⋊Q8×C59—24no—
81C22⋊Q8×C59—22no—
82C22⋊Q8×C59—22no—
83#121×C59—22no—
84C4×Q8×C59—22no—
85C4×Q8×C59—22no—
86C4×Q8×C59—24no—
87C4×D4×C59—24no—
88C4×D4×C59—22no—
89#117×C59—22no—
90#117×C59—21no—
91#117×C59D412yes—
92C2×C42×C59D411yescharacteristic
93#104×C59—22no—
94#104×C59—22no—
95#104×C59—28no—
96#101×C59C2314yes—
97#101×C59C2312yes—
98#100×C59—21no—
99#100×C59—22no—
100#41×C59—22no—
101#41×C59—24no—
102#41×C59—22no—
103#41×C59—24no—
104#41×C59—41no—
105#41×C59—41no—
106#41×C59—42no—
107#41×C59—41no—
108#41×C59—41no—
109#41×C59—41no—
110#41×C59—42no—
111#41×C59—42no—
112#41×C59—44no—
113#41×C59—41no—
114C2×Q8×C59—22no—
115C2×Q8×C59—22no—
116C2×Q8×C59—22no—
117C2×Q8×C59—22no—
118C2×Q8×C59—21no—
119C2×Q8×C59—42no—
120C2×Q8×C59—42no—
121C2×Q8×C59—42no—
122C2×Q8×C59—81no—
123C2×Q8×C59—81no—
124C2×Q8×C59C2×D414yes—
125C2×Q8×C59C2×D411yescharacteristic
126C2×Q8×C59C2411yescharacteristic
127C2×D4×C59—22no—
128C2×D4×C59—21no—
129C2×D4×C59—22no—
130C2×D4×C59—21no—
131C2×D4×C59—42no—
132C2×D4×C59—44no—
133C2×D4×C59—41no—
134C2×D4×C59—42no—
135C2×D4×C59C2×D411yescharacteristic
136C2×D4×C59C2×D412yes—
137C2×D4×C59C2×D411yescharacteristic
138C22×C4×C59—21no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—21no—
142C22×C4×C59—41no—
143C22×C4×C59—42no—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D412yes—
146Q16×C59—48no—
147Q16×C59—42no—
148SD16×C59—48no—
149SD16×C59—42no—
150SD16×C59—42no—
151D8×C59—42no—
152M16×C59—22no—
153M16×C59—24no—
154M16×C59—41no—
155M16×C59—42no—
156C2×C8×C59—42no—
157C2×C8×C59—41no—
158C4⋊C4×C59—22no—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—24no—
166C4⋊C4×C59—24no—
167C22⋊C4×C59—22no—
168C22⋊C4×C59—24no—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—22no—
174C42×C59—24no—
175C42×C59—22no—
176C42×C59—21no—
177C42×C59C2×D411yescharacteristic
178C42×C59C2×D411yescharacteristic
179C42×C59C2×D414yes—
180C23×C59—21no—
181C23×C59—42no—
182C23×C59—41no—
183C23×C59#12012yes—
184C23×C59#12011yescharacteristic
185Q8×C59—22no—
186Q8×C59—24no—
187Q8×C59—41no—
188Q8×C59—44no—
189Q8×C59—41no—
190Q8×C59—44no—
191Q8×C59—41no—
192Q8×C59—42no—
193Q8×C59—42no—
194Q8×C59—82no—
195D4×C59—22no—
196D4×C59—24no—
197D4×C59—41no—
198D4×C59—42no—
199D4×C59—41no—
200D4×C59—44no—
201D4×C59—44no—
202D4×C59—42no—
203D4×C59—42no—
204D4×C59—41no—
205D4×C59—81no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—21no—
211C2×C4×C59—24no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—24no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—44no—
218C2×C4×C59—42no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—42no—
223C2×C4×C59—82no—
224C2×C4×C59—82no—
225C2×C4×C59#12011yescharacteristic
226C2×C4×C59C22×D412yes—
227C2×C4×C59C22×D411yescharacteristic
228C8×C59—42no—
229C8×C59—44no—
230#33571C5911yesmaximal, characteristic, direct product factor, Sylow
231C22×C59—22no—
232C22×C59—21no—
233C22×C59—41no—
234C22×C59—42no—
235C22×C59—42no—
236C22×C59—42no—
237C22×C59—41no—
238C22×C59—162no—
239C22×C59—161no—
240C22×C59C2×#12011yescharacteristic
241C4×C59—22no—
242C4×C59—21no—
243C4×C59—22no—
244C4×C59—41no—
245C4×C59—44no—
246C4×C59—41no—
247C4×C59—44no—
248C4×C59—42no—
249C4×C59B(4,𝔽2)11yescharacteristic
250C4×C59B(4,𝔽2)11yescharacteristic
251#3550C2×C5911yescharacteristic, complemented
252#3022C2×C5912yescomplemented
253#3018C2×C5911yescharacteristic
254#2205C2×C5914yescomplemented
255#2203C2×C5914yescomplemented
256#2135C2×C5912yescomplemented
257#2132C2×C5911yescharacteristic, complemented
258C2×C59—21no—
259C2×C59—42no—
260C2×C59—41no—
261C2×C59—81no—
262C2×C59—82no—
263C2×C59C2×B(4,𝔽2)11yescharacteristic
264#585C22×C5911yescharacteristic
265#575—21no—
266C2×#137—22no—
267#563—21no—
268#563C22×C5912yescomplemented
269#559—21no—
270Q82C22×C5912yescomplemented
271#556—22no—
272D4×Q8—22no—
273#542—24no—
274#536—21no—
275#536C22×C5912yes—
276#533—21no—
277#532C22×C5911yescharacteristic
278#456C22×C5918yes—
279#456C22×C5912yes—
280#455—21no—
281#455—24no—
282#454C22×C5912yes—
283#453—21no—
284#421—22no—
285C2×#104—21no—
286#413C22×C5912yes—
287#413C22×C5911yescharacteristic
288#356C22×C5912yescomplemented
289#356C22×C5918yescomplemented
290#355C22×C5912yescomplemented
291C59#3357111yescharacteristic, direct product factor, Sylow
2922−1+4—41no—
2932−1+4C23×C5912yes—
294H(5,𝔽2)—21no—
295H(5,𝔽2)—22no—
296H(5,𝔽2)—41no—
297C2×#41—22no—
298C2×#41—21no—
299C2×#41—42no—
300C2×#41D4×C5911yescharacteristic
301C2×#41C23×C5911yescharacteristic
302C22×Q8—21no—
303C22×Q8D4×C5912yes—
304#137—28no—
305#137—22no—
306#137—41no—
307#137—44no—
308AGL(1,ℤ/8)—22no—
309AGL(1,ℤ/8)—41no—
310#135—42no—
311C2×Q16—42no—
312C2×SD16—42no—
313C2×M16—22no—
314C2×M16—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—22no—
318C4⋊Q8—24no—
319C4⋊Q8C23×C5911yescharacteristic
320C4⋊Q8C23×C5914yes—
321#127D4×C5911yescharacteristic
322#126—24no—
323#126—24no—
324#125—22no—
325#124—22no—
326#124—24no—
327#124D4×C5911yescharacteristic
328#124D4×C5914yes—
329#124C23×C5911yescharacteristic
330#123—22no—
331#123—22no—
332#123—22no—
333#123—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340#121—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—24no—
344C4×D4—24no—
345C4×D4—22no—
346#117—22no—
347#117—21no—
348#117D4×C5912yes—
349C2×C42D4×C5911yescharacteristic
350#104—22no—
351#104—22no—
352#104—28no—
353#101C23×C5914yes—
354#101C23×C5912yes—
355#100—21no—
356#100—22no—
357#41—22no—
358#41—24no—
359#41—22no—
360#41—24no—
361#41—41no—
362#41—41no—
363#41—42no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—42no—
368#41—42no—
369#41—44no—
370#41—41no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—81no—
380C2×Q8—81no—
381C2×Q8C2×D4×C5914yes—
382C2×Q8C2×D4×C5911yescharacteristic
383C2×Q8C24×C5911yescharacteristic
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4—41no—
392C2×D4C2×D4×C5911yescharacteristic
393C2×D4C2×D4×C5912yes—
394C2×D4C2×D4×C5911yescharacteristic
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4C2×D4×C5911yescharacteristic
402C22×C4C2×D4×C5912yes—
403Q16—48no—
404Q16—42no—
405SD16—48no—
406SD16—42no—
407SD16—42no—
408D8—42no—
409M16—22no—
410M16—24no—
411M16—41no—
412M16—42no—
413C2×C8—42no—
414C2×C8—41no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C42—24no—
432C42—22no—
433C42—21no—
434C42C2×D4×C5911yescharacteristic
435C42C2×D4×C5911yescharacteristic
436C42C2×D4×C5914yes—
437C23—21no—
438C23—42no—
439C23—41no—
440C23#120×C5912yes—
441C23#120×C5911yescharacteristic
442Q8—22no—
443Q8—24no—
444Q8—41no—
445Q8—44no—
446Q8—41no—
447Q8—44no—
448Q8—41no—
449Q8—42no—
450Q8—42no—
451Q8—82no—
452D4—22no—
453D4—24no—
454D4—41no—
455D4—42no—
456D4—41no—
457D4—44no—
458D4—44no—
459D4—42no—
460D4—42no—
461D4—41no—
462D4—81no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4#120×C5911yescharacteristic
483C2×C4C22×D4×C5912yes—
484C2×C4C22×D4×C5911yescharacteristic
485C8—42no—
486C8—44no—
487C22—22no—
488C22—21no—
489C22—41no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—41no—
494C22—162no—
495C22—161no—
496C22C2×#120×C5911yescharacteristic
497C4—22no—
498C4—21no—
499C4—22no—
500C4—41no—
501C4—44no—
502C4—41no—
503C4—44no—
504C4—42no—
505C4C59×B(4,𝔽2)11yescharacteristic
506C4C59×B(4,𝔽2)11yescharacteristic
507C2—21no—
508C2—42no—
509C2—41no—
510C2—81no—
511C2—82no—
512C2C2×C59×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C59C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C59
prime radicalC59×#3018C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C59 ◁ C2×C4×C59 ◁ C2×#41×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3550C2semidirect
2. C59×#3022C2semidirect
4. C59×#2205C2semidirect
5. C59×#2203C2semidirect
6. C59×#2135C2semidirect
7. C59×#2132C2semidirect
12. C59×#563C22semidirect
14. Q82×C59C22semidirect
32. C59×#356C22semidirect
33. C59×#356C22semidirect
34. C59×#355C22semidirect
230. #33571C59direct (N × H)
251. #3550C2×C59semidirect
252. #3022C2×C59semidirect
254. #2205C2×C59semidirect
255. #2203C2×C59semidirect
256. #2135C2×C59semidirect
257. #2132C2×C59semidirect
268. #563C22×C59semidirect
270. Q82C22×C59semidirect
288. #356C22×C59semidirect
289. #356C22×C59semidirect
290. #355C22×C59semidirect
291. C59#33571direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35502
C59×#30222
C59×#30182
C59×#22052
C59×#22032
C59×#21352
C59×#21322
#3357159