Finite Group Databasebeta

← C59×#33570 #28445679

Subgroups — C59×#33570 #28445679

Order 15104.

1212 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3549C211yesmaximal, characteristic, complemented
2C59×#3021C212yesmaximal, complemented
3C59×#3018C211yesmaximal, characteristic, complemented
4C59×#2204C214yesmaximal, complemented
5C59×#2202C214yesmaximal, complemented
6C59×#2134C212yesmaximal, complemented
7C59×#2132C211yesmaximal, characteristic, complemented
8C59×#585C2211yescharacteristic, complemented
9C59×#575—21no—
10C2×AGL(1,ℤ/8)×C59—22no—
11C59×H(3,𝔽4)—21no—
12C59×H(3,𝔽4)C2212yescomplemented
13C59×#555—22no—
14C59×#555C2212yescomplemented
15D42×C59—22no—
16D42×C59—21no—
17C59×#535—21no—
18C59×#535—24no—
19C59×#535C2212yes—
20C59×#532—21no—
21C59×#532C2211yescharacteristic
22C59×#456C2212yescomplemented
23C59×#455—21no—
24C59×#454C2212yescomplemented
25C59×#454C2218yescomplemented
26C59×#453—21no—
27C59×#453—24no—
28C59×#421—22no—
29C2×#104×C59—21no—
30C59×#413C2211yescharacteristic, complemented
31C59×#413C2212yescomplemented
32C59×#356C2212yescomplemented
33C59×#355C2212yescomplemented
34C59×#355C2218yescomplemented
352−1+4×C59—41no—
362−1+4×C59C2312yes—
37H(5,𝔽2)×C59—22no—
38H(5,𝔽2)×C59—21no—
39H(5,𝔽2)×C59—41no—
40C2×#41×C59—22no—
41C2×#41×C59—21no—
42C2×#41×C59—22no—
43C2×#41×C59—42no—
44C2×#41×C59D411yescharacteristic
45C2×#41×C59C2311yescharacteristic
46C22×D4×C59—24no—
47C22×D4×C59—21no—
48C22×D4×C59D412yes—
49#137×C59—22no—
50#137×C59—41no—
51AGL(1,ℤ/8)×C59—28no—
52AGL(1,ℤ/8)×C59—22no—
53AGL(1,ℤ/8)×C59—41no—
54AGL(1,ℤ/8)×C59—44no—
55#135×C59—42no—
56C2×SD16×C59—42no—
57C2×D8×C59—42no—
58C2×M16×C59—21no—
59C2×M16×C59—22no—
60C4⋊Q8×C59—21no—
61C4⋊Q8×C59C2311yescharacteristic
62#127×C59—21no—
63#127×C59—22no—
64#127×C59D411yescharacteristic
65#127×C59D414yes—
66#124×C59—24no—
67#124×C59—22no—
68#124×C59—22no—
69#124×C59—24no—
70#124×C59D411yescharacteristic
71#124×C59C2311yescharacteristic
72#124×C59C2314yes—
73#121×C59—22no—
74#121×C59—22no—
75#121×C59—24no—
76#121×C59—22no—
77#121×C59—24no—
78#121×C59—22no—
79#121×C59—22no—
80#120×C59—22no—
81#120×C59—24no—
82#120×C59—22no—
83#120×C59—22no—
84#120×C59—24no—
85#120×C59—22no—
86C4×D4×C59—22no—
87C4×D4×C59—22no—
88C4×D4×C59—22no—
89#117×C59—21no—
90#117×C59—24no—
91#117×C59D412yes—
92C2×C42×C59D411yescharacteristic
93#104×C59—22no—
94#104×C59—22no—
95#104×C59—28no—
96#101×C59C2312yes—
97#101×C59C2314yes—
98#100×C59—22no—
99#100×C59—21no—
100C24×C59—22no—
101C24×C59—22no—
102#41×C59—22no—
103#41×C59—24no—
104#41×C59—24no—
105#41×C59—22no—
106#41×C59—41no—
107#41×C59—41no—
108#41×C59—42no—
109#41×C59—41no—
110#41×C59—41no—
111#41×C59—41no—
112#41×C59—42no—
113#41×C59—44no—
114#41×C59—42no—
115#41×C59—41no—
116#41×C59—84no—
117C2×Q8×C59—22no—
118C2×Q8×C59—22no—
119C2×Q8×C59—21no—
120C2×Q8×C59—42no—
121C2×Q8×C59C2×D411yescharacteristic
122C2×Q8×C59C2411yescharacteristic
123C2×D4×C59—22no—
124C2×D4×C59—21no—
125C2×D4×C59—22no—
126C2×D4×C59—22no—
127C2×D4×C59—22no—
128C2×D4×C59—22no—
129C2×D4×C59—24no—
130C2×D4×C59—24no—
131C2×D4×C59—21no—
132C2×D4×C59—24no—
133C2×D4×C59—22no—
134C2×D4×C59—42no—
135C2×D4×C59—42no—
136C2×D4×C59—44no—
137C2×D4×C59—42no—
138C2×D4×C59—42no—
139C2×D4×C59—41no—
140C2×D4×C59—88no—
141C2×D4×C59—81no—
142C2×D4×C59—81no—
143C2×D4×C59C2×D411yescharacteristic
144C2×D4×C59C2×D414yes—
145C2×D4×C59C2×D411yescharacteristic
146C2×D4×C59C2×D412yes—
147C22×C4×C59—24no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—41no—
151C22×C4×C59—42no—
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D412yes—
154Q16×C59—42no—
155SD16×C59—42no—
156SD16×C59—48no—
157SD16×C59—42no—
158D8×C59—48no—
159D8×C59—42no—
160M16×C59—24no—
161M16×C59—22no—
162M16×C59—42no—
163M16×C59—41no—
164C2×C8×C59—42no—
165C2×C8×C59—41no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—24no—
177C42×C59—22no—
178C42×C59—21no—
179C42×C59C2×D411yescharacteristic
180C42×C59C2×D411yescharacteristic
181C42×C59C2×D414yes—
182C23×C59—24no—
183C23×C59—21no—
184C23×C59—41no—
185C23×C59—42no—
186C23×C59—82no—
187C23×C59—84no—
188C23×C59—84no—
189C23×C59—82no—
190C23×C59#12012yes—
191C23×C59#12011yescharacteristic
192Q8×C59—24no—
193Q8×C59—22no—
194Q8×C59—41no—
195Q8×C59—41no—
196Q8×C59—41no—
197Q8×C59—82no—
198D4×C59—24no—
199D4×C59—22no—
200D4×C59—42no—
201D4×C59—41no—
202D4×C59—44no—
203D4×C59—41no—
204D4×C59—44no—
205D4×C59—44no—
206D4×C59—42no—
207D4×C59—44no—
208D4×C59—42no—
209D4×C59—41no—
210D4×C59—42no—
211D4×C59—42no—
212D4×C59—81no—
213D4×C59—84no—
214D4×C59—84no—
215D4×C59—82no—
216D4×C59—84no—
217D4×C59—82no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—24no—
226C2×C4×C59—48no—
227C2×C4×C59—42no—
228C2×C4×C59—44no—
229C2×C4×C59—42no—
230C2×C4×C59—41no—
231C2×C4×C59—42no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—42no—
235C2×C4×C59#12011yescharacteristic
236C2×C4×C59C22×D411yescharacteristic
237C2×C4×C59C22×D412yes—
238C8×C59—44no—
239C8×C59—42no—
240#33570C5911yesmaximal, characteristic, direct product factor, Sylow
241C22×C59—21no—
242C22×C59—22no—
243C22×C59—41no—
244C22×C59—42no—
245C22×C59—44no—
246C22×C59—44no—
247C22×C59—42no—
248C22×C59—42no—
249C22×C59—41no—
250C22×C59—44no—
251C22×C59—161no—
252C22×C59—162no—
253C22×C59—164no—
254C22×C59—164no—
255C22×C59C2×#12011yescharacteristic
256C4×C59—22no—
257C4×C59—22no—
258C4×C59—21no—
259C4×C59—44no—
260C4×C59—41no—
261C4×C59—41no—
262C4×C59—42no—
263C4×C59B(4,𝔽2)11yescharacteristic
264C4×C59B(4,𝔽2)11yescharacteristic
265#3549C2×C5911yescharacteristic, complemented
266#3021C2×C5912yescomplemented
267#3018C2×C5911yescharacteristic, complemented
268#2204C2×C5914yescomplemented
269#2202C2×C5914yescomplemented
270#2134C2×C5912yescomplemented
271#2132C2×C5911yescharacteristic, complemented
272C2×C59—21no—
273C2×C59—41no—
274C2×C59—42no—
275C2×C59—84no—
276C2×C59—81no—
277C2×C59—82no—
278C2×C59C2×B(4,𝔽2)11yescharacteristic
279#585C22×C5911yescharacteristic, complemented
280#575—21no—
281C2×AGL(1,ℤ/8)—22no—
282H(3,𝔽4)—21no—
283H(3,𝔽4)C22×C5912yescomplemented
284#555—22no—
285#555C22×C5912yescomplemented
286D42—22no—
287D42—21no—
288#535—21no—
289#535—24no—
290#535C22×C5912yes—
291#532—21no—
292#532C22×C5911yescharacteristic
293#456C22×C5912yescomplemented
294#455—21no—
295#454C22×C5912yescomplemented
296#454C22×C5918yescomplemented
297#453—21no—
298#453—24no—
299#421—22no—
300C2×#104—21no—
301#413C22×C5912yescomplemented
302#413C22×C5911yescharacteristic, complemented
303#356C22×C5912yescomplemented
304#355C22×C5912yescomplemented
305#355C22×C5918yescomplemented
306C59#3357011yescharacteristic, direct product factor, Sylow
3072−1+4—41no—
3082−1+4C23×C5912yes—
309H(5,𝔽2)—22no—
310H(5,𝔽2)—21no—
311H(5,𝔽2)—41no—
312C2×#41—22no—
313C2×#41—22no—
314C2×#41—21no—
315C2×#41—42no—
316C2×#41D4×C5911yescharacteristic
317C2×#41C23×C5911yescharacteristic
318C22×D4—24no—
319C22×D4—21no—
320C22×D4D4×C5912yes—
321#137—22no—
322#137—41no—
323AGL(1,ℤ/8)—28no—
324AGL(1,ℤ/8)—22no—
325AGL(1,ℤ/8)—41no—
326AGL(1,ℤ/8)—44no—
327#135—42no—
328C2×SD16—42no—
329C2×D8—42no—
330C2×M16—22no—
331C2×M16—21no—
332C4⋊Q8—21no—
333C4⋊Q8C23×C5911yescharacteristic
334#127—21no—
335#127—22no—
336#127D4×C5911yescharacteristic
337#127D4×C5914yes—
338#124—24no—
339#124—22no—
340#124—22no—
341#124—24no—
342#124D4×C5911yescharacteristic
343#124C23×C5911yescharacteristic
344#124C23×C5914yes—
345#121—22no—
346#121—22no—
347#121—24no—
348#121—22no—
349#121—24no—
350#121—22no—
351#121—22no—
352#120—22no—
353#120—24no—
354#120—22no—
355#120—22no—
356#120—24no—
357#120—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361#117—21no—
362#117—24no—
363#117D4×C5912yes—
364C2×C42D4×C5911yescharacteristic
365#104—22no—
366#104—22no—
367#104—28no—
368#101C23×C5912yes—
369#101C23×C5914yes—
370#100—22no—
371#100—21no—
372C24—22no—
373C24—22no—
374#41—22no—
375#41—24no—
376#41—24no—
377#41—22no—
378#41—41no—
379#41—41no—
380#41—42no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—42no—
385#41—44no—
386#41—42no—
387#41—41no—
388#41—84no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—42no—
393C2×Q8C2×D4×C5911yescharacteristic
394C2×Q8C24×C5911yescharacteristic
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—41no—
412C2×D4—88no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4C2×D4×C5911yescharacteristic
416C2×D4C2×D4×C5914yes—
417C2×D4C2×D4×C5911yescharacteristic
418C2×D4C2×D4×C5912yes—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4C2×D4×C5911yescharacteristic
425C22×C4C2×D4×C5912yes—
426Q16—42no—
427SD16—42no—
428SD16—48no—
429SD16—42no—
430D8—48no—
431D8—42no—
432M16—24no—
433M16—22no—
434M16—42no—
435M16—41no—
436C2×C8—42no—
437C2×C8—41no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—24no—
449C42—22no—
450C42—21no—
451C42C2×D4×C5911yescharacteristic
452C42C2×D4×C5911yescharacteristic
453C42C2×D4×C5914yes—
454C23—24no—
455C23—21no—
456C23—41no—
457C23—42no—
458C23—82no—
459C23—84no—
460C23—84no—
461C23—82no—
462C23#120×C5912yes—
463C23#120×C5911yescharacteristic
464Q8—24no—
465Q8—22no—
466Q8—41no—
467Q8—41no—
468Q8—41no—
469Q8—82no—
470D4—24no—
471D4—22no—
472D4—42no—
473D4—41no—
474D4—44no—
475D4—41no—
476D4—44no—
477D4—44no—
478D4—42no—
479D4—44no—
480D4—42no—
481D4—41no—
482D4—42no—
483D4—42no—
484D4—81no—
485D4—84no—
486D4—84no—
487D4—82no—
488D4—84no—
489D4—82no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4#120×C5911yescharacteristic
508C2×C4C22×D4×C5911yescharacteristic
509C2×C4C22×D4×C5912yes—
510C8—44no—
511C8—42no—
512C22—21no—
513C22—22no—
514C22—41no—
515C22—42no—
516C22—44no—
517C22—44no—
518C22—42no—
519C22—42no—
520C22—41no—
521C22—44no—
522C22—162no—
523C22—161no—
524C22—164no—
525C22—164no—
526C22C2×#120×C5911yescharacteristic
527C4—22no—
528C4—22no—
529C4—21no—
530C4—44no—
531C4—41no—
532C4—41no—
533C4—42no—
534C4C59×B(4,𝔽2)11yescharacteristic
535C4C59×B(4,𝔽2)11yescharacteristic
536C2—21no—
537C2—42no—
538C2—41no—
539C2—84no—
540C2—81no—
541C2—82no—
542C2C2×C59×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C59C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C59
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×#3549C2semidirect
2. C59×#3021C2semidirect
3. C59×#3018C2semidirect
4. C59×#2204C2semidirect
5. C59×#2202C2semidirect
6. C59×#2134C2semidirect
7. C59×#2132C2semidirect
8. C59×#585C22semidirect
12. C59×H(3,𝔽4)C22semidirect
14. C59×#555C22semidirect
22. C59×#456C22semidirect
24. C59×#454C22semidirect
25. C59×#454C22semidirect
30. C59×#413C22semidirect
31. C59×#413C22semidirect
32. C59×#356C22semidirect
33. C59×#355C22semidirect
34. C59×#355C22semidirect
240. #33570C59direct (N × H)
265. #3549C2×C59semidirect
266. #3021C2×C59semidirect
267. #3018C2×C59semidirect
268. #2204C2×C59semidirect
269. #2202C2×C59semidirect
270. #2134C2×C59semidirect
271. #2132C2×C59semidirect
279. #585C22×C59semidirect
283. H(3,𝔽4)C22×C59semidirect
285. #555C22×C59semidirect
293. #456C22×C59semidirect
295. #454C22×C59semidirect
296. #454C22×C59semidirect
301. #413C22×C59semidirect
302. #413C22×C59semidirect
303. #356C22×C59semidirect
304. #355C22×C59semidirect
305. #355C22×C59semidirect
306. C59#33570direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35492
C59×#30212
C59×#30182
C59×#22042
C59×#22022
C59×#21342
C59×#21322
#3357059