Finite Group Databasebeta

← C59×#33530 #28445639

Subgroups — C59×#33530 #28445639

Order 15104.

1132 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#575C211yesmaximal, characteristic, complemented
2C59×#3414C212yesmaximal, complemented
3C59×#3054C211yesmaximal, characteristic, complemented
4C59×#2191C214yesmaximal, complemented
5C59×#2190C214yesmaximal, complemented
6C59×#2145C212yesmaximal, complemented
7C2×C59×#349C211yesmaximal, characteristic, complemented
8C22×#41×C59—21no—
9C59×#575—24no—
10C59×#575—22no—
11C59×#575C2214yes—
12C2×#137×C59—22no—
13C2×AGL(1,ℤ/8)×C59—22no—
14C2×#135×C59—21no—
15C2×#135×C59C2212yes—
16C2×#135×C59C2211yescharacteristic, complemented
17C22×M16×C59C2211yescharacteristic
18C59×#525—21no—
19C59×#510—22no—
20C59×#510C2214yescomplemented
21C59×#509—22no—
22C59×#509C2214yescomplemented
23C59×#508—24no—
24C2×M32×C59C2212yescomplemented
25C59×#470C2212yescomplemented
26C59×#469C2212yescomplemented
27C59×#468—22no—
28C59×#427C2211yescharacteristic, complemented
29C59×#416—21no—
30C59×#360C2218yescomplemented
31C59×#349C2214yescomplemented
32C2×#41×C59—22no—
33C2×#41×C59—24no—
34C2×#41×C59—24no—
35C2×#41×C59—22no—
36C2×#41×C59—21no—
37C2×#41×C59—41no—
38C2×#41×C59—42no—
39C2×#41×C59—42no—
40C2×#41×C59—41no—
41C22×Q8×C59—21no—
42C22×D4×C59—21no—
43C22×D4×C59—41no—
44C23×C4×C59—41no—
45C23×C4×C59D411yescharacteristic
46#137×C59—28no—
47#137×C59—28no—
48AGL(1,ℤ/8)×C59—28no—
49AGL(1,ℤ/8)×C59—28no—
50#135×C59—24no—
51#135×C59—24no—
52#135×C59—22no—
53#135×C59—22no—
54#135×C59—22no—
55#135×C59—22no—
56#135×C59D414yescomplemented
57#135×C59C2314yes—
58C2×Q16×C59—21no—
59C2×Q16×C59D412yescomplemented
60C2×Q16×C59C2311yescharacteristic, complemented
61C2×SD16×C59—22no—
62C2×SD16×C59—22no—
63C2×SD16×C59—21no—
64C2×D8×C59—21no—
65C2×D8×C59D412yescomplemented
66C2×D8×C59C2311yescharacteristic, complemented
67C2×M16×C59—22no—
68C2×M16×C59—22no—
69C2×M16×C59D412yes—
70C2×M16×C59C2312yes—
71C22×C8×C59D411yescharacteristic
72C22×C8×C59C2311yescharacteristic
73#123×C59—41no—
74C22⋊Q8×C59—22no—
75#121×C59—22no—
76#120×C59—41no—
77C4×D4×C59—42no—
78#117×C59—21no—
79Q32×C59—24no—
80SD32×C59—24no—
81SD32×C59—24no—
82D16×C59—24no—
83M32×C59—22no—
84M32×C59C2314yescomplemented
85C2×C16×C59—22no—
86#107×C59—21no—
87#106×C59C2312yescomplemented
88#103×C59—22no—
89#102×C59—22no—
90C24×C59—41no—
91C24×C59D811yescharacteristic
92#41×C59—28no—
93#41×C59—24no—
94#41×C59—24no—
95#41×C59—44no—
96#41×C59—44no—
97#41×C59—44no—
98#41×C59—44no—
99#41×C59—44no—
100#41×C59—42no—
101#41×C59—42no—
102#41×C59—41no—
103#41×C59—41no—
104#41×C59—41no—
105#41×C59—41no—
106C2×Q8×C59—24no—
107C2×Q8×C59—22no—
108C2×Q8×C59—21no—
109C2×Q8×C59—41no—
110C2×Q8×C59—42no—
111C2×D4×C59—24no—
112C2×D4×C59—22no—
113C2×D4×C59—21no—
114C2×D4×C59—41no—
115C2×D4×C59—42no—
116C2×D4×C59—44no—
117C2×D4×C59—42no—
118C2×D4×C59—44no—
119C2×D4×C59—42no—
120C2×D4×C59—42no—
121C2×D4×C59—41no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—42no—
127C22×C4×C59—44no—
128C22×C4×C59—44no—
129C22×C4×C59—41no—
130C22×C4×C59—42no—
131C22×C4×C59—41no—
132C22×C4×C59D812yes—
133C22×C4×C59D811yescharacteristic
134C22×C4×C59C2×D412yes—
135C22×C4×C59C2×D411yescharacteristic
136Q16×C59—24no—
137Q16×C59—22no—
138Q16×C59—22no—
139Q16×C59C2×D414yes—
140SD16×C59—28no—
141SD16×C59—24no—
142SD16×C59—24no—
143SD16×C59—42no—
144D8×C59—24no—
145D8×C59—22no—
146D8×C59—22no—
147D8×C59C2×D414yes—
148M16×C59—24no—
149M16×C59—24no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59C2×D412yes—
153C2×C8×C59C2×D411yescharacteristic
154C2×C8×C59C2×D412yes—
155C2×C8×C59C2×D412yes—
156C2×C8×C59C2411yescharacteristic
157C4⋊C4×C59—22no—
158C4⋊C4×C59—42no—
159C22⋊C4×C59—41no—
160C22⋊C4×C59—42no—
161C22⋊C4×C59—42no—
162C42×C59—41no—
163C16×C59—24no—
164C23×C59—21no—
165C23×C59—21no—
166C23×C59—44no—
167C23×C59—42no—
168C23×C59—41no—
169C23×C59—81no—
170C23×C59—84no—
171C23×C59C2×D812yes—
172C23×C59C2×D811yescharacteristic
173Q8×C59—24no—
174Q8×C59—44no—
175Q8×C59—42no—
176Q8×C59—41no—
177Q8×C59—41no—
178D4×C59—24no—
179D4×C59—44no—
180D4×C59—48no—
181D4×C59—44no—
182D4×C59—44no—
183D4×C59—41no—
184D4×C59—41no—
185D4×C59—42no—
186D4×C59—84no—
187D4×C59—81no—
188D4×C59—81no—
189C2×C4×C59—21no—
190C2×C4×C59—21no—
191C2×C4×C59—22no—
192C2×C4×C59—22no—
193C2×C4×C59—22no—
194C2×C4×C59—22no—
195C2×C4×C59—48no—
196C2×C4×C59—44no—
197C2×C4×C59—42no—
198C2×C4×C59—44no—
199C2×C4×C59—44no—
200C2×C4×C59—42no—
201C2×C4×C59—41no—
202C2×C4×C59—81no—
203C2×C4×C59—81no—
204C2×C4×C59#12011yescharacteristic
205C2×C4×C59#12011yescharacteristic
206C2×C4×C59C2×D812yes—
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
209C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
210C2×C4×C59C22×D411yescharacteristic
211C8×C59—22no—
212C8×C59#12012yes—
213C8×C59C22×D412yes—
214#33530C5911yesmaximal, characteristic, direct product factor, Sylow
215C22×C59—22no—
216C22×C59—41no—
217C22×C59—42no—
218C22×C59—42no—
219C22×C59—44no—
220C22×C59—81no—
221C22×C59—84no—
222C22×C59—82no—
223C22×C59—88no—
224C22×C59—81no—
225C22×C59#44711yescharacteristic
226C22×C59#44711yescharacteristic
227C22×C59C22×D811yescharacteristic
228C4×C59—22no—
229C4×C59—44no—
230C4×C59—82no—
231C4×C59—81no—
232C4×C59#44711yescharacteristic
233C4×C59#44711yescharacteristic
234C4×C59C2×#12011yescharacteristic
235C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
236C2×#575C2×C5911yescharacteristic, complemented
237#3414C2×C5912yescomplemented
238#3054C2×C5911yescharacteristic, complemented
239#2191C2×C5914yescomplemented
240#2190C2×C5914yescomplemented
241#2145C2×C5912yescomplemented
242C2×#349C2×C5911yescharacteristic, complemented
243C2×C59—21no—
244C2×C59—21no—
245C2×C59—21no—
246C2×C59—42no—
247C2×C59—84no—
248C2×C59—161no—
249C2×C59C2×#44711yescharacteristic
250C22×#41—21no—
251#575—24no—
252#575—22no—
253#575C22×C5914yes—
254C2×#137—22no—
255C2×AGL(1,ℤ/8)—22no—
256C2×#135—21no—
257C2×#135C22×C5912yes—
258C2×#135C22×C5911yescharacteristic, complemented
259C22×M16C22×C5911yescharacteristic
260#525—21no—
261#510—22no—
262#510C22×C5914yescomplemented
263#509—22no—
264#509C22×C5914yescomplemented
265#508—24no—
266C2×M32C22×C5912yescomplemented
267#470C22×C5912yescomplemented
268#469C22×C5912yescomplemented
269#468—22no—
270#427C22×C5911yescharacteristic, complemented
271#416—21no—
272#360C22×C5918yescomplemented
273#349C22×C5914yescomplemented
274C59#3353011yescharacteristic, direct product factor, Sylow
275C2×#41—22no—
276C2×#41—24no—
277C2×#41—22no—
278C2×#41—24no—
279C2×#41—21no—
280C2×#41—41no—
281C2×#41—42no—
282C2×#41—42no—
283C2×#41—41no—
284C22×Q8—21no—
285C22×D4—21no—
286C22×D4—41no—
287C23×C4—41no—
288C23×C4D4×C5911yescharacteristic
289#137—28no—
290#137—28no—
291AGL(1,ℤ/8)—28no—
292AGL(1,ℤ/8)—28no—
293#135—24no—
294#135—24no—
295#135—22no—
296#135—22no—
297#135—22no—
298#135—22no—
299#135D4×C5914yescomplemented
300#135C23×C5914yes—
301C2×Q16—21no—
302C2×Q16D4×C5912yescomplemented
303C2×Q16C23×C5911yescharacteristic, complemented
304C2×SD16—22no—
305C2×SD16—22no—
306C2×SD16—21no—
307C2×D8—21no—
308C2×D8D4×C5912yescomplemented
309C2×D8C23×C5911yescharacteristic, complemented
310C2×M16—22no—
311C2×M16—22no—
312C2×M16D4×C5912yes—
313C2×M16C23×C5912yes—
314C22×C8D4×C5911yescharacteristic
315C22×C8C23×C5911yescharacteristic
316#123—41no—
317C22⋊Q8—22no—
318#121—22no—
319#120—41no—
320C4×D4—42no—
321#117—21no—
322Q32—24no—
323SD32—24no—
324SD32—24no—
325D16—24no—
326M32—22no—
327M32C23×C5914yescomplemented
328C2×C16—22no—
329#107—21no—
330#106C23×C5912yescomplemented
331#103—22no—
332#102—22no—
333C24—41no—
334C24D8×C5911yescharacteristic
335#41—28no—
336#41—24no—
337#41—24no—
338#41—44no—
339#41—44no—
340#41—44no—
341#41—44no—
342#41—44no—
343#41—42no—
344#41—42no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—21no—
352C2×Q8—41no—
353C2×Q8—42no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—41no—
358C2×D4—42no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4D8×C5912yes—
376C22×C4D8×C5911yescharacteristic
377C22×C4C2×D4×C5912yes—
378C22×C4C2×D4×C5911yescharacteristic
379Q16—24no—
380Q16—22no—
381Q16—22no—
382Q16C2×D4×C5914yes—
383SD16—28no—
384SD16—24no—
385SD16—24no—
386SD16—42no—
387D8—24no—
388D8—22no—
389D8—22no—
390D8C2×D4×C5914yes—
391M16—24no—
392M16—24no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8C2×D4×C5912yes—
396C2×C8C2×D4×C5911yescharacteristic
397C2×C8C2×D4×C5912yes—
398C2×C8C2×D4×C5912yes—
399C2×C8C24×C5911yescharacteristic
400C4⋊C4—22no—
401C4⋊C4—42no—
402C22⋊C4—41no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C42—41no—
406C16—24no—
407C23—21no—
408C23—21no—
409C23—44no—
410C23—42no—
411C23—41no—
412C23—81no—
413C23—84no—
414C23C2×D8×C5912yes—
415C23C2×D8×C5911yescharacteristic
416Q8—24no—
417Q8—44no—
418Q8—42no—
419Q8—41no—
420Q8—41no—
421D4—24no—
422D4—44no—
423D4—48no—
424D4—44no—
425D4—44no—
426D4—42no—
427D4—41no—
428D4—41no—
429D4—84no—
430D4—81no—
431D4—81no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—48no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4#120×C5911yescharacteristic
448C2×C4#120×C5911yescharacteristic
449C2×C4C2×D8×C5912yes—
450C2×C4C2×D8×C5911yescharacteristic
451C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
452C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
453C2×C4C22×D4×C5911yescharacteristic
454C8—22no—
455C8#120×C5912yes—
456C8C22×D4×C5912yes—
457C22—22no—
458C22—41no—
459C22—42no—
460C22—42no—
461C22—44no—
462C22—81no—
463C22—84no—
464C22—82no—
465C22—88no—
466C22—81no—
467C22C59×#44711yescharacteristic
468C22C59×#44711yescharacteristic
469C22C22×D8×C5911yescharacteristic
470C4—22no—
471C4—44no—
472C4—82no—
473C4—81no—
474C4C59×#44711yescharacteristic
475C4C59×#44711yescharacteristic
476C4C2×#120×C5911yescharacteristic
477C4C2×AGL(1,ℤ/8)×C5911yescharacteristic
478C2—21no—
479C2—21no—
480C2—21no—
481C2—42no—
482C2—84no—
483C2—161no—
484C2C2×C59×#44711yescharacteristic
subgroupG/H
centerC4×C59#447
commutator, Frattini, cosocleC2×C8C24×C59
socleC2×C59C2×#447
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C59 ◁ C22×C4×C59 ◁ C22×M16×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C59×#575C2semidirect
2. C59×#3414C2semidirect
3. C59×#3054C2semidirect
4. C59×#2191C2semidirect
5. C59×#2190C2semidirect
6. C59×#2145C2semidirect
7. C2×C59×#349C2semidirect
16. C2×#135×C59C22semidirect
20. C59×#510C22semidirect
22. C59×#509C22semidirect
24. C2×M32×C59C22semidirect
25. C59×#470C22semidirect
26. C59×#469C22semidirect
28. C59×#427C22semidirect
30. C59×#360C22semidirect
31. C59×#349C22semidirect
56. #135×C59D4semidirect
59. C2×Q16×C59D4semidirect
60. C2×Q16×C59C23semidirect
65. C2×D8×C59D4semidirect
66. C2×D8×C59C23semidirect
84. M32×C59C23semidirect
87. #106×C59C23semidirect
214. #33530C59direct (N × H)
236. C2×#575C2×C59semidirect
237. #3414C2×C59semidirect
238. #3054C2×C59semidirect
239. #2191C2×C59semidirect
240. #2190C2×C59semidirect
241. #2145C2×C59semidirect
242. C2×#349C2×C59semidirect
258. C2×#135C22×C59semidirect
262. #510C22×C59semidirect
264. #509C22×C59semidirect
266. C2×M32C22×C59semidirect
267. #470C22×C59semidirect
268. #469C22×C59semidirect
270. #427C22×C59semidirect
272. #360C22×C59semidirect
273. #349C22×C59semidirect
274. C59#33530direct (N × H)
299. #135D4×C59semidirect
302. C2×Q16D4×C59semidirect
303. C2×Q16C23×C59semidirect
308. C2×D8D4×C59semidirect
309. C2×D8C23×C59semidirect
327. M32C23×C59semidirect
330. #106C23×C59semidirect

Maximal subgroups

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

subgroupindex
C2×C59×#5752
C59×#34142
C59×#30542
C59×#21912
C59×#21902
C59×#21452
C2×C59×#3492
#3353059