Finite Group Databasebeta

← C59×#18370 #28430479

Subgroups — C59×#18370 #28430479

Order 15104.

1024 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3506C211yesmaximal, characteristic, complemented
2C59×#3215C211yesmaximal, characteristic, complemented
3C59×#3213C211yesmaximal, characteristic, complemented
4C59×#2631C211yesmaximal, characteristic, complemented
5C59×#1606C214yesmaximal, complemented
6C59×#1600C214yesmaximal, complemented
7C59×#1515C211yesmaximal, characteristic, complemented
8C59×#1514C211yesmaximal, characteristic, complemented
9C59×#1466C211yesmaximal, characteristic, complemented
10C59×#559—22no—
11C59×#556—21no—
12C59×#555—21no—
13C59×#553—22no—
14C59×#550—21no—
15C59×#547—21no—
16C59×#538C2214yes—
17C59×#535—21no—
18C59×#533C2211yescharacteristic
19C59×#532C2211yescharacteristic
20C2×#124×C59C2211yescharacteristic
21C59×#525—22no—
22C59×#520—21no—
23C2×C4×D4×C59C2211yescharacteristic
24C59×#497—21no—
25C59×#496—21no—
26C59×#489—21no—
27C59×#489—21no—
28C59×#464—22no—
29C59×#463—22no—
30C59×#451C2212yes—
31C59×#450C2212yes—
32C59×#448C2212yes—
33C59×#447C2212yes—
34C59×#432C2211yescharacteristic
35C59×#432C2211yescharacteristic
36C59×#397—21no—
37C59×#393C2212yes—
38C59×#392—21no—
39C59×#392C2212yes—
40C59×#386—22no—
41C59×#383C2211yescharacteristic
42C59×#328C2214yes—
43C59×#327C2214yes—
44C59×#323C2214yes—
45C2×#41×C59—22no—
46C2×#41×C59—21no—
47C22×Q8×C59C2311yescharacteristic
48C22×D4×C59—22no—
49C22×D4×C59D411yescharacteristic
50C22×D4×C59C2311yescharacteristic
51C23×C4×C59D412yes—
52C2×Q16×C59—22no—
53C2×SD16×C59—22no—
54C2×SD16×C59—22no—
55C2×D8×C59—22no—
56C4⋊Q8×C59D411yescharacteristic, complemented
57C4⋊Q8×C59D411yescharacteristic, complemented
58C4⋊Q8×C59D411yescharacteristic, complemented
59#127×C59—22no—
60#127×C59D411yescharacteristic, complemented
61#126×C59—24no—
62#125×C59—21no—
63#124×C59—22no—
64#124×C59—24no—
65#124×C59—21no—
66#124×C59—21no—
67#124×C59—21no—
68#124×C59—21no—
69#124×C59—21no—
70#123×C59—24no—
71#123×C59—24no—
72C22⋊Q8×C59—22no—
73C22⋊Q8×C59—21no—
74C22⋊Q8×C59C2312yes—
75#121×C59—22no—
76#121×C59—22no—
77#121×C59—22no—
78#121×C59—24no—
79#121×C59—21no—
80#121×C59C2312yes—
81#120×C59—24no—
82C4×Q8×C59—22no—
83C4×Q8×C59—21no—
84C4×Q8×C59D412yescomplemented
85C4×D4×C59—22no—
86C4×D4×C59—24no—
87C4×D4×C59—24no—
88C4×D4×C59—21no—
89C4×D4×C59—21no—
90C4×D4×C59—21no—
91C4×D4×C59—21no—
92C4×D4×C59—21no—
93C4×D4×C59D412yescomplemented
94#117×C59—22no—
95#117×C59—22no—
96#117×C59—21no—
97C2×C4⋊C4×C59—22no—
98C2×C4⋊C4×C59D411yescharacteristic
99C2×C22⋊C4×C59—21no—
100C2×C22⋊C4×C59—21no—
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59—22no—
103C2×C22⋊C4×C59C2312yes—
104C2×C42×C59C2311yescharacteristic
105C4⋊C8×C59—21no—
106C4⋊C8×C59—21no—
107#103×C59—22no—
108#103×C59—22no—
109#102×C59—22no—
110#102×C59—22no—
111C22⋊C8×C59C2312yes—
112C22⋊C8×C59C2312yes—
113#97×C59—21no—
114#97×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—22no—
118#95×C59C2312yes—
119C24×C59—21no—
120C24×C59C2×D412yes—
121#41×C59—44no—
122#41×C59—84no—
123C2×Q8×C59—21no—
124C2×Q8×C59—21no—
125C2×Q8×C59—21no—
126C2×Q8×C59—21no—
127C2×Q8×C59—81no—
128C2×Q8×C59C2×D412yes—
129C2×D4×C59—22no—
130C2×D4×C59—24no—
131C2×D4×C59—24no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—42no—
139C2×D4×C59—44no—
140C2×D4×C59—81no—
141C2×D4×C59—82no—
142C2×D4×C59—82no—
143C2×D4×C59C2×D412yes—
144C22×C4×C59—24no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—22no—
148C22×C4×C59—22no—
149C22×C4×C59—21no—
150C22×C4×C59—82no—
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D412yes—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2411yescharacteristic
155Q16×C59—82no—
156SD16×C59—82no—
157SD16×C59—82no—
158D8×C59—82no—
159C2×C8×C59—22no—
160C2×C8×C59—22no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59C2×D412yes—
171C4⋊C4×C59C2×D412yes—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—41no—
181C22⋊C4×C59—41no—
182C22⋊C4×C59—41no—
183C22⋊C4×C59—41no—
184C22⋊C4×C59—82no—
185C22⋊C4×C59—82no—
186C42×C59—24no—
187C42×C59—22no—
188C42×C59C2×D411yescharacteristic
189C42×C59C2×D411yescharacteristic
190C42×C59C2×D411yescharacteristic
191C42×C59C2×D411yescharacteristic
192C23×C59—22no—
193C23×C59—22no—
194C23×C59—22no—
195C23×C59—22no—
196C23×C59—42no—
197C23×C59—42no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59—42no—
201C23×C59—41no—
202C23×C59—42no—
203C23×C59C22×D411yescharacteristic
204Q8×C59—42no—
205Q8×C59—42no—
206Q8×C59—42no—
207Q8×C59—42no—
208Q8×C59—42no—
209D4×C59—42no—
210D4×C59—42no—
211D4×C59—42no—
212D4×C59—42no—
213D4×C59—42no—
214D4×C59—42no—
215D4×C59—42no—
216D4×C59—44no—
217D4×C59—84no—
218D4×C59—84no—
219D4×C59—88no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—44no—
228C2×C4×C59—44no—
229C2×C4×C59—42no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—41no—
234C2×C4×C59—42no—
235C2×C4×C59—82no—
236C2×C4×C59—84no—
237C2×C4×C59#12011yescharacteristic
238C2×C4×C59#12011yescharacteristic
239C2×C4×C59#12011yescharacteristic
240C2×C4×C59#12011yescharacteristic
241C2×C4×C59C22×D411yescharacteristic
242C2×C4×C59C22×D411yescharacteristic
243C8×C59—42no—
244C8×C59—42no—
245#18370C5911yesmaximal, characteristic, direct product factor, Sylow
246C22×C59—21no—
247C22×C59—21no—
248C22×C59—44no—
249C22×C59—42no—
250C22×C59—42no—
251C22×C59—42no—
252C22×C59—42no—
253C22×C59—42no—
254C22×C59—42no—
255C22×C59—82no—
256C22×C59—82no—
257C22×C59—82no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59C2×#12011yescharacteristic
262C22×C59#57512yes—
263C4×C59—21no—
264C4×C59—21no—
265C4×C59—21no—
266C4×C59—21no—
267C4×C59—21no—
268C4×C59—21no—
269C4×C59—42no—
270C4×C59—42no—
271C4×C59—42no—
272C4×C59—42no—
273C4×C59—82no—
274C4×C59—81no—
275#3506C2×C5911yescharacteristic, complemented
276#3215C2×C5911yescharacteristic, complemented
277#3213C2×C5911yescharacteristic, complemented
278#2631C2×C5911yescharacteristic, complemented
279#1606C2×C5914yescomplemented
280#1600C2×C5914yescomplemented
281#1515C2×C5911yescharacteristic, complemented
282#1514C2×C5911yescharacteristic, complemented
283#1466C2×C5911yescharacteristic, complemented
284C2×C59—22no—
285C2×C59—82no—
286C2×C59—82no—
287C2×C59—161no—
288C2×C59#300511yescharacteristic
289C2×C59#301911yescharacteristic
290C2×C59#302711yescharacteristic
291#559—22no—
292#556—21no—
293#555—21no—
294#553—22no—
295#550—21no—
296#547—21no—
297#538C22×C5914yes—
298#535—21no—
299#533C22×C5911yescharacteristic
300#532C22×C5911yescharacteristic
301C2×#124C22×C5911yescharacteristic
302#525—22no—
303#520—21no—
304C2×C4×D4C22×C5911yescharacteristic
305#497—21no—
306#496—21no—
307#489—21no—
308#489—21no—
309#464—22no—
310#463—22no—
311#451C22×C5912yes—
312#450C22×C5912yes—
313#448C22×C5912yes—
314#447C22×C5912yes—
315#432C22×C5911yescharacteristic
316#432C22×C5911yescharacteristic
317#397—21no—
318#393C22×C5912yes—
319#392—21no—
320#392C22×C5912yes—
321#386—22no—
322#383C22×C5911yescharacteristic
323#328C22×C5914yes—
324#327C22×C5914yes—
325#323C22×C5914yes—
326C59#1837011yescharacteristic, direct product factor, Sylow
327C2×#41—22no—
328C2×#41—21no—
329C22×Q8C23×C5911yescharacteristic
330C22×D4—22no—
331C22×D4D4×C5911yescharacteristic
332C22×D4C23×C5911yescharacteristic
333C23×C4D4×C5912yes—
334C2×Q16—22no—
335C2×SD16—22no—
336C2×SD16—22no—
337C2×D8—22no—
338C4⋊Q8D4×C5911yescharacteristic, complemented
339C4⋊Q8D4×C5911yescharacteristic, complemented
340C4⋊Q8D4×C5911yescharacteristic, complemented
341#127—22no—
342#127D4×C5911yescharacteristic, complemented
343#126—24no—
344#125—21no—
345#124—22no—
346#124—24no—
347#124—21no—
348#124—21no—
349#124—21no—
350#124—21no—
351#124—21no—
352#123—24no—
353#123—24no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8C23×C5912yes—
357#121—22no—
358#121—22no—
359#121—22no—
360#121—24no—
361#121—21no—
362#121C23×C5912yes—
363#120—24no—
364C4×Q8—22no—
365C4×Q8—21no—
366C4×Q8D4×C5912yescomplemented
367C4×D4—22no—
368C4×D4—24no—
369C4×D4—24no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4—21no—
374C4×D4—21no—
375C4×D4D4×C5912yescomplemented
376#117—22no—
377#117—22no—
378#117—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4D4×C5911yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4C23×C5912yes—
386C2×C42C23×C5911yescharacteristic
387C4⋊C8—21no—
388C4⋊C8—21no—
389#103—22no—
390#103—22no—
391#102—22no—
392#102—22no—
393C22⋊C8C23×C5912yes—
394C22⋊C8C23×C5912yes—
395#97—21no—
396#97—21no—
397#95—21no—
398#95—21no—
399#95—22no—
400#95C23×C5912yes—
401C24—21no—
402C24C2×D4×C5912yes—
403#41—44no—
404#41—84no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—81no—
410C2×Q8C2×D4×C5912yes—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—81no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4C2×D4×C5912yes—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—82no—
433C22×C4C2×D4×C5911yescharacteristic
434C22×C4C2×D4×C5912yes—
435C22×C4C2×D4×C5911yescharacteristic
436C22×C4C24×C5911yescharacteristic
437Q16—82no—
438SD16—82no—
439SD16—82no—
440D8—82no—
441C2×C8—22no—
442C2×C8—22no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4C2×D4×C5912yes—
453C4⋊C4C2×D4×C5912yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C42—24no—
469C42—22no—
470C42C2×D4×C5911yescharacteristic
471C42C2×D4×C5911yescharacteristic
472C42C2×D4×C5911yescharacteristic
473C42C2×D4×C5911yescharacteristic
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23C22×D4×C5911yescharacteristic
486Q8—42no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—44no—
499D4—84no—
500D4—84no—
501D4—88no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—82no—
518C2×C4—84no—
519C2×C4#120×C5911yescharacteristic
520C2×C4#120×C5911yescharacteristic
521C2×C4#120×C5911yescharacteristic
522C2×C4#120×C5911yescharacteristic
523C2×C4C22×D4×C5911yescharacteristic
524C2×C4C22×D4×C5911yescharacteristic
525C8—42no—
526C8—42no—
527C22—21no—
528C22—21no—
529C22—44no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22C2×#120×C5911yescharacteristic
543C22C59×#57512yes—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—81no—
556C2—22no—
557C2—82no—
558C2—82no—
559C2—161no—
560C2C59×#300511yescharacteristic
561C2C59×#301911yescharacteristic
562C2C59×#302711yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
LCSG ▷ C22×C4 ▷ 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×#3506C2semidirect
2. C59×#3215C2semidirect
3. C59×#3213C2semidirect
4. C59×#2631C2semidirect
5. C59×#1606C2semidirect
6. C59×#1600C2semidirect
7. C59×#1515C2semidirect
8. C59×#1514C2semidirect
9. C59×#1466C2semidirect
56. C4⋊Q8×C59D4semidirect
57. C4⋊Q8×C59D4semidirect
58. C4⋊Q8×C59D4semidirect
60. #127×C59D4semidirect
84. C4×Q8×C59D4semidirect
93. C4×D4×C59D4semidirect
245. #18370C59direct (N × H)
275. #3506C2×C59semidirect
276. #3215C2×C59semidirect
277. #3213C2×C59semidirect
278. #2631C2×C59semidirect
279. #1606C2×C59semidirect
280. #1600C2×C59semidirect
281. #1515C2×C59semidirect
282. #1514C2×C59semidirect
283. #1466C2×C59semidirect
326. C59#18370direct (N × H)
338. C4⋊Q8D4×C59semidirect
339. C4⋊Q8D4×C59semidirect
340. C4⋊Q8D4×C59semidirect
342. #127D4×C59semidirect
366. C4×Q8D4×C59semidirect
375. C4×D4D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#35062
C59×#32152
C59×#32132
C59×#26312
C59×#16062
C59×#16002
C59×#15152
C59×#15142
C59×#14662
#1837059