Finite Group Databasebeta

← C59×#18150 #28430259

Subgroups — C59×#18150 #28430259

Order 15104.

1014 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×#3460C211yesmaximal, characteristic
2C59×#3092C212yesmaximal, complemented
3C59×#2486C211yesmaximal, characteristic, complemented
4C59×#1612C214yesmaximal, complemented
5C59×#1611C212yesmaximal, complemented
6C59×#1610C212yesmaximal, complemented
7C59×#1513C212yesmaximal, complemented
8C59×#1465C211yesmaximal, characteristic, complemented
9C59×#544C2211yescharacteristic
10C59×#542—22no—
11C59×#540—22no—
12C59×#538—21no—
13C59×#538C2212yes—
14C59×#537C2211yescharacteristic
15C59×#535—21no—
16C59×#532—22no—
17C59×#529—22no—
18C59×#525—21no—
19C59×#525C2212yes—
20C2×#123×C59—21no—
21C4×#41×C59—21no—
22C2×C4×D4×C59C2211yescharacteristic
23C22×C4⋊C4×C59C2211yescharacteristic
24C59×#484—22no—
25C59×#483C2214yes—
26C59×#481—22no—
27C59×#480C2214yes—
28C59×#428—22no—
29C59×#417—22no—
30C59×#407C2212yes—
31C59×#398C2212yes—
32C59×#397C2212yes—
33C59×#389—21no—
34C59×#381—21no—
35C59×#380—21no—
36C59×#380C2211yescharacteristic
37C59×#328C2214yes—
38C59×#327C2214yes—
39C59×#324C2212yes—
40C59×#323C2212yes—
41C2×#41×C59—21no—
42C2×#41×C59—21no—
43C2×#41×C59D412yes—
44C22×D4×C59D411yescharacteristic
45C23×C4×C59—21no—
46C23×C4×C59D411yescharacteristic
47C23×C4×C59C2311yescharacteristic
48C2×M16×C59—22no—
49C4⋊Q8×C59—22no—
50#127×C59—22no—
51#126×C59—22no—
52#126×C59—22no—
53#125×C59—22no—
54#124×C59—22no—
55#124×C59—22no—
56#124×C59—22no—
57#123×C59—22no—
58#123×C59—22no—
59#123×C59—22no—
60#123×C59—22no—
61#123×C59—44no—
62C22⋊Q8×C59—21no—
63C22⋊Q8×C59—21no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59D412yes—
66C22⋊Q8×C59C2312yes—
67#121×C59—21no—
68#121×C59—22no—
69#121×C59—22no—
70#121×C59—22no—
71#121×C59—22no—
72#121×C59—22no—
73#121×C59—21no—
74#121×C59D412yes—
75#121×C59C2312yes—
76#120×C59—22no—
77#120×C59—22no—
78C4×Q8×C59—21no—
79C4×Q8×C59—21no—
80C4×D4×C59—21no—
81C4×D4×C59—21no—
82C4×D4×C59—22no—
83C4×D4×C59—22no—
84C4×D4×C59—22no—
85C4×D4×C59—22no—
86C4×D4×C59—22no—
87C4×D4×C59—22no—
88#117×C59—22no—
89#117×C59—21no—
90#117×C59—21no—
91#117×C59D412yes—
92C2×C4⋊C4×C59—22no—
93C2×C4⋊C4×C59—22no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59—22no—
97C2×C4⋊C4×C59C2312yes—
98C2×C4⋊C4×C59C2311yescharacteristic
99C2×C22⋊C4×C59—22no—
100C2×C22⋊C4×C59—21no—
101C2×C22⋊C4×C59D411yescharacteristic
102C2×C22⋊C4×C59C2311yescharacteristic
103C2×C42×C59—22no—
104C2×C42×C59D411yescharacteristic
105#107×C59—24no—
106#106×C59—24no—
107#103×C59—24no—
108#102×C59—24no—
109C22⋊C8×C59—22no—
110C22⋊C8×C59C2314yes—
111#95×C59—22no—
112#95×C59—22no—
113#95×C59C2312yes—
114C24×C59C2×D411yescharacteristic
115C24×C59C2×D411yescharacteristic
116#41×C59—42no—
117#41×C59—44no—
118#41×C59—42no—
119#41×C59—82no—
120C2×Q8×C59—21no—
121C2×Q8×C59—21no—
122C2×Q8×C59C2×D412yes—
123C2×D4×C59—21no—
124C2×D4×C59—22no—
125C2×D4×C59—22no—
126C2×D4×C59—22no—
127C2×D4×C59—22no—
128C2×D4×C59—21no—
129C2×D4×C59—22no—
130C2×D4×C59—22no—
131C2×D4×C59—22no—
132C2×D4×C59—41no—
133C2×D4×C59C2×D412yes—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—42no—
142C22×C4×C59—42no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—42no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D412yes—
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D412yes—
152C22×C4×C59C2411yescharacteristic
153M16×C59—44no—
154C2×C8×C59—24no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—24no—
165C4⋊C4×C59—42no—
166C4⋊C4×C59—41no—
167C4⋊C4×C59—41no—
168C4⋊C4×C59—44no—
169C4⋊C4×C59C2×D412yes—
170C4⋊C4×C59C2×D412yes—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—22no—
182C22⋊C4×C59—42no—
183C22⋊C4×C59—44no—
184C42×C59—22no—
185C42×C59—22no—
186C42×C59—21no—
187C42×C59—22no—
188C42×C59—21no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—22no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59#12012yes—
200C23×C59C22×D411yescharacteristic
201Q8×C59—42no—
202Q8×C59—42no—
203Q8×C59—81no—
204D4×C59—44no—
205D4×C59—42no—
206D4×C59—42no—
207D4×C59—42no—
208D4×C59—42no—
209D4×C59—82no—
210D4×C59—82no—
211D4×C59—82no—
212D4×C59—81no—
213D4×C59—82no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—22no—
217C2×C4×C59—24no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—42no—
224C2×C4×C59—41no—
225C2×C4×C59—44no—
226C2×C4×C59—42no—
227C2×C4×C59—44no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—81no—
237C2×C4×C59—82no—
238C2×C4×C59—84no—
239C2×C4×C59#12011yescharacteristic
240C2×C4×C59#12011yescharacteristic
241C2×C4×C59AGL(1,ℤ/8)12yes—
242C2×C4×C59C22×D412yes—
243C8×C59—44no—
244#18150C5911yesmaximal, characteristic, direct product factor, Sylow
245C22×C59—21no—
246C22×C59—21no—
247C22×C59—22no—
248C22×C59—22no—
249C22×C59—22no—
250C22×C59—42no—
251C22×C59—42no—
252C22×C59—42no—
253C22×C59—42no—
254C22×C59—41no—
255C22×C59—42no—
256C22×C59—41no—
257C22×C59—41no—
258C22×C59—42no—
259C22×C59—81no—
260C22×C59C2×#12011yescharacteristic
261C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C59#57511yescharacteristic
263C4×C59—22no—
264C4×C59—22no—
265C4×C59—21no—
266C4×C59—42no—
267C4×C59—41no—
268C4×C59—42no—
269C4×C59—41no—
270C4×C59—42no—
271C4×C59—82no—
272C4×C59—82no—
273C4×C59#45812yes—
274#3460C2×C5911yescharacteristic
275#3092C2×C5912yescomplemented
276#2486C2×C5911yescharacteristic, complemented
277#1612C2×C5914yescomplemented
278#1611C2×C5912yescomplemented
279#1610C2×C5912yescomplemented
280#1513C2×C5912yescomplemented
281#1465C2×C5911yescharacteristic, complemented
282C2×C59—21no—
283C2×C59—21no—
284C2×C59—42no—
285C2×C59—82no—
286C2×C59—81no—
287C2×C59#300611yescharacteristic
288C2×C59#302211yescharacteristic
289C2×C59C2×#45811yescharacteristic
290#544C22×C5911yescharacteristic
291#542—22no—
292#540—22no—
293#538—21no—
294#538C22×C5912yes—
295#537C22×C5911yescharacteristic
296#535—21no—
297#532—22no—
298#529—22no—
299#525—21no—
300#525C22×C5912yes—
301C2×#123—21no—
302C4×#41—21no—
303C2×C4×D4C22×C5911yescharacteristic
304C22×C4⋊C4C22×C5911yescharacteristic
305#484—22no—
306#483C22×C5914yes—
307#481—22no—
308#480C22×C5914yes—
309#428—22no—
310#417—22no—
311#407C22×C5912yes—
312#398C22×C5912yes—
313#397C22×C5912yes—
314#389—21no—
315#381—21no—
316#380—21no—
317#380C22×C5911yescharacteristic
318#328C22×C5914yes—
319#327C22×C5914yes—
320#324C22×C5912yes—
321#323C22×C5912yes—
322C59#1815011yescharacteristic, direct product factor, Sylow
323C2×#41—21no—
324C2×#41—21no—
325C2×#41D4×C5912yes—
326C22×D4D4×C5911yescharacteristic
327C23×C4—21no—
328C23×C4D4×C5911yescharacteristic
329C23×C4C23×C5911yescharacteristic
330C2×M16—22no—
331C4⋊Q8—22no—
332#127—22no—
333#126—22no—
334#126—22no—
335#125—22no—
336#124—22no—
337#124—22no—
338#124—22no—
339#123—22no—
340#123—22no—
341#123—22no—
342#123—22no—
343#123—44no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C22⋊Q8—22no—
347C22⋊Q8D4×C5912yes—
348C22⋊Q8C23×C5912yes—
349#121—21no—
350#121—22no—
351#121—22no—
352#121—22no—
353#121—22no—
354#121—22no—
355#121—21no—
356#121D4×C5912yes—
357#121C23×C5912yes—
358#120—22no—
359#120—22no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×D4—21no—
363C4×D4—21no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370#117—22no—
371#117—21no—
372#117—21no—
373#117D4×C5912yes—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4C23×C5912yes—
380C2×C4⋊C4C23×C5911yescharacteristic
381C2×C22⋊C4—22no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4D4×C5911yescharacteristic
384C2×C22⋊C4C23×C5911yescharacteristic
385C2×C42—22no—
386C2×C42D4×C5911yescharacteristic
387#107—24no—
388#106—24no—
389#103—24no—
390#102—24no—
391C22⋊C8—22no—
392C22⋊C8C23×C5914yes—
393#95—22no—
394#95—22no—
395#95C23×C5912yes—
396C24C2×D4×C5911yescharacteristic
397C24C2×D4×C5911yescharacteristic
398#41—42no—
399#41—44no—
400#41—42no—
401#41—82no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8C2×D4×C5912yes—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—41no—
415C2×D4C2×D4×C5912yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4C2×D4×C5911yescharacteristic
429C22×C4C2×D4×C5912yes—
430C22×C4C2×D4×C5911yescharacteristic
431C22×C4C2×D4×C5911yescharacteristic
432C22×C4C2×D4×C5911yescharacteristic
433C22×C4C2×D4×C5912yes—
434C22×C4C24×C5911yescharacteristic
435M16—44no—
436C2×C8—24no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—42no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—44no—
451C4⋊C4C2×D4×C5912yes—
452C4⋊C4C2×D4×C5912yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—22no—
470C42—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23#120×C5912yes—
482C23C22×D4×C5911yescharacteristic
483Q8—42no—
484Q8—42no—
485Q8—81no—
486D4—44no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—82no—
492D4—82no—
493D4—82no—
494D4—81no—
495D4—82no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—84no—
521C2×C4#120×C5911yescharacteristic
522C2×C4#120×C5911yescharacteristic
523C2×C4AGL(1,ℤ/8)×C5912yes—
524C2×C4C22×D4×C5912yes—
525C8—44no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—41no—
539C22—42no—
540C22—81no—
541C22C2×#120×C5911yescharacteristic
542C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
543C22C59×#57511yescharacteristic
544C4—22no—
545C4—22no—
546C4—21no—
547C4—42no—
548C4—41no—
549C4—42no—
550C4—41no—
551C4—42no—
552C4—82no—
553C4—82no—
554C4C59×#45812yes—
555C2—21no—
556C2—21no—
557C2—42no—
558C2—81no—
559C2—82no—
560C2C59×#300611yescharacteristic
561C2C59×#302211yescharacteristic
562C2C2×C59×#45811yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3460C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3092C2semidirect
3. C59×#2486C2semidirect
4. C59×#1612C2semidirect
5. C59×#1611C2semidirect
6. C59×#1610C2semidirect
7. C59×#1513C2semidirect
8. C59×#1465C2semidirect
244. #18150C59direct (N × H)
275. #3092C2×C59semidirect
276. #2486C2×C59semidirect
277. #1612C2×C59semidirect
278. #1611C2×C59semidirect
279. #1610C2×C59semidirect
280. #1513C2×C59semidirect
281. #1465C2×C59semidirect
322. C59#18150direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34602
C59×#30922
C59×#24862
C59×#16122
C59×#16112
C59×#16102
C59×#15132
C59×#14652
#1815059