Finite Group Databasebeta

← C59×#18214 #28430323

Subgroups — C59×#18214 #28430323

Order 15104.

982 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3507C211yesmaximal, characteristic, complemented
2C59×#3101C211yesmaximal, characteristic, complemented
3C59×#3100C211yesmaximal, characteristic
4C59×#2536C211yesmaximal, characteristic, complemented
5C59×#1609C214yesmaximal, complemented
6C59×#1597C214yesmaximal, complemented
7C59×#1518C211yesmaximal, characteristic, complemented
8C59×#1517C211yesmaximal, characteristic, complemented
9C59×#1461C211yesmaximal, characteristic, complemented
10C59×#559—21no—
11C59×#557—21no—
12C59×#556—21no—
13C59×#556—22no—
14C59×#554—21no—
15C59×#552—22no—
16C59×#539C2214yes—
17C59×#536—21no—
18C59×#533C2211yescharacteristic
19C59×#532C2211yescharacteristic
20C59×#529—22no—
21C2×C22⋊Q8×C59—21no—
22C59×#520—21no—
23C2×C4×Q8×C59C2211yescharacteristic
24C2×C4×D4×C59C2211yescharacteristic
25C59×#483C2212yes—
26C59×#480C2212yes—
27C59×#476—21no—
28C59×#474—21no—
29C59×#461—21no—
30C59×#459—21no—
31C59×#450C2212yes—
32C59×#447C2212yes—
33C4×SD16×C59—22no—
34C4×D8×C59—22no—
35C59×#431C2211yescharacteristic
36C59×#431C2211yescharacteristic
37C59×#398C2212yes—
38C59×#394C2212yes—
39C59×#388—22no—
40C59×#382C2211yescharacteristic
41C59×#380—21no—
42C59×#328C2214yes—
43C59×#327C2214yes—
44C59×#324C2214yes—
45C2×#41×C59—21no—
46C22×Q8×C59D411yescharacteristic
47C22×D4×C59C2311yescharacteristic
48C23×C4×C59—21no—
49C2×SD16×C59—22no—
50C2×D8×C59—22no—
51C4⋊Q8×C59—22no—
52C4⋊Q8×C59D411yescharacteristic
53C4⋊Q8×C59D411yescharacteristic
54C4⋊Q8×C59D411yescharacteristic
55#127×C59D411yescharacteristic
56#126×C59—24no—
57#126×C59—24no—
58#125×C59—22no—
59#125×C59—24no—
60#125×C59—21no—
61#124×C59—22no—
62#124×C59—22no—
63#124×C59—21no—
64#123×C59—24no—
65#123×C59—24no—
66C22⋊Q8×C59—22no—
67C22⋊Q8×C59—22no—
68C22⋊Q8×C59—24no—
69C22⋊Q8×C59—21no—
70C22⋊Q8×C59—44no—
71C22⋊Q8×C59C2312yes—
72#121×C59—21no—
73#121×C59C2312yes—
74C4×Q8×C59—22no—
75C4×Q8×C59—22no—
76C4×Q8×C59—21no—
77C4×Q8×C59—21no—
78C4×Q8×C59—21no—
79C4×Q8×C59—21no—
80C4×Q8×C59—21no—
81C4×Q8×C59D412yes—
82C4×D4×C59—22no—
83C4×D4×C59—22no—
84C4×D4×C59—24no—
85C4×D4×C59—21no—
86C4×D4×C59—21no—
87C4×D4×C59—21no—
88C4×D4×C59—21no—
89C4×D4×C59—21no—
90C4×D4×C59D412yes—
91#117×C59—22no—
92#117×C59—24no—
93#117×C59—21no—
94C2×C4⋊C4×C59—22no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59—22no—
97C2×C4⋊C4×C59D411yescharacteristic
98C2×C4⋊C4×C59C2311yescharacteristic
99C2×C4⋊C4×C59C2312yes—
100C2×C22⋊C4×C59—22no—
101C2×C22⋊C4×C59—21no—
102C2×C22⋊C4×C59—22no—
103C2×C42×C59D412yes—
104C2×C42×C59C2311yescharacteristic
105#107×C59—22no—
106#106×C59—22no—
107C4⋊C8×C59—21no—
108C4⋊C8×C59—21no—
109#103×C59—22no—
110#102×C59—22no—
111#102×C59—22no—
112#102×C59—22no—
113C22⋊C8×C59C2312yes—
114C22⋊C8×C59C2312yes—
115C4×C8×C59—21no—
116C4×C8×C59—21no—
117#95×C59—22no—
118#95×C59—21no—
119#95×C59C2312yes—
120C24×C59—21no—
121#41×C59—42no—
122#41×C59—42no—
123C2×Q8×C59—21no—
124C2×Q8×C59—21no—
125C2×Q8×C59—21no—
126C2×Q8×C59—21no—
127C2×Q8×C59—22no—
128C2×Q8×C59—22no—
129C2×Q8×C59C2×D412yes—
130C2×D4×C59—22no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—42no—
136C2×D4×C59C2×D412yes—
137C22×C4×C59—24no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—21no—
141C22×C4×C59—41no—
142C22×C4×C59—41no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59C2×D412yes—
146C22×C4×C59C2×D412yes—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2411yescharacteristic
150SD16×C59—44no—
151D8×C59—44no—
152C2×C8×C59—22no—
153C2×C8×C59—22no—
154C4⋊C4×C59—24no—
155C4⋊C4×C59—24no—
156C4⋊C4×C59—24no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—44no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59C2×D412yes—
172C4⋊C4×C59C2×D412yes—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—44no—
180C22⋊C4×C59—44no—
181C22⋊C4×C59—41no—
182C22⋊C4×C59—41no—
183C42×C59—24no—
184C42×C59—22no—
185C42×C59—22no—
186C42×C59—22no—
187C42×C59C2×D411yescharacteristic
188C42×C59C2×D411yescharacteristic
189C42×C59C2×D411yescharacteristic
190C42×C59C2×D411yescharacteristic
191C23×C59—22no—
192C23×C59—22no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59C22×D411yescharacteristic
200Q8×C59—42no—
201Q8×C59—42no—
202Q8×C59—42no—
203Q8×C59—42no—
204Q8×C59—42no—
205D4×C59—22no—
206D4×C59—22no—
207D4×C59—42no—
208D4×C59—42no—
209D4×C59—41no—
210D4×C59—41no—
211D4×C59—42no—
212D4×C59—42no—
213D4×C59—42no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—44no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—42no—
230C2×C4×C59—41no—
231C2×C4×C59—84no—
232C2×C4×C59—82no—
233C2×C4×C59—82no—
234C2×C4×C59#12011yescharacteristic
235C2×C4×C59#12011yescharacteristic
236C2×C4×C59#12011yescharacteristic
237C2×C4×C59#12011yescharacteristic
238C2×C4×C59#13512yes—
239C2×C4×C59AGL(1,ℤ/8)12yes—
240C2×C4×C59C22×D411yescharacteristic
241C2×C4×C59C22×D411yescharacteristic
242C8×C59—42no—
243C8×C59—42no—
244#18214C5911yesmaximal, characteristic, direct product factor, Sylow
245C22×C59—21no—
246C22×C59—21no—
247C22×C59—42no—
248C22×C59—42no—
249C22×C59—42no—
250C22×C59—42no—
251C22×C59—42no—
252C22×C59—42no—
253C22×C59—81no—
254C22×C59—81no—
255C22×C59—81no—
256C22×C59—81no—
257C22×C59C2×#12011yescharacteristic
258C22×C59C2×#13511yescharacteristic
259C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
260C4×C59—21no—
261C4×C59—21no—
262C4×C59—21no—
263C4×C59—21no—
264C4×C59—42no—
265C4×C59—42no—
266C4×C59—42no—
267C4×C59—42no—
268C4×C59—42no—
269C4×C59—82no—
270C4×C59—81no—
271C4×C59#44912yes—
272C4×C59#44912yes—
273#3507C2×C5911yescharacteristic, complemented
274#3101C2×C5911yescharacteristic, complemented
275#3100C2×C5911yescharacteristic
276#2536C2×C5911yescharacteristic, complemented
277#1609C2×C5914yescomplemented
278#1597C2×C5914yescomplemented
279#1518C2×C5911yescharacteristic, complemented
280#1517C2×C5911yescharacteristic, complemented
281#1461C2×C5911yescharacteristic, complemented
282C2×C59—21no—
283C2×C59—21no—
284C2×C59—82no—
285C2×C59—82no—
286C2×C59C2×#44911yescharacteristic
287C2×C59#302011yescharacteristic
288C2×C59#302511yescharacteristic
289#559—21no—
290#557—21no—
291#556—21no—
292#556—22no—
293#554—21no—
294#552—22no—
295#539C22×C5914yes—
296#536—21no—
297#533C22×C5911yescharacteristic
298#532C22×C5911yescharacteristic
299#529—22no—
300C2×C22⋊Q8—21no—
301#520—21no—
302C2×C4×Q8C22×C5911yescharacteristic
303C2×C4×D4C22×C5911yescharacteristic
304#483C22×C5912yes—
305#480C22×C5912yes—
306#476—21no—
307#474—21no—
308#461—21no—
309#459—21no—
310#450C22×C5912yes—
311#447C22×C5912yes—
312C4×SD16—22no—
313C4×D8—22no—
314#431C22×C5911yescharacteristic
315#431C22×C5911yescharacteristic
316#398C22×C5912yes—
317#394C22×C5912yes—
318#388—22no—
319#382C22×C5911yescharacteristic
320#380—21no—
321#328C22×C5914yes—
322#327C22×C5914yes—
323#324C22×C5914yes—
324C59#1821411yescharacteristic, direct product factor, Sylow
325C2×#41—21no—
326C22×Q8D4×C5911yescharacteristic
327C22×D4C23×C5911yescharacteristic
328C23×C4—21no—
329C2×SD16—22no—
330C2×D8—22no—
331C4⋊Q8—22no—
332C4⋊Q8D4×C5911yescharacteristic
333C4⋊Q8D4×C5911yescharacteristic
334C4⋊Q8D4×C5911yescharacteristic
335#127D4×C5911yescharacteristic
336#126—24no—
337#126—24no—
338#125—22no—
339#125—24no—
340#125—21no—
341#124—22no—
342#124—22no—
343#124—21no—
344#123—24no—
345#123—24no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—24no—
349C22⋊Q8—21no—
350C22⋊Q8—44no—
351C22⋊Q8C23×C5912yes—
352#121—21no—
353#121C23×C5912yes—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8D4×C5912yes—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—24no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4D4×C5912yes—
371#117—22no—
372#117—24no—
373#117—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4D4×C5911yescharacteristic
378C2×C4⋊C4C23×C5911yescharacteristic
379C2×C4⋊C4C23×C5912yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C42D4×C5912yes—
384C2×C42C23×C5911yescharacteristic
385#107—22no—
386#106—22no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389#103—22no—
390#102—22no—
391#102—22no—
392#102—22no—
393C22⋊C8C23×C5912yes—
394C22⋊C8C23×C5912yes—
395C4×C8—21no—
396C4×C8—21no—
397#95—22no—
398#95—21no—
399#95C23×C5912yes—
400C24—21no—
401#41—42no—
402#41—42no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8C2×D4×C5912yes—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4C2×D4×C5912yes—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4C2×D4×C5912yes—
426C22×C4C2×D4×C5912yes—
427C22×C4C2×D4×C5911yescharacteristic
428C22×C4C2×D4×C5911yescharacteristic
429C22×C4C24×C5911yescharacteristic
430SD16—44no—
431D8—44no—
432C2×C8—22no—
433C2×C8—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4C2×D4×C5912yes—
452C4⋊C4C2×D4×C5912yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42C2×D4×C5911yescharacteristic
468C42C2×D4×C5911yescharacteristic
469C42C2×D4×C5911yescharacteristic
470C42C2×D4×C5911yescharacteristic
471C23—22no—
472C23—22no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23C22×D4×C5911yescharacteristic
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485D4—22no—
486D4—22no—
487D4—42no—
488D4—42no—
489D4—41no—
490D4—41no—
491D4—42no—
492D4—42no—
493D4—42no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4#120×C5911yescharacteristic
515C2×C4#120×C5911yescharacteristic
516C2×C4#120×C5911yescharacteristic
517C2×C4#120×C5911yescharacteristic
518C2×C4#135×C5912yes—
519C2×C4AGL(1,ℤ/8)×C5912yes—
520C2×C4C22×D4×C5911yescharacteristic
521C2×C4C22×D4×C5911yescharacteristic
522C8—42no—
523C8—42no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22C2×#120×C5911yescharacteristic
537C22C2×#135×C5911yescharacteristic
538C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—82no—
549C4—81no—
550C4C59×#44912yes—
551C4C59×#44912yes—
552C2—21no—
553C2—21no—
554C2—82no—
555C2—82no—
556C2C2×C59×#44911yescharacteristic
557C2C59×#302011yescharacteristic
558C2C59×#302511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3100C2
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×#3507C2semidirect
2. C59×#3101C2semidirect
4. C59×#2536C2semidirect
5. C59×#1609C2semidirect
6. C59×#1597C2semidirect
7. C59×#1518C2semidirect
8. C59×#1517C2semidirect
9. C59×#1461C2semidirect
244. #18214C59direct (N × H)
273. #3507C2×C59semidirect
274. #3101C2×C59semidirect
276. #2536C2×C59semidirect
277. #1609C2×C59semidirect
278. #1597C2×C59semidirect
279. #1518C2×C59semidirect
280. #1517C2×C59semidirect
281. #1461C2×C59semidirect
324. C59#18214direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35072
C59×#31012
C59×#31002
C59×#25362
C59×#16092
C59×#15972
C59×#15182
C59×#15172
C59×#14612
#1821459