Finite Group Databasebeta

← C59×#18656 #28430765

Subgroups — C59×#18656 #28430765

Order 15104.

960 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3511C211yesmaximal, characteristic
2C59×#3226C211yesmaximal, characteristic, complemented
3C2×C59×#498C211yesmaximal, characteristic, complemented
4C59×#3039C211yesmaximal, characteristic, complemented
5C59×#1654C214yesmaximal, complemented
6C59×#1652C214yesmaximal, complemented
7C59×#1549C211yesmaximal, characteristic, complemented
8C59×#1492C211yesmaximal, characteristic, complemented
9C59×#1476C211yesmaximal, characteristic, complemented
10C2×#137×C59—21no—
11C59×#556C2212yes—
12C59×#555—21no—
13C59×#554—21no—
14C59×#553—22no—
15C59×#548—22no—
16C59×#547C2212yes—
17C59×#542—22no—
18C59×#537—21no—
19C59×#533—21no—
20C59×#532—21no—
21C59×#532C2211yescharacteristic
22C2×C4⋊Q8×C59C2211yescharacteristic
23C2×C4⋊Q8×C59C2211yescharacteristic
24C59×#520—21no—
25C4×#41×C59C2211yescharacteristic
26C59×#500—21no—
27C59×#498—22no—
28C59×#498C2214yes—
29C59×#486—21no—
30C59×#480—22no—
31C59×#474C2214yes—
32C59×#462—21no—
33C59×#462C2212yes—
34C59×#461—21no—
35C59×#461C2212yes—
36C59×#431C2211yescharacteristic
37C2×#106×C59—22no—
38C59×#423C2211yescharacteristic
39C59×#419—21no—
40C2×C4×C8×C59C2211yescharacteristic
41C59×#332C2214yes—
42C59×Q8⋊C8C2212yes—
43C59×D4⋊C8C2212yes—
44C59×D4⋊C8C2214yes—
45C2×#41×C59—22no—
46C2×#41×C59—21no—
47C2×#41×C59D411yescharacteristic
48C22×Q8×C59—21no—
49C22×Q8×C59—21no—
50#137×C59—44no—
51C2×Q16×C59—22no—
52C2×SD16×C59—22no—
53C2×M16×C59—21no—
54C22×C8×C59—21no—
55C4⋊Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59—21no—
58C4⋊Q8×C59—21no—
59C4⋊Q8×C59—21no—
60C4⋊Q8×C59D412yes—
61C4⋊Q8×C59D411yescharacteristic
62C4⋊Q8×C59C2312yes—
63C4⋊Q8×C59C2312yes—
64#127×C59D411yescharacteristic
65#126×C59—24no—
66#125×C59—22no—
67#124×C59—24no—
68#124×C59—22no—
69#124×C59—21no—
70#123×C59—24no—
71C22⋊Q8×C59—22no—
72C22⋊Q8×C59—24no—
73C22⋊Q8×C59—22no—
74C22⋊Q8×C59—22no—
75#121×C59—22no—
76#121×C59—22no—
77#121×C59—24no—
78#121×C59—22no—
79#121×C59—21no—
80#121×C59D412yes—
81C4×Q8×C59—22no—
82C4×Q8×C59—22no—
83C4×Q8×C59D411yescharacteristic
84C4×Q8×C59C2311yescharacteristic
85C4×D4×C59—22no—
86C4×D4×C59—24no—
87C4×D4×C59—22no—
88C4×D4×C59—21no—
89C4×D4×C59—21no—
90C4×D4×C59—21no—
91C4×D4×C59D411yescharacteristic
92C4×D4×C59C2311yescharacteristic
93C4×D4×C59C2312yes—
94#117×C59—22no—
95#117×C59—22no—
96#117×C59—21no—
97#117×C59D411yescharacteristic
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—22no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59D412yes—
103C2×C42×C59—21no—
104C2×C42×C59C2311yescharacteristic
105#107×C59—24no—
106#106×C59—28no—
107C4⋊C8×C59—21no—
108C4⋊C8×C59C2312yes—
109C4⋊C8×C59C2312yes—
110#103×C59—22no—
111#103×C59—22no—
112#102×C59—24no—
113C22⋊C8×C59—21no—
114C4×C8×C59—21no—
115C4×C8×C59—21no—
116C4×C8×C59C2312yes—
117#41×C59—22no—
118#41×C59—22no—
119#41×C59—42no—
120#41×C59—42no—
121#41×C59—84no—
122C2×Q8×C59—21no—
123C2×Q8×C59—22no—
124C2×Q8×C59—22no—
125C2×Q8×C59—22no—
126C2×Q8×C59—41no—
127C2×Q8×C59—42no—
128C2×Q8×C59—42no—
129C2×Q8×C59—82no—
130C2×Q8×C59C2×D411yescharacteristic
131C2×D4×C59—21no—
132C2×D4×C59—22no—
133C2×D4×C59—24no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59C2×D412yes—
137C2×D4×C59C2×D411yescharacteristic
138C22×C4×C59—24no—
139C22×C4×C59—22no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—22no—
144C22×C4×C59—41no—
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148Q16×C59—44no—
149SD16×C59—44no—
150M16×C59—42no—
151C2×C8×C59—24no—
152C2×C8×C59—22no—
153C2×C8×C59—22no—
154C2×C8×C59—22no—
155C4⋊C4×C59—24no—
156C4⋊C4×C59—24no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—22no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—41no—
166C4⋊C4×C59—41no—
167C4⋊C4×C59C2×D411yescharacteristic
168C4⋊C4×C59C2×D411yescharacteristic
169C4⋊C4×C59C2×D414yes—
170C4⋊C4×C59C2×D412yes—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—21no—
175C22⋊C4×C59—21no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—21no—
178C22⋊C4×C59—22no—
179C42×C59—24no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59C2×D411yescharacteristic
184C42×C59C2×D411yescharacteristic
185C42×C59C2×D411yescharacteristic
186C42×C59C2411yescharacteristic
187C23×C59—21no—
188C23×C59—22no—
189C23×C59#12011yescharacteristic
190Q8×C59—21no—
191Q8×C59—21no—
192Q8×C59—44no—
193Q8×C59—42no—
194Q8×C59—42no—
195Q8×C59—42no—
196Q8×C59—82no—
197Q8×C59—82no—
198D4×C59—21no—
199D4×C59—21no—
200D4×C59—24no—
201D4×C59—42no—
202D4×C59—44no—
203D4×C59—42no—
204D4×C59—42no—
205D4×C59—84no—
206C2×C4×C59—22no—
207C2×C4×C59—22no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—22no—
211C2×C4×C59—21no—
212C2×C4×C59—24no—
213C2×C4×C59—42no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—82no—
220C2×C4×C59—84no—
221C2×C4×C59—82no—
222C2×C4×C59#12011yescharacteristic
223C2×C4×C59#12011yescharacteristic
224C2×C4×C59#12011yescharacteristic
225C2×C4×C59#13512yes—
226C2×C4×C59AGL(1,ℤ/8)12yes—
227C2×C4×C59C22×D411yescharacteristic
228C2×C4×C59C22×D411yescharacteristic
229C2×C4×C59C22×D411yescharacteristic
230C8×C59—24no—
231C8×C59—42no—
232C8×C59—42no—
233#18656C5911yesmaximal, characteristic, direct product factor, Sylow
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—41no—
237C22×C59—42no—
238C22×C59—41no—
239C22×C59—42no—
240C22×C59—41no—
241C22×C59—42no—
242C22×C59#44912yes—
243C22×C59C2×#12011yescharacteristic
244C4×C59—21no—
245C4×C59—21no—
246C4×C59—41no—
247C4×C59—42no—
248C4×C59—41no—
249C4×C59—44no—
250C4×C59—41no—
251C4×C59—82no—
252C4×C59#44912yes—
253C4×C59#45612yes—
254C4×C59C2×#13511yescharacteristic
255C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
256C4×C59#57712yes—
257#3511C2×C5911yescharacteristic
258#3226C2×C5911yescharacteristic, complemented
259C2×#498C2×C5911yescharacteristic, complemented
260#3039C2×C5911yescharacteristic, complemented
261#1654C2×C5914yescomplemented
262#1652C2×C5914yescomplemented
263#1549C2×C5911yescharacteristic, complemented
264#1492C2×C5911yescharacteristic, complemented
265#1476C2×C5911yescharacteristic, complemented
266C2×C59—22no—
267C2×C59—81no—
268C2×C59—82no—
269C2×C59C2×#44911yescharacteristic
270C2×C59#300911yescharacteristic
271C2×C59C2×#45611yescharacteristic
272C2×#137—21no—
273#556C22×C5912yes—
274#555—21no—
275#554—21no—
276#553—22no—
277#548—22no—
278#547C22×C5912yes—
279#542—22no—
280#537—21no—
281#533—21no—
282#532—21no—
283#532C22×C5911yescharacteristic
284C2×C4⋊Q8C22×C5911yescharacteristic
285C2×C4⋊Q8C22×C5911yescharacteristic
286#520—21no—
287C4×#41C22×C5911yescharacteristic
288#500—21no—
289#498—22no—
290#498C22×C5914yes—
291#486—21no—
292#480—22no—
293#474C22×C5914yes—
294#462—21no—
295#462C22×C5912yes—
296#461—21no—
297#461C22×C5912yes—
298#431C22×C5911yescharacteristic
299C2×#106—22no—
300#423C22×C5911yescharacteristic
301#419—21no—
302C2×C4×C8C22×C5911yescharacteristic
303#332C22×C5914yes—
304Q8⋊C8C22×C5912yes—
305D4⋊C8C22×C5914yes—
306D4⋊C8C22×C5912yes—
307C59#1865611yescharacteristic, direct product factor, Sylow
308C2×#41—22no—
309C2×#41—21no—
310C2×#41D4×C5911yescharacteristic
311C22×Q8—21no—
312C22×Q8—21no—
313#137—44no—
314C2×Q16—22no—
315C2×SD16—22no—
316C2×M16—21no—
317C22×C8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8D4×C5912yes—
324C4⋊Q8D4×C5911yescharacteristic
325C4⋊Q8C23×C5912yes—
326C4⋊Q8C23×C5912yes—
327#127D4×C5911yescharacteristic
328#126—24no—
329#125—22no—
330#124—24no—
331#124—22no—
332#124—21no—
333#123—24no—
334C22⋊Q8—22no—
335C22⋊Q8—24no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338#121—22no—
339#121—22no—
340#121—24no—
341#121—22no—
342#121—21no—
343#121D4×C5912yes—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8D4×C5911yescharacteristic
347C4×Q8C23×C5911yescharacteristic
348C4×D4—22no—
349C4×D4—24no—
350C4×D4—22no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4D4×C5911yescharacteristic
355C4×D4C23×C5911yescharacteristic
356C4×D4C23×C5912yes—
357#117—22no—
358#117—22no—
359#117—21no—
360#117D4×C5911yescharacteristic
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4D4×C5912yes—
366C2×C42—21no—
367C2×C42C23×C5911yescharacteristic
368#107—24no—
369#106—28no—
370C4⋊C8—21no—
371C4⋊C8C23×C5912yes—
372C4⋊C8C23×C5912yes—
373#103—22no—
374#103—22no—
375#102—24no—
376C22⋊C8—21no—
377C4×C8—21no—
378C4×C8—21no—
379C4×C8C23×C5912yes—
380#41—22no—
381#41—22no—
382#41—42no—
383#41—42no—
384#41—84no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—41no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—82no—
393C2×Q8C2×D4×C5911yescharacteristic
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4C2×D4×C5912yes—
400C2×D4C2×D4×C5911yescharacteristic
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4C2×D4×C5911yescharacteristic
409C22×C4C2×D4×C5911yescharacteristic
410C22×C4C2×D4×C5911yescharacteristic
411Q16—44no—
412SD16—44no—
413M16—42no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4C2×D4×C5911yescharacteristic
431C4⋊C4C2×D4×C5911yescharacteristic
432C4⋊C4C2×D4×C5914yes—
433C4⋊C4C2×D4×C5912yes—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—22no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C42—24no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42C2×D4×C5911yescharacteristic
447C42C2×D4×C5911yescharacteristic
448C42C2×D4×C5911yescharacteristic
449C42C24×C5911yescharacteristic
450C23—21no—
451C23—22no—
452C23#120×C5911yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—42no—
456Q8—44no—
457Q8—42no—
458Q8—42no—
459Q8—82no—
460Q8—82no—
461D4—21no—
462D4—21no—
463D4—24no—
464D4—42no—
465D4—44no—
466D4—42no—
467D4—42no—
468D4—84no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—84no—
484C2×C4—82no—
485C2×C4#120×C5911yescharacteristic
486C2×C4#120×C5911yescharacteristic
487C2×C4#120×C5911yescharacteristic
488C2×C4#135×C5912yes—
489C2×C4AGL(1,ℤ/8)×C5912yes—
490C2×C4C22×D4×C5911yescharacteristic
491C2×C4C22×D4×C5911yescharacteristic
492C2×C4C22×D4×C5911yescharacteristic
493C8—24no—
494C8—42no—
495C8—42no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—42no—
500C22—41no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22C59×#44912yes—
505C22C2×#120×C5911yescharacteristic
506C4—21no—
507C4—21no—
508C4—41no—
509C4—42no—
510C4—41no—
511C4—44no—
512C4—41no—
513C4—82no—
514C4C59×#44912yes—
515C4C59×#45612yes—
516C4C2×#135×C5911yescharacteristic
517C4C2×AGL(1,ℤ/8)×C5911yescharacteristic
518C4C59×#57712yes—
519C2—22no—
520C2—81no—
521C2—82no—
522C2C2×C59×#44911yescharacteristic
523C2C59×#300911yescharacteristic
524C2C2×C59×#45611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC42C24×C59
prime radicalC59×#3511C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3226C2semidirect
3. C2×C59×#498C2semidirect
4. C59×#3039C2semidirect
5. C59×#1654C2semidirect
6. C59×#1652C2semidirect
7. C59×#1549C2semidirect
8. C59×#1492C2semidirect
9. C59×#1476C2semidirect
233. #18656C59direct (N × H)
258. #3226C2×C59semidirect
259. C2×#498C2×C59semidirect
260. #3039C2×C59semidirect
261. #1654C2×C59semidirect
262. #1652C2×C59semidirect
263. #1549C2×C59semidirect
264. #1492C2×C59semidirect
265. #1476C2×C59semidirect
307. C59#18656direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35112
C59×#32262
C2×C59×#4982
C59×#30392
C59×#16542
C59×#16522
C59×#15492
C59×#14922
C59×#14762
#1865659