Finite Group Databasebeta

← C59×#62149 #28474258

Subgroups — C59×#62149 #28474258

Order 15104.

1788 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3503C211yesmaximal, characteristic, complemented
2C59×#3483C211yesmaximal, characteristic
3Q8×Q16×C59C212yesmaximal, complemented
4C59×#3362C212yesmaximal, complemented
5C59×#3357C216yesmaximal, complemented
6C59×#3333C216yesmaximal, complemented
7C59×#3254C213yesmaximal, complemented
8C59×#3226C213yesmaximal, complemented
9C59×#3107C213yesmaximal, complemented
10C59×#3079C213yesmaximal, complemented
11C59×#2991C211yesmaximal, characteristic, complemented
12Q82×C59C2212yes—
13Q82×C59C2211yescharacteristic
14C59×#557—21no—
15C59×#555—23no—
16C59×#555C2213yes—
17C59×#554C2216yes—
18C59×#550—21no—
19C59×#550C2213yes—
20C59×#550C2211yescharacteristic
21D4×Q8×C59—23no—
22C59×#544—23no—
23C59×#533C2213yes—
24C59×#532—23no—
25C59×#532C2213yes—
26C2×C4⋊Q8×C59C2213yes—
27C59×#520—21no—
28C4×#41×C59—21no—
29C4×#41×C59C2213yes—
30C4×#41×C59C2211yescharacteristic
31C59×#500C2216yes—
32C59×#486C2216yes—
33C59×#485C2216yes—
34C59×#480C22112yes—
35C59×#477C2216yes—
36C59×#477C2216yes—
37C59×#476C2216yes—
38C59×#475C2213yes—
39C59×#474C2216yes—
40C59×#474C2213yes—
41C59×#468C22112yes—
42C59×#462C2211yescharacteristic
43C59×#461C2213yes—
44C59×#460C2216yes—
45C59×#452C2216yes—
46C8×Q8×C59C2212yes—
47C4×Q16×C59C2216yes—
48C4×SD16×C59C2216yes—
49C59×#436C2216yes—
50C59×#431C2213yes—
51C59×#428C2216yes—
52C59×#423C2213yes—
53C59×#417C2216yes—
54C2×#41×C59—21no—
55C2×#41×C59—23no—
56C2×#41×C59D411yescharacteristic
57C2×#41×C59C2313yes—
58C22×Q8×C59—23no—
59C2×Q16×C59C2312yes—
60C2×SD16×C59C2316yes—
61C2×M16×C59C2316yes—
62C4⋊Q8×C59—23no—
63C4⋊Q8×C59—26no—
64C4⋊Q8×C59—23no—
65C4⋊Q8×C59—23no—
66C4⋊Q8×C59C2311yescharacteristic
67C4⋊Q8×C59C2313yes—
68C4⋊Q8×C59C2316yes—
69C4⋊Q8×C59C2313yes—
70#127×C59—23no—
71#127×C59C2313yes—
72#125×C59—26no—
73#125×C59—23no—
74#124×C59—26no—
75#124×C59—23no—
76C22⋊Q8×C59—23no—
77C22⋊Q8×C59—212no—
78C22⋊Q8×C59C2316yes—
79#121×C59—23no—
80#121×C59—23no—
81#121×C59—23no—
82#121×C59—26no—
83#121×C59C2316yes—
84C4×Q8×C59—21no—
85C4×Q8×C59—22no—
86C4×Q8×C59—23no—
87C4×Q8×C59D411yescharacteristic
88C4×Q8×C59C2316yes—
89C4×Q8×C59C2313yes—
90C4×Q8×C59C2311yescharacteristic
91C4×Q8×C59C2311yescharacteristic
92C4×D4×C59—26no—
93C4×D4×C59—23no—
94C4×D4×C59—23no—
95C4×D4×C59—23no—
96C4×D4×C59—23no—
97C4×D4×C59D413yes—
98C4×D4×C59C2313yes—
99C4×D4×C59C2313yes—
100C4×D4×C59C2313yes—
101#117×C59—23no—
102#117×C59—23no—
103#117×C59—23no—
104#117×C59D413yes—
105#117×C59C2313yes—
106C2×C4⋊C4×C59—23no—
107C2×C4⋊C4×C59C2316yes—
108C2×C42×C59—23no—
109C2×C42×C59C2313yes—
110#107×C59C23112yes—
111#107×C59C2316yes—
112#106×C59C2316yes—
113C4⋊C8×C59C2313yes—
114C4⋊C8×C59C2316yes—
115C4⋊C8×C59C2311yescharacteristic
116#103×C59C2316yes—
117#103×C59C2312yes—
118#103×C59C2316yes—
119#102×C59C23112yes—
120#102×C59C2316yes—
121C22⋊C8×C59C2316yes—
122C4×C8×C59C2316yes—
123#41×C59—24no—
124#41×C59—26no—
125#41×C59—43no—
126#41×C59—44no—
127#41×C59—86no—
128C2×Q8×C59—23no—
129C2×Q8×C59—26no—
130C2×Q8×C59—23no—
131C2×Q8×C59—86no—
132C2×Q8×C59C2×D411yescharacteristic
133C2×Q8×C59C2412yes—
134C2×Q8×C59C2411yescharacteristic
135C2×D4×C59—23no—
136C2×D4×C59—26no—
137C2×D4×C59—23no—
138C2×D4×C59—23no—
139C2×D4×C59C2×D413yes—
140C2×D4×C59C2413yes—
141C22×C4×C59—26no—
142C22×C4×C59—23no—
143C22×C4×C59—23no—
144C22×C4×C59—23no—
145C22×C4×C59C2×D413yes—
146C22×C4×C59C2×D413yes—
147C22×C4×C59C2413yes—
148Q16×C59—24no—
149SD16×C59—46no—
150M16×C59—46no—
151C2×C8×C59C2412yes—
152C2×C8×C59C2416yes—
153C4⋊C4×C59—23no—
154C4⋊C4×C59—26no—
155C4⋊C4×C59—26no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—23no—
159C4⋊C4×C59—23no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—26no—
162C4⋊C4×C59—26no—
163C4⋊C4×C59C2×D413yes—
164C4⋊C4×C59C2×D413yes—
165C4⋊C4×C59C2416yes—
166C4⋊C4×C59C2411yescharacteristic
167C4⋊C4×C59C2413yes—
168C22⋊C4×C59—26no—
169C22⋊C4×C59—23no—
170C22⋊C4×C59—26no—
171C22⋊C4×C59—23no—
172C22⋊C4×C59—23no—
173C22⋊C4×C59C2×D416yes—
174C42×C59—21no—
175C42×C59—23no—
176C42×C59—23no—
177C42×C59—26no—
178C42×C59C2×D413yes—
179C42×C59C2×D413yes—
180C42×C59C2413yes—
181C42×C59C2411yescharacteristic
182C23×C59—23no—
183C23×C59C22×D413yes—
184Q8×C59—21no—
185Q8×C59—22no—
186Q8×C59—22no—
187Q8×C59—46no—
188Q8×C59—86no—
189Q8×C59—83no—
190Q8×C59#13714yes—
191Q8×C592−1+412yes—
192D4×C59—26no—
193D4×C59—23no—
194D4×C59—23no—
195D4×C59—46no—
196D4×C59—46no—
197D4×C59—83no—
198D4×C59—86no—
199C2×C4×C59—21no—
200C2×C4×C59—26no—
201C2×C4×C59—26no—
202C2×C4×C59—23no—
203C2×C4×C59—23no—
204C2×C4×C59—21no—
205C2×C4×C59—22no—
206C2×C4×C59—23no—
207C2×C4×C59—26no—
208C2×C4×C59—46no—
209C2×C4×C59—46no—
210C2×C4×C59—46no—
211C2×C4×C59—86no—
212C2×C4×C59C22×D413yes—
213C2×C4×C59C22×D411yescharacteristic
214C2×C4×C59C22×D413yes—
215C2×C4×C59C22×D413yes—
216C2×C4×C59C22×D411yescharacteristic
217C2×C4×C59C2511yescharacteristic
218C8×C59—22no—
219C8×C59—26no—
220#62149C5911yesmaximal, characteristic, direct product factor, Sylow
221C22×C59—23no—
222C22×C59—23no—
223C22×C59—23no—
224C22×C59—43no—
225C22×C59—43no—
226C22×C59—43no—
227C22×C59C23×D411yescharacteristic
228C4×C59—23no—
229C4×C59—22no—
230C4×C59—21no—
231C4×C59—23no—
232C4×C59—42no—
233C4×C59—41no—
234C4×C59—43no—
235C4×C59—46no—
236C4×C59C2×#13716yes—
237C4×C59#57612yes—
238C4×C59C2×2−1+411yescharacteristic
239C4×C59#58511yescharacteristic
240#3503C2×C5911yescharacteristic, complemented
241#3483C2×C5911yescharacteristic
242Q8×Q16C2×C5912yescomplemented
243#3362C2×C5912yescomplemented
244#3357C2×C5916yescomplemented
245#3333C2×C5916yescomplemented
246#3254C2×C5913yescomplemented
247#3226C2×C5913yescomplemented
248#3107C2×C5913yescomplemented
249#3079C2×C5913yescomplemented
250#2991C2×C5911yescharacteristic, complemented
251C2×C59—43no—
252C2×C59—83no—
253C2×C59#348911yescharacteristic
254C2×C59C22×#13711yescharacteristic
255C2×C59C2×#57611yescharacteristic
256Q82C22×C5912yes—
257Q82C22×C5911yescharacteristic
258#557—21no—
259#555—23no—
260#555C22×C5913yes—
261#554C22×C5916yes—
262#550—21no—
263#550C22×C5913yes—
264#550C22×C5911yescharacteristic
265D4×Q8—23no—
266#544—23no—
267#533C22×C5913yes—
268#532—23no—
269#532C22×C5913yes—
270C2×C4⋊Q8C22×C5913yes—
271#520—21no—
272C4×#41—21no—
273C4×#41C22×C5913yes—
274C4×#41C22×C5911yescharacteristic
275#500C22×C5916yes—
276#486C22×C5916yes—
277#485C22×C5916yes—
278#480C22×C59112yes—
279#477C22×C5916yes—
280#477C22×C5916yes—
281#476C22×C5916yes—
282#475C22×C5913yes—
283#474C22×C5916yes—
284#474C22×C5913yes—
285#468C22×C59112yes—
286#462C22×C5911yescharacteristic
287#461C22×C5913yes—
288#460C22×C5916yes—
289#452C22×C5916yes—
290C8×Q8C22×C5912yes—
291C4×Q16C22×C5916yes—
292C4×SD16C22×C5916yes—
293#436C22×C5916yes—
294#431C22×C5913yes—
295#428C22×C5916yes—
296#423C22×C5913yes—
297#417C22×C5916yes—
298C59#6214911yescharacteristic, direct product factor, Sylow
299C2×#41—21no—
300C2×#41—23no—
301C2×#41D4×C5911yescharacteristic
302C2×#41C23×C5913yes—
303C22×Q8—23no—
304C2×Q16C23×C5912yes—
305C2×SD16C23×C5916yes—
306C2×M16C23×C5916yes—
307C4⋊Q8—23no—
308C4⋊Q8—26no—
309C4⋊Q8—23no—
310C4⋊Q8—23no—
311C4⋊Q8C23×C5911yescharacteristic
312C4⋊Q8C23×C5913yes—
313C4⋊Q8C23×C5916yes—
314C4⋊Q8C23×C5913yes—
315#127—23no—
316#127C23×C5913yes—
317#125—26no—
318#125—23no—
319#124—26no—
320#124—23no—
321C22⋊Q8—23no—
322C22⋊Q8—212no—
323C22⋊Q8C23×C5916yes—
324#121—23no—
325#121—23no—
326#121—23no—
327#121—26no—
328#121C23×C5916yes—
329C4×Q8—21no—
330C4×Q8—22no—
331C4×Q8—23no—
332C4×Q8D4×C5911yescharacteristic
333C4×Q8C23×C5911yescharacteristic
334C4×Q8C23×C5913yes—
335C4×Q8C23×C5916yes—
336C4×Q8C23×C5911yescharacteristic
337C4×D4—26no—
338C4×D4—23no—
339C4×D4—23no—
340C4×D4—23no—
341C4×D4—23no—
342C4×D4D4×C5913yes—
343C4×D4C23×C5913yes—
344C4×D4C23×C5913yes—
345C4×D4C23×C5913yes—
346#117—23no—
347#117—23no—
348#117—23no—
349#117D4×C5913yes—
350#117C23×C5913yes—
351C2×C4⋊C4—23no—
352C2×C4⋊C4C23×C5916yes—
353C2×C42—23no—
354C2×C42C23×C5913yes—
355#107C23×C59112yes—
356#107C23×C5916yes—
357#106C23×C5916yes—
358C4⋊C8C23×C5913yes—
359C4⋊C8C23×C5916yes—
360C4⋊C8C23×C5911yescharacteristic
361#103C23×C5916yes—
362#103C23×C5912yes—
363#103C23×C5916yes—
364#102C23×C59112yes—
365#102C23×C5916yes—
366C22⋊C8C23×C5916yes—
367C4×C8C23×C5916yes—
368#41—24no—
369#41—26no—
370#41—43no—
371#41—44no—
372#41—86no—
373C2×Q8—23no—
374C2×Q8—26no—
375C2×Q8—23no—
376C2×Q8—86no—
377C2×Q8C2×D4×C5911yescharacteristic
378C2×Q8C24×C5912yes—
379C2×Q8C24×C5911yescharacteristic
380C2×D4—23no—
381C2×D4—26no—
382C2×D4—23no—
383C2×D4—23no—
384C2×D4C2×D4×C5913yes—
385C2×D4C24×C5913yes—
386C22×C4—26no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4—23no—
390C22×C4C2×D4×C5913yes—
391C22×C4C2×D4×C5913yes—
392C22×C4C24×C5913yes—
393Q16—24no—
394SD16—46no—
395M16—46no—
396C2×C8C24×C5912yes—
397C2×C8C24×C5916yes—
398C4⋊C4—23no—
399C4⋊C4—26no—
400C4⋊C4—26no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—23no—
404C4⋊C4—23no—
405C4⋊C4—21no—
406C4⋊C4—26no—
407C4⋊C4—26no—
408C4⋊C4C2×D4×C5913yes—
409C4⋊C4C2×D4×C5913yes—
410C4⋊C4C24×C5916yes—
411C4⋊C4C24×C5911yescharacteristic
412C4⋊C4C24×C5913yes—
413C22⋊C4—26no—
414C22⋊C4—23no—
415C22⋊C4—26no—
416C22⋊C4—23no—
417C22⋊C4—23no—
418C22⋊C4C2×D4×C5916yes—
419C42—21no—
420C42—23no—
421C42—23no—
422C42—26no—
423C42C2×D4×C5913yes—
424C42C2×D4×C5913yes—
425C42C24×C5913yes—
426C42C24×C5911yescharacteristic
427C23—23no—
428C23C22×D4×C5913yes—
429Q8—21no—
430Q8—22no—
431Q8—22no—
432Q8—46no—
433Q8—86no—
434Q8—83no—
435Q8#137×C5914yes—
436Q82−1+4×C5912yes—
437D4—26no—
438D4—23no—
439D4—23no—
440D4—46no—
441D4—46no—
442D4—83no—
443D4—86no—
444C2×C4—21no—
445C2×C4—26no—
446C2×C4—26no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—23no—
452C2×C4—26no—
453C2×C4—46no—
454C2×C4—46no—
455C2×C4—46no—
456C2×C4—86no—
457C2×C4C22×D4×C5913yes—
458C2×C4C22×D4×C5911yescharacteristic
459C2×C4C22×D4×C5913yes—
460C2×C4C22×D4×C5913yes—
461C2×C4C22×D4×C5911yescharacteristic
462C2×C4C25×C5911yescharacteristic
463C8—22no—
464C8—26no—
465C22—23no—
466C22—23no—
467C22—23no—
468C22—43no—
469C22—43no—
470C22—43no—
471C22C23×D4×C5911yescharacteristic
472C4—21no—
473C4—23no—
474C4—23no—
475C4—22no—
476C4—41no—
477C4—43no—
478C4—46no—
479C4—42no—
480C4C2×#137×C5916yes—
481C4C59×#57612yes—
482C4C2×2−1+4×C5911yescharacteristic
483C4C59×#58511yescharacteristic
484C2—43no—
485C2—83no—
486C2C59×#348911yescharacteristic
487C2C22×#137×C5911yescharacteristic
488C2C2×C59×#57611yescharacteristic
subgroupG/H
center, socleC22×C59C23×D4
commutator, Frattini, cosocleC2×C4C25×C59
prime radicalC59×#3483C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C4×#41×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3503C2semidirect
3. Q8×Q16×C59C2semidirect
4. C59×#3362C2semidirect
5. C59×#3357C2semidirect
6. C59×#3333C2semidirect
7. C59×#3254C2semidirect
8. C59×#3226C2semidirect
9. C59×#3107C2semidirect
10. C59×#3079C2semidirect
11. C59×#2991C2semidirect
220. #62149C59direct (N × H)
240. #3503C2×C59semidirect
242. Q8×Q16C2×C59semidirect
243. #3362C2×C59semidirect
244. #3357C2×C59semidirect
245. #3333C2×C59semidirect
246. #3254C2×C59semidirect
247. #3226C2×C59semidirect
248. #3107C2×C59semidirect
249. #3079C2×C59semidirect
250. #2991C2×C59semidirect
298. C59#62149direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35032
C59×#34832
Q8×Q16×C592
C59×#33622
C59×#33572
C59×#33332
C59×#32542
C59×#32262
C59×#31072
C59×#30792
C59×#29912
#6214959