Finite Group Databasebeta

← #28480606

Subgroups — #28480606

Order 15104.

942 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425049C211yesmaximal, characteristic, complemented
2#21424116C211yesmaximal, characteristic, complemented
3#21422773C211yesmaximal, characteristic, complemented
4#21422768C211yesmaximal, characteristic, complemented
5#21422591C211yesmaximal, characteristic, complemented
6#21422483C211yesmaximal, characteristic, complemented
7C59×#3499C211yesmaximal, characteristic
8#16837455C2211yescharacteristic
9#16837426C2211yescharacteristic
10C2×C4⋊(C59⋊C8)C2211yescharacteristic
11C2×#14273654—21no—
12C4×C4⋊Dic59C2211yescharacteristic
13#16837164—22no—
14#16837161—21no—
15#16837148—22no—
16#16837148—21no—
17#16837146—21no—
18#16837136—22no—
19Q8⋊(C59⋊C8)—21no—
20D4⋊(C59⋊C8)—21no—
21C2×2−1+4×C59—21no—
22D4×Q8×C59—21no—
23D4×Q8×C59C412yescomplemented
24C59×#547—22no—
25C59×#547—21no—
26C59×#547—22no—
27C59×#547C412yescomplemented
28C59×#536—22no—
29C59×#536—21no—
30C59×#532—21no—
31C59×#532C2211yescharacteristic
32C2×C4⋊Q8×C59—21no—
33C2×C4⋊Q8×C59C2211yescharacteristic
34C4×#41×C59C2211yescharacteristic
35C2×#11745664—21no—
36C22×C59⋊C8—21no—
37C2×C4⋊Dic59—21no—
38C2×C4×Dic59—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44#14273654—42no—
45C22⋊(C59⋊C8)—21no—
46C4×C59⋊C8—21no—
47C59⋊#95—21no—
482−1+4×C59—42no—
492−1+4×C59—42no—
502−1+4×C59—42no—
512−1+4×C59—42no—
52C2×#41×C59—22no—
53C2×#41×C59—24no—
54C2×#41×C59—22no—
55C2×#41×C59—21no—
56C2×#41×C59D411yescharacteristic
57C22×Q8×C59—22no—
58C22×Q8×C59—22no—
59C22×Q8×C59—21no—
60C4⋊Q8×C59—22no—
61C4⋊Q8×C59—22no—
62C4⋊Q8×C59—22no—
63C4⋊Q8×C59—21no—
64C4⋊Q8×C59—21no—
65C4⋊Q8×C59C2×C412yes—
66C4⋊Q8×C59D411yescharacteristic
67C4⋊Q8×C59D412yes—
68#127×C59D411yescharacteristic
69#124×C59—22no—
70#124×C59—21no—
71#123×C59—24no—
72#123×C59—24no—
73#123×C59—24no—
74C22⋊Q8×C59—24no—
75C22⋊Q8×C59—22no—
76C22⋊Q8×C59—24no—
77C22⋊Q8×C59—22no—
78#121×C59—22no—
79#121×C59—22no—
80#121×C59—21no—
81#121×C59D412yes—
82C4×Q8×C59C2×C411yescharacteristic
83C4×Q8×C59D411yescharacteristic
84C4×D4×C59—22no—
85C4×D4×C59—22no—
86C4×D4×C59—21no—
87C4×D4×C59—21no—
88C4×D4×C59—21no—
89C4×D4×C59C2×C411yescharacteristic
90C4×D4×C59C2×C412yes—
91C4×D4×C59D411yescharacteristic
92#117×C59—21no—
93#117×C59D411yescharacteristic
94C2×C4⋊C4×C59—22no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59—22no—
97C2×C4⋊C4×C59D412yes—
98C2×C42×C59—21no—
99C2×C42×C59C2311yescharacteristic
100#41×C59—21no—
101#41×C59—21no—
102#41×C59—44no—
103#41×C59—44no—
104#41×C59—48no—
105#41×C59—48no—
106#41×C59—44no—
107#41×C59—44no—
108#41×C59—41no—
109#41×C59—41no—
110#41×C59—41no—
111#41×C59—41no—
112#41×C59D812yes—
113#41×C59SD1612yes—
114C2×Q8×C59—24no—
115C2×Q8×C59—22no—
116C2×Q8×C59—22no—
117C2×Q8×C59—21no—
118C2×Q8×C59—44no—
119C2×Q8×C59—44no—
120C2×Q8×C59—44no—
121C2×Q8×C59—44no—
122C2×Q8×C59—41no—
123C2×Q8×C59—41no—
124C2×Q8×C59—41no—
125C2×Q8×C59—41no—
126C2×Q8×C59C22⋊C411yescharacteristic
127C2×D4×C59—21no—
128C2×D4×C59—24no—
129C2×D4×C59—22no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59C22⋊C411yescharacteristic
133C2×D4×C59C22⋊C412yes—
134C22×C4×C59—21no—
135C22×C4×C59—24no—
136C22×C4×C59—24no—
137C22×C4×C59—22no—
138C22×C4×C59—21no—
139C22×C4×C59—22no—
140C22×C4×C59—21no—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×Dic59—41no—
145C4⋊C4×C59—24no—
146C4⋊C4×C59—24no—
147C4⋊C4×C59—22no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—22no—
151C4⋊C4×C59—22no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59C22⋊C412yes—
154C4⋊C4×C59C22⋊C414yes—
155C4⋊C4×C59C22⋊C411yescharacteristic
156C4⋊C4×C59C22⋊C411yescharacteristic
157C22⋊C4×C59—24no—
158C22⋊C4×C59—24no—
159C22⋊C4×C59—21no—
160C22⋊C4×C59—21no—
161C22⋊C4×C59—22no—
162C22⋊C4×C59—21no—
163C22⋊C4×C59—22no—
164C42×C59—21no—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59C22×C411yescharacteristic
168C42×C59C2×D411yescharacteristic
169C42×C59C2×D411yescharacteristic
170C42×C59C2×D411yescharacteristic
171C4⋊Dic59—22no—
172C4⋊Dic59—41no—
173C4×Dic59—41no—
174C4×Dic59—41no—
175#11745664—41no—
176#11745664—41no—
177C2×C59⋊C8—42no—
178C2×C59⋊C8—41no—
179C2×C59⋊C8—41no—
180C2×C59⋊C8—41no—
181C23×C59—21no—
182C23×C59—22no—
183C23×C59#12011yescharacteristic
184Q8×C59—21no—
185Q8×C59—48no—
186Q8×C59—42no—
187Q8×C59—42no—
188Q8×C59—41no—
189Q8×C59—42no—
190Q8×C59—41no—
191Q8×C59—42no—
192Q8×C59#10212yes—
193D4×C59—22no—
194D4×C59—22no—
195D4×C59—21no—
196D4×C59—42no—
197D4×C59—48no—
198D4×C59—42no—
199D4×C59—42no—
200D4×C59—41no—
201D4×C59—41no—
202D4×C59—41no—
203D4×C59—41no—
204D4×C59#10212yes—
205C2×C4×C59—21no—
206C2×C4×C59—22no—
207C2×C4×C59—22no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—24no—
213C2×C4×C59—22no—
214C2×C4×C59—41no—
215C2×C4×C59—42no—
216C2×C4×C59—48no—
217C2×C4×C59—48no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—42no—
221C2×C4×C59—42no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59C2×C22⋊C411yescharacteristic
225C2×C4×C59C2×C22⋊C411yescharacteristic
226C2×C4×C59C2×C22⋊C411yescharacteristic
227C2×C4×C59#12011yescharacteristic
228C2×C4×C59#12011yescharacteristic
229C2×C4×C59#12011yescharacteristic
230C2×C4×C59C2×D811yescharacteristic
231C2×C4×C59C2×SD1611yescharacteristic
232C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
233C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
234C2×Dic59—42no—
235C2×Dic59—81no—
236C2×Dic59—81no—
237C59⋊C8—81no—
238C59⋊C8—81no—
239C59⋊C8—81no—
240#8421—591nomaximal, Sylow
241C22×C59—21no—
242C22×C59—21no—
243C22×C59—22no—
244C22×C59—41no—
245C22×C59—41no—
246C22×C59—42no—
247C22×C59—42no—
248C22×C59—42no—
249C22×C59#37911yescharacteristic
250C22×C59#44711yescharacteristic
251C22×C59#45011yescharacteristic
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—21no—
255C4×C59—22no—
256C4×C59—41no—
257C4×C59—41no—
258C4×C59—44no—
259C4×C59—42no—
260C4×C59C2×#10211yescharacteristic
261C4×C59#41811yescharacteristic
262C4×C59#44711yescharacteristic
263C4×C59#45011yescharacteristic
264C4×C59#45411yescharacteristic
265C4×C59#45611yescharacteristic
266Dic59—82no—
267#3499D5911yescharacteristic
268#1988—591no—
269#1806—591no—
270#1547—591no—
271#1546—591no—
272#1502—591no—
273#1477—591no—
274C2×C59—21no—
275C2×C59—21no—
276C2×C59—42no—
277C2×C59—82no—
278C2×C59#178611yescharacteristic
279C2×C59#179311yescharacteristic
280C2×C59#179611yescharacteristic
281C2×2−1+4—21no—
282D4×Q8—21no—
283D4×Q8Dic5912yescomplemented
284#547—22no—
285#547—21no—
286#547—22no—
287#547Dic5912yescomplemented
288#536—22no—
289#536—21no—
290#532—21no—
291#532C2×D5911yescharacteristic
292C2×C4⋊Q8—21no—
293C2×C4⋊Q8C2×D5911yescharacteristic
294C4×#41C2×D5911yescharacteristic
295#431—591no—
296#423—591no—
297C2×C4⋊C8—591no—
298C2×#104—1181no—
299C4×C4⋊C4—591no—
300#340—1182no—
301#339—1181no—
302#332—1182no—
303#332—1181no—
304#331—1181no—
305#327—1182no—
306Q8⋊C8—1181no—
307D4⋊C8—1181no—
308C59#842111yescharacteristic, complemented, Sylow
3092−1+4—42no—
3102−1+4—42no—
3112−1+4—42no—
3122−1+4—42no—
313C2×#41—22no—
314C2×#41—24no—
315C2×#41—22no—
316C2×#41—21no—
317C2×#41C22⋊D5911yescharacteristic
318C22×Q8—22no—
319C22×Q8—22no—
320C22×Q8—21no—
321C2×M16—1181no—
322C22×C8—1181no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8—22no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8C2×Dic5912yes—
329C4⋊Q8C22⋊D5911yescharacteristic
330C4⋊Q8C22⋊D5912yes—
331#127C22⋊D5911yescharacteristic
332#124—22no—
333#124—21no—
334#123—24no—
335#123—24no—
336#123—24no—
337C22⋊Q8—24no—
338C22⋊Q8—22no—
339C22⋊Q8—24no—
340C22⋊Q8—22no—
341#121—22no—
342#121—22no—
343#121—21no—
344#121C22⋊D5912yes—
345C4×Q8C2×Dic5911yescharacteristic
346C4×Q8C22⋊D5911yescharacteristic
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4C2×Dic5911yescharacteristic
353C4×D4C2×Dic5912yes—
354C4×D4C22⋊D5911yescharacteristic
355#117—21no—
356#117C22⋊D5911yescharacteristic
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—1181no—
361C2×C4⋊C4C22⋊D5912yes—
362C2×C42—21no—
363C2×C42—1181no—
364C2×C42C22×D5911yescharacteristic
365C4⋊C8—1181no—
366C4⋊C8—1181no—
367C4⋊C8—1181no—
368C4⋊C8—1181no—
369C4⋊C8—1181no—
370#104—2362no—
371C22⋊C8—1181no—
372C4×C8—1181no—
373#95—1181no—
374#41—21no—
375#41—21no—
376#41—44no—
377#41—44no—
378#41—48no—
379#41—48no—
380#41—44no—
381#41—44no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41#1174566912yes—
387#41#1174567012yes—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8C22⋊Dic5911yescharacteristic
401C2×D4—21no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C22⋊Dic5911yescharacteristic
407C2×D4C22⋊Dic5912yes—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—2361no—
416C22×C4C2×C22⋊D5911yescharacteristic
417C22×C4C2×C22⋊D5911yescharacteristic
418C22×C4C2×C22⋊D5911yescharacteristic
419M16—2361no—
420M16—2361no—
421C2×C8—2362no—
422C2×C8—2361no—
423C2×C8—2361no—
424C2×C8—2361no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—1182no—
434C4⋊C4—2361no—
435C4⋊C4C22⋊Dic5912yes—
436C4⋊C4C22⋊Dic5914yes—
437C4⋊C4C22⋊Dic5911yescharacteristic
438C4⋊C4C22⋊Dic5911yescharacteristic
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C22⋊C4—21no—
445C22⋊C4—22no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—2361no—
450C42—2361no—
451C42C22×Dic5911yescharacteristic
452C42C2×C22⋊D5911yescharacteristic
453C42C2×C22⋊D5911yescharacteristic
454C42C2×C22⋊D5911yescharacteristic
455C23—21no—
456C23—22no—
457C23#1427369611yescharacteristic
458Q8—21no—
459Q8—48no—
460Q8—42no—
461Q8—42no—
462Q8—41no—
463Q8—42no—
464Q8—41no—
465Q8—42no—
466Q8#1427364812yes—
467D4—22no—
468D4—22no—
469D4—21no—
470D4—42no—
471D4—48no—
472D4—42no—
473D4—42no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4#1427364812yes—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—4721no—
499C2×C4—4721no—
500C2×C4—2362no—
501C2×C4C2×C22⋊Dic5911yescharacteristic
502C2×C4C2×C22⋊Dic5911yescharacteristic
503C2×C4C2×C22⋊Dic5911yescharacteristic
504C2×C4#1427369611yescharacteristic
505C2×C4#1427369611yescharacteristic
506C2×C4#1427369611yescharacteristic
507C2×C4C2×#1174566911yescharacteristic
508C2×C4C2×#1174567011yescharacteristic
509C2×C4#1427375311yescharacteristic
510C2×C4#1427375311yescharacteristic
511C8—4721no—
512C8—4721no—
513C8—4721no—
514C22—21no—
515C22—21no—
516C22—22no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22#1683725811yescharacteristic
523C22#1683753611yescharacteristic
524C22#1683755311yescharacteristic
525C4—22no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—44no—
530C4—42no—
531C4—41no—
532C4—41no—
533C4—4722no—
534C4C2×#1427364811yescharacteristic
535C4#1683740711yescharacteristic
536C4#1683753611yescharacteristic
537C4#1683755311yescharacteristic
538C4#1683757511yescharacteristic
539C4#1683758511yescharacteristic
540C2—21no—
541C2—21no—
542C2—42no—
543C2—82no—
544C2#2142402711yescharacteristic
545C2#2142405511yescharacteristic
546C2#2142407111yescharacteristic
subgroupG/H
centerC22#16837258
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3499C2
socleC22×C59#379
cosocleC2×C42×C59C23
hypercenter#3499D59
hypocenterC59#8421
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #21425049C2semidirect
2. #21424116C2semidirect
3. #21422773C2semidirect
4. #21422768C2semidirect
5. #21422591C2semidirect
6. #21422483C2semidirect
23. D4×Q8×C59C4semidirect
27. C59×#547C4semidirect
283. D4×Q8Dic59semidirect
287. #547Dic59semidirect
308. C59#8421semidirect

Maximal subgroups

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

subgroupindex
#214250492
#214241162
#214227732
#214227682
#214225912
#214224832
C59×#34992
#842159