Finite Group Databasebeta

← #28567527

Subgroups — #28567527

Order 15104.

1170 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435247C211yesmaximal, characteristic, complemented
2#21434104C211yesmaximal, characteristic, complemented
3#21433893C211yesmaximal, characteristic, complemented
4#21423331C214yesmaximal, complemented
5#21423247C214yesmaximal, complemented
6#21422781C211yesmaximal, characteristic, complemented
7#21422521C211yesmaximal, characteristic, complemented
8#21422487C211yesmaximal, characteristic, complemented
9C2×C59×#547C211yesmaximal, characteristic
10#16837959C2211yescharacteristic
11#16837805C2212yescomplemented
12#16837754C2212yescomplemented
13#16837737—22no—
14#16837732—21no—
15#16837702—22no—
16#16837694—22no—
17#16837692C2212yescomplemented
18#16837692C2212yescomplemented
19#16837682C2212yescomplemented
20#16837607C2212yescomplemented
21#16837553—21no—
22C4×#11745670—22no—
23#16837455C2211yescharacteristic
24#16837455C2211yescharacteristic
25#16837442—21no—
26#16837426C2211yescharacteristic
27#16837407—21no—
28#16837148C2214yescomplemented
29D4⋊(C59⋊C8)C2214yescomplemented
30D4⋊(C59⋊C8)C2214yescomplemented
31C22×#41×C59—21no—
32C59×#547—22no—
33C59×#547—22no—
34C59×#547—22no—
35C59×#547C2214yes—
36C2×C4⋊Q8×C59C2211yescharacteristic
37C2×#123×C59—22no—
38C2×C22⋊Q8×C59—21no—
39C2×#121×C59—21no—
40C2×C4×D4×C59C2211yescharacteristic
41C2×C4×D4×C59C2211yescharacteristic
42C22×C4⋊C4×C59—21no—
43C2×#11745670—22no—
44C2×#11745664—21no—
45#14273727C2311yescharacteristic
46#14273727C2311yescharacteristic
47#14273719—21no—
48#14273706—21no—
49#14273706—21no—
50C4×Dic118—21no—
51C4×Dic118—21no—
52#14273686—21no—
53#14273660—22no—
54#14273660—24no—
55#14273658—22no—
56#14273658—22no—
57C4⋊(C59⋊C8)—21no—
58C4⋊(C59⋊C8)—21no—
59C4⋊(C59⋊C8)C2312yes—
60C4⋊(C59⋊C8)C2312yes—
61#14273650—24no—
62#14273650—22no—
63#14273648—22no—
64#14273648—22no—
65#14273648—22no—
66C22⋊(C59⋊C8)—21no—
67C22⋊(C59⋊C8)—21no—
68C4×C59⋊C8—21no—
69C4×C59⋊C8C2312yes—
70C2×#41×C59—24no—
71C2×#41×C59—24no—
72C2×#41×C59—48no—
73C22×Q8×C59—21no—
74C22×D4×C59—22no—
75C22×D4×C59D411yescharacteristic
76C22×D4×C59D411yescharacteristic
77C23×C4×C59—22no—
78C23×C4×C59—21no—
79C23×C4×C59—21no—
80C4⋊Q8×C59—21no—
81C4⋊Q8×C59—21no—
82C4⋊Q8×C59D412yes—
83C4⋊Q8×C59C2312yes—
84#123×C59—28no—
85#123×C59—28no—
86C22⋊Q8×C59—24no—
87C22⋊Q8×C59—24no—
88#121×C59—24no—
89#121×C59—24no—
90C4×D4×C59—21no—
91C4×D4×C59—21no—
92C4×D4×C59—21no—
93C4×D4×C59—21no—
94C4×D4×C59D412yes—
95C4×D4×C59D412yes—
96C4×D4×C59C2312yes—
97C4×D4×C59C2312yes—
98C2×C4⋊C4×C59—22no—
99C2×C4⋊C4×C59—24no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—24no—
102C2×C4⋊C4×C59D411yescharacteristic
103C2×C4⋊C4×C59D411yescharacteristic
104C2×C4⋊C4×C59D412yes—
105C2×C22⋊C4×C59—22no—
106C2×C22⋊C4×C59—21no—
107C2×C22⋊C4×C59—21no—
108C2×C42×C59C2311yescharacteristic
109C24×C59—21no—
110C24×C59—21no—
111#41×C59—416no—
112#41×C59—48no—
113#41×C59—48no—
114C2×Q8×C59—22no—
115C2×Q8×C59—22no—
116C2×Q8×C59—44no—
117C2×D4×C59—24no—
118C2×D4×C59—28no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—22no—
122C2×D4×C59—22no—
123C2×D4×C59—48no—
124C2×D4×C59C2×D412yes—
125C2×D4×C59C2×D412yes—
126C22×C4×C59—28no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—44no—
135C22×C4×C59—48no—
136C22×C4×C59—42no—
137C22×C4×C59—42no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C2×Dic118—21no—
142C2×Dic118—21no—
143C4⋊C4×C59—28no—
144C4⋊C4×C59—21no—
145C4⋊C4×C59—22no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—22no—
148C4⋊C4×C59—22no—
149C4⋊C4×C59C2×D412yes—
150C4⋊C4×C59C2×D412yes—
151C4⋊C4×C59C2×D414yes—
152C22⋊C4×C59—28no—
153C22⋊C4×C59—22no—
154C22⋊C4×C59—22no—
155C22⋊C4×C59—22no—
156C22⋊C4×C59—22no—
157C42×C59C2×D411yescharacteristic
158C42×C59C2×D411yescharacteristic
159C42×C59C2×D411yescharacteristic
160C42×C59C2411yescharacteristic
161C22⋊Dic59—41no—
162#11745670—44no—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167Dic59⋊C4—41no—
168Dic59⋊C4—41no—
169C4×Dic59—41no—
170#11745664—42no—
171C2×C59⋊C8—22no—
172C2×C59⋊C8—22no—
173C2×C59⋊C8—22no—
174C23×C59—24no—
175C23×C59—22no—
176C23×C59—22no—
177C23×C59—42no—
178C23×C59—42no—
179C23×C59—42no—
180C23×C59—42no—
181C23×C59—42no—
182C23×C59#12011yescharacteristic
183Q8×C59—44no—
184Q8×C59—44no—
185D4×C59—24no—
186D4×C59—22no—
187D4×C59—22no—
188D4×C59—22no—
189D4×C59—416no—
190D4×C59—42no—
191D4×C59AGL(1,ℤ/8)14yes—
192C2×C4×C59—22no—
193C2×C4×C59—22no—
194C2×C4×C59—22no—
195C2×C4×C59—24no—
196C2×C4×C59—41no—
197C2×C4×C59—44no—
198C2×C4×C59—42no—
199C2×C4×C59—44no—
200C2×C4×C59—416no—
201C2×C4×C59—42no—
202C2×C4×C59—42no—
203C2×C4×C59—44no—
204C2×C4×C59—42no—
205C2×C4×C59—44no—
206C2×C4×C59#12011yescharacteristic
207C2×C4×C59#12011yescharacteristic
208C2×C4×C59#12011yescharacteristic
209C2×C4×C59#13512yes—
210C2×C4×C59#13712yes—
211C2×C4×C59C22×D411yescharacteristic
212C2×C4×C59C22×D411yescharacteristic
213C2×C4×C59C22×D411yescharacteristic
214C2×Dic59—41no—
215C2×Dic59—41no—
216Dic118—42no—
217Dic118—42no—
218C59⋊C8—42no—
219C59⋊C8—42no—
220C59⋊C8—42no—
221#18783—591nomaximal, Sylow
222C22×C59—21no—
223C22×C59—21no—
224C22×C59—24no—
225C22×C59—48no—
226C22×C59—42no—
227C22×C59—42no—
228C22×C59—42no—
229C22×C59—42no—
230C22×C59—42no—
231C22×C59—84no—
232C22×C59#45212yes—
233C22×C59C2×#12011yescharacteristic
234C4×C59—21no—
235C4×C59—21no—
236C4×C59—21no—
237C4×C59—42no—
238C4×C59—42no—
239C4×C59—44no—
240C4×C59#45212yes—
241C4×C59C2×#13511yescharacteristic
242C4×C59C2×AGL(1,ℤ/8)12yes—
243C4×C59C2×#13711yescharacteristic
244Dic59—81no—
245Dic59—81no—
246C2×#547D5911yescharacteristic
247#3230—591no—
248#3101—591no—
249#3077—591no—
250#1647—594no—
251#1635—594no—
252#1549—591no—
253#1486—591no—
254#1478—591no—
255C2×C59—22no—
256C2×C59—44no—
257C2×C59—82no—
258C2×C59C2×#45211yescharacteristic
259C2×C59#300311yescharacteristic
260C2×C59#302011yescharacteristic
261C22×#41—21no—
262#547—22no—
263#547—22no—
264#547—22no—
265#547C2×D5914yes—
266#533—591no—
267C2×C4⋊Q8C2×D5911yescharacteristic
268C2×#123—22no—
269C2×C22⋊Q8—21no—
270C2×#121—21no—
271C2×C4×D4C2×D5911yescharacteristic
272C2×C4×D4C2×D5911yescharacteristic
273C22×C4⋊C4—21no—
274#500—592no—
275#487—592no—
276#484—1182no—
277#483—1181no—
278#478—1182no—
279#477—1182no—
280#476—592no—
281#476—592no—
282#474—592no—
283#461—592no—
284#450—1181no—
285C4×SD16—1182no—
286#431—591no—
287#431—591no—
288#428—1181no—
289#423—591no—
290#418—1181no—
291#332—594no—
292D4⋊C8—594no—
293D4⋊C8—594no—
294C59#1878311yescharacteristic, complemented, Sylow
295C2×#41—24no—
296C2×#41—24no—
297C2×#41—48no—
298C22×Q8—21no—
299C22×D4—22no—
300C22×D4C22⋊D5911yescharacteristic
301C22×D4C22⋊D5911yescharacteristic
302C23×C4—22no—
303C23×C4—21no—
304C23×C4—21no—
305C2×SD16—1182no—
306C2×M16—1181no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—591no—
310C4⋊Q8—591no—
311C4⋊Q8C22⋊D5912yes—
312C4⋊Q8C22×D5912yes—
313#125—1181no—
314#123—28no—
315#123—28no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—1181no—
319C22⋊Q8—1181no—
320#121—24no—
321#121—24no—
322C4×Q8—1181no—
323C4×Q8—1181no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4C22⋊D5912yes—
329C4×D4C22⋊D5912yes—
330C4×D4C22×D5912yes—
331C4×D4C22×D5912yes—
332#117—1181no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4C22⋊D5911yescharacteristic
338C2×C4⋊C4C22⋊D5911yescharacteristic
339C2×C4⋊C4C22⋊D5912yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C42C22×D5911yescharacteristic
344#107—1182no—
345#107—1184no—
346#106—1182no—
347#106—1182no—
348C4⋊C8—592no—
349C4⋊C8—592no—
350C4⋊C8—1181no—
351C4⋊C8—1181no—
352#103—1182no—
353#103—1184no—
354#102—1182no—
355#102—1182no—
356#102—1182no—
357C22⋊C8—1181no—
358C22⋊C8—1181no—
359C4×C8—592no—
360C4×C8—1181no—
361C24—21no—
362C24—21no—
363#41—416no—
364#41—48no—
365#41—48no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—44no—
369C2×Q8—1181no—
370C2×Q8—1181no—
371C2×D4—24no—
372C2×D4—28no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—48no—
378C2×D4C2×C22⋊D5912yes—
379C2×D4C2×C22⋊D5912yes—
380C22×C4—28no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C2×C22⋊D5911yescharacteristic
393C22×C4C2×C22⋊D5911yescharacteristic
394C22×C4C2×C22⋊D5911yescharacteristic
395SD16—2364no—
396M16—2362no—
397C2×C8—1182no—
398C2×C8—1182no—
399C2×C8—1182no—
400C4⋊C4—28no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—1181no—
407C4⋊C4—1181no—
408C4⋊C4—1181no—
409C4⋊C4—1181no—
410C4⋊C4—2361no—
411C4⋊C4—2361no—
412C4⋊C4C2×C22⋊D5912yes—
413C4⋊C4C2×C22⋊D5914yes—
414C4⋊C4C2×C22⋊D5912yes—
415C22⋊C4—28no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—2361no—
421C42—2361no—
422C42C2×C22⋊D5911yescharacteristic
423C42C2×C22⋊D5911yescharacteristic
424C42C2×C22⋊D5911yescharacteristic
425C42C23×D5911yescharacteristic
426C23—24no—
427C23—22no—
428C23—22no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23#1427369611yescharacteristic
435Q8—44no—
436Q8—44no—
437Q8—2362no—
438Q8—2362no—
439D4—24no—
440D4—22no—
441D4—22no—
442D4—22no—
443D4—416no—
444D4—42no—
445D4#1427375314yes—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—416no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—2361no—
461C2×C4—2361no—
462C2×C4#1427369611yescharacteristic
463C2×C4#1427369611yescharacteristic
464C2×C4#1427369611yescharacteristic
465C2×C4#1427374712yes—
466C2×C4#1427375712yes—
467C2×C4C22×C22⋊D5911yescharacteristic
468C2×C4C22×C22⋊D5911yescharacteristic
469C2×C4C22×C22⋊D5911yescharacteristic
470C8—2362no—
471C8—2362no—
472C8—2362no—
473C22—21no—
474C22—21no—
475C22—24no—
476C22—48no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—84no—
483C22#1683756412yes—
484C22C2×#1427369611yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—44no—
491C4—4721no—
492C4—4721no—
493C4#1683756412yes—
494C4C2×#1427374711yescharacteristic
495C4C2×#1427375312yes—
496C4C2×#1427375711yescharacteristic
497C2—22no—
498C2—44no—
499C2—82no—
500C2C2×#1683756411yescharacteristic
501C2#2143332311yescharacteristic
502C2#2143346011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
Fitting, prime radicalC2×C59×#547C2
socleC22×C59C2×#120
hypercenterC2×#547D59
hypocenterC59#18783
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #21435247C2semidirect
2. #21434104C2semidirect
3. #21433893C2semidirect
4. #21423331C2semidirect
5. #21423247C2semidirect
6. #21422781C2semidirect
7. #21422521C2semidirect
8. #21422487C2semidirect
11. #16837805C22semidirect
12. #16837754C22semidirect
17. #16837692C22semidirect
18. #16837692C22semidirect
19. #16837682C22semidirect
20. #16837607C22semidirect
28. #16837148C22semidirect
29. D4⋊(C59⋊C8)C22semidirect
30. D4⋊(C59⋊C8)C22semidirect
294. C59#18783semidirect

Maximal subgroups

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

subgroupindex
#214352472
#214341042
#214338932
#214233312
#214232472
#214227812
#214225212
#214224872
C2×C59×#5472
#1878359