Finite Group Databasebeta

← #28567269

Subgroups — #28567269

Order 15104.

1170 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435240C211yesmaximal, characteristic, complemented
2#21434104C211yesmaximal, characteristic, complemented
3#21433893C211yesmaximal, characteristic, complemented
4#21423380C214yesmaximal, complemented
5#21423247C214yesmaximal, complemented
6#21422768C211yesmaximal, characteristic, complemented
7#21422521C211yesmaximal, characteristic, complemented
8#21422487C211yesmaximal, characteristic, complemented
9C2×C59×#547C211yesmaximal, characteristic
10#16837959C2211yescharacteristic
11#16837798C2212yescomplemented
12#16837754C2212yescomplemented
13#16837732—22no—
14#16837732—21no—
15#16837702—22no—
16#16837692C2212yescomplemented
17#16837692C2212yescomplemented
18#16837684—22no—
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#14273658—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—44no—
73C2×#41×C59—44no—
74C22×Q8×C59—21no—
75C22×D4×C59—22no—
76C22×D4×C59D411yescharacteristic
77C22×D4×C59D411yescharacteristic
78C23×C4×C59—22no—
79C23×C4×C59—21no—
80C23×C4×C59—21no—
81C4⋊Q8×C59—21no—
82C4⋊Q8×C59—21no—
83C4⋊Q8×C59D412yes—
84C4⋊Q8×C59C2312yes—
85#123×C59—28no—
86#123×C59—28no—
87C22⋊Q8×C59—24no—
88C22⋊Q8×C59—24no—
89#121×C59—24no—
90#121×C59—24no—
91C4×D4×C59—21no—
92C4×D4×C59—21no—
93C4×D4×C59—21no—
94C4×D4×C59—21no—
95C4×D4×C59D412yes—
96C4×D4×C59D412yes—
97C4×D4×C59C2312yes—
98C4×D4×C59C2312yes—
99C2×C4⋊C4×C59—22no—
100C2×C4⋊C4×C59—24no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—24no—
103C2×C4⋊C4×C59D411yescharacteristic
104C2×C4⋊C4×C59D411yescharacteristic
105C2×C4⋊C4×C59D412yes—
106C2×C22⋊C4×C59—22no—
107C2×C22⋊C4×C59—21no—
108C2×C22⋊C4×C59—21no—
109C2×C42×C59C2311yescharacteristic
110C24×C59—21no—
111C24×C59—21no—
112#41×C59—48no—
113#41×C59—416no—
114#41×C59—48no—
115C2×Q8×C59—22no—
116C2×Q8×C59—22no—
117C2×Q8×C59—42no—
118C2×Q8×C59—42no—
119C2×D4×C59—28no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—22no—
123C2×D4×C59—42no—
124C2×D4×C59—48no—
125C2×D4×C59SD1614yescomplemented
126C2×D4×C59C2×D412yes—
127C2×D4×C59C2×D412yes—
128C22×C4×C59—28no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—44no—
137C22×C4×C59—48no—
138C22×C4×C59—42no—
139C22×C4×C59—42no—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C2×Dic118—21no—
144C2×Dic118—21no—
145C4⋊C4×C59—28no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—22no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—22no—
150C4⋊C4×C59—22no—
151C4⋊C4×C59C2×D412yes—
152C4⋊C4×C59C2×D412yes—
153C4⋊C4×C59C2×D414yes—
154C22⋊C4×C59—28no—
155C22⋊C4×C59—22no—
156C22⋊C4×C59—22no—
157C22⋊C4×C59—22no—
158C22⋊C4×C59—22no—
159C42×C59C2×D411yescharacteristic
160C42×C59C2×D411yescharacteristic
161C42×C59C2×D411yescharacteristic
162C42×C59C2411yescharacteristic
163C22⋊Dic59—41no—
164#11745670—44no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169Dic59⋊C4—41no—
170Dic59⋊C4—41no—
171C4×Dic59—41no—
172#11745664—42no—
173C2×C59⋊C8—22no—
174C2×C59⋊C8—22no—
175C2×C59⋊C8—22no—
176C23×C59—22no—
177C23×C59—24no—
178C23×C59—22no—
179C23×C59—42no—
180C23×C59—42no—
181C23×C59—42no—
182C23×C59—42no—
183C23×C59—42no—
184C23×C59#12011yescharacteristic
185Q8×C59—42no—
186Q8×C59—42no—
187Q8×C59—44no—
188D4×C59—22no—
189D4×C59—22no—
190D4×C59—22no—
191D4×C59—24no—
192D4×C59—42no—
193D4×C59—416no—
194D4×C59C2×SD1614yes—
195C2×C4×C59—22no—
196C2×C4×C59—22no—
197C2×C4×C59—22no—
198C2×C4×C59—24no—
199C2×C4×C59—41no—
200C2×C4×C59—44no—
201C2×C4×C59—42no—
202C2×C4×C59—416no—
203C2×C4×C59—42no—
204C2×C4×C59—44no—
205C2×C4×C59—42no—
206C2×C4×C59—42no—
207C2×C4×C59—44no—
208C2×C4×C59—44no—
209C2×C4×C59#12011yescharacteristic
210C2×C4×C59#12011yescharacteristic
211C2×C4×C59#12011yescharacteristic
212C2×C4×C59C2×SD1612yes—
213C2×C4×C59AGL(1,ℤ/8)12yes—
214C2×C4×C59C22×D411yescharacteristic
215C2×C4×C59C22×D411yescharacteristic
216C2×C4×C59C22×D411yescharacteristic
217C2×Dic59—41no—
218C2×Dic59—41no—
219Dic118—42no—
220Dic118—42no—
221C59⋊C8—42no—
222C59⋊C8—42no—
223C59⋊C8—42no—
224#18757—591nomaximal, Sylow
225C22×C59—21no—
226C22×C59—21no—
227C22×C59—24no—
228C22×C59—48no—
229C22×C59—42no—
230C22×C59—42no—
231C22×C59—42no—
232C22×C59—42no—
233C22×C59—42no—
234C22×C59—84no—
235C22×C59#45012yes—
236C22×C59C2×#12011yescharacteristic
237C4×C59—21no—
238C4×C59—21no—
239C4×C59—21no—
240C4×C59—42no—
241C4×C59—44no—
242C4×C59—42no—
243C4×C59#45012yes—
244C4×C59C22×SD1611yescharacteristic
245C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C59#57512yes—
247Dic59—81no—
248Dic59—81no—
249C2×#547D5911yescharacteristic
250#3229—591no—
251#3101—591no—
252#3077—591no—
253#1654—594no—
254#1635—594no—
255#1546—591no—
256#1486—591no—
257#1478—591no—
258C2×C59—22no—
259C2×C59—44no—
260C2×C59—82no—
261C2×C59C2×#45011yescharacteristic
262C2×C59#300511yescharacteristic
263C2×C59#302011yescharacteristic
264C22×#41—21no—
265#547—22no—
266#547—22no—
267#547—22no—
268#547C2×D5914yes—
269#533—591no—
270C2×C4⋊Q8C2×D5911yescharacteristic
271C2×#123—22no—
272C2×C22⋊Q8—21no—
273C2×#121—21no—
274C2×C4×D4C2×D5911yescharacteristic
275C2×C4×D4C2×D5911yescharacteristic
276C22×C4⋊C4—21no—
277#498—592no—
278#487—592no—
279#483—1181no—
280#483—1182no—
281#478—1182no—
282#476—592no—
283#476—592no—
284#475—1182no—
285#474—592no—
286#461—592no—
287#450—1181no—
288C4×SD16—1182no—
289#431—591no—
290#431—591no—
291#428—1181no—
292#423—591no—
293#418—1181no—
294#332—594no—
295D4⋊C8—594no—
296D4⋊C8—594no—
297C59#1875711yescharacteristic, complemented, Sylow
298C2×#41—24no—
299C2×#41—24no—
300C2×#41—44no—
301C2×#41—44no—
302C22×Q8—21no—
303C22×D4—22no—
304C22×D4C22⋊D5911yescharacteristic
305C22×D4C22⋊D5911yescharacteristic
306C23×C4—22no—
307C23×C4—21no—
308C23×C4—21no—
309C2×SD16—1182no—
310C2×M16—1181no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—591no—
314C4⋊Q8—591no—
315C4⋊Q8C22⋊D5912yes—
316C4⋊Q8C22×D5912yes—
317#125—1181no—
318#123—28no—
319#123—28no—
320C22⋊Q8—24no—
321C22⋊Q8—24no—
322C22⋊Q8—1181no—
323C22⋊Q8—1181no—
324#121—24no—
325#121—24no—
326C4×Q8—1181no—
327C4×Q8—1181no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4C22⋊D5912yes—
333C4×D4C22⋊D5912yes—
334C4×D4C22×D5912yes—
335C4×D4C22×D5912yes—
336#117—1181no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4C22⋊D5911yescharacteristic
342C2×C4⋊C4C22⋊D5911yescharacteristic
343C2×C4⋊C4C22⋊D5912yes—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C42C22×D5911yescharacteristic
348#107—1182no—
349#106—1182no—
350#106—1182no—
351#106—1184no—
352C4⋊C8—592no—
353C4⋊C8—592no—
354C4⋊C8—1181no—
355C4⋊C8—1181no—
356#103—1182no—
357#103—1184no—
358#102—1182no—
359#102—1182no—
360#102—1182no—
361C22⋊C8—1181no—
362C22⋊C8—1181no—
363C4×C8—592no—
364C4×C8—1181no—
365C24—21no—
366C24—21no—
367#41—48no—
368#41—416no—
369#41—48no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—1181no—
375C2×Q8—1181no—
376C2×D4—28no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—48no—
382C2×D4#1174567014yescomplemented
383C2×D4C2×C22⋊D5912yes—
384C2×D4C2×C22⋊D5912yes—
385C22×C4—28no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4C2×C22⋊D5911yescharacteristic
398C22×C4C2×C22⋊D5911yescharacteristic
399C22×C4C2×C22⋊D5911yescharacteristic
400SD16—2364no—
401M16—2362no—
402C2×C8—1182no—
403C2×C8—1182no—
404C2×C8—1182no—
405C4⋊C4—28no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—1181no—
412C4⋊C4—1181no—
413C4⋊C4—1181no—
414C4⋊C4—1181no—
415C4⋊C4—2361no—
416C4⋊C4—2361no—
417C4⋊C4C2×C22⋊D5912yes—
418C4⋊C4C2×C22⋊D5914yes—
419C4⋊C4C2×C22⋊D5912yes—
420C22⋊C4—28no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—2361no—
426C42—2361no—
427C42C2×C22⋊D5911yescharacteristic
428C42C2×C22⋊D5911yescharacteristic
429C42C2×C22⋊D5911yescharacteristic
430C42C23×D5911yescharacteristic
431C23—22no—
432C23—24no—
433C23—22no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23#1427369611yescharacteristic
440Q8—42no—
441Q8—42no—
442Q8—44no—
443Q8—2362no—
444Q8—2362no—
445D4—22no—
446D4—22no—
447D4—22no—
448D4—24no—
449D4—42no—
450D4—416no—
451D4C2×#1174567014yes—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—416no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—2361no—
467C2×C4—2361no—
468C2×C4#1427369611yescharacteristic
469C2×C4#1427369611yescharacteristic
470C2×C4#1427369611yescharacteristic
471C2×C4C2×#1174567012yes—
472C2×C4#1427375312yes—
473C2×C4C22×C22⋊D5911yescharacteristic
474C2×C4C22×C22⋊D5911yescharacteristic
475C2×C4C22×C22⋊D5911yescharacteristic
476C8—2362no—
477C8—2362no—
478C8—2362no—
479C22—21no—
480C22—21no—
481C22—24no—
482C22—48no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22#1683755312yes—
490C22C2×#1427369611yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—44no—
496C4—42no—
497C4—4721no—
498C4—4721no—
499C4#1683755312yes—
500C4C22×#1174567011yescharacteristic
501C4C2×#1427375311yescharacteristic
502C4#1683820012yes—
503C2—22no—
504C2—44no—
505C2—82no—
506C2C2×#1683755311yescharacteristic
507C2#2143333811yescharacteristic
508C2#2143346011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
Fitting, prime radicalC2×C59×#547C2
socleC22×C59C2×#120
hypercenterC2×#547D59
hypocenterC59#18757
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. #21435240C2semidirect
2. #21434104C2semidirect
3. #21433893C2semidirect
4. #21423380C2semidirect
5. #21423247C2semidirect
6. #21422768C2semidirect
7. #21422521C2semidirect
8. #21422487C2semidirect
11. #16837798C22semidirect
12. #16837754C22semidirect
16. #16837692C22semidirect
17. #16837692C22semidirect
19. #16837682C22semidirect
20. #16837607C22semidirect
28. #16837148C22semidirect
29. D4⋊(C59⋊C8)C22semidirect
30. D4⋊(C59⋊C8)C22semidirect
125. C2×D4×C59SD16semidirect
297. C59#18757semidirect
382. C2×D4#11745670semidirect

Maximal subgroups

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

subgroupindex
#214352402
#214341042
#214338932
#214233802
#214232472
#214227682
#214225212
#214224872
C2×C59×#5472
#1875759