Finite Group Databasebeta

← #28559831

Subgroups — #28559831

Order 15104.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434165C211yesmaximal, characteristic
2#21433115C211yesmaximal, characteristic
3#21433044C211yesmaximal, characteristic, complemented
4#21422997C211yesmaximal, characteristic, complemented
5#21422986C211yesmaximal, characteristic, complemented
6#21422971C211yesmaximal, characteristic, complemented
7#21422964C211yesmaximal, characteristic, complemented
8#21422941C211yesmaximal, characteristic, complemented
9#21422915C211yesmaximal, characteristic, complemented
10C8×#11745672C211yesmaximal, characteristic, complemented
11#21422880C211yesmaximal, characteristic, complemented
12#21422567C211yesmaximal, characteristic, complemented
13#21422482C211yesmaximal, characteristic
14#21422394C211yesmaximal, characteristic, complemented
15C59×#1584C211yesmaximal, characteristic, complemented
16#16837956C2211yescharacteristic
17C4×#11745689C2211yescharacteristic
18#16837745C411yescharacteristic
19#16837741C411yescharacteristic
20#16837685C411yescharacteristic
21#16837681C411yescharacteristic
22#16837610C411yescharacteristic
23#16837606C411yescharacteristic
24#16837567C411yescharacteristic
25#16837563C411yescharacteristic
26#16837530—21no—
27#16837529—21no—
28#16837529C2211yescharacteristic
29Q8×C59⋊C8C2211yescharacteristic
30Q8×C59⋊C8C2211yescharacteristic
31C8×Dic118C2211yescharacteristic
32C4×#11745672C2211yescharacteristic
33C4×#11745660C2211yescharacteristic
34C4×#11745671C2211yescharacteristic
35#16837481—21no—
36#16837481C2211yescharacteristic
37#16837479—21no—
38#16837479C2211yescharacteristic
39#16837458—21no—
40#16837458C2211yescharacteristic
41#16837453C2211yescharacteristic
42#16837453C2211yescharacteristic
43#16837427—21no—
44C4⋊C8×D59C2211yescharacteristic
45#16837359—21no—
46C4×#11745659—21no—
47C4×#11745659C2211yescharacteristic
48#16837154C2211yescharacteristic
49#16837153C2211yescharacteristic
50#16837152C2211yescharacteristic
51#16837134C2211yescharacteristic
52Q8⋊(C59⋊C8)C2211yescharacteristic
53Q8⋊(C59⋊C8)C2211yescharacteristic
54#16837132C2211yescharacteristic
55#16837131C2211yescharacteristic
56#16837129C2211yescharacteristic
57#16837123C2211yescharacteristic
58#16837123C2211yescharacteristic
59C8×C59⋊C8C2211yescharacteristic
60C59×#446C2211yescharacteristic
61C8×Q8×C59C2211yescharacteristic
62C4×Q16×C59C2211yescharacteristic
63C59×#335C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×Q8⋊C8C2211yescharacteristic
66C59×#322C2211yescharacteristic
67C2×#11745689C2×C411yescharacteristic
68C2×#11745672C2×C411yescharacteristic
69C2×#11745660C2×C411yescharacteristic
70C2×#11745671C2×C411yescharacteristic
71C2×#11745659—21no—
72C2×#11745659—21no—
73C2×#11745659D411yescharacteristic
74C2×#11745659D411yescharacteristic
75C2×C8×D59—21no—
76#14273726C2×C411yescharacteristic
77#14273725C2×C411yescharacteristic
78#14273717—21no—
79#14273707C2×C411yescharacteristic
80#14273705—21no—
81#14273703C2×C411yescharacteristic
82Q8×Dic59C2×C411yescharacteristic
83C4×Dic118—21no—
84C4×Dic118C2311yescharacteristic
85C4×D236—21no—
86C4×D236C2311yescharacteristic
87#14273689—21no—
88#14273689C2×C411yescharacteristic
89#14273685—21no—
90#14273685C2×C411yescharacteristic
91#14273684—21no—
92C42×D59—21no—
93C42×D59C2311yescharacteristic
94#14273660C2×C411yescharacteristic
95#14273659C2×C411yescharacteristic
96#14273658C2×C411yescharacteristic
97C4⋊(C59⋊C8)—21no—
98C4⋊(C59⋊C8)—21no—
99C4⋊(C59⋊C8)C2311yescharacteristic
100C4⋊(C59⋊C8)C2311yescharacteristic
101Dic59⋊C8—21no—
102Dic59⋊C8—21no—
103Dic59⋊C8—21no—
104Dic59⋊C8—21no—
105Dic59⋊C8C2×C411yescharacteristic
106Dic59⋊C8C2×C411yescharacteristic
107#14273652C2×C411yescharacteristic
108#14273651C2×C411yescharacteristic
109#14273651C2×C411yescharacteristic
110#14273650C2×C411yescharacteristic
111#14273649C2×C411yescharacteristic
112#14273647C2×C411yescharacteristic
113(C2×D59)⋊C8—21no—
114(C2×D59)⋊C8—21no—
115(C2×D59)⋊C8—21no—
116(C2×D59)⋊C8—21no—
117(C2×D59)⋊C8C2×C411yescharacteristic
118(C2×D59)⋊C8C2×C411yescharacteristic
119#14273637—21no—
120#14273637—21no—
121#14273637D411yescharacteristic
122#14273637D411yescharacteristic
123C4×C59⋊C8—21no—
124C4×C59⋊C8—21no—
125C4×C59⋊C8C2311yescharacteristic
126C4×C59⋊C8C2311yescharacteristic
127C4×C59⋊C8C2311yescharacteristic
128C8×Dic59—21no—
129C2×Q16×C59C2×C411yescharacteristic
130C4×Q8×C59C2311yescharacteristic
131C4×Q8×C59C2311yescharacteristic
132#107×C59C2×C411yescharacteristic
133C4⋊C8×C59—21no—
134C4⋊C8×C59—21no—
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137#103×C59C2×C411yescharacteristic
138#103×C59C2×C411yescharacteristic
139#97×C59—21no—
140C4×C8×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59C22×C411yescharacteristic
145C2×Q8×C59C22×C411yescharacteristic
146#11745689—22no—
147#11745689—41no—
148C2×D236—21no—
149C2×D236C22×C411yescharacteristic
150C2×C4×D59—21no—
151C2×C4×D59—21no—
152C2×C4×D59—21no—
153C2×C4×D59C22×C411yescharacteristic
154C2×C4×D59C2×D411yescharacteristic
155C2×C4×D59#4111yescharacteristic
156C2×Dic118—21no—
157C2×Dic118C22×C411yescharacteristic
158Q16×C59—22no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C22×C411yescharacteristic
163C2×C8×C59C22×C411yescharacteristic
164C2×C8×C59C2×D411yescharacteristic
165C2×C8×C59C2×D411yescharacteristic
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59C22×C411yescharacteristic
169C4⋊C4×C59C22×C411yescharacteristic
170C42×C59—21no—
171C42×C59—21no—
172C42×C59C2411yescharacteristic
173#11745672—22no—
174#11745671—41no—
175(C2×D59)⋊C4—21no—
176(C2×D59)⋊C4—21no—
177(C2×D59)⋊C4—21no—
178(C2×D59)⋊C4—21no—
179(C2×D59)⋊C4—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59C22×C411yescharacteristic
183C4⋊Dic59C22×C411yescharacteristic
184Dic59⋊C4—21no—
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59C22×C411yescharacteristic
193C4×Dic59C2×D411yescharacteristic
194C4×Dic59#4111yescharacteristic
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C22×C411yescharacteristic
199C2×C59⋊C8C22×C411yescharacteristic
200C2×C59⋊C8C2×D411yescharacteristic
201C2×C59⋊C8C2×D411yescharacteristic
202#11745660—41no—
203#11745659—22no—
204#11745659—41no—
205#11745659—42no—
206#11745659—42no—
207C8×D59—42no—
208C22×D59—21no—
209C22×D59C4×D411yescharacteristic
210Q8×C59—21no—
211Q8×C59—21no—
212Q8×C59—21no—
213Q8×C59#13112yes—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59C23×C411yescharacteristic
219C2×C4×C59C22×D411yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×Dic59—21no—
222C2×Dic59—21no—
223C2×Dic59—21no—
224C2×Dic59C4×D411yescharacteristic
225C2×Dic59C4×D411yescharacteristic
226C2×Dic59C4×D411yescharacteristic
227D236—21no—
228D236—21no—
229D236—42no—
230C4×D59—21no—
231C4×D59—21no—
232C4×D59—21no—
233C4×D59—21no—
234C4×D59—41no—
235C4×D59—42no—
236C4×D59—41no—
237C4×D59—41no—
238C4×D59—81no—
239Dic118—21no—
240Dic118—21no—
241Dic118—81no—
242C8×C59—21no—
243C8×C59—21no—
244C8×C59—21no—
245C8×C59—41no—
246C8×C59—41no—
247C8×C59—41no—
248C8×C59#13712yes—
249C59⋊C8—21no—
250C59⋊C8—21no—
251C59⋊C8—21no—
252C59⋊C8—21no—
253C59⋊C8—41no—
254C59⋊C8—41no—
255C59⋊C8—41no—
256#18013—591nomaximal, Sylow
257C22×C59C2×C4×D411yescharacteristic
258C2×D59—21no—
259C2×D59—21no—
260C2×D59—21no—
261C2×D59—41no—
262C2×D59—41no—
263C2×D59—41no—
264C4×C59—22no—
265C4×C59—21no—
266C4×C59—41no—
267C4×C59—41no—
268C4×C59—41no—
269C4×C59C2×#13111yescharacteristic
270C4×C59#56811yescharacteristic
271C4×C59C2×#13711yescharacteristic
272C4×C59#57511yescharacteristic
273Dic59—21no—
274Dic59—21no—
275Dic59—21no—
276Dic59—41no—
277Dic59—41no—
278Dic59—41no—
279#3107—591no—
280#2971—591no—
281#2962—591no—
282#1592—591no—
283#1590—591no—
284#1588—591no—
285#1587—591no—
286#1584D5911yescharacteristic, complemented
287#1583—591no—
288#1579—591no—
289C8×Q16—591no—
290#1573—591no—
291#1497—591no—
292#1477—591no—
293#1453—591no—
294C2×C59#293311yescharacteristic
295C2×C59C4×#13711yescharacteristic
296C2×C59#295711yescharacteristic
297D59—41no—
298D59—81no—
299#533—591no—
300C4×#41—591no—
301#485—591no—
302#485—591no—
303#475—591no—
304#474—591no—
305#462—591no—
306#461—591no—
307#452—591no—
308#452—591no—
309#446—591no—
310#446—1181no—
311#446—1181no—
312#446C2×D5911yescharacteristic
313C8×Q8—591no—
314C8×Q8—591no—
315C8×Q8—591no—
316C8×Q8C2×D5911yescharacteristic
317C4×Q16—591no—
318C4×Q16C2×D5911yescharacteristic
319C4×SD16—591no—
320C4×SD16—591no—
321#436—591no—
322#436—591no—
323#436—1181no—
324#436—1181no—
325#432—591no—
326#432—1181no—
327#431—591no—
328#431—591no—
329#424—1181no—
330C2×C4⋊C8—591no—
331#405—1181no—
332C4×M16—591no—
333C4×M16—1181no—
334#335—591no—
335#335C2×D5911yescharacteristic
336#334—591no—
337#334—591no—
338Q8⋊C8—591no—
339Q8⋊C8—591no—
340Q8⋊C8—591no—
341Q8⋊C8—591no—
342Q8⋊C8C2×D5911yescharacteristic
343Q8⋊C8C2×D5911yescharacteristic
344D4⋊C8—591no—
345D4⋊C8—591no—
346#322—591no—
347#322—591no—
348#322C2×D5911yescharacteristic
349C82—591no—
350C59#1801311yescharacteristic, complemented, Sylow
351C2×#41—591no—
352C2×Q16—591no—
353C2×Q16C4×D5911yescharacteristic
354C2×SD16—591no—
355C2×SD16—591no—
356C2×M16—591no—
357C2×M16—591no—
358C2×M16—1181no—
359C2×M16—1181no—
360C22×C8—1181no—
361C4⋊Q8—591no—
362C4⋊Q8—591no—
363#125—1181no—
364C22⋊Q8—591no—
365C22⋊Q8—591no—
366C22⋊Q8—1181no—
367C4×Q8—591no—
368C4×Q8—591no—
369C4×Q8—1181no—
370C4×Q8C22×D5911yescharacteristic
371C4×Q8C22×D5911yescharacteristic
372C4×D4—591no—
373C4×D4—591no—
374C4×D4—1181no—
375C4×D4—1181no—
376#117—591no—
377#117—1181no—
378#117—1181no—
379C2×C42—591no—
380C2×C42—1181no—
381#107—591no—
382#107C4×D5911yescharacteristic
383#106—591no—
384#106—591no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—591no—
388C4⋊C8—591no—
389C4⋊C8—591no—
390C4⋊C8—591no—
391C4⋊C8—1181no—
392C4⋊C8—1181no—
393C4⋊C8—1181no—
394C4⋊C8—1181no—
395C4⋊C8—1181no—
396C4⋊C8—1181no—
397C4⋊C8C22×D5911yescharacteristic
398C4⋊C8C22×D5911yescharacteristic
399#103—591no—
400#103—591no—
401#103—591no—
402#103—591no—
403#103C4×D5911yescharacteristic
404#103C4×D5911yescharacteristic
405#102—591no—
406#102—591no—
407C22⋊C8—591no—
408C22⋊C8—591no—
409C22⋊C8—1181no—
410C22⋊C8—1181no—
411C22⋊C8—1181no—
412C22⋊C8—1181no—
413#97—21no—
414#97—591no—
415#97—591no—
416#97—1181no—
417#97—1181no—
418C4×C8—21no—
419C4×C8—591no—
420C4×C8—591no—
421C4×C8—591no—
422C4×C8—1181no—
423C4×C8—1181no—
424C4×C8—1181no—
425C4×C8C22×D5911yescharacteristic
426C4×C8C22×D5911yescharacteristic
427C4×C8C22×D5911yescharacteristic
428#41—1182no—
429#41—2361no—
430C2×Q8—591no—
431C2×Q8—1181no—
432C2×Q8C2×C4×D5911yescharacteristic
433C2×Q8C2×C4×D5911yescharacteristic
434C2×D4—591no—
435C2×D4—1181no—
436C22×C4—591no—
437C22×C4—591no—
438C22×C4—591no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1181no—
442Q16—22no—
443Q16—1182no—
444SD16—2361no—
445SD16—2361no—
446M16—1182no—
447M16—2361no—
448M16—2362no—
449M16—2362no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—591no—
454C2×C8—591no—
455C2×C8—591no—
456C2×C8—591no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—2362no—
461C2×C8C2×C4×D5911yescharacteristic
462C2×C8C2×C4×D5911yescharacteristic
463C2×C8D4×D5911yescharacteristic
464C2×C8D4×D5911yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—591no—
468C4⋊C4—591no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4C2×C4×D5911yescharacteristic
477C4⋊C4C2×C4×D5911yescharacteristic
478C22⋊C4—1181no—
479C22⋊C4—1181no—
480C22⋊C4—1181no—
481C22⋊C4—1181no—
482C22⋊C4—1181no—
483C42—21no—
484C42—21no—
485C42—591no—
486C42—591no—
487C42—591no—
488C42—1181no—
489C42—1181no—
490C42—1181no—
491C42C23×D5911yescharacteristic
492C23—591no—
493C23—1181no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—1181no—
498Q8—1181no—
499Q8—4721no—
500Q8#1427373512yes—
501D4—1181no—
502D4—1181no—
503D4—2362no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—591no—
509C2×C4—591no—
510C2×C4—591no—
511C2×C4—1181no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4—1181no—
517C2×C4—1181no—
518C2×C4—4721no—
519C2×C4—2361no—
520C2×C4—2362no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4C22×C4×D5911yescharacteristic
524C2×C4C2×D4×D5911yescharacteristic
525C2×C4#41×D5911yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—1181no—
533C8—1181no—
534C8—1181no—
535C8—1181no—
536C8—2361no—
537C8—2361no—
538C8—2361no—
539C8#1427375912yes—
540C22—1181no—
541C22—1181no—
542C22—1181no—
543C22—2361no—
544C22—2361no—
545C22—2361no—
546C22C4×D4×D5911yescharacteristic
547C4—21no—
548C4—22no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—1181no—
553C4—1181no—
554C4—1181no—
555C4—2361no—
556C4—2361no—
557C4—2361no—
558C4C2×#1427373511yescharacteristic
559C4#1683816711yescharacteristic
560C4C2×#1427375911yescharacteristic
561C4#1683819911yescharacteristic
562C2—4721no—
563C2—2361no—
564C2#2143282211yescharacteristic
565C2C4×#1427375911yescharacteristic
566C2#2143301011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1584C2
socleC22×C59C2×C4×D4
prime radicalC2×#11745689C2×C4
cosocleC42×C59C24
hypercenter#1584D59
hypocenterC59#18013
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
3. #21433044C2semidirect
4. #21422997C2semidirect
5. #21422986C2semidirect
6. #21422971C2semidirect
7. #21422964C2semidirect
8. #21422941C2semidirect
9. #21422915C2semidirect
10. C8×#11745672C2semidirect
11. #21422880C2semidirect
12. #21422567C2semidirect
14. #21422394C2semidirect
15. C59×#1584C2semidirect
286. #1584D59semidirect
350. C59#18013semidirect

Maximal subgroups

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

subgroupindex
#214341652
#214331152
#214330442
#214229972
#214229862
#214229712
#214229642
#214229412
#214229152
C8×#117456722
#214228802
#214225672
#214224822
#214223942
C59×#15842
#1801359