Finite Group Databasebeta

← #28559165

Subgroups — #28559165

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434133C211yesmaximal, characteristic
2#21433192C211yesmaximal, characteristic
3#21433178C211yesmaximal, characteristic
4#21423003C211yesmaximal, characteristic, complemented
5#21423003C211yesmaximal, characteristic, complemented
6#21422994C211yesmaximal, characteristic, complemented
7#21422970C211yesmaximal, characteristic, complemented
8#21422964C211yesmaximal, characteristic, complemented
9#21422947C211yesmaximal, characteristic, complemented
10#21422945C211yesmaximal, characteristic, complemented
11#21422875C211yesmaximal, characteristic
12#21422541C211yesmaximal, characteristic
13#21422515C211yesmaximal, characteristic
14#21422390C211yesmaximal, characteristic
15C59×#1591C211yesmaximal, characteristic, complemented
16#16837956C2211yescharacteristic
17C4×Q8×D59C2211yescharacteristic
18#16837739C411yescharacteristic
19#16837736C411yescharacteristic
20#16837696C411yescharacteristic
21#16837695C411yescharacteristic
22#16837611C411yescharacteristic
23#16837610C411yescharacteristic
24#16837559C411yescharacteristic
25#16837556C411yescharacteristic
26#16837533C2211yescharacteristic
27#16837532C2211yescharacteristic
28#16837532C2211yescharacteristic
29#16837531—21no—
30#16837530—21no—
31#16837530—21no—
32#16837529—21no—
33#16837529C2211yescharacteristic
34Q8×C59⋊C8C2211yescharacteristic
35#16837526C2211yescharacteristic
36C4×Dic236C2211yescharacteristic
37C4×#11745672C2211yescharacteristic
38C4×#11745672C2211yescharacteristic
39#16837464—21no—
40#16837462—21no—
41#16837458—21no—
42#16837458C2211yescharacteristic
43#16837457—21no—
44#16837454C2211yescharacteristic
45#16837453C2211yescharacteristic
46#16837453C2211yescharacteristic
47C4×#11745659C2211yescharacteristic
48#16837155C2211yescharacteristic
49#16837152C2211yescharacteristic
50#16837152C2211yescharacteristic
51#16837134C2211yescharacteristic
52#16837134C2211yescharacteristic
53Q8⋊(C59⋊C8)C2211yescharacteristic
54Q8⋊(C59⋊C8)C2211yescharacteristic
55#16837132C2211yescharacteristic
56#16837132C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837124C2211yescharacteristic
59#16837123C2211yescharacteristic
60C59×#446C2211yescharacteristic
61C59×#446C2211yescharacteristic
62C4×Q16×C59C2211yescharacteristic
63C59×#335C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×Q8⋊C8C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×Q8×D59C2×C411yescharacteristic
68C2×Dic236C2×C411yescharacteristic
69C2×#11745672C2×C411yescharacteristic
70C2×#11745672C2×C411yescharacteristic
71C2×#11745659—21no—
72#14273726C2×C411yescharacteristic
73#14273725C2×C411yescharacteristic
74#14273717—21no—
75#14273707C2×C411yescharacteristic
76#14273705—21no—
77#14273703C2×C411yescharacteristic
78Q8×Dic59C2×C411yescharacteristic
79C4×Dic118—21no—
80C4×Dic118—21no—
81C4×Dic118C2311yescharacteristic
82C4×Dic118C2311yescharacteristic
83#14273692—21no—
84#14273692C2×C411yescharacteristic
85#14273684—21no—
86C4⋊C4×D59—21no—
87C4⋊C4×D59C2×C411yescharacteristic
88C42×D59—21no—
89C42×D59C2311yescharacteristic
90#14273661C2×C411yescharacteristic
91#14273660C2×C411yescharacteristic
92#14273660C2×C411yescharacteristic
93C4⋊(C59⋊C8)—21no—
94C4⋊(C59⋊C8)—21no—
95C4⋊(C59⋊C8)C2311yescharacteristic
96C4⋊(C59⋊C8)C2311yescharacteristic
97Dic59⋊C8—21no—
98Dic59⋊C8—21no—
99Dic59⋊C8—21no—
100Dic59⋊C8—21no—
101Dic59⋊C8C2×C411yescharacteristic
102Dic59⋊C8C2×C411yescharacteristic
103Dic59⋊C8D411yescharacteristic
104Dic59⋊C8D411yescharacteristic
105#14273652C2×C411yescharacteristic
106#14273652C2×C411yescharacteristic
107#14273651C2×C411yescharacteristic
108#14273651C2×C411yescharacteristic
109#14273650C2×C411yescharacteristic
110#14273650C2×C411yescharacteristic
111(C2×D59)⋊C8—21no—
112(C2×D59)⋊C8—21no—
113(C2×D59)⋊C8C2×C411yescharacteristic
114(C2×D59)⋊C8C2×C411yescharacteristic
115(C2×D59)⋊C8D411yescharacteristic
116(C2×D59)⋊C8D411yescharacteristic
117#14273638—21no—
118#14273637—21no—
119#14273637—21no—
120#14273637—21no—
121#14273637—21no—
122#14273637—21no—
123C4×C59⋊C8—21no—
124C4×C59⋊C8C2311yescharacteristic
125C4×C59⋊C8C2311yescharacteristic
126C4×C59⋊C8C2311yescharacteristic
127C8×Dic59—21no—
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—
140#97×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59C22×C411yescharacteristic
145C2×Q8×C59C22×C411yescharacteristic
146Q8×D59—41no—
147Q8×D59—41no—
148C2×C4×D59—21no—
149C2×C4×D59—21no—
150C2×C4×D59C22×C411yescharacteristic
151C2×C4×D59C2×D411yescharacteristic
152C2×C4×D59#4111yescharacteristic
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118C22×C411yescharacteristic
156C2×Dic118C22×C411yescharacteristic
157Q16×C59—41no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59C22×C411yescharacteristic
162C2×C8×C59C22×C411yescharacteristic
163C2×C8×C59C2×D411yescharacteristic
164C2×C8×C59C2×D411yescharacteristic
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59C22×C411yescharacteristic
168C4⋊C4×C59C22×C411yescharacteristic
169C42×C59—21no—
170C42×C59—21no—
171C42×C59C2411yescharacteristic
172#11745672—41no—
173#11745672—41no—
174(C2×D59)⋊C4—21no—
175(C2×D59)⋊C4—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59C22×C411yescharacteristic
179C4⋊Dic59C22×C411yescharacteristic
180Dic59⋊C4—21no—
181Dic59⋊C4—21no—
182Dic59⋊C4—21no—
183Dic59⋊C4—21no—
184Dic59⋊C4—21no—
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188C4×Dic59—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
202Dic236—41no—
203#11745659—42no—
204C22×D59C4×D411yescharacteristic
205Q8×C59—21no—
206Q8×C59—21no—
207Q8×C59—21no—
208Q8×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59C23×C411yescharacteristic
214C2×C4×C59C22×D411yescharacteristic
215C2×C4×C59C2×#4111yescharacteristic
216C2×Dic59—21no—
217C2×Dic59—21no—
218C2×Dic59—21no—
219C2×Dic59—21no—
220C2×Dic59C4×D411yescharacteristic
221C2×Dic59C4×D411yescharacteristic
222C2×Dic59C4×D411yescharacteristic
223C4×D59—21no—
224C4×D59—21no—
225C4×D59—21no—
226C4×D59—21no—
227C4×D59—41no—
228C4×D59—42no—
229C4×D59—81no—
230Dic118—21no—
231Dic118—21no—
232Dic118—21no—
233Dic118—21no—
234Dic118—42no—
235Dic118—81no—
236C8×C59—22no—
237C8×C59—21no—
238C8×C59—21no—
239C8×C59—21no—
240C8×C59—21no—
241C8×C59—41no—
242C8×C59—41no—
243C59⋊C8—21no—
244C59⋊C8—21no—
245C59⋊C8—21no—
246C59⋊C8—21no—
247C59⋊C8—41no—
248C59⋊C8—41no—
249C59⋊C8—41no—
250#17953—591nomaximal, Sylow
251C22×C59C2×C4×D411yescharacteristic
252C2×D59—21no—
253C2×D59—21no—
254C2×D59—21no—
255C4×C59—21no—
256C4×C59—41no—
257C4×C59—41no—
258C4×C59—41no—
259C4×C59—41no—
260C4×C59#56811yescharacteristic
261C4×C59#56811yescharacteristic
262C4×C59#57811yescharacteristic
263C4×C59#57811yescharacteristic
264Dic59—21no—
265Dic59—21no—
266Dic59—21no—
267Dic59—41no—
268Dic59—41no—
269Dic59—41no—
270Dic59—41no—
271#3104—591no—
272#2985—591no—
273#2983—591no—
274#1593—591no—
275#1593—591no—
276#1591—591no—
277#1591D5911yescharacteristic, complemented
278#1587—591no—
279#1587—591no—
280#1584—591no—
281#1584—591no—
282#1572—591no—
283#1491—591no—
284#1485—591no—
285#1453—591no—
286C2×C59#293011yescharacteristic
287C2×C59#295011yescharacteristic
288C2×C59#295611yescharacteristic
289D59—41no—
290#533—591no—
291C2×C4×Q8—591no—
292#484—591no—
293#484—591no—
294#477—591no—
295#477—591no—
296#462—591no—
297#462—591no—
298#451—591no—
299#451—591no—
300#446—591no—
301#446—591no—
302#446—591no—
303#446—591no—
304#446—1181no—
305#446—1181no—
306#446—1181no—
307#446—1181no—
308#446C2×D5911yescharacteristic
309#446C2×D5911yescharacteristic
310C8×Q8—591no—
311C8×Q8—591no—
312C4×Q16—591no—
313C4×Q16—591no—
314C4×Q16—591no—
315C4×Q16C2×D5911yescharacteristic
316#433—1181no—
317#433—1181no—
318#432—591no—
319#432—1181no—
320#432—1181no—
321#431—591no—
322#431—591no—
323#431—591no—
324C4×M16—591no—
325#335—591no—
326#335C2×D5911yescharacteristic
327#334—591no—
328#334—591no—
329Q8⋊C8—591no—
330Q8⋊C8—591no—
331Q8⋊C8—591no—
332Q8⋊C8—591no—
333Q8⋊C8—591no—
334Q8⋊C8—591no—
335Q8⋊C8C2×D5911yescharacteristic
336Q8⋊C8C2×D5911yescharacteristic
337#322—591no—
338#322—591no—
339#322—591no—
340C82C2×D5911yescharacteristic
341C59#1795311yescharacteristic, complemented, Sylow
342C22×Q8—591no—
343C2×Q16—591no—
344C2×Q16—591no—
345C2×Q16—591no—
346C2×Q16C4×D5911yescharacteristic
347C2×M16—1181no—
348C4⋊Q8—591no—
349C4⋊Q8—591no—
350#125—1181no—
351C22⋊Q8—591no—
352C22⋊Q8—591no—
353C22⋊Q8—1181no—
354C4×Q8—591no—
355C4×Q8—591no—
356C4×Q8—591no—
357C4×Q8—591no—
358C4×Q8—1181no—
359C4×Q8—1181no—
360C4×Q8—1181no—
361C4×Q8C22×D5911yescharacteristic
362C4×Q8C22×D5911yescharacteristic
363#117—1181no—
364C2×C4⋊C4—591no—
365C2×C4⋊C4—1181no—
366C2×C42—591no—
367C2×C42—1181no—
368#107—591no—
369#107—591no—
370#107—591no—
371#107C4×D5911yescharacteristic
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—591no—
375C4⋊C8—591no—
376C4⋊C8—591no—
377C4⋊C8—591no—
378C4⋊C8—591no—
379C4⋊C8—591no—
380C4⋊C8—1181no—
381C4⋊C8—1181no—
382C4⋊C8—1181no—
383C4⋊C8—1181no—
384C4⋊C8—1181no—
385C4⋊C8—1181no—
386C4⋊C8C22×D5911yescharacteristic
387C4⋊C8C22×D5911yescharacteristic
388#103—591no—
389#103—591no—
390#103—591no—
391#103—591no—
392#103—591no—
393#103—591no—
394#103C4×D5911yescharacteristic
395#103C4×D5911yescharacteristic
396C22⋊C8—591no—
397C22⋊C8—591no—
398C22⋊C8—591no—
399C22⋊C8—591no—
400C22⋊C8—1181no—
401C22⋊C8—1181no—
402#97—21no—
403#97—21no—
404#97—1181no—
405#97—1181no—
406#97—1181no—
407#97—1181no—
408#97—1181no—
409#97—1181no—
410C4×C8—591no—
411C4×C8—591no—
412C4×C8—591no—
413C4×C8—1181no—
414C4×C8—1181no—
415C4×C8—1181no—
416C4×C8C22×D5911yescharacteristic
417C4×C8C22×D5911yescharacteristic
418C4×C8C22×D5911yescharacteristic
419C2×Q8—591no—
420C2×Q8—591no—
421C2×Q8—1181no—
422C2×Q8—1181no—
423C2×Q8—2361no—
424C2×Q8—2361no—
425C2×Q8C2×C4×D5911yescharacteristic
426C2×Q8C2×C4×D5911yescharacteristic
427C22×C4—591no—
428C22×C4—591no—
429C22×C4—591no—
430C22×C4—1181no—
431C22×C4—1181no—
432Q16—41no—
433Q16—2361no—
434Q16—2361no—
435Q16—2361no—
436M16—2362no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—591no—
441C2×C8—591no—
442C2×C8—591no—
443C2×C8—591no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8C2×C4×D5911yescharacteristic
448C2×C8C2×C4×D5911yescharacteristic
449C2×C8D4×D5911yescharacteristic
450C2×C8D4×D5911yescharacteristic
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—591no—
454C4⋊C4—591no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4C2×C4×D5911yescharacteristic
466C4⋊C4C2×C4×D5911yescharacteristic
467C22⋊C4—1181no—
468C22⋊C4—1181no—
469C42—21no—
470C42—21no—
471C42—591no—
472C42—591no—
473C42—591no—
474C42—1181no—
475C42—1181no—
476C42—1181no—
477C42—1181no—
478C42C23×D5911yescharacteristic
479C23—591no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—21no—
484Q8—1181no—
485Q8—1181no—
486Q8—1181no—
487Q8—1181no—
488Q8—4721no—
489Q8—2362no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—591no—
495C2×C4—591no—
496C2×C4—591no—
497C2×C4—1181no—
498C2×C4—1181no—
499C2×C4—1181no—
500C2×C4—1181no—
501C2×C4—1181no—
502C2×C4—1181no—
503C2×C4—1181no—
504C2×C4—1181no—
505C2×C4—4721no—
506C2×C4—2361no—
507C2×C4—2362no—
508C2×C4C22×C4×D5911yescharacteristic
509C2×C4C2×D4×D5911yescharacteristic
510C2×C4#41×D5911yescharacteristic
511C8—22no—
512C8—21no—
513C8—21no—
514C8—21no—
515C8—21no—
516C8—41no—
517C8—41no—
518C8—1181no—
519C8—1181no—
520C8—1181no—
521C8—1181no—
522C8—2361no—
523C8—2361no—
524C8—2361no—
525C22—1181no—
526C22—1181no—
527C22—1181no—
528C22C4×D4×D5911yescharacteristic
529C4—21no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—1181no—
535C4—1181no—
536C4—1181no—
537C4—2361no—
538C4—2361no—
539C4—2361no—
540C4—2361no—
541C4#1683816711yescharacteristic
542C4#1683816811yescharacteristic
543C4#1683821511yescharacteristic
544C4#1683821611yescharacteristic
545C2—2361no—
546C2#2143279611yescharacteristic
547C2#2143295911yescharacteristic
548C2#2143300111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1591C2
socleC22×C59C2×C4×D4
prime radicalC22×D59C4×D4
cosocleC42×C59C24
hypercenter#1591D59
hypocenterC59#17953
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1591

Complements

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

NH ≅ G/Ntype
4. #21423003C2semidirect
5. #21423003C2semidirect
6. #21422994C2semidirect
7. #21422970C2semidirect
8. #21422964C2semidirect
9. #21422947C2semidirect
10. #21422945C2semidirect
15. C59×#1591C2semidirect
277. #1591D59semidirect
341. C59#17953semidirect

Maximal subgroups

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

subgroupindex
#214341332
#214331922
#214331782
#214230032
#214229942
#214229702
#214229642
#214229472
#214229452
#214228752
#214225412
#214225152
#214223902
C59×#15912
#1795359