Finite Group Databasebeta

← #28646853

Subgroups — #28646853

Order 15104.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429113C211yesmaximal, characteristic, complemented
2#21427068C211yesmaximal, characteristic
3#21425482C211yesmaximal, characteristic
4#21425470C211yesmaximal, characteristic
5#21425240C211yesmaximal, characteristic
6#21425052C211yesmaximal, characteristic, complemented
7#21425049C211yesmaximal, characteristic, complemented
8#21424867C211yesmaximal, characteristic, complemented
9#21424813C211yesmaximal, characteristic, complemented
10#21424689C211yesmaximal, characteristic, complemented
11#21424679C211yesmaximal, characteristic, complemented
12#21424225C211yesmaximal, characteristic, complemented
13#21424104C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#2023C211yesmaximal, characteristic
16C2×#14273718C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18C2×#14273661C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26#16837332—21no—
27#16837321—21no—
28#16837309—21no—
29#16837305C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837298C2211yescharacteristic
32#16837297—21no—
33#16837296—21no—
34#16837290—21no—
35#16837267C2211yescharacteristic
36#16837267C2211yescharacteristic
37#16837263—21no—
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4×Dic59⋊C4C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42C22⋊C4×Dic59—21no—
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×C4⋊Q8×C59C2211yescharacteristic
53C2×#102×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C59×#390C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62#14273718—22no—
63#14273718—41no—
64D4×Dic59—22no—
65D4×Dic59Q814yescomplemented
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4D411yescharacteristic
75C2×Dic59⋊C4D411yescharacteristic
76C2×C22⋊Dic59—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59C2311yescharacteristic
83C2×C4×Dic59C2311yescharacteristic
84#14273661—22no—
85#14273658—22no—
86C4⋊(C59⋊C8)—41no—
87Dic59⋊C8—22no—
88Dic59⋊C8—41no—
89#14273648—22no—
90#14273648—41no—
91C8×Dic59—22no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95D411yescharacteristic
102C59⋊#95D411yescharacteristic
103C22×Q8×C59—21no—
104C22×D4×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C4⋊Q8×C59—22no—
108C4⋊Q8×C59—41no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C42×C59C2311yescharacteristic
116#102×C59—22no—
117#102×C59—41no—
118C24×C59—21no—
119C2×Q8×C59—42no—
120C2×Q8×C59—42no—
121C2×Q8×C59—81no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59C2×Q812yes—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2411yescharacteristic
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—24no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—81no—
140C22×Dic59—81no—
141C22×Dic59C2×D411yescharacteristic
142C22×Dic59C2×D411yescharacteristic
143C22×Dic59C2×D411yescharacteristic
144C22×Dic59C2×D411yescharacteristic
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—42no—
157C4⋊C4×C59#4112yes—
158C22⋊C4×C59—42no—
159C22⋊C4×C59—81no—
160C42×C59—21no—
161C42×C59—21no—
162C22⋊Dic59—24no—
163C22⋊Dic59—42no—
164C22⋊Dic59—81no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—42no—
172C4⋊Dic59C2×Q812yes—
173C4⋊Dic59#4112yes—
174Dic59⋊C4—22no—
175Dic59⋊C4—41no—
176Dic59⋊C4—42no—
177Dic59⋊C4—42no—
178Dic59⋊C4—81no—
179C4×Dic59—21no—
180C4×Dic59—21no—
181C4×Dic59—81no—
182C4×Dic59—81no—
183C4×Dic59—81no—
184C4×Dic59C2×Q812yes—
185C4×Dic59#4112yes—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C23×C59—22no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59C22×D411yescharacteristic
200Q8×C59—84no—
201D4×C59—22no—
202D4×C59—22no—
203D4×C59—41no—
204D4×C59—41no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—22no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59#13512yes—
220C2×C4×C59C22×Q811yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C59C2×#4111yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C59H(5,𝔽2)11yescharacteristic
225C2×C4×C592−1+411yescharacteristic
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—22no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—22no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—84no—
249C2×Dic59C22⋊Q812yes—
250C2×Dic59C22⋊Q812yes—
251C8×C59—41no—
252C8×C59—41no—
253C59⋊C8—41no—
254C59⋊C8—41no—
255#26028—591nomaximal, Sylow
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59C2×C22⋊Q811yescharacteristic
264C22×C59#54611yescharacteristic
265C22×C59#54811yescharacteristic
266C22×C59C2×#13511yescharacteristic
267C22×C59#57511yescharacteristic
268C22×C59#57611yescharacteristic
269C22×C59#57711yescharacteristic
270C4×C59—22no—
271C4×C59—42no—
272C4×C59—41no—
273C4×C59—81no—
274C4×C59—81no—
275C4×C59#53311yescharacteristic
276C4×C59#54411yescharacteristic
277C4×C59#55511yescharacteristic
278C4×C59#55611yescharacteristic
279Dic59—22no—
280Dic59—22no—
281Dic59—42no—
282Dic59—42no—
283Dic59—41no—
284Dic59—41no—
285Dic59—81no—
286Dic59—81no—
287#2550—591no—
288#2357—591no—
289#2067—591no—
290#2065—591no—
291#2025—591no—
292#2023D5911yescharacteristic
293#1989—591no—
294#1988—591no—
295#1949—591no—
296#1939—591no—
297#1921—591no—
298#1920—591no—
299#1835—591no—
300#1804—591no—
301C4×#102—591no—
302C2×C59—82no—
303C2×C59#243911yescharacteristic
304C2×C59#307611yescharacteristic
305C2×C59#308111yescharacteristic
306C2×C59#329711yescharacteristic
307C2×C59#332611yescharacteristic
308C2×C59#336511yescharacteristic
309C2×C59#337011yescharacteristic
310C2×C4⋊Q8C2×D5911yescharacteristic
311C2×#125—591no—
312C2×C4×D4—591no—
313C2×#107—591no—
314C2×#106—591no—
315C2×C4⋊C8—591no—
316C2×C4⋊C8—591no—
317C2×C4⋊C8—591no—
318C2×#102—591no—
319C2×#102—591no—
320C2×#102C2×D5911yescharacteristic
321C2×#102C2×D5911yescharacteristic
322C2×C4×C8—591no—
323#398—1181no—
324#395—1181no—
325#391—1181no—
326#390—591no—
327#390C2×D5911yescharacteristic
328#389—591no—
329#389—591no—
330#389—1181no—
331#388—1181no—
332#388—1181no—
333#382—591no—
334#382—591no—
335#381—1181no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4—591no—
340C4×C22⋊C4—1181no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340C2×D5911yescharacteristic
350#340C2×D5911yescharacteristic
351#336—591no—
352#336C2×D5911yescharacteristic
353C59#2602811yescharacteristic, complemented, Sylow
354C22×Q8—21no—
355C22×D4C22×D5911yescharacteristic
356C23×C4—1181no—
357C22×C8—591no—
358C22×C8—591no—
359C22×C8C22×D5911yescharacteristic
360C22×C8C22×D5911yescharacteristic
361C4⋊Q8—22no—
362C4⋊Q8—41no—
363#125—1182no—
364#125—2361no—
365C4×D4—594no—
366C4×D4—1182no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1181no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C42—591no—
387C2×C42—591no—
388C2×C42—1181no—
389C2×C42—1181no—
390C2×C42—1181no—
391C2×C42C22×D5911yescharacteristic
392#107—1182no—
393#106—1182no—
394C4⋊C8—1182no—
395C4⋊C8—2361no—
396C4⋊C8—2361no—
397#102—22no—
398#102—41no—
399#102—1182no—
400#102—2361no—
401C4×C8—1182no—
402#95—591no—
403#95—591no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413C24—21no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—81no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4Q8×D5912yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—591no—
427C22×C4—591no—
428C22×C4—591no—
429C22×C4—591no—
430C22×C4—1184no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—4721no—
436C22×C4—4721no—
437C22×C4D4×D5911yescharacteristic
438C22×C4D4×D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—592no—
459C4⋊C4—592no—
460C4⋊C4—1182no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—4721no—
468C4⋊C4—2361no—
469C4⋊C4—2362no—
470C4⋊C4—2362no—
471C4⋊C4—2362no—
472C4⋊C4#1174568712yes—
473C22⋊C4—42no—
474C22⋊C4—81no—
475C22⋊C4—1184no—
476C22⋊C4—4721no—
477C22⋊C4—2362no—
478C42—21no—
479C42—21no—
480C42—592no—
481C42—592no—
482C42—1181no—
483C42—1181no—
484C42—4721no—
485C42—4721no—
486C42—4721no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×D4×D5911yescharacteristic
495Q8—84no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—592no—
515C2×C4—592no—
516C2×C4—1181no—
517C2×C4—1181no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1182no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1182no—
528C2×C4—4724no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4#1427374612yes—
540C2×C4C2×Q8×D5911yescharacteristic
541C2×C4#41×D5911yescharacteristic
542C2×C4#41×D5911yescharacteristic
543C2×C4C2×#1174568711yescharacteristic
544C2×C4#1427377211yescharacteristic
545C2×C4#1427377411yescharacteristic
546C8—41no—
547C8—41no—
548C8—2361no—
549C8—2361no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C22⋊Q8×D5911yescharacteristic
558C22#1683803911yescharacteristic
559C22#1683806011yescharacteristic
560C22C2×#1427374611yescharacteristic
561C22#1683820111yescharacteristic
562C22#1683820611yescharacteristic
563C22#1683820811yescharacteristic
564C4—22no—
565C4—42no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—1182no—
570C4—1182no—
571C4—4721no—
572C4—4721no—
573C4—2361no—
574C4—2362no—
575C4—2362no—
576C4—2361no—
577C4#1683795411yescharacteristic
578C4#1683803111yescharacteristic
579C4#1683811411yescharacteristic
580C4#1683812111yescharacteristic
581C2—82no—
582C2#2142778011yescharacteristic
583C2#2143387711yescharacteristic
584C2#2143392211yescharacteristic
585C2#2143595011yescharacteristic
586C2#2143635311yescharacteristic
587C2#2143679711yescharacteristic
588C2#2143684811yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2023C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2023D59
hypocenterC59#26028
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. #21429113C2semidirect
6. #21425052C2semidirect
7. #21425049C2semidirect
8. #21424867C2semidirect
9. #21424813C2semidirect
10. #21424689C2semidirect
11. #21424679C2semidirect
12. #21424225C2semidirect
65. D4×Dic59Q8semidirect
353. C59#26028semidirect

Maximal subgroups

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

subgroupindex
#214291132
#214270682
#214254822
#214254702
#214252402
#214250522
#214250492
#214248672
#214248132
#214246892
#214246792
#214242252
#214241042
#102×Dic592
C59×#20232
#2602859