Finite Group Databasebeta

← #28647739

Subgroups — #28647739

Order 15104.

678 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429328C211yesmaximal, characteristic, complemented
2#21427068C211yesmaximal, characteristic
3#21425622C211yesmaximal, characteristic
4#21425470C211yesmaximal, characteristic
5#21425466C211yesmaximal, characteristic
6#21424865C211yesmaximal, characteristic, complemented
7#21424862C211yesmaximal, characteristic, complemented
8#21424856C211yesmaximal, characteristic, complemented
9#21424813C211yesmaximal, characteristic, complemented
10#21424690C211yesmaximal, characteristic, complemented
11#21424668C211yesmaximal, characteristic, complemented
12#21424259C211yesmaximal, characteristic, complemented
13#21424104C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#2092C211yesmaximal, characteristic
16C2×D4×Dic59C2211yescharacteristic
17C2×#14273661C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27#16837343—21no—
28#16837336—21no—
29#16837305C2211yescharacteristic
30#16837301C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837296—21no—
33#16837290—21no—
34#16837287—21no—
35#16837284—21no—
36#16837271C2211yescharacteristic
37#16837267C2211yescharacteristic
38#16837267C2211yescharacteristic
39#16837263—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×Dic59⋊C4C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44C22⋊C4×Dic59—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×#107×C59C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C59×#390C2211yescharacteristic
57C59×#382C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62D4×Dic59—22no—
63D4×Dic59Q814yes—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59C2311yescharacteristic
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×Dic59⋊C4—21no—
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4D411yescharacteristic
73C2×Dic59⋊C4D411yescharacteristic
74C2×C22⋊Dic59—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59C2311yescharacteristic
81C2×C4×Dic59C2311yescharacteristic
82#14273661—22no—
83#14273660—41no—
84#14273658—41no—
85#14273658—41no—
86C4⋊(C59⋊C8)—22no—
87Dic59⋊C8—22no—
88Dic59⋊C8—22no—
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×D4×C59C2311yescharacteristic
104C22×C8×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C42×C59C2311yescharacteristic
112#107×C59—22no—
113#106×C59—41no—
114#102×C59—22no—
115#102×C59—41no—
116#95×C59—21no—
117#95×C59—21no—
118C24×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59C2×Q812yes—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2411yescharacteristic
131C22×Dic59—21no—
132C22×Dic59—24no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—81no—
137C22×Dic59—81no—
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59#4112yes—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C22⋊C4×C59—42no—
155C22⋊C4×C59—81no—
156C42×C59—21no—
157C42×C59—21no—
158C22⋊Dic59—24no—
159C22⋊Dic59—42no—
160C22⋊Dic59—81no—
161C4⋊Dic59—21no—
162C4⋊Dic59—21no—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—42no—
169C4⋊Dic59C2×Q812yes—
170Dic59⋊C4—22no—
171Dic59⋊C4—41no—
172Dic59⋊C4—42no—
173Dic59⋊C4—81no—
174Dic59⋊C4—81no—
175C4×Dic59—21no—
176C4×Dic59—21no—
177C4×Dic59—81no—
178C4×Dic59—81no—
179C4×Dic59—81no—
180C4×Dic59C2×Q812yes—
181C4×Dic59#4112yes—
182C2×C59⋊C8—21no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8#4112yes—
188C23×C59—22no—
189C23×C59—41no—
190C23×C59—41no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59C22×D411yescharacteristic
196D4×C59—22no—
197D4×C59—22no—
198D4×C59—41no—
199D4×C59—41no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59C22×Q811yescharacteristic
216C2×C4×C59C2×#4111yescharacteristic
217C2×C4×C59C2×#4111yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×C4×C59H(5,𝔽2)11yescharacteristic
220C2×C4×C592−1+411yescharacteristic
221C2×Dic59—21no—
222C2×Dic59—21no—
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—22no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—84no—
243C2×Dic59C22⋊Q812yes—
244C2×Dic59C22⋊Q812yes—
245C2×Dic59#13512yes—
246C8×C59—22no—
247C8×C59—41no—
248C59⋊C8—22no—
249C59⋊C8—41no—
250#26094—591nomaximal, Sylow
251C22×C59—42no—
252C22×C59—42no—
253C22×C59—42no—
254C22×C59—81no—
255C22×C59—81no—
256C22×C59—81no—
257C22×C59—81no—
258C22×C59C2×C22⋊Q811yescharacteristic
259C22×C59#54711yescharacteristic
260C22×C59#55011yescharacteristic
261C22×C59C2×#13511yescharacteristic
262C22×C59#57511yescharacteristic
263C22×C59#57611yescharacteristic
264C22×C59#57711yescharacteristic
265C4×C59—41no—
266C4×C59—41no—
267C4×C59—81no—
268C4×C59—81no—
269C4×C59—81no—
270C4×C59#53311yescharacteristic
271C4×C59#54411yescharacteristic
272C4×C59#55211yescharacteristic
273C4×C59#55311yescharacteristic
274Dic59—22no—
275Dic59—22no—
276Dic59—22no—
277Dic59—42no—
278Dic59—41no—
279Dic59—81no—
280Dic59—81no—
281Dic59—81no—
282#2570—591no—
283#2357—591no—
284#2093—591no—
285#2092D5911yescharacteristic
286#2065—591no—
287#2065—591no—
288#1949—591no—
289#1949—591no—
290#1948—591no—
291#1939—591no—
292#1921—591no—
293#1918—591no—
294#1846—591no—
295#1804—591no—
296C4×#102—591no—
297C2×C59—82no—
298C2×C59#244111yescharacteristic
299C2×C59#307611yescharacteristic
300C2×C59#308111yescharacteristic
301C2×C59#333711yescharacteristic
302C2×C59#335511yescharacteristic
303C2×C59#339611yescharacteristic
304C2×C59#340111yescharacteristic
305C2×C4×D4—591no—
306C2×#107—591no—
307C2×#107—591no—
308C2×#107C2×D5911yescharacteristic
309C2×#106—591no—
310C2×#106—591no—
311C2×#106C2×D5911yescharacteristic
312C2×C4⋊C8—591no—
313C2×C4⋊C8—591no—
314C2×C4⋊C8—591no—
315C2×#102—591no—
316C2×#102—591no—
317C2×#102C2×D5911yescharacteristic
318C2×#102C2×D5911yescharacteristic
319C2×C4×C8—591no—
320#400—1181no—
321#398—1181no—
322#390—591no—
323#390C2×D5911yescharacteristic
324#389—591no—
325#389—591no—
326#388—1181no—
327#388—1181no—
328#387—1181no—
329#387—1181no—
330#384—591no—
331#382—591no—
332#382—591no—
333#382C2×D5911yescharacteristic
334#381—1181no—
335C4×C4⋊C4—591no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339C4×C22⋊C4—1181no—
340#340—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#336—591no—
347#336C2×D5911yescharacteristic
348C59#2609411yescharacteristic, complemented, Sylow
349C22×D4C22×D5911yescharacteristic
350C23×C4—1181no—
351C22×C8—591no—
352C22×C8—591no—
353C22×C8C22×D5911yescharacteristic
354C22×C8C22×D5911yescharacteristic
355C4×D4—594no—
356C4×D4—1182no—
357C2×C4⋊C4—591no—
358C2×C4⋊C4—591no—
359C2×C4⋊C4—591no—
360C2×C4⋊C4—591no—
361C2×C4⋊C4—591no—
362C2×C4⋊C4—591no—
363C2×C4⋊C4—1181no—
364C2×C4⋊C4—1181no—
365C2×C4⋊C4—1181no—
366C2×C4⋊C4—1181no—
367C2×C4⋊C4C22×D5911yescharacteristic
368C2×C4⋊C4C22×D5911yescharacteristic
369C2×C4⋊C4C22×D5911yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1181no—
373C2×C22⋊C4—1181no—
374C2×C22⋊C4—1181no—
375C2×C42—591no—
376C2×C42—591no—
377C2×C42—1181no—
378C2×C42—1181no—
379C2×C42—1181no—
380C2×C42C22×D5911yescharacteristic
381#107—22no—
382#107—1182no—
383#107—2361no—
384#106—41no—
385#106—2361no—
386#106—2361no—
387C4⋊C8—1182no—
388C4⋊C8—1182no—
389C4⋊C8—1182no—
390#102—22no—
391#102—41no—
392#102—1182no—
393#102—2361no—
394C4×C8—1182no—
395#95—21no—
396#95—21no—
397#95—591no—
398#95—591no—
399#95—1181no—
400#95—1181no—
401#95—1181no—
402#95—1181no—
403#95—1181no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408C24—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4Q8×D5912yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—591no—
419C22×C4—591no—
420C22×C4—591no—
421C22×C4—591no—
422C22×C4—1184no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4—1181no—
427C22×C4—4721no—
428C22×C4—4721no—
429C22×C4D4×D5911yescharacteristic
430C22×C4D4×D5911yescharacteristic
431C22×C4C23×D5911yescharacteristic
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—592no—
438C2×C8—1181no—
439C2×C8—1181no—
440C2×C8—1181no—
441C2×C8—1181no—
442C2×C8—1181no—
443C2×C8#1174568712yes—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—592no—
451C4⋊C4—1182no—
452C4⋊C4—1181no—
453C4⋊C4—1181no—
454C4⋊C4—1181no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—4721no—
460C4⋊C4—4721no—
461C4⋊C4—2361no—
462C4⋊C4—2362no—
463C4⋊C4—2362no—
464C22⋊C4—42no—
465C22⋊C4—81no—
466C22⋊C4—1184no—
467C22⋊C4—4721no—
468C22⋊C4—2362no—
469C42—21no—
470C42—21no—
471C42—592no—
472C42—592no—
473C42—1181no—
474C42—1181no—
475C42—4721no—
476C42—4721no—
477C42—4721no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C2×D4×D5911yescharacteristic
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—592no—
506C2×C4—592no—
507C2×C4—592no—
508C2×C4—1181no—
509C2×C4—1181no—
510C2×C4—1181no—
511C2×C4—1181no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1182no—
515C2×C4—1181no—
516C2×C4—1181no—
517C2×C4—1181no—
518C2×C4—4724no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4C2×Q8×D5911yescharacteristic
531C2×C4#41×D5911yescharacteristic
532C2×C4#41×D5911yescharacteristic
533C2×C4C2×#1174568711yescharacteristic
534C2×C4#1427377211yescharacteristic
535C2×C4#1427377411yescharacteristic
536C8—22no—
537C8—41no—
538C8—1182no—
539C8—2361no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C22⋊Q8×D5911yescharacteristic
548C22#1683805011yescharacteristic
549C22#1683807211yescharacteristic
550C22#135×D5911yescharacteristic
551C22#1683819811yescharacteristic
552C22#1683820411yescharacteristic
553C22#1683821211yescharacteristic
554C4—41no—
555C4—41no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—1182no—
560C4—1182no—
561C4—1182no—
562C4—4721no—
563C4—4721no—
564C4—4721no—
565C4—2361no—
566C4—2362no—
567C4#1683795411yescharacteristic
568C4#1683803111yescharacteristic
569C4#1683808911yescharacteristic
570C4#1683809711yescharacteristic
571C2—82no—
572C2#2142780211yescharacteristic
573C2#2143388111yescharacteristic
574C2#2143392411yescharacteristic
575C2#2143646811yescharacteristic
576C2#2143669511yescharacteristic
577C2#2143710511yescharacteristic
578C2#2143715711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2092C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2092D59
hypocenterC59#26094
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2092

Complements

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

NH ≅ G/Ntype
1. #21429328C2semidirect
6. #21424865C2semidirect
7. #21424862C2semidirect
8. #21424856C2semidirect
9. #21424813C2semidirect
10. #21424690C2semidirect
11. #21424668C2semidirect
12. #21424259C2semidirect
348. C59#26094semidirect

Maximal subgroups

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

subgroupindex
#214293282
#214270682
#214256222
#214254702
#214254662
#214248652
#214248622
#214248562
#214248132
#214246902
#214246682
#214242592
#214241042
#102×Dic592
C59×#20922
#2609459