Finite Group Databasebeta

← #28646860

Subgroups — #28646860

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#21429111C211yesmaximal, characteristic, complemented
2#21427069C211yesmaximal, characteristic
3#21425480C211yesmaximal, characteristic
4#21425466C211yesmaximal, characteristic
5#21425240C211yesmaximal, characteristic
6#21425230C211yesmaximal, characteristic
7#21425052C211yesmaximal, characteristic, complemented
8#21425046C211yesmaximal, characteristic, complemented
9#21424864C211yesmaximal, characteristic, complemented
10#21424814C211yesmaximal, characteristic, complemented
11#21424691C211yesmaximal, characteristic, complemented
12#21424684C211yesmaximal, characteristic, complemented
13#21424227C211yesmaximal, characteristic, complemented
14#21424106C211yesmaximal, characteristic
15C4×#102×C59C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×#14273718C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25#16837334—21no—
26#16837323—21no—
27#16837308—21no—
28#16837305C2211yescharacteristic
29#16837305C2211yescharacteristic
30#16837301C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837300—21no—
33#16837295—21no—
34#16837295—21no—
35#16837268C2211yescharacteristic
36#16837268C2211yescharacteristic
37#16837265—21no—
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C22⋊Dic59—21no—
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×C4×D4×C59C2211yescharacteristic
53C2×#102×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C22×Dic118—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62#14273726—22no—
63#14273726—41no—
64#14273718—22no—
65#14273718—41no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59D411yescharacteristic
70C2×C4⋊Dic59D411yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59C2311yescharacteristic
84C2×C4×Dic59C2311yescharacteristic
85#14273660—22no—
86#14273658—22no—
87C4⋊(C59⋊C8)—22no—
88Dic59⋊C8—41no—
89Dic59⋊C8—41no—
90#14273648—41no—
91#14273648—41no—
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⋊#95D411yescharacteristic
101C59⋊#95D411yescharacteristic
102C22×D4×C59C2311yescharacteristic
103C23×C4×C59—21no—
104C22×C8×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C4×D4×C59—22no—
107C4×D4×C59Q814yescomplemented
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C42×C59—21no—
113C2×C42×C59C2311yescharacteristic
114#102×C59—22no—
115#102×C59—22no—
116C4×C8×C59—22no—
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×C59—24no—
129C22×C4×C59—81no—
130C22×C4×C59—81no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2411yescharacteristic
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C2×Dic118—42no—
143C2×Dic118—42no—
144C2×Dic118—81no—
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×C59C2×Q812yes—
155C4⋊C4×C59#4112yes—
156C4⋊C4×C59#4112yes—
157C22⋊C4×C59—24no—
158C42×C59—81no—
159C42×C59C2×Q812yes—
160C42×C59#4112yes—
161C22⋊Dic59—42no—
162C22⋊Dic59—42no—
163C22⋊Dic59—81no—
164C22⋊Dic59—81no—
165C4⋊Dic59—22no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—41no—
175C4⋊Dic59—42no—
176C4⋊Dic59—42no—
177C4⋊Dic59—81no—
178Dic59⋊C4—42no—
179Dic59⋊C4—42no—
180C4×Dic59—21no—
181C4×Dic59—21no—
182C4×Dic59—21no—
183C4×Dic59—21no—
184C4×Dic59—81no—
185C4×Dic59—81no—
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
200D4×C59—22no—
201D4×C59—22no—
202D4×C59—41no—
203D4×C59—41no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
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—84no—
218C2×C4×C59C22⋊Q812yes—
219C2×C4×C59C22⋊Q812yes—
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—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—41no—
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—41no—
249C2×Dic59#13512yes—
250Dic118—84no—
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—22no—
272C4×C59—42no—
273C4×C59—42no—
274C4×C59—42no—
275C4×C59#53311yescharacteristic
276C4×C59#54411yescharacteristic
277C4×C59#55511yescharacteristic
278C4×C59#55611yescharacteristic
279Dic59—22no—
280Dic59—41no—
281Dic59—41no—
282Dic59—41no—
283Dic59—81no—
284Dic59—81no—
285Dic59—81no—
286Dic59—81no—
287#2550—591no—
288#2357—591no—
289#2067—591no—
290#2065—591no—
291#2025—591no—
292#2023—591no—
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×#102D5911yescharacteristic
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⋊Q8—591no—
311C2×#125—591no—
312C2×C4×D4C2×D5911yescharacteristic
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×C8C2×D5911yescharacteristic
323#398—1181no—
324#395—1181no—
325#391—1181no—
326#390—591no—
327#390—591no—
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⋊C4C2×D5911yescharacteristic
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#336—591no—
353C59#2602811yescharacteristic, complemented, Sylow
354C22×Q8—1181no—
355C22×D4C22×D5911yescharacteristic
356C23×C4—21no—
357C22×C8—591no—
358C22×C8—591no—
359C22×C8C22×D5911yescharacteristic
360C22×C8C22×D5911yescharacteristic
361C4⋊Q8—1182no—
362C4⋊Q8—2361no—
363#125—1182no—
364#125—2361no—
365C4×D4—22no—
366C4×D4Dic11814yescomplemented
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—1181no—
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—1181no—
383C2×C22⋊C4—1181no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C42—21no—
387C2×C42—591no—
388C2×C42—591no—
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—22no—
399#102—2361no—
400#102—2361no—
401C4×C8—22no—
402#95—21no—
403#95—591no—
404#95—591no—
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—4721no—
415C2×Q8—2362no—
416C2×Q8—2362no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Dic11812yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—591no—
430C22×C4—591no—
431C22×C4—591no—
432C22×C4—591no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
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—1182no—
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—4721no—
465C4⋊C4—2361no—
466C4⋊C4—2362no—
467C4⋊C4—2362no—
468C4⋊C4—2362no—
469C4⋊C4—2362no—
470C4⋊C4C2×Dic11812yes—
471C4⋊C4#1174568512yes—
472C4⋊C4#1174568512yes—
473C22⋊C4—24no—
474C22⋊C4—4721no—
475C22⋊C4—4721no—
476C22⋊C4—2362no—
477C22⋊C4—2362no—
478C42—81no—
479C42—1181no—
480C42—1181no—
481C42—1181no—
482C42—1181no—
483C42—4721no—
484C42—4721no—
485C42C2×Dic11812yes—
486C42#1174568712yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×D4×D5911yescharacteristic
495Q8—4724no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—592no—
515C2×C4—1181no—
516C2×C4—1181no—
517C2×C4—1181no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
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#1427370412yes—
539C2×C4#1427370412yes—
540C2×C4C22×Dic11811yescharacteristic
541C2×C4C2×#1174568711yescharacteristic
542C2×C4C2×#1174568511yescharacteristic
543C2×C4C2×#1174568511yescharacteristic
544C2×C4#1427377311yescharacteristic
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—
557C22C2×#1427370411yescharacteristic
558C22#1683804811yescharacteristic
559C22#1683806111yescharacteristic
560C22#135×D5911yescharacteristic
561C22#1683819911yescharacteristic
562C22#1683820311yescharacteristic
563C22#1683820811yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—1182no—
570C4—4721no—
571C4—4721no—
572C4—4721no—
573C4—4721no—
574C4—2361no—
575C4—2361no—
576C4—2361no—
577C4#1683795911yescharacteristic
578C4#1683803011yescharacteristic
579C4#1683811711yescharacteristic
580C4#1683812211yescharacteristic
581C2—82no—
582C2#2142778711yescharacteristic
583C2#2143388011yescharacteristic
584C2#2143392311yescharacteristic
585C2#2143595111yescharacteristic
586C2#2143635411yescharacteristic
587C2#2143679311yescharacteristic
588C2#2143684511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC4×#102×C59C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenterC4×#102D59
hypocenterC59#26028
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
1. #21429111C2semidirect
7. #21425052C2semidirect
8. #21425046C2semidirect
9. #21424864C2semidirect
10. #21424814C2semidirect
11. #21424691C2semidirect
12. #21424684C2semidirect
13. #21424227C2semidirect
107. C4×D4×C59Q8semidirect
353. C59#26028semidirect
366. C4×D4Dic118semidirect

Maximal subgroups

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

subgroupindex
#214291112
#214270692
#214254802
#214254662
#214252402
#214252302
#214250522
#214250462
#214248642
#214248142
#214246912
#214246842
#214242272
#214241062
C4×#102×C592
#2602859