Finite Group Databasebeta

← #28661521

Subgroups — #28661521

Order 15104.

678 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428503C211yesmaximal, characteristic
2#21427127C211yesmaximal, characteristic
3#21425517C211yesmaximal, characteristic, complemented
4#21425475C211yesmaximal, characteristic, complemented
5#21425296C211yesmaximal, characteristic
6#21424865C211yesmaximal, characteristic, complemented
7#21424825C211yesmaximal, characteristic, complemented
8#21424705C211yesmaximal, characteristic
9#21424687C211yesmaximal, characteristic, complemented
10#21424670C211yesmaximal, characteristic, complemented
11#21424352C211yesmaximal, characteristic, complemented
12#21424297C211yesmaximal, characteristic
13#21424188C211yesmaximal, characteristic
14#103×Dic59C211yesmaximal, characteristic, complemented
15C59×#2035C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18C2×#14273661C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26#16837344—21no—
27#16837327—21no—
28#16837309C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837301C2211yescharacteristic
31#16837298C2211yescharacteristic
32#16837296—21no—
33#16837295—21no—
34#16837295—21no—
35#16837288—21no—
36#16837269—21no—
37#16837267C2211yescharacteristic
38#16837262C2211yescharacteristic
39C4⋊C4×Dic59—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×C22⋊Dic59C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×C22⋊Q8×C59C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C2×#103×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C22×C59⋊C8C2311yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61Q8×Dic59—22no—
62Q8×Dic59—41no—
63#14273686—22no—
64#14273686—41no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×C22⋊Dic59—21no—
74C2×C22⋊Dic59—21no—
75C2×C22⋊Dic59D411yescharacteristic
76C2×C22⋊Dic59D411yescharacteristic
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59C2311yescharacteristic
82C2×C4×Dic59C2311yescharacteristic
83#14273661—22no—
84#14273660—22no—
85#14273659—41no—
86Dic59⋊C8—22no—
87#14273651—22no—
88#14273651—41no—
89C22⋊(C59⋊C8)—41no—
90C8×Dic59—41no—
91C59⋊#95—21no—
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×C59C2311yescharacteristic
104C23×C4×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22⋊Q8×C59—22no—
108C22⋊Q8×C59—22no—
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—
115#106×C59—22no—
116#103×C59—22no—
117#103×C59—22no—
118C22⋊C8×C59—22no—
119#95×C59—21no—
120C24×C59C2×D411yescharacteristic
121C2×Q8×C59—21no—
122C2×Q8×C59—21no—
123C2×Q8×C59—21no—
124C2×Q8×C59—21no—
125C2×Q8×C59—21no—
126C2×Q8×C59#4112yes—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59C2×D411yescharacteristic
144C22×Dic59C2×D411yescharacteristic
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59#4112yes—
153C4⋊C4×C59—24no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59#4112yes—
160C22⋊C4×C59—24no—
161C22⋊C4×C59—81no—
162C22⋊Dic59—22no—
163C22⋊Dic59—41no—
164C22⋊Dic59—42no—
165C22⋊Dic59—42no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—42no—
174C4⋊Dic59—42no—
175C4⋊Dic59—81no—
176C4⋊Dic59#4112yes—
177Dic59⋊C4—81no—
178C4×Dic59—21no—
179C4×Dic59—21no—
180C4×Dic59—21no—
181C4×Dic59—21no—
182C4×Dic59—42no—
183C4×Dic59—42no—
184C4×Dic59—42no—
185C4×Dic59—42no—
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—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59C22×D411yescharacteristic
200Q8×C59—22no—
201Q8×C59—22no—
202Q8×C59—41no—
203Q8×C59—41no—
204C2×C4×C59—22no—
205C2×C4×C59—22no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
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×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C59C2×#4111yescharacteristic
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59H(5,𝔽2)11yescharacteristic
228C2×C4×C592−1+411yescharacteristic
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—22no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
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—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59#13512yes—
253C8×C59—22no—
254C8×C59—41no—
255C59⋊C8—41no—
256C59⋊C8—41no—
257#27042—591nomaximal, Sylow
258C22×C59—41no—
259C22×C59—41no—
260C22×C59—41no—
261C22×C59—41no—
262C22×C59—41no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59#52511yescharacteristic
266C22×C59#54611yescharacteristic
267C22×C59#54711yescharacteristic
268C22×C59C2×#13511yescharacteristic
269C22×C59#57511yescharacteristic
270C22×C59#57711yescharacteristic
271C22×C59#57811yescharacteristic
272C4×C59—42no—
273C4×C59—41no—
274C4×C59—42no—
275C4×C59—81no—
276C4×C59—81no—
277C4×C59#52911yescharacteristic
278C4×C59#53911yescharacteristic
279C4×C59#55311yescharacteristic
280C4×C59#55611yescharacteristic
281Dic59—22no—
282Dic59—41no—
283Dic59—41no—
284Dic59—41no—
285Dic59—42no—
286Dic59—81no—
287Dic59—81no—
288Dic59—81no—
289#2495—591no—
290#2362—591no—
291#2074—591no—
292#2066—591no—
293#2036—591no—
294#2035D5911yescharacteristic
295#1949—591no—
296#1941—591no—
297#1923—591no—
298#1921—591no—
299#1918—591no—
300#1863—591no—
301#1854—591no—
302#1825—591no—
303C4×#103—591no—
304C2×C59—81no—
305C2×C59#246611yescharacteristic
306C2×C59#310211yescharacteristic
307C2×C59#312111yescharacteristic
308C2×C59#330011yescharacteristic
309C2×C59#330711yescharacteristic
310C2×C59#334011yescharacteristic
311C2×C59#335211yescharacteristic
312C2×C22⋊Q8C2×D5911yescharacteristic
313C2×C4×Q8—591no—
314C2×#117—591no—
315C2×#107—591no—
316C2×#107—591no—
317C2×#106—591no—
318C2×#106C2×D5911yescharacteristic
319C2×C4⋊C8—591no—
320C2×#103—591no—
321C2×#103—591no—
322C2×#103C2×D5911yescharacteristic
323C2×#103C2×D5911yescharacteristic
324C2×C22⋊C8—591no—
325C2×C22⋊C8C2×D5911yescharacteristic
326C2×C4×C8—591no—
327#400—1181no—
328#397—1181no—
329#391—591no—
330#389—591no—
331#389—591no—
332#389—591no—
333#388—1181no—
334#388—1181no—
335#388—1181no—
336#387—1181no—
337#383—1181no—
338#382—591no—
339#380—591no—
340#380C2×D5911yescharacteristic
341C4×C4⋊C4—591no—
342C4×C4⋊C4—591no—
343C4×C4⋊C4—1181no—
344C4×C22⋊C4—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340C2×D5911yescharacteristic
353#336—591no—
354#336—591no—
355C59#2704211yescharacteristic, complemented, Sylow
356C22×Q8C22×D5911yescharacteristic
357C23×C4C22×D5911yescharacteristic
358C22×C8—591no—
359C22×C8—591no—
360C22×C8C22×D5911yescharacteristic
361C22×C8C22×D5911yescharacteristic
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C4×Q8—1182no—
365C4×Q8—2361no—
366#117—1182no—
367#117—2361no—
368C2×C4⋊C4—21no—
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⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—591no—
383C2×C22⋊C4—591no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C42—591no—
387C2×C42—591no—
388C2×C42—1181no—
389C2×C42—1181no—
390C2×C42—1181no—
391C2×C42—1181no—
392#107—1182no—
393#107—1182no—
394#106—22no—
395#106—2361no—
396C4⋊C8—1182no—
397#103—22no—
398#103—22no—
399#103—1182no—
400#103—2361no—
401C22⋊C8—22no—
402C22⋊C8—2361no—
403C4×C8—2361no—
404#95—21no—
405#95—591no—
406#95—591no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417C24D4×D5911yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8#1174568712yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—591no—
435C22×C4—591no—
436C22×C4—591no—
437C22×C4—591no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4D4×D5911yescharacteristic
443C22×C4C23×D5911yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8#1174568712yes—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—592no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—4721no—
471C4⋊C4—4721no—
472C4⋊C4—2362no—
473C4⋊C4—2362no—
474C4⋊C4#1174568712yes—
475C22⋊C4—24no—
476C22⋊C4—81no—
477C22⋊C4—1182no—
478C22⋊C4—2361no—
479C22⋊C4—2362no—
480C22⋊C4—2362no—
481C42—1181no—
482C42—1181no—
483C42—1181no—
484C42—1181no—
485C42—2362no—
486C42—2362no—
487C42—2362no—
488C42—2362no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23C2×D4×D5911yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—41no—
500Q8—41no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
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—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—592no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1182no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4#41×D5911yescharacteristic
545C2×C4C2×#1174568711yescharacteristic
546C2×C4C2×#1174568711yescharacteristic
547C2×C4C2×#1174568711yescharacteristic
548C2×C4#1427377311yescharacteristic
549C2×C4#1427377411yescharacteristic
550C8—22no—
551C8—41no—
552C8—2361no—
553C8—2361no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#1683790811yescharacteristic
562C22#1683804411yescharacteristic
563C22#1683805211yescharacteristic
564C22#135×D5911yescharacteristic
565C22#1683820111yescharacteristic
566C22#1683820811yescharacteristic
567C22#1683821811yescharacteristic
568C4—42no—
569C4—41no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—1182no—
574C4—4721no—
575C4—4721no—
576C4—4721no—
577C4—2361no—
578C4—2361no—
579C4—2362no—
580C4—2361no—
581C4#1683793311yescharacteristic
582C4#1683799611yescharacteristic
583C4#1683810411yescharacteristic
584C4#1683812111yescharacteristic
585C2—81no—
586C2#2142812711yescharacteristic
587C2#2143411511yescharacteristic
588C2#2143431811yescharacteristic
589C2#2143598611yescharacteristic
590C2#2143607511yescharacteristic
591C2#2143650411yescharacteristic
592C2#2143666611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2035C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2035D59
hypocenterC59#27042
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #21425517C2semidirect
4. #21425475C2semidirect
6. #21424865C2semidirect
7. #21424825C2semidirect
9. #21424687C2semidirect
10. #21424670C2semidirect
11. #21424352C2semidirect
14. #103×Dic59C2semidirect
355. C59#27042semidirect

Maximal subgroups

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

subgroupindex
#214285032
#214271272
#214255172
#214254752
#214252962
#214248652
#214248252
#214247052
#214246872
#214246702
#214243522
#214242972
#214241882
#103×Dic592
C59×#20352
#2704259