Finite Group Databasebeta

← #28647796

Subgroups — #28647796

Order 15104.

708 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429330C211yesmaximal, characteristic, complemented
2#21427043C211yesmaximal, characteristic
3#21425623C211yesmaximal, characteristic
4#21425616C211yesmaximal, characteristic
5#21425483C211yesmaximal, characteristic
6#21425469C211yesmaximal, characteristic
7#21424866C211yesmaximal, characteristic, complemented
8#21424855C211yesmaximal, characteristic, complemented
9#21424855C211yesmaximal, characteristic, complemented
10#21424689C211yesmaximal, characteristic, complemented
11#21424682C211yesmaximal, characteristic, complemented
12#21424264C211yesmaximal, characteristic, complemented
13#21424107C211yesmaximal, characteristic
14#21423922C211yesmaximal, characteristic
15C59×#1940C211yesmaximal, characteristic, complemented
16C2×C4×D236C2211yescharacteristic
17C2×#14273685—21no—
18C2×#14273661C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273661C2211yescharacteristic
21C2×#14273661C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273649C2211yescharacteristic
28#16837341—21no—
29#16837335—21no—
30#16837304C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837294—21no—
35#16837294—21no—
36#16837289—21no—
37#16837285—21no—
38#16837272C2211yescharacteristic
39#16837272C2211yescharacteristic
40#16837268C2211yescharacteristic
41#16837268C2211yescharacteristic
42#16837259—21no—
43C4×C4⋊Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45C4×C4⋊Dic59C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52C2×C4⋊C8×C59C2211yescharacteristic
53C2×C4⋊C8×C59C2211yescharacteristic
54C2×C4⋊C8×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C4×D59—21no—
61C4×D236—22no—
62C4×D236Q814yes—
63#14273685—44no—
64C2×C4⋊Dic59—21no—
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×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85#14273661—22no—
86#14273661—41no—
87#14273661—41no—
88#14273661—41no—
89#14273659—22no—
90#14273659—41no—
91#14273649—22no—
92#14273649—22no—
93#14273649—41no—
94#14273649—41no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C22×C8×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112C2×C42×C59C2311yescharacteristic
113C4⋊C8×C59—22no—
114C4⋊C8×C59—22no—
115C4⋊C8×C59—22no—
116#97×C59—22no—
117#95×C59D411yescharacteristic
118#95×C59D411yescharacteristic
119C23×D59—21no—
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2411yescharacteristic
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59—21no—
132C22×Dic59—21no—
133C22×Dic59—21no—
134C2×D236—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236C2×Q812yes—
140C2×C4×D59—24no—
141C2×C4×D59—42no—
142C2×C4×D59—42no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59#4112yes—
154C2×C8×C59#4112yes—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—41no—
157C42×C59—21no—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59C2×Q812yes—
162C42×C59#4112yes—
163(C2×D59)⋊C4—24no—
164(C2×D59)⋊C4—42no—
165(C2×D59)⋊C4—42no—
166(C2×D59)⋊C4—81no—
167(C2×D59)⋊C4—81no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—42no—
182C4⋊Dic59—42no—
183C4⋊Dic59—81no—
184C4⋊Dic59—81no—
185C4⋊Dic59C2×Q812yes—
186Dic59⋊C4—42no—
187Dic59⋊C4—81no—
188C4×Dic59—42no—
189C4×Dic59—42no—
190C4×Dic59—81no—
191C23×C59C22×D411yescharacteristic
192C22×D59—22no—
193C22×D59—41no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C2×C4×C59—21no—
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—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59C22⋊Q812yes—
215C2×C4×C59C22⋊Q812yes—
216C2×C4×C59AGL(1,ℤ/8)12yes—
217C2×C4×C59C22×Q811yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C59H(5,𝔽2)11yescharacteristic
222C2×C4×C592−1+411yescharacteristic
223C2×Dic59—22no—
224C2×Dic59—41no—
225C2×Dic59—41no—
226C2×Dic59—41no—
227C2×Dic59—41no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230C2×Dic59—41no—
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—41no—
243C2×Dic59—41no—
244D236—22no—
245D236—22no—
246D236—41no—
247D236—41no—
248C4×D59—48no—
249C8×C59—22no—
250C8×C59—22no—
251C8×C59—41no—
252C8×C59—41no—
253#26098—591nomaximal, Sylow
254C22×C59C2×C22⋊Q811yescharacteristic
255C22×C59#54711yescharacteristic
256C22×C59#55011yescharacteristic
257C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C59#57511yescharacteristic
259C22×C59#57611yescharacteristic
260C22×C59#57811yescharacteristic
261C2×D59—42no—
262C2×D59—42no—
263C2×D59—42no—
264C2×D59—81no—
265C2×D59—81no—
266C2×D59—81no—
267C2×D59—81no—
268C4×C59—22no—
269C4×C59—22no—
270C4×C59—22no—
271C4×C59—41no—
272C4×C59—41no—
273C4×C59—41no—
274C4×C59#53311yescharacteristic
275C4×C59#54411yescharacteristic
276C4×C59#55211yescharacteristic
277C4×C59#55311yescharacteristic
278Dic59—42no—
279Dic59—81no—
280Dic59—81no—
281Dic59—81no—
282Dic59—81no—
283Dic59—81no—
284Dic59—81no—
285#2570—591no—
286#2354—591no—
287#2093—591no—
288#2092—591no—
289#2067—591no—
290#2065—591no—
291#1949—591no—
292#1947—591no—
293#1947—591no—
294#1940D5911yescharacteristic, complemented
295#1921—591no—
296#1920—591no—
297#1847—591no—
298#1804—591no—
299#1762—591no—
300C2×C59#244111yescharacteristic
301C2×C59#307711yescharacteristic
302C2×C59#308311yescharacteristic
303C2×C59#334311yescharacteristic
304C2×C59#334711yescharacteristic
305C2×C59#339911yescharacteristic
306C2×C59#340211yescharacteristic
307D59—82no—
308C2×C4×D4—591no—
309C2×#117—1181no—
310C2×#107—591no—
311C2×#107—591no—
312C2×#107—591no—
313C2×#107—591no—
314C2×#106—591no—
315C2×#106—591no—
316C2×C4⋊C8C2×D5911yescharacteristic
317C2×C4⋊C8C2×D5911yescharacteristic
318C2×C4⋊C8C2×D5911yescharacteristic
319C2×#102—591no—
320C2×#102—591no—
321C2×#102—591no—
322C2×#102—591no—
323C2×#97C2×D5911yescharacteristic
324#400—1181no—
325#398—1181no—
326#390—591no—
327#390—591no—
328#389—591no—
329#389—591no—
330#388—1181no—
331#388—1181no—
332#387—1181no—
333#387—1181no—
334#384—591no—
335#384—591no—
336#382—591no—
337#382—591no—
338#380—1181no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4—591no—
342C4×C4⋊C4C2×D5911yescharacteristic
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#336C2×D5911yescharacteristic
350#336C2×D5911yescharacteristic
351C59#2609811yescharacteristic, complemented, Sylow
352C22×D4—591no—
353C23×C4—1181no—
354C22×C8C22×D5911yescharacteristic
355C22×C8C22×D5911yescharacteristic
356C22×C8C22×D5911yescharacteristic
357C22×C8C22×D5911yescharacteristic
358C4×D4—594no—
359C4×D4—1182no—
360#117—2364no—
361C2×C4⋊C4—591no—
362C2×C4⋊C4—591no—
363C2×C4⋊C4—591no—
364C2×C4⋊C4—591no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—1181no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4D23611yescharacteristic
375C2×C4⋊C4D23611yescharacteristic
376C2×C22⋊C4—1181no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C42—1181no—
382C2×C42—1181no—
383C2×C42—1181no—
384C2×C42C22×D5911yescharacteristic
385C2×C42C22×D5911yescharacteristic
386C2×C42C22×D5911yescharacteristic
387#107—1182no—
388#107—2361no—
389#107—2361no—
390#107—2361no—
391#106—1182no—
392#106—2361no—
393C4⋊C8—22no—
394C4⋊C8—22no—
395C4⋊C8—22no—
396#102—1182no—
397#102—1182no—
398#102—2361no—
399#102—2361no—
400#97—22no—
401#95—1181no—
402#95—1181no—
403#95—1181no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95D23611yescharacteristic
411#95D23611yescharacteristic
412C24—1181no—
413C2×D4—592no—
414C2×D4—1181no—
415C2×D4—1181no—
416C2×D4—1181no—
417C2×D4—1181no—
418C2×D4—1181no—
419C22×C4—1181no—
420C22×C4—1184no—
421C22×C4—1181no—
422C22×C4—1181no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4—1181no—
427C22×C4—2362no—
428C22×C4—2362no—
429C22×C4C2×D23611yescharacteristic
430C22×C4C2×D23611yescharacteristic
431C22×C4C2×D23611yescharacteristic
432C22×C4C2×D23611yescharacteristic
433C22×C4C2×D23611yescharacteristic
434C22×C4C2×D23611yescharacteristic
435C22×C4C23×D5911yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8#1174568712yes—
447C2×C8#1174568912yes—
448C4⋊C4—22no—
449C4⋊C4—41no—
450C4⋊C4—592no—
451C4⋊C4—1181no—
452C4⋊C4—1181no—
453C4⋊C4—1181no—
454C4⋊C4—1181no—
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—4721no—
465C4⋊C4—4721no—
466C4⋊C4—4721no—
467C4⋊C4—2362no—
468C4⋊C4—2362no—
469C4⋊C4—2362no—
470C22⋊C4—1184no—
471C22⋊C4—4721no—
472C22⋊C4—4721no—
473C22⋊C4—2362no—
474C22⋊C4—2362no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—4721no—
480C42—2362no—
481C42—2362no—
482C42#1174568712yes—
483C42Q8×D5912yes—
484C23—1182no—
485C23—2361no—
486C23—2361no—
487C23—2361no—
488C23—2361no—
489C23—2361no—
490C23—2361no—
491C23C22×D23611yescharacteristic
492D4—1182no—
493D4—1182no—
494D4—2361no—
495D4—2361no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—1182no—
512C2×C4—2368no—
513C2×C4—2361no—
514C2×C4—2361no—
515C2×C4—2361no—
516C2×C4—2361no—
517C2×C4—2361no—
518C2×C4—2361no—
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×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4#1427370712yes—
534C2×C4#1427370712yes—
535C2×C4#1427375512yes—
536C2×C4C2×Q8×D5911yescharacteristic
537C2×C4C2×#1174568711yescharacteristic
538C2×C4C2×#1174568711yescharacteristic
539C2×C4C2×#1174568911yescharacteristic
540C2×C4#1427377311yescharacteristic
541C2×C4#1427377411yescharacteristic
542C8—22no—
543C8—22no—
544C8—41no—
545C8—41no—
546C22—4721no—
547C22—4721no—
548C22—4721no—
549C22—4721no—
550C22—2362no—
551C22—2362no—
552C22—2362no—
553C22C2×#1427370711yescharacteristic
554C22#1683805611yescharacteristic
555C22#1683807411yescharacteristic
556C22C2×#1427375511yescharacteristic
557C22#1683820211yescharacteristic
558C22#1683820711yescharacteristic
559C22#1683821911yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—4721no—
567C4—4721no—
568C4—4721no—
569C4—4721no—
570C4—4721no—
571C4—4721no—
572C4—2362no—
573C4#1683795711yescharacteristic
574C4#1683803211yescharacteristic
575C4#1683809411yescharacteristic
576C4#1683810411yescharacteristic
577C2—4722no—
578C2#2142780711yescharacteristic
579C2#2143389511yescharacteristic
580C2#2143394911yescharacteristic
581C2#2143653811yescharacteristic
582C2#2143659811yescharacteristic
583C2#2143714011yescharacteristic
584C2#2143717711yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1940C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#1940D59
hypocenterC59#26098
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
1. #21429330C2semidirect
7. #21424866C2semidirect
8. #21424855C2semidirect
9. #21424855C2semidirect
10. #21424689C2semidirect
11. #21424682C2semidirect
12. #21424264C2semidirect
15. C59×#1940C2semidirect
294. #1940D59semidirect
351. C59#26098semidirect

Maximal subgroups

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

subgroupindex
#214293302
#214270432
#214256232
#214256162
#214254832
#214254692
#214248662
#214248552
#214246892
#214246822
#214242642
#214241072
#214239222
C59×#19402
#2609859