Finite Group Databasebeta

← #28647794

Subgroups — #28647794

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#21427043C211yesmaximal, characteristic
2#21425625C211yesmaximal, characteristic
3#21425620C211yesmaximal, characteristic
4#21425481C211yesmaximal, characteristic
5#21425467C211yesmaximal, characteristic
6#21424862C211yesmaximal, characteristic, complemented
7#21424854C211yesmaximal, characteristic, complemented
8#21424852C211yesmaximal, characteristic, complemented
9#21424820C211yesmaximal, characteristic, complemented
10#21424685C211yesmaximal, characteristic, complemented
11#21424678C211yesmaximal, characteristic, complemented
12#21424265C211yesmaximal, characteristic, complemented
13#21424103C211yesmaximal, characteristic
14#21423924C211yesmaximal, characteristic
15C59×#2570C211yesmaximal, characteristic, complemented
16C2×C4×D236C2211yescharacteristic
17C2×#14273685—21no—
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×#14273647C2211yescharacteristic
31C2×#14273638C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837304C2211yescharacteristic
34#16837294—21no—
35#16837294—21no—
36#16837272C2211yescharacteristic
37#16837259—21no—
38C4×C4⋊Dic59C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C59×#400—21no—
49C59×#398—21no—
50C59×#389C2211yescharacteristic
51C59×#389C2211yescharacteristic
52C59×#387—21no—
53C59×#387—21no—
54C59×#384C2211yescharacteristic
55C59×#382C2211yescharacteristic
56C59×#382C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C4×D59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C4×D236—22no—
66C4×D236Q814yes—
67#14273685—44no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×(C2×D59)⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—22no—
81#14273660—41no—
82#14273660—41no—
83#14273660—41no—
84#14273658—22no—
85#14273658—41no—
86C4⋊(C59⋊C8)—22no—
87C4⋊(C59⋊C8)—22no—
88C4⋊(C59⋊C8)—22no—
89#14273647—22no—
90#14273647—22no—
91#14273647—41no—
92#14273647—41no—
93#14273638—22no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59D411yescharacteristic
101C2×C4⋊C4×C59D411yescharacteristic
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C42×C59—21no—
107C2×C42×C59C2311yescharacteristic
108C2×C42×C59C2311yescharacteristic
109C2×C42×C59C2311yescharacteristic
110#95×C59—21no—
111#95×C59—21no—
112#95×C59—21no—
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59D411yescharacteristic
118#95×C59D411yescharacteristic
119C23×D59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2411yescharacteristic
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—
143C4⋊C4×C59—22no—
144C4⋊C4×C59—21no—
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—41no—
153C4⋊C4×C59—42no—
154C4⋊C4×C59—42no—
155C4⋊C4×C59—81no—
156C4⋊C4×C59—81no—
157C42×C59—21no—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59—81no—
162C42×C59C2×Q812yes—
163C42×C59#4112yes—
164(C2×D59)⋊C4—24no—
165(C2×D59)⋊C4—42no—
166(C2×D59)⋊C4—42no—
167(C2×D59)⋊C4—81no—
168(C2×D59)⋊C4—81no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—42no—
175C4⋊Dic59—81no—
176C4⋊Dic59C2×Q812yes—
177C4×Dic59—42no—
178C4×Dic59—42no—
179C2×C59⋊C8—21no—
180C2×C59⋊C8—21no—
181C2×C59⋊C8—21no—
182C2×C59⋊C8—21no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8#4112yes—
190C2×C59⋊C8#4112yes—
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×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×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59C22⋊Q812yes—
227C2×C4×C59C22⋊Q812yes—
228C2×C4×C59AGL(1,ℤ/8)12yes—
229C2×C4×C59C22×Q811yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59H(5,𝔽2)11yescharacteristic
234C2×C4×C592−1+411yescharacteristic
235C2×Dic59—22no—
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—
249C59⋊C8—22no—
250C59⋊C8—22no—
251C59⋊C8—41no—
252C59⋊C8—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—81no—
275C4×C59—81no—
276C4×C59—81no—
277C4×C59—81no—
278C4×C59#53311yescharacteristic
279C4×C59#54411yescharacteristic
280C4×C59#55211yescharacteristic
281C4×C59#55311yescharacteristic
282Dic59—42no—
283Dic59—81no—
284Dic59—81no—
285#2570D5911yescharacteristic, complemented
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#1940—591no—
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⋊C8—591no—
317C2×C4⋊C8—591no—
318C2×C4⋊C8—591no—
319C2×#102—591no—
320C2×#102—591no—
321C2×#102—591no—
322C2×#102—591no—
323C2×#97—591no—
324#400—21no—
325#398—21no—
326#390—591no—
327#390—591no—
328#389C2×D5911yescharacteristic
329#389C2×D5911yescharacteristic
330#388—1181no—
331#388—1181no—
332#387—21no—
333#387—21no—
334#384—591no—
335#384C2×D5911yescharacteristic
336#382C2×D5911yescharacteristic
337#382C2×D5911yescharacteristic
338#380—1181no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4C2×D5911yescharacteristic
342C4×C4⋊C4C2×D5911yescharacteristic
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#336—591no—
350#336—591no—
351C59#2609811yescharacteristic, complemented, Sylow
352C22×D4—591no—
353C23×C4—1181no—
354C22×C8—591no—
355C22×C8—591no—
356C22×C8—591no—
357C22×C8—591no—
358C4×D4—594no—
359C4×D4—1182no—
360#117—2364no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—1181no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4C22⋊D5911yescharacteristic
371C2×C4⋊C4C22⋊D5911yescharacteristic
372C2×C4⋊C4C22×D5911yescharacteristic
373C2×C4⋊C4C22×D5911yescharacteristic
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C4⋊C4C22×D5911yescharacteristic
376C2×C22⋊C4—1181no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C42—21no—
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—1182no—
394C4⋊C8—1182no—
395C4⋊C8—1182no—
396#102—1182no—
397#102—1182no—
398#102—2361no—
399#102—2361no—
400#97—1182no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—1181no—
409#95—1181no—
410#95C22⋊D5911yescharacteristic
411#95C22⋊D5911yescharacteristic
412C24—1181no—
413C2×D4—592no—
414C2×D4—1181no—
415C2×D4—1181no—
416C2×D4—1181no—
417C2×D4—1181no—
418C2×D4—1181no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—1181no—
424C22×C4—1184no—
425C22×C4—1181no—
426C22×C4—1181no—
427C22×C4—2362no—
428C22×C4—2362no—
429C22×C4C2×C22⋊D5911yescharacteristic
430C22×C4C2×C22⋊D5911yescharacteristic
431C22×C4C2×C22⋊D5911yescharacteristic
432C22×C4C2×C22⋊D5911yescharacteristic
433C22×C4C2×C22⋊D5911yescharacteristic
434C22×C4C2×C22⋊D5911yescharacteristic
435C22×C4C23×D5911yescharacteristic
436C2×C8—592no—
437C2×C8—592no—
438C2×C8—1181no—
439C2×C8—1181no—
440C2×C8—1181no—
441C2×C8—1181no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—592no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—4721no—
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—81no—
480C42—2362no—
481C42—2362no—
482C42Q8×D5912yes—
483C42#1174568912yes—
484C23—1182no—
485C23—2361no—
486C23—2361no—
487C23—2361no—
488C23—2361no—
489C23—2361no—
490C23—2361no—
491C23C22×C22⋊D5911yescharacteristic
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—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—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—1182no—
524C2×C4—2368no—
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#1427370312yes—
534C2×C4#1427370312yes—
535C2×C4#1427375212yes—
536C2×C4C2×Q8×D5911yescharacteristic
537C2×C4#41×D5911yescharacteristic
538C2×C4#41×D5911yescharacteristic
539C2×C4C2×#1174568911yescharacteristic
540C2×C4#1427377311yescharacteristic
541C2×C4#1427377411yescharacteristic
542C8—1182no—
543C8—1182no—
544C8—2361no—
545C8—2361no—
546C22—4721no—
547C22—4721no—
548C22—4721no—
549C22—4721no—
550C22—2362no—
551C22—2362no—
552C22—2362no—
553C22C2×#1427370311yescharacteristic
554C22#1683805111yescharacteristic
555C22#1683807111yescharacteristic
556C22C2×#1427375211yescharacteristic
557C22#1683820011yescharacteristic
558C22#1683820511yescharacteristic
559C22#1683821711yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—4721no—
571C4—4721no—
572C4—2362no—
573C4#1683795811yescharacteristic
574C4#1683803211yescharacteristic
575C4#1683809011yescharacteristic
576C4#1683810511yescharacteristic
577C2—4722no—
578C2#2142780511yescharacteristic
579C2#2143388911yescharacteristic
580C2#2143394311yescharacteristic
581C2#2143653611yescharacteristic
582C2#2143659011yescharacteristic
583C2#2143713411yescharacteristic
584C2#2143717111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2570C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2570D59
hypocenterC59#26098
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2570

Complements

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

NH ≅ G/Ntype
6. #21424862C2semidirect
7. #21424854C2semidirect
8. #21424852C2semidirect
9. #21424820C2semidirect
10. #21424685C2semidirect
11. #21424678C2semidirect
12. #21424265C2semidirect
15. C59×#2570C2semidirect
285. #2570D59semidirect
351. C59#26098semidirect

Maximal subgroups

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

subgroupindex
#214270432
#214256252
#214256202
#214254812
#214254672
#214248622
#214248542
#214248522
#214248202
#214246852
#214246782
#214242652
#214241032
#214239242
C59×#25702
#2609859