Finite Group Databasebeta

← #28657179

Subgroups — #28657179

Order 15104.

762 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428448C211yesmaximal, characteristic
2#21425411C211yesmaximal, characteristic, complemented
3#21425289C211yesmaximal, characteristic, complemented
4#21425247C211yesmaximal, characteristic, complemented
5#21425097C211yesmaximal, characteristic
6#21425053C211yesmaximal, characteristic, complemented
7#21424792C211yesmaximal, characteristic, complemented
8#21424723C211yesmaximal, characteristic, complemented
9#21424711C211yesmaximal, characteristic, complemented
10#21424678C211yesmaximal, characteristic, complemented
11#21424451C211yesmaximal, characteristic, complemented
12#21424418C211yesmaximal, characteristic
13#21424191C211yesmaximal, characteristic, complemented
14#21423926C211yesmaximal, characteristic, complemented
15C59×#2362C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×#14273701C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30#16837331—21no—
31#16837317—21no—
32#16837304C2211yescharacteristic
33#16837304C2211yescharacteristic
34#16837302—21no—
35#16837291—21no—
36#16837262C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C22⋊Q8×C59C2211yescharacteristic
49C2×C4×Q8×C59C2211yescharacteristic
50C2×#117×C59C2211yescharacteristic
51C59×#391C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#388—21no—
54C59×#388—21no—
55C59×#387—21no—
56C4×C4⋊C4×C59—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C22⋊D59—21no—
61C2×#11745671—22no—
62C2×#11745671—41no—
63C2×#11745671—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273719—22no—
69#14273719—41no—
70#14273701—22no—
71#14273701—41no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×C4×Dic59—21no—
84#14273658—22no—
85C4⋊(C59⋊C8)—41no—
86#14273651—22no—
87#14273651—22no—
88#14273651—41no—
89#14273647—22no—
90#14273647—22no—
91#14273647—41no—
92C22⋊(C59⋊C8)—41no—
93C22⋊(C59⋊C8)—41no—
94#14273638—41no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×Q8×C59C2311yescharacteristic
98C23×C4×C59C2311yescharacteristic
99C22⋊Q8×C59—22no—
100C22⋊Q8×C59—22no—
101C4×Q8×C59—22no—
102C4×Q8×C59—41no—
103#117×C59—22no—
104#117×C59—41no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59C2×D411yescharacteristic
127C23×D59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59—21no—
133C2×Q8×C59#4112yes—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2411yescharacteristic
150C2×C22⋊D59—42no—
151C2×C22⋊D59—42no—
152C2×C22⋊D59—81no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236—21no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59#4112yes—
170C4⋊C4×C59#4112yes—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—41no—
173C22⋊C4×C59—41no—
174C22⋊C4×C59—42no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—42no—
180C42×C59—42no—
181C42×C59—42no—
182C22⋊Dic59—81no—
183#11745671—44no—
184(C2×D59)⋊C4—42no—
185(C2×D59)⋊C4—42no—
186(C2×D59)⋊C4—81no—
187(C2×D59)⋊C4—81no—
188(C2×D59)⋊C4—81no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59#4112yes—
195Dic59⋊C4—42no—
196Dic59⋊C4—42no—
197C4×Dic59—81no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59C22×D411yescharacteristic
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225Q8×C59—22no—
226Q8×C59—22no—
227Q8×C59—41no—
228Q8×C59—41no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59H(5,𝔽2)11yescharacteristic
265C2×C4×C59H(5,𝔽2)11yescharacteristic
266C22⋊D59—84no—
267C2×Dic59—22no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276D236—41no—
277D236—41no—
278D236—41no—
279D236—41no—
280C59⋊C8—41no—
281C59⋊C8—41no—
282C59⋊C8—41no—
283C59⋊C8—41no—
284#26751—591nomaximal, Sylow
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59#52511yescharacteristic
293C22×C59#54611yescharacteristic
294C22×C59#54611yescharacteristic
295C22×C59#57511yescharacteristic
296C22×C59#57511yescharacteristic
297C22×C59#57711yescharacteristic
298C22×C59#57711yescharacteristic
299C2×D59—81no—
300C2×D59—81no—
301C2×D59—81no—
302C2×D59—81no—
303C2×D59—81no—
304C2×D59—81no—
305C2×D59—81no—
306C4×C59—42no—
307C4×C59—41no—
308C4×C59—42no—
309C4×C59—41no—
310C4×C59—42no—
311C4×C59—41no—
312C4×C59—41no—
313C4×C59—41no—
314C4×C59—81no—
315C4×C59#52911yescharacteristic
316C4×C59#53811yescharacteristic
317C4×C59#55511yescharacteristic
318C4×C59#55911yescharacteristic
319Dic59—42no—
320Dic59—81no—
321Dic59—81no—
322#2490—591no—
323#2362D5911yescharacteristic, complemented
324#2055—591no—
325#2035—591no—
326#2026—591no—
327#2000—591no—
328#1989—591no—
329#1936—591no—
330#1926—591no—
331#1924—591no—
332#1920—591no—
333#1880—591no—
334#1875—591no—
335#1825—591no—
336#1763—591no—
337C2×C59—81no—
338C2×C59#245911yescharacteristic
339C2×C59#311111yescharacteristic
340C2×C59#312611yescharacteristic
341C2×C59#330711yescharacteristic
342C2×C59#331111yescharacteristic
343C2×C59#331711yescharacteristic
344C2×C59#332611yescharacteristic
345D59—161no—
346C22×SD16—591no—
347C2×#125—591no—
348C2×C22⋊Q8C2×D5911yescharacteristic
349C2×#121—591no—
350C2×C4×Q8C2×D5911yescharacteristic
351C2×#117C2×D5911yescharacteristic
352C2×#106—591no—
353C2×C4⋊C8—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×C22⋊C8—591no—
361C2×C22⋊C8—591no—
362C2×#97—591no—
363#397—1181no—
364#394—1181no—
365#391C2×D5911yescharacteristic
366#390—591no—
367#390—591no—
368#390—1181no—
369#389C2×D5911yescharacteristic
370#388—21no—
371#388—21no—
372#388—1181no—
373#387—21no—
374#380—591no—
375C4×C4⋊C4—21no—
376C4×C4⋊C4—591no—
377C4×C4⋊C4C2×D5911yescharacteristic
378C4×C22⋊C4C2×D5911yescharacteristic
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#336—591no—
388#336—591no—
389C59#2675111yescharacteristic, complemented, Sylow
390C22×Q8C22×D5911yescharacteristic
391C22×D4—591no—
392C22×D4—1181no—
393C23×C4C22×D5911yescharacteristic
394C2×SD16—1182no—
395C2×SD16—2361no—
396C2×SD16—2361no—
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8—591no—
401#125—1182no—
402#125—2361no—
403C22⋊Q8—22no—
404C22⋊Q8—22no—
405#121—1182no—
406#121—2361no—
407C4×Q8—22no—
408C4×Q8—41no—
409#117—22no—
410#117—41no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—1181no—
418C2×C4⋊C4—1181no—
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C4⋊C4C22×D5911yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4—1181no—
426C2×C22⋊C4—1181no—
427C2×C22⋊C4—1181no—
428C2×C22⋊C4—1181no—
429C2×C22⋊C4—1181no—
430C2×C22⋊C4C22⋊D5911yescharacteristic
431C2×C22⋊C4C22⋊D5911yescharacteristic
432C2×C42—21no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—1181no—
436C2×C42C22×D5911yescharacteristic
437C2×C42C22×D5911yescharacteristic
438#106—1182no—
439C4⋊C8—2361no—
440#103—1182no—
441#103—1182no—
442#103—2361no—
443#102—1182no—
444#102—1182no—
445#102—2361no—
446C22⋊C8—2361no—
447C22⋊C8—2361no—
448#97—2361no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—1181no—
454#95—1181no—
455#95C22⋊D5911yescharacteristic
456#95C22⋊D5911yescharacteristic
457C24—1181no—
458C24C2×C22⋊D5911yescharacteristic
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8#1174568512yes—
465C2×D4—1181no—
466C2×D4—1181no—
467C2×D4—1181no—
468C2×D4—1181no—
469C2×D4—1181no—
470C2×D4—1181no—
471C2×D4—4721no—
472C2×D4—2362no—
473C2×D4—2362no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C23×D5911yescharacteristic
493SD16—2364no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—592no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—2362no—
520C4⋊C4—2362no—
521C4⋊C4#1174568512yes—
522C4⋊C4#1174568712yes—
523C22⋊C4—24no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—42no—
527C22⋊C4—4721no—
528C22⋊C4—4721no—
529C22⋊C4—4721no—
530C22⋊C4—4721no—
531C22⋊C4—2362no—
532C22⋊C4—2362no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—4721no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—2361no—
549C23—2361no—
550C23—2361no—
551C23—2361no—
552C23—2361no—
553C23—2361no—
554C23—2361no—
555C23C22×C22⋊D5911yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560D4—4724no—
561D4—2361no—
562D4—2361no—
563D4—2361no—
564D4—2361no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—1182no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4#41×D5911yescharacteristic
606C2×C4C2×#1174568711yescharacteristic
607C2×C4C2×#1174568511yescharacteristic
608C2×C4C2×#1174568511yescharacteristic
609C2×C4#1427377211yescharacteristic
610C2×C4#1427377311yescharacteristic
611C8—2361no—
612C8—2361no—
613C8—2361no—
614C8—2361no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—4721no—
623C22—4721no—
624C22—4721no—
625C22—4721no—
626C22—4721no—
627C22—4721no—
628C22—4721no—
629C22#1683791311yescharacteristic
630C22#1683804011yescharacteristic
631C22#1683804611yescharacteristic
632C22#1683820011yescharacteristic
633C22#1683820011yescharacteristic
634C22#1683821011yescharacteristic
635C22#1683821011yescharacteristic
636C4—41no—
637C4—41no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—81no—
645C4—4721no—
646C4—4721no—
647C4—2362no—
648C4#1683793211yescharacteristic
649C4#1683798511yescharacteristic
650C4#1683811711yescharacteristic
651C4#1683813511yescharacteristic
652C2—81no—
653C2—9441no—
654C2#2142802711yescharacteristic
655C2#2143420511yescharacteristic
656C2#2143438711yescharacteristic
657C2#2143607211yescharacteristic
658C2#2143612611yescharacteristic
659C2#2143621611yescharacteristic
660C2#2143634911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2362C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenter#2362D59
hypocenterC59#26751
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #21425411C2semidirect
3. #21425289C2semidirect
4. #21425247C2semidirect
6. #21425053C2semidirect
7. #21424792C2semidirect
8. #21424723C2semidirect
9. #21424711C2semidirect
10. #21424678C2semidirect
11. #21424451C2semidirect
13. #21424191C2semidirect
14. #21423926C2semidirect
15. C59×#2362C2semidirect
323. #2362D59semidirect
389. C59#26751semidirect

Maximal subgroups

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

subgroupindex
#214284482
#214254112
#214252892
#214252472
#214250972
#214250532
#214247922
#214247232
#214247112
#214246782
#214244512
#214244182
#214241912
#214239262
C59×#23622
#2675159