Finite Group Databasebeta

← #28651559

Subgroups — #28651559

Order 15104.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837702C211yesmaximal, characteristic, complemented
2C2×#16837445C211yesmaximal, characteristic, complemented
3#21427820C211yesmaximal, characteristic, complemented
4#21425617C211yesmaximal, characteristic, complemented
5#21425597C211yesmaximal, characteristic, complemented
6#21425470C211yesmaximal, characteristic, complemented
7#21425434C211yesmaximal, characteristic, complemented
8#21424864C211yesmaximal, characteristic, complemented
9#21424667C211yesmaximal, characteristic, complemented
10#21424535C211yesmaximal, characteristic
11#21424414C211yesmaximal, characteristic
12#21424343C211yesmaximal, characteristic, complemented
13#21424295C211yesmaximal, characteristic, complemented
14#21424281C211yesmaximal, characteristic, complemented
15C59×#2363C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19#16837702—24no—
20#16837445—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×C4⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34#16837344—21no—
35#16837336—21no—
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837274—21no—
39#16837268C2211yescharacteristic
40#16837262C2211yescharacteristic
41#16837260—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#121×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C59×#390—21no—
54C59×#390C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#388—21no—
57C59×#386—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59—21no—
60C4×C22⋊C4×C59C2211yescharacteristic
61C23×Dic59—21no—
62C2×#11745664—22no—
63C2×#11745664—22no—
64C2×#11745664—22no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273719—22no—
70#14273719—41no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—41no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83#14273660—22no—
84#14273660—41no—
85#14273660D414yescomplemented
86#14273658—22no—
87#14273658—22no—
88#14273658—22no—
89C4⋊(C59⋊C8)—22no—
90C4⋊(C59⋊C8)—22no—
91#14273648—22no—
92#14273648—22no—
93#14273648—22no—
94#14273648—41no—
95C22⋊(C59⋊C8)—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×D4×C59—21no—
102C22×D4×C59C2311yescharacteristic
103C23×C4×C59—21no—
104C23×C4×C59C2311yescharacteristic
105#121×C59—22no—
106#121×C59Q814yes—
107C4×D4×C59—22no—
108C4×D4×C59—41no—
109#117×C59—22no—
110#117×C59—41no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59D411yescharacteristic
121C2×C22⋊C4×C59D411yescharacteristic
122C2×C42×C59—21no—
123C2×C42×C59—21no—
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126#95×C59—21no—
127#95×C59—21no—
128#95×C59D411yescharacteristic
129#95×C59D411yescharacteristic
130C24×C59—21no—
131C24×C59C2×D411yescharacteristic
132C2×D4×C59—24no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—81no—
139C2×D4×C59—81no—
140C2×D4×C59C2×Q812yes—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—42no—
149C22×C4×C59—42no—
150C22×C4×C59—81no—
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×Q812yes—
157C22×C4×C59#4112yes—
158C22×C4×C59C2411yescharacteristic
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—42no—
164C22×Dic59—42no—
165C22×Dic59—81no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—81no—
171C4⋊C4×C59C2×D412yes—
172C4⋊C4×C59C2×Q812yes—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—41no—
175C22⋊C4×C59—41no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—42no—
179C22⋊C4×C59—81no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—42no—
185C42×C59—81no—
186C22⋊Dic59—42no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—42no—
195C4⋊Dic59C2×D412yes—
196Dic59⋊C4—42no—
197Dic59⋊C4—42no—
198Dic59⋊C4—81no—
199Dic59⋊C4—81no—
200#11745664—28no—
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—
210C2×C59⋊C8—21no—
211C2×C59⋊C8C2×D412yes—
212C2×C59⋊C8#4112yes—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—22no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59C22⋊Q812yes—
227C23×C59C22×D411yescharacteristic
228D4×C59—22no—
229D4×C59—22no—
230D4×C59—41no—
231D4×C59—41no—
232D4×C59—84no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—22no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
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×C59—84no—
261C2×C4×C59C22⋊Q812yes—
262C2×C4×C59C22×D411yescharacteristic
263C2×C4×C59C22×Q811yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59H(5,𝔽2)11yescharacteristic
267C2×C4×C592−1+411yescharacteristic
268C2×Dic59—22no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—84no—
281C59⋊C8—22no—
282C59⋊C8—22no—
283C59⋊C8—41no—
284C59⋊C8—41no—
285#26363—591nomaximal, Sylow
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—22no—
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—42no—
294C22×C59—42no—
295C22×C59—42no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59—81no—
299C22×C59—81no—
300C22×C59C2×C22⋊Q811yescharacteristic
301C22×C59D4211yescharacteristic
302C22×C59#54711yescharacteristic
303C22×C59#57511yescharacteristic
304C22×C59#57511yescharacteristic
305C22×C59#57611yescharacteristic
306C22×C59#57711yescharacteristic
307C4×C59—22no—
308C4×C59—42no—
309C4×C59—41no—
310C4×C59—41no—
311C4×C59—41no—
312C4×C59—41no—
313C4×C59—42no—
314C4×C59—81no—
315C4×C59#52511yescharacteristic
316C4×C59#53911yescharacteristic
317C4×C59D4×Q811yescharacteristic
318C4×C59#55411yescharacteristic
319Dic59—42no—
320Dic59—81no—
321Dic59—81no—
322Dic59—81no—
323C2×#478—591no—
324C2×#428—591no—
325#2442—591no—
326#2363D5911yescharacteristic
327#2092—591no—
328#2088—591no—
329#2065—591no—
330#2059—591no—
331#1949—591no—
332#1918—591no—
333#1895—591no—
334#1874—591no—
335#1862—591no—
336#1854—591no—
337#1851—591no—
338C2×C59—42no—
339C2×C59—82no—
340C2×C59#244911yescharacteristic
341C2×C59#308011yescharacteristic
342C2×C59#308111yescharacteristic
343C2×C59#327111yescharacteristic
344C2×C59#327311yescharacteristic
345C2×C59#334811yescharacteristic
346C2×C59#335511yescharacteristic
347C22×M16—591no—
348C2×#125—591no—
349C2×#121C2×D5911yescharacteristic
350C2×C4×D4C2×D5911yescharacteristic
351C2×#117C2×D5911yescharacteristic
352C22×C4⋊C4—591no—
353#478—1184no—
354#428—1184no—
355C2×#107—591no—
356C2×#107—591no—
357C2×#107—591no—
358C2×#106—591no—
359C2×#106—591no—
360C2×#106—591no—
361C2×C4⋊C8—591no—
362C2×C4⋊C8—591no—
363C2×#102—591no—
364C2×#102—591no—
365C2×#102—591no—
366C2×#102—591no—
367C2×C22⋊C8—591no—
368#400—1181no—
369#398—1181no—
370#390—21no—
371#390C2×D5911yescharacteristic
372#389—591no—
373#389—591no—
374#389C2×D5911yescharacteristic
375#388—21no—
376#386—21no—
377#385—1181no—
378#382—591no—
379#380—591no—
380#380—1181no—
381C4×C4⋊C4C2×D5911yescharacteristic
382C4×C22⋊C4—21no—
383C4×C22⋊C4C2×D5911yescharacteristic
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#340—591no—
390#336—591no—
391#336—591no—
392C59#2636311yescharacteristic, complemented, Sylow
393C22×D4—21no—
394C22×D4C22×D5911yescharacteristic
395C23×C4—21no—
396C23×C4—1181no—
397C23×C4C22×D5911yescharacteristic
398C2×M16—1182no—
399C2×M16—1182no—
400C2×M16—1182no—
401C22×C8—591no—
402C22×C8—591no—
403C22×C8—591no—
404C22×C8—591no—
405#125—1182no—
406#125—2361no—
407#121—22no—
408#121Dic11814yes—
409C4×D4—22no—
410C4×D4—41no—
411#117—22no—
412#117—41no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—1182no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—2361no—
425C2×C4⋊C4C22×D5911yescharacteristic
426C2×C4⋊C4C22×D5911yescharacteristic
427C2×C4⋊C4C22×D5911yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1181no—
434C2×C22⋊C4C22⋊D5911yescharacteristic
435C2×C22⋊C4C22⋊D5911yescharacteristic
436C2×C42—21no—
437C2×C42—21no—
438C2×C42C22×D5911yescharacteristic
439C2×C42C22×D5911yescharacteristic
440#107—594no—
441#107—1182no—
442#107—2361no—
443#106—1182no—
444#106—1182no—
445#106—1182no—
446C4⋊C8—1182no—
447C4⋊C8—1182no—
448#102—1182no—
449#102—1182no—
450#102—1182no—
451#102—2361no—
452C22⋊C8—1182no—
453#95—21no—
454#95—21no—
455#95—1181no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460#95C22⋊D5911yescharacteristic
461#95C22⋊D5911yescharacteristic
462C24—21no—
463C24C2×C22⋊D5911yescharacteristic
464C2×D4—24no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—81no—
471C2×D4—81no—
472C2×D4C2×Dic11812yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4—4721no—
488C22×C4—2362no—
489C22×C4—2362no—
490C22×C4C2×Dic11812yes—
491C22×C4#1174568512yes—
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C2×C22⋊D5911yescharacteristic
497C22×C4C23×D5911yescharacteristic
498M16—1188no—
499C2×C8—592no—
500C2×C8—592no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—81no—
516C4⋊C4—592no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—4721no—
525C4⋊C4—4721no—
526C4⋊C4—2362no—
527C4⋊C4—2362no—
528C4⋊C4—2362no—
529C4⋊C4C2×Dic11812yes—
530C4⋊C4D4×D5912yes—
531C22⋊C4—24no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—81no—
538C22⋊C4—2362no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—42no—
544C42—81no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23#1427370612yes—
559C23C22×C22⋊D5911yescharacteristic
560D4—22no—
561D4—22no—
562D4—41no—
563D4—41no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
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—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—84no—
593C2×C4—1182no—
594C2×C4—4724no—
595C2×C4—2361no—
596C2×C4—2361no—
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—2361no—
606C2×C4#1427370612yes—
607C2×C4C2×D4×D5911yescharacteristic
608C2×C4C22×Dic11811yescharacteristic
609C2×C4#41×D5911yescharacteristic
610C2×C4C2×#1174568511yescharacteristic
611C2×C4#1427377211yescharacteristic
612C2×C4#1427377411yescharacteristic
613C8—1182no—
614C8—1182no—
615C8—2361no—
616C8—2361no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×#1427370611yescharacteristic
632C22D4×C22⋊D5911yescharacteristic
633C22#1683805111yescharacteristic
634C22#1683820011yescharacteristic
635C22#1683820011yescharacteristic
636C22#1683820511yescharacteristic
637C22#1683821311yescharacteristic
638C4—22no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—41no—
645C4—81no—
646C4—4721no—
647C4—4721no—
648C4—4721no—
649C4—2362no—
650C4#1683791111yescharacteristic
651C4#1683799411yescharacteristic
652C4D4×Dic11811yescharacteristic
653C4#1683811111yescharacteristic
654C2—42no—
655C2—82no—
656C2#2142789911yescharacteristic
657C2#2143391811yescharacteristic
658C2#2143392711yescharacteristic
659C2#2143567711yescharacteristic
660C2#2143570311yescharacteristic
661C2#2143661111yescharacteristic
662C2#2143669611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2363C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2363D59
hypocenterC59#26363
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. C2×#16837702C2semidirect
2. C2×#16837445C2semidirect
3. #21427820C2semidirect
4. #21425617C2semidirect
5. #21425597C2semidirect
6. #21425470C2semidirect
7. #21425434C2semidirect
8. #21424864C2semidirect
9. #21424667C2semidirect
12. #21424343C2semidirect
13. #21424295C2semidirect
14. #21424281C2semidirect
85. #14273660D4semidirect
392. C59#26363semidirect

Maximal subgroups

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

subgroupindex
C2×#168377022
C2×#168374452
#214278202
#214256172
#214255972
#214254702
#214254342
#214248642
#214246672
#214245352
#214244142
#214243432
#214242952
#214242812
C59×#23632
#2636359