Finite Group Databasebeta

← #28623001

Subgroups — #28623001

Order 15104.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837689C211yesmaximal, characteristic, complemented
2C2×#16837604C211yesmaximal, characteristic, complemented
3C4×#14273701C211yesmaximal, characteristic
4#21425289C212yesmaximal, complemented
5#21425097C212yesmaximal
6#21425045C211yesmaximal, characteristic, complemented
7#21425020C211yesmaximal, characteristic, complemented
8#21424451C212yesmaximal, complemented
9#21424364C212yesmaximal, complemented
10#21424076C211yesmaximal, characteristic, complemented
11C2×C59×#533C211yesmaximal, characteristic, complemented
12C22×#11745671C2212yes—
13C2×#14273701C2211yescharacteristic
14C2×C4×D236C2211yescharacteristic
15C2×C4×C22⋊D59—21no—
16#16837689—24no—
17#16837604—24no—
18C2×#14273658C2212yes—
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2212yes—
22C2×#14273651C2212yes—
23C2×#14273651C2212yes—
24C2×#14273647C2212yes—
25C2×C22⋊(C59⋊C8)C2212yes—
26#16837304C2211yescharacteristic
27#16837300C2211yescharacteristic
28#16837293—21no—
29#16837281—21no—
30#16837262C2211yescharacteristic
31C4×C4⋊Dic59C2211yescharacteristic
32C4×(C2×D59)⋊C4—21no—
33#16837164C2212yes—
34#16837164C2212yes—
35#16837157C2212yes—
36C59×#533—24no—
37C59×#533—24no—
38C2×C4⋊Q8×C59C2211yescharacteristic
39C2×C4⋊Q8×C59C2211yescharacteristic
40C2×#125×C59—21no—
41C2×C22⋊Q8×C59—21no—
42C2×C22⋊Q8×C59C2212yes—
43C2×C4×Q8×C59—21no—
44C2×C4×Q8×C59C2212yes—
45C2×#117×C59—21no—
46C22×C42×C59C2211yescharacteristic
47C22×D236C2311yescharacteristic
48C22×C22⋊D59—21no—
49C22×C4×D59—21no—
50C2×#11745671—24no—
51C2×#11745671—24no—
52C2×#11745671—42no—
53C22×C59⋊C8C2312yes—
54C22×C59⋊C8C2312yes—
55#14273701—22no—
56#14273701—22no—
57C4×D236—22no—
58C4×D236—22no—
59C4×C22⋊D59—44no—
60C2×C4⋊Dic59C2311yescharacteristic
61C2×C4⋊Dic59C2311yescharacteristic
62C2×C4⋊Dic59C2311yescharacteristic
63C2×Dic59⋊C4—21no—
64C2×C22⋊Dic59—21no—
65C2×(C2×D59)⋊C4—21no—
66C2×(C2×D59)⋊C4—21no—
67C2×(C2×D59)⋊C4—21no—
68C2×(C2×D59)⋊C4—21no—
69C2×C4×Dic59—21no—
70#14273658—24no—
71C4⋊(C59⋊C8)—22no—
72C4⋊(C59⋊C8)—22no—
73#14273651—24no—
74#14273651—24no—
75#14273651—42no—
76#14273647—24no—
77C22⋊(C59⋊C8)—42no—
78C59⋊#95—21no—
79C59⋊#95—21no—
80C59⋊#95—21no—
81C22×Q8×C59—21no—
82C22×Q8×C59C2312yes—
83C23×C4×C59D411yescharacteristic
84C23×C4×C59D411yescharacteristic
85C23×C4×C59C2311yescharacteristic
86C4⋊Q8×C59—22no—
87C4⋊Q8×C59—22no—
88C4⋊Q8×C59D414yescomplemented
89C4⋊Q8×C59D414yescomplemented
90#125×C59—28no—
91C22⋊Q8×C59—28no—
92C22⋊Q8×C59—24no—
93C22⋊Q8×C59—24no—
94C4×Q8×C59—28no—
95C4×Q8×C59—24no—
96C4×Q8×C59—24no—
97#117×C59—28no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—22no—
102C2×C4⋊C4×C59—22no—
103C2×C4⋊C4×C59C2312yes—
104C2×C22⋊C4×C59—22no—
105C2×C42×C59—22no—
106C2×C42×C59—22no—
107C2×C42×C59—22no—
108C2×C42×C59D411yescharacteristic
109C2×C42×C59D411yescharacteristic
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112C24×C59C2×D411yescharacteristic
113C23×D59—21no—
114C2×Q8×C59—24no—
115C2×Q8×C59—22no—
116C2×Q8×C59—22no—
117C2×Q8×C59—22no—
118C2×Q8×C59—22no—
119C2×Q8×C59—22no—
120C2×Q8×C59—42no—
121C2×Q8×C59—42no—
122C2×Q8×C59C2×D414yes—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
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—41no—
135C22×C4×C59—41no—
136C22×C4×C59SD1614yescomplemented
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2411yescharacteristic
143C2×C22⋊D59—24no—
144C2×C22⋊D59—42no—
145C2×C22⋊D59—42no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236#4112yes—
155C2×C4×D59—42no—
156C2×C4×D59—42no—
157C2×C4×D59—42no—
158C4⋊C4×C59—28no—
159C4⋊C4×C59—28no—
160C4⋊C4×C59—24no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59C2×D414yes—
165C22⋊C4×C59—28no—
166C42×C59—28no—
167C42×C59—22no—
168C42×C59—22no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59C2×D412yes—
172C42×C59C2×D412yes—
173C22⋊Dic59—42no—
174#11745671—48no—
175(C2×D59)⋊C4—24no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—42no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59#4112yes—
185Dic59⋊C4—42no—
186C4×Dic59—42no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8—22no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59C2×SD1612yes—
200C23×C59C22×D411yescharacteristic
201C22×D59—22no—
202C22×D59—41no—
203C22×D59—41no—
204C22×D59—41no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—41no—
208Q8×C59—24no—
209Q8×C59—24no—
210Q8×C59—48no—
211Q8×C59—42no—
212Q8×C59—42no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—28no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—24no—
221C2×C4×C59—24no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—44no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—42no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59#12012yes—
236C2×C4×C59#12012yes—
237C2×C4×C59C2×SD1612yes—
238C2×C4×C59C2×SD1612yes—
239C2×C4×C59#13712yes—
240C2×C4×C59C22×D411yescharacteristic
241C2×C4×C59C22×D411yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59H(5,𝔽2)11yescharacteristic
245C2×C4×C592−1+411yescharacteristic
246C22⋊D59—48no—
247C2×Dic59—22no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256D236—22no—
257D236—22no—
258D236—41no—
259D236—41no—
260C4×D59—84no—
261C59⋊C8—42no—
262C59⋊C8—42no—
263#24151—591nomaximal, Sylow
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59C2×#12011yescharacteristic
272C22×C59#52511yescharacteristic
273C22×C59#53711yescharacteristic
274C22×C59C22×SD1611yescharacteristic
275C22×C59C2×#13711yescharacteristic
276C22×C59#57511yescharacteristic
277C22×C59#57511yescharacteristic
278C2×D59—42no—
279C2×D59—42no—
280C2×D59—42no—
281C2×D59—81no—
282C2×D59—81no—
283C2×D59—81no—
284C2×D59—81no—
285C4×C59—22no—
286C4×C59—22no—
287C4×C59—41no—
288C4×C59—44no—
289C4×C59—41no—
290C4×C59—44no—
291C4×C59#44814yes—
292C4×C59#54611yescharacteristic
293C4×C59#54711yescharacteristic
294C4×C59#54811yescharacteristic
295C4×C59#55011yescharacteristic
296Dic59—42no—
297Dic59—81no—
298Dic59—81no—
299C2×#533D5911yescharacteristic, complemented
300C2×#476—591no—
301C2×#461—591no—
302C4×#121—591no—
303#2035—592no—
304#2000—592no—
305#1987—591no—
306#1981—591no—
307#1880—592no—
308#1865—592no—
309#1797—591no—
310C2×C59—42no—
311C2×C59#240711yescharacteristic
312C2×C59C2×#44811yescharacteristic
313C2×C59#300511yescharacteristic
314C2×C59#310111yescharacteristic
315C2×C59#311611yescharacteristic
316C2×C59#324611yescharacteristic
317C2×C59#325211yescharacteristic
318D59—82no—
319C22×SD16—592no—
320#533—24no—
321#533—24no—
322C2×C4⋊Q8C2×D5911yescharacteristic
323C2×C4⋊Q8C2×D5911yescharacteristic
324C2×#125—21no—
325C2×C22⋊Q8—21no—
326C2×C22⋊Q8C2×D5912yes—
327C2×#121—591no—
328C2×C4×Q8—21no—
329C2×C4×Q8C2×D5912yes—
330C2×C4×D4—591no—
331C2×C4×D4—1181no—
332C2×#117—21no—
333C22×C42C2×D5911yescharacteristic
334#476—1184no—
335#461—1184no—
336C2×#106—592no—
337C2×C4⋊C8—591no—
338C2×C4⋊C8—591no—
339C2×#103—592no—
340C2×#103—592no—
341C2×#103—592no—
342C2×#102—592no—
343C2×C22⋊C8—592no—
344#390—591no—
345#389—591no—
346#388—1181no—
347#386—1181no—
348#380—591no—
349C4×C4⋊C4—591no—
350C4×C22⋊C4—1181no—
351#340—592no—
352#340—592no—
353#336—592no—
354C59#2415111yescharacteristic, complemented, Sylow
355C22×Q8—21no—
356C22×Q8C22×D5912yes—
357C22×D4—591no—
358C22×D4—1181no—
359C23×C4—1181no—
360C23×C4C22⋊D5911yescharacteristic
361C23×C4C22⋊D5911yescharacteristic
362C23×C4C22×D5911yescharacteristic
363C2×SD16—1184no—
364C2×SD16—1184no—
365C2×SD16—2362no—
366C22×C8—592no—
367C22×C8—592no—
368C4⋊Q8—22no—
369C4⋊Q8—22no—
370C4⋊Q8C22⋊D5914yescomplemented
371C4⋊Q8C22⋊D5914yescomplemented
372#125—28no—
373C22⋊Q8—28no—
374C22⋊Q8—24no—
375C22⋊Q8—24no—
376#121—1182no—
377#121—1182no—
378C4×Q8—28no—
379C4×Q8—24no—
380C4×Q8—24no—
381C4×D4—1182no—
382C4×D4—1182no—
383C4×D4—2364no—
384#117—28no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4C22×D5912yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—1181no—
397C2×C22⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—1181no—
405C2×C42C22⋊D5911yescharacteristic
406C2×C42C22⋊D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#106—1184no—
410C4⋊C8—1182no—
411C4⋊C8—1182no—
412#103—1184no—
413#103—1184no—
414#103—2362no—
415#102—1184no—
416C22⋊C8—2362no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420C24—1181no—
421C24C2×C22⋊D5911yescharacteristic
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8C2×C22⋊D5914yes—
431C2×D4—592no—
432C2×D4—1184no—
433C2×D4—1181no—
434C2×D4—1181no—
435C2×D4—1181no—
436C2×D4—1181no—
437C2×D4—1181no—
438C2×D4—2362no—
439C2×D4—2362no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—2362no—
457C22×C4—2362no—
458C22×C4—2362no—
459C22×C4#1174567114yescomplemented
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466SD16—2368no—
467C2×C8—1182no—
468C2×C8—1182no—
469C2×C8—1182no—
470C2×C8—1182no—
471C2×C8—1182no—
472C2×C8—1182no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—592no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—2362no—
486C4⋊C4C2×C22⋊D5914yes—
487C22⋊C4—28no—
488C22⋊C4—1184no—
489C22⋊C4—2362no—
490C22⋊C4—2362no—
491C22⋊C4—2362no—
492C22⋊C4—2362no—
493C42—28no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—2362no—
499C42C2×C22⋊D5912yes—
500C42C2×C22⋊D5912yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—1182no—
508C23—2361no—
509C23—2361no—
510C23—2361no—
511C23—2361no—
512C23—2361no—
513C23—2361no—
514C23C2×#1174567112yes—
515C23C22×C22⋊D5911yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—48no—
519Q8—42no—
520Q8—42no—
521D4—1182no—
522D4—1182no—
523D4—2368no—
524D4—2361no—
525D4—2361no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—1182no—
549C2×C4—4724no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4#1427369612yes—
559C2×C4#1427369612yes—
560C2×C4C2×#1174567112yes—
561C2×C4C2×#1174567112yes—
562C2×C4#1427375812yes—
563C2×C4C22×C22⋊D5911yescharacteristic
564C2×C4C22×C22⋊D5911yescharacteristic
565C2×C4#41×D5911yescharacteristic
566C2×C4#41×D5911yescharacteristic
567C2×C4#1427377311yescharacteristic
568C2×C4#1427377411yescharacteristic
569C8—2362no—
570C8—2362no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—4721no—
579C22—4721no—
580C22—4721no—
581C22—4721no—
582C22—2362no—
583C22—2362no—
584C22—2362no—
585C22C2×#1427369611yescharacteristic
586C22#1683790911yescharacteristic
587C22#1683798011yescharacteristic
588C22C22×#1174567111yescharacteristic
589C22C2×#1427375811yescharacteristic
590C22#1683820011yescharacteristic
591C22#1683820011yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—44no—
596C4—41no—
597C4—44no—
598C4—4721no—
599C4—4721no—
600C4—2362no—
601C4#1683754114yes—
602C4#1683804011yescharacteristic
603C4#1683805111yescharacteristic
604C4#1683805911yescharacteristic
605C4#1683807111yescharacteristic
606C2—42no—
607C2—4722no—
608C2#2142743511yescharacteristic
609C2C2×#1683754111yescharacteristic
610C2#2143333811yescharacteristic
611C2#2143410011yescharacteristic
612C2#2143425211yescharacteristic
613C2#2143540811yescharacteristic
614C2#2143546311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#533C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenterC2×#533D59
hypocenterC59#24151
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#16837689C2semidirect
2. C2×#16837604C2semidirect
4. #21425289C2semidirect
6. #21425045C2semidirect
7. #21425020C2semidirect
8. #21424451C2semidirect
9. #21424364C2semidirect
10. #21424076C2semidirect
11. C2×C59×#533C2semidirect
88. C4⋊Q8×C59D4semidirect
89. C4⋊Q8×C59D4semidirect
136. C22×C4×C59SD16semidirect
299. C2×#533D59semidirect
354. C59#24151semidirect
370. C4⋊Q8C22⋊D59semidirect
371. C4⋊Q8C22⋊D59semidirect
459. C22×C4#11745671semidirect

Maximal subgroups

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

subgroupindex
C2×#168376892
C2×#168376042
C4×#142737012
#214252892
#214250972
#214250452
#214250202
#214244512
#214243642
#214240762
C2×C59×#5332
#2415159