Finite Group Databasebeta

← #28605364

Subgroups — #28605364

Order 15104.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838111C211yesmaximal, characteristic
2C2×#16837529C211yesmaximal, characteristic, complemented
3#21427048C211yesmaximal, characteristic, complemented
4#21425067C211yesmaximal, characteristic, complemented
5#21425063C211yesmaximal, characteristic, complemented
6#21425060C211yesmaximal, characteristic, complemented
7#21425052C211yesmaximal, characteristic, complemented
8#21425051C211yesmaximal, characteristic, complemented
9#21425046C211yesmaximal, characteristic, complemented
10#21424104C211yesmaximal, characteristic
11#21424094C211yesmaximal, characteristic
12#21423923C211yesmaximal, characteristic
13#21423923C211yesmaximal, characteristic
14C4×#14273648C211yesmaximal, characteristic
15C59×#1804C211yesmaximal, characteristic
16#16838111—24no—
17#16838111C418yescomplemented
18C2×#14273726C2211yescharacteristic
19C2×#14273719C2211yescharacteristic
20C2×#14273718C2211yescharacteristic
21C2×#14273706—21no—
22C2×#14273704—21no—
23C2×#14273704—21no—
24C2×C4×Dic118C2211yescharacteristic
25C2×D4×Dic59C2211yescharacteristic
26C2×D4×Dic59C2211yescharacteristic
27C22×Dic59⋊C4—21no—
28#16837529—24no—
29C2×Dic59⋊C8C2211yescharacteristic
30C2×Dic59⋊C8C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×#14273637C2211yescharacteristic
34C2×#14273637C2211yescharacteristic
35C2×C4×C59⋊C8C2211yescharacteristic
36#16837306—21no—
37#16837287—21no—
38#16837284—21no—
39#16837271C2211yescharacteristic
40#16837267C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44#16837253—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837164C2211yescharacteristic
55C2×C4×D4×C59C2211yescharacteristic
56C2×C4⋊C8×C59C2211yescharacteristic
57C2×#102×C59C2211yescharacteristic
58C2×#102×C59C2211yescharacteristic
59C59×#382C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#340C2211yescharacteristic
62C22×Dic118—21no—
63C22×Dic118D411yescharacteristic
64C23×Dic59—21no—
65C23×Dic59—21no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273726—22no—
69#14273726C2×C414yes—
70#14273719—22no—
71#14273719C2×C414yes—
72#14273718—22no—
73#14273718C2×C414yes—
74#14273706—28no—
75#14273704—28no—
76#14273704—28no—
77C4×Dic118C2×C414yes—
78C4×Dic118D414yes—
79D4×Dic59—22no—
80D4×Dic59—22no—
81D4×Dic59C2×C414yes—
82D4×Dic59C2×C414yes—
83C2×C4⋊Dic59—21no—
84C2×C4⋊Dic59D411yescharacteristic
85C2×C4⋊Dic59C2311yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×C4⋊Dic59C2311yescharacteristic
88C2×C4⋊Dic59C2311yescharacteristic
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×Dic59⋊C4—21no—
92C2×Dic59⋊C4—21no—
93C2×Dic59⋊C4—28no—
94C2×Dic59⋊C4—21no—
95C2×Dic59⋊C4—21no—
96C2×Dic59⋊C4D411yescharacteristic
97C2×Dic59⋊C4D411yescharacteristic
98C2×C22⋊Dic59—21no—
99C2×C22⋊Dic59—21no—
100C2×C4×Dic59—21no—
101C2×C4×Dic59—21no—
102C2×C4×Dic59—21no—
103C2×C4×Dic59C2311yescharacteristic
104C2×C4×Dic59C2311yescharacteristic
105Dic59⋊C8—22no—
106Dic59⋊C8—22no—
107#14273648—22no—
108#14273648Q814yescomplemented
109#14273637—22no—
110#14273637—22no—
111C4×C59⋊C8—22no—
112C59⋊#95—21no—
113C59⋊#95—21no—
114C59⋊#95—21no—
115C59⋊#95—21no—
116C59⋊#95—21no—
117C22×D4×C59C2311yescharacteristic
118C23×C4×C59—21no—
119C22×C8×C59C2311yescharacteristic
120C22×C8×C59C2311yescharacteristic
121C4×D4×C59—22no—
122C4×D4×C59C2×C414yes—
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C4⋊C4×C59C2311yescharacteristic
126C2×C22⋊C4×C59—21no—
127C2×C42×C59C2311yescharacteristic
128C4⋊C8×C59—22no—
129#102×C59—22no—
130#102×C59—22no—
131#95×C59—21no—
132#95×C59—21no—
133C24×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59C22×C412yes—
138C2×D4×C59C2×Q812yes—
139C2×D4×C59#4112yes—
140C22×C4×C59—21no—
141C22×C4×C59—24no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—81no—
145C22×C4×C59—81no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—24no—
153C22×Dic59—24no—
154C22×Dic59—21no—
155C22×Dic59—42no—
156C22×Dic59—42no—
157C22×Dic59—81no—
158C22×Dic59—81no—
159C22×Dic59C2×D411yescharacteristic
160C22×Dic59C2×D411yescharacteristic
161C22×Dic59C2×D411yescharacteristic
162C22×Dic59C2×D411yescharacteristic
163C2×Dic118—24no—
164C2×Dic118—41no—
165C2×Dic118—41no—
166C2×Dic118—81no—
167C2×Dic118—81no—
168C2×Dic118C22⋊C414yes—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59C22×C412yes—
181C22⋊C4×C59—24no—
182C42×C59C22×C412yes—
183C42×C59C2×D412yes—
184C22⋊Dic59—24no—
185C22⋊Dic59—24no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—24no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59C22⋊C414yes—
193C4⋊Dic59C22×C412yes—
194C4⋊Dic59C22×C412yes—
195C4⋊Dic59C2×Q812yes—
196Dic59⋊C4—24no—
197Dic59⋊C4—24no—
198Dic59⋊C4—24no—
199Dic59⋊C4—24no—
200Dic59⋊C4—42no—
201Dic59⋊C4—81no—
202Dic59⋊C4C22⋊C414yes—
203Dic59⋊C4C22⋊C414yes—
204C4×Dic59—42no—
205C4×Dic59—81no—
206C4×Dic59—81no—
207C4×Dic59C22×C412yes—
208C4×Dic59C22×C412yes—
209C4×Dic59C2×D412yes—
210C4×Dic59C2×D412yes—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8C2×Q812yes—
217C23×C59—22no—
218C23×C59—22no—
219C23×C59—22no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59C22×D411yescharacteristic
225D4×C59—22no—
226D4×C59—22no—
227D4×C59—22no—
228D4×C59C4×Q814yes—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—22no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—84no—
243C2×C4×C59C2×C22⋊C412yes—
244C2×C4×C59C2×C22⋊C412yes—
245C2×C4×C59C23×C411yescharacteristic
246C2×C4×C59C22×D411yescharacteristic
247C2×C4×C59C22×Q811yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C592−1+411yescharacteristic
250C2×C4×C592−1+411yescharacteristic
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—22no—
260C2×Dic59—22no—
261C2×Dic59—22no—
262C2×Dic59—48no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—84no—
273C2×Dic59C2×C22⋊C412yes—
274C2×Dic59C2×C22⋊C412yes—
275C2×Dic59C2×C22⋊C412yes—
276C2×Dic59C2×C22⋊C412yes—
277Dic118—44no—
278Dic118—84no—
279C8×C59—41no—
280C8×C59—41no—
281C59⋊C8—22no—
282C59⋊C8—41no—
283#22526—591nomaximal, Sylow
284C22×C59—24no—
285C22×C59—42no—
286C22×C59—42no—
287C22×C59—42no—
288C22×C59—42no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59C22×C22⋊C411yescharacteristic
292C22×C59#54811yescharacteristic
293C22×C59D4×Q811yescharacteristic
294C22×C59#57511yescharacteristic
295C22×C59#57511yescharacteristic
296C22×C59#57611yescharacteristic
297C22×C59#57711yescharacteristic
298C4×C59—22no—
299C4×C59—22no—
300C4×C59—42no—
301C4×C59—81no—
302C4×C59—81no—
303C4×C59C2×C4×Q811yescharacteristic
304C4×C59#51911yescharacteristic
305C4×C59#54811yescharacteristic
306C4×C59D4×Q811yescharacteristic
307Dic59—22no—
308Dic59—22no—
309Dic59—22no—
310Dic59—22no—
311Dic59—42no—
312Dic59—42no—
313Dic59—42no—
314Dic59—81no—
315C2×#554—591no—
316C2×#446—591no—
317#2355—591no—
318#1993—591no—
319#1992—591no—
320#1991—591no—
321#1989—591no—
322#1989—591no—
323#1988—591no—
324#1804—591no—
325#1804D5911yescharacteristic
326#1802—591no—
327#1762—591no—
328#1762—591no—
329C4×#102—591no—
330C2×C59—44no—
331C2×C59Q8×C22⋊C411yescharacteristic
332C2×C59#289611yescharacteristic
333C2×C59#289711yescharacteristic
334C2×C59#337011yescharacteristic
335C2×C59#337611yescharacteristic
336C2×C59#338811yescharacteristic
337C2×C59#338911yescharacteristic
338#554—598no—
339#554—1184no—
340C2×C4⋊Q8—591no—
341C2×#125—591no—
342C2×#125—591no—
343C2×C22⋊Q8—1181no—
344C2×C22⋊Q8—1181no—
345C2×C22⋊Q8—1181no—
346C2×C4×Q8—591no—
347C2×C4×D4—591no—
348C2×C4×D4—591no—
349C2×C4×D4C2×D5911yescharacteristic
350C22×C4⋊C4—1181no—
351#446—1184no—
352C2×C4⋊C8—591no—
353C2×C4⋊C8—591no—
354C2×C4⋊C8C2×D5911yescharacteristic
355C2×#102—591no—
356C2×#102—591no—
357C2×#102C2×D5911yescharacteristic
358C2×#102C2×D5911yescharacteristic
359C2×#97—591no—
360C2×#97—591no—
361C2×C4×C8—591no—
362#391—1181no—
363#387—1181no—
364#387—1181no—
365#384—591no—
366#382—591no—
367#382C2×D5911yescharacteristic
368C4×C4⋊C4—591no—
369C4×C4⋊C4—591no—
370C4×C4⋊C4—591no—
371C4×C4⋊C4—1181no—
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340C2×D5911yescharacteristic
383#340C2×D5911yescharacteristic
384C59#2252611yescharacteristic, complemented, Sylow
385C22×Q8—591no—
386C22×Q8—1181no—
387C22×D4C22×D5911yescharacteristic
388C23×C4—21no—
389C23×C4—1181no—
390C23×C4—1181no—
391C22×C8—591no—
392C22×C8—591no—
393C22×C8C22×D5911yescharacteristic
394C22×C8C22×D5911yescharacteristic
395C4⋊Q8—594no—
396C4⋊Q8—1182no—
397#125—594no—
398#125—594no—
399#125—1182no—
400#125—1182no—
401C22⋊Q8—1188no—
402C22⋊Q8—1188no—
403C22⋊Q8—1188no—
404C4×Q8—594no—
405C4×Q8—594no—
406C4×D4—22no—
407C4×D4—594no—
408C4×D4—594no—
409C4×D4—1182no—
410C4×D4—1182no—
411C4×D4C4×D5914yes—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—1188no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4C22×D5911yescharacteristic
428C2×C4⋊C4C22×D5911yescharacteristic
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4—1181no—
433C2×C42—591no—
434C2×C42—591no—
435C2×C42—1181no—
436C2×C42—1181no—
437C2×C42—1181no—
438C2×C42C22×D5911yescharacteristic
439C4⋊C8—22no—
440C4⋊C8—1182no—
441C4⋊C8—1182no—
442#102—22no—
443#102—22no—
444#102—594no—
445#102—1182no—
446#97—1182no—
447#97—1182no—
448C4×C8—1182no—
449#95—21no—
450#95—21no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456C24—21no—
457C2×Q8—594no—
458C2×Q8—1184no—
459C2×Q8—4721no—
460C2×Q8—4721no—
461C2×Q8—2361no—
462C2×Q8—2361no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4C2×C4×D5912yes—
467C2×D4#1174568712yes—
468C2×D4Q8×D5912yes—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—591no—
476C22×C4—591no—
477C22×C4—591no—
478C22×C4—591no—
479C22×C4—1184no—
480C22×C4—1184no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4—4721no—
486C22×C4—4721no—
487C22×C4—2362no—
488C22×C4—2362no—
489C22×C4D4×D5911yescharacteristic
490C22×C4D4×D5911yescharacteristic
491C22×C4C23×D5911yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—592no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—594no—
510C4⋊C4—594no—
511C4⋊C4—594no—
512C4⋊C4—592no—
513C4⋊C4—592no—
514C4⋊C4—592no—
515C4⋊C4—1184no—
516C4⋊C4—1184no—
517C4⋊C4—1184no—
518C4⋊C4—1184no—
519C4⋊C4—1184no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—4721no—
526C4⋊C4—2362no—
527C4⋊C4C2×C4×D5912yes—
528C22⋊C4—24no—
529C22⋊C4—1184no—
530C22⋊C4—1184no—
531C42—592no—
532C42—592no—
533C42—592no—
534C42—592no—
535C42—4721no—
536C42—4721no—
537C42—2362no—
538C42C2×C4×D5912yes—
539C42D4×D5912yes—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C2×D4×D5911yescharacteristic
548Q8—4724no—
549Q8—2364no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4#1427369214yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—592no—
569C2×C4—592no—
570C2×C4—592no—
571C2×C4—592no—
572C2×C4—1181no—
573C2×C4—1181no—
574C2×C4—1181no—
575C2×C4—1181no—
576C2×C4—1181no—
577C2×C4—1181no—
578C2×C4—1181no—
579C2×C4—1181no—
580C2×C4—1182no—
581C2×C4—1182no—
582C2×C4—1182no—
583C2×C4—4724no—
584C2×C4—2368no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4C22⋊C4×D5912yes—
595C2×C4C22⋊C4×D5912yes—
596C2×C4C22×C4×D5911yescharacteristic
597C2×C4C2×D4×D5911yescharacteristic
598C2×C4C2×Q8×D5911yescharacteristic
599C2×C4C2×#1174568711yescharacteristic
600C2×C4#1427377411yescharacteristic
601C2×C4#1427377411yescharacteristic
602C8—41no—
603C8—41no—
604C8—1182no—
605C8—2361no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C2×C22⋊C4×D5911yescharacteristic
614C22#1683805811yescharacteristic
615C22#1683806511yescharacteristic
616C22#1683819911yescharacteristic
617C22#1683819911yescharacteristic
618C22#1683820611yescharacteristic
619C22#1683821211yescharacteristic
620C4—22no—
621C4—22no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—1182no—
626C4—1182no—
627C4—1182no—
628C4—1182no—
629C4—4721no—
630C4—2362no—
631C4—2362no—
632C4—2362no—
633C4C2×#1427369211yescharacteristic
634C4#1683787911yescharacteristic
635C4#1683805811yescharacteristic
636C4#1683806511yescharacteristic
637C2—44no—
638C2#2142692811yescharacteristic
639C2#2143261111yescharacteristic
640C2#2143262111yescharacteristic
641C2#2143685311yescharacteristic
642C2#2143692711yescharacteristic
643C2#2143704511yescharacteristic
644C2#2143705411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1804C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1804D59
hypocenterC59#22526
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#16837529C2semidirect
3. #21427048C2semidirect
4. #21425067C2semidirect
5. #21425063C2semidirect
6. #21425060C2semidirect
7. #21425052C2semidirect
8. #21425051C2semidirect
9. #21425046C2semidirect
17. #16838111C4semidirect
108. #14273648Q8semidirect
384. C59#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#168381112
C2×#168375292
#214270482
#214250672
#214250632
#214250602
#214250522
#214250512
#214250462
#214241042
#214240942
#214239232
C4×#142736482
C59×#18042
#2252659