Finite Group Databasebeta

← #28592170

Subgroups — #28592170

Order 15104.

1022 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837443C211yesmaximal, characteristic
2C2×#16837441C211yesmaximal, characteristic
3C2×#16837424C211yesmaximal, characteristic
4#382×D59C211yesmaximal, characteristic
5C4×#14273685C211yesmaximal, characteristic
6#21424867C211yesmaximal, characteristic, complemented
7#21424866C211yesmaximal, characteristic, complemented
8#21424865C211yesmaximal, characteristic, complemented
9#21424864C211yesmaximal, characteristic, complemented
10#21424863C211yesmaximal, characteristic, complemented
11#21424862C211yesmaximal, characteristic, complemented
12#21424861C211yesmaximal, characteristic, complemented
13#21423814C211yesmaximal, characteristic
14#21423814C211yesmaximal, characteristic
15C59×#1949C211yesmaximal, characteristic, complemented
16C22×#11745659C2211yescharacteristic
17C22×#11745659C2211yescharacteristic
18C2×#14273685C2211yescharacteristic
19C2×#14273685C2211yescharacteristic
20C2×C4⋊C4×D59C2211yescharacteristic
21C2×C4⋊C4×D59C2211yescharacteristic
22C2×C42×D59C2211yescharacteristic
23#16837443—24no—
24#16837441—24no—
25C2×#14273661C2211yescharacteristic
26C2×#14273660C2211yescharacteristic
27C2×#14273660C2211yescharacteristic
28C2×#14273659C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273658C2211yescharacteristic
31#16837424C418yescomplemented
32C2×C4⋊(C59⋊C8)C2211yescharacteristic
33C2×Dic59⋊C8C2211yescharacteristic
34C2×Dic59⋊C8C2211yescharacteristic
35#16837268C2211yescharacteristic
36#16837267C2211yescharacteristic
37#16837267C2211yescharacteristic
38#16837266—21no—
39#16837266—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×(C2×D59)⋊C4—21no—
44#16837249—21no—
45#16837247—21no—
46#95×D59—21no—
47#95×D59—21no—
48C42×Dic59—21no—
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837164C2211yescharacteristic
55C2×#107×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×C4⋊C8×C59C2211yescharacteristic
58C59×#382C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#340C2211yescharacteristic
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64C22×C4×D59D411yescharacteristic
65C22×C4×D59D411yescharacteristic
66C22×C4×D59C2311yescharacteristic
67C2×#11745659—22no—
68C2×#11745659—22no—
69C2×#11745659—22no—
70C2×#11745659—22no—
71C2×#11745659C2×C414yes—
72C2×#11745659C2×C414yes—
73C22×C59⋊C8C2311yescharacteristic
74C22×C59⋊C8C2311yescharacteristic
75#14273685—22no—
76#14273685—22no—
77#14273685D414yes—
78#14273685Q814yes—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C4⋊C4×D59—22no—
86C4⋊C4×D59—22no—
87C4⋊C4×D59—22no—
88C4⋊C4×D59—22no—
89C2×(C2×D59)⋊C4—21no—
90C2×(C2×D59)⋊C4—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59—21no—
95C2×C4×Dic59—21no—
96C2×C4×Dic59D411yescharacteristic
97C2×C4×Dic59D411yescharacteristic
98C2×C4×Dic59C2311yescharacteristic
99C42×D59C2×C414yes—
100C42×D59Q814yes—
101#14273661—22no—
102#14273660—22no—
103#14273660—22no—
104#14273659—22no—
105#14273658—22no—
106#14273658—22no—
107C4⋊(C59⋊C8)C2×C414yescomplemented
108Dic59⋊C8C2×C414yescomplemented
109Dic59⋊C8C2×C414yescomplemented
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C59⋊#95—21no—
114C59⋊#95—21no—
115C59⋊#95—21no—
116C59⋊#95—21no—
117C59⋊#95—21no—
118C59⋊#95—21no—
119C22×C8×C59C2311yescharacteristic
120C22×C8×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C42×C59—21no—
126C2×C42×C59C2311yescharacteristic
127#107×C59—22no—
128#106×C59—22no—
129C4⋊C8×C59C2×C414yescomplemented
130#95×C59—21no—
131#95×C59—21no—
132#95×C59—21no—
133C23×D59C2×D411yescharacteristic
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C2×C4×D59—41no—
149C2×C4×D59—41no—
150C2×C4×D59—41no—
151C2×C4×D59—41no—
152C2×C4×D59—42no—
153C2×C4×D59—42no—
154C2×C4×D59—42no—
155C2×C4×D59—42no—
156C2×C4×D59—42no—
157C2×C4×D59—42no—
158C2×C4×D59C4⋊C414yes—
159C2×C4×D59C4⋊C414yes—
160C2×C4×D59C22×C412yes—
161C2×C4×D59C2×D412yes—
162C2×C4×D59C2×Q812yes—
163C2×C4×D59C2×Q812yes—
164C2×C4×D59#4112yes—
165C2×C4×D59#4112yes—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59C22×C412yes—
171C2×C8×C59C22×C412yes—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59C2×D412yes—
179C4⋊C4×C59C2×Q812yes—
180C42×C59—42no—
181C42×C59C22×C412yes—
182C42×C59C2×Q812yes—
183(C2×D59)⋊C4—24no—
184(C2×D59)⋊C4—24no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59C2×D412yes—
192C4⋊Dic59C2×Q812yes—
193Dic59⋊C4—42no—
194Dic59⋊C4—42no—
195C4×Dic59—24no—
196C4×Dic59—24no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C4×Dic59—42no—
200C4×Dic59C4⋊C414yes—
201C4×Dic59C4⋊C414yes—
202C4×Dic59C22×C412yes—
203C4×Dic59C2×Q812yes—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8C22×C412yes—
209C2×C59⋊C8C22×C412yes—
210#11745659—28no—
211#11745659—28no—
212C23×C59C22×D411yescharacteristic
213C22×D59—21no—
214C22×D59—21no—
215C22×D59—21no—
216C22×D59—21no—
217C22×D59C2×C4⋊C412yes—
218C22×D59#12112yes—
219C22×D59C22⋊Q812yes—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59C2×C4⋊C412yes—
237C2×C4×C59C2×C4⋊C412yes—
238C2×C4×C59C23×C411yescharacteristic
239C2×C4×C59C22×D411yescharacteristic
240C2×C4×C59C22×Q811yescharacteristic
241C2×C4×C59C22×Q811yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—22no—
249C2×Dic59—22no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59C2×C4⋊C412yes—
261C2×Dic59C2×C4⋊C412yes—
262C2×Dic59C2×C4⋊C412yes—
263C2×Dic59#12112yes—
264C2×Dic59C22⋊Q812yes—
265C4×D59—44no—
266C4×D59—44no—
267C4×D59—84no—
268C4×D59—84no—
269C4×D59C4×D414yes—
270C4×D59C4×Q814yes—
271C4×D59#12514yes—
272C4×D59C4⋊Q814yes—
273C8×C59—22no—
274C8×C59—22no—
275C59⋊C8—22no—
276C59⋊C8—22no—
277#21172—591nomaximal, Sylow
278C22×C59C22×C4⋊C411yescharacteristic
279C22×C59C2×#12111yescharacteristic
280C22×C59C2×C22⋊Q811yescharacteristic
281C22×C59#57511yescharacteristic
282C22×C59#57511yescharacteristic
283C22×C59#57511yescharacteristic
284C22×C59#57511yescharacteristic
285C2×D59—22no—
286C2×D59—22no—
287C2×D59—22no—
288C2×D59—22no—
289C2×D59—22no—
290C2×D59—22no—
291C2×D59#38914yes—
292C4×C59—22no—
293C4×C59—22no—
294C4×C59—42no—
295C4×C59—42no—
296C4×C59—81no—
297C4×C59—81no—
298C4×C59C2×C4×D411yescharacteristic
299C4×C59C2×C4×Q811yescharacteristic
300C4×C59C2×#12511yescharacteristic
301C4×C59C2×C4⋊Q811yescharacteristic
302Dic59—22no—
303Dic59—22no—
304Dic59—42no—
305Dic59—42no—
306Dic59—81no—
307Dic59—81no—
308Dic59#38914yes—
309C2×#428—591no—
310C2×#428—591no—
311C2×#423—591no—
312C2×#382—591no—
313C4×#117—591no—
314#1949—591no—
315#1949—591no—
316#1949—591no—
317#1949—591no—
318#1949—591no—
319#1949—591no—
320#1949—591no—
321#1949D5911yescharacteristic, complemented
322#1736—591no—
323#1736—591no—
324C2×C59C2×#38911yescharacteristic
325C2×C59#291111yescharacteristic
326C2×C59#291111yescharacteristic
327C2×C59#304011yescharacteristic
328C2×C59#304011yescharacteristic
329C2×C59#308011yescharacteristic
330C2×C59#308011yescharacteristic
331D59—24no—
332C22×M16—591no—
333C22×M16—591no—
334C2×#117—591no—
335C2×#117—591no—
336C22×C4⋊C4—591no—
337C22×C4⋊C4—591no—
338C22×C42—591no—
339#428—1184no—
340#428—1184no—
341C2×#107—591no—
342C2×#107—591no—
343C2×#107—591no—
344C2×#107C2×D5911yescharacteristic
345C2×#106—591no—
346C2×#106—591no—
347C2×#106—591no—
348C2×#106C2×D5911yescharacteristic
349#423—598no—
350C2×C4⋊C8—591no—
351C2×C4⋊C8—591no—
352C2×C4⋊C8—591no—
353C2×C4⋊C8C2×D5911yescharacteristic
354#382—591no—
355#382—591no—
356#382—591no—
357#382—1181no—
358#382—1181no—
359#382C2×D5911yescharacteristic
360C4×C4⋊C4—591no—
361C4×C4⋊C4—591no—
362C4×C4⋊C4—591no—
363C4×C4⋊C4C2×D5911yescharacteristic
364C4×C22⋊C4—1181no—
365C4×C22⋊C4—1181no—
366#376—1181no—
367C2×#95—1181no—
368C2×#95—1181no—
369C43—1181no—
370#340—591no—
371#340—591no—
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340C2×D5911yescharacteristic
377#340C2×D5911yescharacteristic
378C59#2117211yescharacteristic, complemented, Sylow
379C23×C4—591no—
380C23×C4—591no—
381C23×C4—591no—
382C23×C4—1181no—
383C23×C4—1181no—
384C2×M16—594no—
385C2×M16—594no—
386C2×M16—1182no—
387C2×M16—1182no—
388C2×M16—1182no—
389C2×M16—1182no—
390C22×C8—591no—
391C22×C8—591no—
392C22×C8C22×D5911yescharacteristic
393C22×C8C22×D5911yescharacteristic
394#117—594no—
395#117—594no—
396#117—1182no—
397#117—1182no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—1182no—
403C2×C4⋊C4—1182no—
404C2×C4⋊C4—1182no—
405C2×C4⋊C4—1182no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C4⋊C4C22×D5911yescharacteristic
410C2×C4⋊C4C22×D5911yescharacteristic
411C2×C4⋊C4C22×D5911yescharacteristic
412C2×C22⋊C4—1181no—
413C2×C22⋊C4—1181no—
414C2×C42—21no—
415C2×C42—591no—
416C2×C42—591no—
417C2×C42—591no—
418C2×C42—594no—
419C2×C42—594no—
420C2×C42—1181no—
421C2×C42—1181no—
422C2×C42—1181no—
423C2×C42—1181no—
424C2×C42—1181no—
425C2×C42C22×D5911yescharacteristic
426#107—22no—
427#107—1182no—
428#107—1182no—
429#107—1182no—
430#106—22no—
431#106—1182no—
432#106—1182no—
433#106—1182no—
434C4⋊C8—594no—
435C4⋊C8—594no—
436C4⋊C8—594no—
437C4⋊C8C4×D5914yescomplemented
438#95—21no—
439#95—21no—
440#95—21no—
441#95—1181no—
442#95—1181no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450C24—591no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—591no—
456C22×C4—592no—
457C22×C4—592no—
458C22×C4—592no—
459C22×C4—592no—
460C22×C4—592no—
461C22×C4—592no—
462C22×C4—591no—
463C22×C4—591no—
464C22×C4—594no—
465C22×C4—594no—
466C22×C4—1181no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—2361no—
471C22×C4—2361no—
472C22×C4—2361no—
473C22×C4—2361no—
474C22×C4—2362no—
475C22×C4—2362no—
476C22×C4—2362no—
477C22×C4—2362no—
478C22×C4—2362no—
479C22×C4—2362no—
480C22×C4D4×D5911yescharacteristic
481C22×C4D4×D5911yescharacteristic
482C22×C4C23×D5911yescharacteristic
483M16—1188no—
484M16—1188no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—592no—
490C2×C8—592no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8C2×C4×D5912yes—
496C2×C8C2×C4×D5912yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—592no—
504C4⋊C4—592no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—2362no—
512C4⋊C4—2362no—
513C4⋊C4D4×D5912yes—
514C4⋊C4Q8×D5912yes—
515C22⋊C4—1184no—
516C22⋊C4—1184no—
517C42—42no—
518C42—594no—
519C42—594no—
520C42—592no—
521C42—592no—
522C42—1184no—
523C42—1184no—
524C42—2362no—
525C42—2362no—
526C42—2362no—
527C42C2×C4×D5912yes—
528C42Q8×D5912yes—
529C23—592no—
530C23—592no—
531C23—592no—
532C23—1181no—
533C23—1181no—
534C23—1181no—
535C23—1181no—
536C23C2×D4×D5911yescharacteristic
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—592no—
554C2×C4—592no—
555C2×C4—592no—
556C2×C4—594no—
557C2×C4—594no—
558C2×C4—594no—
559C2×C4—594no—
560C2×C4—592no—
561C2×C4—592no—
562C2×C4—1181no—
563C2×C4—1181no—
564C2×C4—1181no—
565C2×C4—1181no—
566C2×C4—1182no—
567C2×C4—1182no—
568C2×C4—4724no—
569C2×C4—4724no—
570C2×C4—2364no—
571C2×C4—2364no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4C4⋊C4×D5912yes—
583C2×C4C4⋊C4×D5912yes—
584C2×C4C22×C4×D5911yescharacteristic
585C2×C4C2×D4×D5911yescharacteristic
586C2×C4C2×Q8×D5911yescharacteristic
587C2×C4C2×Q8×D5911yescharacteristic
588C2×C4#41×D5911yescharacteristic
589C2×C4#41×D5911yescharacteristic
590C8—22no—
591C8—22no—
592C8—1182no—
593C8—1182no—
594C22—594no—
595C22—1182no—
596C22—1182no—
597C22—1182no—
598C22—1182no—
599C22—1182no—
600C22—1182no—
601C22C2×C4⋊C4×D5911yescharacteristic
602C22#121×D5911yescharacteristic
603C22C22⋊Q8×D5911yescharacteristic
604C22#1683819811yescharacteristic
605C22#1683819911yescharacteristic
606C22#1683819911yescharacteristic
607C22#1683820111yescharacteristic
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4—594no—
615C4—1182no—
616C4—1182no—
617C4—4721no—
618C4—4721no—
619C4—2362no—
620C4—2362no—
621C4C4×D4×D5911yescharacteristic
622C4C4×Q8×D5911yescharacteristic
623C4#125×D5911yescharacteristic
624C4C4⋊Q8×D5911yescharacteristic
625C2—1184no—
626C2#389×D5911yescharacteristic
627C2#2143268311yescharacteristic
628C2#2143268411yescharacteristic
629C2#2143358511yescharacteristic
630C2#2143358711yescharacteristic
631C2#2143391311yescharacteristic
632C2#2143391511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1949C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1949D59
hypocenterC59#21172
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
6. #21424867C2semidirect
7. #21424866C2semidirect
8. #21424865C2semidirect
9. #21424864C2semidirect
10. #21424863C2semidirect
11. #21424862C2semidirect
12. #21424861C2semidirect
15. C59×#1949C2semidirect
31. #16837424C4semidirect
107. C4⋊(C59⋊C8)C2×C4semidirect
108. Dic59⋊C8C2×C4semidirect
109. Dic59⋊C8C2×C4semidirect
129. C4⋊C8×C59C2×C4semidirect
321. #1949D59semidirect
378. C59#21172semidirect
437. C4⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168374432
C2×#168374412
C2×#168374242
#382×D592
C4×#142736852
#214248672
#214248662
#214248652
#214248642
#214248632
#214248622
#214248612
#214238142
C59×#19492
#2117259