Finite Group Databasebeta

← #28592200

Subgroups — #28592200

Order 15104.

990 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837443C211yesmaximal, characteristic
2C2×#16837437C211yesmaximal, characteristic
3C2×#16837427C211yesmaximal, characteristic
4#21426721C211yesmaximal, characteristic
5#21426450C211yesmaximal, characteristic
6#21424865C211yesmaximal, characteristic, complemented
7#21424863C211yesmaximal, characteristic, complemented
8#21424862C211yesmaximal, characteristic, complemented
9#21424860C211yesmaximal, characteristic, complemented
10#21424858C211yesmaximal, characteristic, complemented
11#21424854C211yesmaximal, characteristic, complemented
12#21424851C211yesmaximal, characteristic, complemented
13#21423814C211yesmaximal, characteristic
14#340×D59C211yesmaximal, characteristic
15C59×#1949C211yesmaximal, characteristic, complemented
16C22×#11745659C2211yescharacteristic
17C22×C8×D59C2211yescharacteristic
18C2×#14273685C2211yescharacteristic
19C2×#14273685C2211yescharacteristic
20C2×#14273684C2211yescharacteristic
21C2×C4⋊C4×D59C2211yescharacteristic
22C2×C4⋊C4×D59C2211yescharacteristic
23#16837443—24no—
24#16837437—24no—
25C2×#14273660C2211yescharacteristic
26C2×#14273660C2211yescharacteristic
27C2×#14273660C2211yescharacteristic
28C2×#14273659C2211yescharacteristic
29C2×#14273659C2211yescharacteristic
30C2×#14273658C2211yescharacteristic
31#16837427C418yescomplemented
32C2×C4⋊(C59⋊C8)C2211yescharacteristic
33C2×Dic59⋊C8C2211yescharacteristic
34C2×Dic59⋊C8C2211yescharacteristic
35#16837299—21no—
36#16837299—21no—
37#16837276—21no—
38#16837276—21no—
39#16837272C2211yescharacteristic
40#16837271C2211yescharacteristic
41#16837267C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4×Dic59⋊C4—21no—
45C4×C4⋊Dic59C2211yescharacteristic
46#16837253—21no—
47C4×(C2×D59)⋊C4—21no—
48#16837249—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×D59C2311yescharacteristic
65C2×#11745659—22no—
66C2×#11745659—22no—
67C2×#11745659C2×C414yes—
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C2×C8×D59—22no—
71C2×C8×D59—22no—
72C2×C8×D59C2×C414yes—
73#14273685—22no—
74#14273685—22no—
75#14273685—22no—
76#14273685D414yes—
77#14273684C2×C414yes—
78#14273684Q814yes—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59—21no—
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×C4⋊Dic59C2311yescharacteristic
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4D411yescharacteristic
90C2×Dic59⋊C4D411yescharacteristic
91C4⋊C4×D59—22no—
92C4⋊C4×D59—22no—
93C4⋊C4×D59—22no—
94C4⋊C4×D59Q814yes—
95C2×(C2×D59)⋊C4—21no—
96C2×(C2×D59)⋊C4—21no—
97C2×(C2×D59)⋊C4D411yescharacteristic
98C2×(C2×D59)⋊C4D411yescharacteristic
99C2×C4×Dic59—21no—
100C2×C4×Dic59—21no—
101C2×C4×Dic59—21no—
102C2×C4×Dic59—21no—
103C2×C4×Dic59—21no—
104C2×C4×Dic59C2311yescharacteristic
105#14273660—22no—
106#14273660—22no—
107#14273660—22no—
108#14273659—22no—
109#14273659—22no—
110#14273658—22no—
111C4⋊(C59⋊C8)C2×C414yescomplemented
112Dic59⋊C8C2×C414yescomplemented
113Dic59⋊C8C2×C414yescomplemented
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—24no—
149C2×C4×D59—42no—
150C2×C4×D59—81no—
151C2×C4×D59—81no—
152C2×C4×D59—81no—
153C2×C4×D59—81no—
154C2×C4×D59C22×C412yes—
155C2×C4×D59C2×D412yes—
156C2×C4×D59C2×Q812yes—
157C2×C4×D59C2×Q812yes—
158C2×C4×D59#4112yes—
159C2×C4×D59#4112yes—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59C22×C412yes—
165C2×C8×C59C22×C412yes—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59C2×D412yes—
173C4⋊C4×C59C2×Q812yes—
174C42×C59—81no—
175C42×C59C22×C412yes—
176C42×C59C2×Q812yes—
177(C2×D59)⋊C4—24no—
178(C2×D59)⋊C4—42no—
179(C2×D59)⋊C4C4⋊C414yes—
180(C2×D59)⋊C4C4⋊C414yes—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—81no—
188C4⋊Dic59—81no—
189C4⋊Dic59C2×D412yes—
190C4⋊Dic59C2×Q812yes—
191Dic59⋊C4—24no—
192Dic59⋊C4—42no—
193Dic59⋊C4—81no—
194Dic59⋊C4—81no—
195Dic59⋊C4C4⋊C414yes—
196Dic59⋊C4C4⋊C414yes—
197C4×Dic59—24no—
198C4×Dic59—42no—
199C4×Dic59—81no—
200C4×Dic59—81no—
201C4×Dic59—81no—
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—
211C8×D59—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—84no—
266C4×D59—84no—
267C4×D59C4×D414yes—
268C4×D59C4×Q814yes—
269C4×D59#12514yes—
270C4×D59C4⋊Q814yes—
271C8×C59—22no—
272C8×C59—22no—
273C59⋊C8—22no—
274C59⋊C8—22no—
275#21175—591nomaximal, Sylow
276C22×C59C22×C4⋊C411yescharacteristic
277C22×C59C2×#12111yescharacteristic
278C22×C59C2×C22⋊Q811yescharacteristic
279C22×C59#57611yescharacteristic
280C22×C59#57711yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57811yescharacteristic
283C2×D59—22no—
284C2×D59—22no—
285C2×D59—22no—
286C2×D59—22no—
287C2×D59—22no—
288C2×D59—22no—
289C2×D59#38914yes—
290C4×C59—22no—
291C4×C59—22no—
292C4×C59—42no—
293C4×C59—42no—
294C4×C59—81no—
295C4×C59—81no—
296C4×C59C2×C4×D411yescharacteristic
297C4×C59C2×C4×Q811yescharacteristic
298C4×C59C2×#12511yescharacteristic
299C4×C59C2×C4⋊Q811yescharacteristic
300Dic59—22no—
301Dic59—22no—
302Dic59—42no—
303Dic59—42no—
304Dic59—81no—
305Dic59—81no—
306Dic59#38914yes—
307C2×#428—591no—
308C2×#427—591no—
309C2×#424—591no—
310#2317—591no—
311#2274—591no—
312#1949—591no—
313#1949—591no—
314#1949—591no—
315#1949D5911yescharacteristic, complemented
316#1948—591no—
317#1948—591no—
318#1947—591no—
319#1947—591no—
320#1736—591no—
321C2×#340—591no—
322C2×C59C2×#38911yescharacteristic
323C2×C59#291211yescharacteristic
324C2×C59#291311yescharacteristic
325C2×C59#304511yescharacteristic
326C2×C59#304611yescharacteristic
327C2×C59#308111yescharacteristic
328C2×C59#308211yescharacteristic
329D59—24no—
330C22×M16—591no—
331C23×C8—591no—
332C2×#117—591no—
333C2×#117—591no—
334C2×#117—591no—
335C22×C4⋊C4—591no—
336C22×C4⋊C4—591no—
337#428—1184no—
338#427—1184no—
339C2×#107—591no—
340C2×#107—591no—
341C2×#107—591no—
342C2×#107C2×D5911yescharacteristic
343C2×#106—591no—
344C2×#106—591no—
345C2×#106—591no—
346C2×#106C2×D5911yescharacteristic
347#424—598no—
348C2×C4⋊C8—591no—
349C2×C4⋊C8—591no—
350C2×C4⋊C8—591no—
351C2×C4⋊C8C2×D5911yescharacteristic
352#389—1181no—
353#389—1181no—
354#385—1181no—
355#385—1181no—
356#384—591no—
357#384—591no—
358#382—591no—
359#382C2×D5911yescharacteristic
360C4×C4⋊C4—591no—
361C4×C4⋊C4—591no—
362C4×C4⋊C4—591no—
363C4×C4⋊C4—1181no—
364C4×C4⋊C4—1181no—
365C4×C4⋊C4C2×D5911yescharacteristic
366C4×C22⋊C4—1181no—
367C4×C22⋊C4—1181no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#340—591no—
373#340—591no—
374#340C2×D5911yescharacteristic
375#340C2×D5911yescharacteristic
376C59#2117511yescharacteristic, complemented, Sylow
377C23×C4—591no—
378C23×C4—1181no—
379C23×C4—1181no—
380C2×M16—594no—
381C2×M16—1182no—
382C2×M16—1182no—
383C22×C8—591no—
384C22×C8—591no—
385C22×C8—594no—
386C22×C8—1182no—
387C22×C8—1182no—
388C22×C8C22×D5911yescharacteristic
389C22×C8C22×D5911yescharacteristic
390#117—594no—
391#117—594no—
392#117—594no—
393#117—1182no—
394#117—1182no—
395#117—1182no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—594no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—1182no—
404C2×C4⋊C4—1182no—
405C2×C4⋊C4—1182no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4C22×D5911yescharacteristic
413C2×C4⋊C4C22×D5911yescharacteristic
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C22⋊C4—591no—
417C2×C22⋊C4—591no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C42—21no—
421C2×C42—591no—
422C2×C42—1181no—
423C2×C42—1181no—
424C2×C42—1181no—
425C2×C42—1181no—
426C2×C42—1181no—
427C2×C42C22×D5911yescharacteristic
428#107—22no—
429#107—1182no—
430#107—1182no—
431#107—1182no—
432#106—22no—
433#106—1182no—
434#106—1182no—
435#106—1182no—
436C4⋊C8—594no—
437C4⋊C8—594no—
438C4⋊C8—594no—
439C4⋊C8C4×D5914yescomplemented
440#95—21no—
441#95—21no—
442#95—21no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448C24—591no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—591no—
454C22×C4—592no—
455C22×C4—592no—
456C22×C4—592no—
457C22×C4—592no—
458C22×C4—592no—
459C22×C4—592no—
460C22×C4—591no—
461C22×C4—591no—
462C22×C4—1184no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—1181no—
466C22×C4—1181no—
467C22×C4—4721no—
468C22×C4—4721no—
469C22×C4—4721no—
470C22×C4—4721no—
471C22×C4—2362no—
472C22×C4D4×D5911yescharacteristic
473C22×C4D4×D5911yescharacteristic
474C22×C4C23×D5911yescharacteristic
475M16—1188no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—592no—
481C2×C8—592no—
482C2×C8—1188no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8C2×C4×D5912yes—
488C2×C8C2×C4×D5912yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—594no—
496C4⋊C4—594no—
497C4⋊C4—592no—
498C4⋊C4—592no—
499C4⋊C4—1184no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—4721no—
507C4⋊C4—4721no—
508C4⋊C4—4721no—
509C4⋊C4—4721no—
510C4⋊C4—2362no—
511C4⋊C4D4×D5912yes—
512C4⋊C4Q8×D5912yes—
513C22⋊C4—594no—
514C22⋊C4—594no—
515C22⋊C4—1184no—
516C22⋊C4—2362no—
517C42—81no—
518C42—592no—
519C42—592no—
520C42—1184no—
521C42—4721no—
522C42—4721no—
523C42—4721no—
524C42—2362no—
525C42C2×C4×D5912yes—
526C42Q8×D5912yes—
527C23—592no—
528C23—592no—
529C23—592no—
530C23—1181no—
531C23—1181no—
532C23—1181no—
533C23—1181no—
534C23C2×D4×D5911yescharacteristic
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
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—592no—
552C2×C4—592no—
553C2×C4—592no—
554C2×C4—594no—
555C2×C4—594no—
556C2×C4—594no—
557C2×C4—594no—
558C2×C4—592no—
559C2×C4—592no—
560C2×C4—1181no—
561C2×C4—1181no—
562C2×C4—1181no—
563C2×C4—1181no—
564C2×C4—1182no—
565C2×C4—1182no—
566C2×C4—4724no—
567C2×C4—4724no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4C4⋊C4×D5912yes—
579C2×C4C4⋊C4×D5912yes—
580C2×C4C22×C4×D5911yescharacteristic
581C2×C4C2×D4×D5911yescharacteristic
582C2×C4C2×Q8×D5911yescharacteristic
583C2×C4C2×Q8×D5911yescharacteristic
584C2×C4#41×D5911yescharacteristic
585C2×C4#41×D5911yescharacteristic
586C8—22no—
587C8—22no—
588C8—1182no—
589C8—1182no—
590C22—594no—
591C22—1182no—
592C22—1182no—
593C22—1182no—
594C22—1182no—
595C22—1182no—
596C22—1182no—
597C22C2×C4⋊C4×D5911yescharacteristic
598C22#121×D5911yescharacteristic
599C22C22⋊Q8×D5911yescharacteristic
600C22#1683820611yescharacteristic
601C22#1683820911yescharacteristic
602C22#1683820911yescharacteristic
603C22#1683821811yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—594no—
611C4—1182no—
612C4—1182no—
613C4—4721no—
614C4—4721no—
615C4—2362no—
616C4—2362no—
617C4C4×D4×D5911yescharacteristic
618C4C4×Q8×D5911yescharacteristic
619C4#125×D5911yescharacteristic
620C4C4⋊Q8×D5911yescharacteristic
621C2—1184no—
622C2#389×D5911yescharacteristic
623C2#2143268811yescharacteristic
624C2#2143269511yescharacteristic
625C2#2143363611yescharacteristic
626C2#2143364911yescharacteristic
627C2#2143392611yescharacteristic
628C2#2143393511yescharacteristic
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#21175
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. #21424865C2semidirect
7. #21424863C2semidirect
8. #21424862C2semidirect
9. #21424860C2semidirect
10. #21424858C2semidirect
11. #21424854C2semidirect
12. #21424851C2semidirect
15. C59×#1949C2semidirect
31. #16837427C4semidirect
111. C4⋊(C59⋊C8)C2×C4semidirect
112. Dic59⋊C8C2×C4semidirect
113. Dic59⋊C8C2×C4semidirect
129. C4⋊C8×C59C2×C4semidirect
315. #1949D59semidirect
376. C59#21175semidirect
439. 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×#168374372
C2×#168374272
#214267212
#214264502
#214248652
#214248632
#214248622
#214248602
#214248582
#214248542
#214248512
#214238142
#340×D592
C59×#19492
#2117559