Finite Group Databasebeta

← #28607106

Subgroups — #28607106

Order 15104.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11745671C211yesmaximal, characteristic
2C2×#16837487C211yesmaximal, characteristic
3C2×#16837487C211yesmaximal, characteristic
4#21427014C211yesmaximal, characteristic, complemented
5#21427005C211yesmaximal, characteristic
6#21424823C211yesmaximal, characteristic, complemented
7#21424821C211yesmaximal, characteristic, complemented
8#21424352C211yesmaximal, characteristic, complemented
9#21424346C211yesmaximal, characteristic
10#21424340C211yesmaximal, characteristic
11#21424259C211yesmaximal, characteristic, complemented
12#21424109C211yesmaximal, characteristic, complemented
13#21424105C211yesmaximal, characteristic
14#21423957C211yesmaximal, characteristic, complemented
15C59×#1863C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C2×Q8×Dic59C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20C2×#14273689C2211yescharacteristic
21C2×#14273689C2211yescharacteristic
22C4×#11745671—24no—
23#16837487—24no—
24#16837487C418yescomplemented
25C2×#14273661C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×Dic59⋊C8C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273649C2211yescharacteristic
32C2×#14273649C2211yescharacteristic
33C2×#14273647C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35C2×C8×Dic59C2211yescharacteristic
36C2×C8×Dic59C2211yescharacteristic
37#16837301C2211yescharacteristic
38#16837300—21no—
39#16837300C2211yescharacteristic
40#16837298—21no—
41#16837298C2211yescharacteristic
42#16837276—21no—
43#16837273—21no—
44#16837267—21no—
45#16837267C2211yescharacteristic
46#16837267C2211yescharacteristic
47#16837259—21no—
48C4×Dic59⋊C4—21no—
49C4×Dic59⋊C4C2211yescharacteristic
50#95×D59—21no—
51#16837165C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837156C2211yescharacteristic
55C2×C4×Q8×C59C2211yescharacteristic
56C2×#107×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C59×#389C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#336C2211yescharacteristic
62C22×D236C2311yescharacteristic
63C22×C4×D59—21no—
64C22×C4×D59—21no—
65C22×C4×D59—21no—
66C2×#11745671C2×C414yes—
67C2×#11745671D414yes—
68C2×#11745671Q814yes—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71Q8×Dic59—22no—
72Q8×Dic59—22no—
73Q8×Dic59—41no—
74Q8×Dic59C2×C414yescomplemented
75C4×D236—22no—
76C4×D236—41no—
77#14273689—22no—
78#14273689—22no—
79#14273689—41no—
80#14273689C2×C414yes—
81C2×C4⋊Dic59—21no—
82C2×C4⋊Dic59—21no—
83C2×C4⋊Dic59—21no—
84C2×C4⋊Dic59C2311yescharacteristic
85C2×C4⋊Dic59C2311yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4D411yescharacteristic
90C2×Dic59⋊C4D411yescharacteristic
91C2×(C2×D59)⋊C4—21no—
92C2×(C2×D59)⋊C4—21no—
93C2×(C2×D59)⋊C4—21no—
94C2×C4×Dic59—21no—
95C2×C4×Dic59—21no—
96C2×C4×Dic59C2311yescharacteristic
97C2×C4×Dic59C2311yescharacteristic
98#14273661—41no—
99#14273658—22no—
100#14273658—22no—
101#14273658C2×C414yescomplemented
102Dic59⋊C8—22no—
103#14273651—22no—
104#14273649—22no—
105#14273649C2×C414yes—
106#14273647—22no—
107C4×C59⋊C8—22no—
108C8×Dic59—22no—
109C8×Dic59C2×C414yescomplemented
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C59⋊#95—21no—
114C59⋊#95—21no—
115C59⋊#95—21no—
116C59⋊#95—21no—
117C59⋊#95D411yescharacteristic
118C59⋊#95D411yescharacteristic
119C22×Q8×C59C2311yescharacteristic
120C22×C8×C59C2311yescharacteristic
121C22×C8×C59C2311yescharacteristic
122C4×Q8×C59—22no—
123C4×Q8×C59—41no—
124C2×C4⋊C4×C59—21no—
125C2×C4⋊C4×C59—21no—
126C2×C4⋊C4×C59C2311yescharacteristic
127C2×C4⋊C4×C59C2311yescharacteristic
128C2×C4⋊C4×C59C2311yescharacteristic
129C2×C42×C59—21no—
130C2×C42×C59C2311yescharacteristic
131#107×C59—41no—
132#103×C59—22no—
133#103×C59C2×C414yescomplemented
134#95×C59—21no—
135C23×D59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59C22×C412yes—
140C2×Q8×C59C2×D412yes—
141C2×Q8×C59C2×Q812yes—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236C22×C412yes—
160C2×D236C2×D412yes—
161C2×D236C2×Q812yes—
162C2×C4×D59—24no—
163C2×C4×D59—42no—
164C2×C4×D59—42no—
165C2×C4×D59—42no—
166C2×C4×D59—42no—
167C2×C4×D59—81no—
168C2×C4×D59—81no—
169C2×C4×D59—81no—
170C2×C4×D59—81no—
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59C22×C412yes—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—42no—
183C4⋊C4×C59—81no—
184C4⋊C4×C59C22×C412yes—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—42no—
188#11745671C4⋊C4116yescomplemented
189(C2×D59)⋊C4—24no—
190(C2×D59)⋊C4—42no—
191(C2×D59)⋊C4—42no—
192C4⋊Dic59—21no—
193C4⋊Dic59—24no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—42no—
199C4⋊Dic59—81no—
200C4⋊Dic59C22×C412yes—
201Dic59⋊C4—22no—
202Dic59⋊C4—22no—
203Dic59⋊C4—42no—
204Dic59⋊C4—42no—
205C4×Dic59—21no—
206C4×Dic59—21no—
207C4×Dic59—24no—
208C4×Dic59—42no—
209C4×Dic59C22×C412yes—
210C4×Dic59#4112yes—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8C22×C412yes—
215C2×C59⋊C8C2×D412yes—
216C2×C59⋊C8C2×Q812yes—
217C23×C59C22×D411yescharacteristic
218C22×D59—22no—
219C22×D59—22no—
220C22×D59—22no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225Q8×C59—22no—
226Q8×C59—22no—
227Q8×C59—22no—
228Q8×C59C2×C4⋊C414yes—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—21no—
232C2×C4×C59—22no—
233C2×C4×C59—21no—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59#13512yes—
247C2×C4×C59C23×C411yescharacteristic
248C2×C4×C59C22×D411yescharacteristic
249C2×C4×C59C22×Q811yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59H(5,𝔽2)11yescharacteristic
252C2×C4×C592−1+411yescharacteristic
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—22no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
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×Dic59C4×D412yes—
272C2×Dic59C4×D412yes—
273C2×Dic59#13512yes—
274D236—22no—
275D236—22no—
276D236—22no—
277D236C2×C4⋊C414yes—
278C4×D59—84no—
279C4×D59—84no—
280C4×D59—84no—
281C8×C59—22no—
282C8×C59—41no—
283C59⋊C8—41no—
284C59⋊C8C2×C4⋊C414yes—
285#22683—591nomaximal, Sylow
286C22×C59C2×C4×D411yescharacteristic
287C22×C59D4211yescharacteristic
288C22×C59D4×Q811yescharacteristic
289C22×C59C2×#13511yescharacteristic
290C22×C59C2×#13511yescharacteristic
291C22×C59#57711yescharacteristic
292C22×C59#57711yescharacteristic
293C2×D59—24no—
294C2×D59—42no—
295C2×D59—42no—
296C2×D59—42no—
297C2×D59—42no—
298C2×D59—42no—
299C2×D59—42no—
300C4×C59—24no—
301C4×C59—22no—
302C4×C59—42no—
303C4×C59—41no—
304C4×C59—81no—
305C4×C59—81no—
306C4×C59C22×C4⋊C411yescharacteristic
307C4×C59#52011yescharacteristic
308C4×C59#54711yescharacteristic
309C4×C59#55411yescharacteristic
310Dic59—22no—
311Dic59—22no—
312Dic59—22no—
313Dic59—42no—
314Dic59—41no—
315Dic59—81no—
316Dic59—81no—
317C2×C4×SD16—591no—
318C2×C4×SD16—591no—
319C2×C4×SD16—591no—
320#2351—591no—
321#2349—591no—
322#1941—591no—
323#1941—591no—
324#1863—591no—
325#1863D5911yescharacteristic, complemented
326#1862—591no—
327#1862—591no—
328#1846—591no—
329#1805—591no—
330#1804—591no—
331C8×C4⋊C4—591no—
332C2×C59D4×C4⋊C411yescharacteristic
333C2×C59C4×#13511yescharacteristic
334C2×C59#294911yescharacteristic
335C2×C59#328211yescharacteristic
336C2×C59#328211yescharacteristic
337C2×C59#338111yescharacteristic
338C2×C59#338111yescharacteristic
339D59—44no—
340C22×SD16—591no—
341C2×C4×Q8—591no—
342C2×C4×Q8—591no—
343C2×C4×Q8C2×D5911yescharacteristic
344C2×C4×D4—591no—
345C2×C4×D4—591no—
346C2×C4×D4—591no—
347C4×SD16—598no—
348C4×SD16—1184no—
349C4×SD16—1184no—
350C2×#107—591no—
351C2×#107C2×D5911yescharacteristic
352C2×#106—591no—
353C2×#106—591no—
354C2×#106—591no—
355C2×C4⋊C8—591no—
356C2×#103—591no—
357C2×#103C2×D5911yescharacteristic
358C2×#103C2×D5911yescharacteristic
359C2×#102—591no—
360C2×#102—591no—
361C2×#102—591no—
362C2×C4×C8—591no—
363C2×C4×C8—591no—
364C2×C4×C8—591no—
365#389—591no—
366#389—591no—
367#389—591no—
368#389—1181no—
369#389—1181no—
370#389C2×D5911yescharacteristic
371#385—1181no—
372#385—1181no—
373#382—591no—
374#382—591no—
375#382—1181no—
376#380—1181no—
377C4×C4⋊C4—591no—
378C4×C4⋊C4—1181no—
379C2×#95—1181no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340C2×D5911yescharacteristic
384#336—591no—
385#336C2×D5911yescharacteristic
386C59#2268311yescharacteristic, complemented, Sylow
387C22×Q8C22×D5911yescharacteristic
388C22×D4—591no—
389C23×C4—1181no—
390C23×C4—1181no—
391C23×C4—1181no—
392C2×SD16—594no—
393C2×SD16—594no—
394C2×SD16—594no—
395C22×C8—591no—
396C22×C8—591no—
397C22×C8C22×D5911yescharacteristic
398C22×C8C22×D5911yescharacteristic
399C4×Q8—22no—
400C4×Q8—41no—
401C4×Q8—594no—
402C4×Q8—1182no—
403C4×Q8—1182no—
404C4×Q8—2361no—
405C4×D4—594no—
406C4×D4—1182no—
407C4×D4—1182no—
408C4×D4—1182no—
409C4×D4—2361no—
410C4×D4—2361no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—1181no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4C22×D5911yescharacteristic
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C4⋊C4C22×D5911yescharacteristic
426C2×C22⋊C4—1181no—
427C2×C22⋊C4—1181no—
428C2×C22⋊C4—1181no—
429C2×C42—21no—
430C2×C42—591no—
431C2×C42—591no—
432C2×C42—1181no—
433C2×C42—1181no—
434C2×C42C22×D5911yescharacteristic
435#107—41no—
436#107—2361no—
437#106—594no—
438#106—1182no—
439#106—1182no—
440C4⋊C8—1182no—
441#103—22no—
442#103—1182no—
443#103C4×D5914yescomplemented
444#102—594no—
445#102—1182no—
446#102—1182no—
447C4×C8—594no—
448C4×C8—1182no—
449C4×C8—1182no—
450#95—21no—
451#95—591no—
452#95—591no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460C24—1181no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×C4×D5912yes—
465C2×Q8D4×D5912yes—
466C2×Q8Q8×D5912yes—
467C2×D4—592no—
468C2×D4—592no—
469C2×D4—592no—
470C2×D4—1181no—
471C2×D4—1181no—
472C2×D4—1181no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—591no—
478C22×C4—591no—
479C22×C4—591no—
480C22×C4—591no—
481C22×C4—1184no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4—4721no—
486C22×C4—4721no—
487C22×C4—4721no—
488C22×C4—4721no—
489C22×C4—2362no—
490C22×C4—2362no—
491C22×C4—2362no—
492C22×C4—2362no—
493C22×C4D4×D5911yescharacteristic
494C22×C4D4×D5911yescharacteristic
495C22×C4C23×D5911yescharacteristic
496SD16—5916no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—592no—
503C2×C8—592no—
504C2×C8—592no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8C2×C4×D5912yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—592no—
517C4⋊C4—1182no—
518C4⋊C4—1184no—
519C4⋊C4—1182no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—4721no—
526C4⋊C4—2362no—
527C4⋊C4—2362no—
528C4⋊C4—2362no—
529C4⋊C4C2×C4×D5912yes—
530C22⋊C4—1184no—
531C22⋊C4—2362no—
532C22⋊C4—2362no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—592no—
537C42—592no—
538C42—1184no—
539C42—1181no—
540C42—1181no—
541C42—2362no—
542C23—1182no—
543C23—1182no—
544C23—1182no—
545C23—2361no—
546C23—2361no—
547C23—2361no—
548C23—2361no—
549C23C2×D4×D5911yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8C4⋊C4×D5914yes—
554D4—594no—
555D4—1182no—
556D4—1182no—
557D4—1182no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—592no—
576C2×C4—592no—
577C2×C4—592no—
578C2×C4—1181no—
579C2×C4—1181no—
580C2×C4—1181no—
581C2×C4—1181no—
582C2×C4—1181no—
583C2×C4—1181no—
584C2×C4—1181no—
585C2×C4—1181no—
586C2×C4—1181no—
587C2×C4—1182no—
588C2×C4—4724no—
589C2×C4—4724no—
590C2×C4—4724no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4#1427375012yes—
600C2×C4C22×C4×D5911yescharacteristic
601C2×C4C2×D4×D5911yescharacteristic
602C2×C4C2×Q8×D5911yescharacteristic
603C2×C4#41×D5911yescharacteristic
604C2×C4#1427377211yescharacteristic
605C2×C4#1427377411yescharacteristic
606C8—22no—
607C8—41no—
608C8—594no—
609C8—2361no—
610C22—1184no—
611C22—2362no—
612C22—2362no—
613C22—2362no—
614C22—2362no—
615C22—2362no—
616C22—2362no—
617C22C4×D4×D5911yescharacteristic
618C22#1683803411yescharacteristic
619C22#1683806511yescharacteristic
620C22#135×D5911yescharacteristic
621C22C2×#1427375011yescharacteristic
622C22#1683821111yescharacteristic
623C22#1683821211yescharacteristic
624C4—24no—
625C4—22no—
626C4—42no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—1182no—
631C4—1182no—
632C4—1182no—
633C4—4721no—
634C4—4721no—
635C4—2362no—
636C4—2361no—
637C4C2×C4⋊C4×D5911yescharacteristic
638C4#1683788111yescharacteristic
639C4#1683805011yescharacteristic
640C4#1683810611yescharacteristic
641C2—2364no—
642C2#2142699411yescharacteristic
643C2C4×#1427375011yescharacteristic
644C2#2143295211yescharacteristic
645C2#2143579611yescharacteristic
646C2#2143580611yescharacteristic
647C2#2143697011yescharacteristic
648C2#2143697411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1863C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1863D59
hypocenterC59#22683
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
4. #21427014C2semidirect
6. #21424823C2semidirect
7. #21424821C2semidirect
8. #21424352C2semidirect
11. #21424259C2semidirect
12. #21424109C2semidirect
14. #21423957C2semidirect
15. C59×#1863C2semidirect
24. #16837487C4semidirect
74. Q8×Dic59C2×C4semidirect
101. #14273658C2×C4semidirect
109. C8×Dic59C2×C4semidirect
133. #103×C59C2×C4semidirect
188. #11745671C4⋊C4semidirect
325. #1863D59semidirect
386. C59#22683semidirect
443. #103C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117456712
C2×#168374872
#214270142
#214270052
#214248232
#214248212
#214243522
#214243462
#214243402
#214242592
#214241092
#214241052
#214239572
C59×#18632
#2268359