Finite Group Databasebeta

← #28607060

Subgroups — #28607060

Order 15104.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837499C211yesmaximal, characteristic
2C2×#16837499C211yesmaximal, characteristic
3C2×C4×#11745670C211yesmaximal, characteristic
4#21427010C211yesmaximal, characteristic, complemented
5#21427001C211yesmaximal, characteristic
6#21424833C211yesmaximal, characteristic, complemented
7#21424826C211yesmaximal, characteristic, complemented
8#21424361C211yesmaximal, characteristic, complemented
9#21424353C211yesmaximal, characteristic, complemented
10#21424345C211yesmaximal, characteristic
11#21424261C211yesmaximal, characteristic, complemented
12#21424110C211yesmaximal, characteristic, complemented
13#21424094C211yesmaximal, characteristic
14#21423967C211yesmaximal, characteristic, complemented
15C59×#1864C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273692C2211yescharacteristic
19C2×#14273692C2211yescharacteristic
20C2×D4×Dic59C2211yescharacteristic
21C2×D4×Dic59C2211yescharacteristic
22#16837499—24no—
23#16837499C418yescomplemented
24C4×#11745670—24no—
25C2×#14273661C2211yescharacteristic
26C2×#14273660C2211yescharacteristic
27C2×#14273660C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×Dic59⋊C8C2211yescharacteristic
30C2×#14273652C2211yescharacteristic
31C2×#14273652C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×#14273648C2211yescharacteristic
34C2×#14273637C2211yescharacteristic
35C2×#14273637C2211yescharacteristic
36C2×C4×C59⋊C8C2211yescharacteristic
37#16837301C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837299—21no—
40#16837298C2211yescharacteristic
41#16837297—21no—
42#16837275—21no—
43#16837274—21no—
44#16837271—21no—
45#16837271C2211yescharacteristic
46#16837267C2211yescharacteristic
47#16837260—21no—
48C4×Dic59⋊C4C2211yescharacteristic
49#16837253—21no—
50C2×C59⋊#95—21no—
51#16837165C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837156C2211yescharacteristic
55C2×C4×D4×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×#102×C59C2211yescharacteristic
58C2×#102×C59C2211yescharacteristic
59C59×#389C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#336C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C23×Dic59—21no—
64C23×Dic59—21no—
65C2×#11745670C2×C414yes—
66C2×#11745670D414yes—
67C2×#11745670Q814yes—
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C4×Dic118—22no—
71C4×Dic118—41no—
72#14273692—22no—
73#14273692—22no—
74#14273692—41no—
75#14273692C2×C414yescomplemented
76D4×Dic59—22no—
77D4×Dic59—22no—
78D4×Dic59—41no—
79D4×Dic59C2×C414yes—
80C2×C4⋊Dic59—21no—
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×Dic59⋊C4D411yescharacteristic
92C2×Dic59⋊C4D411yescharacteristic
93C2×C22⋊Dic59—21no—
94C2×C22⋊Dic59—21no—
95C2×C4×Dic59—21no—
96C2×C4×Dic59—21no—
97C2×C4×Dic59—21no—
98C2×C4×Dic59C2311yescharacteristic
99C2×C4×Dic59C2311yescharacteristic
100#14273661—41no—
101#14273660—22no—
102#14273660C2×C414yescomplemented
103#14273658—22no—
104Dic59⋊C8—22no—
105#14273652—22no—
106#14273652C2×C414yescomplemented
107#14273650—22no—
108#14273648—22no—
109#14273637—22no—
110#14273637C2×C414yescomplemented
111C4×C59⋊C8—22no—
112C59⋊#95—21no—
113C59⋊#95—21no—
114C59⋊#95—21no—
115C59⋊#95—21no—
116C59⋊#95—21no—
117C59⋊#95D411yescharacteristic
118C59⋊#95D411yescharacteristic
119C22×D4×C59C2311yescharacteristic
120C23×C4×C59—21no—
121C22×C8×C59C2311yescharacteristic
122C22×C8×C59C2311yescharacteristic
123C4×D4×C59—22no—
124C4×D4×C59—41no—
125C2×C4⋊C4×C59—21no—
126C2×C4⋊C4×C59C2311yescharacteristic
127C2×C4⋊C4×C59C2311yescharacteristic
128C2×C4⋊C4×C59C2311yescharacteristic
129C2×C22⋊C4×C59—21no—
130C2×C42×C59C2311yescharacteristic
131#106×C59—41no—
132#102×C59—22no—
133#102×C59C2×C414yes—
134#95×C59—21no—
135C24×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59C22×C412yes—
140C2×D4×C59C2×D412yes—
141C2×D4×C59C2×Q812yes—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—42no—
146C22×C4×C59—42no—
147C22×C4×C59—81no—
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2411yescharacteristic
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—24no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—42no—
157C22×Dic59—42no—
158C22×Dic59—81no—
159C22×Dic59—81no—
160C22×Dic59—81no—
161C22×Dic59C2×D411yescharacteristic
162C22×Dic59C2×D411yescharacteristic
163C22×Dic59C2×D411yescharacteristic
164C22×Dic59C2×D411yescharacteristic
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118C22×C412yes—
169C2×Dic118C2×D412yes—
170C2×Dic118C2×Q812yes—
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—81no—
183C4⋊C4×C59C22×C412yes—
184C22⋊C4×C59—42no—
185C42×C59—21no—
186C42×C59—21no—
187C22⋊Dic59—24no—
188C22⋊Dic59—42no—
189#11745670C4⋊C4116yescomplemented
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—81no—
196C4⋊Dic59C22×C412yes—
197Dic59⋊C4—22no—
198Dic59⋊C4—24no—
199Dic59⋊C4—41no—
200Dic59⋊C4—42no—
201Dic59⋊C4—42no—
202Dic59⋊C4—42no—
203Dic59⋊C4—81no—
204Dic59⋊C4—81no—
205Dic59⋊C4—81no—
206C4×Dic59—21no—
207C4×Dic59—21no—
208C4×Dic59—24no—
209C4×Dic59—42no—
210C4×Dic59—42no—
211C4×Dic59C22×C412yes—
212C4×Dic59#4112yes—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8C22×C412yes—
217C2×C59⋊C8C2×D412yes—
218C2×C59⋊C8C2×Q812yes—
219C23×C59—22no—
220C23×C59—22no—
221C23×C59—22no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59C22×D411yescharacteristic
227D4×C59—22no—
228D4×C59—22no—
229D4×C59—22no—
230D4×C59C2×C4⋊C414yes—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—22no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—41no—
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—84no—
246C2×C4×C59#13712yes—
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—22no—
254C2×Dic59—22no—
255C2×Dic59—22no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—22no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—21no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—84no—
276C2×Dic59—84no—
277C2×Dic59C4×D412yes—
278C2×Dic59C4×D412yes—
279Dic118—22no—
280Dic118—22no—
281Dic118—22no—
282Dic118C2×C4⋊C414yes—
283C8×C59—22no—
284C8×C59—41no—
285C59⋊C8—41no—
286C59⋊C8C2×C4⋊C414yes—
287#22678—591nomaximal, Sylow
288C22×C59—24no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—42no—
292C22×C59—42no—
293C22×C59—42no—
294C22×C59—42no—
295C22×C59C2×C4×D411yescharacteristic
296C22×C59D4211yescharacteristic
297C22×C59D4×Q811yescharacteristic
298C22×C59C2×#13711yescharacteristic
299C22×C59#57511yescharacteristic
300C22×C59#57611yescharacteristic
301C22×C59#57711yescharacteristic
302C4×C59—22no—
303C4×C59—42no—
304C4×C59—41no—
305C4×C59—81no—
306C4×C59—81no—
307C4×C59C22×C4⋊C411yescharacteristic
308C4×C59#52011yescharacteristic
309C4×C59#54711yescharacteristic
310C4×C59#55411yescharacteristic
311Dic59—24no—
312Dic59—22no—
313Dic59—22no—
314Dic59—42no—
315Dic59—41no—
316Dic59—41no—
317Dic59—81no—
318Dic59—81no—
319C2×#440—591no—
320C2×#440—591no—
321C2×C4×SD16—591no—
322Q8×C4⋊C4—591no—
323#2349—591no—
324#1943—591no—
325#1942—591no—
326#1865—591no—
327#1864D5911yescharacteristic
328#1863—591no—
329#1862—591no—
330#1847—591no—
331#1805—591no—
332#1802—591no—
333#1771—591no—
334C2×C59—44no—
335C2×C59D4×C4⋊C411yescharacteristic
336C2×C59C4×#13711yescharacteristic
337C2×C59#294611yescharacteristic
338C2×C59#326611yescharacteristic
339C2×C59#326811yescharacteristic
340C2×C59#338511yescharacteristic
341C2×C59#338811yescharacteristic
342C22×SD16—591no—
343C2×C4×Q8—591no—
344C2×C4×Q8—591no—
345C2×C4×Q8—591no—
346C2×C4×D4—591no—
347C2×C4×D4—591no—
348C2×C4×D4C2×D5911yescharacteristic
349#440—598no—
350#440—1184no—
351C4×SD16—1184no—
352C2×#107—591no—
353C2×#107—591no—
354C2×#107—591no—
355C2×#106—591no—
356C2×#106C2×D5911yescharacteristic
357C2×C4⋊C8—591no—
358C2×#103—591no—
359C2×#103—591no—
360C2×#103—591no—
361C2×#102—591no—
362C2×#102C2×D5911yescharacteristic
363C2×#102C2×D5911yescharacteristic
364C2×#97—591no—
365C2×#97—591no—
366C2×C4×C8—591no—
367#389—591no—
368#389—591no—
369#389—591no—
370#389—1181no—
371#389—1181no—
372#389C2×D5911yescharacteristic
373#385—1181no—
374#385—1181no—
375#384—591no—
376#384—1181no—
377#382—591no—
378#380—1181no—
379C4×C4⋊C4—591no—
380C4×C4⋊C4—1181no—
381C2×#95—1181no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340C2×D5911yescharacteristic
386#336—591no—
387#336C2×D5911yescharacteristic
388C59#2267811yescharacteristic, complemented, Sylow
389C22×Q8—591no—
390C22×D4C22×D5911yescharacteristic
391C23×C4—21no—
392C23×C4—1181no—
393C23×C4—1181no—
394C2×SD16—594no—
395C2×SD16—594no—
396C2×SD16—594no—
397C22×C8—591no—
398C22×C8—591no—
399C22×C8C22×D5911yescharacteristic
400C22×C8C22×D5911yescharacteristic
401C4×Q8—594no—
402C4×Q8—1182no—
403C4×Q8—1182no—
404C4×Q8—1182no—
405C4×Q8—2361no—
406C4×Q8—2361no—
407C4×D4—22no—
408C4×D4—41no—
409C4×D4—594no—
410C4×D4—1182no—
411C4×D4—1182no—
412C4×D4—2361no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—1181no—
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
439#107—594no—
440#107—1182no—
441#107—2361no—
442#106—41no—
443#106—1182no—
444C4⋊C8—1182no—
445#103—594no—
446#103—1182no—
447#103—1182no—
448#102—22no—
449#102—1182no—
450#102C4×D5914yes—
451#97—594no—
452#97—1182no—
453C4×C8—1182no—
454#95—21no—
455#95—591no—
456#95—591no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462C24—21no—
463C2×Q8—592no—
464C2×Q8—592no—
465C2×Q8—592no—
466C2×Q8—1181no—
467C2×Q8—1181no—
468C2×Q8—1181no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D5912yes—
473C2×D4D4×D5912yes—
474C2×D4Q8×D5912yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—81no—
481C22×C4—591no—
482C22×C4—591no—
483C22×C4—591no—
484C22×C4—591no—
485C22×C4—1184no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—1181no—
490C22×C4—4721no—
491C22×C4—4721no—
492C22×C4—4721no—
493C22×C4—2362no—
494C22×C4—2362no—
495C22×C4D4×D5911yescharacteristic
496C22×C4D4×D5911yescharacteristic
497C22×C4C23×D5911yescharacteristic
498SD16—5916no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—592no—
505C2×C8—592no—
506C2×C8—592no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8C2×C4×D5912yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—81no—
517C4⋊C4—592no—
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—4721no—
527C4⋊C4—4721no—
528C4⋊C4—4721no—
529C4⋊C4—2361no—
530C4⋊C4—2362no—
531C4⋊C4—2362no—
532C4⋊C4—2362no—
533C4⋊C4C2×C4×D5912yes—
534C22⋊C4—42no—
535C22⋊C4—1184no—
536C22⋊C4—2362no—
537C42—21no—
538C42—21no—
539C42—592no—
540C42—592no—
541C42—1184no—
542C42—1181no—
543C42—1181no—
544C42—2362no—
545C42—2362no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C2×D4×D5911yescharacteristic
554Q8—594no—
555Q8—1182no—
556Q8—1182no—
557Q8—1182no—
558D4—22no—
559D4—22no—
560D4—22no—
561D4C4⋊C4×D5914yes—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—592no—
578C2×C4—592no—
579C2×C4—1181no—
580C2×C4—1181no—
581C2×C4—1181no—
582C2×C4—1181no—
583C2×C4—1182no—
584C2×C4—1182no—
585C2×C4—1182no—
586C2×C4—1181no—
587C2×C4—1181no—
588C2×C4—1181no—
589C2×C4—1181no—
590C2×C4—1181no—
591C2×C4—1181no—
592C2×C4—1182no—
593C2×C4—4724no—
594C2×C4—4724no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4#1427375612yes—
604C2×C4C22×C4×D5911yescharacteristic
605C2×C4C2×D4×D5911yescharacteristic
606C2×C4C2×Q8×D5911yescharacteristic
607C2×C4#41×D5911yescharacteristic
608C2×C4#1427377211yescharacteristic
609C2×C4#1427377411yescharacteristic
610C8—22no—
611C8—41no—
612C8—594no—
613C8—2361no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C4×D4×D5911yescharacteristic
622C22#1683803411yescharacteristic
623C22#1683806511yescharacteristic
624C22C2×#1427375611yescharacteristic
625C22#1683819911yescharacteristic
626C22#1683820611yescharacteristic
627C22#1683821211yescharacteristic
628C4—22no—
629C4—42no—
630C4—41no—
631C4—81no—
632C4—81no—
633C4—1184no—
634C4—1182no—
635C4—1182no—
636C4—4721no—
637C4—4721no—
638C4—2362no—
639C4—2361no—
640C4—2361no—
641C4C2×C4⋊C4×D5911yescharacteristic
642C4#1683788111yescharacteristic
643C4#1683805011yescharacteristic
644C4#1683810611yescharacteristic
645C2—44no—
646C2#2142699411yescharacteristic
647C2C4×#1427375611yescharacteristic
648C2#2143292511yescharacteristic
649C2#2143561011yescharacteristic
650C2#2143564011yescharacteristic
651C2#2143701011yescharacteristic
652C2#2143704511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1864C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1864D59
hypocenterC59#22678
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
4. #21427010C2semidirect
6. #21424833C2semidirect
7. #21424826C2semidirect
8. #21424361C2semidirect
9. #21424353C2semidirect
11. #21424261C2semidirect
12. #21424110C2semidirect
14. #21423967C2semidirect
23. #16837499C4semidirect
75. #14273692C2×C4semidirect
102. #14273660C2×C4semidirect
106. #14273652C2×C4semidirect
110. #14273637C2×C4semidirect
189. #11745670C4⋊C4semidirect
388. C59#22678semidirect

Maximal subgroups

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

subgroupindex
C2×#168374992
C2×C4×#117456702
#214270102
#214270012
#214248332
#214248262
#214243612
#214243532
#214243452
#214242612
#214241102
#214240942
#214239672
C59×#18642
#2267859