Finite Group Databasebeta

← #28611323

Subgroups — #28611323

Order 15104.

938 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×#16837641C211yesmaximal, characteristic
2C2×#16837494C211yesmaximal, characteristic, complemented
3C2×#16837438C211yesmaximal, characteristic
4#21427127C211yesmaximal, characteristic
5#21427123C211yesmaximal, characteristic
6#21424954C211yesmaximal, characteristic, complemented
7#21424828C211yesmaximal, characteristic, complemented
8#21424733C211yesmaximal, characteristic, complemented
9#21424728C211yesmaximal, characteristic, complemented
10#21424406C211yesmaximal, characteristic
11#21424365C211yesmaximal, characteristic, complemented
12#21424362C211yesmaximal, characteristic, complemented
13#21423988C211yesmaximal, characteristic, complemented
14#21423870C211yesmaximal, characteristic
15C59×#1873C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C23×C59⋊C8C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×Q8×Dic59C2211yescharacteristic
20C2×#14273692C2211yescharacteristic
21C2×#14273686C2211yescharacteristic
22#16837641C418yescomplemented
23#16837494—24no—
24#16837438—24no—
25C2×#14273660C2211yescharacteristic
26C2×#14273660C2211yescharacteristic
27C2×#14273660C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273652C2211yescharacteristic
31C2×#14273652C2211yescharacteristic
32C2×#14273651C2211yescharacteristic
33C2×#14273650C2211yescharacteristic
34C2×C8×Dic59C2211yescharacteristic
35C2×C8×Dic59C2211yescharacteristic
36#16837309C2211yescharacteristic
37#16837307C2211yescharacteristic
38#16837301C2211yescharacteristic
39#16837298C2211yescharacteristic
40#16837296—21no—
41#16837295—21no—
42#16837295—21no—
43#16837290—21no—
44#16837288—21no—
45#16837286—21no—
46C4⋊C4×Dic59—21no—
47C4⋊C4×Dic59C2211yescharacteristic
48C4⋊C4×Dic59C2211yescharacteristic
49C4×Dic59⋊C4—21no—
50C4×C22⋊Dic59C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53#16837156C2211yescharacteristic
54#16837156C2211yescharacteristic
55C2×C22⋊Q8×C59C2211yescharacteristic
56C2×#117×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C2×C22⋊C8×C59C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#340C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672C2×C414yes—
66C22×C59⋊C8—22no—
67C22×C59⋊C8—22no—
68C22×C59⋊C8C2×C414yes—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71#14273706—22no—
72#14273706C2×C414yes—
73Q8×Dic59—22no—
74Q8×Dic59—41no—
75#14273692—22no—
76#14273692—41no—
77#14273686—22no—
78#14273686—41no—
79C2×C4⋊Dic59—21no—
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×C22⋊Dic59—21no—
88C2×C22⋊Dic59—21no—
89C2×C22⋊Dic59D411yescharacteristic
90C2×C22⋊Dic59D411yescharacteristic
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59—21no—
95C2×C4×Dic59—21no—
96C2×C4×Dic59C2311yescharacteristic
97C2×C4×Dic59C2311yescharacteristic
98#14273660—22no—
99#14273660C2×C414yes—
100#14273660D414yes—
101#14273658—22no—
102#14273658—22no—
103#14273652—22no—
104#14273652—41no—
105#14273651C2×C414yes—
106#14273650C2×C414yes—
107C8×Dic59—22no—
108C8×Dic59—22no—
109C59⋊#95—21no—
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
120C23×C4×C59C2311yescharacteristic
121C22×C8×C59C2311yescharacteristic
122C22×C8×C59C2311yescharacteristic
123C22⋊Q8×C59—22no—
124C22⋊Q8×C59C2×C414yes—
125#117×C59—22no—
126#117×C59—41no—
127C2×C4⋊C4×C59—21no—
128C2×C4⋊C4×C59C2311yescharacteristic
129C2×C4⋊C4×C59C2311yescharacteristic
130C2×C4⋊C4×C59C2311yescharacteristic
131C2×C22⋊C4×C59—21no—
132C2×C22⋊C4×C59—21no—
133C2×C42×C59—21no—
134#103×C59—22no—
135#103×C59—41no—
136C22⋊C8×C59—22no—
137C24×C59C2×D411yescharacteristic
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59C22×C412yes—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59C22×C412yes—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59#4112yes—
155C22×C4×C59C2411yescharacteristic
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59C2×D411yescharacteristic
161C22×Dic59C2×D411yescharacteristic
162C22×Dic59C2×D411yescharacteristic
163C22×Dic59C2×D411yescharacteristic
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118C22×C412yes—
170C2×C8×C59—21no—
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C4⋊C4×C59—24no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59C22×C412yes—
182C4⋊C4×C59C2×D412yes—
183C22⋊C4×C59—24no—
184C22⋊C4×C59—42no—
185C42×C59—42no—
186C22⋊Dic59—24no—
187C22⋊Dic59—22no—
188C22⋊Dic59—22no—
189C22⋊Dic59—42no—
190#11745672—28no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—42no—
196C4⋊Dic59—42no—
197C4⋊Dic59C22×C412yes—
198C4⋊Dic59C2×D412yes—
199Dic59⋊C4—24no—
200Dic59⋊C4—42no—
201Dic59⋊C4—42no—
202C4×Dic59—21no—
203C4×Dic59—21no—
204C4×Dic59—21no—
205C4×Dic59—21no—
206C4×Dic59—42no—
207C4×Dic59—42no—
208C4×Dic59—42no—
209C4×Dic59—42no—
210C4×Dic59—42no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—28no—
214C2×C59⋊C8C22×C412yes—
215C2×C59⋊C8C22×C412yes—
216C2×C59⋊C8C2×D412yes—
217C2×C59⋊C8#4112yes—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59C4×D412yes—
225C23×C59C22×D411yescharacteristic
226Q8×C59—22no—
227Q8×C59—22no—
228Q8×C59—41no—
229Q8×C59—41no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
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—41no—
247C2×C4×C59—41no—
248C2×C4×C59C4×D412yes—
249C2×C4×C59C23×C411yescharacteristic
250C2×C4×C59C22×D411yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59H(5,𝔽2)11yescharacteristic
254C2×C4×C592−1+411yescharacteristic
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—22no—
260C2×Dic59—22no—
261C2×Dic59—21no—
262C2×Dic59—22no—
263C2×Dic59—21no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—21no—
267C2×Dic59—21no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59#13512yes—
278C2×Dic59#13512yes—
279Dic118—22no—
280Dic118—22no—
281Dic118—41no—
282Dic118—41no—
283C8×C59—41no—
284C8×C59—41no—
285C59⋊C8—22no—
286C59⋊C8C4×D414yes—
287#23026—591nomaximal, Sylow
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59C2×C4×D411yescharacteristic
296C22×C59D4211yescharacteristic
297C22×C59#54711yescharacteristic
298C22×C59C2×#13511yescharacteristic
299C22×C59C2×#13511yescharacteristic
300C22×C59#57811yescharacteristic
301C22×C59#57811yescharacteristic
302C4×C59—22no—
303C4×C59—42no—
304C4×C59—42no—
305C4×C59—42no—
306C4×C59—81no—
307C4×C59C2×C4×D411yescharacteristic
308C4×C59#52011yescharacteristic
309C4×C59#54811yescharacteristic
310C4×C59#55311yescharacteristic
311Dic59—22no—
312Dic59—22no—
313Dic59—41no—
314Dic59—42no—
315Dic59—42no—
316Dic59—41no—
317Dic59—42no—
318Dic59—81no—
319C2×#467—591no—
320C2×C4×Q16—591no—
321C2×#427—591no—
322#2362—591no—
323#2362—591no—
324#1969—591no—
325#1942—591no—
326#1927—591no—
327#1927—591no—
328#1873—591no—
329#1873D5911yescharacteristic
330#1865—591no—
331#1865—591no—
332C4×#107—591no—
333C8×C22⋊C4—591no—
334C2×C59—42no—
335C2×C59#236111yescharacteristic
336C2×C59C4×#13511yescharacteristic
337C2×C59#295011yescharacteristic
338C2×C59#328311yescharacteristic
339C2×C59#328511yescharacteristic
340C2×C59#333611yescharacteristic
341C2×C59#334211yescharacteristic
342C22×Q16—591no—
343C23×C8—591no—
344C2×C22⋊Q8—591no—
345C2×C22⋊Q8C2×D5911yescharacteristic
346C2×C4×Q8—591no—
347C2×C4×Q8—591no—
348C2×#117—591no—
349C2×#117C2×D5911yescharacteristic
350#467—598no—
351C4×Q16—1184no—
352#427—1184no—
353C2×#107—591no—
354C2×#107—591no—
355C2×#107—591no—
356C2×#106—591no—
357C2×#106—591no—
358C2×#103—591no—
359C2×#103—591no—
360C2×#103—591no—
361C2×#103—591no—
362C2×#103C2×D5911yescharacteristic
363C2×#103C2×D5911yescharacteristic
364C2×C22⋊C8C2×D5911yescharacteristic
365C2×C4×C8—591no—
366C2×C4×C8—591no—
367#391—591no—
368#391—591no—
369#389—591no—
370#389—591no—
371#388—1181no—
372#388—1181no—
373#388—1181no—
374#388—1181no—
375#387—1181no—
376#387—1181no—
377C4×C4⋊C4—591no—
378C4×C4⋊C4—591no—
379C4×C4⋊C4—1181no—
380C4×C4⋊C4—1181no—
381C4×C22⋊C4—591no—
382#340—591no—
383#340—591no—
384#340C2×D5911yescharacteristic
385#340C2×D5911yescharacteristic
386#336—591no—
387#336—591no—
388C59#2302611yescharacteristic, complemented, Sylow
389C22×Q8—591no—
390C22×Q8C22×D5911yescharacteristic
391C23×C4C22×D5911yescharacteristic
392C2×Q16—594no—
393C2×Q16—1182no—
394C2×Q16—1182no—
395C22×C8—591no—
396C22×C8—594no—
397C22×C8—591no—
398C22×C8—1182no—
399C22×C8—1182no—
400C22×C8C22×D5911yescharacteristic
401C22×C8C22×D5911yescharacteristic
402C22⋊Q8—22no—
403C22⋊Q8—594no—
404C22⋊Q8—1182no—
405C22⋊Q8C4×D5914yes—
406C4×Q8—1182no—
407C4×Q8—1182no—
408C4×Q8—2361no—
409C4×Q8—2361no—
410#117—22no—
411#117—41no—
412#117—1182no—
413#117—2361no—
414C2×C4⋊C4—21no—
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—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—591no—
429C2×C22⋊C4—591no—
430C2×C22⋊C4—1181no—
431C2×C22⋊C4—1181no—
432C2×C42—21no—
433C2×C42—591no—
434C2×C42—591no—
435C2×C42—1181no—
436C2×C42—1181no—
437C2×C42—1181no—
438C2×C42—1181no—
439C2×C42—1181no—
440#107—594no—
441#107—594no—
442#107—1182no—
443#106—1182no—
444#106—1182no—
445#103—22no—
446#103—41no—
447#103—594no—
448#103—594no—
449#103—1182no—
450#103—2361no—
451C22⋊C8—22no—
452C4×C8—1182no—
453C4×C8—1182no—
454#95—591no—
455#95—591no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464C24D4×D5911yescharacteristic
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—592no—
471C2×Q8—1181no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8C2×C4×D5912yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—591no—
486C22×C4—591no—
487C22×C4—591no—
488C22×C4—591no—
489C22×C4—1181no—
490C22×C4—1181no—
491C22×C4—1181no—
492C22×C4—1181no—
493C22×C4C2×C4×D5912yes—
494C22×C4D4×D5911yescharacteristic
495C22×C4#1174568712yes—
496C22×C4C23×D5911yescharacteristic
497Q16—1188no—
498C2×C8—21no—
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—592no—
508C2×C8—1188no—
509C2×C8—1181no—
510C2×C8—1181no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—592no—
517C4⋊C4—592no—
518C4⋊C4—1184no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—2362no—
524C4⋊C4—2362no—
525C4⋊C4—2362no—
526C4⋊C4—2362no—
527C4⋊C4C2×C4×D5912yes—
528C4⋊C4D4×D5912yes—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—1182no—
532C22⋊C4—1182no—
533C22⋊C4—1184no—
534C22⋊C4—2362no—
535C42—42no—
536C42—1181no—
537C42—1181no—
538C42—1181no—
539C42—1181no—
540C42—2362no—
541C42—2362no—
542C42—2362no—
543C42—2362no—
544C42—2362no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23(C22⋊D59)⋊C412yes—
552C23C2×D4×D5911yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—41no—
556Q8—41no—
557Q8—1182no—
558Q8—1182no—
559Q8—2361no—
560Q8—2361no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—592no—
580C2×C4—592no—
581C2×C4—1181no—
582C2×C4—1181no—
583C2×C4—1181no—
584C2×C4—1181no—
585C2×C4—1181no—
586C2×C4—1181no—
587C2×C4—1181no—
588C2×C4—1181no—
589C2×C4—1181no—
590C2×C4—1182no—
591C2×C4—1181no—
592C2×C4—1182no—
593C2×C4—1182no—
594C2×C4—2361no—
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(C22⋊D59)⋊C412yes—
604C2×C4C22×C4×D5911yescharacteristic
605C2×C4C2×D4×D5911yescharacteristic
606C2×C4#41×D5911yescharacteristic
607C2×C4C2×#1174568711yescharacteristic
608C2×C4#1427377211yescharacteristic
609C2×C4#1427377411yescharacteristic
610C8—41no—
611C8—41no—
612C8—594no—
613C8—1182no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22C2×(C22⋊D59)⋊C411yescharacteristic
622C22#1683803411yescharacteristic
623C22#1683805011yescharacteristic
624C22#135×D5911yescharacteristic
625C22#135×D5911yescharacteristic
626C22#1683821811yescharacteristic
627C22#1683821811yescharacteristic
628C4—22no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—1182no—
634C4—1182no—
635C4—4721no—
636C4—2362no—
637C4—2361no—
638C4—2361no—
639C4—2362no—
640C4—2362no—
641C4C4×D4×D5911yescharacteristic
642C4#1683788111yescharacteristic
643C4#1683805811yescharacteristic
644C4#1683809511yescharacteristic
645C2—42no—
646C2#2142710711yescharacteristic
647C2#2143286411yescharacteristic
648C2#2143296011yescharacteristic
649C2#2143581111yescharacteristic
650C2#2143583511yescharacteristic
651C2#2143645711yescharacteristic
652C2#2143652111yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1873C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1873D59
hypocenterC59#23026
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
2. C2×#16837494C2semidirect
6. #21424954C2semidirect
7. #21424828C2semidirect
8. #21424733C2semidirect
9. #21424728C2semidirect
11. #21424365C2semidirect
12. #21424362C2semidirect
13. #21423988C2semidirect
22. #16837641C4semidirect
388. C59#23026semidirect

Maximal subgroups

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

subgroupindex
C2×#168376412
C2×#168374942
C2×#168374382
#214271272
#214271232
#214249542
#214248282
#214247332
#214247282
#214244062
#214243652
#214243622
#214239882
#214238702
C59×#18732
#2302659