Finite Group Databasebeta

← #28591714

Subgroups — #28591714

Order 15104.

1042 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837512C211yesmaximal, characteristic
2C42×D236C211yesmaximal, characteristic, complemented
3#21424720C211yesmaximal, characteristic, complemented
4#21424713C211yesmaximal, characteristic, complemented
5#21424678C211yesmaximal, characteristic, complemented
6#21424671C211yesmaximal, characteristic, complemented
7#21424447C211yesmaximal, characteristic
8#21424447C211yesmaximal, characteristic
9#21424356C211yesmaximal, characteristic, complemented
10#21424342C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12#21423924C211yesmaximal, characteristic, complemented
13#21423921C211yesmaximal, characteristic, complemented
14#21423845C211yesmaximal, characteristic, complemented
15C59×#2293C211yesmaximal, characteristic, complemented
16C22×#11745669C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20C2×C42×D59—21no—
21#16837512C418yescomplemented
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×#14273647C2211yescharacteristic
31C2×#14273638C2211yescharacteristic
32C2×#14273638C2211yescharacteristic
33C4×C4⋊Dic59C2211yescharacteristic
34C4×C4⋊Dic59C2211yescharacteristic
35C4×C4⋊Dic59C2211yescharacteristic
36C4×(C2×D59)⋊C4—21no—
37C4×(C2×D59)⋊C4—21no—
38C4×(C2×D59)⋊C4—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×#117×C59—21no—
51C59×#390C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#388—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C43×C59C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64C2×#11745669—22no—
65C2×#11745669—22no—
66C2×#11745669C2×C414yes—
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C4×D236—22no—
72C4×D236—22no—
73C4×D236—22no—
74C4×D236—22no—
75C4×D236—22no—
76C4×D236C2×C414yes—
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C42×D59—44no—
87#14273660—22no—
88#14273658—22no—
89#14273658C2×C414yescomplemented
90#14273648—22no—
91#14273648—22no—
92#14273648C2×C414yescomplemented
93#14273647—22no—
94#14273647—22no—
95#14273647C2×C414yes—
96#14273638—22no—
97#14273638C2×C414yescomplemented
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×D4×C59C2311yescharacteristic
102C23×C4×C59—21no—
103C4×D4×C59—22no—
104C4×D4×C59C2×C414yescomplemented
105#117×C59—44no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59D411yescharacteristic
118C2×C42×C59D411yescharacteristic
119C2×C42×C59D411yescharacteristic
120C2×C42×C59D411yescharacteristic
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127C24×C59—21no—
128C23×D59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59C22×C412yes—
135C22×C4×C59—21no—
136C22×C4×C59—24no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—42no—
140C22×C4×C59—42no—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2411yescharacteristic
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236C22×C412yes—
155C2×D236#4112yes—
156C2×D236#4112yes—
157C2×C4×D59—24no—
158C2×C4×D59—24no—
159C2×C4×D59—24no—
160C2×C4×D59—42no—
161C2×C4×D59—42no—
162C2×C4×D59—42no—
163C2×C4×D59—42no—
164C2×C4×D59—42no—
165C2×C4×D59—42no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59C22×C412yes—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—42no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—42no—
179C42×C59—22no—
180C42×C59—22no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—41no—
186C42×C59—41no—
187C42×C59—42no—
188C42×C59C22×C412yes—
189C42×C59#4112yes—
190#11745669—28no—
191(C2×D59)⋊C4—24no—
192(C2×D59)⋊C4—24no—
193(C2×D59)⋊C4—24no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59C22×C412yes—
198C4⋊Dic59#4112yes—
199C4⋊Dic59#4112yes—
200C4×Dic59—42no—
201C4×Dic59—42no—
202C4×Dic59—42no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8C22×C412yes—
214C2×C59⋊C8C22×C412yes—
215C23×C59—22no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59C22×D411yescharacteristic
223C22×D59—22no—
224C22×D59—22no—
225C22×D59—22no—
226C22×D59—41no—
227C22×D59—41no—
228C22×D59—41no—
229C22×D59—41no—
230D4×C59—22no—
231D4×C59—22no—
232D4×C59—41no—
233D4×C59—41no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—22no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—48no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59C4×D412yes—
259C2×C4×C59C4×D412yes—
260C2×C4×C59AGL(1,ℤ/8)12yes—
261C2×C4×C59AGL(1,ℤ/8)12yes—
262C2×C4×C59C23×C411yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×Dic59—22no—
269C2×Dic59—22no—
270C2×Dic59—22no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277D236—22no—
278D236—22no—
279D236—22no—
280D236#11714yescomplemented
281C4×D59—48no—
282C4×D59—48no—
283C4×D59—48no—
284C59⋊C8—22no—
285C59⋊C8—22no—
286C59⋊C8—41no—
287C59⋊C8—41no—
288#21122—591nomaximal, Sylow
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—42no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59—81no—
296C22×C59C2×C4×D411yescharacteristic
297C22×C59#52511yescharacteristic
298C22×C59#52511yescharacteristic
299C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C59#57511yescharacteristic
302C22×C59#57511yescharacteristic
303C2×D59—24no—
304C2×D59—42no—
305C2×D59—42no—
306C2×D59—42no—
307C2×D59—42no—
308C2×D59—42no—
309C2×D59—42no—
310C4×C59—22no—
311C4×C59—22no—
312C4×C59—22no—
313C4×C59—22no—
314C4×C59—41no—
315C4×C59—41no—
316C4×C59—42no—
317C4×C59—81no—
318C4×C59—81no—
319C4×C59C2×#11711yescharacteristic
320C4×C59C4×#4111yescharacteristic
321C4×C59#52911yescharacteristic
322C4×C59#52911yescharacteristic
323Dic59—42no—
324Dic59—42no—
325Dic59—42no—
326C2×#442—591no—
327#2293D5911yescharacteristic, complemented
328C42×D4—591no—
329#1926—591no—
330#1925—591no—
331#1920—591no—
332#1919—591no—
333#1879—591no—
334#1879—591no—
335#1864—591no—
336#1862—591no—
337#1776—591no—
338#1762—591no—
339#1762—591no—
340#1745—591no—
341C2×C59—82no—
342C2×C59#229011yescharacteristic
343C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
344C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
345C2×C59#310911yescharacteristic
346C2×C59#311111yescharacteristic
347C2×C59#311411yescharacteristic
348C2×C59#311511yescharacteristic
349D59—44no—
350C22×D8—591no—
351C2×C4×D4—591no—
352C2×C4×D4—591no—
353C2×C4×D4—591no—
354C2×C4×D4C2×D5911yescharacteristic
355C2×#117—21no—
356C22×C42—1181no—
357#442—598no—
358C2×#107—591no—
359C2×#106—591no—
360C2×#106—591no—
361C2×#102—591no—
362C2×#102—591no—
363C2×#102—591no—
364C2×#102—591no—
365C2×#102—591no—
366C2×#102—591no—
367C2×#97—591no—
368C2×#97—591no—
369#390C2×D5911yescharacteristic
370#390C2×D5911yescharacteristic
371#389C2×D5911yescharacteristic
372#389C2×D5911yescharacteristic
373#388—21no—
374#388—21no—
375C4×C4⋊C4—591no—
376C4×C4⋊C4—591no—
377C4×C4⋊C4—591no—
378C4×C4⋊C4C2×D5911yescharacteristic
379C4×C22⋊C4—21no—
380C4×C22⋊C4—1181no—
381C4×C22⋊C4—1181no—
382C4×C22⋊C4—1181no—
383C43C2×D5911yescharacteristic
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#340—591no—
390#336—591no—
391#336—591no—
392#336—591no—
393#336—591no—
394C59#2112211yescharacteristic, complemented, Sylow
395C22×D4—591no—
396C22×D4C22×D5911yescharacteristic
397C23×C4—21no—
398C23×C4—1181no—
399C23×C4—1181no—
400C23×C4—1181no—
401C2×D8—594no—
402C2×D8—1182no—
403C2×D8—1182no—
404C22×C8—591no—
405C22×C8—591no—
406C22×C8—591no—
407C22×C8—591no—
408C4×D4—22no—
409C4×D4—594no—
410C4×D4—1182no—
411C4×D4—1182no—
412C4×D4—1182no—
413C4×D4—1182no—
414C4×D4—1182no—
415C4×D4C4×D5914yescomplemented
416#117—44no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—591no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4C22×D5911yescharacteristic
423C2×C4⋊C4C22×D5911yescharacteristic
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—1181no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4—1181no—
433C2×C42—21no—
434C2×C42—1181no—
435C2×C42—1181no—
436C2×C42—1181no—
437C2×C42—2364no—
438C2×C42C22⋊D5911yescharacteristic
439C2×C42C22⋊D5911yescharacteristic
440C2×C42C22⋊D5911yescharacteristic
441C2×C42C22⋊D5911yescharacteristic
442C2×C42C22×D5911yescharacteristic
443C2×C42C22×D5911yescharacteristic
444C2×C42C22×D5911yescharacteristic
445#107—1182no—
446#106—594no—
447#106—1182no—
448#102—594no—
449#102—594no—
450#102—1182no—
451#102—1182no—
452#102—1182no—
453#102—1182no—
454#97—594no—
455#97—1182no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462C24—21no—
463C24—1181no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—592no—
470C2×D4—592no—
471C2×D4—592no—
472C2×D4—1181no—
473C2×D4—1181no—
474C2×D4—1181no—
475C2×D4C2×C4×D5912yes—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—1184no—
483C22×C4—1184no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4—1184no—
488C22×C4—2362no—
489C22×C4—2362no—
490C22×C4—2362no—
491C22×C4—2362no—
492C22×C4—2362no—
493C22×C4—2362no—
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C2×C22⋊D5911yescharacteristic
497C22×C4C2×C22⋊D5911yescharacteristic
498C22×C4C2×C22⋊D5911yescharacteristic
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C23×D5911yescharacteristic
501D8—1188no—
502C2×C8—592no—
503C2×C8—592no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—592no—
522C4⋊C4—592no—
523C4⋊C4—592no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4C2×C4×D5912yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—1184no—
534C22⋊C4—1184no—
535C22⋊C4—1184no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—41no—
543C42—41no—
544C42—42no—
545C42—2362no—
546C42—2362no—
547C42—2362no—
548C42C2×C4×D5912yes—
549C42#1174568512yes—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—1182no—
558C23—1182no—
559C23—1182no—
560C23—2361no—
561C23—2361no—
562C23—2361no—
563C23—2361no—
564C23C22×C22⋊D5911yescharacteristic
565D4—22no—
566D4—22no—
567D4—41no—
568D4—41no—
569D4—594no—
570D4—1182no—
571D4—1182no—
572D4—1182no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—1182no—
598C2×C4—1182no—
599C2×C4—1182no—
600C2×C4—2368no—
601C2×C4—2368no—
602C2×C4—2368no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4C4×C22⋊D5912yes—
610C2×C4C4×C22⋊D5912yes—
611C2×C4#1427375212yes—
612C2×C4#1427375312yes—
613C2×C4C22×C4×D5911yescharacteristic
614C2×C4#41×D5911yescharacteristic
615C2×C4#41×D5911yescharacteristic
616C2×C4#41×D5911yescharacteristic
617C2×C4#41×D5911yescharacteristic
618C2×C4C2×#1174568511yescharacteristic
619C8—1182no—
620C8—1182no—
621C8—2361no—
622C8—2361no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—1184no—
631C22—2362no—
632C22—2362no—
633C22—2362no—
634C22—2362no—
635C22—2362no—
636C22—2362no—
637C22C2×C4×C22⋊D5911yescharacteristic
638C22#1683790911yescharacteristic
639C22#1683790911yescharacteristic
640C22C2×#1427375211yescharacteristic
641C22C2×#1427375311yescharacteristic
642C22#1683820011yescharacteristic
643C22#1683820011yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—81no—
652C4—81no—
653C4—2362no—
654C4—2362no—
655C4—2362no—
656C4#117×D5911yescharacteristic
657C4#1683787011yescharacteristic
658C4#1683793011yescharacteristic
659C4#1683793011yescharacteristic
660C2—82no—
661C2—2364no—
662C2#2142653311yescharacteristic
663C2C4×#1427375211yescharacteristic
664C2C4×#1427375311yescharacteristic
665C2#2143418311yescharacteristic
666C2#2143419911yescharacteristic
667C2#2143423011yescharacteristic
668C2#2143424111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2293C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
cosocleC22×C4×C59C24
hypercenter#2293D59
hypocenterC59#21122
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. C42×D236C2semidirect
3. #21424720C2semidirect
4. #21424713C2semidirect
5. #21424678C2semidirect
6. #21424671C2semidirect
9. #21424356C2semidirect
10. #21424342C2semidirect
11. #21423994C2semidirect
12. #21423924C2semidirect
13. #21423921C2semidirect
14. #21423845C2semidirect
15. C59×#2293C2semidirect
21. #16837512C4semidirect
89. #14273658C2×C4semidirect
92. #14273648C2×C4semidirect
97. #14273638C2×C4semidirect
104. C4×D4×C59C2×C4semidirect
280. D236#117semidirect
327. #2293D59semidirect
394. C59#21122semidirect
415. C4×D4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168375122
C42×D2362
#214247202
#214247132
#214246782
#214246712
#214244472
#214243562
#214243422
#214239942
#214239242
#214239212
#214238452
C59×#22932
#2112259