Finite Group Databasebeta

← #28611482

Subgroups — #28611482

Order 15104.

942 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837660C211yesmaximal, characteristic
2C2×#16837510C211yesmaximal, characteristic, complemented
3C2×#16837444C211yesmaximal, characteristic
4#21427121C211yesmaximal, characteristic
5#21427099C211yesmaximal, characteristic
6#21424957C211yesmaximal, characteristic, complemented
7#21424839C211yesmaximal, characteristic, complemented
8#21424738C211yesmaximal, characteristic, complemented
9#21424731C211yesmaximal, characteristic, complemented
10#21424406C211yesmaximal, characteristic
11#21424401C211yesmaximal, characteristic
12#21424367C211yesmaximal, characteristic, complemented
13#21424353C211yesmaximal, characteristic, complemented
14C4×#14273659C211yesmaximal, characteristic, complemented
15C59×#1753C211yesmaximal, characteristic
16C22×Dic236C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20C2×C4×Dic118C2211yescharacteristic
21C2×#14273686C2211yescharacteristic
22C22×C4⋊Dic59C2211yescharacteristic
23#16837660C418yescomplemented
24#16837510—24no—
25#16837444—24no—
26C2×#14273661C2211yescharacteristic
27C2×#14273661C2211yescharacteristic
28C2×#14273659C2211yescharacteristic
29C2×#14273659C2211yescharacteristic
30C2×#14273659C2211yescharacteristic
31C2×#14273652C2211yescharacteristic
32C2×#14273652C2211yescharacteristic
33C2×#14273652C2211yescharacteristic
34C2×#14273652C2211yescharacteristic
35C2×#14273652C2211yescharacteristic
36C2×#14273652C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837308C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837300C2211yescharacteristic
41#16837299—21no—
42#16837296—21no—
43#16837290—21no—
44#16837289—21no—
45#16837284—21no—
46#16837274—21no—
47C4×C4⋊Dic59C2211yescharacteristic
48C4×C4⋊Dic59C2211yescharacteristic
49#16837253—21no—
50C22⋊C4×Dic59—21no—
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53#16837165C2211yescharacteristic
54#16837165C2211yescharacteristic
55C22×M16×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C2×#97×C59C2211yescharacteristic
58C2×C4×C8×C59C2211yescharacteristic
59C4×C22⋊C4×C59C2211yescharacteristic
60C59×#336C2211yescharacteristic
61C59×#336C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C22×Dic118C2311yescharacteristic
64C23×Dic59—21no—
65C2×Dic236—22no—
66C2×Dic236—22no—
67C2×Dic236C2×C414yescomplemented
68#14273706—22no—
69#14273706—22no—
70#14273706C2×C414yes—
71#14273706C2×C414yes—
72C4×Dic118—22no—
73C4×Dic118—22no—
74C4×Dic118—41no—
75C4×Dic118—41no—
76#14273686—22no—
77#14273686—41no—
78C2×C4⋊Dic59—22no—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59—41no—
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×C4⋊Dic59C2311yescharacteristic
85C2×C4⋊Dic59C2311yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×Dic59⋊C4—21no—
92C2×Dic59⋊C4—21no—
93C2×C22⋊Dic59—21no—
94C2×C22⋊Dic59—21no—
95C2×C22⋊Dic59—21no—
96C2×C4×Dic59—21no—
97C2×C4×Dic59—21no—
98C2×C4×Dic59—21no—
99C2×C4×Dic59—21no—
100C2×C4×Dic59—21no—
101#14273661—22no—
102#14273661—22no—
103#14273659—22no—
104#14273659C2×C414yescomplemented
105#14273659D414yes—
106#14273652—22no—
107#14273652—22no—
108#14273652—41no—
109#14273652—41no—
110#14273652C2×C414yescomplemented
111#14273652C2×C414yescomplemented
112C59⋊#95—21no—
113C59⋊#95—21no—
114C59⋊#95—21no—
115C59⋊#95—21no—
116C59⋊#95—21no—
117C59⋊#95—21no—
118C59⋊#95—21no—
119C59⋊#95—21no—
120C23×C4×C59C2311yescharacteristic
121C2×M16×C59—22no—
122C2×M16×C59—22no—
123C2×M16×C59C2×C414yes—
124C22×C8×C59C2311yescharacteristic
125C22×C8×C59C2311yescharacteristic
126C22×C8×C59C2311yescharacteristic
127C22×C8×C59C2311yescharacteristic
128C2×C22⋊C4×C59D411yescharacteristic
129C2×C22⋊C4×C59D411yescharacteristic
130C2×C42×C59C2311yescharacteristic
131C2×C42×C59C2311yescharacteristic
132C22⋊C8×C59—22no—
133#97×C59—22no—
134C4×C8×C59—22no—
135#95×C59D411yescharacteristic
136#95×C59D411yescharacteristic
137C24×C59C2×D411yescharacteristic
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59C22×C412yes—
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59#4112yes—
149C22×C4×C59C2411yescharacteristic
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—42no—
159C22×Dic59—42no—
160C22×Dic59—81no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118C22×C412yes—
172C2×Dic118C22×C412yes—
173M16×C59—28no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C2×C8×C59—21no—
178C2×C8×C59—21no—
179C2×C8×C59—21no—
180C2×C8×C59—21no—
181C2×C8×C59—21no—
182C2×C8×C59C22×C412yes—
183C2×C8×C59C22×C412yes—
184C2×C8×C59C2×D412yes—
185C2×C8×C59#4112yes—
186C22⋊C4×C59—22no—
187C22⋊C4×C59—41no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—21no—
191C42×C59—21no—
192C22⋊Dic59—24no—
193C22⋊Dic59—24no—
194C22⋊Dic59—42no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—42no—
204C4⋊Dic59C22×C412yes—
205C4⋊Dic59C22×C412yes—
206C4⋊Dic59C2×D412yes—
207C4⋊Dic59C2×D412yes—
208Dic59⋊C4—24no—
209Dic59⋊C4—24no—
210Dic59⋊C4—42no—
211Dic59⋊C4—42no—
212Dic59⋊C4—42no—
213Dic59⋊C4—81no—
214C4×Dic59—42no—
215C4×Dic59—42no—
216C4×Dic59—42no—
217C4×Dic59—42no—
218C4×Dic59—81no—
219Dic236—28no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59C4×D412yes—
227C23×C59C22×D411yescharacteristic
228C2×C4×C59—21no—
229C2×C4×C59—21no—
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—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59C4×D412yes—
246C2×C4×C59#13712yes—
247C2×C4×C59C23×C411yescharacteristic
248C2×C4×C59C22×D411yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
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—22no—
257C2×Dic59—22no—
258C2×Dic59—22no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
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×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—84no—
278Dic118—22no—
279Dic118—22no—
280Dic118—22no—
281Dic118—22no—
282Dic118—41no—
283Dic118—41no—
284Dic118—41no—
285Dic118—41no—
286C8×C59—22no—
287C8×C59—41no—
288C8×C59—41no—
289C8×C59C4×D414yes—
290#23038—591nomaximal, Sylow
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59—41no—
298C22×C59C2×C4×D411yescharacteristic
299C22×C59D4211yescharacteristic
300C22×C59#54711yescharacteristic
301C22×C59C2×#13711yescharacteristic
302C22×C59#57511yescharacteristic
303C22×C59#57711yescharacteristic
304C22×C59#57811yescharacteristic
305C4×C59—22no—
306C4×C59—22no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—41no—
310C4×C59C2×C4×D411yescharacteristic
311C4×C59#52011yescharacteristic
312C4×C59#54811yescharacteristic
313C4×C59#55311yescharacteristic
314Dic59—42no—
315Dic59—42no—
316Dic59—42no—
317Dic59—42no—
318Dic59—42no—
319Dic59—42no—
320Dic59—81no—
321Dic59—81no—
322C2×#470—591no—
323C2×#441—591no—
324C2×#428—591no—
325#2362—591no—
326#2360—591no—
327#1969—591no—
328#1944—591no—
329#1928—591no—
330#1927—591no—
331#1873—591no—
332#1872—591no—
333#1865—591no—
334#1863—591no—
335C4×#106—591no—
336#1753D5911yescharacteristic
337C2×C59—42no—
338C2×C59#236111yescharacteristic
339C2×C59C4×#13711yescharacteristic
340C2×C59#294711yescharacteristic
341C2×C59#327511yescharacteristic
342C2×C59#327611yescharacteristic
343C2×C59#334411yescharacteristic
344C2×C59#334911yescharacteristic
345C22×Q16—591no—
346C22×M16C2×D5911yescharacteristic
347C2×C22⋊Q8—591no—
348C2×C22⋊Q8—591no—
349C2×C4×Q8—591no—
350C2×C4×Q8—591no—
351C2×#117—591no—
352C22×C4⋊C4—591no—
353#470—598no—
354#441—1184no—
355#428—1184no—
356C2×#107—591no—
357C2×#107—591no—
358C2×#106—591no—
359C2×#106—591no—
360C2×#106—591no—
361C2×#103—591no—
362C2×#103—591no—
363C2×#103—591no—
364C2×#103—591no—
365C2×#103—591no—
366C2×#103—591no—
367C2×C22⋊C8C2×D5911yescharacteristic
368C2×#97C2×D5911yescharacteristic
369C2×C4×C8C2×D5911yescharacteristic
370#391—591no—
371#391—591no—
372#389—591no—
373#389—591no—
374#389—1181no—
375#388—1181no—
376#388—1181no—
377#387—1181no—
378#387—1181no—
379#385—1181no—
380C4×C4⋊C4—591no—
381C4×C4⋊C4—591no—
382C4×C4⋊C4—1181no—
383C4×C22⋊C4—1181no—
384C4×C22⋊C4C2×D5911yescharacteristic
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#336C2×D5911yescharacteristic
390#336C2×D5911yescharacteristic
391C59#2303811yescharacteristic, complemented, Sylow
392C22×Q8—591no—
393C22×Q8—591no—
394C23×C4—1181no—
395C23×C4C22×D5911yescharacteristic
396C2×Q16—594no—
397C2×Q16—1182no—
398C2×Q16—1182no—
399C2×M16—22no—
400C2×M16—22no—
401C2×M16C4×D5914yes—
402C22×C8C22×D5911yescharacteristic
403C22×C8C22×D5911yescharacteristic
404C22×C8C22×D5911yescharacteristic
405C22×C8C22×D5911yescharacteristic
406C22⋊Q8—594no—
407C22⋊Q8—594no—
408C22⋊Q8—1182no—
409C22⋊Q8—1182no—
410C4×Q8—1182no—
411C4×Q8—1182no—
412C4×Q8—2361no—
413C4×Q8—2361no—
414#117—1182no—
415#117—2361no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—591no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—1182no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—2361no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4—1181no—
433C2×C22⋊C4—1181no—
434C2×C22⋊C4D23611yescharacteristic
435C2×C22⋊C4D23611yescharacteristic
436C2×C42—1181no—
437C2×C42—1181no—
438C2×C42—1181no—
439C2×C42—1181no—
440C2×C42—1181no—
441C2×C42C22×D5911yescharacteristic
442C2×C42C22×D5911yescharacteristic
443#107—1182no—
444#107—1182no—
445#106—594no—
446#106—594no—
447#106—1182no—
448#103—594no—
449#103—594no—
450#103—1182no—
451#103—1182no—
452#103—2361no—
453#103—2361no—
454C22⋊C8—22no—
455#97—22no—
456C4×C8—22no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464#95—1181no—
465#95D23611yescharacteristic
466#95D23611yescharacteristic
467C24C2×D23611yescharacteristic
468C2×Q8—592no—
469C2×Q8—592no—
470C2×Q8—1181no—
471C2×Q8—1181no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—1181no—
490C22×C4—1181no—
491C22×C4—1181no—
492C22×C4—4721no—
493C22×C4—2362no—
494C22×C4—2362no—
495C22×C4C2×C4×D5912yes—
496C22×C4C2×D23611yescharacteristic
497C22×C4C2×D23611yescharacteristic
498C22×C4C2×D23611yescharacteristic
499C22×C4C2×D23611yescharacteristic
500C22×C4C2×D23611yescharacteristic
501C22×C4#1174568512yes—
502C22×C4C23×D5911yescharacteristic
503Q16—1188no—
504M16—28no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8C2×C4×D5912yes—
514C2×C8C2×C4×D5912yes—
515C2×C8D4×D5912yes—
516C2×C8#1174568712yes—
517C4⋊C4—592no—
518C4⋊C4—592no—
519C4⋊C4—592no—
520C4⋊C4—592no—
521C4⋊C4—1184no—
522C4⋊C4—1184no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—4721no—
532C4⋊C4—2362no—
533C4⋊C4—2362no—
534C4⋊C4—2362no—
535C4⋊C4—2362no—
536C22⋊C4—22no—
537C22⋊C4—41no—
538C22⋊C4—1184no—
539C22⋊C4—1184no—
540C22⋊C4—2362no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—4721no—
546C42—2362no—
547C42—2362no—
548C42—2362no—
549C42—2362no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23C4×D23612yes—
557C23C22×D23611yescharacteristic
558Q8—1182no—
559Q8—1182no—
560Q8—1182no—
561Q8—1182no—
562Q8—2361no—
563Q8—2361no—
564Q8—2361no—
565Q8—2361no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
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—1182no—
584C2×C4—1182no—
585C2×C4—1182no—
586C2×C4—1182no—
587C2×C4—1182no—
588C2×C4—1182no—
589C2×C4—4724no—
590C2×C4—2361no—
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—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4C4×D23612yes—
609C2×C4#1427376012yes—
610C2×C4C22×C4×D5911yescharacteristic
611C2×C4C2×D4×D5911yescharacteristic
612C2×C4C2×#1174568711yescharacteristic
613C2×C4C2×#1174568511yescharacteristic
614C2×C4#1427377311yescharacteristic
615C2×C4#1427377411yescharacteristic
616C8—22no—
617C8—41no—
618C8—41no—
619C8(C22⋊D59)⋊C414yes—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22C2×C4×D23611yescharacteristic
628C22D4×D23611yescharacteristic
629C22#1683805611yescharacteristic
630C22C2×#1427376011yescharacteristic
631C22#1683820211yescharacteristic
632C22#1683821411yescharacteristic
633C22#1683821911yescharacteristic
634C4—22no—
635C4—22no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—4721no—
640C4—4721no—
641C4—2362no—
642C4—2362no—
643C4—2362no—
644C4—2362no—
645C4—2362no—
646C4—2362no—
647C4C2×(C22⋊D59)⋊C411yescharacteristic
648C4#1683788211yescharacteristic
649C4#1683806111yescharacteristic
650C4#1683809811yescharacteristic
651C2—42no—
652C2#2142711011yescharacteristic
653C2C4×#1427376011yescharacteristic
654C2#2143294111yescharacteristic
655C2#2143573211yescharacteristic
656C2#2143574411yescharacteristic
657C2#2143655311yescharacteristic
658C2#2143662811yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1753C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1753D59
hypocenterC59#23038
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
2. C2×#16837510C2semidirect
6. #21424957C2semidirect
7. #21424839C2semidirect
8. #21424738C2semidirect
9. #21424731C2semidirect
12. #21424367C2semidirect
13. #21424353C2semidirect
14. C4×#14273659C2semidirect
23. #16837660C4semidirect
67. C2×Dic236C2×C4semidirect
104. #14273659C2×C4semidirect
110. #14273652C2×C4semidirect
111. #14273652C2×C4semidirect
391. C59#23038semidirect

Maximal subgroups

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

subgroupindex
C2×#168376602
C2×#168375102
C2×#168374442
#214271212
#214270992
#214249572
#214248392
#214247382
#214247312
#214244062
#214244012
#214243672
#214243532
C4×#142736592
C59×#17532
#2303859