Finite Group Databasebeta

← #28607044

Subgroups — #28607044

Order 15104.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837504C211yesmaximal, characteristic
2C2×#16837504C211yesmaximal, characteristic
3C2×C4×#11745660C211yesmaximal, characteristic
4#21427016C211yesmaximal, characteristic, complemented
5C4⋊C4×D236C211yesmaximal, characteristic
6#21424834C211yesmaximal, characteristic, complemented
7#21424829C211yesmaximal, characteristic, complemented
8#21424367C211yesmaximal, characteristic, complemented
9#21424360C211yesmaximal, characteristic
10#21424353C211yesmaximal, characteristic, complemented
11#21424346C211yesmaximal, characteristic
12#21424264C211yesmaximal, characteristic, complemented
13#21424107C211yesmaximal, characteristic
14#21424102C211yesmaximal, characteristic, complemented
15C59×#1771C211yesmaximal, characteristic, complemented
16C22×#11745660C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20C2×C4×D236C2211yescharacteristic
21C2×C4×D236C2211yescharacteristic
22C2×C4×D236C2211yescharacteristic
23C2×C4⋊C4×D59—21no—
24#16837504—24no—
25#16837504C418yescomplemented
26C4×#11745660—24no—
27C2×#14273661C2211yescharacteristic
28C2×#14273661C2211yescharacteristic
29C2×#14273661C2211yescharacteristic
30C2×#14273659C2211yescharacteristic
31C2×#14273659C2211yescharacteristic
32C2×#14273652C2211yescharacteristic
33C2×#14273652C2211yescharacteristic
34C2×#14273652C2211yescharacteristic
35C2×#14273649C2211yescharacteristic
36C2×#14273649C2211yescharacteristic
37C2×#14273649C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837300C2211yescharacteristic
41#16837300C2211yescharacteristic
42#16837297—21no—
43#16837297—21no—
44#16837276—21no—
45#16837276—21no—
46#16837272C2211yescharacteristic
47#16837268C2211yescharacteristic
48#16837266—21no—
49#16837259—21no—
50#16837253—21no—
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53#16837165C2211yescharacteristic
54#16837165C2211yescharacteristic
55C2×C4⋊C8×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C2×#97×C59C2211yescharacteristic
58C2×C4×C8×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C59×#336C2211yescharacteristic
61C59×#336C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C22×D236C2311yescharacteristic
64C22×C4×D59—21no—
65C22×C4×D59—21no—
66C22×C4×D59—21no—
67C2×#11745660C2×C414yes—
68C2×#11745660D414yes—
69C2×#11745660Q814yes—
70C4×Dic118—22no—
71C4×Dic118—22no—
72C4×Dic118—22no—
73C4×Dic118—41no—
74C4×Dic118—41no—
75C4×Dic118C2×C414yescomplemented
76C4×D236—22no—
77C4×D236—22no—
78C4×D236—22no—
79C4×D236—41no—
80C4×D236—41no—
81C4×D236C2×C414yes—
82C2×C4⋊Dic59—21no—
83C2×C4⋊Dic59—21no—
84C2×C4⋊Dic59C2311yescharacteristic
85C2×C4⋊Dic59C2311yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×C4⋊Dic59C2311yescharacteristic
88C2×C4⋊Dic59C2311yescharacteristic
89C2×C4⋊Dic59C2311yescharacteristic
90C2×Dic59⋊C4—21no—
91C2×Dic59⋊C4—21no—
92C2×Dic59⋊C4—21no—
93C2×Dic59⋊C4—21no—
94C2×Dic59⋊C4—21no—
95C2×Dic59⋊C4—21no—
96C2×Dic59⋊C4—21no—
97C4⋊C4×D59—44no—
98C2×(C2×D59)⋊C4—21no—
99C2×(C2×D59)⋊C4—21no—
100C2×(C2×D59)⋊C4—21no—
101C2×C4×Dic59—21no—
102C2×C4×Dic59—21no—
103C2×C4×Dic59—21no—
104#14273661—22no—
105#14273661—41no—
106#14273661C2×C414yescomplemented
107#14273659—22no—
108#14273659—41no—
109#14273652—22no—
110#14273652—22no—
111#14273652C2×C414yescomplemented
112#14273649—22no—
113#14273649—22no—
114#14273649C2×C414yes—
115C59⋊#95—21no—
116C59⋊#95—21no—
117C59⋊#95—21no—
118C59⋊#95—21no—
119C59⋊#95—21no—
120C59⋊#95—21no—
121C22×C8×C59C2311yescharacteristic
122C22×C8×C59C2311yescharacteristic
123C22×C8×C59C2311yescharacteristic
124C22×C8×C59C2311yescharacteristic
125C2×C4⋊C4×C59D411yescharacteristic
126C2×C4⋊C4×C59D411yescharacteristic
127C2×C42×C59C2311yescharacteristic
128C2×C42×C59C2311yescharacteristic
129C2×C42×C59C2311yescharacteristic
130C4⋊C8×C59—22no—
131#97×C59—22no—
132#97×C59C2×C414yescomplemented
133C4×C8×C59—22no—
134#95×C59D411yescharacteristic
135#95×C59D411yescharacteristic
136C23×D59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236C22×C412yes—
155C2×D236C2×D412yes—
156C2×D236C2×Q812yes—
157C2×C4×D59—24no—
158C2×C4×D59—42no—
159C2×C4×D59—42no—
160C2×C4×D59—42no—
161C2×C4×D59—42no—
162C2×C4×D59—42no—
163C2×C4×D59—42no—
164C2×C4×D59—81no—
165C2×C4×D59—81no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118C22×C412yes—
170C2×Dic118C2×D412yes—
171C2×Dic118C2×Q812yes—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
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×C59C22×C412yes—
181C2×C8×C59C22×C412yes—
182C2×C8×C59C2×D412yes—
183C2×C8×C59C2×Q812yes—
184C4⋊C4×C59—22no—
185C4⋊C4×C59—41no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59C22×C412yes—
191C42×C59#4112yes—
192(C2×D59)⋊C4—24no—
193(C2×D59)⋊C4—42no—
194(C2×D59)⋊C4—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—21no—
204C4⋊Dic59—21no—
205C4⋊Dic59—42no—
206C4⋊Dic59—81no—
207C4⋊Dic59C22×C412yes—
208C4⋊Dic59C22×C412yes—
209Dic59⋊C4—24no—
210Dic59⋊C4—42no—
211Dic59⋊C4—42no—
212Dic59⋊C4—42no—
213Dic59⋊C4—42no—
214Dic59⋊C4—42no—
215Dic59⋊C4—42no—
216C4×Dic59—24no—
217C4×Dic59—42no—
218C4×Dic59—42no—
219#11745660C4⋊C4116yescomplemented
220C23×C59C22×D411yescharacteristic
221C22×D59—22no—
222C22×D59—22no—
223C22×D59—22no—
224C22×D59—41no—
225C22×D59—41no—
226C22×D59—41no—
227C22×D59—41no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
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—21no—
243C2×C4×C59C4×D412yes—
244C2×C4×C59C4×D412yes—
245C2×C4×C59AGL(1,ℤ/8)12yes—
246C2×C4×C59C23×C411yescharacteristic
247C2×C4×C59C22×D411yescharacteristic
248C2×C4×C59C22×Q811yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59H(5,𝔽2)11yescharacteristic
251C2×C4×C592−1+411yescharacteristic
252C2×Dic59—22no—
253C2×Dic59—22no—
254C2×Dic59—22no—
255C2×Dic59—22no—
256C2×Dic59—22no—
257C2×Dic59—41no—
258C2×Dic59—41no—
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—
273D236—22no—
274D236—22no—
275D236—22no—
276D236C2×C4⋊C414yes—
277C4×D59—48no—
278C4×D59—84no—
279C4×D59—84no—
280Dic118—22no—
281Dic118—22no—
282Dic118—22no—
283Dic118C2×C4⋊C414yescomplemented
284C8×C59—22no—
285C8×C59—41no—
286C8×C59—41no—
287C8×C59C2×C4⋊C414yescomplemented
288#22677—591nomaximal, Sylow
289C22×C59C2×C4×D411yescharacteristic
290C22×C59D4211yescharacteristic
291C22×C59D4×Q811yescharacteristic
292C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C59#57511yescharacteristic
294C22×C59#57711yescharacteristic
295C22×C59#57811yescharacteristic
296C2×D59—24no—
297C2×D59—42no—
298C2×D59—42no—
299C2×D59—42no—
300C2×D59—42no—
301C2×D59—42no—
302C2×D59—42no—
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—22no—
306C4×C59—41no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59C22×C4⋊C411yescharacteristic
310C4×C59#52011yescharacteristic
311C4×C59#54711yescharacteristic
312C4×C59#55411yescharacteristic
313Dic59—24no—
314Dic59—42no—
315Dic59—42no—
316Dic59—81no—
317Dic59—81no—
318Dic59—81no—
319Dic59—81no—
320C2×#440—591no—
321C2×#440—591no—
322C2×C4×SD16—591no—
323#2351—591no—
324D4×C4⋊C4—591no—
325#1943—591no—
326#1942—591no—
327#1865—591no—
328#1864—591no—
329#1863—591no—
330#1862—591no—
331#1847—591no—
332#1804—591no—
333#1803—591no—
334#1771D5911yescharacteristic, complemented
335C2×C59D4×C4⋊C411yescharacteristic
336C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
337C2×C59#294711yescharacteristic
338C2×C59#326511yescharacteristic
339C2×C59#326911yescharacteristic
340C2×C59#338611yescharacteristic
341C2×C59#338811yescharacteristic
342D59—44no—
343C22×SD16—591no—
344C2×C4×Q8—591no—
345C2×C4×Q8—591no—
346C2×C4×Q8—591no—
347C2×C4×D4—591no—
348C2×C4×D4—591no—
349C2×C4×D4—591no—
350C22×C4⋊C4—1181no—
351#440—598no—
352#440—1184no—
353C4×SD16—1184no—
354C2×#107—591no—
355C2×#107—591no—
356C2×#107—591no—
357C2×#106—591no—
358C2×#106—591no—
359C2×C4⋊C8C2×D5911yescharacteristic
360C2×#103—591no—
361C2×#103—591no—
362C2×#103—591no—
363C2×#102—591no—
364C2×#102—591no—
365C2×#102—591no—
366C2×#97C2×D5911yescharacteristic
367C2×#97C2×D5911yescharacteristic
368C2×C4×C8C2×D5911yescharacteristic
369#389—591no—
370#389—591no—
371#389—591no—
372#389—591no—
373#389—1181no—
374#389—1181no—
375#385—1181no—
376#385—1181no—
377#384—591no—
378#382—591no—
379#382—1181no—
380#380—1181no—
381C4×C4⋊C4—1181no—
382C4×C4⋊C4C2×D5911yescharacteristic
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#336C2×D5911yescharacteristic
388#336C2×D5911yescharacteristic
389C59#2267711yescharacteristic, complemented, Sylow
390C22×Q8—591no—
391C22×D4—591no—
392C23×C4—1181no—
393C23×C4—1181no—
394C23×C4—1181no—
395C2×SD16—594no—
396C2×SD16—594no—
397C2×SD16—594no—
398C22×C8C22×D5911yescharacteristic
399C22×C8C22×D5911yescharacteristic
400C22×C8C22×D5911yescharacteristic
401C22×C8C22×D5911yescharacteristic
402C4×Q8—594no—
403C4×Q8—1182no—
404C4×Q8—1182no—
405C4×Q8—1182no—
406C4×Q8—2361no—
407C4×Q8—2361no—
408C4×D4—594no—
409C4×D4—1182no—
410C4×D4—1182no—
411C4×D4—1182no—
412C4×D4—2361no—
413C4×D4—2361no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—591no—
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⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—2364no—
430C2×C4⋊C4D23611yescharacteristic
431C2×C4⋊C4D23611yescharacteristic
432C2×C22⋊C4—1181no—
433C2×C22⋊C4—1181no—
434C2×C22⋊C4—1181no—
435C2×C42—1181no—
436C2×C42—1181no—
437C2×C42—1181no—
438C2×C42C22×D5911yescharacteristic
439C2×C42C22×D5911yescharacteristic
440C2×C42C22×D5911yescharacteristic
441#107—594no—
442#107—1182no—
443#107—2361no—
444#106—1182no—
445#106—2361no—
446C4⋊C8—22no—
447#103—594no—
448#103—1182no—
449#103—1182no—
450#102—594no—
451#102—1182no—
452#102—1182no—
453#97—22no—
454#97C4×D5914yescomplemented
455C4×C8—22no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462#95D23611yescharacteristic
463#95D23611yescharacteristic
464C24—1181no—
465C2×Q8—592no—
466C2×Q8—592no—
467C2×Q8—592no—
468C2×Q8—1181no—
469C2×Q8—1181no—
470C2×Q8—1181no—
471C2×D4—592no—
472C2×D4—592no—
473C2×D4—592no—
474C2×D4—1181no—
475C2×D4—1181no—
476C2×D4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—1184no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4—4721no—
486C22×C4—4721no—
487C22×C4—2362no—
488C22×C4—2362no—
489C22×C4—2362no—
490C22×C4—2362no—
491C22×C4—2362no—
492C22×C4—2362no—
493C22×C4C2×D23611yescharacteristic
494C22×C4C2×D23611yescharacteristic
495C22×C4C2×D23611yescharacteristic
496C22×C4C2×D23611yescharacteristic
497C22×C4C2×D23611yescharacteristic
498C22×C4C2×D23611yescharacteristic
499C22×C4C23×D5911yescharacteristic
500SD16—5916no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8C2×C4×D5912yes—
510C2×C8C2×C4×D5912yes—
511C2×C8D4×D5912yes—
512C2×C8Q8×D5912yes—
513C4⋊C4—22no—
514C4⋊C4—41no—
515C4⋊C4—592no—
516C4⋊C4—592no—
517C4⋊C4—1184no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—1181no—
528C4⋊C4—4721no—
529C4⋊C4—2362no—
530C4⋊C4—2362no—
531C4⋊C4—2362no—
532C4⋊C4—2362no—
533C4⋊C4—2362no—
534C4⋊C4—2362no—
535C4⋊C4—2362no—
536C22⋊C4—1184no—
537C22⋊C4—2362no—
538C22⋊C4—2362no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—1184no—
544C42—2362no—
545C42—2362no—
546C42C2×C4×D5912yes—
547C42#1174568512yes—
548C23—1182no—
549C23—1182no—
550C23—1182no—
551C23—2361no—
552C23—2361no—
553C23—2361no—
554C23—2361no—
555C23C22×D23611yescharacteristic
556Q8—594no—
557Q8—1182no—
558Q8—1182no—
559Q8—1182no—
560D4—594no—
561D4—1182no—
562D4—1182no—
563D4—1182no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—1182no—
580C2×C4—1182no—
581C2×C4—1182no—
582C2×C4—1182no—
583C2×C4—1182no—
584C2×C4—4724no—
585C2×C4—4724no—
586C2×C4—2368no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
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×C4C4×D23612yes—
604C2×C4C4×D23612yes—
605C2×C4#1427375512yes—
606C2×C4C22×C4×D5911yescharacteristic
607C2×C4C2×D4×D5911yescharacteristic
608C2×C4C2×Q8×D5911yescharacteristic
609C2×C4C2×#1174568511yescharacteristic
610C2×C4#1427377311yescharacteristic
611C2×C4#1427377411yescharacteristic
612C8—22no—
613C8—41no—
614C8—41no—
615C8C4⋊C4×D5914yescomplemented
616C22—1184no—
617C22—2362no—
618C22—2362no—
619C22—2362no—
620C22—2362no—
621C22—2362no—
622C22—2362no—
623C22C2×C4×D23611yescharacteristic
624C22D4×D23611yescharacteristic
625C22Q8×D23611yescharacteristic
626C22C2×#1427375511yescharacteristic
627C22#1683820211yescharacteristic
628C22#1683821411yescharacteristic
629C22#1683821911yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—1184no—
637C4—4721no—
638C4—4721no—
639C4—4721no—
640C4—4721no—
641C4—2362no—
642C4—2362no—
643C4C2×C4⋊C4×D5911yescharacteristic
644C4#1683788211yescharacteristic
645C4#1683805711yescharacteristic
646C4#1683811211yescharacteristic
647C2—2364no—
648C2C4⋊C4×D23611yescharacteristic
649C2C4×#1427375511yescharacteristic
650C2#2143294111yescharacteristic
651C2#2143559611yescharacteristic
652C2#2143565611yescharacteristic
653C2#2143701911yescharacteristic
654C2#2143704311yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1771C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1771D59
hypocenterC59#22677
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
4. #21427016C2semidirect
6. #21424834C2semidirect
7. #21424829C2semidirect
8. #21424367C2semidirect
10. #21424353C2semidirect
12. #21424264C2semidirect
14. #21424102C2semidirect
15. C59×#1771C2semidirect
25. #16837504C4semidirect
75. C4×Dic118C2×C4semidirect
106. #14273661C2×C4semidirect
111. #14273652C2×C4semidirect
132. #97×C59C2×C4semidirect
219. #11745660C4⋊C4semidirect
283. Dic118C2×C4⋊C4semidirect
287. C8×C59C2×C4⋊C4semidirect
334. #1771D59semidirect
389. C59#22677semidirect
454. #97C4×D59semidirect
615. C8C4⋊C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168375042
C2×C4×#117456602
#214270162
C4⋊C4×D2362
#214248342
#214248292
#214243672
#214243602
#214243532
#214243462
#214242642
#214241072
#214241022
C59×#17712
#2267759