Finite Group Databasebeta

← #28611963

Subgroups — #28611963

Order 15104.

806 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837689C211yesmaximal, characteristic
2#21427067C211yesmaximal, characteristic
3#21425070C211yesmaximal, characteristic, complemented
4#21424864C211yesmaximal, characteristic, complemented
5#21424792C211yesmaximal, characteristic
6#21424778C211yesmaximal, characteristic
7#21424723C211yesmaximal, characteristic
8#21424716C211yesmaximal, characteristic
9#21424678C211yesmaximal, characteristic, complemented
10#21424671C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12C4×#14273658C211yesmaximal, characteristic, complemented
13#21423971C211yesmaximal, characteristic, complemented
14#21423924C211yesmaximal, characteristic
15C59×#2365C211yesmaximal, characteristic, complemented
16C2×C4×D236C2211yescharacteristic
17#16837689C418yescomplemented
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273647C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837304C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837292—21no—
33#16837292—21no—
34#16837268C2211yescharacteristic
35#16837264—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837249—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4⋊Q8×C59C2211yescharacteristic
50C2×C4×Q8×C59—21no—
51C59×#391—21no—
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#387—21no—
55C59×#387—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C4×D236—22no—
67C4×D236C2×C414yes—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×(C2×D59)⋊C4—21no—
72C2×(C2×D59)⋊C4—21no—
73C2×(C2×D59)⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59—21no—
78#14273660—22no—
79#14273658—22no—
80#14273658C2×C414yes—
81#14273658C2×C414yes—
82C4⋊(C59⋊C8)C2×C414yes—
83#14273647—22no—
84#14273647—22no—
85#14273647C2×C414yes—
86#14273647C2×C414yes—
87#14273638—22no—
88#14273638—22no—
89C4×C59⋊C8—22no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C22×Q8×C59—21no—
95C4⋊Q8×C59—22no—
96C4⋊Q8×C59C2×C414yes—
97C4×Q8×C59—44no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59D411yescharacteristic
102C2×C4⋊C4×C59D411yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C42×C59—21no—
108C2×C42×C59—21no—
109C2×C42×C59—21no—
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112C2×C42×C59C2311yescharacteristic
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—21no—
118#95×C59D411yescharacteristic
119#95×C59D411yescharacteristic
120C23×D59—21no—
121C2×Q8×C59—24no—
122C2×Q8×C59—42no—
123C2×Q8×C59—42no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2411yescharacteristic
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C2×D236—21no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236C22×C412yes—
144C2×C4×D59—24no—
145C2×C4×D59—81no—
146C2×C4×D59—81no—
147C4⋊C4×C59—24no—
148C4⋊C4×C59—22no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—41no—
155C4⋊C4×C59—42no—
156C4⋊C4×C59—42no—
157C4⋊C4×C59C22×C412yes—
158C4⋊C4×C59C22×C412yes—
159C4⋊C4×C59#4112yes—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—42no—
165C42×C59—42no—
166C42×C59—42no—
167C42×C59C22×C412yes—
168C42×C59#4112yes—
169(C2×D59)⋊C4—24no—
170(C2×D59)⋊C4—42no—
171(C2×D59)⋊C4—42no—
172(C2×D59)⋊C4—81no—
173(C2×D59)⋊C4—81no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59C22×C412yes—
180C4×Dic59—81no—
181C4×Dic59—81no—
182C2×C59⋊C8—21no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8C22×C412yes—
192C2×C59⋊C8C22×C412yes—
193C2×C59⋊C8#4112yes—
194C23×C59C22×D411yescharacteristic
195C22×D59—22no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C22×D59—41no—
200C22×D59—41no—
201C22×D59—41no—
202Q8×C59—48no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—22no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59C4×D412yes—
231C2×C4×C59C4×D412yes—
232C2×C4×C59AGL(1,ℤ/8)12yes—
233C2×C4×C59C23×C411yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C592−1+411yescharacteristic
238C2×C4×C592−1+411yescharacteristic
239C2×Dic59—22no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248D236—22no—
249D236—22no—
250D236—41no—
251D236—41no—
252C4×D59—84no—
253C59⋊C8—22no—
254C59⋊C8—22no—
255C59⋊C8—22no—
256C59⋊C8—41no—
257#23075—591nomaximal, Sylow
258C22×C59C2×C4×D411yescharacteristic
259C22×C59#54711yescharacteristic
260C22×C59#54811yescharacteristic
261C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C59#57511yescharacteristic
263C22×C59#57711yescharacteristic
264C22×C59#57711yescharacteristic
265C2×D59—42no—
266C2×D59—42no—
267C2×D59—42no—
268C2×D59—81no—
269C2×D59—81no—
270C2×D59—81no—
271C2×D59—81no—
272C4×C59—22no—
273C4×C59—22no—
274C4×C59—22no—
275C4×C59—41no—
276C4×C59—41no—
277C4×C59—41no—
278C4×C59—42no—
279C4×C59—42no—
280C4×C59—42no—
281C4×C59—81no—
282C4×C59C4×#4111yescharacteristic
283C4×C59#51911yescharacteristic
284C4×C59#55211yescharacteristic
285C4×C59#55611yescharacteristic
286Dic59—42no—
287Dic59—81no—
288Dic59—81no—
289C2×#476—591no—
290#2365D5911yescharacteristic, complemented
291#2357—591no—
292#1993—591no—
293#1949—591no—
294#1936—591no—
295#1934—591no—
296#1926—591no—
297#1925—591no—
298#1920—591no—
299#1919—591no—
300#1776—591no—
301C4×#106—591no—
302#1771—591no—
303#1762—591no—
304C2×C59#236211yescharacteristic
305C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
306C2×C59#294911yescharacteristic
307C2×C59#334511yescharacteristic
308C2×C59#335011yescharacteristic
309C2×C59#337111yescharacteristic
310C2×C59#337311yescharacteristic
311D59—82no—
312C2×C4⋊Q8C2×D5911yescharacteristic
313C2×C4×Q8—21no—
314C2×C4×D4—591no—
315#476—598no—
316C2×#107—591no—
317C2×#106—591no—
318C2×#106—591no—
319C2×#106—591no—
320C2×C4⋊C8—591no—
321C2×#102—591no—
322C2×#102—591no—
323C2×#102—591no—
324C2×#102—591no—
325C2×#97—591no—
326C2×#97—591no—
327C2×C4×C8—591no—
328#391—21no—
329#390—591no—
330#390—591no—
331#389C2×D5911yescharacteristic
332#389C2×D5911yescharacteristic
333#388—1181no—
334#388—1181no—
335#387—21no—
336#387—21no—
337#382—591no—
338#381—1181no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4C2×D5911yescharacteristic
342C4×C4⋊C4C2×D5911yescharacteristic
343C4×C4⋊C4C2×D5911yescharacteristic
344C4×C4⋊C4C2×D5911yescharacteristic
345C4×C22⋊C4—1181no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#336—591no—
355#336—591no—
356C59#2307511yescharacteristic, complemented, Sylow
357C22×Q8—21no—
358C22×D4—591no—
359C23×C4—1181no—
360C22×C8—591no—
361C22×C8—591no—
362C22×C8—591no—
363C22×C8—591no—
364C4⋊Q8—22no—
365C4⋊Q8C4×D5914yes—
366C4×Q8—44no—
367C4×D4—594no—
368C4×D4—1182no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4C22⋊D5911yescharacteristic
376C2×C4⋊C4C22⋊D5911yescharacteristic
377C2×C4⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C22⋊C4—1181no—
382C2×C22⋊C4—1181no—
383C2×C22⋊C4—1181no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—1181no—
390C2×C42—1181no—
391C2×C42C22×D5911yescharacteristic
392C2×C42C22×D5911yescharacteristic
393C2×C42C22×D5911yescharacteristic
394#107—1182no—
395#106—594no—
396#106—594no—
397#106—1182no—
398C4⋊C8—594no—
399#102—594no—
400#102—594no—
401#102—1182no—
402#102—1182no—
403#97—1182no—
404#97—1182no—
405C4×C8—1182no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95C22⋊D5911yescharacteristic
416#95C22⋊D5911yescharacteristic
417C24—1181no—
418C2×Q8—24no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×D4—592no—
422C2×D4—1181no—
423C2×D4—1181no—
424C2×D4—1181no—
425C2×D4—1181no—
426C2×D4—1181no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—1181no—
432C22×C4—1184no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—4721no—
436C22×C4—4721no—
437C22×C4C2×C22⋊D5911yescharacteristic
438C22×C4C2×C22⋊D5911yescharacteristic
439C22×C4C2×C22⋊D5911yescharacteristic
440C22×C4C2×C22⋊D5911yescharacteristic
441C22×C4C2×C22⋊D5911yescharacteristic
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C23×D5911yescharacteristic
444C2×C8—592no—
445C2×C8—592no—
446C2×C8—592no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—592no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4C2×C4×D5912yes—
473C4⋊C4C2×C4×D5912yes—
474C4⋊C4#1174568912yes—
475C22⋊C4—1184no—
476C22⋊C4—4721no—
477C22⋊C4—4721no—
478C22⋊C4—2362no—
479C22⋊C4—2362no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—4721no—
488C42—4721no—
489C42C2×C4×D5912yes—
490C42#1174568512yes—
491C23—1182no—
492C23—2361no—
493C23—2361no—
494C23—2361no—
495C23—2361no—
496C23—2361no—
497C23—2361no—
498C23C22×C22⋊D5911yescharacteristic
499Q8—48no—
500D4—1182no—
501D4—1182no—
502D4—2361no—
503D4—2361no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—1182no—
532C2×C4—4724no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4C4×C22⋊D5912yes—
542C2×C4C4×C22⋊D5912yes—
543C2×C4#1427375312yes—
544C2×C4C22×C4×D5911yescharacteristic
545C2×C4#41×D5911yescharacteristic
546C2×C4C2×#1174568511yescharacteristic
547C2×C4C2×#1174568911yescharacteristic
548C2×C4#1427377411yescharacteristic
549C2×C4#1427377511yescharacteristic
550C8—1182no—
551C8—1182no—
552C8—1182no—
553C8—2361no—
554C22—4721no—
555C22—4721no—
556C22—4721no—
557C22—4721no—
558C22—2362no—
559C22—2362no—
560C22—2362no—
561C22C2×C4×C22⋊D5911yescharacteristic
562C22#1683805111yescharacteristic
563C22#1683806311yescharacteristic
564C22C2×#1427375311yescharacteristic
565C22#1683820011yescharacteristic
566C22#1683821011yescharacteristic
567C22#1683821011yescharacteristic
568C4—22no—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—42no—
573C4—41no—
574C4—42no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—4721no—
579C4—4721no—
580C4—2362no—
581C4C4×#1174568911yescharacteristic
582C4#1683787811yescharacteristic
583C4#1683809311yescharacteristic
584C4#1683812511yescharacteristic
585C2—4722no—
586C2#2142711911yescharacteristic
587C2C4×#1427375311yescharacteristic
588C2#2143295511yescharacteristic
589C2#2143656611yescharacteristic
590C2#2143664111yescharacteristic
591C2#2143686411yescharacteristic
592C2#2143688811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2365C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#2365D59
hypocenterC59#23075
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
3. #21425070C2semidirect
4. #21424864C2semidirect
9. #21424678C2semidirect
10. #21424671C2semidirect
11. #21423994C2semidirect
12. C4×#14273658C2semidirect
13. #21423971C2semidirect
15. C59×#2365C2semidirect
17. #16837689C4semidirect
290. #2365D59semidirect
356. C59#23075semidirect

Maximal subgroups

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

subgroupindex
C2×#168376892
#214270672
#214250702
#214248642
#214247922
#214247782
#214247232
#214247162
#214246782
#214246712
#214239942
C4×#142736582
#214239712
#214239242
C59×#23652
#2307559