Finite Group Databasebeta

← #28640482

Subgroups — #28640482

Order 15104.

756 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837812C211yesmaximal, characteristic, complemented
2C2×#16837699C211yesmaximal, characteristic
3#21429265C211yesmaximal, characteristic
4#21425577C211yesmaximal, characteristic, complemented
5#21425577C211yesmaximal, characteristic, complemented
6#21425481C211yesmaximal, characteristic
7#21425467C211yesmaximal, characteristic
8#21425048C211yesmaximal, characteristic, complemented
9#21424864C211yesmaximal, characteristic, complemented
10#21424864C211yesmaximal, characteristic, complemented
11#21424820C211yesmaximal, characteristic, complemented
12#21424723C211yesmaximal, characteristic
13#21424716C211yesmaximal, characteristic
14#21423924C211yesmaximal, characteristic
15C59×#2370C211yesmaximal, characteristic, complemented
16C2×#14273727C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18#16837812—24no—
19#16837699—24no—
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×C4⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×#14273647C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×#14273638C2211yescharacteristic
34#16837315—21no—
35#16837315—21no—
36#16837304C2211yescharacteristic
37#16837304C2211yescharacteristic
38#16837268C2211yescharacteristic
39#16837268C2211yescharacteristic
40#16837259—21no—
41#16837259—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#125×C59C2211yescharacteristic
51C59×#389—21no—
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#387—21no—
55C59×#387—21no—
56C4×C4⋊C4×C59—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118—21no—
62C22×D236C2311yescharacteristic
63C22×C4×D59—21no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273727—22no—
69#14273727—22no—
70C4×D236—22no—
71C4×D236—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81#14273660—22no—
82#14273660—22no—
83#14273660—22no—
84#14273658—22no—
85#14273658—22no—
86#14273658—22no—
87C4⋊(C59⋊C8)—22no—
88C4⋊(C59⋊C8)—22no—
89C4⋊(C59⋊C8)—22no—
90#14273647—22no—
91#14273647—22no—
92#14273647—22no—
93#14273647Q814yescomplemented
94#14273638—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99#125×C59—22no—
100#125×C59D414yes—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59D411yescharacteristic
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C42×C59—21no—
112C2×C42×C59—21no—
113C2×C42×C59—21no—
114C2×C42×C59C2311yescharacteristic
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117#95×C59—21no—
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59D411yescharacteristic
123#95×C59D411yescharacteristic
124C23×D59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2411yescharacteristic
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236C2×Q812yes—
145C2×C4×D59—42no—
146C2×C4×D59—42no—
147C2×C4×D59—42no—
148C2×Dic118—42no—
149C2×Dic118—81no—
150C2×Dic118—81no—
151C4⋊C4×C59—24no—
152C4⋊C4×C59—24no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—41no—
159C4⋊C4×C59—41no—
160C4⋊C4×C59—42no—
161C4⋊C4×C59—81no—
162C4⋊C4×C59C2×D412yes—
163C4⋊C4×C59C2×D412yes—
164C4⋊C4×C59C2×Q812yes—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—42no—
170C42×C59—81no—
171C42×C59—81no—
172C42×C59C2×D412yes—
173C42×C59#4112yes—
174(C2×D59)⋊C4—42no—
175(C2×D59)⋊C4—81no—
176(C2×D59)⋊C4—81no—
177(C2×D59)⋊C4—81no—
178(C2×D59)⋊C4—81no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—42no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8C2×Q812yes—
197C2×C59⋊C8#4112yes—
198C23×C59C22×D411yescharacteristic
199C22×D59—22no—
200C22×D59—41no—
201C22×D59—41no—
202C22×D59—41no—
203C22×D59—41no—
204C22×D59—41no—
205C22×D59—41no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—22no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59#12112yes—
235C2×C4×C59#12112yes—
236C2×C4×C59C22×D411yescharacteristic
237C2×C4×C59C22×Q811yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C592−1+411yescharacteristic
241C2×C4×C592−1+411yescharacteristic
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251D236—22no—
252D236—22no—
253D236—41no—
254D236—41no—
255C4×D59—84no—
256Dic118—84no—
257C59⋊C8—22no—
258C59⋊C8—22no—
259C59⋊C8—41no—
260C59⋊C8—41no—
261#25466—591nomaximal, Sylow
262C22×C59C2×#12111yescharacteristic
263C22×C59#54711yescharacteristic
264C22×C59D4×Q811yescharacteristic
265C22×C59#57511yescharacteristic
266C22×C59#57511yescharacteristic
267C22×C59#57611yescharacteristic
268C22×C59#57711yescharacteristic
269C2×D59—42no—
270C2×D59—42no—
271C2×D59—42no—
272C2×D59—81no—
273C2×D59—81no—
274C2×D59—81no—
275C2×D59—81no—
276C4×C59—22no—
277C4×C59—22no—
278C4×C59—41no—
279C4×C59—41no—
280C4×C59—41no—
281C4×C59—41no—
282C4×C59—42no—
283C4×C59—42no—
284C4×C59—42no—
285C4×C59—81no—
286C4×C59#53311yescharacteristic
287C4×C59#54111yescharacteristic
288C4×C59#54811yescharacteristic
289C4×C59D4×Q811yescharacteristic
290Dic59—81no—
291Dic59—81no—
292Dic59—81no—
293C2×#501—591no—
294C2×#478—591no—
295#2563—591no—
296#2370D5911yescharacteristic, complemented
297#2085—591no—
298#2085—591no—
299#2067—591no—
300#2065—591no—
301#1988—591no—
302#1949—591no—
303#1949—591no—
304#1940—591no—
305#1926—591no—
306#1925—591no—
307#1762—591no—
308C2×C59#242911yescharacteristic
309C2×C59#304011yescharacteristic
310C2×C59#304511yescharacteristic
311C2×C59#334711yescharacteristic
312C2×C59#334811yescharacteristic
313C2×C59#338811yescharacteristic
314C2×C59#338911yescharacteristic
315D59—82no—
316C2×C4⋊Q8—591no—
317C2×#125C2×D5911yescharacteristic
318C2×C4×D4—591no—
319#501—1184no—
320#478—1184no—
321C2×#107—591no—
322C2×#107—591no—
323C2×#107—591no—
324C2×#106—591no—
325C2×#106—591no—
326C2×#106—591no—
327C2×C4⋊C8—591no—
328C2×C4⋊C8—591no—
329C2×C4⋊C8—591no—
330C2×#102—591no—
331C2×#102—591no—
332C2×#102—591no—
333C2×#102—591no—
334C2×#97—591no—
335#393—1181no—
336#393—1181no—
337#390—591no—
338#390—591no—
339#389—21no—
340#389C2×D5911yescharacteristic
341#389C2×D5911yescharacteristic
342#387—21no—
343#387—21no—
344#382—591no—
345#382—591no—
346#380—1181no—
347#380—1181no—
348C4×C4⋊C4—21no—
349C4×C4⋊C4C2×D5911yescharacteristic
350C4×C4⋊C4C2×D5911yescharacteristic
351C4×C4⋊C4C2×D5911yescharacteristic
352C4×C4⋊C4C2×D5911yescharacteristic
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#336—591no—
360#336—591no—
361C59#2546611yescharacteristic, complemented, Sylow
362C22×Q8—1181no—
363C22×D4—591no—
364C23×C4—1181no—
365C22×C8—591no—
366C22×C8—591no—
367C22×C8—591no—
368C22×C8—591no—
369C4⋊Q8—1182no—
370C4⋊Q8—1182no—
371#125—22no—
372#125D23614yes—
373C4×D4—1182no—
374C4×D4—1182no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4C22⋊D5911yescharacteristic
384C2×C4⋊C4C22⋊D5911yescharacteristic
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C4⋊C4C22×D5911yescharacteristic
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C22⋊C4—1181no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C22⋊C4—1181no—
393C2×C22⋊C4—1181no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42C22×D5911yescharacteristic
398C2×C42C22×D5911yescharacteristic
399C2×C42C22×D5911yescharacteristic
400#107—1182no—
401#107—1182no—
402#107—1182no—
403#106—1182no—
404#106—1182no—
405#106—1182no—
406C4⋊C8—1182no—
407C4⋊C8—1182no—
408C4⋊C8—1182no—
409#102—594no—
410#102—1182no—
411#102—1182no—
412#102—1182no—
413#97—1182no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95C22⋊D5911yescharacteristic
424#95C22⋊D5911yescharacteristic
425C24—1181no—
426C2×Q8—4721no—
427C2×Q8—4721no—
428C2×Q8—2362no—
429C2×D4—592no—
430C2×D4—1181no—
431C2×D4—1181no—
432C2×D4—1181no—
433C2×D4—1181no—
434C2×D4—1181no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4—2362no—
443C22×C4—2362no—
444C22×C4—2362no—
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C2×C22⋊D5911yescharacteristic
447C22×C4C2×C22⋊D5911yescharacteristic
448C22×C4C2×C22⋊D5911yescharacteristic
449C22×C4C2×C22⋊D5911yescharacteristic
450C22×C4C2×C22⋊D5911yescharacteristic
451C22×C4C23×D5911yescharacteristic
452C2×C8—592no—
453C2×C8—592no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—81no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—2362no—
482C4⋊C4C2×D23612yes—
483C4⋊C4C2×D23612yes—
484C4⋊C4Q8×D5912yes—
485C22⋊C4—4721no—
486C22⋊C4—4721no—
487C22⋊C4—4721no—
488C22⋊C4—4721no—
489C22⋊C4—2362no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—42no—
495C42—81no—
496C42—81no—
497C42C2×D23612yes—
498C42#1174568512yes—
499C23—1182no—
500C23—2361no—
501C23—2361no—
502C23—2361no—
503C23—2361no—
504C23—2361no—
505C23—2361no—
506C23C22×C22⋊D5911yescharacteristic
507Q8—4724no—
508D4—1182no—
509D4—1182no—
510D4—2361no—
511D4—2361no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—4724no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4#1427370112yes—
551C2×C4#1427370112yes—
552C2×C4C22×D23611yescharacteristic
553C2×C4C2×Q8×D5911yescharacteristic
554C2×C4#41×D5911yescharacteristic
555C2×C4C2×#1174568511yescharacteristic
556C2×C4#1427377411yescharacteristic
557C2×C4#1427377511yescharacteristic
558C8—1182no—
559C8—1182no—
560C8—2361no—
561C8—2361no—
562C22—4721no—
563C22—4721no—
564C22—4721no—
565C22—4721no—
566C22—2362no—
567C22—2362no—
568C22—2362no—
569C22C2×#1427370111yescharacteristic
570C22#1683805111yescharacteristic
571C22Q8×C22⋊D5911yescharacteristic
572C22#1683820011yescharacteristic
573C22#1683820011yescharacteristic
574C22#1683820511yescharacteristic
575C22#1683821011yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—81no—
586C4—4721no—
587C4—4721no—
588C4—4721no—
589C4#1683795611yescharacteristic
590C4#1683801011yescharacteristic
591C4#1683806411yescharacteristic
592C4Q8×D23611yescharacteristic
593C2—4722no—
594C2#2142768911yescharacteristic
595C2#2143359011yescharacteristic
596C2#2143363511yescharacteristic
597C2#2143659011yescharacteristic
598C2#2143660511yescharacteristic
599C2#2143703511yescharacteristic
600C2#2143705011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2370C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2370D59
hypocenterC59#25466
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
1. C2×#16837812C2semidirect
4. #21425577C2semidirect
5. #21425577C2semidirect
8. #21425048C2semidirect
9. #21424864C2semidirect
10. #21424864C2semidirect
11. #21424820C2semidirect
15. C59×#2370C2semidirect
93. #14273647Q8semidirect
296. #2370D59semidirect
361. C59#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#168378122
C2×#168376992
#214292652
#214255772
#214254812
#214254672
#214250482
#214248642
#214248202
#214247232
#214247162
#214239242
C59×#23702
#2546659