Finite Group Databasebeta

← #28640494

Subgroups — #28640494

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×#16837699C211yesmaximal, characteristic
2#21429270C211yesmaximal, characteristic
3#21427196C211yesmaximal, characteristic, complemented
4#21425581C211yesmaximal, characteristic, complemented
5#21425580C211yesmaximal, characteristic, complemented
6#21425479C211yesmaximal, characteristic
7#21425465C211yesmaximal, characteristic
8#21425049C211yesmaximal, characteristic, complemented
9#21424865C211yesmaximal, characteristic, complemented
10#21424861C211yesmaximal, characteristic, complemented
11#21424819C211yesmaximal, characteristic, complemented
12#21424722C211yesmaximal, characteristic
13#21424715C211yesmaximal, characteristic
14#21423922C211yesmaximal, characteristic
15C2×C59×#501C211yesmaximal, characteristic, complemented
16C2×C4×D236C2211yescharacteristic
17#16837699—24no—
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273649C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27#16837313—21no—
28#16837313—21no—
29#16837303C2211yescharacteristic
30#16837303C2211yescharacteristic
31#16837301—21no—
32#16837301C2211yescharacteristic
33#16837301C2211yescharacteristic
34#16837288—21no—
35#16837288—21no—
36#16837267C2211yescharacteristic
37#16837267C2211yescharacteristic
38#16837259—21no—
39#16837259—21no—
40C4⋊C4×Dic59—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4⋊Q8×C59C2211yescharacteristic
54C2×#125×C59C2211yescharacteristic
55C59×#501—24no—
56C2×#107×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×#106×C59C2211yescharacteristic
59C2×#106×C59C2211yescharacteristic
60C2×#97×C59C2211yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C4×D236—22no—
66C4×D236—22no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59D411yescharacteristic
70C2×C4⋊Dic59D411yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×(C2×D59)⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59C2311yescharacteristic
83C2×C4×Dic59C2311yescharacteristic
84#14273660—22no—
85#14273658—22no—
86C4⋊(C59⋊C8)—22no—
87Dic59⋊C8—22no—
88Dic59⋊C8—22no—
89#14273649—22no—
90#14273649Q814yescomplemented
91#14273647—22no—
92#14273647—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C22×Q8×C59—21no—
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C4⋊Q8×C59—22no—
108C4⋊Q8×C59—22no—
109#125×C59—22no—
110#125×C59D414yes—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119#107×C59—22no—
120#107×C59—22no—
121#106×C59—22no—
122#106×C59—22no—
123#97×C59—22no—
124C23×D59—21no—
125C2×Q8×C59—42no—
126C2×Q8×C59—81no—
127C2×Q8×C59—81no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2411yescharacteristic
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C2×D236—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236—21no—
147C2×D236C2×Q812yes—
148C2×C4×D59—42no—
149C2×C4×D59—42no—
150C2×C4×D59—42no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59C2×Q812yes—
156C2×C8×C59#4112yes—
157C4⋊C4×C59—24no—
158C4⋊C4×C59—24no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—42no—
166C4⋊C4×C59C2×D412yes—
167C4⋊C4×C59C2×D412yes—
168C42×C59C2×D412yes—
169C42×C59#4112yes—
170(C2×D59)⋊C4—42no—
171(C2×D59)⋊C4—81no—
172(C2×D59)⋊C4—81no—
173(C2×D59)⋊C4—81no—
174(C2×D59)⋊C4—81no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—41no—
181C4⋊Dic59—41no—
182C4⋊Dic59—42no—
183C4⋊Dic59—81no—
184C4⋊Dic59C2×Q812yes—
185C4×Dic59—21no—
186C4×Dic59—21no—
187C4×Dic59—21no—
188C4×Dic59—21no—
189C4×Dic59—42no—
190C4×Dic59—81no—
191C4×Dic59—81no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C23×C59C22×D411yescharacteristic
199C22×D59—22no—
200C22×D59—41no—
201C22×D59—41no—
202C22×D59—41no—
203C22×D59—41no—
204C22×D59—41no—
205C22×D59—41no—
206Q8×C59—84no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59#12112yes—
224C2×C4×C59#12112yes—
225C2×C4×C59C22×D411yescharacteristic
226C2×C4×C59C22×Q811yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C592−1+411yescharacteristic
230C2×C4×C592−1+411yescharacteristic
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—22no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252D236—22no—
253D236—22no—
254D236—41no—
255D236—41no—
256C4×D59—84no—
257C8×C59—22no—
258C8×C59—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—42no—
279C4×C59—42no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59#53311yescharacteristic
283C4×C59#54111yescharacteristic
284C4×C59#54811yescharacteristic
285C4×C59D4×Q811yescharacteristic
286Dic59—41no—
287Dic59—41no—
288Dic59—41no—
289Dic59—41no—
290Dic59—42no—
291Dic59—81no—
292Dic59—81no—
293C2×#501D5911yescharacteristic, complemented
294C2×#478—591no—
295#2563—591no—
296#2370—591no—
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⋊Q8C2×D5911yescharacteristic
317C2×#125C2×D5911yescharacteristic
318C2×C4×D4—591no—
319#501—24no—
320#478—1184no—
321C2×#107—591no—
322C2×#107C2×D5911yescharacteristic
323C2×#107C2×D5911yescharacteristic
324C2×#106—591no—
325C2×#106C2×D5911yescharacteristic
326C2×#106C2×D5911yescharacteristic
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×#97C2×D5911yescharacteristic
335#393—1181no—
336#393—1181no—
337#390—591no—
338#390—591no—
339#389—591no—
340#389—591no—
341#389—1181no—
342#387—1181no—
343#387—1181no—
344#382—591no—
345#382—591no—
346#380—1181no—
347#380—1181no—
348C4×C4⋊C4—591no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—1181no—
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—21no—
363C22×D4—591no—
364C23×C4—1181no—
365C22×C8—591no—
366C22×C8—591no—
367C22×C8C22×D5911yescharacteristic
368C22×C8C22×D5911yescharacteristic
369C4⋊Q8—22no—
370C4⋊Q8—22no—
371#125—22no—
372#125C22⋊D5914yes—
373C4×D4—1182no—
374C4×D4—1182no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4—1181no—
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—591no—
395C2×C42—591no—
396C2×C42—1181no—
397C2×C42—1181no—
398C2×C42—1181no—
399C2×C42C22×D5911yescharacteristic
400#107—22no—
401#107—22no—
402#107—1182no—
403#106—22no—
404#106—22no—
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—22no—
414#95—591no—
415#95—591no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425C24—1181no—
426C2×Q8—42no—
427C2×Q8—81no—
428C2×Q8—81no—
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—591no—
440C22×C4—591no—
441C22×C4—591no—
442C22×C4—591no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—2362no—
447C22×C4—2362no—
448C22×C4—2362no—
449C22×C4D4×D5911yescharacteristic
450C22×C4D4×D5911yescharacteristic
451C22×C4C23×D5911yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8Q8×D5912yes—
463C2×C8#1174568912yes—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—592no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—4721no—
480C4⋊C4—2361no—
481C4⋊C4—2361no—
482C4⋊C4—2362no—
483C4⋊C4C2×C22⋊D5912yes—
484C4⋊C4C2×C22⋊D5912yes—
485C22⋊C4—4721no—
486C22⋊C4—4721no—
487C22⋊C4—4721no—
488C22⋊C4—4721no—
489C22⋊C4—2362no—
490C42—1181no—
491C42—1181no—
492C42—1181no—
493C42—1181no—
494C42—4721no—
495C42—4721no—
496C42—2362no—
497C42#1174568712yes—
498C42C2×C22⋊D5912yes—
499C23—1182no—
500C23—2361no—
501C23—2361no—
502C23—2361no—
503C23—2361no—
504C23—2361no—
505C23—2361no—
506C23C2×D4×D5911yescharacteristic
507Q8—84no—
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—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1182no—
541C2×C4—4724no—
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#1427369912yes—
551C2×C4#1427369912yes—
552C2×C4C22×C22⋊D5911yescharacteristic
553C2×C4C2×Q8×D5911yescharacteristic
554C2×C4C2×#1174568711yescharacteristic
555C2×C4C2×#1174568911yescharacteristic
556C2×C4#1427377511yescharacteristic
557C2×C4#1427377511yescharacteristic
558C8—22no—
559C8—22no—
560C8—2361no—
561C8—2361no—
562C22—4721no—
563C22—4721no—
564C22—4721no—
565C22—4721no—
566C22—2362no—
567C22—2362no—
568C22—2362no—
569C22C2×#1427369911yescharacteristic
570C22#1683805511yescharacteristic
571C22#1683806711yescharacteristic
572C22#1683820111yescharacteristic
573C22#1683820111yescharacteristic
574C22#1683820611yescharacteristic
575C22#1683821211yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—4721no—
583C4—4721no—
584C4—2362no—
585C4—2361no—
586C4—2361no—
587C4—2361no—
588C4—2361no—
589C4#1683795711yescharacteristic
590C4#1683801311yescharacteristic
591C4#1683806311yescharacteristic
592C4Q8×C22⋊D5911yescharacteristic
593C2—4722no—
594C2#2142769811yescharacteristic
595C2#2143359211yescharacteristic
596C2#2143363911yescharacteristic
597C2#2143659511yescharacteristic
598C2#2143661011yescharacteristic
599C2#2143704011yescharacteristic
600C2#2143705111yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#501C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenterC2×#501D59
hypocenterC59#25466
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #21427196C2semidirect
4. #21425581C2semidirect
5. #21425580C2semidirect
8. #21425049C2semidirect
9. #21424865C2semidirect
10. #21424861C2semidirect
11. #21424819C2semidirect
15. C2×C59×#501C2semidirect
90. #14273649Q8semidirect
293. C2×#501D59semidirect
361. C59#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#168376992
#214292702
#214271962
#214255812
#214255802
#214254792
#214254652
#214250492
#214248652
#214248612
#214248192
#214247222
#214247152
#214239222
C2×C59×#5012
#2546659