Finite Group Databasebeta

← #28637181

Subgroups — #28637181

Order 15104.

880 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837713C211yesmaximal, characteristic
2C2×#16837646C211yesmaximal, characteristic
3#21428574C211yesmaximal, characteristic, complemented
4#21425567C211yesmaximal, characteristic, complemented
5#21425499C211yesmaximal, characteristic, complemented
6#21425488C211yesmaximal, characteristic, complemented
7#21425413C211yesmaximal, characteristic, complemented
8#21424854C211yesmaximal, characteristic, complemented
9#21424737C211yesmaximal, characteristic, complemented
10#21424430C211yesmaximal, characteristic, complemented
11#21424408C211yesmaximal, characteristic, complemented
12#21424281C211yesmaximal, characteristic, complemented
13#21424096C211yesmaximal, characteristic, complemented
14#21423817C211yesmaximal, characteristic, complemented
15C59×#2363C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C22×#11745664C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837713—24no—
21#16837646—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34#16837334—21no—
35#16837334—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837277—21no—
39#16837277—21no—
40#16837272C2211yescharacteristic
41#16837272C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#121×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C59×#390—21no—
54C59×#390C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#388—21no—
57C59×#386—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59—21no—
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C23×Dic59—21no—
63C2×#11745670—22no—
64C2×#11745670—22no—
65C2×#11745670—41no—
66C2×#11745664—22no—
67C2×#11745664—22no—
68C2×#11745664—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273706—22no—
74#14273706—22no—
75C2×C4⋊Dic59—22no—
76C2×C4⋊Dic59—22no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59—21no—
81C2×C4⋊Dic59—21no—
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×C4⋊Dic59C2311yescharacteristic
85C2×Dic59⋊C4—21no—
86C2×C22⋊Dic59—21no—
87#14273660—22no—
88#14273660—22no—
89#14273660—22no—
90C4⋊(C59⋊C8)—22no—
91C4⋊(C59⋊C8)—41no—
92#14273650—22no—
93#14273650—22no—
94#14273650—41no—
95#14273648—22no—
96#14273648—22no—
97#14273648—22no—
98C22⋊(C59⋊C8)—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×D4×C59—21no—
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59—21no—
106C23×C4×C59C2311yescharacteristic
107#121×C59—22no—
108#121×C59D414yes—
109C4×D4×C59—22no—
110C4×D4×C59—41no—
111#117×C59—22no—
112#117×C59—41no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59D411yescharacteristic
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C42×C59—21no—
125C2×C42×C59—21no—
126C2×C42×C59C2311yescharacteristic
127C2×C42×C59C2311yescharacteristic
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59—21no—
133C24×C59C2×D411yescharacteristic
134C2×D4×C59—24no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59—81no—
141C2×D4×C59—81no—
142C2×D4×C59C2×D412yes—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—42no—
151C22×C4×C59—42no—
152C22×C4×C59—81no—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D412yes—
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59#4112yes—
160C22×C4×C59C2411yescharacteristic
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C22×Dic59—42no—
166C22×Dic59—81no—
167C22×Dic59—81no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—81no—
179C4⋊C4×C59C2×D412yes—
180C4⋊C4×C59#4112yes—
181C22⋊C4×C59—24no—
182C22⋊C4×C59—22no—
183C22⋊C4×C59—22no—
184C22⋊C4×C59—42no—
185C22⋊C4×C59—42no—
186C22⋊C4×C59—42no—
187C22⋊C4×C59—81no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—21no—
191C42×C59—21no—
192C42×C59—42no—
193C42×C59—81no—
194C22⋊Dic59—42no—
195#11745670—44no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—42no—
203C4⋊Dic59—81no—
204C4⋊Dic59—81no—
205C4⋊Dic59—81no—
206C4⋊Dic59—81no—
207Dic59⋊C4—42no—
208#11745664—44no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8—21no—
220C2×C59⋊C8#4112yes—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59—21no—
227C23×C59—22no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59—41no—
231C23×C59—41no—
232C23×C59—41no—
233C23×C59—41no—
234C23×C59#12112yes—
235C23×C59C22×D411yescharacteristic
236D4×C59—22no—
237D4×C59—22no—
238D4×C59—41no—
239D4×C59—41no—
240D4×C59—84no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—22no—
248C2×C4×C59—22no—
249C2×C4×C59—22no—
250C2×C4×C59—22no—
251C2×C4×C59—22no—
252C2×C4×C59—21no—
253C2×C4×C59—22no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—21no—
257C2×C4×C59—21no—
258C2×C4×C59—21no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—84no—
267C2×C4×C59#12112yes—
268C2×C4×C59#13712yes—
269C2×C4×C59#13712yes—
270C2×C4×C59C22×D411yescharacteristic
271C2×C4×C59C2×#4111yescharacteristic
272C2×C4×C59C2×#4111yescharacteristic
273C2×C4×C59C2×#4111yescharacteristic
274C2×C4×C59H(5,𝔽2)11yescharacteristic
275C2×C4×C592−1+411yescharacteristic
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C2×Dic59—41no—
285C2×Dic59—41no—
286C2×Dic59—41no—
287C2×Dic59—41no—
288C2×Dic59—84no—
289Dic118—41no—
290Dic118—41no—
291Dic118—41no—
292Dic118—41no—
293C59⋊C8—22no—
294C59⋊C8—41no—
295C59⋊C8—41no—
296C59⋊C8—41no—
297#25223—591nomaximal, Sylow
298C22×C59—22no—
299C22×C59—22no—
300C22×C59—22no—
301C22×C59—41no—
302C22×C59—41no—
303C22×C59—41no—
304C22×C59—41no—
305C22×C59—42no—
306C22×C59—42no—
307C22×C59—42no—
308C22×C59—81no—
309C22×C59—81no—
310C22×C59—81no—
311C22×C59—81no—
312C22×C59C2×#12111yescharacteristic
313C22×C59#54611yescharacteristic
314C22×C59#54711yescharacteristic
315C22×C59C2×#13711yescharacteristic
316C22×C59C2×#13711yescharacteristic
317C22×C59#57611yescharacteristic
318C22×C59#57611yescharacteristic
319C4×C59—22no—
320C4×C59—22no—
321C4×C59—22no—
322C4×C59—41no—
323C4×C59—41no—
324C4×C59—42no—
325C4×C59—42no—
326C4×C59—81no—
327C4×C59#52911yescharacteristic
328C4×C59#53911yescharacteristic
329C4×C59#54611yescharacteristic
330C4×C59#54711yescharacteristic
331Dic59—81no—
332Dic59—81no—
333Dic59—81no—
334Dic59—81no—
335C2×#480—591no—
336C2×#468—591no—
337#2502—591no—
338#2363D5911yescharacteristic
339#2083—591no—
340#2071—591no—
341#2068—591no—
342#2055—591no—
343#1947—591no—
344#1928—591no—
345#1877—591no—
346#1873—591no—
347#1851—591no—
348#1802—591no—
349#1736—591no—
350C2×C59—42no—
351C2×C59—82no—
352C2×C59#242511yescharacteristic
353C2×C59#303911yescharacteristic
354C2×C59#304111yescharacteristic
355C2×C59#331411yescharacteristic
356C2×C59#333311yescharacteristic
357C2×C59#334611yescharacteristic
358C2×C59#335711yescharacteristic
359C22×SD16—591no—
360C22×M16—591no—
361C2×C22⋊Q8—591no—
362C2×#121C2×D5911yescharacteristic
363C2×C4×D4C2×D5911yescharacteristic
364C2×#117C2×D5911yescharacteristic
365C22×C4⋊C4—591no—
366#480—1184no—
367#468—1184no—
368C2×#107—591no—
369C2×#107—591no—
370C2×#107—591no—
371C2×C4⋊C8—591no—
372C2×C4⋊C8—591no—
373C2×#103—591no—
374C2×#103—591no—
375C2×#103—591no—
376C2×#102—591no—
377C2×#102—591no—
378C2×#102—591no—
379C2×C22⋊C8—591no—
380#398—1181no—
381#398—1181no—
382#391—591no—
383#391—591no—
384#390—21no—
385#390C2×D5911yescharacteristic
386#389C2×D5911yescharacteristic
387#388—21no—
388#386—21no—
389#385—1181no—
390#385—1181no—
391#384—591no—
392#384—591no—
393C4×C4⋊C4C2×D5911yescharacteristic
394C4×C22⋊C4—21no—
395C4×C22⋊C4C2×D5911yescharacteristic
396#340—591no—
397#340—591no—
398#340—591no—
399#340—591no—
400#340—591no—
401#340—591no—
402#336—591no—
403#336—591no—
404C59#2522311yescharacteristic, complemented, Sylow
405C22×Q8—591no—
406C22×D4—21no—
407C22×D4C22×D5911yescharacteristic
408C23×C4—21no—
409C23×C4—1181no—
410C23×C4C22×D5911yescharacteristic
411C2×SD16—1182no—
412C2×SD16—1182no—
413C2×SD16—2361no—
414C2×M16—1182no—
415C2×M16—1182no—
416C2×M16—2361no—
417C22×C8—591no—
418C22×C8—591no—
419C22×C8—591no—
420C22×C8—591no—
421C22⋊Q8—1182no—
422C22⋊Q8—1182no—
423#121—22no—
424#121D23614yes—
425C4×D4—22no—
426C4×D4—41no—
427#117—22no—
428#117—41no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—591no—
431C2×C4⋊C4—591no—
432C2×C4⋊C4—591no—
433C2×C4⋊C4—1182no—
434C2×C4⋊C4—1182no—
435C2×C4⋊C4—1181no—
436C2×C4⋊C4—1181no—
437C2×C4⋊C4—1181no—
438C2×C4⋊C4—1181no—
439C2×C4⋊C4—1181no—
440C2×C4⋊C4—1181no—
441C2×C4⋊C4C22×D5911yescharacteristic
442C2×C4⋊C4C22×D5911yescharacteristic
443C2×C4⋊C4C22×D5911yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1181no—
450C2×C22⋊C4C22⋊D5911yescharacteristic
451C2×C22⋊C4C22⋊D5911yescharacteristic
452C2×C42—21no—
453C2×C42—21no—
454C2×C42C22×D5911yescharacteristic
455C2×C42C22×D5911yescharacteristic
456#107—1182no—
457#107—1182no—
458#107—1182no—
459C4⋊C8—1182no—
460C4⋊C8—2361no—
461#103—1182no—
462#103—1182no—
463#103—2361no—
464#102—1182no—
465#102—1182no—
466#102—1182no—
467C22⋊C8—1182no—
468#95—21no—
469#95—21no—
470#95—1181no—
471#95—1181no—
472#95—1181no—
473#95—1181no—
474#95C22⋊D5911yescharacteristic
475#95C22⋊D5911yescharacteristic
476C24—21no—
477C24C2×C22⋊D5911yescharacteristic
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×Q8—1181no—
481C2×Q8—1181no—
482C2×Q8—1181no—
483C2×Q8—1181no—
484C2×D4—24no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—81no—
491C2×D4—81no—
492C2×D4C2×D23612yes—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—81no—
503C22×C4—1181no—
504C22×C4—1181no—
505C22×C4—1181no—
506C22×C4—1181no—
507C22×C4—4721no—
508C22×C4—4721no—
509C22×C4—2362no—
510C22×C4C2×D23612yes—
511C22×C4#1174568512yes—
512C22×C4C2×C22⋊D5911yescharacteristic
513C22×C4C2×C22⋊D5911yescharacteristic
514C22×C4C2×C22⋊D5911yescharacteristic
515C22×C4C2×C22⋊D5911yescharacteristic
516C22×C4C2×C22⋊D5911yescharacteristic
517C22×C4C23×D5911yescharacteristic
518SD16—2364no—
519M16—2364no—
520C2×C8—592no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C2×C8—1181no—
527C2×C8—1181no—
528C2×C8—1181no—
529C2×C8—1181no—
530C2×C8—1181no—
531C2×C8—1181no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—81no—
537C4⋊C4—1181no—
538C4⋊C4—1181no—
539C4⋊C4—1181no—
540C4⋊C4—1181no—
541C4⋊C4—1181no—
542C4⋊C4—1181no—
543C4⋊C4—4721no—
544C4⋊C4—4721no—
545C4⋊C4—4721no—
546C4⋊C4—4721no—
547C4⋊C4—2362no—
548C4⋊C4—2362no—
549C4⋊C4C2×D23612yes—
550C4⋊C4#1174568712yes—
551C22⋊C4—24no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—81no—
558C22⋊C4—2362no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—42no—
564C42—81no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—22no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23#1427370112yes—
579C23C22×C22⋊D5911yescharacteristic
580Q8—2361no—
581Q8—2361no—
582Q8—2361no—
583Q8—2361no—
584D4—22no—
585D4—22no—
586D4—41no—
587D4—41no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—84no—
615C2×C4—4724no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4—2361no—
621C2×C4—2361no—
622C2×C4—2361no—
623C2×C4—2361no—
624C2×C4—2361no—
625C2×C4—2361no—
626C2×C4—2361no—
627C2×C4—2361no—
628C2×C4#1427370112yes—
629C2×C4#1427375712yes—
630C2×C4#1427375712yes—
631C2×C4C22×D23611yescharacteristic
632C2×C4#41×D5911yescharacteristic
633C2×C4C2×#1174568711yescharacteristic
634C2×C4C2×#1174568511yescharacteristic
635C2×C4#1427377211yescharacteristic
636C2×C4#1427377411yescharacteristic
637C8—1182no—
638C8—2361no—
639C8—2361no—
640C8—2361no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22C2×#1427370111yescharacteristic
656C22#1683804611yescharacteristic
657C22#1683805111yescharacteristic
658C22C2×#1427375711yescharacteristic
659C22C2×#1427375711yescharacteristic
660C22#1683820511yescharacteristic
661C22#1683820511yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—42no—
666C4—41no—
667C4—41no—
668C4—42no—
669C4—81no—
670C4—4721no—
671C4—4721no—
672C4—4721no—
673C4—4721no—
674C4#1683793211yescharacteristic
675C4#1683799411yescharacteristic
676C4#1683804711yescharacteristic
677C4#1683805611yescharacteristic
678C2—42no—
679C2—82no—
680C2#2142764111yescharacteristic
681C2#2143358311yescharacteristic
682C2#2143359911yescharacteristic
683C2#2143617711yescharacteristic
684C2#2143643411yescharacteristic
685C2#2143658111yescharacteristic
686C2#2143671611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2363C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2363D59
hypocenterC59#25223
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
3. #21428574C2semidirect
4. #21425567C2semidirect
5. #21425499C2semidirect
6. #21425488C2semidirect
7. #21425413C2semidirect
8. #21424854C2semidirect
9. #21424737C2semidirect
10. #21424430C2semidirect
11. #21424408C2semidirect
12. #21424281C2semidirect
13. #21424096C2semidirect
14. #21423817C2semidirect
404. C59#25223semidirect

Maximal subgroups

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

subgroupindex
C2×#168377132
C2×#168376462
#214285742
#214255672
#214254992
#214254882
#214254132
#214248542
#214247372
#214244302
#214244082
#214242812
#214240962
#214238172
C59×#23632
#2522359