Finite Group Databasebeta

← #28646288

Subgroups — #28646288

Order 15104.

802 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426963C211yesmaximal, characteristic
2#21425653C211yesmaximal, characteristic, complemented
3#21425587C211yesmaximal, characteristic, complemented
4#21425587C211yesmaximal, characteristic, complemented
5#21425421C211yesmaximal, characteristic, complemented
6#21425049C211yesmaximal, characteristic, complemented
7#21424852C211yesmaximal, characteristic, complemented
8#21424664C211yesmaximal, characteristic, complemented
9#21424664C211yesmaximal, characteristic, complemented
10#21424284C211yesmaximal, characteristic
11#21424276C211yesmaximal, characteristic
12#21424180C211yesmaximal, characteristic
13#21424123C211yesmaximal, characteristic
14C2×#16837164C211yesmaximal, characteristic
15C59×#2440C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C2×#14273683—21no—
18C22×C4⋊Dic59C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C22⋊(C59⋊C8)C2211yescharacteristic
26#16837300C2211yescharacteristic
27#16837300C2211yescharacteristic
28#16837287—21no—
29#16837287—21no—
30#16837263—21no—
31#16837262C2211yescharacteristic
32C4×C4⋊Dic59C2211yescharacteristic
33C4×C4⋊Dic59C2211yescharacteristic
34#16837164C2211yescharacteristic
35#16837164C2211yescharacteristic
36#16837164C2211yescharacteristic
37#16837164C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4⋊Q8×C59C2211yescharacteristic
49C2×C22⋊Q8×C59—21no—
50C22×C4⋊C4×C59C2211yescharacteristic
51C59×#398—21no—
52C59×#395—21no—
53C59×#389C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#385—21no—
56C59×#384C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8—22no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273683—44no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59Q814yescomplemented
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—22no—
81#14273660—22no—
82#14273660—41no—
83#14273660—41no—
84C4⋊(C59⋊C8)—22no—
85C4⋊(C59⋊C8)—41no—
86C22⋊(C59⋊C8)—22no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C22×Q8×C59—21no—
93C23×C4×C59—21no—
94C23×C4×C59C2311yescharacteristic
95C4⋊Q8×C59—22no—
96C4⋊Q8×C59—41no—
97C22⋊Q8×C59—44no—
98C2×C4⋊C4×C59—22no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—41no—
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59D411yescharacteristic
112C2×C22⋊C4×C59D411yescharacteristic
113C2×C42×C59C2311yescharacteristic
114C2×C42×C59C2311yescharacteristic
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—21no—
118#95×C59D411yescharacteristic
119#95×C59D411yescharacteristic
120C24×C59C2×D411yescharacteristic
121C2×Q8×C59—42no—
122C2×Q8×C59—42no—
123C2×Q8×C59—81no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—42no—
133C22×C4×C59—42no—
134C22×C4×C59—81no—
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×Q812yes—
141C22×C4×C59#4112yes—
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—24no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—42no—
149C22×Dic59—42no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—42no—
158C4⋊C4×C59—42no—
159C4⋊C4×C59—42no—
160C4⋊C4×C59—42no—
161C4⋊C4×C59—42no—
162C4⋊C4×C59—81no—
163C4⋊C4×C59#4112yes—
164C22⋊C4×C59—22no—
165C22⋊C4×C59—42no—
166C22⋊C4×C59D814yes—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59—21no—
171C22⋊Dic59—42no—
172C4⋊Dic59—24no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59C2×Q812yes—
180C4⋊Dic59C2×Q812yes—
181Dic59⋊C4—42no—
182Dic59⋊C4—42no—
183C4×Dic59—42no—
184C4×Dic59—42no—
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⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—44no—
197C2×C59⋊C8#4112yes—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59C22⋊Q812yes—
203C23×C59C2×D812yes—
204C23×C59#13712yes—
205C23×C59C22×D411yescharacteristic
206Q8×C59—84no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—21no—
218C2×C4×C59—22no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
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×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—84no—
236C2×C4×C59C22⋊Q812yes—
237C2×C4×C59C2×D812yes—
238C2×C4×C59C2×D812yes—
239C2×C4×C59C22×Q811yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59H(5,𝔽2)11yescharacteristic
244C2×C4×C592−1+411yescharacteristic
245C2×Dic59—22no—
246C2×Dic59—22no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—48no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C59⋊C8—22no—
259C59⋊C8—41no—
260C59⋊C8—41no—
261C59⋊C8—41no—
262#25988—591nomaximal, Sylow
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—22no—
269C22×C59#47414yes—
270C22×C59C2×C22⋊Q811yescharacteristic
271C22×C59#54611yescharacteristic
272C22×C59#54711yescharacteristic
273C22×C59C22×D811yescharacteristic
274C22×C59C2×#13711yescharacteristic
275C22×C59#57711yescharacteristic
276C22×C59#57811yescharacteristic
277C4×C59—22no—
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—41no—
281C4×C59—42no—
282C4×C59—41no—
283C4×C59—81no—
284C4×C59—81no—
285C4×C59—81no—
286C4×C59#53311yescharacteristic
287C4×C59#54411yescharacteristic
288C4×C59#55111yescharacteristic
289C4×C59#55211yescharacteristic
290Dic59—42no—
291Dic59—42no—
292Dic59—81no—
293Dic59—81no—
294#2440D5911yescharacteristic
295#2344—591no—
296#2099—591no—
297#2086—591no—
298#2086—591no—
299#2057—591no—
300#1988—591no—
301#1947—591no—
302#1918—591no—
303#1918—591no—
304#1852—591no—
305#1850—591no—
306#1823—591no—
307#1808—591no—
308C2×#340—591no—
309C2×C59—24no—
310C2×C59#243811yescharacteristic
311C2×C59C2×#47411yescharacteristic
312C2×C59#308211yescharacteristic
313C2×C59#329411yescharacteristic
314C2×C59#331211yescharacteristic
315C2×C59#333411yescharacteristic
316C2×C59#334411yescharacteristic
317C23×C8—591no—
318C2×C4⋊Q8C2×D5911yescharacteristic
319C2×C22⋊Q8—21no—
320C2×#117—1181no—
321C22×C4⋊C4—591no—
322C22×C4⋊C4C2×D5911yescharacteristic
323C2×#107—591no—
324C2×#107—591no—
325C2×#107—591no—
326C2×#107—591no—
327C2×C4⋊C8—591no—
328C2×C4⋊C8—591no—
329C2×C22⋊C8—591no—
330#398—21no—
331#395—21no—
332#389—591no—
333#389—591no—
334#389C2×D5911yescharacteristic
335#389C2×D5911yescharacteristic
336#387—1181no—
337#387—1181no—
338#385—21no—
339#384C2×D5911yescharacteristic
340#381—1181no—
341#380—591no—
342#380C2×D5911yescharacteristic
343C4×C4⋊C4—591no—
344C4×C4⋊C4—591no—
345C4×C22⋊C4C2×D5911yescharacteristic
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#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#336—591no—
359#336—591no—
360C59#2598811yescharacteristic, complemented, Sylow
361C22×Q8—21no—
362C23×C4—21no—
363C23×C4—1181no—
364C23×C4C22×D5911yescharacteristic
365C22×C8—591no—
366C22×C8—591no—
367C22×C8—591no—
368C22×C8—591no—
369C22×C8—1182no—
370C22×C8—1182no—
371C22×C8—2361no—
372C4⋊Q8—22no—
373C4⋊Q8—41no—
374C22⋊Q8—44no—
375#117—2364no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—41no—
384C2×C4⋊C4—594no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1182no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4C22⋊D5911yescharacteristic
400C2×C22⋊C4C22⋊D5911yescharacteristic
401C2×C42—1181no—
402C2×C42—1181no—
403C2×C42C22×D5911yescharacteristic
404C2×C42C22×D5911yescharacteristic
405#107—1182no—
406#107—1182no—
407#107—2361no—
408#107—2361no—
409C4⋊C8—1182no—
410C4⋊C8—2361no—
411C22⋊C8—1182no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95C22⋊D5911yescharacteristic
421#95C22⋊D5911yescharacteristic
422C24C2×C22⋊D5911yescharacteristic
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—81no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4—1184no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4—2362no—
443C22×C4—2362no—
444C22×C4Q8×D5912yes—
445C22×C4#1174568912yes—
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—1181no—
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—
464C2×C8—2364no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4—592no—
479C4⋊C4—592no—
480C4⋊C4—1184no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—2362no—
488C4⋊C4—2362no—
489C4⋊C4#1174568712yes—
490C22⋊C4—22no—
491C22⋊C4—42no—
492C22⋊C4—2362no—
493C22⋊C4#1174566914yes—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—2362no—
499C42—2362no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23#1427370312yes—
505C23C2×#1174566912yes—
506C23#1427375812yes—
507C23C22×C22⋊D5911yescharacteristic
508Q8—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—1182no—
539C2×C4—1182no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2368no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4#1427370312yes—
552C2×C4C2×#1174566912yes—
553C2×C4C2×#1174566912yes—
554C2×C4C2×Q8×D5911yescharacteristic
555C2×C4#41×D5911yescharacteristic
556C2×C4C2×#1174568711yescharacteristic
557C2×C4C2×#1174568911yescharacteristic
558C2×C4#1427377211yescharacteristic
559C2×C4#1427377411yescharacteristic
560C8—1182no—
561C8—2361no—
562C8—2361no—
563C8—2361no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22#1683767914yes—
571C22C2×#1427370311yescharacteristic
572C22#1683804611yescharacteristic
573C22#1683805111yescharacteristic
574C22C22×#1174566911yescharacteristic
575C22C2×#1427375811yescharacteristic
576C22#1683821311yescharacteristic
577C22#1683821711yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—4721no—
588C4—4721no—
589C4—2362no—
590C4—2362no—
591C4#1683795411yescharacteristic
592C4#1683803111yescharacteristic
593C4#1683807911yescharacteristic
594C4#1683809411yescharacteristic
595C2—24no—
596C2#2142776811yescharacteristic
597C2C2×#1683767911yescharacteristic
598C2#2143393211yescharacteristic
599C2#2143590911yescharacteristic
600C2#2143614711yescharacteristic
601C2#2143643911yescharacteristic
602C2#2143654511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2440C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2440D59
hypocenterC59#25988
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
2. #21425653C2semidirect
3. #21425587C2semidirect
4. #21425587C2semidirect
5. #21425421C2semidirect
6. #21425049C2semidirect
7. #21424852C2semidirect
8. #21424664C2semidirect
9. #21424664C2semidirect
70. C2×C4⋊Dic59Q8semidirect
360. C59#25988semidirect

Maximal subgroups

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

subgroupindex
#214269632
#214256532
#214255872
#214254212
#214250492
#214248522
#214246642
#214242842
#214242762
#214241802
#214241232
C2×#168371642
C59×#24402
#2598859