Finite Group Databasebeta

← #28646856

Subgroups — #28646856

Order 15104.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427067C211yesmaximal, characteristic
2#21425481C211yesmaximal, characteristic
3#21425467C211yesmaximal, characteristic
4#21425239C211yesmaximal, characteristic
5#21425229C211yesmaximal, characteristic
6#21425054C211yesmaximal, characteristic, complemented
7#21425049C211yesmaximal, characteristic, complemented
8#21424864C211yesmaximal, characteristic, complemented
9#21424815C211yesmaximal, characteristic, complemented
10#21424685C211yesmaximal, characteristic, complemented
11#21424678C211yesmaximal, characteristic, complemented
12#21424226C211yesmaximal, characteristic, complemented
13#21424103C211yesmaximal, characteristic
14C4×#14273647C211yesmaximal, characteristic
15C59×#2550C211yesmaximal, characteristic, complemented
16C2×C4×D236C2211yescharacteristic
17C2×#14273660C2211yescharacteristic
18C2×#14273658C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273647C2211yescharacteristic
23C2×#14273647C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×C4×C59⋊C8C2211yescharacteristic
27#16837304C2211yescharacteristic
28#16837304C2211yescharacteristic
29#16837292—21no—
30#16837292—21no—
31#16837268C2211yescharacteristic
32#16837264—21no—
33C4×C4⋊Dic59C2211yescharacteristic
34C4×C4⋊Dic59C2211yescharacteristic
35#16837249—21no—
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×#125×C59C2211yescharacteristic
50C59×#398—21no—
51C59×#395—21no—
52C59×#391—21no—
53C59×#389—21no—
54C59×#389C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#382C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C4×D59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C4×D236—22no—
66C4×D236Q814yescomplemented
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×(C2×D59)⋊C4—21no—
71C2×(C2×D59)⋊C4—21no—
72C2×(C2×D59)⋊C4—21no—
73C2×(C2×D59)⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77#14273660—22no—
78#14273658—22no—
79C4⋊(C59⋊C8)—22no—
80C4⋊(C59⋊C8)—41no—
81C4⋊(C59⋊C8)—41no—
82#14273647—22no—
83#14273647—22no—
84#14273647—41no—
85#14273647—41no—
86C4×C59⋊C8—22no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C22×Q8×C59—21no—
92C4⋊Q8×C59—22no—
93C4⋊Q8×C59—41no—
94#125×C59—22no—
95#125×C59—41no—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59—21no—
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×C59C2311yescharacteristic
109C2×C42×C59C2311yescharacteristic
110C2×C42×C59C2311yescharacteristic
111#95×C59—21no—
112#95×C59—21no—
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59D411yescharacteristic
117#95×C59D411yescharacteristic
118C23×D59—21no—
119C2×Q8×C59—42no—
120C2×Q8×C59—42no—
121C2×Q8×C59—81no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2411yescharacteristic
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236C2×Q812yes—
142C2×C4×D59—24no—
143C2×C4×D59—81no—
144C2×C4×D59—81no—
145C4⋊C4×C59—22no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—41no—
153C4⋊C4×C59—42no—
154C4⋊C4×C59—42no—
155C4⋊C4×C59—42no—
156C4⋊C4×C59—42no—
157C4⋊C4×C59—81no—
158C4⋊C4×C59#4112yes—
159C4⋊C4×C59#4112yes—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—81no—
165C42×C59C2×Q812yes—
166C42×C59#4112yes—
167(C2×D59)⋊C4—24no—
168(C2×D59)⋊C4—42no—
169(C2×D59)⋊C4—42no—
170(C2×D59)⋊C4—81no—
171(C2×D59)⋊C4—81no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59C2×Q812yes—
178C4×Dic59—81no—
179C4×Dic59—81no—
180C2×C59⋊C8—21no—
181C2×C59⋊C8—21no—
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⋊C8—21no—
192C23×C59C22×D411yescharacteristic
193C22×D59—22no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C22×D59—41no—
200Q8×C59—84no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
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—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—21no—
217C2×C4×C59—21no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
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×C59C22⋊Q812yes—
229C2×C4×C59C22⋊Q812yes—
230C2×C4×C59#13512yes—
231C2×C4×C59C22×Q811yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59H(5,𝔽2)11yescharacteristic
236C2×C4×C592−1+411yescharacteristic
237C2×Dic59—22no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246D236—22no—
247D236—22no—
248D236—41no—
249D236—41no—
250C4×D59—84no—
251C59⋊C8—41no—
252C59⋊C8—41no—
253C59⋊C8—41no—
254C59⋊C8—41no—
255#26028—591nomaximal, Sylow
256C22×C59C2×C22⋊Q811yescharacteristic
257C22×C59#54611yescharacteristic
258C22×C59#54811yescharacteristic
259C22×C59C2×#13511yescharacteristic
260C22×C59#57511yescharacteristic
261C22×C59#57611yescharacteristic
262C22×C59#57711yescharacteristic
263C2×D59—42no—
264C2×D59—42no—
265C2×D59—42no—
266C2×D59—81no—
267C2×D59—81no—
268C2×D59—81no—
269C2×D59—81no—
270C4×C59—22no—
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—42no—
274C4×C59—41no—
275C4×C59—42no—
276C4×C59—41no—
277C4×C59—41no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59#53311yescharacteristic
281C4×C59#54411yescharacteristic
282C4×C59#55511yescharacteristic
283C4×C59#55611yescharacteristic
284Dic59—42no—
285Dic59—81no—
286Dic59—81no—
287#2550D5911yescharacteristic, complemented
288#2357—591no—
289#2067—591no—
290#2065—591no—
291#2025—591no—
292#2023—591no—
293#1989—591no—
294#1988—591no—
295#1949—591no—
296#1939—591no—
297#1921—591no—
298#1920—591no—
299#1835—591no—
300#1804—591no—
301C4×#102—591no—
302C2×C59#243911yescharacteristic
303C2×C59#307611yescharacteristic
304C2×C59#308111yescharacteristic
305C2×C59#329711yescharacteristic
306C2×C59#332611yescharacteristic
307C2×C59#336511yescharacteristic
308C2×C59#337011yescharacteristic
309D59—82no—
310C2×C4⋊Q8C2×D5911yescharacteristic
311C2×#125C2×D5911yescharacteristic
312C2×C4×D4—591no—
313C2×#107—591no—
314C2×#106—591no—
315C2×C4⋊C8—591no—
316C2×C4⋊C8—591no—
317C2×C4⋊C8—591no—
318C2×#102—591no—
319C2×#102—591no—
320C2×#102—591no—
321C2×#102—591no—
322C2×C4×C8—591no—
323#398—21no—
324#395—21no—
325#391—21no—
326#390—591no—
327#390—591no—
328#389—21no—
329#389C2×D5911yescharacteristic
330#389C2×D5911yescharacteristic
331#388—1181no—
332#388—1181no—
333#382—591no—
334#382C2×D5911yescharacteristic
335#381—1181no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4C2×D5911yescharacteristic
339C4×C4⋊C4C2×D5911yescharacteristic
340C4×C22⋊C4—1181no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#336—591no—
352#336—591no—
353C59#2602811yescharacteristic, complemented, Sylow
354C22×Q8—21no—
355C22×D4—591no—
356C23×C4—1181no—
357C22×C8—591no—
358C22×C8—591no—
359C22×C8—591no—
360C22×C8—591no—
361C4⋊Q8—22no—
362C4⋊Q8—41no—
363#125—22no—
364#125—41no—
365C4×D4—594no—
366C4×D4—1182no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
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—1181no—
388C2×C42—1181no—
389C2×C42C22×D5911yescharacteristic
390C2×C42C22×D5911yescharacteristic
391C2×C42C22×D5911yescharacteristic
392#107—1182no—
393#106—1182no—
394C4⋊C8—1182no—
395C4⋊C8—2361no—
396C4⋊C8—2361no—
397#102—1182no—
398#102—1182no—
399#102—2361no—
400#102—2361no—
401C4×C8—1182no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95C22⋊D5911yescharacteristic
412#95C22⋊D5911yescharacteristic
413C24—1181no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—81no—
417C2×D4—592no—
418C2×D4—1181no—
419C2×D4—1181no—
420C2×D4—1181no—
421C2×D4—1181no—
422C2×D4—1181no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—1181no—
428C22×C4—1184no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—4721no—
432C22×C4—4721no—
433C22×C4C2×C22⋊D5911yescharacteristic
434C22×C4C2×C22⋊D5911yescharacteristic
435C22×C4C2×C22⋊D5911yescharacteristic
436C22×C4C2×C22⋊D5911yescharacteristic
437C22×C4C2×C22⋊D5911yescharacteristic
438C22×C4C2×C22⋊D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—1181no—
441C2×C8—1181no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4—592no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4#1174568712yes—
472C4⋊C4#1174568912yes—
473C22⋊C4—1184no—
474C22⋊C4—4721no—
475C22⋊C4—4721no—
476C22⋊C4—2362no—
477C22⋊C4—2362no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—81no—
483C42—4721no—
484C42—4721no—
485C42Q8×D5912yes—
486C42#1174568912yes—
487C23—1182no—
488C23—2361no—
489C23—2361no—
490C23—2361no—
491C23—2361no—
492C23—2361no—
493C23—2361no—
494C23C22×C22⋊D5911yescharacteristic
495Q8—84no—
496D4—1182no—
497D4—1182no—
498D4—2361no—
499D4—2361no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
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—22no—
516C2×C4—22no—
517C2×C4—41no—
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—1182no—
528C2×C4—4724no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4#1427370312yes—
538C2×C4#1427370312yes—
539C2×C4#1427374712yes—
540C2×C4C2×Q8×D5911yescharacteristic
541C2×C4C2×#1174568711yescharacteristic
542C2×C4C2×#1174568911yescharacteristic
543C2×C4C2×#1174568911yescharacteristic
544C2×C4#1427377211yescharacteristic
545C2×C4#1427377511yescharacteristic
546C8—2361no—
547C8—2361no—
548C8—2361no—
549C8—2361no—
550C22—4721no—
551C22—4721no—
552C22—4721no—
553C22—4721no—
554C22—2362no—
555C22—2362no—
556C22—2362no—
557C22C2×#1427370311yescharacteristic
558C22#1683804611yescharacteristic
559C22#1683806311yescharacteristic
560C22C2×#1427374711yescharacteristic
561C22#1683820011yescharacteristic
562C22#1683820511yescharacteristic
563C22#1683821011yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—42no—
569C4—42no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—4721no—
575C4—4721no—
576C4—2362no—
577C4#1683795711yescharacteristic
578C4#1683802911yescharacteristic
579C4#1683811811yescharacteristic
580C4#1683812611yescharacteristic
581C2—4722no—
582C2#2142778311yescharacteristic
583C2#2143387811yescharacteristic
584C2#2143392111yescharacteristic
585C2#2143594611yescharacteristic
586C2#2143634911yescharacteristic
587C2#2143679211yescharacteristic
588C2#2143684911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2550C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2550D59
hypocenterC59#26028
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2550

Complements

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

NH ≅ G/Ntype
6. #21425054C2semidirect
7. #21425049C2semidirect
8. #21424864C2semidirect
9. #21424815C2semidirect
10. #21424685C2semidirect
11. #21424678C2semidirect
12. #21424226C2semidirect
15. C59×#2550C2semidirect
66. C4×D236Q8semidirect
287. #2550D59semidirect
353. C59#26028semidirect

Maximal subgroups

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

subgroupindex
#214270672
#214254812
#214254672
#214252392
#214252292
#214250542
#214250492
#214248642
#214248152
#214246852
#214246782
#214242262
#214241032
C4×#142736472
C59×#25502
#2602859