Finite Group Databasebeta

← #28646884

Subgroups — #28646884

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#21429117C211yesmaximal, characteristic, complemented
2#21425484C211yesmaximal, characteristic
3#21425470C211yesmaximal, characteristic
4#21425242C211yesmaximal, characteristic
5#21425232C211yesmaximal, characteristic
6#21425053C211yesmaximal, characteristic, complemented
7#21425044C211yesmaximal, characteristic, complemented
8#21424862C211yesmaximal, characteristic, complemented
9#21424815C211yesmaximal, characteristic, complemented
10#21424674C211yesmaximal, characteristic, complemented
11#21424667C211yesmaximal, characteristic, complemented
12#21424226C211yesmaximal, characteristic, complemented
13#21424106C211yesmaximal, characteristic
14C4×#14273648C211yesmaximal, characteristic
15C59×#2357C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×C4×C59⋊C8C2211yescharacteristic
28#16837335—21no—
29#16837322—21no—
30#16837306—21no—
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837299—21no—
34#16837268C2211yescharacteristic
35C4×C4⋊Dic59C2211yescharacteristic
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×D4×C59C2211yescharacteristic
49C59×#390C2211yescharacteristic
50C59×#390C2211yescharacteristic
51C59×#388—21no—
52C59×#388—21no—
53C59×#382C2211yescharacteristic
54C59×#381—21no—
55C4×C4⋊C4×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C22×Dic118—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273727—22no—
65#14273727—41no—
66#14273719—22no—
67#14273719—41no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C4×Dic59—21no—
78#14273660—22no—
79#14273658—22no—
80C4⋊(C59⋊C8)—22no—
81C4⋊(C59⋊C8)—41no—
82C4⋊(C59⋊C8)—41no—
83#14273648—22no—
84#14273648—22no—
85#14273648—41no—
86#14273648—41no—
87C4×C59⋊C8—22no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C22×D4×C59C2311yescharacteristic
94C23×C4×C59—21no—
95C4×D4×C59—22no—
96C4×D4×C59Q814yescomplemented
97C2×C4⋊C4×C59D411yescharacteristic
98C2×C4⋊C4×C59D411yescharacteristic
99C2×C4⋊C4×C59C2311yescharacteristic
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C22⋊C4×C59—21no—
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—21no—
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C42×C59—21no—
108C2×C42×C59—21no—
109C2×C42×C59C2311yescharacteristic
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112#95×C59—21no—
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59D411yescharacteristic
117#95×C59D411yescharacteristic
118C24×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59C2×Q812yes—
125C22×C4×C59—21no—
126C22×C4×C59—24no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—81no—
130C22×C4×C59—81no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C2×Dic118—42no—
143C2×Dic118—42no—
144C2×Dic118—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—41no—
152C4⋊C4×C59C2×Q812yes—
153C22⋊C4×C59—24no—
154C22⋊C4×C59—42no—
155C22⋊C4×C59—42no—
156C22⋊C4×C59—81no—
157C22⋊C4×C59—81no—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—81no—
163C42×C59—81no—
164C42×C59C2×Q812yes—
165C42×C59#4112yes—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—42no—
173C4⋊Dic59#4112yes—
174C4⋊Dic59#4112yes—
175Dic59⋊C4—42no—
176Dic59⋊C4—42no—
177Dic59⋊C4—42no—
178Dic59⋊C4—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×C59—22no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59C22×D411yescharacteristic
200D4×C59—22no—
201D4×C59—22no—
202D4×C59—41no—
203D4×C59—41no—
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—21no—
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—21no—
219C2×C4×C59—21no—
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×C59—84no—
229C2×C4×C59C22⋊Q812yes—
230C2×C4×C59C22⋊Q812yes—
231C2×C4×C59#13512yes—
232C2×C4×C59C22×Q811yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59H(5,𝔽2)11yescharacteristic
237C2×C4×C592−1+411yescharacteristic
238C2×Dic59—22no—
239C2×Dic59—22no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250Dic118—84no—
251C59⋊C8—41no—
252C59⋊C8—41no—
253C59⋊C8—41no—
254C59⋊C8—41no—
255#26030—591nomaximal, Sylow
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59C2×C22⋊Q811yescharacteristic
264C22×C59#54611yescharacteristic
265C22×C59#54811yescharacteristic
266C22×C59C2×#13511yescharacteristic
267C22×C59#57511yescharacteristic
268C22×C59#57611yescharacteristic
269C22×C59#57711yescharacteristic
270C4×C59—22no—
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—42no—
274C4×C59—41no—
275C4×C59—41no—
276C4×C59—41no—
277C4×C59—81no—
278C4×C59—81no—
279C4×C59#53311yescharacteristic
280C4×C59#54411yescharacteristic
281C4×C59#55511yescharacteristic
282C4×C59#55611yescharacteristic
283Dic59—42no—
284Dic59—42no—
285Dic59—81no—
286Dic59—81no—
287#2550—591no—
288#2357D5911yescharacteristic
289#2067—591no—
290#2065—591no—
291#2025—591no—
292#2023—591no—
293#1989—591no—
294#1987—591no—
295#1949—591no—
296#1939—591no—
297#1919—591no—
298#1918—591no—
299#1835—591no—
300#1804—591no—
301C4×#102—591no—
302C2×C59—82no—
303C2×C59#243911yescharacteristic
304C2×C59#307611yescharacteristic
305C2×C59#308111yescharacteristic
306C2×C59#330111yescharacteristic
307C2×C59#332211yescharacteristic
308C2×C59#336111yescharacteristic
309C2×C59#337111yescharacteristic
310C2×C4⋊Q8—591no—
311C2×#125—591no—
312C2×C4×D4C2×D5911yescharacteristic
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—1181no—
324#395—1181no—
325#391—1181no—
326#390C2×D5911yescharacteristic
327#390C2×D5911yescharacteristic
328#389—591no—
329#389—591no—
330#389—1181no—
331#388—21no—
332#388—21no—
333#382—591no—
334#382C2×D5911yescharacteristic
335#381—21no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4C2×D5911yescharacteristic
338C4×C4⋊C4C2×D5911yescharacteristic
339C4×C4⋊C4C2×D5911yescharacteristic
340C4×C22⋊C4—21no—
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#2603011yescharacteristic, complemented, Sylow
354C22×Q8—1181no—
355C22×D4C22×D5911yescharacteristic
356C23×C4—21no—
357C22×C8—591no—
358C22×C8—591no—
359C22×C8—591no—
360C22×C8—591no—
361C4⋊Q8—1182no—
362C4⋊Q8—2361no—
363#125—1182no—
364#125—2361no—
365C4×D4—22no—
366C4×D4Dic11814yescomplemented
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4C22⋊D5911yescharacteristic
377C2×C4⋊C4C22⋊D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42—21no—
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—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95C22⋊D5911yescharacteristic
412#95C22⋊D5911yescharacteristic
413C24—21no—
414C2×Q8—4721no—
415C2×Q8—2362no—
416C2×Q8—2362no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Dic11812yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4—1181no—
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—41no—
459C4⋊C4—592no—
460C4⋊C4—592no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—4721no—
468C4⋊C4—2362no—
469C4⋊C4—2362no—
470C4⋊C4—2362no—
471C4⋊C4—2362no—
472C4⋊C4C2×Dic11812yes—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—81no—
483C42—81no—
484C42—4721no—
485C42C2×Dic11812yes—
486C42#1174568512yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C22×C22⋊D5911yescharacteristic
495Q8—4724no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
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—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—84no—
525C2×C4—1182no—
526C2×C4—1182no—
527C2×C4—2361no—
528C2×C4—2361no—
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#1427370612yes—
538C2×C4#1427370612yes—
539C2×C4#1427374712yes—
540C2×C4C22×Dic11811yescharacteristic
541C2×C4#41×D5911yescharacteristic
542C2×C4#41×D5911yescharacteristic
543C2×C4C2×#1174568511yescharacteristic
544C2×C4#1427377311yescharacteristic
545C2×C4#1427377411yescharacteristic
546C8—2361no—
547C8—2361no—
548C8—2361no—
549C8—2361no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1427370611yescharacteristic
558C22#1683804011yescharacteristic
559C22#1683805911yescharacteristic
560C22C2×#1427374711yescharacteristic
561C22#1683820011yescharacteristic
562C22#1683820511yescharacteristic
563C22#1683821311yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—4721no—
574C4—4721no—
575C4—2362no—
576C4—2362no—
577C4#1683795511yescharacteristic
578C4#1683803011yescharacteristic
579C4#1683811511yescharacteristic
580C4#1683812011yescharacteristic
581C2—82no—
582C2#2142778111yescharacteristic
583C2#2143388411yescharacteristic
584C2#2143392711yescharacteristic
585C2#2143599211yescharacteristic
586C2#2143629111yescharacteristic
587C2#2143675211yescharacteristic
588C2#2143685811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2357C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2357D59
hypocenterC59#26030
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #21429117C2semidirect
6. #21425053C2semidirect
7. #21425044C2semidirect
8. #21424862C2semidirect
9. #21424815C2semidirect
10. #21424674C2semidirect
11. #21424667C2semidirect
12. #21424226C2semidirect
96. C4×D4×C59Q8semidirect
353. C59#26030semidirect
366. C4×D4Dic118semidirect

Maximal subgroups

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

subgroupindex
#214291172
#214254842
#214254702
#214252422
#214252322
#214250532
#214250442
#214248622
#214248152
#214246742
#214246672
#214242262
#214241062
C4×#142736482
C59×#23572
#2603059