Finite Group Databasebeta

← #28661446

Subgroups — #28661446

Order 15104.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428492C211yesmaximal, characteristic
2#21427105C211yesmaximal, characteristic
3#21425517C211yesmaximal, characteristic, complemented
4#21425475C211yesmaximal, characteristic, complemented
5#21425296C211yesmaximal, characteristic
6#21424865C211yesmaximal, characteristic, complemented
7#21424825C211yesmaximal, characteristic, complemented
8#21424705C211yesmaximal, characteristic
9#21424691C211yesmaximal, characteristic, complemented
10#21424680C211yesmaximal, characteristic, complemented
11#21424366C211yesmaximal, characteristic, complemented
12#21424291C211yesmaximal, characteristic
13#21424180C211yesmaximal, characteristic
14#103×Dic59C211yesmaximal, characteristic, complemented
15C59×#2035C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273659C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26C59⋊#401—21no—
27#16837330—21no—
28#16837309C2211yescharacteristic
29#16837301—21no—
30#16837301C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837298C2211yescharacteristic
33#16837295—21no—
34#16837288—21no—
35#16837287—21no—
36#16837277—21no—
37#16837267C2211yescharacteristic
38#16837262C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C22⋊C4×Dic59—21no—
42C4×C22⋊Dic59C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×C22⋊Q8×C59C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C2×#103×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62Q8×Dic59—22no—
63Q8×Dic59—41no—
64C2×C4⋊Dic59—22no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—41no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59D411yescharacteristic
77C2×C22⋊Dic59D411yescharacteristic
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59C2311yescharacteristic
82C2×C4×Dic59C2311yescharacteristic
83#14273660—22no—
84#14273659—22no—
85#14273659—41no—
86Dic59⋊C8—22no—
87#14273651—22no—
88#14273651—41no—
89C22⋊(C59⋊C8)—41no—
90C8×Dic59—41no—
91C59⋊#95—21no—
92C59⋊#95—21no—
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⋊#95D411yescharacteristic
102C59⋊#95D411yescharacteristic
103C22×Q8×C59C2311yescharacteristic
104C23×C4×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22⋊Q8×C59—22no—
108C22⋊Q8×C59—22no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115#106×C59—22no—
116#103×C59—22no—
117#103×C59—22no—
118C22⋊C8×C59—22no—
119#95×C59—21no—
120C24×C59C2×D411yescharacteristic
121C2×Q8×C59—21no—
122C2×Q8×C59—21no—
123C2×Q8×C59—21no—
124C2×Q8×C59—21no—
125C2×Q8×C59—21no—
126C2×Q8×C59#4112yes—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—42no—
144C22×Dic59—42no—
145C22×Dic59—81no—
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59#4112yes—
156C4⋊C4×C59—24no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59#4112yes—
163C22⋊C4×C59—24no—
164C22⋊C4×C59—42no—
165C22⋊Dic59—22no—
166C22⋊Dic59—22no—
167C22⋊Dic59—81no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—42no—
176C4⋊Dic59—42no—
177C4⋊Dic59—42no—
178C4⋊Dic59—81no—
179C4⋊Dic59#4112yes—
180Dic59⋊C4—81no—
181C4×Dic59—21no—
182C4×Dic59—21no—
183C4×Dic59—21no—
184C4×Dic59—21no—
185C4×Dic59—42no—
186C4×Dic59—81no—
187C4×Dic59—81no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59#13512yes—
201C23×C59C22×D411yescharacteristic
202Q8×C59—22no—
203Q8×C59—22no—
204Q8×C59—41no—
205Q8×C59—41no—
206C2×C4×C59—22no—
207C2×C4×C59—22no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
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—41no—
224C2×C4×C59—41no—
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59H(5,𝔽2)11yescharacteristic
230C2×C4×C592−1+411yescharacteristic
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—22no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—84no—
255C2×Dic59#13712yes—
256C8×C59—22no—
257C8×C59—41no—
258C59⋊C8—41no—
259C59⋊C8—41no—
260#27037—591nomaximal, Sylow
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—41no—
265C22×C59—41no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59#52511yescharacteristic
269C22×C59#54611yescharacteristic
270C22×C59#54711yescharacteristic
271C22×C59C2×#13511yescharacteristic
272C22×C59C2×#13711yescharacteristic
273C22×C59#57711yescharacteristic
274C22×C59#57711yescharacteristic
275C4×C59—42no—
276C4×C59—41no—
277C4×C59—42no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59#52911yescharacteristic
281C4×C59#53911yescharacteristic
282C4×C59#55311yescharacteristic
283C4×C59#55611yescharacteristic
284Dic59—22no—
285Dic59—41no—
286Dic59—41no—
287Dic59—41no—
288Dic59—42no—
289Dic59—81no—
290Dic59—81no—
291Dic59—81no—
292#2494—591no—
293#2360—591no—
294#2074—591no—
295#2066—591no—
296#2036—591no—
297#2035D5911yescharacteristic
298#1949—591no—
299#1941—591no—
300#1923—591no—
301#1921—591no—
302#1920—591no—
303#1865—591no—
304#1853—591no—
305#1823—591no—
306C4×#103—591no—
307C2×C59—42no—
308C2×C59#246611yescharacteristic
309C2×C59#310811yescharacteristic
310C2×C59#312511yescharacteristic
311C2×C59#329611yescharacteristic
312C2×C59#332011yescharacteristic
313C2×C59#333711yescharacteristic
314C2×C59#335611yescharacteristic
315C2×C22⋊Q8C2×D5911yescharacteristic
316C2×C4×Q8—591no—
317C22×C4⋊C4—591no—
318C2×#107—591no—
319C2×#106—591no—
320C2×#106—591no—
321C2×#106C2×D5911yescharacteristic
322C2×C4⋊C8—591no—
323C2×#103—591no—
324C2×#103—591no—
325C2×#103C2×D5911yescharacteristic
326C2×#103C2×D5911yescharacteristic
327C2×C22⋊C8—591no—
328C2×C22⋊C8C2×D5911yescharacteristic
329C2×C4×C8—591no—
330#401—1181no—
331#397—1181no—
332#391—591no—
333#389—591no—
334#389—591no—
335#389—591no—
336#389—1181no—
337#388—1181no—
338#387—1181no—
339#387—1181no—
340#385—1181no—
341#382—591no—
342#380—591no—
343#380C2×D5911yescharacteristic
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C22⋊C4—591no—
347C4×C22⋊C4—1181no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340C2×D5911yescharacteristic
356#336—591no—
357#336—591no—
358C59#2703711yescharacteristic, complemented, Sylow
359C22×Q8C22×D5911yescharacteristic
360C23×C4—1181no—
361C23×C4C22×D5911yescharacteristic
362C22×C8—591no—
363C22×C8—591no—
364C22×C8C22×D5911yescharacteristic
365C22×C8C22×D5911yescharacteristic
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C4×Q8—1182no—
369C4×Q8—2361no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—1182no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—2361no—
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—591no—
388C2×C22⋊C4—591no—
389C2×C22⋊C4—1181no—
390C2×C42—591no—
391C2×C42—591no—
392C2×C42—1181no—
393C2×C42—1181no—
394C2×C42—1181no—
395#107—1182no—
396#106—22no—
397#106—1182no—
398#106—2361no—
399C4⋊C8—1182no—
400#103—22no—
401#103—22no—
402#103—1182no—
403#103—2361no—
404C22⋊C8—22no—
405C22⋊C8—2361no—
406C4×C8—2361no—
407#95—21no—
408#95—591no—
409#95—591no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420C24D4×D5911yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1174568712yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—591no—
438C22×C4—591no—
439C22×C4—591no—
440C22×C4—591no—
441C22×C4—1181no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—4721no—
446C22×C4—2362no—
447C22×C4—2362no—
448C22×C4D4×D5911yescharacteristic
449C22×C4C23×D5911yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8#1174568712yes—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—592no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—4721no—
477C4⋊C4—4721no—
478C4⋊C4—2362no—
479C4⋊C4—2362no—
480C4⋊C4—2362no—
481C4⋊C4#1174568712yes—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—1182no—
485C22⋊C4—1182no—
486C22⋊C4—4721no—
487C42—1181no—
488C42—1181no—
489C42—1181no—
490C42—1181no—
491C42—4721no—
492C42—4721no—
493C42—2362no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1427374812yes—
501C23C2×D4×D5911yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
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—41no—
525C2×C4—592no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1182no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—4724no—
539C2×C4—2361no—
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—2361no—
549C2×C4—2361no—
550C2×C4#41×D5911yescharacteristic
551C2×C4C2×#1174568711yescharacteristic
552C2×C4C2×#1174568711yescharacteristic
553C2×C4C2×#1174568711yescharacteristic
554C2×C4#1427377311yescharacteristic
555C2×C4#1427377411yescharacteristic
556C8—22no—
557C8—41no—
558C8—2361no—
559C8—2361no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1683790811yescharacteristic
568C22#1683804411yescharacteristic
569C22#1683805211yescharacteristic
570C22C2×#1427374811yescharacteristic
571C22#137×D5911yescharacteristic
572C22#1683820911yescharacteristic
573C22#1683821111yescharacteristic
574C4—41no—
575C4—42no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—1182no—
580C4—4721no—
581C4—4721no—
582C4—4721no—
583C4—2361no—
584C4—2362no—
585C4—2361no—
586C4—2361no—
587C4#1683793311yescharacteristic
588C4#1683799611yescharacteristic
589C4#1683810411yescharacteristic
590C4#1683812111yescharacteristic
591C2—42no—
592C2#2142812711yescharacteristic
593C2#2143417111yescharacteristic
594C2#2143436611yescharacteristic
595C2#2143592911yescharacteristic
596C2#2143627011yescharacteristic
597C2#2143648111yescharacteristic
598C2#2143670611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2035C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2035D59
hypocenterC59#27037
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #21425517C2semidirect
4. #21425475C2semidirect
6. #21424865C2semidirect
7. #21424825C2semidirect
9. #21424691C2semidirect
10. #21424680C2semidirect
11. #21424366C2semidirect
14. #103×Dic59C2semidirect
358. C59#27037semidirect

Maximal subgroups

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

subgroupindex
#214284922
#214271052
#214255172
#214254752
#214252962
#214248652
#214248252
#214247052
#214246912
#214246802
#214243662
#214242912
#214241802
#103×Dic592
C59×#20352
#2703759