Finite Group Databasebeta

← #28620958

Subgroups — #28620958

Order 15104.

1278 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837557C211yesmaximal, characteristic, complemented
2C22×#14273650C211yesmaximal, characteristic, complemented
3#21427461C211yesmaximal, characteristic
4#21425234C212yesmaximal, complemented
5#21425104C212yesmaximal, complemented
6#21424520C211yesmaximal, characteristic, complemented
7#21424437C212yesmaximal, complemented
8#21424403C212yesmaximal, complemented
9#21424389C211yesmaximal, characteristic
10#21424186C211yesmaximal, characteristic, complemented
11C2×C59×#554C211yesmaximal, characteristic
12C23×Dic118C2211yescharacteristic
13C22×#11745672C2212yes—
14C23×C59⋊C8C2211yescharacteristic
15C2×#14273706C2212yes—
16#16837557—24no—
17C2×#14273651C2212yes—
18C2×#14273650—24no—
19C2×#14273650C2212yes—
20C2×#14273650C2212yes—
21C2×#14273650C2212yes—
22C2×C22⋊(C59⋊C8)C2211yescharacteristic
23C2×C22⋊(C59⋊C8)C2212yes—
24#16837322—21no—
25#16837314—21no—
26#16837308C2212yes—
27#16837282—21no—
28#16837274—21no—
29#16837262C2211yescharacteristic
30#16837164C2212yes—
31#16837164C2212yes—
32#16837164C2212yes—
33#16837157C2212yes—
34C59×#554—24no—
35C59×#554—24no—
36C2×C4⋊Q8×C59C2211yescharacteristic
37C2×#125×C59—21no—
38C2×C22⋊Q8×C59—21no—
39C2×C22⋊Q8×C59—21no—
40C2×C22⋊Q8×C59C2211yescharacteristic
41C2×C22⋊Q8×C59C2211yescharacteristic
42C2×C4×Q8×C59C2211yescharacteristic
43C2×C4×D4×C59—21no—
44C2×C4×D4×C59C2211yescharacteristic
45C22×C4⋊C4×C59C2211yescharacteristic
46C22×C4⋊C4×C59C2211yescharacteristic
47C22×Dic118—22no—
48C22×Dic118—24no—
49C22×Dic118—22no—
50C22×Dic118—21no—
51C22×Dic118—42no—
52C22×Dic118C2312yes—
53C23×Dic59—21no—
54C2×#11745672—24no—
55C2×#11745672—24no—
56C2×#11745672—42no—
57C22×C59⋊C8—22no—
58C22×C59⋊C8—42no—
59C22×C59⋊C8C2312yes—
60C22×C59⋊C8C2312yes—
61#14273706—24no—
62#14273706—42no—
63C2×C4⋊Dic59C2312yes—
64C2×Dic59⋊C4—22no—
65C2×C22⋊Dic59—22no—
66C2×C22⋊Dic59—21no—
67#14273651—42no—
68#14273650—24no—
69#14273650—24no—
70#14273650—24no—
71C22⋊(C59⋊C8)—22no—
72C22⋊(C59⋊C8)—42no—
73C59⋊#95—21no—
74C59⋊#95—22no—
75C59⋊#95—22no—
76C22×Q8×C59—21no—
77C22×Q8×C59C2311yescharacteristic
78C22×D4×C59D411yescharacteristic
79C23×C4×C59—21no—
80C23×C4×C59—21no—
81C23×C4×C59C2311yescharacteristic
82C23×C4×C59C2311yescharacteristic
83C4⋊Q8×C59—22no—
84C4⋊Q8×C59—22no—
85#125×C59—28no—
86C22⋊Q8×C59—28no—
87C22⋊Q8×C59—28no—
88C22⋊Q8×C59—22no—
89C22⋊Q8×C59—22no—
90C22⋊Q8×C59—22no—
91C22⋊Q8×C59D414yes—
92C4×Q8×C59—22no—
93C4×Q8×C59—22no—
94C4×D4×C59—28no—
95C4×D4×C59—22no—
96C4×D4×C59—22no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—22no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—22no—
105C2×C4⋊C4×C59—22no—
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59D414yes—
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117C24×C59C2×D411yescharacteristic
118C24×C59C2×D411yescharacteristic
119C2×Q8×C59—24no—
120C2×Q8×C59—21no—
121C2×Q8×C59—22no—
122C2×Q8×C59—22no—
123C2×Q8×C59—82no—
124C2×Q8×C59C2×D412yes—
125C2×D4×C59—22no—
126C2×D4×C59—22no—
127C2×D4×C59Q1614yescomplemented
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—24no—
135C22×C4×C59—24no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—82no—
141C22×C4×C59—82no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D412yes—
146C22×C4×C59C2×D412yes—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—22no—
150C22×Dic59—22no—
151C22×Dic59—44no—
152C22×Dic59—42no—
153C2×Dic118—28no—
154C2×Dic118—22no—
155C2×Dic118—22no—
156C2×Dic118—22no—
157C2×Dic118—22no—
158C2×Dic118—22no—
159C2×Dic118—48no—
160C2×Dic118—44no—
161C2×Dic118—44no—
162C2×Dic118—42no—
163C2×Dic118#4114yes—
164C4⋊C4×C59—24no—
165C4⋊C4×C59—24no—
166C4⋊C4×C59—24no—
167C4⋊C4×C59—24no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—24no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59C2×D412yes—
175C4⋊C4×C59C2×D412yes—
176C4⋊C4×C59C2×D412yes—
177C22⋊C4×C59—24no—
178C22⋊C4×C59—24no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—22no—
181C42×C59—24no—
182C42×C59—21no—
183C42×C59—21no—
184C22⋊Dic59—44no—
185C22⋊Dic59—42no—
186#11745672—48no—
187C4⋊Dic59—22no—
188C4⋊Dic59—22no—
189Dic59⋊C4—44no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C2×C59⋊C8—22no—
194C2×C59⋊C8—22no—
195C2×C59⋊C8—22no—
196C2×C59⋊C8—44no—
197C23×C59—22no—
198C23×C59—22no—
199C23×C59—21no—
200C23×C59—22no—
201C23×C59—21no—
202C23×C59—22no—
203C23×C59#12012yes—
204C23×C59C2×Q1612yes—
205C23×C59C2×Q1612yes—
206C23×C59#13712yes—
207C23×C59C22×D411yescharacteristic
208Q8×C59—24no—
209Q8×C59—41no—
210Q8×C59—41no—
211Q8×C59—84no—
212D4×C59—28no—
213C2×C4×C59—21no—
214C2×C4×C59—22no—
215C2×C4×C59—21no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—24no—
219C2×C4×C59—22no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—42no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—42no—
234C2×C4×C59—84no—
235C2×C4×C59—84no—
236C2×C4×C59#12012yes—
237C2×C4×C59C2×Q1612yes—
238C2×C4×C59C22×D411yescharacteristic
239C2×C4×C59C22×D411yescharacteristic
240C2×C4×C59C2×#4112yes—
241C2×C4×C59H(5,𝔽2)12yes—
242C2×Dic59—24no—
243C2×Dic59—48no—
244C2×Dic59—42no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248C2×Dic59—42no—
249Dic118—24no—
250Dic118—24no—
251Dic118—48no—
252Dic118—42no—
253Dic118—42no—
254C59⋊C8—42no—
255C59⋊C8—42no—
256#23970—591nomaximal, Sylow
257C22×C59—22no—
258C22×C59—22no—
259C22×C59—24no—
260C22×C59—24no—
261C22×C59—22no—
262C22×C59—24no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59—42no—
266C22×C59#45114yes—
267C22×C59C2×#12011yescharacteristic
268C22×C59#54612yes—
269C22×C59C22×Q1611yescharacteristic
270C22×C59C2×#13711yescharacteristic
271C22×C59#57712yes—
272C4×C59—22no—
273C4×C59—22no—
274C4×C59—41no—
275C4×C59—42no—
276C4×C59—41no—
277C4×C59—42no—
278C4×C59—42no—
279C4×C59—42no—
280C4×C59#52511yescharacteristic
281C4×C59#53511yescharacteristic
282C4×C59#54612yes—
283Dic59—44no—
284Dic59—82no—
285C2×#554D5911yescharacteristic
286C2×#451—591no—
287C22×#103—591no—
288#2409—591no—
289#2024—592no—
290#2001—592no—
291#1892—591no—
292#1878—592no—
293#1872—592no—
294#1869—591no—
295#1824—591no—
296C2×C59—24no—
297C2×C59—42no—
298C2×C59#240411yescharacteristic
299C2×C59C2×#45111yescharacteristic
300C2×C59#300911yescharacteristic
301C2×C59#329912yes—
302C2×C59#331612yes—
303C23×Q8—591no—
304C22×Q16—592no—
305C23×C8—591no—
306#554—24no—
307#554—24no—
308C2×C4⋊Q8C2×D5911yescharacteristic
309C2×#125—21no—
310C2×C22⋊Q8—21no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8—592no—
313C2×C22⋊Q8C2×D5911yescharacteristic
314C2×C22⋊Q8C2×D5911yescharacteristic
315C2×C4×Q8C2×D5911yescharacteristic
316C2×C4×D4—21no—
317C2×C4×D4C2×D5911yescharacteristic
318C22×C4⋊C4C2×D5911yescharacteristic
319C22×C4⋊C4C2×D5911yescharacteristic
320#451—1184no—
321C2×#103—592no—
322C2×#103—592no—
323C2×#103—592no—
324C2×#103—592no—
325C2×#103—1184no—
326C2×C22⋊C8—591no—
327C2×C22⋊C8—592no—
328#395—1181no—
329#393—1181no—
330#391—592no—
331#386—1181no—
332#385—1181no—
333#380—591no—
334#340—592no—
335#340—592no—
336#340—592no—
337#336—592no—
338C59#2397011yescharacteristic, complemented, Sylow
339C22×Q8—21no—
340C22×Q8—592no—
341C22×Q8—1184no—
342C22×Q8—1181no—
343C22×Q8—1182no—
344C22×Q8—1182no—
345C22×Q8—2362no—
346C22×Q8C22×D5911yescharacteristic
347C22×D4C22⋊D5911yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—1181no—
351C23×C4C22×D5911yescharacteristic
352C23×C4C22×D5911yescharacteristic
353C2×Q16—1184no—
354C2×Q16—1184no—
355C2×Q16—2362no—
356C22×C8—592no—
357C22×C8—592no—
358C22×C8—1182no—
359C22×C8—2362no—
360C4⋊Q8—22no—
361C4⋊Q8—22no—
362#125—28no—
363C22⋊Q8—28no—
364C22⋊Q8—28no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—1184no—
369C22⋊Q8—2362no—
370C22⋊Q8C22⋊D5914yes—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×D4—28no—
374C4×D4—22no—
375C4×D4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—592no—
386C2×C4⋊C4—1182no—
387C2×C4⋊C4C22⋊D5911yescharacteristic
388C2×C4⋊C4C22⋊D5914yes—
389C2×C4⋊C4C22×D5911yescharacteristic
390C2×C4⋊C4C22×D5911yescharacteristic
391C2×C4⋊C4C22×D5911yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1182no—
395C2×C22⋊C4—1181no—
396C2×C22⋊C4C22⋊D5911yescharacteristic
397C2×C22⋊C4C22⋊D5911yescharacteristic
398C2×C42—21no—
399C2×C42C22×D5911yescharacteristic
400#103—1184no—
401#103—1184no—
402#103—1184no—
403#103—2362no—
404C22⋊C8—1182no—
405C22⋊C8—2362no—
406#95—1181no—
407#95—1182no—
408#95—1182no—
409C24C2×C22⋊D5911yescharacteristic
410C24C2×C22⋊D5911yescharacteristic
411C2×Q8—24no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—82no—
416C2×Q8—594no—
417C2×Q8—1188no—
418C2×Q8—1182no—
419C2×Q8—1182no—
420C2×Q8—1182no—
421C2×Q8—1182no—
422C2×Q8—1182no—
423C2×Q8—2364no—
424C2×Q8—2368no—
425C2×Q8—2364no—
426C2×Q8—2362no—
427C2×Q8C2×C22⋊D5912yes—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4#1174567214yescomplemented
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—1182no—
446C22×C4—1182no—
447C22×C4—2364no—
448C22×C4—2362no—
449C22×C4C2×C22⋊D5911yescharacteristic
450C22×C4C2×C22⋊D5911yescharacteristic
451C22×C4C2×C22⋊D5911yescharacteristic
452C22×C4C2×C22⋊D5912yes—
453C22×C4C2×C22⋊D5912yes—
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C23×D5911yescharacteristic
456Q16—2368no—
457C2×C8—1182no—
458C2×C8—1182no—
459C2×C8—1182no—
460C2×C8—1182no—
461C2×C8—1182no—
462C2×C8—1182no—
463C2×C8—2364no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—1182no—
475C4⋊C4—1182no—
476C4⋊C4—2364no—
477C4⋊C4C2×C22⋊D5912yes—
478C4⋊C4C2×C22⋊D5912yes—
479C4⋊C4C2×C22⋊D5912yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—2364no—
485C22⋊C4—2362no—
486C42—24no—
487C42—21no—
488C42—21no—
489C23—22no—
490C23—22no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—22no—
495C23#1427369612yes—
496C23C2×#1174567212yes—
497C23C2×#1174567212yes—
498C23#1427375812yes—
499C23C22×C22⋊D5911yescharacteristic
500Q8—24no—
501Q8—41no—
502Q8—41no—
503Q8—84no—
504Q8—1184no—
505Q8—1184no—
506Q8—2368no—
507Q8—2362no—
508Q8—2362no—
509D4—28no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—1184no—
534C2×C4—2362no—
535C2×C4—2362no—
536C2×C4—2362no—
537C2×C4—2368no—
538C2×C4—2362no—
539C2×C4—2362no—
540C2×C4#1427369612yes—
541C2×C4C2×#1174567212yes—
542C2×C4C22×C22⋊D5911yescharacteristic
543C2×C4C22×C22⋊D5911yescharacteristic
544C2×C4#41×D5912yes—
545C2×C4#1427377312yes—
546C8—2362no—
547C8—2362no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—24no—
553C22—22no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22#1683755814yes—
558C22C2×#1427369611yescharacteristic
559C22#1683804012yes—
560C22C22×#1174567211yescharacteristic
561C22C2×#1427375811yescharacteristic
562C22#1683821312yes—
563C4—22no—
564C4—22no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—4722no—
572C4—2364no—
573C4#1683790911yescharacteristic
574C4#1683796611yescharacteristic
575C4#1683804012yes—
576C2—24no—
577C2—42no—
578C2#2142741111yescharacteristic
579C2C2×#1683755811yescharacteristic
580C2#2143337311yescharacteristic
581C2#2143596612yes—
582C2#2143620112yes—
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#554C2
socleC23×C59C22×D4
prime radicalC22×D4×C59D4
hypercenterC2×#554D59
hypocenterC59#23970
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16837557C2semidirect
2. C22×#14273650C2semidirect
4. #21425234C2semidirect
5. #21425104C2semidirect
6. #21424520C2semidirect
7. #21424437C2semidirect
8. #21424403C2semidirect
10. #21424186C2semidirect
127. C2×D4×C59Q16semidirect
338. C59#23970semidirect
430. C2×D4#11745672semidirect

Maximal subgroups

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

subgroupindex
C2×#168375572
C22×#142736502
#214274612
#214252342
#214251042
#214245202
#214244372
#214244032
#214243892
#214241862
C2×C59×#5542
#2397059