Finite Group Databasebeta

← #28618264

Subgroups — #28618264

Order 15104.

874 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837507C211yesmaximal, characteristic, complemented
2C2×#16837459C211yesmaximal, characteristic
3C4×#14273706C211yesmaximal, characteristic
4#21425598C211yesmaximal, characteristic, complemented
5#21425532C211yesmaximal, characteristic
6#21425488C211yesmaximal, characteristic, complemented
7#21425477C211yesmaximal, characteristic, complemented
8#21425418C211yesmaximal, characteristic, complemented
9#21425296C211yesmaximal, characteristic
10#21425292C211yesmaximal, characteristic
11#21425104C211yesmaximal, characteristic
12#21423994C211yesmaximal, characteristic, complemented
13#21423929C211yesmaximal, characteristic, complemented
14#21423926C211yesmaximal, characteristic, complemented
15C4×C22⋊Q8×C59C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19#16837507—24no—
20#16837459—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×C22⋊(C59⋊C8)C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34C2×#14273638C2211yescharacteristic
35C2×#14273638C2211yescharacteristic
36#16837308C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837286—21no—
39#16837275—21no—
40#16837262C2211yescharacteristic
41C4×Dic59⋊C4—21no—
42C4×C4⋊Dic59C2211yescharacteristic
43C4×C22⋊Dic59—21no—
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50C2×C22⋊Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C22×C42×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#387—21no—
56C59×#385—21no—
57C59×#380C2211yescharacteristic
58C4×C4⋊C4×C59—21no—
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C22⋊C4×C59—21no—
61C22×Dic118C2311yescharacteristic
62C2×#11745672—22no—
63C2×#11745672—22no—
64C2×#11745672—22no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273706—22no—
70#14273706—22no—
71C4×Dic118—22no—
72C4×Dic118—22no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83#14273660—22no—
84#14273658—22no—
85#14273658—22no—
86C4⋊(C59⋊C8)—22no—
87C4⋊(C59⋊C8)—22no—
88#14273651—22no—
89#14273651—22no—
90#14273651—22no—
91#14273650—22no—
92#14273650—22no—
93#14273650—22no—
94C22⋊(C59⋊C8)—22no—
95C22⋊(C59⋊C8)—22no—
96#14273638—22no—
97#14273638D414yes—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59C2311yescharacteristic
104C23×C4×C59D411yescharacteristic
105C23×C4×C59D411yescharacteristic
106C23×C4×C59C2311yescharacteristic
107C22⋊Q8×C59—22no—
108C22⋊Q8×C59—22no—
109C4×Q8×C59—22no—
110C4×Q8×C59—22no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C42×C59—22no—
120C2×C42×C59—22no—
121C2×C42×C59—21no—
122C2×C42×C59—21no—
123C2×C42×C59D411yescharacteristic
124C2×C42×C59D411yescharacteristic
125C2×C42×C59C2311yescharacteristic
126C2×C42×C59C2311yescharacteristic
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132C24×C59C2×D411yescharacteristic
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59#4112yes—
139C22×C4×C59—22no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59—41no—
154C22×C4×C59—41no—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2411yescharacteristic
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118#4112yes—
171C4⋊C4×C59—24no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59#4112yes—
180C22⋊C4×C59—24no—
181C22⋊C4×C59—42no—
182C42×C59—22no—
183C42×C59—22no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—42no—
187C42×C59—42no—
188C42×C59C2×D412yes—
189C42×C59#4112yes—
190C22⋊Dic59—24no—
191C22⋊Dic59—42no—
192#11745672—28no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59#4112yes—
199Dic59⋊C4—24no—
200Dic59⋊C4—42no—
201Dic59⋊C4—42no—
202C4×Dic59—42no—
203C4×Dic59—42no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8C2×D412yes—
215C2×C59⋊C8C2×D412yes—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59#13712yes—
223C23×C59C22×D411yescharacteristic
224Q8×C59—22no—
225Q8×C59—22no—
226Q8×C59—41no—
227Q8×C59—41no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—44no—
243C2×C4×C59—44no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59#12112yes—
259C2×C4×C59#12112yes—
260C2×C4×C59#13712yes—
261C2×C4×C59C22×D411yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×Dic59—22no—
268C2×Dic59—22no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279Dic118—22no—
280Dic118—22no—
281Dic118—41no—
282Dic118—41no—
283C59⋊C8—22no—
284C59⋊C8—22no—
285C59⋊C8—41no—
286C59⋊C8—41no—
287#23709—591nomaximal, Sylow
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59C2×#12111yescharacteristic
296C22×C59#52511yescharacteristic
297C22×C59#52511yescharacteristic
298C22×C59C2×#13711yescharacteristic
299C22×C59C2×#13711yescharacteristic
300C22×C59#57511yescharacteristic
301C22×C59#57511yescharacteristic
302C4×C59—22no—
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—42no—
306C4×C59—41no—
307C4×C59—42no—
308C4×C59—41no—
309C4×C59—81no—
310C4×C59—81no—
311C4×C59#52511yescharacteristic
312C4×C59#52911yescharacteristic
313C4×C59#52911yescharacteristic
314C4×C59#53211yescharacteristic
315Dic59—42no—
316Dic59—42no—
317Dic59—81no—
318Dic59—81no—
319C2×#441—591no—
320C2×#432—591no—
321C4×C22⋊Q8—591no—
322C4×C22⋊Q8D5911yescharacteristic
323#2088—591no—
324#2077—591no—
325#2068—591no—
326#2066—591no—
327#2056—591no—
328#2036—591no—
329#2035—591no—
330#2001—591no—
331#1776—591no—
332#1763—591no—
333#1763—591no—
334C2×C59—42no—
335C2×C59#239211yescharacteristic
336C2×C59#305611yescharacteristic
337C2×C59#305611yescharacteristic
338C2×C59#311111yescharacteristic
339C2×C59#311311yescharacteristic
340C2×C59#311611yescharacteristic
341C2×C59#311711yescharacteristic
342C22×Q16—591no—
343C2×C22⋊Q8—591no—
344C2×C22⋊Q8C2×D5911yescharacteristic
345C2×C4×Q8—591no—
346C2×C4×Q8C2×D5911yescharacteristic
347C22×C42C2×D5911yescharacteristic
348#441—1184no—
349#432—1184no—
350C2×#107—591no—
351C2×#106—591no—
352C2×#106—591no—
353C2×C4⋊C8—591no—
354C2×C4⋊C8—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#103—591no—
358C2×#103—591no—
359C2×#103—591no—
360C2×#103—591no—
361C2×C22⋊C8—591no—
362C2×C22⋊C8—591no—
363C2×#97—591no—
364C2×#97—591no—
365#391—591no—
366#391C2×D5911yescharacteristic
367#389—591no—
368#389C2×D5911yescharacteristic
369#387—21no—
370#387—1181no—
371#385—21no—
372#385—1181no—
373#380—591no—
374#380C2×D5911yescharacteristic
375C4×C4⋊C4—21no—
376C4×C4⋊C4—591no—
377C4×C4⋊C4—1181no—
378C4×C4⋊C4C2×D5911yescharacteristic
379C4×C22⋊C4—21no—
380C4×C22⋊C4—1181no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387C59#2370911yescharacteristic, complemented, Sylow
388C22×Q8—591no—
389C22×Q8C22×D5911yescharacteristic
390C23×C4C22⋊D5911yescharacteristic
391C23×C4C22⋊D5911yescharacteristic
392C23×C4C22×D5911yescharacteristic
393C2×Q16—1182no—
394C2×Q16—1182no—
395C2×Q16—1182no—
396C22×C8—591no—
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—1182no—
403C22⋊Q8—1182no—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—1182no—
407C4×Q8—1182no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1181no—
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1181no—
423C2×C22⋊C4—1181no—
424C2×C42—22no—
425C2×C42—22no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—1181no—
429C2×C42—1181no—
430C2×C42C22⋊D5911yescharacteristic
431C2×C42C22⋊D5911yescharacteristic
432C2×C42C22×D5911yescharacteristic
433C2×C42C22×D5911yescharacteristic
434#107—1182no—
435#106—1182no—
436#106—1182no—
437C4⋊C8—1182no—
438C4⋊C8—1182no—
439#103—1182no—
440#103—1182no—
441#103—1182no—
442#103—1182no—
443#103—1182no—
444#103—1182no—
445C22⋊C8—1182no—
446C22⋊C8—1182no—
447#97—594no—
448#97—1182no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1181no—
455#95—1181no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459C24C2×C22⋊D5911yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—592no—
466C2×Q8—1181no—
467C2×Q8—1181no—
468C2×Q8—1181no—
469C2×Q8—1181no—
470C2×Q8—1181no—
471C2×Q8#1174568512yes—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—1181no—
489C22×C4—1181no—
490C22×C4—1181no—
491C22×C4—1181no—
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C2×C22⋊D5911yescharacteristic
497C22×C4C23×D5911yescharacteristic
498Q16—1188no—
499C2×C8—592no—
500C2×C8—592no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—592no—
520C4⋊C4—1184no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—2362no—
527C4⋊C4—2362no—
528C4⋊C4#1174568512yes—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—1184no—
532C22⋊C4—2362no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—42no—
538C42—42no—
539C42—2362no—
540C42—2362no—
541C42D4×D5912yes—
542C42#1174568712yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23#1427375712yes—
550C23C22×C22⋊D5911yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—41no—
554Q8—41no—
555Q8—1182no—
556Q8—1182no—
557Q8—2361no—
558Q8—2361no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—1182no—
590C2×C4—1182no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4#1427369912yes—
602C2×C4#1427369912yes—
603C2×C4#1427375812yes—
604C2×C4C2×D4×D5911yescharacteristic
605C2×C4#41×D5911yescharacteristic
606C2×C4#41×D5911yescharacteristic
607C2×C4C2×#1174568711yescharacteristic
608C2×C4C2×#1174568511yescharacteristic
609C2×C4C2×#1174568511yescharacteristic
610C8—1182no—
611C8—1182no—
612C8—2361no—
613C8—2361no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22C2×#1427369911yescharacteristic
622C22#1683790911yescharacteristic
623C22#1683791311yescharacteristic
624C22C2×#1427375711yescharacteristic
625C22C2×#1427375811yescharacteristic
626C22#1683820011yescharacteristic
627C22#1683820011yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—4721no—
638C4—4721no—
639C4—2362no—
640C4—2362no—
641C4#1683791111yescharacteristic
642C4#1683793211yescharacteristic
643C4#1683793411yescharacteristic
644C4#1683794911yescharacteristic
645C2—42no—
646C2#2142733811yescharacteristic
647C2#2143372211yescharacteristic
648C2#2143372811yescharacteristic
649C2#2143420511yescharacteristic
650C2#2143422611yescharacteristic
651C2#2143425211yescharacteristic
652C2#2143426311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC4×C22⋊Q8×C59C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenterC4×C22⋊Q8D59
hypocenterC59#23709
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16837507C2semidirect
4. #21425598C2semidirect
6. #21425488C2semidirect
7. #21425477C2semidirect
8. #21425418C2semidirect
12. #21423994C2semidirect
13. #21423929C2semidirect
14. #21423926C2semidirect
387. C59#23709semidirect

Maximal subgroups

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

subgroupindex
C2×#168375072
C2×#168374592
C4×#142737062
#214255982
#214255322
#214254882
#214254772
#214254182
#214252962
#214252922
#214251042
#214239942
#214239292
#214239262
C4×C22⋊Q8×C592
#2370959