Finite Group Databasebeta

← #28648711

Subgroups — #28648711

Order 15104.

840 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837624C211yesmaximal, characteristic
2C2×C4×#11745671C211yesmaximal, characteristic
3#21427883C211yesmaximal, characteristic
4#21425640C211yesmaximal, characteristic, complemented
5#21425597C211yesmaximal, characteristic, complemented
6#21425500C211yesmaximal, characteristic
7#21425337C211yesmaximal, characteristic
8#21424771C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424451C211yesmaximal, characteristic
11#21424437C211yesmaximal, characteristic, complemented
12#21424437C211yesmaximal, characteristic, complemented
13#21424364C211yesmaximal, characteristic, complemented
14#21424356C211yesmaximal, characteristic
15Q8×C4⋊C4×C59C211yesmaximal, characteristic, complemented
16C22×#11745672C2211yescharacteristic
17C22×#11745671C2211yescharacteristic
18C2×#14273715C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20#16837624—24no—
21C4×#11745671—24no—
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273647C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837331—21no—
35#16837328—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837304C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837273—21no—
41#16837264—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×C4×Q8×C59C2211yescharacteristic
53C59×#389—21no—
54C59×#389—21no—
55C59×#389C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#384—21no—
58C59×#384C2211yescharacteristic
59C4×C4⋊C4×C59—21no—
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C22×D236C2311yescharacteristic
63C22×C4×D59—21no—
64C2×#11745672—22no—
65C2×#11745672—22no—
66C2×#11745672—22no—
67C2×#11745671—22no—
68C2×#11745671—22no—
69C2×#11745671D414yes—
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73C22×C59⋊C8C2311yescharacteristic
74#14273715—22no—
75#14273715—41no—
76C4×D236—22no—
77C4×D236—41no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×Dic59⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86#14273658—22no—
87#14273658—41no—
88C4⋊(C59⋊C8)—22no—
89#14273651—22no—
90#14273651—22no—
91#14273650—22no—
92#14273650—22no—
93#14273650—41no—
94#14273647—22no—
95#14273647—22no—
96#14273647—22no—
97C4×C59⋊C8—22no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59C2311yescharacteristic
104C4×Q8×C59—22no—
105C4×Q8×C59—22no—
106C4×Q8×C59—22no—
107C4×Q8×C59—22no—
108C4×Q8×C59—41no—
109C4×Q8×C59—41no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59D411yescharacteristic
118C2×C4⋊C4×C59D411yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C42×C59—21no—
123C2×C42×C59—21no—
124C2×C42×C59—21no—
125C2×C42×C59C2311yescharacteristic
126C2×C42×C59C2311yescharacteristic
127C2×C42×C59C2311yescharacteristic
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C23×D59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59C2×D412yes—
137C2×Q8×C59#4112yes—
138C2×Q8×C59#4112yes—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2411yescharacteristic
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236—21no—
156C2×D236—21no—
157C2×D236—21no—
158C2×D236C2×D412yes—
159C2×C4×D59—42no—
160C2×C4×D59—42no—
161C2×C4×D59—81no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59—81no—
180C4⋊C4×C59—81no—
181C4⋊C4×C59—81no—
182C4⋊C4×C59#4112yes—
183C42×C59—24no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—42no—
191C42×C59—42no—
192#11745672—44no—
193#11745671—28no—
194(C2×D59)⋊C4—42no—
195(C2×D59)⋊C4—42no—
196(C2×D59)⋊C4—42no—
197(C2×D59)⋊C4—42no—
198(C2×D59)⋊C4—81no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203Dic59⋊C4—81no—
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⋊C8—21no—
215C2×C59⋊C8C2×D412yes—
216C23×C59C22×D411yescharacteristic
217C22×D59—22no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224Q8×C59—22no—
225Q8×C59—22no—
226Q8×C59—22no—
227Q8×C59#12314yes—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—21no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—22no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
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#13512yes—
257C2×C4×C59#13712yes—
258C2×C4×C59C22×D411yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59H(5,𝔽2)11yescharacteristic
263C2×C4×C592−1+411yescharacteristic
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273D236—22no—
274D236—22no—
275D236—41no—
276D236—41no—
277C4×D59—84no—
278Dic118—41no—
279Dic118—41no—
280Dic118—41no—
281Dic118—41no—
282C59⋊C8—22no—
283C59⋊C8—41no—
284C59⋊C8—41no—
285C59⋊C8—41no—
286#26161—591nomaximal, Sylow
287C22×C59#52511yescharacteristic
288C22×C59D4211yescharacteristic
289C22×C59#54811yescharacteristic
290C22×C59C2×#13511yescharacteristic
291C22×C59C2×#13711yescharacteristic
292C22×C59#57711yescharacteristic
293C22×C59#57711yescharacteristic
294C2×D59—42no—
295C2×D59—42no—
296C2×D59—42no—
297C2×D59—81no—
298C2×D59—81no—
299C2×D59—81no—
300C2×D59—81no—
301C4×C59—22no—
302C4×C59—24no—
303C4×C59—22no—
304C4×C59—41no—
305C4×C59—42no—
306C4×C59—41no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—81no—
310C4×C59—81no—
311C4×C59C2×#12311yescharacteristic
312C4×C59#53911yescharacteristic
313C4×C59#54711yescharacteristic
314C4×C59#55311yescharacteristic
315Dic59—81no—
316Dic59—81no—
317Dic59—81no—
318C2×#464—591no—
319C2×C4×SD16—591no—
320#2447—591no—
321Q8×C4⋊C4D5911yescharacteristic, complemented
322#2096—591no—
323#2088—591no—
324#2071—591no—
325#2042—591no—
326#1933—591no—
327#1916—591no—
328#1880—591no—
329#1878—591no—
330#1878—591no—
331#1865—591no—
332#1864—591no—
333C2×C59#244311yescharacteristic
334C2×C59#310811yescharacteristic
335C2×C59#312511yescharacteristic
336C2×C59#326611yescharacteristic
337C2×C59#328411yescharacteristic
338C2×C59#336511yescharacteristic
339C2×C59#336911yescharacteristic
340D59—82no—
341C22×Q16—591no—
342C22×SD16—591no—
343C2×#124—591no—
344C2×C4×Q8C2×D5911yescharacteristic
345C2×C4×Q8C2×D5911yescharacteristic
346C2×C4×Q8C2×D5911yescharacteristic
347C2×C4×D4—591no—
348#464—1184no—
349C4×SD16—1184no—
350C2×#106—591no—
351C2×#106—591no—
352C2×C4⋊C8—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#103—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×C4×C8—591no—
362#397—1181no—
363#397—1181no—
364#391—591no—
365#391—591no—
366#390—591no—
367#389—21no—
368#389—21no—
369#389—591no—
370#389C2×D5911yescharacteristic
371#389C2×D5911yescharacteristic
372#385—1181no—
373#384—21no—
374#384C2×D5911yescharacteristic
375#381—1181no—
376C4×C4⋊C4—21no—
377C4×C4⋊C4C2×D5911yescharacteristic
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#336—591no—
383#336—591no—
384#336—591no—
385#336—591no—
386C59#2616111yescharacteristic, complemented, Sylow
387C22×Q8—591no—
388C22×Q8C22×D5911yescharacteristic
389C22×D4—591no—
390C23×C4—1181no—
391C2×Q16—1182no—
392C2×Q16—1182no—
393C2×Q16—1182no—
394C2×SD16—594no—
395C2×SD16—1182no—
396C2×SD16—1182no—
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8—591no—
401#124—1182no—
402#124—2361no—
403C4×Q8—22no—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—22no—
407C4×Q8—41no—
408C4×Q8—41no—
409C4×D4—1182no—
410C4×D4—2361no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—591no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4C22⋊D5911yescharacteristic
422C2×C4⋊C4C22⋊D5911yescharacteristic
423C2×C4⋊C4C22×D5911yescharacteristic
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C4⋊C4C22×D5911yescharacteristic
426C2×C22⋊C4—1181no—
427C2×C22⋊C4—1181no—
428C2×C22⋊C4—1181no—
429C2×C22⋊C4—1181no—
430C2×C22⋊C4—1181no—
431C2×C42—21no—
432C2×C42—21no—
433C2×C42—21no—
434C2×C42C22×D5911yescharacteristic
435C2×C42C22×D5911yescharacteristic
436C2×C42C22×D5911yescharacteristic
437#106—1182no—
438#106—2361no—
439C4⋊C8—1182no—
440#103—1182no—
441#103—1182no—
442#103—1182no—
443#103—1182no—
444#103—2361no—
445#102—1182no—
446#102—1182no—
447#102—1182no—
448C4×C8—1182no—
449#95—21no—
450#95—21no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456#95C22⋊D5911yescharacteristic
457#95C22⋊D5911yescharacteristic
458C24—1181no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—1181no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C2×Q8—1181no—
467C2×Q8—1181no—
468C2×Q8#1174568512yes—
469C2×Q8D4×D5912yes—
470C2×Q8#1174568912yes—
471C2×D4—592no—
472C2×D4—1181no—
473C2×D4—1181no—
474C2×D4—1181no—
475C2×D4—1181no—
476C2×D4—1181no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—4721no—
485C22×C4—2362no—
486C22×C4—2362no—
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C23×D5911yescharacteristic
494Q16—2364no—
495SD16—1188no—
496C2×C8—592no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—4721no—
527C4⋊C4#1174568512yes—
528C22⋊C4—4721no—
529C22⋊C4—2362no—
530C22⋊C4—2362no—
531C22⋊C4—2362no—
532C22⋊C4—2362no—
533C42—24no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—42no—
542C23—1182no—
543C23—2361no—
544C23—2361no—
545C23—2361no—
546C23—2361no—
547C23—2361no—
548C23—2361no—
549C23C22×C22⋊D5911yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8—2361no—
554Q8—2361no—
555Q8—2361no—
556Q8—2361no—
557Q8#1427371214yes—
558D4—1182no—
559D4—1182no—
560D4—2361no—
561D4—2361no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—4724no—
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#1427374712yes—
601C2×C4#1427375812yes—
602C2×C4C2×D4×D5911yescharacteristic
603C2×C4C2×#1174568511yescharacteristic
604C2×C4C2×#1174568511yescharacteristic
605C2×C4C2×#1174568911yescharacteristic
606C2×C4#1427377211yescharacteristic
607C2×C4#1427377511yescharacteristic
608C8—1182no—
609C8—2361no—
610C8—2361no—
611C8—2361no—
612C22—4721no—
613C22—4721no—
614C22—4721no—
615C22—4721no—
616C22—2362no—
617C22—2362no—
618C22—2362no—
619C22#1683791311yescharacteristic
620C22D4×C22⋊D5911yescharacteristic
621C22#1683806311yescharacteristic
622C22C2×#1427374711yescharacteristic
623C22C2×#1427375811yescharacteristic
624C22#1683821011yescharacteristic
625C22#1683821311yescharacteristic
626C4—22no—
627C4—24no—
628C4—22no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—4721no—
637C4—4721no—
638C4—4721no—
639C4C2×#1427371211yescharacteristic
640C4#1683799911yescharacteristic
641C4#1683805311yescharacteristic
642C4#1683810011yescharacteristic
643C2—4722no—
644C2#2142783111yescharacteristic
645C2#2143417611yescharacteristic
646C2#2143437311yescharacteristic
647C2#2143560711yescharacteristic
648C2#2143582611yescharacteristic
649C2#2143679211yescharacteristic
650C2#2143683611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingQ8×C4⋊C4×C59C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenterQ8×C4⋊C4D59
hypocenterC59#26161
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #21425640C2semidirect
5. #21425597C2semidirect
8. #21424771C2semidirect
9. #21424655C2semidirect
11. #21424437C2semidirect
12. #21424437C2semidirect
13. #21424364C2semidirect
15. Q8×C4⋊C4×C59C2semidirect
321. Q8×C4⋊C4D59semidirect
386. C59#26161semidirect

Maximal subgroups

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

subgroupindex
C2×#168376242
C2×C4×#117456712
#214278832
#214256402
#214255972
#214255002
#214253372
#214247712
#214246552
#214244512
#214244372
#214243642
#214243562
Q8×C4⋊C4×C592
#2616159