Finite Group Databasebeta

← #28617990

Subgroups — #28617990

Order 15104.

844 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837500C211yesmaximal, characteristic
2C2×C4×#11745671C211yesmaximal, characteristic
3#21426740C211yesmaximal, characteristic
4#21425578C211yesmaximal, characteristic, complemented
5#21425510C211yesmaximal, characteristic
6#21425491C211yesmaximal, characteristic, complemented
7#21425477C211yesmaximal, characteristic, complemented
8#21425411C211yesmaximal, characteristic
9#21425411C211yesmaximal, characteristic
10#21425339C211yesmaximal, characteristic, complemented
11#21425179C211yesmaximal, characteristic
12#21423953C211yesmaximal, characteristic, complemented
13#21423926C211yesmaximal, characteristic, complemented
14C4×#14273651C211yesmaximal, characteristic, complemented
15C59×#2327C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C22×#11745671C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19#16837500—24no—
20C4×#11745671—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×#14273638C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35#16837304C2211yescharacteristic
36#16837304C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837291—21no—
40#16837291—21no—
41#16837264—21no—
42#16837261—21no—
43C4×C4⋊Dic59C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4⋊Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×C4×Q8×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#391C2211yescharacteristic
55C59×#387—21no—
56C59×#387—21no—
57C59×#387—21no—
58C59×#387—21no—
59C59×#382C2211yescharacteristic
60C59×#376C2211yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C2×#11745671—22no—
64C2×#11745671—22no—
65C2×#11745671—22no—
66C2×#11745671—22no—
67C2×#11745671—22no—
68C2×#11745671—22no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73C4×D236—22no—
74C4×D236D414yes—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×C4×Dic59—21no—
86#14273660—22no—
87#14273658—22no—
88C4⋊(C59⋊C8)—22no—
89C4⋊(C59⋊C8)—22no—
90#14273651—22no—
91#14273651—22no—
92#14273651—22no—
93#14273651—22no—
94#14273651—22no—
95#14273651—22no—
96#14273647—22no—
97#14273647—22no—
98#14273638—22no—
99C4×C59⋊C8—22no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59C2311yescharacteristic
104C22×Q8×C59C2311yescharacteristic
105C4⋊Q8×C59—22no—
106C4⋊Q8×C59—22no—
107C4×Q8×C59—22no—
108C4×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×C59—21no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C42×C59—21no—
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59D411yescharacteristic
129#95×C59D411yescharacteristic
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C23×D59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59#4112yes—
144C2×Q8×C59#4112yes—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2411yescharacteristic
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236—21no—
162C2×D236—21no—
163C2×D236—21no—
164C2×D236C2×D412yes—
165C2×C4×D59—24no—
166C2×C4×D59—81no—
167C2×C4×D59—81no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—24no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59—42no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—42no—
181C42×C59—42no—
182C42×C59C2×D412yes—
183C42×C59#4112yes—
184#11745671—28no—
185#11745671—28no—
186(C2×D59)⋊C4—24no—
187(C2×D59)⋊C4—81no—
188(C2×D59)⋊C4—81no—
189(C2×D59)⋊C4—81no—
190(C2×D59)⋊C4—81no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59C2×D412yes—
197Dic59⋊C4—81no—
198Dic59⋊C4—81no—
199C4×Dic59—81no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
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#4112yes—
211C2×C59⋊C8#4112yes—
212C23×C59C22×D411yescharacteristic
213C22×D59—22no—
214C22×D59—41no—
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220Q8×C59—22no—
221Q8×C59—22no—
222Q8×C59—22no—
223Q8×C59—22no—
224Q8×C59—41no—
225Q8×C59—41no—
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—22no—
235C2×C4×C59—22no—
236C2×C4×C59—21no—
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—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#12112yes—
257C2×C4×C59#12112yes—
258C2×C4×C59C22×D411yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×Dic59—22no—
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—
278C59⋊C8—22no—
279C59⋊C8—22no—
280C59⋊C8—41no—
281C59⋊C8—41no—
282#23689—591nomaximal, Sylow
283C22×C59C2×#12111yescharacteristic
284C22×C59#52511yescharacteristic
285C22×C59#53211yescharacteristic
286C22×C59#57711yescharacteristic
287C22×C59#57711yescharacteristic
288C22×C59#57711yescharacteristic
289C22×C59#57811yescharacteristic
290C2×D59—42no—
291C2×D59—42no—
292C2×D59—42no—
293C2×D59—81no—
294C2×D59—81no—
295C2×D59—81no—
296C2×D59—81no—
297C4×C59—22no—
298C4×C59—22no—
299C4×C59—41no—
300C4×C59—41no—
301C4×C59—42no—
302C4×C59—42no—
303C4×C59—41no—
304C4×C59—41no—
305C4×C59—81no—
306C4×C59—81no—
307C4×C59#52511yescharacteristic
308C4×C59#52511yescharacteristic
309C4×C59#52911yescharacteristic
310C4×C59#52911yescharacteristic
311Dic59—42no—
312Dic59—81no—
313Dic59—81no—
314C2×#440—591no—
315C2×C4×SD16—591no—
316#2327D5911yescharacteristic, complemented
317#2319—591no—
318#2085—591no—
319#2073—591no—
320#2068—591no—
321#2066—591no—
322#2055—591no—
323#2055—591no—
324#2043—591no—
325#2013—591no—
326#1768—591no—
327#1763—591no—
328C4×#103—591no—
329C2×C59#239211yescharacteristic
330C2×C59#304211yescharacteristic
331C2×C59#304611yescharacteristic
332C2×C59#312111yescharacteristic
333C2×C59#312611yescharacteristic
334C2×C59#317411yescharacteristic
335C2×C59#317811yescharacteristic
336D59—82no—
337C22×SD16—591no—
338C22×SD16—591no—
339C2×C4⋊Q8C2×D5911yescharacteristic
340C2×C4×Q8C2×D5911yescharacteristic
341C2×C4×Q8C2×D5911yescharacteristic
342C2×C4×D4—591no—
343#440—1184no—
344C4×SD16—1184no—
345C2×#107—591no—
346C2×#106—591no—
347C2×C4⋊C8—591no—
348C2×C4⋊C8—591no—
349C2×#103—591no—
350C2×#103—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×#97—591no—
358C2×C4×C8—591no—
359#391C2×D5911yescharacteristic
360#391C2×D5911yescharacteristic
361#390—591no—
362#390—591no—
363#389—591no—
364#389—591no—
365#388—1181no—
366#388—1181no—
367#387—21no—
368#387—21no—
369#387—21no—
370#387—21no—
371#382C2×D5911yescharacteristic
372#381—1181no—
373#380—1181no—
374C4×C4⋊C4—591no—
375#376C2×D5911yescharacteristic
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#336—591no—
381#336—591no—
382C59#2368911yescharacteristic, complemented, Sylow
383C22×Q8C22×D5911yescharacteristic
384C22×Q8C22×D5911yescharacteristic
385C22×D4—591no—
386C23×C4—1181no—
387C2×SD16—1182no—
388C2×SD16—1182no—
389C2×SD16—1182no—
390C2×SD16—1182no—
391C2×SD16—1182no—
392C2×SD16—1182no—
393C22×C8—591no—
394C22×C8—591no—
395C22×C8—591no—
396C22×C8—591no—
397C4⋊Q8—22no—
398C4⋊Q8—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×D4—594no—
404C4×D4—1182no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C22⋊C4—1181no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42—1181no—
424C2×C42C22×D5911yescharacteristic
425C2×C42C22×D5911yescharacteristic
426C2×C42C22×D5911yescharacteristic
427#107—1182no—
428#106—1182no—
429C4⋊C8—1182no—
430C4⋊C8—1182no—
431#103—1182no—
432#103—1182no—
433#103—1182no—
434#103—1182no—
435#103—1182no—
436#103—1182no—
437#102—1182no—
438#102—1182no—
439#97—1182no—
440C4×C8—1182no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95C22⋊D5911yescharacteristic
451#95C22⋊D5911yescharacteristic
452#95C22⋊D5911yescharacteristic
453#95C22⋊D5911yescharacteristic
454C24—1181no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8#1174568512yes—
466C2×Q8#1174568512yes—
467C2×D4—592no—
468C2×D4—1181no—
469C2×D4—1181no—
470C2×D4—1181no—
471C2×D4—1181no—
472C2×D4—1181no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—1181no—
478C22×C4—1184no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4—4721no—
482C22×C4—4721no—
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C23×D5911yescharacteristic
490SD16—1188no—
491SD16—1188no—
492C2×C8—592no—
493C2×C8—592no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—592no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—4721no—
519C4⋊C4—4721no—
520C22⋊C4—1184no—
521C22⋊C4—4721no—
522C22⋊C4—4721no—
523C22⋊C4—4721no—
524C22⋊C4—4721no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—42no—
530C42—42no—
531C42—4721no—
532C42D4×D5912yes—
533C42#1174568712yes—
534C23—1182no—
535C23—2361no—
536C23—2361no—
537C23—2361no—
538C23—2361no—
539C23—2361no—
540C23—2361no—
541C23C22×C22⋊D5911yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—22no—
545Q8—22no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550D4—1182no—
551D4—1182no—
552D4—2361no—
553D4—2361no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—1182no—
583C2×C4—4724no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4#1427369812yes—
593C2×C4#1427369812yes—
594C2×C4C2×D4×D5911yescharacteristic
595C2×C4#41×D5911yescharacteristic
596C2×C4#41×D5911yescharacteristic
597C2×C4C2×#1174568711yescharacteristic
598C2×C4C2×#1174568511yescharacteristic
599C2×C4C2×#1174568511yescharacteristic
600C8—1182no—
601C8—1182no—
602C8—2361no—
603C8—2361no—
604C22—4721no—
605C22—4721no—
606C22—4721no—
607C22—4721no—
608C22—2362no—
609C22—2362no—
610C22—2362no—
611C22C2×#1427369811yescharacteristic
612C22#1683791311yescharacteristic
613C22#1683794811yescharacteristic
614C22#1683821011yescharacteristic
615C22#1683821011yescharacteristic
616C22#1683821011yescharacteristic
617C22#1683821711yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—4721no—
629C4—4721no—
630C4—2362no—
631C4#1683791111yescharacteristic
632C4#1683791111yescharacteristic
633C4#1683793211yescharacteristic
634C4#1683793211yescharacteristic
635C2—4722no—
636C2#2142733411yescharacteristic
637C2#2143360411yescharacteristic
638C2#2143364211yescharacteristic
639C2#2143432511yescharacteristic
640C2#2143438711yescharacteristic
641C2#2143472511yescharacteristic
642C2#2143476611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2327C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2327D59
hypocenterC59#23689
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2327

Complements

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

NH ≅ G/Ntype
4. #21425578C2semidirect
6. #21425491C2semidirect
7. #21425477C2semidirect
10. #21425339C2semidirect
12. #21423953C2semidirect
13. #21423926C2semidirect
14. C4×#14273651C2semidirect
15. C59×#2327C2semidirect
316. #2327D59semidirect
382. C59#23689semidirect

Maximal subgroups

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

subgroupindex
C2×#168375002
C2×C4×#117456712
#214267402
#214255782
#214255102
#214254912
#214254772
#214254112
#214253392
#214251792
#214239532
#214239262
C4×#142736512
C59×#23272
#2368959