Finite Group Databasebeta

← #28651715

Subgroups — #28651715

Order 15104.

864 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×#16837687C211yesmaximal, characteristic, complemented
2C2×C4×#11745671C211yesmaximal, characteristic
3#21427946C211yesmaximal, characteristic
4#21425630C211yesmaximal, characteristic
5#21425573C211yesmaximal, characteristic, complemented
6#21425467C211yesmaximal, characteristic
7#21425429C211yesmaximal, characteristic, complemented
8#21424821C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424451C211yesmaximal, characteristic
11#21424451C211yesmaximal, characteristic
12#21424364C211yesmaximal, characteristic, complemented
13#21424364C211yesmaximal, characteristic, complemented
14#21424356C211yesmaximal, characteristic
15Q8×C4⋊C4×C59C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×#14273727C2211yescharacteristic
18C2×#14273705—21no—
19C2×C4×D236C2211yescharacteristic
20#16837687—24no—
21C4×#11745671—24no—
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×#14273647C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35#16837324—21no—
36#16837315—21no—
37#16837304C2211yescharacteristic
38#16837304C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837300C2211yescharacteristic
41#16837276—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×Dic118—21no—
62C22×D236C2311yescharacteristic
63C22×C4×D59—21no—
64C2×#11745671—22no—
65C2×#11745671—22no—
66C2×#11745671Q814yescomplemented
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71#14273727—22no—
72#14273727—41no—
73#14273705—44no—
74C4×D236—22no—
75C4×D236—41no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87#14273658—22no—
88#14273658—22no—
89#14273658—22no—
90#14273658—22no—
91#14273658—41no—
92C4⋊(C59⋊C8)—22no—
93#14273651—22no—
94#14273651—22no—
95#14273651D414yes—
96#14273647—22no—
97#14273647—22no—
98#14273647—41no—
99C4×C59⋊C8—22no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C22×Q8×C59C2311yescharacteristic
103C4×Q8×C59—22no—
104C4×Q8×C59—22no—
105C4×Q8×C59—22no—
106C4×Q8×C59—22no—
107C4×Q8×C59—41no—
108C4×Q8×C59—41no—
109C2×C4⋊C4×C59—21no—
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×C59D411yescharacteristic
117C2×C4⋊C4×C59D411yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C42×C59—21no—
122C2×C42×C59—21no—
123C2×C42×C59—21no—
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126C2×C42×C59C2311yescharacteristic
127#95×C59—21no—
128#95×C59—21no—
129#95×C59D411yescharacteristic
130#95×C59D411yescharacteristic
131C23×D59—21no—
132C2×Q8×C59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59C2×D412yes—
136C2×Q8×C59C2×Q812yes—
137C2×Q8×C59#4112yes—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236—21no—
156C2×D236—21no—
157C2×D236C2×Q812yes—
158C2×C4×D59—42no—
159C2×C4×D59—42no—
160C2×C4×D59—81no—
161C2×Dic118—42no—
162C2×Dic118—42no—
163C2×Dic118—81no—
164C4⋊C4×C59—24no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—41no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59—42no—
174C4⋊C4×C59—81no—
175C4⋊C4×C59—81no—
176C4⋊C4×C59—81no—
177C4⋊C4×C59SD1614yescomplemented
178C4⋊C4×C59#4112yes—
179C42×C59—24no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—42no—
187C42×C59—42no—
188#11745671—28no—
189(C2×D59)⋊C4—42no—
190(C2×D59)⋊C4—42no—
191(C2×D59)⋊C4—42no—
192(C2×D59)⋊C4—42no—
193(C2×D59)⋊C4—81no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—42no—
200C4⋊Dic59C2×D412yes—
201Dic59⋊C4—42no—
202Dic59⋊C4—42no—
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—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8C2×D412yes—
214C2×C59⋊C8C2×Q812yes—
215C23×C59C22×D411yescharacteristic
216C22×D59—22no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223Q8×C59—22no—
224Q8×C59—22no—
225Q8×C59—22no—
226Q8×C59C22⋊Q814yescomplemented
227C2×C4×C59—21no—
228C2×C4×C59—22no—
229C2×C4×C59—21no—
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—21no—
237C2×C4×C59—22no—
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×C59C2×SD1612yes—
255C2×C4×C59C2×SD1612yes—
256C2×C4×C59C2×SD1612yes—
257C2×C4×C59C22×D411yescharacteristic
258C2×C4×C59C22×Q811yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59H(5,𝔽2)11yescharacteristic
262C2×C4×C592−1+411yescharacteristic
263C2×Dic59—22no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272D236—22no—
273D236—22no—
274D236—41no—
275D236—41no—
276C4×D59—84no—
277Dic118—84no—
278C59⋊C8—22no—
279C59⋊C8—22no—
280C59⋊C8—41no—
281C59⋊C8—41no—
282#26373—591nomaximal, Sylow
283C22×C59#52511yescharacteristic
284C22×C59D4211yescharacteristic
285C22×C59D4×Q811yescharacteristic
286C22×C59C22×SD1611yescharacteristic
287C22×C59#57511yescharacteristic
288C22×C59#57711yescharacteristic
289C22×C59#57711yescharacteristic
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—24no—
299C4×C59—22no—
300C4×C59—22no—
301C4×C59—41no—
302C4×C59—42no—
303C4×C59—41no—
304C4×C59—41no—
305C4×C59—81no—
306C4×C59—81no—
307C4×C59C2×C22⋊Q811yescharacteristic
308C4×C59#53911yescharacteristic
309C4×C59#54711yescharacteristic
310C4×C59#55411yescharacteristic
311Dic59—42no—
312Dic59—81no—
313Dic59—81no—
314C2×#475—591no—
315C2×C4×SD16—591no—
316#2452—591no—
317Q8×C4⋊C4D5911yescharacteristic, complemented
318#2094—591no—
319#2084—591no—
320#2065—591no—
321#2058—591no—
322#1941—591no—
323#1916—591no—
324#1880—591no—
325#1880—591no—
326#1865—591no—
327#1865—591no—
328#1864—591no—
329C2×C59#244911yescharacteristic
330C2×C59#310311yescharacteristic
331C2×C59#312511yescharacteristic
332C2×C59#327111yescharacteristic
333C2×C59D4×SD1611yescharacteristic
334C2×C59Q8×SD1611yescharacteristic
335C2×C59#338811yescharacteristic
336D59—82no—
337C22×SD16—591no—
338C2×C4⋊Q8—591no—
339C2×C22⋊Q8—1181no—
340C2×C4×Q8C2×D5911yescharacteristic
341C2×C4×Q8C2×D5911yescharacteristic
342C2×C4×Q8C2×D5911yescharacteristic
343C2×C4×D4—591no—
344#475—1184no—
345C4×SD16—1184no—
346C2×#106—591no—
347C2×#106—591no—
348C2×#106—591no—
349C2×#106—591no—
350C2×#106—591no—
351C2×C4⋊C8—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×#102—591no—
358C2×C4×C8—591no—
359#396—1181no—
360#393—1181no—
361#390—591no—
362#390—591no—
363#389—21no—
364#389—21no—
365#389—591no—
366#389—591no—
367#389C2×D5911yescharacteristic
368#389C2×D5911yescharacteristic
369#385—1181no—
370#384—21no—
371#384C2×D5911yescharacteristic
372C4×C4⋊C4—21no—
373C4×C4⋊C4C2×D5911yescharacteristic
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#336—591no—
379#336—591no—
380#336—591no—
381#336—591no—
382C59#2637311yescharacteristic, complemented, Sylow
383C22×Q8—1181no—
384C22×Q8C22×D5911yescharacteristic
385C22×D4—591no—
386C23×C4—1181no—
387C2×SD16—594no—
388C2×SD16—1182no—
389C2×SD16—1182no—
390C22×C8—591no—
391C22×C8—591no—
392C22×C8—591no—
393C22×C8—591no—
394C4⋊Q8—1182no—
395C4⋊Q8—2361no—
396C22⋊Q8—2364no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—41no—
402C4×Q8—41no—
403C4×D4—1182no—
404C4×D4—2361no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1181no—
417C2×C4⋊C4—1181no—
418C2×C4⋊C4C22⋊D5911yescharacteristic
419C2×C4⋊C4C22⋊D5911yescharacteristic
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C4⋊C4C22×D5911yescharacteristic
422C2×C4⋊C4C22×D5911yescharacteristic
423C2×C22⋊C4—1181no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4—1181no—
426C2×C22⋊C4—1181no—
427C2×C22⋊C4—1181no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42C22×D5911yescharacteristic
432C2×C42C22×D5911yescharacteristic
433C2×C42C22×D5911yescharacteristic
434#106—1182no—
435#106—1182no—
436#106—1182no—
437#106—1182no—
438#106—2361no—
439C4⋊C8—1182no—
440#103—594no—
441#103—1182no—
442#103—1182no—
443#102—1182no—
444#102—1182no—
445#102—2361no—
446C4×C8—1182no—
447#95—21no—
448#95—21no—
449#95—1181no—
450#95—1181no—
451#95C22⋊D5911yescharacteristic
452#95C22⋊D5911yescharacteristic
453C24—1181no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—4721no—
458C2×Q8—2362no—
459C2×Q8—2362no—
460C2×Q8#1174568512yes—
461C2×Q8D4×D5912yes—
462C2×Q8Q8×D5912yes—
463C2×D4—592no—
464C2×D4—1181no—
465C2×D4—1181no—
466C2×D4—1181no—
467C2×D4—1181no—
468C2×D4—1181no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—1181no—
476C22×C4—4721no—
477C22×C4—2362no—
478C22×C4—2362no—
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C2×C22⋊D5911yescharacteristic
481C22×C4C2×C22⋊D5911yescharacteristic
482C22×C4C2×C22⋊D5911yescharacteristic
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C23×D5911yescharacteristic
486SD16—1188no—
487C2×C8—592no—
488C2×C8—592no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—592no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—2362no—
519C4⋊C4—2362no—
520C4⋊C4—2362no—
521C4⋊C4#1174567114yescomplemented
522C4⋊C4#1174568512yes—
523C22⋊C4—4721no—
524C22⋊C4—2362no—
525C22⋊C4—2362no—
526C22⋊C4—2362no—
527C22⋊C4—2362no—
528C42—24no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—42no—
537C23—1182no—
538C23—2361no—
539C23—2361no—
540C23—2361no—
541C23—2361no—
542C23—2361no—
543C23—2361no—
544C23C22×C22⋊D5911yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—4724no—
549Q8#1427370514yescomplemented
550D4—1182no—
551D4—1182no—
552D4—2361no—
553D4—2361no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
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—1182no—
582C2×C4—4724no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4C2×#1174567112yes—
592C2×C4C2×#1174567112yes—
593C2×C4C2×#1174567112yes—
594C2×C4C2×D4×D5911yescharacteristic
595C2×C4C2×Q8×D5911yescharacteristic
596C2×C4C2×#1174568511yescharacteristic
597C2×C4C2×#1174568511yescharacteristic
598C2×C4#1427377211yescharacteristic
599C2×C4#1427377511yescharacteristic
600C8—1182no—
601C8—1182no—
602C8—2361no—
603C8—2361no—
604C22—4721no—
605C22—4721no—
606C22—4721no—
607C22—4721no—
608C22—2362no—
609C22—2362no—
610C22—2362no—
611C22#1683791311yescharacteristic
612C22D4×C22⋊D5911yescharacteristic
613C22Q8×C22⋊D5911yescharacteristic
614C22C22×#1174567111yescharacteristic
615C22#1683820011yescharacteristic
616C22#1683821011yescharacteristic
617C22#1683821311yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—24no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—4721no—
629C4—4721no—
630C4—2362no—
631C4C2×#1427370511yescharacteristic
632C4#1683799911yescharacteristic
633C4#1683805311yescharacteristic
634C4#1683810711yescharacteristic
635C2—4722no—
636C2#2142790511yescharacteristic
637C2#2143412711yescharacteristic
638C2#2143437311yescharacteristic
639C2#2143567711yescharacteristic
640C2D4×#1174567111yescharacteristic
641C2Q8×#1174567111yescharacteristic
642C2#2143703511yescharacteristic
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#26373
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
1. C2×#16837687C2semidirect
5. #21425573C2semidirect
7. #21425429C2semidirect
8. #21424821C2semidirect
9. #21424655C2semidirect
12. #21424364C2semidirect
13. #21424364C2semidirect
15. Q8×C4⋊C4×C59C2semidirect
66. C2×#11745671Q8semidirect
177. C4⋊C4×C59SD16semidirect
226. Q8×C59C22⋊Q8semidirect
317. Q8×C4⋊C4D59semidirect
382. C59#26373semidirect
521. C4⋊C4#11745671semidirect
549. Q8#14273705semidirect

Maximal subgroups

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

subgroupindex
C2×#168376872
C2×C4×#117456712
#214279462
#214256302
#214255732
#214254672
#214254292
#214248212
#214246552
#214244512
#214243642
#214243562
Q8×C4⋊C4×C592
#2637359