Finite Group Databasebeta

← #28648831

Subgroups — #28648831

Order 15104.

826 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837624C211yesmaximal, characteristic
2C2×C4×#11745672C211yesmaximal, characteristic, complemented
3#21427894C211yesmaximal, characteristic
4#21425625C211yesmaximal, characteristic
5#21425587C211yesmaximal, characteristic, complemented
6#21425505C211yesmaximal, characteristic, complemented
7#21425337C211yesmaximal, characteristic
8#21424778C211yesmaximal, characteristic
9#21424655C211yesmaximal, characteristic, complemented
10#21424451C211yesmaximal, characteristic
11#21424451C211yesmaximal, characteristic
12#21424437C211yesmaximal, characteristic, complemented
13#21424350C211yesmaximal, characteristic, complemented
14#21424349C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C59C211yesmaximal, characteristic, complemented
16C22×#11745672C2211yescharacteristic
17C22×#11745671C2211yescharacteristic
18C2×#14273715C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837624—24no—
21C4×#11745672—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
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#16837337—21no—
35#16837331—21no—
36#16837308C2211yescharacteristic
37#16837304C2211yescharacteristic
38#16837304C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837292—21no—
41#16837291—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
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672D414yes—
66C2×#11745671—22no—
67C2×#11745671—22no—
68C2×#11745671—22no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273715—22no—
74#14273715—41no—
75C4×Dic118—22no—
76C4×Dic118—41no—
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
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—
86C2×(C2×D59)⋊C4—21no—
87C2×C4×Dic59—21no—
88#14273660—22no—
89#14273660—41no—
90C4⋊(C59⋊C8)—22no—
91#14273651—22no—
92#14273651—22no—
93#14273650—22no—
94#14273650—22no—
95#14273650—22no—
96#14273647—22no—
97#14273647—22no—
98#14273647—41no—
99C4×C59⋊C8—22no—
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×D236—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118C2×D412yes—
165C4⋊C4×C59—24no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—42no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59—42no—
176C4⋊C4×C59—81no—
177C4⋊C4×C59—81no—
178C4⋊C4×C59—81no—
179C4⋊C4×C59#4112yes—
180C42×C59—24no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—42no—
188C42×C59—42no—
189#11745672—28no—
190#11745671—44no—
191(C2×D59)⋊C4—42no—
192(C2×D59)⋊C4—42no—
193(C2×D59)⋊C4—42no—
194(C2×D59)⋊C4—42no—
195(C2×D59)⋊C4—81no—
196(C2×D59)⋊C4—81no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201Dic59⋊C4—42no—
202Dic59⋊C4—81no—
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⋊C8—21no—
215C2×C59⋊C8C2×D412yes—
216C23×C59C22×D411yescharacteristic
217C22×D59—41no—
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—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—41no—
274D236—41no—
275D236—41no—
276D236—41no—
277Dic118—22no—
278Dic118—22no—
279Dic118—41no—
280Dic118—41no—
281C59⋊C8—22no—
282C59⋊C8—41no—
283C59⋊C8—41no—
284C59⋊C8—41no—
285#26169—591nomaximal, Sylow
286C22×C59#52511yescharacteristic
287C22×C59D4211yescharacteristic
288C22×C59#54811yescharacteristic
289C22×C59C2×#13511yescharacteristic
290C22×C59C2×#13711yescharacteristic
291C22×C59#57611yescharacteristic
292C22×C59#57811yescharacteristic
293C2×D59—81no—
294C2×D59—81no—
295C2×D59—81no—
296C2×D59—81no—
297C2×D59—81no—
298C2×D59—81no—
299C2×D59—81no—
300C4×C59—22no—
301C4×C59—24no—
302C4×C59—22no—
303C4×C59—41no—
304C4×C59—42no—
305C4×C59—41no—
306C4×C59—41no—
307C4×C59—41no—
308C4×C59—81no—
309C4×C59—81no—
310C4×C59C2×#12311yescharacteristic
311C4×C59#53911yescharacteristic
312C4×C59#54711yescharacteristic
313C4×C59#55311yescharacteristic
314Dic59—42no—
315Dic59—81no—
316Dic59—81no—
317C2×#464—591no—
318C2×C4×Q16—591no—
319#2448—591no—
320Q8×C4⋊C4D5911yescharacteristic, complemented
321#2093—591no—
322#2086—591no—
323#2072—591no—
324#2042—591no—
325#1934—591no—
326#1916—591no—
327#1880—591no—
328#1880—591no—
329#1878—591no—
330#1863—591no—
331#1863—591no—
332C2×C59#244311yescharacteristic
333C2×C59#310811yescharacteristic
334C2×C59#312411yescharacteristic
335C2×C59#327011yescharacteristic
336C2×C59#328711yescharacteristic
337C2×C59#336411yescharacteristic
338C2×C59#337511yescharacteristic
339D59—161no—
340C22×Q16—591no—
341C22×SD16—591no—
342C2×#124—591no—
343C2×C4×Q8—591no—
344C2×C4×Q8C2×D5911yescharacteristic
345C2×C4×Q8C2×D5911yescharacteristic
346C2×C4×Q8C2×D5911yescharacteristic
347#464—1184no—
348C4×Q16—1184no—
349C2×#107—591no—
350C2×#107—591no—
351C2×C4⋊C8—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×C4×C8—591no—
361#399—1181no—
362#397—1181no—
363#391—591no—
364#390—591no—
365#390—591no—
366#389—21no—
367#389—21no—
368#389—591no—
369#389C2×D5911yescharacteristic
370#389C2×D5911yescharacteristic
371#388—1181no—
372#388—1181no—
373#384—21no—
374#384C2×D5911yescharacteristic
375C4×C4⋊C4—21no—
376C4×C4⋊C4C2×D5911yescharacteristic
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#336—591no—
382#336—591no—
383#336—591no—
384#336—591no—
385C59#2616911yescharacteristic, complemented, Sylow
386C22×Q8—591no—
387C22×Q8C22×D5911yescharacteristic
388C22×D4—591no—
389C2×Q16—594no—
390C2×Q16—1182no—
391C2×Q16—1182no—
392C2×SD16—1182no—
393C2×SD16—1182no—
394C2×SD16—1182no—
395C22×C8—591no—
396C22×C8—591no—
397C22×C8—591no—
398C22×C8—591no—
399#124—1182no—
400#124—2361no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—22no—
405C4×Q8—41no—
406C4×Q8—41no—
407C4×Q8—1182no—
408C4×Q8—2361no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—1181no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4C22⋊D5911yescharacteristic
421C2×C4⋊C4C22⋊D5911yescharacteristic
422C2×C4⋊C4C22×D5911yescharacteristic
423C2×C4⋊C4C22×D5911yescharacteristic
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C22⋊C4—1181no—
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×C42—1181no—
435C2×C42C22×D5911yescharacteristic
436C2×C42C22×D5911yescharacteristic
437C2×C42C22×D5911yescharacteristic
438#107—1182no—
439#107—2361no—
440C4⋊C8—1182no—
441#103—1182no—
442#103—1182no—
443#103—1182no—
444#103—1182no—
445#103—1182no—
446#102—1182no—
447#102—1182no—
448#102—2361no—
449C4×C8—1182no—
450#95—21no—
451#95—21no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95C22⋊D5911yescharacteristic
456#95C22⋊D5911yescharacteristic
457C24—1181no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—592no—
462C2×Q8—1181no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C2×Q8—1181no—
467C2×Q8#1174568512yes—
468C2×Q8D4×D5912yes—
469C2×Q8#1174568912yes—
470C2×D4—1181no—
471C2×D4—1181no—
472C2×D4—1181no—
473C2×D4—1181no—
474C2×D4—1181no—
475C2×D4—1181no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—1181no—
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
490Q16—1188no—
491SD16—2364no—
492C2×C8—592no—
493C2×C8—1181no—
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—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—4721no—
523C4⋊C4—2362no—
524C4⋊C4#1174568512yes—
525C22⋊C4—4721no—
526C22⋊C4—4721no—
527C22⋊C4—2362no—
528C22⋊C4—2362no—
529C22⋊C4—2362no—
530C22⋊C4—2362no—
531C42—24no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—42no—
539C42—42no—
540C42—2362no—
541C23—2361no—
542C23—2361no—
543C23—2361no—
544C23—2361no—
545C23—2361no—
546C23—2361no—
547C23—2361no—
548C23C22×C22⋊D5911yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—22no—
552Q8—1182no—
553Q8—1182no—
554Q8—2361no—
555Q8—2361no—
556Q8#1427371214yes—
557D4—2361no—
558D4—2361no—
559D4—2361no—
560D4—2361no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
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—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—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4#1427374712yes—
599C2×C4#1427375812yes—
600C2×C4C2×D4×D5911yescharacteristic
601C2×C4C2×#1174568511yescharacteristic
602C2×C4C2×#1174568511yescharacteristic
603C2×C4C2×#1174568911yescharacteristic
604C2×C4#1427377211yescharacteristic
605C2×C4#1427377511yescharacteristic
606C8—1182no—
607C8—2361no—
608C8—2361no—
609C8—2361no—
610C22—4721no—
611C22—4721no—
612C22—4721no—
613C22—4721no—
614C22—4721no—
615C22—4721no—
616C22—4721no—
617C22#1683791311yescharacteristic
618C22D4×C22⋊D5911yescharacteristic
619C22#1683806311yescharacteristic
620C22C2×#1427374711yescharacteristic
621C22C2×#1427375811yescharacteristic
622C22#1683820511yescharacteristic
623C22#1683821711yescharacteristic
624C4—22no—
625C4—22no—
626C4—24no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—4721no—
635C4—4721no—
636C4—2362no—
637C4C2×#1427371211yescharacteristic
638C4#1683799911yescharacteristic
639C4#1683805311yescharacteristic
640C4#1683810011yescharacteristic
641C2—9441no—
642C2#2142783111yescharacteristic
643C2#2143417611yescharacteristic
644C2#2143435811yescharacteristic
645C2#2143566711yescharacteristic
646C2#2143585211yescharacteristic
647C2#2143678111yescharacteristic
648C2#2143691011yescharacteristic
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#26169
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
2. C2×C4×#11745672C2semidirect
5. #21425587C2semidirect
6. #21425505C2semidirect
9. #21424655C2semidirect
12. #21424437C2semidirect
13. #21424350C2semidirect
14. #21424349C2semidirect
15. Q8×C4⋊C4×C59C2semidirect
320. Q8×C4⋊C4D59semidirect
385. C59#26169semidirect

Maximal subgroups

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

subgroupindex
C2×#168376242
C2×C4×#117456722
#214278942
#214256252
#214255872
#214255052
#214253372
#214247782
#214246552
#214244512
#214244372
#214243502
#214243492
Q8×C4⋊C4×C592
#2616959