Finite Group Databasebeta

← #27311057

Subgroups — #27311057

Order 13568.

1558 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810522C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×Dic53C211yesmaximal, characteristic
3#20272983C211yesmaximal, characteristic, complemented
4#20272972C211yesmaximal, characteristic, complemented
5#20272968C212yesmaximal, complemented
6#20272960C211yesmaximal, characteristic
7#20272960C212yesmaximal
8#20272947C212yesmaximal, complemented
9#20272940C211yesmaximal, characteristic, complemented
10#20272937C212yesmaximal
11C2×C53×#399C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2212yes—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C22⋊Dic53C2211yescharacteristic
16C22×C22⋊Dic53C2212yes—
17C22×C4×Dic53C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810522—24no—
20C22⋊C4×Dic53—24no—
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2212yes—
34C22×C22⋊C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2212yes—
36C53×#399C418yescomplemented
37C2×#95×C53C2211yescharacteristic
38C23×Dic53—22no—
39C23×Dic53—22no—
40C23×Dic53—42no—
41C23×Dic53—42no—
42C23×Dic53Q814yes—
43C23×Dic53C2312yes—
44C23×Dic53C2312yes—
45C23×Dic53C2311yescharacteristic
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2311yescharacteristic
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—24no—
51C2×C4⋊Dic53—42no—
52C2×C4⋊Dic53—42no—
53C2×C4⋊Dic53—42no—
54C2×C22⋊Dic53—22no—
55C2×C22⋊Dic53—24no—
56C2×C22⋊Dic53—22no—
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53—42no—
59C2×C22⋊Dic53—42no—
60C2×C4×Dic53—22no—
61C2×C4×Dic53—42no—
62C2×C4×Dic53—42no—
63C2×C4×Dic53D414yes—
64#13133124—22no—
65#13133124—22no—
66#13133124—24no—
67#13133124—22no—
68#13133124—24no—
69#13133124—24no—
70#13133124—42no—
71#13133124—42no—
72#13133124—42no—
73#13133124—42no—
74#13133124—42no—
75C25×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2312yes—
78C23×C4×C53C2312yes—
79C23×C4×C53C2311yescharacteristic
80C2×C4⋊C4×C53—22no—
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53—42no—
83C2×C4⋊C4×C53—42no—
84C2×C4⋊C4×C53C2×C414yescomplemented
85C2×C4⋊C4×C53C2×C418yescomplemented
86C2×C22⋊C4×C53—22no—
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53—42no—
89C2×C22⋊C4×C53—42no—
90C2×C22⋊C4×C53C2×C414yes—
91C2×C22⋊C4×C53C2×C418yes—
92#95×C53C2×C414yescomplemented
93C24×C53—21no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53—21no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53C22×C412yes—
102C24×C53C2×Q812yes—
103C24×C53#4112yes—
104C24×C53C2411yescharacteristic
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53C22×C414yes—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C22×C414yes—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C2×D412yes—
128C22×C4×C53#4112yes—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—28no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—48no—
159C22×Dic53—44no—
160C22×Dic53—44no—
161C22×Dic53—48no—
162C22×Dic53C2×D412yes—
163C22×Dic53C2×D412yes—
164C22×Dic53C2×Q812yes—
165C22×Dic53C2×Q812yes—
166C22×Dic53#4112yes—
167C22×Dic53#4112yes—
168C4⋊C4×C53—28no—
169C4⋊C4×C53—48no—
170C22⋊C4×C53—28no—
171C22⋊C4×C53—48no—
172C22⋊Dic53—28no—
173C22⋊Dic53—48no—
174C4⋊Dic53—48no—
175C4⋊Dic53—44no—
176C4×Dic53—44no—
177C4×Dic53—44no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—24no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53C4×Q814yes—
201C23×C53C23×C411yescharacteristic
202C23×C53C22×D411yescharacteristic
203C23×C53C22×Q811yescharacteristic
204C23×C53C2×#4111yescharacteristic
205C23×C53C2×#4111yescharacteristic
206C23×C53C2×#4111yescharacteristic
207C23×C53C2×#4111yescharacteristic
208C23×C53H(5,𝔽2)12yes—
209C23×C53H(5,𝔽2)12yes—
210C23×C53H(5,𝔽2)12yes—
211C23×C532−1+412yes—
212C2×C4×C53—24no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—24no—
216C2×C4×C53—22no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—42no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53C4×D414yes—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—416no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53C22⋊Q814yes—
271#14317—531nomaximal, Sylow
272C22×C53—28no—
273C22×C53—24no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—44no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—44no—
282C22×C53—44no—
283C22×C53C2×C4×D411yescharacteristic
284C22×C53C2×C4×Q811yescharacteristic
285C22×C53C4×#4111yescharacteristic
286C22×C53#51812yes—
287C22×C53#52012yes—
288C22×C53C2×C22⋊Q811yescharacteristic
289C22×C53#52511yescharacteristic
290C22×C53#52911yescharacteristic
291C22×C53#53311yescharacteristic
292C22×C53#53411yescharacteristic
293C22×C53#53511yescharacteristic
294C22×C53#53511yescharacteristic
295C22×C53#53611yescharacteristic
296C22×C53#53812yes—
297C22×C53#53812yes—
298C22×C53#53912yes—
299C22×C53#53912yes—
300C22×C53#53912yes—
301C22×C53#54012yes—
302C22×C53#54012yes—
303C22×C53#54212yes—
304C22×C53#54312yes—
305C22×C53#54412yes—
306C4×C53—24no—
307C4×C53—42no—
308C4×C53—44no—
309C4×C53—44no—
310Dic53—24no—
311Dic53—42no—
312Dic53—42no—
313Dic53—42no—
314Dic53—82no—
315Dic53—82no—
316C2×#399D5311yescharacteristic
317C2×#389—531no—
318C2×C4×C22⋊C4—531no—
319#1445—531no—
320#1442—531no—
321#1441—532no—
322#1439—532no—
323#1439—531no—
324#1436—532no—
325#1434—531no—
326C22⋊#95—532no—
327C2×C53—44no—
328C2×C53C4×C22⋊Q811yescharacteristic
329C2×C53#232211yescharacteristic
330C2×C53#232811yescharacteristic
331C2×C53#232912yes—
332C2×C53#233512yes—
333C2×C53#262611yescharacteristic
334C2×C53#262811yescharacteristic
335C2×C53#263311yescharacteristic
336C2×C53#263511yescharacteristic
337C2×C53#264912yes—
338C2×C53#266712yes—
339C24×C4—531no—
340C22×C4⋊C4—532no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4C2×D5311yescharacteristic
343C22×C4⋊C4C2×D5312yes—
344C22×C22⋊C4—531no—
345C22×C22⋊C4—532no—
346C22×C22⋊C4C2×D5311yescharacteristic
347C22×C22⋊C4C2×D5312yes—
348C22×C42—531no—
349C22×C42—531no—
350#399Dic5318yescomplemented
351#389—1064no—
352C4×C22⋊C4—1064no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95—531no—
356C2×#95—532no—
357C2×#95—531no—
358C2×#95—531no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95C2×D5311yescharacteristic
365C53#1431711yescharacteristic, complemented, Sylow
366C25C22×D5311yescharacteristic
367C23×C4—531no—
368C23×C4—531no—
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—534no—
372C23×C4—532no—
373C23×C4—532no—
374C23×C4—1062no—
375C23×C4—1062no—
376C23×C4—2122no—
377C23×C4—2122no—
378C23×C4C22×D5312yes—
379C23×C4C22×D5311yescharacteristic
380C23×C4C22×D5312yes—
381C23×C4C22×D5311yescharacteristic
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—42no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—2122no—
389C2×C4⋊C4—2122no—
390C2×C4⋊C4—2122no—
391C2×C4⋊C4C2×Dic5314yescomplemented
392C2×C4⋊C4C2×Dic5318yescomplemented
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—2122no—
402C2×C22⋊C4—2122no—
403C2×C22⋊C4C2×Dic5314yes—
404C2×C22⋊C4C2×Dic5318yes—
405C2×C42—534no—
406C2×C42—1062no—
407C2×C42—2122no—
408C2×C42—2122no—
409#95—1064no—
410#95—1064no—
411#95—1062no—
412#95—1064no—
413#95—1062no—
414#95—1062no—
415#95—2122no—
416#95—2122no—
417#95—2122no—
418#95—2122no—
419#95—2122no—
420#95C2×Dic5314yescomplemented
421C24—21no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24Q8×D5312yes—
430C24#1171994912yes—
431C24C22×Dic5312yes—
432C24C23×D5311yescharacteristic
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—532no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—1068no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—2128no—
487C22×C4—2124no—
488C22×C4—2124no—
489C22×C4—2128no—
490C22×C4D4×D5312yes—
491C22×C4#1171994712yes—
492C22×C4C22×Dic5314yes—
493C22×C4C22×Dic5312yes—
494C22×C4C22×Dic5314yes—
495C22×C4C22×Dic5312yes—
496C4⋊C4—28no—
497C4⋊C4—48no—
498C4⋊C4—2124no—
499C4⋊C4—2128no—
500C22⋊C4—28no—
501C22⋊C4—48no—
502C22⋊C4—1068no—
503C22⋊C4—2128no—
504C42—2124no—
505C42—2124no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23Q8×Dic5314yes—
529C23C23×Dic5311yescharacteristic
530C23C2×D4×D5311yescharacteristic
531C23C2×Q8×D5311yescharacteristic
532C23#41×D5311yescharacteristic
533C23#41×D5311yescharacteristic
534C23C2×#1171994711yescharacteristic
535C23C2×#1171994911yescharacteristic
536C23#1313326212yes—
537C23#1313326212yes—
538C23#1313326312yes—
539C23#1313326412yes—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—534no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1064no—
567C2×C4—1064no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—21216no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4D4×Dic5314yes—
599C22—24no—
600C22—28no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22C2×D4×Dic5311yescharacteristic
611C22C2×Q8×Dic5311yescharacteristic
612C22#41×Dic5311yescharacteristic
613C22#1681109812yes—
614C22#1681110612yes—
615C22C22⋊Q8×D5311yescharacteristic
616C22#1681113611yescharacteristic
617C22#1681115911yescharacteristic
618C22#1681117511yescharacteristic
619C22#1681118411yescharacteristic
620C22#1681118611yescharacteristic
621C22#1681118611yescharacteristic
622C22#1681119511yescharacteristic
623C22#1681120912yes—
624C22#1681121312yes—
625C22#1681121612yes—
626C22#1681121712yes—
627C22#1681122212yes—
628C22#1681122512yes—
629C22#1681122712yes—
630C22#1681124112yes—
631C22#1681124812yes—
632C22#1681125212yes—
633C4—24no—
634C4—44no—
635C4—42no—
636C4—44no—
637C4—1064no—
638C4—4242no—
639C4—4242no—
640C4—2122no—
641C4—2122no—
642C4—2122no—
643C2—44no—
644C2C22⋊Q8×Dic5311yescharacteristic
645C2#2027738711yescharacteristic
646C2#2027744111yescharacteristic
647C2#2027745012yes—
648C2#2027750912yes—
649C2#2028052111yescharacteristic
650C2#2028054111yescharacteristic
651C2#2028058911yescharacteristic
652C2#2028060511yescharacteristic
653C2#2028072112yes—
654C2#2028090812yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#399C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×#399D53
hypocenterC53#14317
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C2×#16810522C2semidirect
3. #20272983C2semidirect
4. #20272972C2semidirect
5. #20272968C2semidirect
8. #20272947C2semidirect
9. #20272940C2semidirect
36. C53×#399C4semidirect
84. C2×C4⋊C4×C53C2×C4semidirect
85. C2×C4⋊C4×C53C2×C4semidirect
92. #95×C53C2×C4semidirect
350. #399Dic53semidirect
365. C53#14317semidirect
391. C2×C4⋊C4C2×Dic53semidirect
392. C2×C4⋊C4C2×Dic53semidirect
420. #95C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105222
C2×C22⋊C4×Dic532
#202729832
#202729722
#202729682
#202729602
#202729472
#202729402
#202729372
C2×C53×#3992
#1431753