Finite Group Databasebeta

← #27401024

Subgroups — #27401024

Order 13568.

1212 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810673C211yesmaximal, characteristic, complemented
2C2×#16810587C211yesmaximal, characteristic
3C2×C4×C22⋊Dic53C211yesmaximal, characteristic
4#20275325C212yesmaximal, complemented
5#20275324C212yesmaximal, complemented
6#20274897C212yesmaximal, complemented
7#20274781C211yesmaximal, characteristic, complemented
8#20274693C211yesmaximal, characteristic, complemented
9#20274501C212yesmaximal, complemented
10#20274479C211yesmaximal, characteristic, complemented
11C53×#1779C211yesmaximal, characteristic
12C22×#11719917C2212yes—
13C22×C22⋊Dic53C411yescharacteristic
14C22×C22⋊Dic53C411yescharacteristic
15C22×C4×Dic53C2211yescharacteristic
16C22×C4×Dic53C2211yescharacteristic
17#16810673—24no—
18C2×#13133146C2212yescomplemented
19C2×#13133146C2212yescomplemented
20#16810587—24no—
21C2×C22⋊#88322C2211yescharacteristic
22C2×C22⋊#88322C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133127C2212yescomplemented
25C2×C8×Dic53C2212yescomplemented
26C4×C22⋊Dic53C411yescharacteristic
27C4×C22⋊Dic53C411yescharacteristic
28C4×C22⋊Dic53C411yescharacteristic
29C4×C22⋊Dic53C411yescharacteristic
30C4×C22⋊Dic53C2211yescharacteristic
31C4×C22⋊Dic53C2211yescharacteristic
32C4×C22⋊Dic53C2211yescharacteristic
33C4×C22⋊Dic53C2211yescharacteristic
34C2×#13133124C411yescharacteristic
35C2×#13133124C411yescharacteristic
36#16810377C2212yescomplemented
37#16810377C2212yescomplemented
38#16810377C2212yescomplemented
39#16810377C2212yescomplemented
40C24×C4×C53C2211yescharacteristic
41C2×C22⋊C8×C53C2211yescharacteristic
42C2×C22⋊C8×C53C2211yescharacteristic
43C2×C22⋊C8×C53C2212yes—
44C2×C22⋊C8×C53C2212yes—
45C23×Dic53C2×C411yescharacteristic
46C23×Dic53C2×C411yescharacteristic
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C2×#11719917—24no—
50C2×#11719917—24no—
51C2×#11719917—42no—
52C22×#88322C2312yes—
53C22×#88322C2312yes—
54C2×C22⋊Dic53—24no—
55C2×C22⋊Dic53—24no—
56C2×C22⋊Dic53C2×C411yescharacteristic
57C2×C22⋊Dic53C2×C411yescharacteristic
58C2×C22⋊Dic53C2×C411yescharacteristic
59C2×C22⋊Dic53C2×C411yescharacteristic
60C2×C22⋊Dic53C2×C411yescharacteristic
61C2×C22⋊Dic53C2×C411yescharacteristic
62C2×C22⋊Dic53C2×C411yescharacteristic
63C2×C22⋊Dic53C2×C411yescharacteristic
64C2×C4×Dic53—22no—
65C2×C4×Dic53—22no—
66C2×C4×Dic53—42no—
67C2×C4×Dic53C2×C411yescharacteristic
68C2×C4×Dic53C2×C411yescharacteristic
69C2×C4×Dic53C2×C411yescharacteristic
70C2×C4×Dic53C2×C411yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72C2×C4×Dic53C2311yescharacteristic
73C2×C4×Dic53C2311yescharacteristic
74C2×C4×Dic53C2311yescharacteristic
75#13133146—24no—
76#13133146—42no—
77C22⋊#88322—24no—
78C22⋊#88322—22no—
79C22⋊#88322D414yes—
80#13133127—42no—
81C8×Dic53—24no—
82#13133124C2×C411yescharacteristic
83#13133124C2×C411yescharacteristic
84#13133124C2×C411yescharacteristic
85#13133124C2×C411yescharacteristic
86#13133124C2×C411yescharacteristic
87#13133124C2×C411yescharacteristic
88#13133124C2×C411yescharacteristic
89#13133124C2×C411yescharacteristic
90C25×C53C2×C411yescharacteristic
91C23×C4×C53—22no—
92C23×C4×C53—24no—
93C23×C4×C53—22no—
94C23×C4×C53—42no—
95C23×C4×C53C2×C411yescharacteristic
96C23×C4×C53C2×C411yescharacteristic
97C23×C4×C53C2×C411yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C23×C4×C53C2311yescharacteristic
100C23×C4×C53C2311yescharacteristic
101C22×C8×C53C2312yes—
102C22×C8×C53C2312yes—
103C22⋊C8×C53—22no—
104C22⋊C8×C53—24no—
105C22⋊C8×C53—22no—
106C22⋊C8×C53—42no—
107C24×C53—22no—
108C24×C53—22no—
109C24×C53—22no—
110C24×C53—22no—
111C24×C53—22no—
112C24×C53—41no—
113C24×C53—41no—
114C24×C53C22×C411yescharacteristic
115C24×C53C22×C411yescharacteristic
116C24×C53C22×C411yescharacteristic
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—22no—
132C22×C4×C53—41no—
133C22×C4×C53—41no—
134C22×C4×C53—41no—
135C22×C4×C53—48no—
136C22×C4×C53—48no—
137C22×C4×C53C22×C411yescharacteristic
138C22×C4×C53C22×C411yescharacteristic
139C22×C4×C53C22×C411yescharacteristic
140C22×C4×C53C2×D412yes—
141C22×C4×C53#4114yes—
142C22×C4×C53#4112yes—
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—41no—
152C22×Dic53—41no—
153C22×Dic53—41no—
154C22×Dic53—41no—
155C22×Dic53M1614yescomplemented
156C22×Dic53C22×C411yescharacteristic
157C22×Dic53C22×C411yescharacteristic
158C22×Dic53C22×C411yescharacteristic
159C22×Dic53C22×C411yescharacteristic
160C22×Dic53C22×C411yescharacteristic
161C22×Dic53C22×C411yescharacteristic
162C22×Dic53C22×C411yescharacteristic
163C22×Dic53C22×C411yescharacteristic
164C2×C8×C53—22no—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C22⋊Dic53—22no—
171C22⋊Dic53—22no—
172C22⋊Dic53—22no—
173C22⋊Dic53—22no—
174C22⋊Dic53—22no—
175C22⋊Dic53—22no—
176C22⋊Dic53—22no—
177C22⋊Dic53M1614yescomplemented
178C4×Dic53—22no—
179C4×Dic53—22no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—21no—
183C4×Dic53—21no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—41no—
187C4×Dic53—41no—
188#11719917—48no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322C2×D414yes—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—28no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—44no—
213C23×C53—44no—
214C23×C53—44no—
215C23×C53—44no—
216C23×C53—44no—
217C23×C53—44no—
218C23×C53#11712yes—
219C23×C53C4×D412yes—
220C23×C53C2×M1612yes—
221C23×C53C2×M1612yes—
222C23×C53#13112yes—
223C23×C53C23×C411yescharacteristic
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—44no—
236C2×C4×C53—44no—
237C2×C4×C53—44no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53#11712yes—
241C2×C4×C53C4×D412yes—
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C2×#4112yes—
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53H(5,𝔽2)12yes—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—28no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—44no—
268C2×Dic53—44no—
269C2×Dic53—44no—
270C2×Dic53C2×M1612yes—
271C2×Dic53C2×M1612yes—
272C2×Dic53C2×M1612yes—
273C2×Dic53C2×M1612yes—
274C8×C53—42no—
275C8×C53—42no—
276#88322—24no—
277#88322—42no—
278#22553—531nomaximal, Sylow
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—416no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53#43514yes—
301C22×C53C2×#11711yescharacteristic
302C22×C53C2×C4×D411yescharacteristic
303C22×C53#51811yescharacteristic
304C22×C53C22×M1611yescharacteristic
305C22×C53C22×M1611yescharacteristic
306C22×C53C2×#13111yescharacteristic
307C22×C53#56811yescharacteristic
308C4×C53—22no—
309C4×C53—22no—
310C4×C53—41no—
311C4×C53#54612yes—
312C4×C53#55112yes—
313Dic53—22no—
314Dic53—22no—
315Dic53—22no—
316Dic53—22no—
317Dic53—41no—
318Dic53—41no—
319Dic53—41no—
320Dic53—41no—
321C2×#431—531no—
322C2×#407—531no—
323C2×C4×C22⋊C4—531no—
324#1924—532no—
325#1923—532no—
326#1851—532no—
327#1817—531no—
328#1797—531no—
329#1779D5311yescharacteristic
330#1753—532no—
331#1749—531no—
332C2×C53—24no—
333C2×C53—42no—
334C2×C53—42no—
335C2×C53#234211yescharacteristic
336C2×C53#292011yescharacteristic
337C2×C53#292411yescharacteristic
338C2×C53C2×#43511yescharacteristic
339C2×C53D4×M1611yescharacteristic
340C2×C53#297411yescharacteristic
341C2×C53#297511yescharacteristic
342C24×C4C2×D5311yescharacteristic
343C22×M16—532no—
344C22×C22⋊C4—531no—
345C22×C22⋊C4—531no—
346C22×C42—531no—
347C22×C42—531no—
348#431—1064no—
349C2×C4⋊C8—532no—
350C2×C4⋊C8—532no—
351#407—1064no—
352C2×C22⋊C8—531no—
353C2×C22⋊C8—531no—
354C2×C22⋊C8—532no—
355C2×C22⋊C8C2×D5311yescharacteristic
356C2×C22⋊C8C2×D5311yescharacteristic
357C2×C22⋊C8C2×D5312yes—
358C2×C22⋊C8C2×D5312yes—
359C2×#97—532no—
360C2×C4×C8—532no—
361C4×C22⋊C4—531no—
362C4×C22⋊C4—531no—
363C4×C22⋊C4—531no—
364C4×C22⋊C4—531no—
365C4×C22⋊C4—531no—
366C4×C22⋊C4—531no—
367C4×C22⋊C4—531no—
368C4×C22⋊C4—531no—
369C2×#95—531no—
370C2×#95—531no—
371#336—532no—
372#336—532no—
373#336—532no—
374#336—532no—
375C53#2255311yescharacteristic, complemented, Sylow
376C25C4×D5311yescharacteristic
377C23×C4—22no—
378C23×C4—24no—
379C23×C4—22no—
380C23×C4—42no—
381C23×C4—531no—
382C23×C4—531no—
383C23×C4—531no—
384C23×C4—531no—
385C23×C4C4×D5311yescharacteristic
386C23×C4C4×D5311yescharacteristic
387C23×C4C4×D5311yescharacteristic
388C23×C4C22×D5311yescharacteristic
389C23×C4C22×D5311yescharacteristic
390C23×C4C22×D5311yescharacteristic
391C2×M16—1064no—
392C2×M16—1064no—
393C2×M16—2122no—
394C22×C8—532no—
395C22×C8—532no—
396C22×C8C22×D5312yes—
397C22×C8C22×D5312yes—
398C2×C22⋊C4—531no—
399C2×C22⋊C4—531no—
400C2×C22⋊C4—531no—
401C2×C22⋊C4—531no—
402C2×C22⋊C4—531no—
403C2×C22⋊C4—531no—
404C2×C22⋊C4—531no—
405C2×C22⋊C4—531no—
406C2×C22⋊C4—1064no—
407C2×C22⋊C4—1064no—
408C2×C42—531no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—1062no—
417C2×C42—1062no—
418C2×C42—2122no—
419C4⋊C8—1064no—
420C4⋊C8—2122no—
421C22⋊C8—22no—
422C22⋊C8—24no—
423C22⋊C8—22no—
424C22⋊C8—42no—
425C22⋊C8—534no—
426C22⋊C8—1064no—
427C22⋊C8—1062no—
428#97—2122no—
429C4×C8—1064no—
430#95—531no—
431#95—531no—
432#95—531no—
433#95—531no—
434#95—531no—
435#95—531no—
436#95—531no—
437#95—531no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—41no—
444C24—41no—
445C24C2×C4×D5311yescharacteristic
446C24C2×C4×D5311yescharacteristic
447C24C2×C4×D5311yescharacteristic
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—534no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—2121no—
485C22×C4—2121no—
486C22×C4—2121no—
487C22×C4—2121no—
488C22×C4C2×C4×D5311yescharacteristic
489C22×C4C2×C4×D5311yescharacteristic
490C22×C4C2×C4×D5311yescharacteristic
491C22×C4D4×D5312yes—
492C22×C4#1171994714yes—
493C22×C4#1171994712yes—
494C22×C4C23×D5311yescharacteristic
495M16—2128no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—534no—
503C2×C8—1062no—
504C2×C8—1062no—
505C2×C8—1062no—
506C2×C8—1062no—
507C2×C8—1062no—
508C22⋊C4—534no—
509C22⋊C4—1062no—
510C22⋊C4—1062no—
511C22⋊C4—1062no—
512C22⋊C4—1062no—
513C22⋊C4—1062no—
514C22⋊C4—1062no—
515C22⋊C4—1062no—
516C42—1062no—
517C42—1062no—
518C42—1061no—
519C42—1061no—
520C42—1061no—
521C42—1061no—
522C42—1062no—
523C42—1062no—
524C42—2121no—
525C42—2121no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—28no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23#1313317312yes—
550C23#1313317712yes—
551C23M16×D5312yes—
552C23M16×D5312yes—
553C23#1313322312yes—
554C23C22×C4×D5311yescharacteristic
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—532no—
572C2×C4—532no—
573C2×C4—532no—
574C2×C4—532no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1068no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—2124no—
597C2×C4—2124no—
598C2×C4—2124no—
599C2×C4#1313317312yes—
600C2×C4#1313317712yes—
601C2×C4C2×D4×D5311yescharacteristic
602C2×C4C2×#1171994711yescharacteristic
603C2×C4C2×#1171994712yes—
604C2×C4#1313326212yes—
605C8—42no—
606C8—42no—
607C8—1064no—
608C8—2122no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—416no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22#1681069714yes—
631C22C2×#1313317311yescharacteristic
632C22C2×#1313317711yescharacteristic
633C22#1681109611yescharacteristic
634C22C2×M16×D5311yescharacteristic
635C22C2×M16×D5311yescharacteristic
636C22C2×#1313322311yescharacteristic
637C22#1681138911yescharacteristic
638C4—22no—
639C4—22no—
640C4—41no—
641C4—1062no—
642C4—1062no—
643C4—1062no—
644C4—1062no—
645C4—2121no—
646C4—2121no—
647C4—2121no—
648C4—2121no—
649C4#1681126012yes—
650C4#1681130412yes—
651C2—24no—
652C2—42no—
653C2—42no—
654C2#2027756611yescharacteristic
655C2#2028334411yescharacteristic
656C2#2028336911yescharacteristic
657C2C2×#1681069711yescharacteristic
658C2#2028344711yescharacteristic
659C2#2028374611yescharacteristic
660C2#2028375411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D53
commutator, socleC23×C53C23×C4
FittingC53×#1779C2
prime radicalC25×C53C2×C4
cosocleC22×C4×C53C24
hypercenter#1779D53
hypocenterC53#22553
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
1. C2×#16810673C2semidirect
4. #20275325C2semidirect
5. #20275324C2semidirect
6. #20274897C2semidirect
7. #20274781C2semidirect
8. #20274693C2semidirect
9. #20274501C2semidirect
10. #20274479C2semidirect
18. C2×#13133146C22semidirect
19. C2×#13133146C22semidirect
24. C2×#13133127C22semidirect
25. C2×C8×Dic53C22semidirect
36. #16810377C22semidirect
37. #16810377C22semidirect
38. #16810377C22semidirect
39. #16810377C22semidirect
155. C22×Dic53M16semidirect
177. C22⋊Dic53M16semidirect
375. C53#22553semidirect

Maximal subgroups

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

subgroupindex
C2×#168106732
C2×#168105872
C2×C4×C22⋊Dic532
#202753252
#202753242
#202748972
#202747812
#202746932
#202745012
#202744792
C53×#17792
#2255353