Finite Group Databasebeta

← #27389079

Subgroups — #27389079

Order 13568.

1186 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811153C211yesmaximal, characteristic
2C2×#16810673C211yesmaximal, characteristic, complemented
3C2×M16×Dic53C211yesmaximal, characteristic, complemented
4#20275649C212yesmaximal, complemented
5#20275507C212yesmaximal, complemented
6#20275013C211yesmaximal, characteristic, complemented
7#20275012C211yesmaximal, characteristic, complemented
8#20275010C211yesmaximal, characteristic, complemented
9#20274483C212yesmaximal, complemented
10#20274480C212yesmaximal, complemented
11C53×#1870C211yesmaximal, characteristic, complemented
12C22×#11719917C2211yescharacteristic
13#16811153—24no—
14#16811153—24no—
15C2×#13133212C411yescharacteristic
16C2×#13133210C411yescharacteristic
17C2×#13133207C411yescharacteristic
18C2×#13133203C411yescharacteristic
19C2×#13133201C411yescharacteristic
20C2×#13133198C411yescharacteristic
21C2×#13133194C411yescharacteristic
22C2×#13133189C411yescharacteristic
23C2×C4×Dic106C2211yescharacteristic
24C2×C4×C22⋊D53C2211yescharacteristic
25C2×C4×C22⋊D53C2211yescharacteristic
26C2×#13133176C2211yescharacteristic
27C2×#13133174C2211yescharacteristic
28C22×C4×Dic53C2211yescharacteristic
29#16810673—24no—
30C2×#13133146C2212yes—
31C2×C22⋊#88322C2212yes—
32C2×C22⋊#88322C2211yescharacteristic
33C2×#13133129C2212yes—
34C2×#13133129C2212yes—
35M16×Dic53—24no—
36C2×#13133127C2212yes—
37C2×C8×Dic53C2212yes—
38C2×C8×Dic53C2212yes—
39#16810378C2212yes—
40#16810377C2212yes—
41#16810377C2212yes—
42C22×M16×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×C22⋊C8×C53C2212yes—
45C2×C22⋊C8×C53C2211yescharacteristic
46C53×#336C2212yes—
47C22×Dic106C2×C411yescharacteristic
48C22×C22⋊D53C2×C411yescharacteristic
49C22×C22⋊D53C2×C411yescharacteristic
50C23×Dic53C2×C411yescharacteristic
51C23×Dic53C2×C411yescharacteristic
52C22×C4×D53C2311yescharacteristic
53C2×#11719917—24no—
54C2×#11719917—22no—
55C22×#88322C2312yes—
56C22×#88322C2312yes—
57#13133212—24no—
58#13133210—24no—
59#13133207—24no—
60#13133203—24no—
61#13133201—24no—
62#13133198—24no—
63#13133194—24no—
64#13133189—24no—
65C4×Dic106—22no—
66C4×Dic106—22no—
67C4×C22⋊D53—22no—
68C4×C22⋊D53—22no—
69C4×C22⋊D53—22no—
70C4×C22⋊D53—22no—
71#13133176—22no—
72#13133176—22no—
73#13133174—22no—
74#13133174—22no—
75C2×C4⋊Dic53C2×C411yescharacteristic
76C2×C4⋊Dic53C2×C411yescharacteristic
77C2×C4⋊Dic53C2×C411yescharacteristic
78C2×#11719919C2×C411yescharacteristic
79C2×#11719919C2×C411yescharacteristic
80C2×#11719919C2×C411yescharacteristic
81C2×#11719919C2×C411yescharacteristic
82C2×#11719919C2×C411yescharacteristic
83C2×#11719919C2×C411yescharacteristic
84C2×C22⋊Dic53C2×C411yescharacteristic
85C2×C22⋊Dic53C2×C411yescharacteristic
86C2×C22⋊Dic53C2×C411yescharacteristic
87C2×C22⋊Dic53C2×C411yescharacteristic
88C2×#11719921C2×C411yescharacteristic
89C2×#11719921C2×C411yescharacteristic
90C2×#11719921C2×C411yescharacteristic
91C2×#11719921C2×C411yescharacteristic
92C2×C4×Dic53C2×C411yescharacteristic
93C2×C4×Dic53C2×C411yescharacteristic
94C2×C4×Dic53D414yes—
95C2×C4×Dic53D414yes—
96C2×C4×Dic53C2311yescharacteristic
97C2×C4×Dic53C2311yescharacteristic
98C2×C4×Dic53C2311yescharacteristic
99#13133146—24no—
100C22⋊#88322—24no—
101C22⋊#88322—22no—
102#13133129—24no—
103#13133129—24no—
104#13133127—24no—
105C8×Dic53—24no—
106C8×Dic53—24no—
107C22×D4×C53C2×C411yescharacteristic
108C23×C4×C53C2311yescharacteristic
109C23×C4×C53C2311yescharacteristic
110C2×M16×C53—24no—
111C2×M16×C53—22no—
112C22×C8×C53C2312yes—
113C22×C8×C53C2312yes—
114C4×D4×C53—22no—
115C4×D4×C53—22no—
116C2×C4⋊C4×C53C2×C411yescharacteristic
117C2×C22⋊C4×C53C2×C411yescharacteristic
118C2×C22⋊C4×C53C2×C411yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C22⋊C8×C53—24no—
121C22⋊C8×C53—22no—
122C24×C53C22×C411yescharacteristic
123C24×C53C22×C411yescharacteristic
124C23×D53C22×C411yescharacteristic
125C2×D4×C53—22no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53C22×C411yescharacteristic
135C22×C4×C53C22×C411yescharacteristic
136C22×C4×C53C22×C411yescharacteristic
137C22×C4×C53C22×C411yescharacteristic
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2411yescharacteristic
141C2×C22⋊D53—22no—
142C2×C22⋊D53—22no—
143C2×C22⋊D53—24no—
144C2×C22⋊D53—41no—
145C2×C22⋊D53—41no—
146C22×Dic53—24no—
147C22×Dic53—41no—
148C22×Dic53—41no—
149C22×Dic53C22⋊C414yes—
150C22×Dic53C22⋊C414yes—
151C22×Dic53C22×C411yescharacteristic
152C22×Dic53C22×C411yescharacteristic
153C22×Dic53C22×C411yescharacteristic
154C22×Dic53C22×C411yescharacteristic
155C22×Dic53C22×C411yescharacteristic
156C22×Dic53C22×C411yescharacteristic
157C22×Dic53C22×C411yescharacteristic
158C2×C4×D53—21no—
159C2×C4×D53—21no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×Dic106—22no—
163C2×Dic106—24no—
164M16×C53—28no—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C2×C8×C53#4114yes—
171C4⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C42×C53—21no—
175C42×C53—21no—
176C22⋊Dic53—22no—
177C22⋊Dic53—22no—
178C22⋊Dic53—22no—
179C22⋊Dic53—22no—
180#11719921—22no—
181#11719921—22no—
182#11719921—22no—
183#11719921—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53—22no—
187#11719919—22no—
188#11719919—22no—
189#11719919—22no—
190#11719919—22no—
191#11719919—22no—
192#11719919—22no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53C22⋊C414yes—
196C4×Dic53C22⋊C414yes—
197C4×Dic53C2×D412yes—
198C4×Dic53C2×D412yes—
199C4×Dic53C2×D412yes—
200C4×Dic53C2×D412yes—
201#11719917—28no—
202C2×#88322—22no—
203C2×#88322—22no—
204C2×#88322—22no—
205C2×#88322—22no—
206C2×#88322—22no—
207C2×#88322#4114yes—
208C23×C53—22no—
209C23×C53—22no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—22no—
214C23×C53—22no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53C2×C22⋊C412yes—
218C23×C53C23×C411yescharacteristic
219C22×D53—21no—
220C22×D53—22no—
221C22×D53—22no—
222C22×D53—21no—
223C22×D53—21no—
224D4×C53—44no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—42no—
239C2×C4×C53C2×C22⋊C412yes—
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C2×#4112yes—
243C2×C4×C53C2×#4112yes—
244C22⋊D53—44no—
245C22⋊D53—44no—
246C2×Dic53—22no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—28no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—44no—
260C2×Dic53C2×C22⋊C412yes—
261C2×Dic53C2×C22⋊C412yes—
262C2×Dic53C2×C22⋊C412yes—
263C2×Dic53C2×C22⋊C412yes—
264C2×Dic53#13114yes—
265C4×D53—41no—
266C4×D53—41no—
267C4×D53—42no—
268Dic106—44no—
269C8×C53—24no—
270C8×C53—42no—
271#88322—24no—
272#88322—42no—
273#21378—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—41no—
284C22×C53C22×C22⋊C411yescharacteristic
285C22×C53C4×#4112yes—
286C22×C53C2×#13112yes—
287C22×C53#56811yescharacteristic
288C22×C53#56811yescharacteristic
289C2×D53—41no—
290C2×D53—41no—
291C2×D53—42no—
292C2×D53—42no—
293C2×D53—41no—
294C4×C53—22no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53#52511yescharacteristic
299C4×C53#52511yescharacteristic
300C4×C53#53212yes—
301Dic53—22no—
302Dic53—22no—
303Dic53—22no—
304Dic53—41no—
305Dic53—41no—
306Dic53—41no—
307Dic53#40814yes—
308C2×#529—531no—
309C2×#431—531no—
310C2×C4×M16—531no—
311#1984—532no—
312#1954—532no—
313#1870—531no—
314#1870—531no—
315#1870—531no—
316#1870D5311yescharacteristic, complemented
317#1750—532no—
318#1750—532no—
319C2×C53—42no—
320C2×C53—81no—
321C2×C53#229711yescharacteristic
322C2×C53C2×#40811yescharacteristic
323C2×C53#287911yescharacteristic
324C2×C53#296912yes—
325C2×C53#296912yes—
326D53—81no—
327C22×M16—531no—
328C22×M16C2×D5311yescharacteristic
329#529—1064no—
330#529—1064no—
331C2×#126—531no—
332C2×#126—531no—
333C2×#125—531no—
334C2×#124—531no—
335C2×#123—531no—
336C2×#123—531no—
337C2×C22⋊Q8—531no—
338C2×#121—531no—
339C2×C4×Q8—531no—
340C2×C4×D4—531no—
341C2×C4×D4—531no—
342C2×C4×D4C2×D5311yescharacteristic
343C2×#117—531no—
344C2×#117—531no—
345C22×C42—531no—
346#431—1064no—
347C2×C4⋊C8—532no—
348C2×C22⋊C8—531no—
349C2×C22⋊C8—532no—
350C2×C22⋊C8—532no—
351C2×C22⋊C8—532no—
352C2×C22⋊C8C2×D5311yescharacteristic
353C2×C22⋊C8C2×D5312yes—
354C4×M16—1064no—
355C2×#97—532no—
356C2×C4×C8—532no—
357C2×C4×C8—532no—
358#336—532no—
359#336—532no—
360#336—532no—
361#336C2×D5312yes—
362C53#2137811yescharacteristic, complemented, Sylow
363C22×Q8—531no—
364C22×D4—531no—
365C22×D4—531no—
366C22×D4C4×D5311yescharacteristic
367C23×C4—531no—
368C23×C4—531no—
369C23×C4—531no—
370C23×C4C22×D5311yescharacteristic
371C23×C4C22×D5311yescharacteristic
372C2×M16—22no—
373C2×M16—24no—
374C2×M16—1064no—
375C2×M16—1062no—
376C22×C8—532no—
377C22×C8—532no—
378C22×C8C22×D5312yes—
379C22×C8C22×D5312yes—
380#126—1064no—
381#126—1064no—
382#125—1064no—
383#124—1064no—
384#123—1064no—
385#123—1064no—
386C22⋊Q8—1064no—
387#121—1064no—
388C4×Q8—1062no—
389C4×Q8—1062no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—1062no—
393C4×D4—1062no—
394C4×D4—1062no—
395C4×D4—1062no—
396#117—1062no—
397#117—1062no—
398#117—1062no—
399#117—1062no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4C4×D5311yescharacteristic
410C2×C22⋊C4—531no—
411C2×C22⋊C4—531no—
412C2×C22⋊C4—531no—
413C2×C22⋊C4—531no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—531no—
417C2×C22⋊C4—531no—
418C2×C22⋊C4C4×D5311yescharacteristic
419C2×C22⋊C4C4×D5311yescharacteristic
420C2×C42—531no—
421C2×C42—531no—
422C2×C42—534no—
423C2×C42—534no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42C22×D5311yescharacteristic
428C4⋊C8—1064no—
429C22⋊C8—22no—
430C22⋊C8—24no—
431C22⋊C8—1064no—
432C22⋊C8—1064no—
433C22⋊C8—1062no—
434C22⋊C8—1064no—
435#97—1064no—
436C4×C8—1064no—
437C4×C8—1064no—
438C24—531no—
439C24C2×C4×D5311yescharacteristic
440C24C2×C4×D5311yescharacteristic
441C2×Q8—1062no—
442C2×Q8—1064no—
443C2×D4—22no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—1062no—
447C2×D4—1064no—
448C2×D4—1062no—
449C2×D4—2121no—
450C2×D4—2121no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—531no—
458C22×C4—534no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—534no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—1064no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—2121no—
472C22×C4—2121no—
473C22×C4C2×C4×D5311yescharacteristic
474C22×C4C2×C4×D5311yescharacteristic
475C22×C4C2×C4×D5311yescharacteristic
476C22×C4C2×C4×D5311yescharacteristic
477C22×C4D4×D5312yes—
478C22×C4D4×D5312yes—
479C22×C4C23×D5311yescharacteristic
480M16—28no—
481M16—1068no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—534no—
488C2×C8—1062no—
489C2×C8—1062no—
490C2×C8—1062no—
491C2×C8—1062no—
492C2×C8—1062no—
493C2×C8#1171994714yes—
494C4⋊C4—22no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—1062no—
498C4⋊C4—1062no—
499C4⋊C4—1062no—
500C4⋊C4—1062no—
501C4⋊C4—1062no—
502C4⋊C4—1062no—
503C4⋊C4—1062no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—1062no—
507C22⋊C4—1062no—
508C22⋊C4—1062no—
509C22⋊C4—1062no—
510C22⋊C4—1062no—
511C22⋊C4—1062no—
512C22⋊C4—1062no—
513C22⋊C4—1062no—
514C42—21no—
515C42—21no—
516C42—534no—
517C42—534no—
518C42—532no—
519C42—532no—
520C42—532no—
521C42—532no—
522C42—1061no—
523C42—1061no—
524C23—22no—
525C23—22no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—21no—
532C23—21no—
533C23—1061no—
534C23—1062no—
535C23—1062no—
536C23—1061no—
537C23—1061no—
538C23C22⋊C4×D5312yes—
539C23C22×C4×D5311yescharacteristic
540Q8—2124no—
541D4—44no—
542D4—2124no—
543D4—2124no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—532no—
559C2×C4—534no—
560C2×C4—532no—
561C2×C4—532no—
562C2×C4—532no—
563C2×C4—1068no—
564C2×C4—1062no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—2124no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2122no—
580C2×C4C22⋊C4×D5312yes—
581C2×C4C2×D4×D5311yescharacteristic
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4#41×D5312yes—
584C2×C4C2×#1171994712yes—
585C8—24no—
586C8—42no—
587C8—1064no—
588C8—2122no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—42no—
598C22—41no—
599C22—2121no—
600C22—2121no—
601C22—2122no—
602C22—2122no—
603C22—2121no—
604C22C2×C22⋊C4×D5311yescharacteristic
605C22C4×#1171994712yes—
606C22#131×D5312yes—
607C22#1681138911yescharacteristic
608C22#1681138911yescharacteristic
609C4—22no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—534no—
614C4—1062no—
615C4—1062no—
616C4—1062no—
617C4—2121no—
618C4—2121no—
619C4—2121no—
620C4#1681112911yescharacteristic
621C4#1681113111yescharacteristic
622C4#1681116812yes—
623C2—42no—
624C2—81no—
625C2—4241no—
626C2#2027720011yescharacteristic
627C2#408×D5311yescharacteristic
628C2#2028315811yescharacteristic
629C2#2028371212yes—
630C2#2028371412yes—
subgroupG/H
center, FrattiniC22×C4C23×D53
commutator, socleC23×C53C23×C4
FittingC53×#1870C2
prime radicalC2×#13133189C4
cosocleC22×C4×C53C24
hypercenter#1870D53
hypocenterC53#21378
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. C2×#16810673C2semidirect
3. C2×M16×Dic53C2semidirect
4. #20275649C2semidirect
5. #20275507C2semidirect
6. #20275013C2semidirect
7. #20275012C2semidirect
8. #20275010C2semidirect
9. #20274483C2semidirect
10. #20274480C2semidirect
11. C53×#1870C2semidirect
316. #1870D53semidirect
362. C53#21378semidirect

Maximal subgroups

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

subgroupindex
C2×#168111532
C2×#168106732
C2×M16×Dic532
#202756492
#202755072
#202750132
#202750122
#202750102
#202744832
#202744802
C53×#18702
#2137853