Finite Group Databasebeta

← #27393764

Subgroups — #27393764

Order 13568.

1502 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×#16811168C211yesmaximal, characteristic
2C2×#16810673C211yesmaximal, characteristic, complemented
3C2×#16810492C211yesmaximal, characteristic, complemented
4#20275674C212yesmaximal, complemented
5#20275519C212yesmaximal, complemented
6#20275033C211yesmaximal, characteristic, complemented
7#20275032C211yesmaximal, characteristic, complemented
8#20275030C211yesmaximal, characteristic, complemented
9#20274714C212yesmaximal, complemented
10#20274713C212yesmaximal, complemented
11C53×#1874C211yesmaximal, characteristic, complemented
12C22×#11719947—21no—
13#16811168—24no—
14#16811168C418yescomplemented
15C2×#13133216C2211yescharacteristic
16C2×#13133213C2211yescharacteristic
17C2×#13133203—21no—
18C2×#13133191C2211yescharacteristic
19C2×#13133189—21no—
20C2×D4×Dic53C2211yescharacteristic
21C2×#13133179C2211yescharacteristic
22C2×#13133177—21no—
23C22×C4⋊Dic53C2211yescharacteristic
24C22×#11719919—21no—
25C22×#11719919—21no—
26C22×C4×Dic53C2211yescharacteristic
27#16810673—24no—
28C2×#13133146C2212yes—
29C2×#13133139C2212yes—
30C2×#13133138C2212yes—
31C2×#13133137C2212yes—
32C2×C22⋊#88322C2211yescharacteristic
33C2×C8×Dic53C2212yes—
34#16810492—21no—
35#16810492—21no—
36#16810492C2212yes—
37#16810492C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×#121×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C2×#102×C53C2212yes—
44C2×C22⋊C8×C53C2211yescharacteristic
45C53×#340C2212yes—
46C2×#11719947—28no—
47C2×#11719947—44no—
48C2×#11719947—44no—
49C22×Dic106—21no—
50C22×D212C2311yescharacteristic
51C22×C22⋊D53—21no—
52C22×C22⋊D53—21no—
53C22×C22⋊D53—21no—
54C23×Dic53—21no—
55C23×Dic53—21no—
56C23×Dic53D411yescharacteristic
57C23×Dic53D411yescharacteristic
58C22×C4×D53—21no—
59C22×#88322C2312yes—
60#13133216—22no—
61#13133216C2×C414yescomplemented
62#13133213—22no—
63#13133213C2×C414yescomplemented
64#13133203—28no—
65#13133191—22no—
66#13133191C2×C414yes—
67#13133189—28no—
68D4×Dic53—22no—
69D4×Dic53C2×C414yescomplemented
70#13133179—22no—
71#13133179C2×C414yescomplemented
72#13133177—28no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—42no—
75C2×C4⋊Dic53C2312yes—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—22no—
79C2×#11719919—22no—
80C2×#11719919—22no—
81C2×#11719919—22no—
82C2×#11719919—44no—
83C2×#11719919—44no—
84C2×C22⋊Dic53—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×C4×Dic53C2×C414yes—
88C2×C4×Dic53D414yes—
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133146—24no—
94#13133139—42no—
95#13133138—42no—
96#13133137—42no—
97C22⋊#88322—22no—
98C8×Dic53—24no—
99C22×D4×C53—21no—
100C22×D4×C53C2311yescharacteristic
101C23×C4×C53—21no—
102C23×C4×C53C2311yescharacteristic
103C22×C8×C53C2312yes—
104#121×C53—22no—
105#121×C53C2×C414yes—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—42no—
108C2×C4⋊C4×C53C2312yes—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111#102×C53—42no—
112C22⋊C8×C53—22no—
113C24×C53—21no—
114C24×C53C2×D411yescharacteristic
115C23×D53—21no—
116C2×D4×C53—24no—
117C2×D4×C53—21no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53—42no—
121C2×D4×C53—42no—
122C2×D4×C53C22×C412yes—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2411yescharacteristic
134C2×C22⋊D53—24no—
135C2×C22⋊D53—24no—
136C2×C22⋊D53—24no—
137C2×C22⋊D53—42no—
138C2×C22⋊D53—42no—
139C2×C22⋊D53—42no—
140C2×C22⋊D53—42no—
141C2×C22⋊D53—42no—
142C2×C22⋊D53—42no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—21no—
146C22×Dic53—24no—
147C22×Dic53—22no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53C22⋊C414yes—
154C22×Dic53C22⋊C414yes—
155C22×Dic53D814yes—
156C22×Dic53SD1614yes—
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161#11719947—432no—
162C2×D212—22no—
163C2×D212—22no—
164C2×D212—21no—
165C2×D212C22×C412yes—
166C2×C4×D53—24no—
167C2×C4×D53—42no—
168C2×C4×D53—42no—
169C2×Dic106—24no—
170C2×Dic106—42no—
171C2×Dic106—42no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—42no—
179C4⋊C4×C53C22×C412yes—
180C22⋊C4×C53—24no—
181C22⋊Dic53—24no—
182#11719921—24no—
183#11719921—24no—
184C4⋊Dic53—21no—
185C4⋊Dic53—22no—
186C4⋊Dic53—22no—
187C4⋊Dic53—42no—
188C4⋊Dic53C22×C412yes—
189#11719919—24no—
190#11719919—44no—
191#11719919—44no—
192#11719919—44no—
193#11719919—44no—
194C4×Dic53C22⋊C414yes—
195C4×Dic53C22⋊C414yes—
196C4×Dic53C22×C412yes—
197C4×Dic53C22×C412yes—
198C4×Dic53C2×D412yes—
199C4×Dic53C2×D412yes—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—22no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53C2×C22⋊C412yes—
215C23×C53C2×D812yes—
216C23×C53C2×SD1612yes—
217C23×C53C22×D411yescharacteristic
218C22×D53—22no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225D4×C53—24no—
226D4×C53—48no—
227D4×C53—42no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—22no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—84no—
239C2×C4×C53C2×C22⋊C412yes—
240C2×C4×C53C23×C411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53H(5,𝔽2)12yes—
243C2×C4×C532−1+412yes—
244C22⋊D53—48no—
245C22⋊D53—48no—
246C22⋊D53—48no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—28no—
250C2×Dic53—22no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—48no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—84no—
258C2×Dic53#102116yescomplemented
259C2×Dic53C2×C22⋊C412yes—
260C2×Dic53C2×C22⋊C412yes—
261C2×Dic53C2×C22⋊C412yes—
262C2×Dic53C2×C22⋊C412yes—
263C2×Dic53C2×D812yes—
264C2×Dic53C2×D812yes—
265C2×Dic53C2×SD1612yes—
266C2×Dic53C2×SD1612yes—
267C2×Dic53AGL(1,ℤ/8)12yes—
268C2×Dic53#13712yes—
269D212—24no—
270D212—42no—
271C4×D53—48no—
272Dic106—48no—
273C8×C53—42no—
274#88322—42no—
275#21859—531nomaximal, Sylow
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—81no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53—81no—
289C22×C53C2×#10214yes—
290C22×C53C22×C22⋊C411yescharacteristic
291C22×C53#53511yescharacteristic
292C22×C53#53611yescharacteristic
293C22×C53C22×D811yescharacteristic
294C22×C53C22×SD1611yescharacteristic
295C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C53C2×#13711yescharacteristic
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—81no—
301C2×D53—81no—
302C2×D53—81no—
303C2×D53—81no—
304C4×C53—22no—
305C4×C53—42no—
306C4×C53—82no—
307C4×C53#52012yes—
308C4×C53#53712yes—
309Dic53—22no—
310Dic53—42no—
311Dic53—82no—
312Dic53C2×#10214yes—
313Dic53C2×#10214yes—
314Dic53#41714yes—
315C2×#532—531no—
316C2×#431—531no—
317C2×#384—531no—
318#1990—532no—
319#1957—532no—
320#1874—531no—
321#1874—531no—
322#1874—531no—
323#1874D5311yescharacteristic, complemented
324#1802—532no—
325#1802—532no—
326C2×C53—24no—
327C2×C53—82no—
328C2×C53#231111yescharacteristic
329C2×C53C22×#10211yescharacteristic
330C2×C53C2×#41711yescharacteristic
331C2×C53#320511yescharacteristic
332C2×C53#320611yescharacteristic
333C2×C53#322611yescharacteristic
334C2×C53#322711yescharacteristic
335D53—82no—
336C22×#41—1061no—
337#532—538no—
338#532—1064no—
339C2×C4⋊Q8—531no—
340C2×#127—531no—
341C2×#124—1061no—
342C2×#121—531no—
343C2×#121—1061no—
344C2×#121C2×D5311yescharacteristic
345C2×C4×D4—531no—
346C2×C4×D4—531no—
347C2×C4×D4—1061no—
348C22×C4⋊C4—531no—
349C22×C4⋊C4—1061no—
350C22×C4⋊C4—1061no—
351C22×C4⋊C4C2×D5311yescharacteristic
352C22×C42—531no—
353#431—1064no—
354C2×C4⋊C8—532no—
355C2×#102—532no—
356C2×#102—532no—
357C2×#102—532no—
358C2×#102C2×D5312yes—
359C2×C22⋊C8—531no—
360C2×C22⋊C8C2×D5311yescharacteristic
361C2×C4×C8—532no—
362#384—532no—
363#384—532no—
364#384—1061no—
365#384—1061no—
366#340—532no—
367#340—532no—
368#340—532no—
369#340C2×D5312yes—
370C53#2185911yescharacteristic, complemented, Sylow
371C2×#41—1068no—
372C2×#41—2124no—
373C2×#41—2124no—
374C22×Q8—1061no—
375C22×D4—21no—
376C22×D4—531no—
377C22×D4—1061no—
378C22×D4—1061no—
379C22×D4—1061no—
380C22×D4C22×D5311yescharacteristic
381C23×C4—21no—
382C23×C4—531no—
383C23×C4—531no—
384C23×C4—1061no—
385C23×C4—1061no—
386C23×C4—1061no—
387C23×C4C22×D5311yescharacteristic
388C22×C8—532no—
389C22×C8C22×D5312yes—
390C4⋊Q8—534no—
391C4⋊Q8—1062no—
392#127—534no—
393#127—1062no—
394#124—1068no—
395#121—22no—
396#121—534no—
397#121—1068no—
398#121—1062no—
399#121C4×D5314yes—
400C4×D4—534no—
401C4×D4—534no—
402C4×D4—1068no—
403C4×D4—1062no—
404C4×D4—1062no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—42no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—532no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—2122no—
416C2×C4⋊C4—2124no—
417C2×C4⋊C4—2124no—
418C2×C4⋊C4C22×D5312yes—
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C42—534no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—534no—
428C2×C42—531no—
429C2×C42—531no—
430C4⋊C8—1064no—
431#102—42no—
432#102—2122no—
433#102—2122no—
434#102—2122no—
435C22⋊C8—22no—
436C22⋊C8—1062no—
437C4×C8—1064no—
438C24—21no—
439C24—1061no—
440C24D4×D5311yescharacteristic
441#41—21232no—
442C2×Q8—1064no—
443C2×Q8—2122no—
444C2×Q8—2122no—
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—532no—
452C2×D4—1064no—
453C2×D4—1064no—
454C2×D4—1064no—
455C2×D4—1061no—
456C2×D4—1062no—
457C2×D4—1062no—
458C2×D4—2122no—
459C2×D4—2122no—
460C2×D4—2122no—
461C2×D4—2122no—
462C2×D4—2122no—
463C2×D4—2122no—
464C2×D4C2×C4×D5312yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—534no—
473C22×C4—534no—
474C22×C4—534no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—534no—
479C22×C4—531no—
480C22×C4—1064no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1064no—
484C22×C4—1061no—
485C22×C4—1062no—
486C22×C4—2122no—
487C22×C4—2122no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4C2×C4×D5312yes—
494C22×C4D4×D5311yescharacteristic
495C22×C4D4×D5312yes—
496C22×C4C23×D5311yescharacteristic
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—1062no—
501C2×C8—1062no—
502C2×C8—1062no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—42no—
507C4⋊C4—532no—
508C4⋊C4—1064no—
509C4⋊C4—1061no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C4⋊C4—2122no—
513C4⋊C4—2124no—
514C4⋊C4—2124no—
515C4⋊C4—2124no—
516C4⋊C4—2124no—
517C4⋊C4C2×C4×D5312yes—
518C22⋊C4—24no—
519C22⋊C4—1064no—
520C22⋊C4—1064no—
521C22⋊C4—1064no—
522C42—534no—
523C42—534no—
524C42—532no—
525C42—532no—
526C42—532no—
527C42—532no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—1062no—
540C23—2121no—
541C23—2121no—
542C23—2121no—
543C23—2121no—
544C23—2121no—
545C23—2121no—
546C23C22⋊C4×D5312yes—
547C23D8×D5312yes—
548C23SD16×D5312yes—
549C23C2×D4×D5311yescharacteristic
550Q8—2128no—
551D4—24no—
552D4—48no—
553D4—42no—
554D4—1064no—
555D4—2128no—
556D4—2128no—
557D4—2128no—
558D4—2122no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—84no—
570C2×C4—532no—
571C2×C4—532no—
572C2×C4—532no—
573C2×C4—5316no—
574C2×C4—532no—
575C2×C4—532no—
576C2×C4—532no—
577C2×C4—532no—
578C2×C4—532no—
579C2×C4—532no—
580C2×C4—532no—
581C2×C4—1068no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1062no—
585C2×C4—4244no—
586C2×C4—2128no—
587C2×C4—2128no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4C22⋊C4×D5312yes—
594C2×C4C22×C4×D5311yescharacteristic
595C2×C4C2×D4×D5311yescharacteristic
596C2×C4#1313326212yes—
597C2×C4#1313326412yes—
598C8—42no—
599C8—2122no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—4241no—
614C22—4241no—
615C22—4241no—
616C22—4241no—
617C22—2122no—
618C22—2122no—
619C22—2122no—
620C22#102×D5314yes—
621C22C2×C22⋊C4×D5311yescharacteristic
622C22#1681118811yescharacteristic
623C22#1681119511yescharacteristic
624C22C2×D8×D5311yescharacteristic
625C22C2×SD16×D5311yescharacteristic
626C22AGL(1,ℤ/8)×D5311yescharacteristic
627C22#137×D5311yescharacteristic
628C4—22no—
629C4—42no—
630C4—82no—
631C4—534no—
632C4—534no—
633C4—534no—
634C4—1062no—
635C4—4242no—
636C4—2122no—
637C4#1681110212yes—
638C4#1681120012yes—
639C2—24no—
640C2—82no—
641C2—4242no—
642C2#2027730211yescharacteristic
643C2C2×#102×D5311yescharacteristic
644C2#417×D5311yescharacteristic
645C2#2028563111yescharacteristic
646C2#2028563811yescharacteristic
647C2#2028583411yescharacteristic
648C2#2028584111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1874C2
socleC23×C53C22×D4
prime radicalC2×#16811168C2
cosocleC22×C4×C53C24
hypercenter#1874D53
hypocenterC53#21859
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#16810673C2semidirect
3. C2×#16810492C2semidirect
4. #20275674C2semidirect
5. #20275519C2semidirect
6. #20275033C2semidirect
7. #20275032C2semidirect
8. #20275030C2semidirect
9. #20274714C2semidirect
10. #20274713C2semidirect
11. C53×#1874C2semidirect
14. #16811168C4semidirect
61. #13133216C2×C4semidirect
63. #13133213C2×C4semidirect
69. D4×Dic53C2×C4semidirect
71. #13133179C2×C4semidirect
258. C2×Dic53#102semidirect
323. #1874D53semidirect
370. C53#21859semidirect

Maximal subgroups

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

subgroupindex
C2×#168111682
C2×#168106732
C2×#168104922
#202756742
#202755192
#202750332
#202750322
#202750302
#202747142
#202747132
C53×#18742
#2185953