Finite Group Databasebeta

← #27383584

Subgroups — #27383584

Order 13568.

1598 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719949C211yesmaximal, characteristic
2C2×#16810645C211yesmaximal, characteristic, complemented
3C2×#16810623C212yesmaximal
4#384×D53C211yesmaximal, characteristic, complemented
5#20274720C211yesmaximal, characteristic, complemented
6#20274718C212yesmaximal, complemented
7#20274716C211yesmaximal, characteristic, complemented
8#20274714C212yesmaximal, complemented
9#20274712C211yesmaximal, characteristic, complemented
10#20274433C212yesmaximal, complemented
11C53×#1803C211yesmaximal, characteristic, complemented
12C22×#11719949C2211yescharacteristic
13C22×#11719912C2212yes—
14C4×#11719949—24no—
15C4×#11719949C418yescomplemented
16C2×Q8×Dic53C2211yescharacteristic
17C2×C4×D212—21no—
18C2×C4×D212C2211yescharacteristic
19C2×#13133179—21no—
20C2×#13133179C2211yescharacteristic
21C2×#13133175—21no—
22C2×#13133175C2211yescharacteristic
23C2×C4⋊C4×D53—21no—
24C2×C4⋊C4×D53—21no—
25C2×C4⋊C4×D53C2212yes—
26C2×C42×D53—21no—
27C2×C42×D53C2211yescharacteristic
28#16810645—24no—
29C2×C4⋊#88322C2211yescharacteristic
30C2×#13133146C2212yes—
31#16810623—28no—
32C2×#13133141C2212yes—
33C2×#13133139C2212yes—
34C2×#13133137C2212yes—
35#16810493C2211yescharacteristic
36#16810492—21no—
37#16810492—21no—
38#16810492C2212yes—
39#16810386C2212yes—
40#16810385C2212yes—
41#16810385C2212yes—
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4⋊C8×C53C2211yescharacteristic
44C2×#103×C53C2212yes—
45C53×#384C2211yescharacteristic
46C53×#340C2212yes—
47C2×#11719949—22no—
48C2×#11719949—22no—
49C2×#11719949—22no—
50C2×#11719949C2×C414yes—
51C2×#11719949D414yes—
52C2×#11719949Q814yes—
53C22×D212—21no—
54C22×D212C2311yescharacteristic
55C22×C4×D53—21no—
56C22×C4×D53—21no—
57C22×C4×D53—21no—
58C22×C4×D53—22no—
59C22×C4×D53D411yescharacteristic
60C22×C4×D53D411yescharacteristic
61C22×C4×D53C2311yescharacteristic
62C2×#11719912—24no—
63C2×#11719912—24no—
64C2×#11719912—24no—
65C22×#88322C2312yes—
66Q8×Dic53—22no—
67Q8×Dic53C2×C414yescomplemented
68C4×D212—22no—
69C4×D212—28no—
70C4×D212C2×C414yescomplemented
71#13133179—22no—
72#13133179—28no—
73#13133179C2×C414yescomplemented
74#13133175—28no—
75#13133175—22no—
76#13133175C2×C414yes—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53C2312yes—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—22no—
83C2×#11719919—22no—
84C2×#11719919—22no—
85C4⋊C4×D53—24no—
86C4⋊C4×D53—24no—
87C4⋊C4×D53—44no—
88C4⋊C4×D53—44no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53D411yescharacteristic
95C2×C4×Dic53D411yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97C42×D53—28no—
98C42×D53C2×C414yes—
99C42×D53D414yes—
100C4⋊#88322—22no—
101#13133146—24no—
102#13133141—24no—
103#13133139—24no—
104#13133137—24no—
105C22×Q8×C53C2311yescharacteristic
106C22×C8×C53C2312yes—
107C4×Q8×C53—22no—
108C4×Q8×C53C2×C414yescomplemented
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53C2312yes—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C4⋊C8×C53—22no—
117#103×C53—24no—
118C23×D53—21no—
119C23×D53C2×D411yescharacteristic
120C2×Q8×C53—21no—
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53C22×C412yes—
124C2×Q8×C53C2×D412yes—
125C2×Q8×C53C2×Q812yes—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—22no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137#11719949—28no—
138#11719949—28no—
139#11719949—28no—
140#11719949C4⋊C4116yescomplemented
141C2×D212—24no—
142C2×D212—24no—
143C2×D212—24no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212C22×C412yes—
148C2×D212C2×D412yes—
149C2×D212C2×Q812yes—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—24no—
153C2×C4×D53—24no—
154C2×C4×D53—24no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—24no—
158C2×C4×D53—24no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53—42no—
162C2×C4×D53—42no—
163C2×C4×D53—44no—
164C2×C4×D53—44no—
165C2×C4×D53—44no—
166C2×C4×D53C22⋊C414yes—
167C2×C4×D53C22⋊C414yes—
168C2×C4×D53C22×C412yes—
169C2×C4×D53C2×D412yes—
170C2×C4×D53C2×D412yes—
171C2×C4×D53C2×Q812yes—
172C2×C4×D53#4112yes—
173C2×C4×D53#4112yes—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C4⋊C4×C53—24no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—22no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—42no—
182C4⋊C4×C53—42no—
183C4⋊C4×C53C22×C412yes—
184C42×C53—24no—
185C42×C53C22×C412yes—
186C42×C53C2×D412yes—
187#11719921—24no—
188#11719921—24no—
189#11719921—24no—
190C4⋊Dic53—21no—
191C4⋊Dic53—24no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—42no—
195C4⋊Dic53—42no—
196C4⋊Dic53C22×C412yes—
197#11719919—44no—
198#11719919—44no—
199#11719919—44no—
200C4×Dic53—24no—
201C4×Dic53—24no—
202C4×Dic53C22⋊C414yes—
203C4×Dic53C22⋊C414yes—
204C4×Dic53C22×C412yes—
205C4×Dic53C2×D412yes—
206C2×#88322—22no—
207C2×#88322—22no—
208C2×#88322—22no—
209#11719912—48no—
210C23×C53C22×D411yescharacteristic
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—22no—
218C22×D53—41no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53C2×C22⋊C412yes—
223C22×D53#12112yes—
224C22×D53C22⋊Q812yes—
225Q8×C53—22no—
226Q8×C53—22no—
227Q8×C53—22no—
228Q8×C53C2×C4⋊C414yescomplemented
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53C2×C22⋊C412yes—
241C2×C4×C53C2×C22⋊C412yes—
242C2×C4×C53AGL(1,ℤ/8)12yes—
243C2×C4×C53#13712yes—
244C2×C4×C53C23×C411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C22×Q811yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—22no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53C2×C22⋊C412yes—
259C2×Dic53C2×C22⋊C412yes—
260C2×Dic53C2×C22⋊C412yes—
261C2×Dic53#12112yes—
262C2×Dic53C22⋊Q812yes—
263C2×Dic53AGL(1,ℤ/8)12yes—
264C2×Dic53#13712yes—
265D212—216no—
266D212—22no—
267D212—22no—
268D212—22no—
269D212C2×C4⋊C414yescomplemented
270C4×D53—28no—
271C4×D53—216no—
272C4×D53—28no—
273C4×D53—48no—
274C4×D53—48no—
275C4×D53—88no—
276C4×D53C2×C4⋊C414yes—
277C4×D53#11714yes—
278C4×D53#12114yes—
279C4×D53C22⋊Q814yes—
280C8×C53—42no—
281#88322—42no—
282#20648—531nomaximal, Sylow
283C22×C53C22×C22⋊C411yescharacteristic
284C22×C53C2×#12111yescharacteristic
285C22×C53C2×C22⋊Q811yescharacteristic
286C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53C2×#13711yescharacteristic
290C2×D53—22no—
291C2×D53—22no—
292C2×D53—22no—
293C2×D53—22no—
294C2×D53—22no—
295C2×D53—22no—
296C2×D53—24no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53#38014yes—
304C4×C53—22no—
305C4×C53—24no—
306C4×C53—42no—
307C4×C53—82no—
308C4×C53#41714yescomplemented
309C4×C53C22×C4⋊C411yescharacteristic
310C4×C53C2×#11711yescharacteristic
311C4×C53C2×#12111yescharacteristic
312C4×C53C2×C22⋊Q811yescharacteristic
313Dic53—22no—
314Dic53—42no—
315Dic53—82no—
316Dic53#38014yes—
317Dic53#41714yescomplemented
318C2×C4×#41—531no—
319C2×#423—531no—
320C2×#417—532no—
321C2×#384—531no—
322#1803—532no—
323#1803—531no—
324#1803D5311yescharacteristic, complemented
325#1802—531no—
326#1802—531no—
327#1802—532no—
328#1736—532no—
329C2×C53C2×#38011yescharacteristic
330C2×C53C2×#41711yescharacteristic
331C2×C53C2×#41711yescharacteristic
332C2×C53#303811yescharacteristic
333C2×C53#303911yescharacteristic
334C2×C53#307911yescharacteristic
335C2×C53#307911yescharacteristic
336D53—24no—
337D53—44no—
338C22×#41—531no—
339C22×M16—532no—
340C4×#41—538no—
341C4×#41—1064no—
342C2×C4×Q8—531no—
343C2×C4×Q8C2×D5311yescharacteristic
344C2×C4×D4—531no—
345C2×C4×D4—531no—
346C2×C4×D4—1061no—
347C2×C4×D4—1061no—
348C2×#117—531no—
349C2×#117—1061no—
350C22×C4⋊C4—532no—
351C22×C4⋊C4—1061no—
352C22×C4⋊C4—1061no—
353C22×C42—531no—
354C22×C42—1061no—
355#423—1064no—
356C2×C4⋊C8—532no—
357C2×C4⋊C8—531no—
358C2×C4⋊C8C2×D5311yescharacteristic
359#417—1068no—
360C2×#103—532no—
361C2×#103C2×D5312yes—
362C2×#102—532no—
363C2×#102—532no—
364#384—532no—
365#384—531no—
366#384—1061no—
367#384—1061no—
368#384C2×D5311yescharacteristic
369#340—532no—
370#340—532no—
371#340—532no—
372#340C2×D5312yes—
373C53#2064811yescharacteristic, complemented, Sylow
374C2×#41—534no—
375C2×#41—534no—
376C2×#41—534no—
377C2×#41—1062no—
378C2×#41—1062no—
379C2×#41—1062no—
380C22×Q8C22×D5311yescharacteristic
381C22×D4—531no—
382C22×D4—1061no—
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—1061no—
387C23×C4—1061no—
388C23×C4—1062no—
389C23×C4—1061no—
390C2×M16—1064no—
391C2×M16—1064no—
392C2×M16—1064no—
393C22×C8—532no—
394C22×C8C22×D5312yes—
395C4×Q8—22no—
396C4×Q8—534no—
397C4×Q8—1062no—
398C4×Q8C4×D5314yescomplemented
399C4×D4—534no—
400C4×D4—534no—
401C4×D4—1068no—
402C4×D4—1062no—
403C4×D4—1068no—
404C4×D4—1062no—
405#117—534no—
406#117—1068no—
407#117—1062no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—532no—
413C2×C4⋊C4—1064no—
414C2×C4⋊C4—1064no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1062no—
417C2×C4⋊C4—1062no—
418C2×C4⋊C4—1062no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—2124no—
422C2×C4⋊C4—2124no—
423C2×C4⋊C4C22×D5312yes—
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C42—21no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—534no—
433C2×C42—534no—
434C2×C42—1068no—
435C2×C42—1061no—
436C2×C42—1061no—
437C2×C42C22×D5311yescharacteristic
438C4⋊C8—22no—
439C4⋊C8—1064no—
440C4⋊C8—1062no—
441#103—24no—
442#103—1064no—
443#102—1064no—
444#102—1064no—
445C24—531no—
446C24—1061no—
447#41—5316no—
448#41—1068no—
449#41—1068no—
450#41—1068no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8C2×C4×D5312yes—
455C2×Q8D4×D5312yes—
456C2×Q8Q8×D5312yes—
457C2×D4—532no—
458C2×D4—532no—
459C2×D4—532no—
460C2×D4—1064no—
461C2×D4—1064no—
462C2×D4—1064no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—1061no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—532no—
472C22×C4—532no—
473C22×C4—532no—
474C22×C4—532no—
475C22×C4—532no—
476C22×C4—532no—
477C22×C4—531no—
478C22×C4—534no—
479C22×C4—534no—
480C22×C4—1064no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1064no—
484C22×C4—1064no—
485C22×C4—1064no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1061no—
489C22×C4—1064no—
490C22×C4—1062no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4—2124no—
494C22×C4—2124no—
495C22×C4—2124no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4D4×D5311yescharacteristic
499C22×C4D4×D5311yescharacteristic
500C22×C4C23×D5311yescharacteristic
501M16—2128no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—1062no—
506C2×C8—1062no—
507C2×C8—1062no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—532no—
515C4⋊C4—1064no—
516C4⋊C4—1061no—
517C4⋊C4—1062no—
518C4⋊C4—1062no—
519C4⋊C4—2122no—
520C4⋊C4—2124no—
521C4⋊C4—2124no—
522C4⋊C4—2124no—
523C4⋊C4—2122no—
524C4⋊C4C2×C4×D5312yes—
525C22⋊C4—1064no—
526C22⋊C4—1064no—
527C22⋊C4—1064no—
528C42—24no—
529C42—534no—
530C42—534no—
531C42—532no—
532C42—532no—
533C42—1064no—
534C42—1064no—
535C42C2×C4×D5312yes—
536C42D4×D5312yes—
537C23—532no—
538C23—532no—
539C23—532no—
540C23—1061no—
541C23—1061no—
542C23—1061no—
543C23—1061no—
544C23—1062no—
545C23—1062no—
546C23—1062no—
547C23—2121no—
548C23—2121no—
549C23—2121no—
550C23—2121no—
551C23C2×D4×D5311yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8C4⋊C4×D5314yescomplemented
556D4—534no—
557D4—10616no—
558D4—1062no—
559D4—1062no—
560D4—1062no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—532no—
573C2×C4—532no—
574C2×C4—532no—
575C2×C4—532no—
576C2×C4—532no—
577C2×C4—532no—
578C2×C4—534no—
579C2×C4—534no—
580C2×C4—534no—
581C2×C4—534no—
582C2×C4—532no—
583C2×C4—1068no—
584C2×C4—10616no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1068no—
588C2×C4—1062no—
589C2×C4—4248no—
590C2×C4—2128no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2128no—
597C2×C4C22⋊C4×D5312yes—
598C2×C4C22⋊C4×D5312yes—
599C2×C4#1313324412yes—
600C2×C4#1313324912yes—
601C2×C4C22×C4×D5311yescharacteristic
602C2×C4C2×D4×D5311yescharacteristic
603C2×C4C2×D4×D5311yescharacteristic
604C2×C4C2×Q8×D5311yescharacteristic
605C2×C4#41×D5311yescharacteristic
606C2×C4#41×D5311yescharacteristic
607C8—42no—
608C8—2122no—
609C22—534no—
610C22—1062no—
611C22—1062no—
612C22—1062no—
613C22—1062no—
614C22—1062no—
615C22—1062no—
616C22—1064no—
617C22—2122no—
618C22—2122no—
619C22—2122no—
620C22—2122no—
621C22—2122no—
622C22—2122no—
623C22C2×C22⋊C4×D5311yescharacteristic
624C22#121×D5311yescharacteristic
625C22C22⋊Q8×D5311yescharacteristic
626C22AGL(1,ℤ/8)×D5311yescharacteristic
627C22C2×#1313324411yescharacteristic
628C22#137×D5311yescharacteristic
629C22C2×#1313324911yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—82no—
634C4—534no—
635C4—534no—
636C4—1062no—
637C4—4242no—
638C4—2122no—
639C4#1681062314yescomplemented
640C4C2×C4⋊C4×D5311yescharacteristic
641C4#117×D5311yescharacteristic
642C4#121×D5311yescharacteristic
643C4C22⋊Q8×D5311yescharacteristic
644C2—1064no—
645C2—2124no—
646C2#380×D5311yescharacteristic
647C2#417×D5311yescharacteristic
648C2C2×#1681062311yescharacteristic
649C2#2028419211yescharacteristic
650C2#2028419911yescharacteristic
651C2#2028452511yescharacteristic
652C2#2028452911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC22×#11719949C22
cosocleC22×C4×C53C24
hypercenter#1803D53
hypocenterC53#20648
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#16810645C2semidirect
4. #384×D53C2semidirect
5. #20274720C2semidirect
6. #20274718C2semidirect
7. #20274716C2semidirect
8. #20274714C2semidirect
9. #20274712C2semidirect
10. #20274433C2semidirect
11. C53×#1803C2semidirect
15. C4×#11719949C4semidirect
67. Q8×Dic53C2×C4semidirect
70. C4×D212C2×C4semidirect
73. #13133179C2×C4semidirect
108. C4×Q8×C53C2×C4semidirect
140. #11719949C4⋊C4semidirect
228. Q8×C53C2×C4⋊C4semidirect
269. D212C2×C4⋊C4semidirect
308. C4×C53#417semidirect
317. Dic53#417semidirect
324. #1803D53semidirect
373. C53#20648semidirect
398. C4×Q8C4×D53semidirect
555. Q8C4⋊C4×D53semidirect
639. C4#16810623semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199492
C2×#168106452
C2×#168106232
#384×D532
#202747202
#202747182
#202747162
#202747142
#202747122
#202744332
C53×#18032
#2064853