Finite Group Databasebeta

← C4×#16810549 #27305916

Subgroups — C4×#16810549 #27305916

Order 13568.

1790 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810549C211yesmaximal, characteristic
2C2×C4×#11719921C212yesmaximal
3C2×C4×#11719921C211yesmaximal, characteristic
4#20272983C211yesmaximal, characteristic, complemented
5#20272976C212yesmaximal, complemented
6#20272974C211yesmaximal, characteristic, complemented
7#20272963C212yesmaximal
8#20272957C211yesmaximal, characteristic
9C4×#13133124C212yesmaximal, complemented
10C4×#13133124C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C53C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×#11719921C2212yes—
14C22×#11719921C2211yescharacteristic
15C22×#11719921C2211yescharacteristic
16C22×#11719921C2212yes—
17C22×C4×Dic53C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19#16810549C418yesdirect product factor
20C4×#11719921—28no—
21C4×#11719921—24no—
22C2×#13133124C2212yes—
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C4⋊C4×C53C2211yescharacteristic
35C22×C42×C53C2212yes—
36C22×C42×C53C2211yescharacteristic
37C4×C4⋊C4×C53—24no—
38C2×#95×C53C2212yes—
39C24×D53C2311yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C22×C4×D53—22no—
46C22×C4×D53—22no—
47C22×C4×D53—24no—
48C22×C4×D53—24no—
49C22×C4×D53D414yes—
50C2×#11719921—24no—
51C2×#11719921—24no—
52C2×#11719921—22no—
53C2×#11719921—24no—
54C2×#11719921—22no—
55C2×#11719921—22no—
56C2×#11719921—24no—
57C2×#11719921—22no—
58C2×#11719921—24no—
59C2×#11719921—22no—
60C2×#11719921C2×C418yes—
61C2×#11719921C2×C414yes—
62C2×C4×Dic53—24no—
63C2×C4×Dic53—22no—
64C2×C4×Dic53—24no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67#13133124—22no—
68#13133124—22no—
69#13133124—24no—
70#13133124—22no—
71#13133124—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124—24no—
75#13133124—24no—
76#13133124C2×C418yescomplemented
77#13133124C2×C414yescomplemented
78C23×C4×C53C2312yes—
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2312yes—
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—24no—
86C2×C4⋊C4×C53—22no—
87C2×C4⋊C4×C53C2×C414yescomplemented
88C2×C42×C53—24no—
89C2×C42×C53—24no—
90C2×C42×C53—24no—
91C2×C42×C53—24no—
92C2×C42×C53D414yes—
93#95×C53—24no—
94C24×C53C2411yescharacteristic
95C23×D53—21no—
96C23×D53—21no—
97C23×D53—22no—
98C23×D53—22no—
99C23×D53—21no—
100C23×D53—21no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53C22×C412yes—
104C23×D53C2×D412yes—
105C23×D53#4112yes—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
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—21no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53C22×C414yes—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D412yes—
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53#4114yes—
131C22×C4×C53#4112yes—
132C22×C4×C53#4112yes—
133C22×C4×C53#4114yes—
134C22×C4×C53#4114yes—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
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—21no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—21no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53C22×C414yes—
160C22×Dic53C22×C412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53#4112yes—
163C22×Dic53#4114yes—
164C2×C4×D53—28no—
165C2×C4×D53—216no—
166C2×C4×D53—44no—
167C2×C4×D53—44no—
168C2×C4×D53—48no—
169C4⋊C4×C53—28no—
170C4⋊C4×C53—28no—
171C42×C53—28no—
172C42×C53—48no—
173#11719921—28no—
174#11719921—216no—
175#11719921—216no—
176#11719921—28no—
177C4×Dic53—48no—
178C4×Dic53—44no—
179C23×C53C23×C411yescharacteristic
180C23×C53C22×D411yescharacteristic
181C23×C53C22×D411yescharacteristic
182C23×C53C2×#4111yescharacteristic
183C23×C53C2×#4111yescharacteristic
184C23×C53C2×#4112yes—
185C23×C53C2×#4111yescharacteristic
186C23×C53C2×#4111yescharacteristic
187C23×C53C2×#4112yes—
188C23×C53C2×#4112yes—
189C23×C53C2×#4112yes—
190C22×D53—22no—
191C22×D53—22no—
192C22×D53—24no—
193C22×D53—24no—
194C22×D53—22no—
195C22×D53—24no—
196C22×D53—24no—
197C22×D53—22no—
198C22×D53—24no—
199C22×D53—22no—
200C22×D53—24no—
201C22×D53—22no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—42no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—41no—
212C22×D53C4×D414yes—
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—22no—
226C2×C4×C53—42no—
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×C53—42no—
236C2×C4×C53C4×D414yes—
237C2×C4×C53#12114yes—
238C2×C4×C53#12314yes—
239C2×C4×C53#12318yes—
240C2×C4×C53#12414yes—
241C2×C4×C53#12618yes—
242C2×Dic53—24no—
243C2×Dic53—22no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—22no—
248C2×Dic53—24no—
249C2×Dic53—22no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—41no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—41no—
270C2×Dic53—42no—
271C4×D53—416no—
272C4×#397—531nomaximal, Sylow
273C22×C53C2×C4×D411yescharacteristic
274C22×C53C2×C4×D411yescharacteristic
275C22×C53C4×#4111yescharacteristic
276C22×C53C4×#4112yes—
277C22×C53C4×#4112yes—
278C22×C53C2×#12111yescharacteristic
279C22×C53C2×#12311yescharacteristic
280C22×C53C2×#12312yes—
281C22×C53#52511yescharacteristic
282C22×C53#52512yes—
283C22×C53#52511yescharacteristic
284C22×C53C2×#12411yescharacteristic
285C22×C53C2×#12612yes—
286C22×C53#52912yes—
287C22×C53#52912yes—
288C22×C53#52912yes—
289C22×C53#52911yescharacteristic
290C22×C53#52912yes—
291C22×C53#52912yes—
292C22×C53#52912yes—
293C22×C53#52912yes—
294C22×C53#53211yescharacteristic
295C22×C53#53211yescharacteristic
296C2×D53—24no—
297C2×D53—28no—
298C2×D53—44no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—44no—
303C2×D53—42no—
304C2×D53—44no—
305C2×D53—42no—
306C2×D53—42no—
307C4×C53—24no—
308C4×C53—44no—
309C4×C53—42no—
310C4×C53—44no—
311C4×C53#39718yescomplemented
312Dic53—42no—
313Dic53—44no—
314Dic53—42no—
315Dic53—44no—
316Dic53—42no—
317C2×#397—531no—
318C2×C4×C4⋊C4D5311yescharacteristic, complemented
319C2×C4×C22⋊C4—531no—
320C2×C4×C22⋊C4—532no—
321#1445—531no—
322#1443—532no—
323#1442—531no—
324#1440—532no—
325#1439—531no—
326C4×#95—531no—
327C4×#95—532no—
328C2×C53C4×#12111yescharacteristic
329C2×C53C4×#12311yescharacteristic
330C2×C53C4×#12312yes—
331C2×C53C4×#12411yescharacteristic
332C2×C53C4×#12612yes—
333C2×C53C2×#39711yescharacteristic
334C2×C53#239211yescharacteristic
335C2×C53#239511yescharacteristic
336C2×C53#239512yes—
337C2×C53#239811yescharacteristic
338C2×C53#240112yes—
339D53—44no—
340C24×C4—531no—
341C22×C4⋊C4C2×D5311yescharacteristic
342C22×C4⋊C4C2×D5311yescharacteristic
343C22×C22⋊C4—531no—
344C22×C22⋊C4—532no—
345C22×C22⋊C4—532no—
346C22×C22⋊C4—531no—
347C22×C42—532no—
348C22×C42—531no—
349C22×C42C2×D5312yes—
350C22×C42C2×D5311yescharacteristic
351#397—538no—
352C4×C4⋊C4—24no—
353C4×C22⋊C4—1068no—
354C4×C22⋊C4—1064no—
355C2×#95—531no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—531no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—531no—
364C2×#95—531no—
365C2×#95—532no—
366C2×#95C2×D5312yes—
367C53C4×#39711yescharacteristic, complemented, Sylow
368C25—531no—
369C23×C4—532no—
370C23×C4—531no—
371C23×C4—532no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—534no—
375C23×C4—1064no—
376C23×C4—1062no—
377C23×C4—1064no—
378C23×C4—1062no—
379C23×C4C22×D5312yes—
380C23×C4C22×D5312yes—
381C23×C4C22×D5311yescharacteristic
382C23×C4C22×D5311yescharacteristic
383C23×C4C22×D5311yescharacteristic
384C2×C4⋊C4—24no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4C4×D5314yescomplemented
389C2×C22⋊C4—538no—
390C2×C22⋊C4—534no—
391C2×C22⋊C4—1064no—
392C2×C22⋊C4—1062no—
393C2×C22⋊C4—1064no—
394C2×C22⋊C4—1062no—
395C2×C22⋊C4—1064no—
396C2×C22⋊C4—1064no—
397C2×C22⋊C4—1064no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1062no—
401C2×C42—24no—
402C2×C42—24no—
403C2×C42—24no—
404C2×C42—24no—
405C2×C42—1064no—
406C2×C42—1064no—
407C2×C42—1064no—
408C2×C42—1064no—
409C2×C42—1062no—
410C2×C42C22⋊D5314yes—
411#95—24no—
412#95—538no—
413#95—534no—
414#95—1064no—
415#95—1064no—
416#95—1064no—
417#95—1064no—
418#95—1064no—
419#95—1064no—
420#95—1062no—
421#95—1062no—
422#95—1062no—
423C24—532no—
424C24—532no—
425C24—532no—
426C24—1061no—
427C24—1061no—
428C24—1062no—
429C24—1061no—
430C24—1061no—
431C24—1062no—
432C24—1062no—
433C24—1062no—
434C24C23×D5311yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—534no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—534no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1061no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1068no—
483C22×C4—10616no—
484C22×C4—2124no—
485C22×C4—2124no—
486C22×C4—2128no—
487C22×C4C2×C4×D5314yes—
488C22×C4C2×C4×D5312yes—
489C22×C4#1171994514yes—
490C22×C4#1171994512yes—
491C22×C4#1171994514yes—
492C22×C4D4×D5312yes—
493C22×C4#1171994712yes—
494C22×C4#1171994914yes—
495C22×C4C2×C22⋊D5312yes—
496C22×C4C2×C22⋊D5312yes—
497C22×C4C2×C22⋊D5312yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C22⋊C4—10616no—
501C22⋊C4—1068no—
502C22⋊C4—10616no—
503C22⋊C4—1068no—
504C42—28no—
505C42—48no—
506C42—2128no—
507C42—2124no—
508C23—534no—
509C23—1062no—
510C23—1062no—
511C23—1064no—
512C23—1064no—
513C23—1062no—
514C23—1064no—
515C23—1062no—
516C23—1064no—
517C23—1064no—
518C23—1062no—
519C23—1064no—
520C23—1062no—
521C23—2121no—
522C23—2121no—
523C23—2121no—
524C23—2122no—
525C23—2122no—
526C23—2122no—
527C23—2122no—
528C23—2122no—
529C23—2122no—
530C23—2121no—
531C23C22×C4×D5311yescharacteristic
532C23C2×D4×D5311yescharacteristic
533C23C22×C22⋊D5311yescharacteristic
534C23#41×D5311yescharacteristic
535C23#41×D5311yescharacteristic
536C23#41×D5312yes—
537C23C2×#1171994711yescharacteristic
538C23C2×#1171994511yescharacteristic
539C23C2×#1171994512yes—
540C23C2×#1171994512yes—
541C23C2×#1171994912yes—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—1064no—
566C2×C4—1062no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1062no—
571C2×C4—1064no—
572C2×C4—1062no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—2121no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2121no—
593C2×C4—2122no—
594C2×C4—21216no—
595C2×C4C4×C22⋊D5314yes—
596C2×C4#1313318814yes—
597C2×C4#1313320014yes—
598C2×C4#1313320218yes—
599C2×C4#1313320614yes—
600C2×C4#1313321218yes—
601C22—1064no—
602C22—1068no—
603C22—2124no—
604C22—2122no—
605C22—2124no—
606C22—2122no—
607C22—2122no—
608C22—2122no—
609C22—2124no—
610C22—2122no—
611C22—2122no—
612C22C4×D4×D5311yescharacteristic
613C22C2×C4×C22⋊D5311yescharacteristic
614C22C4×#1171994711yescharacteristic
615C22C4×#1171994512yes—
616C22C4×#1171994912yes—
617C22C2×#1313318811yescharacteristic
618C22C2×#1313320011yescharacteristic
619C22C2×#1313320212yes—
620C22#1681113011yescharacteristic
621C22#1681113212yes—
622C22#1681113411yescharacteristic
623C22C2×#1313320611yescharacteristic
624C22C2×#1313321212yes—
625C22#1681115112yes—
626C22#1681115112yes—
627C22#1681115312yes—
628C22#1681115511yescharacteristic
629C22#1681115712yes—
630C22#1681116012yes—
631C22#1681116012yes—
632C22#1681116012yes—
633C22#1681116911yescharacteristic
634C22#1681117011yescharacteristic
635C4—24no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—2122no—
640C4—2122no—
641C4—2124no—
642C4—2122no—
643C4—2124no—
644C4#1681054918yesdirect product factor
645C2—2124no—
646C2C4×#1313318811yescharacteristic
647C2C4×#1313320011yescharacteristic
648C2C4×#1313320212yes—
649C2C4×#1313320611yescharacteristic
650C2C4×#1313321212yes—
651C2C2×#1681054911yescharacteristic
652C2#2027795311yescharacteristic
653C2#2027797211yescharacteristic
654C2#2027797812yes—
655C2#2027799811yescharacteristic
656C2#2027801612yes—
subgroupG/H
centerC23×C4C22×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC2×C4×C4⋊C4×C53C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC2×C4×C4⋊C4D53
hypocenterC53C4×#397
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23×C4 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #20272983C2semidirect
5. #20272976C2semidirect
6. #20272974C2semidirect
9. C4×#13133124C2semidirect
10. C4×#13133124C2semidirect
11. C2×C4×C4⋊C4×C53C2semidirect
19. #16810549C4direct (N × H)
76. #13133124C2×C4semidirect
77. #13133124C2×C4semidirect
87. C2×C4⋊C4×C53C2×C4semidirect
311. C4×C53#397semidirect
318. C2×C4×C4⋊C4D53semidirect
367. C53C4×#397semidirect
388. C2×C4⋊C4C4×D53semidirect
644. C4#16810549direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168105492
C2×C4×#117199212
#202729832
#202729762
#202729742
#202729632
#202729572
C4×#131331242
C2×C4×C4⋊C4×C532
C4×#39753