Finite Group Databasebeta

← C4×#16810776 #27389676

Subgroups — C4×#16810776 #27389676

Order 13568.

1880 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D212C211yesmaximal, characteristic
2C2×#16810776C211yesmaximal, characteristic
3C2×C4×#11719913C212yesmaximal, complemented
4C4×#13133196C211yesmaximal, characteristic, complemented
5#20275048C212yesmaximal, complemented
6#20275020C211yesmaximal, characteristic, complemented
7#20274979C212yesmaximal, complemented
8#20274910C211yesmaximal, characteristic, complemented
9#20274647C211yesmaximal, characteristic
10C4×#13133139C212yesmaximal, complemented
11C4×C22⋊C8×C53C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C23×C4×D53—21no—
14C22×#11719913C2212yes—
15C2×#13133196C2211yescharacteristic
16C2×C4×Dic106C2211yescharacteristic, complemented
17C2×C4×D212—28no—
18C2×C4×D212—21no—
19C2×C4×D212—21no—
20C2×C4×D212C2212yes—
21C2×C4×D212C2212yes—
22C22×C4⋊Dic53C2211yescharacteristic
23C22×#11719921—21no—
24#16810776C418yesdirect product factor
25C4×#11719913—28no—
26C2×#13133149C2212yescomplemented
27C2×#13133142C2212yescomplemented
28C2×#13133139C2212yes—
29C2×#13133139C2212yes—
30C2×#13133139C2212yes—
31#16810529C2211yescharacteristic, complemented
32#16810521C2211yescharacteristic, complemented
33#16810507—21no—
34#16810496—21no—
35#16810483C2211yescharacteristic
36C4×#11719919—21no—
37C4×C4⋊Dic53C2211yescharacteristic, complemented
38C4×C22⋊Dic53—21no—
39#16810386C2212yescomplemented
40#16810386C2212yescomplemented
41C22×C42×C53C2211yescharacteristic
42C2×C22⋊C8×C53C2211yescharacteristic
43C2×C22⋊C8×C53C2211yescharacteristic
44C2×C4×C8×C53C2212yescomplemented
45C53×#336C2212yescomplemented
46C24×D53—21no—
47C22×Dic106C2311yescharacteristic
48C22×D212—21no—
49C22×D212—22no—
50C22×D212—24no—
51C22×D212—24no—
52C22×D212C2×C414yes—
53C22×D212C2312yes—
54C23×Dic53—21no—
55C22×C4×D53—28no—
56C22×C4×D53—22no—
57C22×C4×D53—22no—
58C22×C4×D53—44no—
59C22×C4×D53—44no—
60C2×#11719913—24no—
61C2×#11719913—24no—
62C2×#11719913C2×C418yescomplemented
63#13133196—22no—
64#13133196C2×C414yescomplemented
65C4×Dic106—24no—
66C4×D212—28no—
67C4×D212—28no—
68C4×D212—24no—
69C4×D212—24no—
70C4×D212—24no—
71C4×D212D418yes—
72C2×C4⋊Dic53—24no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2312yes—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×#11719921—28no—
84C2×#11719921—22no—
85C2×#11719921—22no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88#13133149—24no—
89#13133142—24no—
90#13133139—24no—
91#13133139—24no—
92#13133139C2×C418yescomplemented
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98C23×C4×C53D411yescharacteristic
99C23×C4×C53D411yescharacteristic
100C23×C4×C53C2311yescharacteristic
101C22×C8×C53C2312yes—
102C22×C8×C53C2312yes—
103C2×C42×C53—24no—
104C2×C42×C53D411yescharacteristic
105C2×C42×C53D411yescharacteristic
106C2×C42×C53C2311yescharacteristic
107C2×C42×C53C2311yescharacteristic
108C22⋊C8×C53—22no—
109C22⋊C8×C53C2×C414yescomplemented
110C4×C8×C53—24no—
111C24×C53C2×D411yescharacteristic
112C23×D53—24no—
113C23×D53—24no—
114C23×D53—24no—
115C23×D53—22no—
116C23×D53—42no—
117C23×D53—42no—
118C23×D53—42no—
119C23×D53—42no—
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—41no—
127C22×C4×C53SD1614yescomplemented
128C22×C4×C53C22×C412yes—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53#4112yes—
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—24no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C2×D212—24no—
145C2×D212—24no—
146C2×D212—216no—
147C2×D212—24no—
148C2×D212—22no—
149C2×D212—22no—
150C2×D212—22no—
151C2×D212—216no—
152C2×D212C22×C414yes—
153C2×D212C2×D414yes—
154C2×D212#4114yes—
155C2×C4×D53—28no—
156C2×C4×D53—28no—
157C2×C4×D53—44no—
158C2×C4×D53—432no—
159C2×C4×D53—44no—
160C2×C4×D53—44no—
161C2×C4×D53—44no—
162C2×Dic106—22no—
163C2×Dic106—22no—
164C2×Dic106—21no—
165C2×Dic106C22×C412yescomplemented
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C2×C8×C53C22×C414yescomplemented
172C42×C53—22no—
173C42×C53—22no—
174C42×C53—22no—
175C42×C53C2×D414yes—
176C22⋊Dic53—24no—
177C22⋊Dic53—42no—
178#11719921—28no—
179#11719921—28no—
180C4⋊Dic53—24no—
181C4⋊Dic53—22no—
182C4⋊Dic53—21no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53C22×C412yescomplemented
186C4⋊Dic53C2×D414yes—
187#11719919—24no—
188#11719919—42no—
189#11719919—42no—
190C4×Dic53—42no—
191C4×Dic53—42no—
192#11719913—216no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53C4×D412yes—
198C23×C53C2×SD1612yes—
199C23×C53#13512yes—
200C23×C53C22×D411yescharacteristic
201C22×D53—216no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—48no—
206C22×D53—48no—
207C22×D53—48no—
208C22×D53—48no—
209C22×D53—48no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—48no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—28no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—44no—
228C2×C4×C53C4×D412yes—
229C2×C4×C53#12012yes—
230C2×C4×C53#12012yes—
231C2×C4×C53C2×SD1612yes—
232C2×C4×C53C2×SD1612yes—
233C2×C4×C53AGL(1,ℤ/8)12yes—
234C2×C4×C53C23×C411yescharacteristic
235C2×C4×C53C22×D412yes—
236C2×C4×C53C2×#4112yes—
237C2×C4×C53C2×#4111yescharacteristic
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—24no—
241C2×Dic53—48no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254D212—216no—
255D212—24no—
256D212—24no—
257D212—24no—
258D212C4×D418yes—
259C4×D53—416no—
260C4×D53—416no—
261Dic106—24no—
262Dic106—42no—
263C8×C53—24no—
264C8×C53—42no—
265C4×#450—531nomaximal, Sylow
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53C4×SD1614yescomplemented
273C22×C53C2×C4×D411yescharacteristic
274C22×C53C2×#12011yescharacteristic
275C22×C53#52511yescharacteristic
276C22×C53C22×SD1611yescharacteristic
277C22×C53C2×#13511yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53#57511yescharacteristic
280C2×D53—28no—
281C2×D53—432no—
282C2×D53—44no—
283C2×D53—44no—
284C2×D53—44no—
285C2×D53—44no—
286C2×D53—44no—
287C2×D53—44no—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—41no—
292C4×C53#45014yescomplemented
293C4×C53C2×C4×D412yes—
294C4×C53#52512yes—
295Dic53—42no—
296Dic53—44no—
297Dic53—42no—
298Dic53—81no—
299Dic53—81no—
300C22×C4×D4—531no—
301C2×#450—531no—
302C2×C4×SD16—532no—
303C4×C22⋊Q8—531no—
304#1877—532no—
305#1872—531no—
306#1864—532no—
307#1853—531no—
308#1786—531no—
309C4×#102—532no—
310C4×C22⋊C8D5311yescharacteristic, complemented
311C2×C53—24no—
312C2×C53C4×#12011yescharacteristic
313C2×C53C2×C4×SD1611yescharacteristic
314C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C53C2×#45011yescharacteristic
316C2×C53#300211yescharacteristic
317C2×C53#310111yescharacteristic
318C2×C53#310511yescharacteristic
319D53—48no—
320C23×D4—531no—
321C24×C4—1061no—
322C22×SD16—532no—
323C2×C22⋊Q8—531no—
324C2×C4×Q8—531no—
325C2×C4×D4—532no—
326C2×C4×D4—532no—
327C2×C4×D4—1068no—
328C2×C4×D4—1061no—
329C2×C4×D4—1061no—
330C22×C4⋊C4—531no—
331C22×C22⋊C4—1061no—
332C22×C42C2×D5311yescharacteristic
333#450—538no—
334C4×SD16—1068no—
335C2×#106—532no—
336C2×#103—532no—
337C2×#102—532no—
338C2×#102—532no—
339C2×#102—532no—
340C2×C22⋊C8C2×D5311yescharacteristic
341C2×C22⋊C8C2×D5311yescharacteristic
342C2×C4×C8C2×D5312yescomplemented
343#391—531no—
344#389—531no—
345#387—1061no—
346#385—1061no—
347#380—531no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—1061no—
350C4×C22⋊C4—1061no—
351#340—532no—
352#340—532no—
353#336C2×D5312yescomplemented
354C53C4×#45011yescharacteristic, complemented, Sylow
355C25—1061no—
356C22×Q8—531no—
357C22×D4—534no—
358C22×D4—532no—
359C22×D4—1064no—
360C22×D4—1064no—
361C22×D4—1061no—
362C22×D4—1062no—
363C23×C4—1068no—
364C23×C4—1061no—
365C23×C4—1062no—
366C23×C4—1062no—
367C23×C4—2124no—
368C23×C4—2124no—
369C23×C4D21211yescharacteristic
370C23×C4D21211yescharacteristic
371C23×C4C22×D5311yescharacteristic
372C2×SD16—538no—
373C2×SD16—1064no—
374C2×SD16—1064no—
375C22×C8C22×D5312yes—
376C22×C8C22×D5312yes—
377C22⋊Q8—534no—
378C22⋊Q8—1062no—
379C4×Q8—1064no—
380C4×D4—538no—
381C4×D4—1064no—
382C4×D4—1064no—
383C4×D4—1068no—
384C4×D4—1068no—
385C4×D4—1064no—
386C2×C4⋊C4—532no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C22⋊C4—1068no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1061no—
400C2×C42—24no—
401C2×C42—1061no—
402C2×C42—1061no—
403C2×C42D21211yescharacteristic
404C2×C42D21211yescharacteristic
405C2×C42C22×D5311yescharacteristic
406C2×C42C22×D5311yescharacteristic
407#106—1064no—
408#103—1064no—
409#102—538no—
410#102—1064no—
411#102—1064no—
412C22⋊C8—22no—
413C22⋊C8C4×D5314yescomplemented
414C4×C8—24no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420C24—1064no—
421C24—1064no—
422C24—1064no—
423C24—1062no—
424C24—2122no—
425C24—2122no—
426C24—2122no—
427C24—2122no—
428C24C2×D21211yescharacteristic
429C2×Q8—532no—
430C2×Q8—1061no—
431C2×Q8—1062no—
432C2×Q8—1062no—
433C2×D4—534no—
434C2×D4—534no—
435C2×D4—534no—
436C2×D4—10616no—
437C2×D4—1064no—
438C2×D4—1064no—
439C2×D4—10616no—
440C2×D4—1064no—
441C2×D4—1062no—
442C2×D4—1062no—
443C2×D4—1062no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—41no—
451C22×C4—1068no—
452C22×C4—1064no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1062no—
456C22×C4—1068no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—2124no—
460C22×C4—21232no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4—2124no—
464C22×C4—2124no—
465C22×C4—2124no—
466C22×C4#1171991314yescomplemented
467C22×C4C2×C4×D5312yes—
468C22×C4C2×D21211yescharacteristic
469C22×C4C2×D21211yescharacteristic
470C22×C4C2×D21211yescharacteristic
471C22×C4C2×D21211yescharacteristic
472C22×C4C2×D21211yescharacteristic
473C22×C4#1171994512yes—
474C22×C4C23×D5311yescharacteristic
475SD16—10616no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8C2×C4×D5314yescomplemented
482C4⋊C4—532no—
483C4⋊C4—534no—
484C4⋊C4—1064no—
485C4⋊C4—1064no—
486C4⋊C4—1062no—
487C4⋊C4—1061no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—2122no—
491C4⋊C4—2122no—
492C22⋊C4—1068no—
493C22⋊C4—1064no—
494C22⋊C4—1068no—
495C22⋊C4—2122no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—2122no—
500C42—2122no—
501C42D4×D5314yes—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—10616no—
507C23—1064no—
508C23—1064no—
509C23—1064no—
510C23—2128no—
511C23—2128no—
512C23—2128no—
513C23—2128no—
514C23—2128no—
515C23—2122no—
516C23—2122no—
517C23—2122no—
518C23—2122no—
519C23—2128no—
520C23C4×D21212yes—
521C23C2×#1171991312yes—
522C23#1313323912yes—
523C23C22×D21211yescharacteristic
524Q8—1064no—
525Q8—2122no—
526D4—538no—
527D4—10616no—
528D4—1064no—
529D4—1064no—
530D4—1064no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—44no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1064no—
547C2×C4—21216no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2128no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2122no—
558C2×C4—2121no—
559C2×C4—2122no—
560C2×C4—2121no—
561C2×C4—21216no—
562C2×C4C4×D21212yes—
563C2×C4#1313318712yes—
564C2×C4#1313318712yes—
565C2×C4C2×#1171991312yes—
566C2×C4C2×#1171991312yes—
567C2×C4#1313324512yes—
568C2×C4C22×C4×D5311yescharacteristic
569C2×C4C2×D4×D5312yes—
570C2×C4#41×D5312yes—
571C2×C4C2×#1171994511yescharacteristic
572C8—24no—
573C8—42no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—1068no—
581C22—21232no—
582C22—2124no—
583C22—2124no—
584C22—2124no—
585C22—2124no—
586C22—2124no—
587C22—2124no—
588C22C4×#1171991314yescomplemented
589C22C2×C4×D21211yescharacteristic
590C22C2×#1313318711yescharacteristic
591C22#1681113511yescharacteristic
592C22C22×#1171991311yescharacteristic
593C22C2×#1313323911yescharacteristic
594C22C2×#1313324511yescharacteristic
595C22#1681142311yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—4241no—
601C4—4241no—
602C4—2122no—
603C4—2122no—
604C4—2124no—
605C4#1681077614yesdirect product factor
606C4C4×D4×D5312yes—
607C4#1681113212yes—
608C2—24no—
609C2—2128no—
610C2C4×#1313318711yescharacteristic
611C2C2×C4×#1171991311yescharacteristic
612C2C4×#1313324511yescharacteristic
613C2C2×#1681077611yescharacteristic
614C2#2028393711yescharacteristic
615C2#2028472411yescharacteristic
616C2#2028476411yescharacteristic
subgroupG/H
centerC22×C4C2×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×C22⋊C8×C53C2
socleC23×C53C22×D4
prime radicalC23×D212C22
cosocleC22×C4×C53C24
hypercenterC4×C22⋊C8D53
hypocenterC53C4×#450
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×C4×#11719913C2semidirect
4. C4×#13133196C2semidirect
5. #20275048C2semidirect
6. #20275020C2semidirect
7. #20274979C2semidirect
8. #20274910C2semidirect
10. C4×#13133139C2semidirect
11. C4×C22⋊C8×C53C2semidirect
16. C2×C4×Dic106C22semidirect
24. #16810776C4direct (N × H)
26. C2×#13133149C22semidirect
27. C2×#13133142C22semidirect
31. #16810529C22semidirect
32. #16810521C22semidirect
37. C4×C4⋊Dic53C22semidirect
39. #16810386C22semidirect
40. #16810386C22semidirect
44. C2×C4×C8×C53C22semidirect
45. C53×#336C22semidirect
62. C2×#11719913C2×C4semidirect
64. #13133196C2×C4semidirect
92. #13133139C2×C4semidirect
109. C22⋊C8×C53C2×C4semidirect
127. C22×C4×C53SD16semidirect
165. C2×Dic106C22×C4semidirect
171. C2×C8×C53C22×C4semidirect
185. C4⋊Dic53C22×C4semidirect
272. C22×C53C4×SD16semidirect
292. C4×C53#450semidirect
310. C4×C22⋊C8D53semidirect
342. C2×C4×C8C2×D53semidirect
353. #336C2×D53semidirect
354. C53C4×#450semidirect
413. C22⋊C8C4×D53semidirect
466. C22×C4#11719913semidirect
481. C2×C8C2×C4×D53semidirect
588. C22C4×#11719913semidirect
605. C4#16810776direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D2122
C2×#168107762
C2×C4×#117199132
C4×#131331962
#202750482
#202750202
#202749792
#202749102
#202746472
C4×#131331392
C4×C22⋊C8×C532
C4×#45053