Finite Group Databasebeta

← C4×#16810757 #27389661

Subgroups — C4×#16810757 #27389661

Order 13568.

1964 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×#16810757C211yesmaximal, characteristic
2C2×C4×#11719922C212yesmaximal, complemented
3C4×#13133191C211yesmaximal, characteristic, complemented
4#20275063C212yesmaximal, complemented
5#20275032C211yesmaximal, characteristic, complemented
6#20274962C212yesmaximal, complemented
7#20274903C211yesmaximal, characteristic, complemented
8#20274646C211yesmaximal, characteristic, complemented
9C4×#13133138C212yesmaximal, complemented
10C4×C22⋊#88322C211yesmaximal, characteristic, complemented
11C22×C4×D4×C53C211yesmaximal, characteristic, complemented
12C22×#11719922C2212yes—
13C2×#13133191C2211yescharacteristic
14C2×C4×D212C2211yescharacteristic, complemented
15C2×C4×C22⋊D53—21no—
16#16810757C418yesdirect product factor
17C4×#11719922—28no—
18C2×#13133150C2212yescomplemented
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×#13133137C2212yescomplemented
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C4×#88322C2212yescomplemented
26#16810525C2211yescharacteristic, complemented
27#16810521C2211yescharacteristic, complemented
28#16810514—21no—
29#16810502—21no—
30#16810483C2211yescharacteristic
31C4×C4⋊Dic53C2211yescharacteristic, complemented
32C4×#11719921—21no—
33#16810385C2212yescomplemented
34#16810385C2212yescomplemented
35#16810378C2212yescomplemented
36C23×D4×C53C2211yescharacteristic
37C24×C4×C53—21no—
38C2×C4×D4×C53—28no—
39C2×C4×D4×C53—21no—
40C2×C4×D4×C53—21no—
41C2×C4×D4×C53C2212yes—
42C2×C4×D4×C53C2212yes—
43C22×C4⋊C4×C53C2211yescharacteristic
44C22×C22⋊C4×C53—21no—
45C22×C42×C53C2211yescharacteristic
46C22×D212C2311yescharacteristic
47C22×C22⋊D53—21no—
48C22×C4×D53—21no—
49C2×#11719922—24no—
50C2×#11719922—24no—
51C2×#11719922C2×C418yescomplemented
52C22×#88322C2312yes—
53C22×#88322C2312yes—
54#13133191—22no—
55#13133191C2×C414yescomplemented
56C4×D212—24no—
57C4×C22⋊D53—44no—
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2311yescharacteristic
60C2×C4⋊Dic53C2311yescharacteristic
61C2×#11719919—21no—
62C2×C22⋊Dic53—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66C2×#11719921—21no—
67C2×C4×Dic53—21no—
68#13133150—24no—
69#13133138—24no—
70#13133138—24no—
71#13133138C2×C418yescomplemented
72#13133137—24no—
73C22⋊#88322—22no—
74C22⋊#88322C2×C414yescomplemented
75C4×#88322—24no—
76#13133124—21no—
77#13133124—21no—
78#13133124—21no—
79C25×C53—21no—
80C22×D4×C53—24no—
81C22×D4×C53—24no—
82C22×D4×C53—21no—
83C22×D4×C53—22no—
84C22×D4×C53C2×C414yescomplemented
85C22×D4×C53C2312yes—
86C23×C4×C53—28no—
87C23×C4×C53—21no—
88C23×C4×C53—22no—
89C23×C4×C53—22no—
90C23×C4×C53—44no—
91C23×C4×C53—44no—
92C23×C4×C53D411yescharacteristic
93C23×C4×C53D411yescharacteristic
94C23×C4×C53C2311yescharacteristic
95C4×D4×C53—28no—
96C4×D4×C53—28no—
97C4×D4×C53—24no—
98C4×D4×C53—24no—
99C4×D4×C53—24no—
100C4×D4×C53D418yescomplemented
101C2×C4⋊C4×C53—24no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2312yes—
104C2×C22⋊C4×C53—28no—
105C2×C22⋊C4×C53—22no—
106C2×C22⋊C4×C53—22no—
107C2×C42×C53—24no—
108C2×C42×C53D411yescharacteristic
109C2×C42×C53D411yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C24×C53—24no—
113C24×C53—24no—
114C24×C53—24no—
115C24×C53—22no—
116C24×C53—42no—
117C24×C53—42no—
118C24×C53—42no—
119C24×C53—42no—
120C24×C53C2×D411yescharacteristic
121C23×D53—21no—
122C2×D4×C53—24no—
123C2×D4×C53—24no—
124C2×D4×C53—216no—
125C2×D4×C53—216no—
126C2×D4×C53—24no—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—22no—
130C2×D4×C53C22×C414yes—
131C2×D4×C53C2×D414yes—
132C2×D4×C53#4114yes—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—24no—
138C22×C4×C53—28no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—28no—
143C22×C4×C53—41no—
144C22×C4×C53—44no—
145C22×C4×C53—432no—
146C22×C4×C53—42no—
147C22×C4×C53—42no—
148C22×C4×C53—44no—
149C22×C4×C53—44no—
150C22×C4×C53—44no—
151C22×C4×C53D814yescomplemented
152C22×C4×C53C22×C412yes—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53#4112yes—
159C22×C4×C53C2411yescharacteristic
160C2×C22⋊D53—24no—
161C2×C22⋊D53—42no—
162C2×C22⋊D53—42no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C2×D212—22no—
167C2×D212—22no—
168C2×D212—21no—
169C2×D212C22×C412yescomplemented
170C2×C4×D53—42no—
171C2×C4×D53—42no—
172C2×C4×D53—42no—
173C4⋊C4×C53—24no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53C2×D414yes—
176C22⋊C4×C53—28no—
177C22⋊C4×C53—28no—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53—22no—
181C42×C53C2×D414yes—
182C22⋊Dic53—42no—
183#11719922—216no—
184#11719921—24no—
185#11719921—42no—
186#11719921—42no—
187#11719921—42no—
188C4⋊Dic53—21no—
189C4⋊Dic53—22no—
190C4⋊Dic53—22no—
191C4⋊Dic53C22×C412yescomplemented
192#11719919—42no—
193C4×Dic53—42no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322C22×C414yescomplemented
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—216no—
205C23×C53—24no—
206C23×C53—24no—
207C23×C53—24no—
208C23×C53—48no—
209C23×C53—48no—
210C23×C53—48no—
211C23×C53—48no—
212C23×C53—48no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—48no—
218C23×C53C4×D412yes—
219C23×C53C2×D812yes—
220C23×C53#13512yes—
221C23×C53C22×D411yescharacteristic
222C22×D53—22no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229D4×C53—216no—
230D4×C53—24no—
231D4×C53—24no—
232D4×C53—24no—
233D4×C53C4×D418yescomplemented
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—28no—
243C2×C4×C53—24no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—44no—
248C2×C4×C53—416no—
249C2×C4×C53—48no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—416no—
253C2×C4×C53C4×D412yes—
254C2×C4×C53#12012yes—
255C2×C4×C53#12012yes—
256C2×C4×C53C2×D812yes—
257C2×C4×C53C2×D812yes—
258C2×C4×C53AGL(1,ℤ/8)12yes—
259C2×C4×C53C23×C411yescharacteristic
260C2×C4×C53C22×D412yes—
261C2×C4×C53C2×#4112yes—
262C2×C4×C53C2×#4111yescharacteristic
263C22⋊D53—48no—
264C2×Dic53—22no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273D212—24no—
274D212—42no—
275C4×D53—84no—
276#88322—24no—
277#88322—42no—
278C4×#447—531nomaximal, Sylow
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—28no—
286C22×C53—432no—
287C22×C53—44no—
288C22×C53—44no—
289C22×C53—44no—
290C22×C53—44no—
291C22×C53—44no—
292C22×C53—44no—
293C22×C53C4×D814yescomplemented
294C22×C53C2×C4×D411yescharacteristic
295C22×C53C2×#12011yescharacteristic
296C22×C53#52511yescharacteristic
297C22×C53C22×D811yescharacteristic
298C22×C53C2×#13511yescharacteristic
299C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C53#57511yescharacteristic
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C2×D53—81no—
308C4×C53—22no—
309C4×C53—22no—
310C4×C53—22no—
311C4×C53—44no—
312C4×C53—41no—
313C4×C53#44714yescomplemented
314C4×C53C2×C4×D412yes—
315C4×C53#52512yes—
316Dic53—42no—
317Dic53—81no—
318Dic53—81no—
319C22×C4×D4D5311yescharacteristic, complemented
320C2×#447—531no—
321C2×C4×D8—532no—
322C4×#121—531no—
323#1879—532no—
324#1874—531no—
325#1862—532no—
326#1852—531no—
327#1786—531no—
328C4×#102—532no—
329C4×C22⋊C8—531no—
330C2×C53—24no—
331C2×C53—48no—
332C2×C53C4×#12011yescharacteristic
333C2×C53C2×C4×D811yescharacteristic
334C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
335C2×C53C2×#44711yescharacteristic
336C2×C53#300211yescharacteristic
337C2×C53#310011yescharacteristic
338C2×C53#310511yescharacteristic
339D53—82no—
340C23×D4C2×D5311yescharacteristic
341C24×C4—21no—
342C22×D8—532no—
343C2×#121—531no—
344C2×C4×D4—28no—
345C2×C4×D4—21no—
346C2×C4×D4—21no—
347C2×C4×D4—531no—
348C2×C4×D4—1061no—
349C2×C4×D4C2×D5312yes—
350C2×C4×D4C2×D5312yes—
351C22×C4⋊C4C2×D5311yescharacteristic
352C22×C22⋊C4—21no—
353C22×C42C2×D5311yescharacteristic
354#447—538no—
355C4×D8—1068no—
356C2×#107—532no—
357C2×#102—532no—
358C2×#102—532no—
359C2×#102—532no—
360C2×#102—532no—
361C2×C22⋊C8—531no—
362C2×C22⋊C8—531no—
363C2×C4×C8—532no—
364#390—531no—
365#389—531no—
366#388—1061no—
367#386—1061no—
368#380—531no—
369C4×C4⋊C4—531no—
370C4×C22⋊C4—1061no—
371#340—532no—
372#340—532no—
373#336—532no—
374C53C4×#44711yescharacteristic, complemented, Sylow
375C25—21no—
376C22×D4—24no—
377C22×D4—24no—
378C22×D4—22no—
379C22×D4—21no—
380C22×D4—531no—
381C22×D4—1061no—
382C22×D4C4×D5314yescomplemented
383C22×D4C22×D5312yes—
384C23×C4—28no—
385C23×C4—21no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—44no—
389C23×C4—44no—
390C23×C4—1061no—
391C23×C4C22⋊D5311yescharacteristic
392C23×C4C22⋊D5311yescharacteristic
393C23×C4C22×D5311yescharacteristic
394C2×D8—538no—
395C2×D8—1064no—
396C2×D8—1064no—
397C22×C8—532no—
398C22×C8—532no—
399#121—534no—
400#121—1062no—
401C4×D4—28no—
402C4×D4—28no—
403C4×D4—24no—
404C4×D4—24no—
405C4×D4—24no—
406C4×D4—1064no—
407C4×D4—2124no—
408C4×D4C22⋊D5318yescomplemented
409C2×C4⋊C4—21no—
410C2×C4⋊C4—24no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4C22×D5312yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C42—24no—
425C2×C42—1061no—
426C2×C42C22⋊D5311yescharacteristic
427C2×C42C22⋊D5311yescharacteristic
428C2×C42C22×D5311yescharacteristic
429C2×C42C22×D5311yescharacteristic
430#107—1064no—
431#102—538no—
432#102—1064no—
433#102—1064no—
434#102—1064no—
435C22⋊C8—534no—
436C22⋊C8—1062no—
437C4×C8—1064no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441C24—24no—
442C24—24no—
443C24—24no—
444C24—22no—
445C24—42no—
446C24—42no—
447C24—42no—
448C24—42no—
449C24—1061no—
450C24C2×C22⋊D5311yescharacteristic
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—216no—
454C2×D4—216no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—532no—
460C2×D4—1064no—
461C2×D4—1062no—
462C2×D4—1062no—
463C2×D4—1061no—
464C2×D4—2122no—
465C2×D4—2122no—
466C2×D4C2×C4×D5314yes—
467C2×D4#1171994514yes—
468C2×D4C2×C22⋊D5314yes—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—28no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—41no—
480C22×C4—44no—
481C22×C4—432no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4#1171992214yescomplemented
494C22×C4C2×C4×D5312yes—
495C22×C4#1171994512yes—
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C2×C22⋊D5311yescharacteristic
498C22×C4C2×C22⋊D5311yescharacteristic
499C22×C4C2×C22⋊D5311yescharacteristic
500C22×C4C2×C22⋊D5311yescharacteristic
501C22×C4C23×D5311yescharacteristic
502D8—10616no—
503C2×C8—534no—
504C2×C8—1062no—
505C2×C8—1062no—
506C2×C8—1062no—
507C2×C8—1062no—
508C2×C8—1062no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—532no—
512C4⋊C4—1061no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—2122no—
516C4⋊C4C2×C22⋊D5314yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—1064no—
520C22⋊C4—2122no—
521C22⋊C4—2122no—
522C22⋊C4—2122no—
523C22⋊C4—2122no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—2122no—
528C42C2×C22⋊D5314yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—216no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—48no—
547C23—1062no—
548C23—2121no—
549C23—2121no—
550C23—2121no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23C4×C22⋊D5312yes—
555C23C2×#1171992212yes—
556C23#1313323712yes—
557C23C22×C22⋊D5311yescharacteristic
558D4—216no—
559D4—24no—
560D4—24no—
561D4—24no—
562D4—1064no—
563D4—2128no—
564D4—2122no—
565D4C4×C22⋊D5318yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—44no—
580C2×C4—416no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—416no—
585C2×C4—1062no—
586C2×C4—4244no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4C4×C22⋊D5312yes—
596C2×C4#1313318612yes—
597C2×C4#1313318612yes—
598C2×C4C2×#1171992212yes—
599C2×C4C2×#1171992212yes—
600C2×C4#1313324312yes—
601C2×C4C22×C4×D5311yescharacteristic
602C2×C4C22×C22⋊D5312yes—
603C2×C4C2×#1171994512yes—
604C2×C4C2×#1171994511yescharacteristic
605C8—1064no—
606C8—2122no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—432no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—4241no—
622C22—4241no—
623C22—4241no—
624C22—4241no—
625C22—2122no—
626C22—2122no—
627C22—2122no—
628C22C4×#1171992214yescomplemented
629C22C2×C4×C22⋊D5311yescharacteristic
630C22C2×#1313318611yescharacteristic
631C22#1681113411yescharacteristic
632C22C22×#1171992211yescharacteristic
633C22C2×#1313323711yescharacteristic
634C22C2×#1313324311yescharacteristic
635C22#1681142111yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—44no—
641C4—4241no—
642C4—4241no—
643C4—2122no—
644C4#1681075714yesdirect product factor
645C4C2×C4×C22⋊D5312yes—
646C4#1681113412yes—
647C2—24no—
648C2—48no—
649C2—4242no—
650C2C4×#1313318611yescharacteristic
651C2C2×C4×#1171992211yescharacteristic
652C2C4×#1313324311yescharacteristic
653C2C2×#1681075711yescharacteristic
654C2#2028393311yescharacteristic
655C2#2028471511yescharacteristic
656C2#2028476211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×C4×D4×C53C2
socleC23×C53C22×D4
prime radicalC2×#16810757C2
cosocleC22×C4×C53C24
hypercenterC22×C4×D4D53
hypocenterC53C4×#447
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719922C2semidirect
3. C4×#13133191C2semidirect
4. #20275063C2semidirect
5. #20275032C2semidirect
6. #20274962C2semidirect
7. #20274903C2semidirect
8. #20274646C2semidirect
9. C4×#13133138C2semidirect
10. C4×C22⋊#88322C2semidirect
11. C22×C4×D4×C53C2semidirect
14. C2×C4×D212C22semidirect
16. #16810757C4direct (N × H)
18. C2×#13133150C22semidirect
22. C2×#13133137C22semidirect
25. C2×C4×#88322C22semidirect
26. #16810525C22semidirect
27. #16810521C22semidirect
31. C4×C4⋊Dic53C22semidirect
33. #16810385C22semidirect
34. #16810385C22semidirect
35. #16810378C22semidirect
51. C2×#11719922C2×C4semidirect
55. #13133191C2×C4semidirect
71. #13133138C2×C4semidirect
74. C22⋊#88322C2×C4semidirect
84. C22×D4×C53C2×C4semidirect
100. C4×D4×C53D4semidirect
151. C22×C4×C53D8semidirect
169. C2×D212C22×C4semidirect
191. C4⋊Dic53C22×C4semidirect
199. C2×#88322C22×C4semidirect
233. D4×C53C4×D4semidirect
293. C22×C53C4×D8semidirect
313. C4×C53#447semidirect
319. C22×C4×D4D53semidirect
374. C53C4×#447semidirect
382. C22×D4C4×D53semidirect
408. C4×D4C22⋊D53semidirect
493. C22×C4#11719922semidirect
565. D4C4×C22⋊D53semidirect
628. C22C4×#11719922semidirect
644. C4#16810757direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168107572
C2×C4×#117199222
C4×#131331912
#202750632
#202750322
#202749622
#202749032
#202746462
C4×#131331382
C4×C22⋊#883222
C22×C4×D4×C532
C4×#44753