Finite Group Databasebeta

← C4×#16811014 #27392639

Subgroups — C4×#16811014 #27392639

Order 13568.

1094 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
1C2×#16811014C211yesmaximal, characteristic
2C2×C4×Dic212C211yesmaximal, characteristic, complemented
3C2×C4×#11719925C211yesmaximal, characteristic, complemented
4C2×C4×#11719923C211yesmaximal, characteristic
5C4×#13133215C211yesmaximal, characteristic, complemented
6C4×#13133204C211yesmaximal, characteristic
7#20275598C211yesmaximal, characteristic
8#20275597C211yesmaximal, characteristic
9#20275584C211yesmaximal, characteristic, complemented
10#20275583C211yesmaximal, characteristic, complemented
11#20275533C211yesmaximal, characteristic, complemented
12#20275530C211yesmaximal, characteristic
13#20274880C211yesmaximal, characteristic, complemented
14C4×#13133127C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C53C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17C22×#11719925C2211yescharacteristic
18C22×#11719923C2211yescharacteristic
19C2×#13133215C2211yescharacteristic
20C2×#13133204C2211yescharacteristic
21C2×C4×Dic106C2211yescharacteristic
22C2×C4×Dic106C2211yescharacteristic
23#16811014C418yesdirect product factor
24C4×Dic212—24no—
25C4×#11719925—24no—
26C4×#11719923—24no—
27C2×#13133151C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133141C2211yescharacteristic
33C2×#13133140C2211yescharacteristic
34C2×#13133140C2211yescharacteristic
35C2×#13133138C2211yescharacteristic
36C2×#13133127C2211yescharacteristic
37C2×#13133127C2211yescharacteristic
38C2×#13133127C2211yescharacteristic
39C2×#13133127C2211yescharacteristic
40C2×C4×#88322C2211yescharacteristic
41#16810530C2211yescharacteristic
42#16810528C2211yescharacteristic
43#16810528C2211yescharacteristic
44#16810526C2211yescharacteristic
45#16810518—22no—
46#16810517—22no—
47#16810492C2211yescharacteristic
48#16810488C2211yescharacteristic
49C4×#11719919—22no—
50C4×C22⋊Dic53—22no—
51C42×Dic53C2211yescharacteristic
52C22×SD16×C53C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C2×C4×D4×C53C2211yescharacteristic
55C4×SD16×C53—24no—
56C2×#106×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C2×C4×C8×C53C2211yescharacteristic
60C22×Dic106C2311yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×Dic212—22no—
63C2×Dic212—22no—
64C2×Dic212C2×C414yescomplemented
65C2×#11719925—22no—
66C2×#11719925—22no—
67C2×#11719925C2×C414yescomplemented
68C2×#11719923—22no—
69C2×#11719923—22no—
70C2×#11719923C2×C414yes—
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73#13133215—22no—
74#13133215C2×C414yescomplemented
75#13133204—22no—
76#13133204C2×C414yes—
77C4×Dic106—22no—
78C4×Dic106—22no—
79C4×Dic106—22no—
80C4×Dic106—22no—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—22no—
86C2×#11719919—22no—
87C2×C22⋊Dic53—22no—
88C2×C22⋊Dic53—22no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53D412yes—
92C2×C4×Dic53D412yes—
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95#13133151—22no—
96#13133150—22no—
97#13133148—22no—
98#13133142—22no—
99#13133142—22no—
100#13133141—22no—
101#13133140—22no—
102#13133140—22no—
103#13133138—22no—
104#13133127—22no—
105#13133127—22no—
106#13133127—22no—
107#13133127C2×C414yescomplemented
108C4×#88322D414yes—
109#13133124—22no—
110#13133124—22no—
111#13133124—22no—
112#13133124—22no—
113C22×Q8×C53C2311yescharacteristic
114C22×D4×C53C2311yescharacteristic
115C23×C4×C53—21no—
116C2×SD16×C53—22no—
117C2×SD16×C53—22no—
118C2×SD16×C53C2×C414yes—
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C4×Q8×C53—22no—
122C4×Q8×C53—22no—
123C4×D4×C53—22no—
124C4×D4×C53—22no—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C22⋊C4×C53—21no—
129C2×C42×C53—21no—
130C2×C42×C53C2311yescharacteristic
131#106×C53—22no—
132#103×C53—22no—
133#102×C53—22no—
134C4×C8×C53D414yes—
135C24×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53C22×C412yes—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—21no—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53C22×C412yes—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—42no—
152C22×C4×C53—44no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171C2×Dic106—21no—
172C2×Dic106C22×C412yes—
173C2×Dic106C22×C412yes—
174SD16×C53—28no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53C22×C412yes—
178C2×C8×C53C2×D412yes—
179C2×C8×C53C2×D412yes—
180C2×C8×C53#4112yes—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—42no—
186C22⋊C4×C53—42no—
187C42×C53—42no—
188C42×C53C2×D412yes—
189C42×C53C2×D412yes—
190C22⋊Dic53—28no—
191C22⋊Dic53—44no—
192#11719925—28no—
193#11719923—28no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198#11719919—28no—
199#11719919—44no—
200#11719919—42no—
201#11719919—42no—
202C4×Dic53—24no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—42no—
206C4×Dic53—42no—
207C4×Dic53—42no—
208C4×Dic53C22×C412yes—
209C4×Dic53#4112yes—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322C22×C412yes—
213C2×#88322C2×D412yes—
214C2×#88322C2×D412yes—
215C2×#88322#4112yes—
216Dic212—28no—
217C23×C53—22no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—42no—
221C23×C53—42no—
222C23×C53C22×D411yescharacteristic
223Q8×C53—22no—
224Q8×C53—22no—
225Q8×C53—41no—
226Q8×C53—41no—
227D4×C53—22no—
228D4×C53—22no—
229D4×C53—41no—
230D4×C53—41no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—84no—
240C2×C4×C53#12712yes—
241C2×C4×C53#12712yes—
242C2×C4×C53#13714yes—
243C2×C4×C53C23×C411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53C4×D414yes—
265Dic106—22no—
266Dic106—22no—
267Dic106—22no—
268Dic106—22no—
269Dic106—41no—
270Dic106—41no—
271Dic106—41no—
272Dic106—41no—
273C8×C53—22no—
274C8×C53C4×D414yes—
275#88322—22no—
276#88322C4×D414yes—
277C4×#497—531nomaximal, Sylow
278C22×C53—42no—
279C22×C53—44no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53—82no—
283C22×C53C2×C4×D411yescharacteristic
284C22×C53C2×#12711yescharacteristic
285C22×C53#53211yescharacteristic
286C22×C53C2×#13712yes—
287C22×C53#57512yes—
288C4×C53—22no—
289C4×C53—42no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53#49714yescomplemented
293C4×C53C2×C4×D411yescharacteristic
294C4×C53C2×C4×D411yescharacteristic
295C4×C53#53211yescharacteristic
296C4×C53#53211yescharacteristic
297Dic53—24no—
298Dic53—42no—
299Dic53—42no—
300Dic53—42no—
301Dic53—81no—
302Dic53—81no—
303C2×#497—531no—
304C2×C4×Q16—531no—
305C2×C4×Q16—531no—
306C2×C4×SD16—531no—
307C2×C4×SD16D5311yescharacteristic
308C4×C4⋊Q8—531no—
309C4×#124—531no—
310#1973—531no—
311#1973—531no—
312#1971—531no—
313#1971—531no—
314#1960—531no—
315#1959—531no—
316#1847—531no—
317C4×#97—531no—
318C2×C53—82no—
319C2×C53C4×#12711yescharacteristic
320C2×C53C4×#13712yes—
321C2×C53C2×#49711yescharacteristic
322C2×C53#315011yescharacteristic
323C2×C53#317212yes—
324C22×Q16—531no—
325C22×Q16—531no—
326C22×SD16—531no—
327C22×SD16C2×D5311yescharacteristic
328C2×C4⋊Q8—531no—
329C2×#124—531no—
330C2×C4×Q8—531no—
331C2×C4×Q8—531no—
332C2×C4×Q8C2×D5311yescharacteristic
333C2×C4×D4C2×D5311yescharacteristic
334#497—538no—
335C4×Q16—1064no—
336C4×Q16—1064no—
337C4×SD16—24no—
338C4×SD16—1064no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#106—531no—
342C2×#106C2×D5311yescharacteristic
343C2×#103—531no—
344C2×#103—531no—
345C2×#103—531no—
346C2×#103—531no—
347C2×#103—531no—
348C2×#103C2×D5311yescharacteristic
349C2×#102—531no—
350C2×#102C2×D5311yescharacteristic
351C2×#97—531no—
352C2×#97—531no—
353C2×#97—531no—
354C2×#97—531no—
355C2×C4×C8—531no—
356C2×C4×C8C2×D5311yescharacteristic
357#391—531no—
358#391—531no—
359#391—531no—
360#390—531no—
361#389—1062no—
362#388—1062no—
363#384—531no—
364#382—531no—
365C4×C4⋊C4—1062no—
366C4×C22⋊C4—1062no—
367C43—531no—
368C53C4×#49711yescharacteristic, complemented, Sylow
369C22×Q8—531no—
370C22×Q8—531no—
371C22×Q8C22×D5311yescharacteristic
372C22×D4C22×D5311yescharacteristic
373C23×C4—21no—
374C2×Q16—534no—
375C2×Q16—534no—
376C2×Q16—1062no—
377C2×Q16—1062no—
378C2×Q16—1062no—
379C2×Q16—1062no—
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16—534no—
383C2×SD16—1062no—
384C2×SD16—1062no—
385C2×SD16C4×D5314yes—
386C22×C8—531no—
387C22×C8—531no—
388C22×C8C22×D5311yescharacteristic
389C22×C8C22×D5311yescharacteristic
390C4⋊Q8—534no—
391C4⋊Q8—1062no—
392#124—534no—
393#124—1062no—
394C4×Q8—22no—
395C4×Q8—22no—
396C4×Q8—1062no—
397C4×Q8—1062no—
398C4×Q8—1062no—
399C4×Q8—1062no—
400C4×D4—22no—
401C4×D4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1062no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1062no—
413C2×C22⋊C4—1062no—
414C2×C42—21no—
415C2×C42—531no—
416C2×C42—532no—
417C2×C42—531no—
418C2×C42—532no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42C22×D5311yescharacteristic
422#107—1062no—
423#107—1062no—
424#106—22no—
425#106—1062no—
426#103—22no—
427#103—1062no—
428#103—1062no—
429#103—1062no—
430#103—1062no—
431#103—1062no—
432#102—22no—
433#102—1062no—
434#97—534no—
435#97—1062no—
436#97—1062no—
437#97—1062no—
438C4×C8—534no—
439C4×C8C22⋊D5314yes—
440#95—1062no—
441#95—1062no—
442#95—1062no—
443#95—1062no—
444C24—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—532no—
451C2×Q8—532no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C2×Q8—1061no—
456C2×Q8—1061no—
457C2×Q8—1061no—
458C2×Q8—1061no—
459C2×Q8—1061no—
460C2×Q8—1061no—
461C2×Q8—1061no—
462C2×Q8C2×C4×D5312yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×C4×D5312yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—532no—
475C22×C4—532no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4D4×D5311yescharacteristic
481C22×C4D4×D5311yescharacteristic
482C22×C4C23×D5311yescharacteristic
483Q16—1068no—
484Q16—1068no—
485SD16—28no—
486SD16—1068no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—532no—
490C2×C8—532no—
491C2×C8—532no—
492C2×C8—532no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8C2×C4×D5312yes—
496C2×C8#1171994512yes—
497C2×C8C2×C22⋊D5312yes—
498C2×C8C2×C22⋊D5312yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—1068no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—2124no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C22⋊C4—42no—
513C22⋊C4—1068no—
514C22⋊C4—2124no—
515C42—42no—
516C42—532no—
517C42—532no—
518C42—1064no—
519C42—1061no—
520C42—1061no—
521C42—2122no—
522C42—2122no—
523C42—2122no—
524C42D4×D5312yes—
525C42C2×C22⋊D5312yes—
526C23—22no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23C2×D4×D5311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—1062no—
537Q8—1062no—
538Q8—1062no—
539Q8—1062no—
540Q8—2121no—
541Q8—2121no—
542Q8—2121no—
543Q8—2121no—
544D4—22no—
545D4—22no—
546D4—41no—
547D4—41no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—534no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4#1313321312yes—
574C2×C4#1313321312yes—
575C2×C4#1313324614yes—
576C2×C4C22×C4×D5311yescharacteristic
577C2×C4C2×D4×D5311yescharacteristic
578C2×C4C22×C22⋊D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4C2×#1171994511yescharacteristic
582C8—22no—
583C8—534no—
584C8—1062no—
585C8C4×C22⋊D5314yes—
586C22—42no—
587C22—44no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22C4×D4×D5311yescharacteristic
592C22C2×#1313321311yescharacteristic
593C22#1681117011yescharacteristic
594C22C2×#1313324612yes—
595C22#1681142012yes—
596C4—22no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—1064no—
601C4—4241no—
602C4—4241no—
603C4—2122no—
604C4—2122no—
605C4—2122no—
606C4#1681101414yesdirect product factor
607C4C4×D4×D5311yescharacteristic
608C4C2×C4×C22⋊D5311yescharacteristic
609C4#1681116911yescharacteristic
610C4#1681117011yescharacteristic
611C2—82no—
612C2C4×#1313321311yescharacteristic
613C2C4×#1313324612yes—
614C2C2×#1681101411yescharacteristic
615C2#2028515811yescharacteristic
616C2#2028531712yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C4×SD16×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenterC2×C4×SD16D53
hypocenterC53C4×#497
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic212C2semidirect
3. C2×C4×#11719925C2semidirect
5. C4×#13133215C2semidirect
9. #20275584C2semidirect
10. #20275583C2semidirect
11. #20275533C2semidirect
13. #20274880C2semidirect
14. C4×#13133127C2semidirect
23. #16811014C4direct (N × H)
64. C2×Dic212C2×C4semidirect
67. C2×#11719925C2×C4semidirect
74. #13133215C2×C4semidirect
107. #13133127C2×C4semidirect
292. C4×C53#497semidirect
368. C53C4×#497semidirect
606. C4#16811014direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110142
C2×C4×Dic2122
C2×C4×#117199252
C2×C4×#117199232
C4×#131332152
C4×#131332042
#202755982
#202755972
#202755842
#202755832
#202755332
#202755302
#202748802
C4×#131331272
C2×C4×SD16×C532
C4×#49753