Finite Group Databasebeta

← C2×#20274923 #27384893

Subgroups — C2×#20274923 #27384893

Order 13568.

1706 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811097C211yesmaximal, characteristic
2C2×#16810663C211yesmaximal, characteristic, complemented
3C22×#13133133C211yesmaximal, characteristic
4C2×#16810524C211yesmaximal, characteristic
5#20274923C218yesmaximal, direct product factor
6C2×#16810384C211yesmaximal, characteristic, complemented
7C2×#16810382C211yesmaximal, characteristic, complemented
8C2×C53×#428C211yesmaximal, characteristic, complemented
9C23×D212C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11#16811097—22no—
12#16811097—22no—
13#16811097—22no—
14#16811097C2214yes—
15C2×C4×D212—21no—
16C2×C4×D212—21no—
17C2×#13133179—21no—
18C2×#13133179—21no—
19C2×#13133176C2211yescharacteristic
20C22×#11719921—22no—
21C2×C22⋊C4×D53—22no—
22C22×C4×Dic53C2211yescharacteristic
23#16810663—22no—
24#16810663C2214yescomplemented
25C2×#13133150—21no—
26C2×#13133148—21no—
27C2×#13133133C418yescomplemented
28C2×#13133133C2214yes—
29#16810524—22no—
30#16810524C2214yes—
31#16810384C2214yescomplemented
32#16810382C2214yescomplemented
33C22×M16×C53C2211yescharacteristic
34C2×#117×C53C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C53×#428—22no—
37C53×#428C2214yescomplemented
38C2×#107×C53—21no—
39C2×#106×C53—21no—
40C24×D53D412yes—
41C22×D212—21no—
42C22×D212—21no—
43C22×D212—21no—
44C22×D212—21no—
45C22×D212—22no—
46C22×D212—22no—
47C22×D212C2×C414yes—
48C22×D212D412yes—
49C22×D212Q814yes—
50C22×D212C2312yes—
51C23×Dic53—21no—
52C22×C4×D53—22no—
53C22×C4×D53—22no—
54C2×#11719917—21no—
55C2×#11719917—22no—
56C2×#11719917C2×C414yescomplemented
57C2×#11719917C2312yes—
58C22×#88322—21no—
59C4×D212—24no—
60C4×D212—24no—
61C4×D212—24no—
62C4×D212—24no—
63#13133179—24no—
64#13133179—24no—
65#13133179—24no—
66#13133179—24no—
67#13133176—21no—
68#13133176—21no—
69#13133176D412yes—
70#13133176C2312yes—
71C2×C4⋊Dic53D411yescharacteristic
72C2×C4⋊Dic53D411yescharacteristic
73C2×C22⋊Dic53—21no—
74C2×#11719921—22no—
75C2×#11719921—22no—
76C2×#11719921—28no—
77C2×#11719921—48no—
78C22⋊C4×D53—28no—
79C22⋊C4×D53—28no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53C2312yes—
85#13133150—24no—
86#13133148—24no—
87#13133133C2×C4116yescomplemented
88C23×C4×C53—21no—
89C23×C4×C53C2311yescharacteristic
90C2×M16×C53—21no—
91C2×M16×C53—22no—
92C2×M16×C53C2×C414yescomplemented
93C2×M16×C53C2312yes—
94C22×C8×C53—21no—
95#117×C53—21no—
96#117×C53—21no—
97#117×C53D412yes—
98#117×C53C2312yes—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53C2312yes—
107C2×C22⋊C4×C53—21no—
108C2×C42×C53—21no—
109#107×C53—24no—
110#106×C53—24no—
111C24×C53C2×D411yescharacteristic
112C23×D53—22no—
113C23×D53—22no—
114C23×D53—24no—
115C23×D53—44no—
116C23×D53—44no—
117C23×D53C4⋊C418yes—
118C23×D53C2×D414yes—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—44no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×Q812yes—
132C22×C4×C53#4112yes—
133C22×C4×C53#4112yes—
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—24no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—42no—
140C22×Dic53—42no—
141C2×D212—24no—
142C2×D212—24no—
143C2×D212—24no—
144C2×D212—24no—
145C2×D212—22no—
146C2×D212—22no—
147C2×D212—24no—
148C2×D212—24no—
149C2×D212—24no—
150C2×D212—24no—
151C2×D212—24no—
152C2×D212—22no—
153C2×D212—22no—
154C2×D212—22no—
155C2×D212—22no—
156C2×D212C4⋊C418yes—
157C2×D212C22×C414yes—
158C2×D212C2×Q814yes—
159C2×C4×D53—28no—
160C2×C4×D53—28no—
161C2×C4×D53—84no—
162C2×C4×D53—84no—
163M16×C53—24no—
164M16×C53—22no—
165M16×C53C22×C414yescomplemented
166C2×C8×C53—24no—
167C2×C8×C53—22no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—41no—
173C4⋊C4×C53—41no—
174C4⋊C4×C53C2×D412yes—
175C4⋊C4×C53C2×D412yes—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C42×C53—22no—
179C42×C53—22no—
180C22⋊Dic53—22no—
181C22⋊Dic53—22no—
182#11719921—28no—
183#11719921—28no—
184#11719921—416no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53C2×D412yes—
188C4⋊Dic53C2×D412yes—
189C4×Dic53—22no—
190C4×Dic53—22no—
191C4×Dic53—22no—
192C4×Dic53—22no—
193C4×Dic53—42no—
194#11719917—24no—
195#11719917—22no—
196#11719917C22×C414yescomplemented
197C2×#88322—24no—
198C2×#88322—22no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53C2×C4⋊C412yes—
202C23×C53#12011yescharacteristic
203C23×C53#12011yescharacteristic
204C23×C53#12312yes—
205C23×C53C22×D411yescharacteristic
206C22×D53—24no—
207C22×D53—28no—
208C22×D53—24no—
209C22×D53—28no—
210C22×D53—44no—
211C22×D53—44no—
212C22×D53—416no—
213C22×D53—44no—
214C22×D53—44no—
215C22×D53—88no—
216C22×D53—82no—
217C22×D53—82no—
218C22×D53C2×C4⋊C418yes—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—44no—
224C2×C4×C53—42no—
225C2×C4×C53—82no—
226C2×C4×C53—82no—
227C2×C4×C53C2×C4⋊C412yes—
228C2×C4×C53C4×D412yes—
229C2×C4×C53C4×D412yes—
230C2×C4×C53C4×D412yes—
231C2×C4×C53C4×D412yes—
232C2×C4×C53#12011yescharacteristic
233C2×C4×C53#12011yescharacteristic
234C2×C4×C53C22⋊Q812yes—
235C2×C4×C53C22⋊Q812yes—
236C2×C4×C53C22⋊Q812yes—
237C2×C4×C53C22⋊Q812yes—
238C2×C4×C53#12312yes—
239C2×C4×C53C23×C411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×Q811yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×Dic53—24no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—48no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253D212—24no—
254D212—24no—
255D212—24no—
256D212—24no—
257D212—28no—
258D212—28no—
259C4×D53—88no—
260C4×D53—88no—
261C8×C53—24no—
262#88322—24no—
263C2×#1856—531nomaximal, Sylow
264C22×C53—24no—
265C22×C53—24no—
266C22×C53—21no—
267C22×C53—21no—
268C22×C53—21no—
269C22×C53—21no—
270C22×C53#38512yes—
271C22×C53#38512yes—
272C22×C53#45312yes—
273C22×C53#45512yes—
274C22×C53C22×C4⋊C411yescharacteristic
275C22×C53C2×#12011yescharacteristic
276C22×C53C2×#12311yescharacteristic
277C2×D53—28no—
278C2×D53—416no—
279C2×D53—44no—
280C2×D53—44no—
281C2×D53—48no—
282C2×D53—84no—
283C2×D53—88no—
284C4×C53—42no—
285C4×C53—42no—
286C4×C53—82no—
287C4×C53#38512yes—
288C4×C53#38512yes—
289C4×C53C2×C4×D411yescharacteristic
290C4×C53C2×C4×D411yescharacteristic
291C4×C53C2×C22⋊Q811yescharacteristic
292C4×C53C2×C22⋊Q811yescharacteristic
293Dic53—44no—
294Dic53—42no—
295Dic53—42no—
296C2×#518—531no—
297C2×#428—531no—
298C2×#428D5311yescharacteristic, complemented
299C22×#100—531no—
300C2×#390—531no—
301#1856—538no—
302C2×#339—531no—
303C2×#339—531no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53—22no—
307C2×C53#185614yescomplemented
308C2×C53C2×#38511yescharacteristic
309C2×C53C2×#45311yescharacteristic
310C2×C53C2×#45511yescharacteristic
311D53—48no—
312C23×D4—531no—
313C22×M16—531no—
314C22×M16C2×D5311yescharacteristic
315#518—534no—
316#518—1062no—
317#518—1062no—
318#518—1062no—
319C2×C4×D4—1061no—
320C2×C4×D4—1061no—
321C2×C4×D4—1061no—
322C2×C4×D4—1061no—
323C2×#117—531no—
324C2×#117C2×D5311yescharacteristic
325C22×C4⋊C4C2×D5311yescharacteristic
326C22×C22⋊C4—1062no—
327C22×C22⋊C4—1062no—
328C22×C42—531no—
329#428—22no—
330#428—534no—
331#428—1062no—
332#428C2×D5314yescomplemented
333C2×#107—21no—
334C2×#107—1061no—
335C2×#106—21no—
336C2×#106—1061no—
337C2×#100—538no—
338C2×#100—534no—
339#390—534no—
340#390—1062no—
341#339—534no—
342#339—534no—
343C53C2×#185611yescharacteristic, complemented, Sylow
344C25—532no—
345C22×D4—534no—
346C22×D4—532no—
347C22×D4—534no—
348C22×D4—532no—
349C22×D4—1062no—
350C22×D4—1061no—
351C22×D4—1061no—
352C22×D4—1062no—
353C22×D4—1061no—
354C22×D4—1061no—
355C23×C4—21no—
356C23×C4—1062no—
357C23×C4—1062no—
358C23×C4—1061no—
359C23×C4C22×D5311yescharacteristic
360C2×M16—22no—
361C2×M16—21no—
362C2×M16—534no—
363C2×M16—532no—
364C2×M16—1061no—
365C2×M16—1062no—
366C2×M16C4×D5314yescomplemented
367C2×M16C22×D5312yes—
368C22×C8—21no—
369C22×C8—1061no—
370C4×D4—1064no—
371C4×D4—1064no—
372C4×D4—1064no—
373C4×D4—1064no—
374C4×D4—1064no—
375C4×D4—1064no—
376C4×D4—1064no—
377C4×D4—1064no—
378#117—21no—
379#117—21no—
380#117—532no—
381#117—532no—
382#117—1061no—
383#117—1061no—
384#117C22⋊D5312yes—
385#117C22×D5312yes—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4C22⋊D5311yescharacteristic
394C2×C4⋊C4C22⋊D5311yescharacteristic
395C2×C4⋊C4C22×D5312yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1068no—
398C2×C22⋊C4—1068no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1062no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1068no—
403C2×C22⋊C4—2128no—
404C2×C42—21no—
405C2×C42—532no—
406C2×C42—1061no—
407C2×C42—1062no—
408C2×C42—1062no—
409C2×C42—1061no—
410#107—24no—
411#107—1064no—
412#106—24no—
413#106—1064no—
414#100—5316no—
415C24—538no—
416C24—534no—
417C24—1062no—
418C24—1064no—
419C24—1062no—
420C24—2124no—
421C24—2124no—
422C24D4×D5311yescharacteristic
423C2×D4—538no—
424C2×D4—534no—
425C2×D4—534no—
426C2×D4—1064no—
427C2×D4—1064no—
428C2×D4—1064no—
429C2×D4—1064no—
430C2×D4—1062no—
431C2×D4—1062no—
432C2×D4—1064no—
433C2×D4—1064no—
434C2×D4—1064no—
435C2×D4—1064no—
436C2×D4—1064no—
437C2×D4—1062no—
438C2×D4—1062no—
439C2×D4—1062no—
440C2×D4—1062no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—1068no—
448C22×C4—1068no—
449C22×C4—1064no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1062no—
453C22×C4—4244no—
454C22×C4—4244no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4C2×C4×D5312yes—
458C22×C4#1171994512yes—
459C22×C4D4×D5311yescharacteristic
460C22×C4D4×D5311yescharacteristic
461C22×C4D4×D5311yescharacteristic
462C22×C4Q8×D5312yes—
463C22×C4#1171994912yes—
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C23×D5311yescharacteristic
467M16—24no—
468M16—22no—
469M16—534no—
470M16—1064no—
471M16—1062no—
472M16C2×C4×D5314yescomplemented
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—1064no—
476C2×C8—1062no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—532no—
484C4⋊C4—532no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4C2×C22⋊D5312yes—
488C4⋊C4C2×C22⋊D5312yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—1068no—
492C22⋊C4—1068no—
493C22⋊C4—1062no—
494C22⋊C4—1062no—
495C22⋊C4—21216no—
496C42—22no—
497C42—22no—
498C42—1062no—
499C42—1062no—
500C42—1062no—
501C42—1062no—
502C42—2122no—
503C23—22no—
504C23—22no—
505C23—538no—
506C23—1064no—
507C23—1068no—
508C23—1064no—
509C23—1068no—
510C23—4242no—
511C23—4242no—
512C23—4248no—
513C23—2124no—
514C23—2124no—
515C23—21216no—
516C23—2124no—
517C23—2124no—
518C23C4⋊C4×D5312yes—
519C23#1313318511yescharacteristic
520C23#1313318511yescharacteristic
521C23#1313320212yes—
522C23C2×D4×D5311yescharacteristic
523D4—1064no—
524D4—1064no—
525D4—1064no—
526D4—1064no—
527D4—1068no—
528D4—1068no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—1064no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—4248no—
541C2×C4—4248no—
542C2×C4—2122no—
543C2×C4—2128no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4C4⋊C4×D5312yes—
548C2×C4C4×C22⋊D5312yes—
549C2×C4C4×C22⋊D5312yes—
550C2×C4#1313317912yes—
551C2×C4#1313317912yes—
552C2×C4#1313318511yescharacteristic
553C2×C4#1313318511yescharacteristic
554C2×C4#1313319312yes—
555C2×C4#1313319312yes—
556C2×C4#1313319512yes—
557C2×C4#1313319512yes—
558C2×C4#1313320212yes—
559C2×C4C22×C4×D5311yescharacteristic
560C2×C4C2×D4×D5311yescharacteristic
561C2×C4C22×C22⋊D5311yescharacteristic
562C2×C4C2×Q8×D5311yescharacteristic
563C2×C4C2×#1171994511yescharacteristic
564C2×C4C2×#1171994911yescharacteristic
565C8—24no—
566C8—1064no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—24no—
572C22—24no—
573C22—1068no—
574C22—4244no—
575C22—4248no—
576C22—2128no—
577C22—2124no—
578C22—2124no—
579C22—21216no—
580C22#1681049412yes—
581C22#1681049412yes—
582C22#1681078912yes—
583C22#1681079912yes—
584C22C2×C4⋊C4×D5311yescharacteristic
585C22C2×#1313318511yescharacteristic
586C22C2×#1313320211yescharacteristic
587C4—42no—
588C4—42no—
589C4—82no—
590C4—2124no—
591C4—2122no—
592C4—2122no—
593C4#1681049412yes—
594C4#1681049412yes—
595C4C2×C4×C22⋊D5311yescharacteristic
596C4C2×#1313317911yescharacteristic
597C4C2×#1313319311yescharacteristic
598C4C2×#1313319511yescharacteristic
599C2—21no—
600C2—22no—
601C2—21no—
602C2—2128no—
603C2#2027492314yesdirect product factor
604C2C2×#1681049411yescharacteristic
605C2C2×#1681078911yescharacteristic
606C2C2×#1681079911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC23×C53#120
prime radicalC23×D212C22
cosocleC22×C4×C53C24
hypercenterC2×#428D53
hypocenterC53C2×#1856
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#16810663C2semidirect
5. #20274923C2direct (N × H)
6. C2×#16810384C2semidirect
7. C2×#16810382C2semidirect
8. C2×C53×#428C2semidirect
24. #16810663C22semidirect
27. C2×#13133133C4semidirect
31. #16810384C22semidirect
32. #16810382C22semidirect
37. C53×#428C22semidirect
56. C2×#11719917C2×C4semidirect
87. #13133133C2×C4semidirect
92. C2×M16×C53C2×C4semidirect
165. M16×C53C22×C4semidirect
196. #11719917C22×C4semidirect
298. C2×#428D53semidirect
307. C2×C53#1856semidirect
332. #428C2×D53semidirect
343. C53C2×#1856semidirect
366. C2×M16C4×D53semidirect
472. M16C2×C4×D53semidirect
603. C2#20274923direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110972
C2×#168106632
C22×#131331332
C2×#168105242
#202749232
C2×#168103842
C2×#168103822
C2×C53×#4282
C2×#185653