Finite Group Databasebeta

← C2×#20275081 #27385579

Subgroups — C2×#20275081 #27385579

Order 13568.

1802 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133248C211yesmaximal, characteristic
2C2×#16811103C211yesmaximal, characteristic
3C22×#13133143C211yesmaximal, characteristic
4C2×#16810530C211yesmaximal, characteristic, complemented
5#20275081C218yesmaximal, direct product factor
6C2×#16810391C211yesmaximal, characteristic, complemented
7C2×#16810382C211yesmaximal, characteristic, complemented
8C2×C53×#419C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C2×#13133248—24no—
11C2×#13133248—22no—
12C2×#13133248C418yescomplemented
13C2×#13133248C2214yes—
14C22×#11719925—21no—
15C22×#11719924—21no—
16C22×#11719917C2211yescharacteristic
17#16811103—22no—
18#16811103—22no—
19#16811103—22no—
20#16811103C2214yes—
21C2×C4×Dic106—21no—
22C2×C4×D212—21no—
23C2×#13133177—22no—
24C2×#13133175—21no—
25C22×C4⋊Dic53C2211yescharacteristic
26C2×C4⋊C4×D53—21no—
27C22×C4×Dic53C2211yescharacteristic
28C2×#13133143—24no—
29C2×#13133143C2214yes—
30#16810530—22no—
31#16810530C2214yescomplemented
32C2×#13133124—22no—
33#16810391C2214yescomplemented
34#16810382C2214yescomplemented
35C23×Q8×C53C2211yescharacteristic
36C22×M16×C53C2211yescharacteristic
37C2×#117×C53C2211yescharacteristic
38C53×#419—22no—
39C53×#419C2214yescomplemented
40C2×#103×C53—21no—
41C2×#103×C53—21no—
42C2×#11719945—21no—
43C2×#11719945—21no—
44C2×#11719945—22no—
45C2×#11719945—22no—
46C2×#11719945C2×C414yes—
47C2×#11719945D414yes—
48C2×#11719945D412yes—
49C2×#11719945C2312yes—
50C22×Dic106D411yescharacteristic
51C22×D212D411yescharacteristic
52C22×C22⋊D53—21no—
53C23×Dic53—21no—
54C23×Dic53D412yes—
55C22×C4×D53—21no—
56C22×C4×D53—22no—
57#13133248—216no—
58#13133248—28no—
59#13133248C2×C4116yescomplemented
60C2×#11719925—28no—
61C2×#11719925—24no—
62C2×#11719924—28no—
63C2×#11719924—24no—
64C2×#11719917—21no—
65C2×#11719917—22no—
66C2×#11719917C2×C414yescomplemented
67C2×#11719917C2312yes—
68C22×#88322—21no—
69C4×Dic106—24no—
70C4×Dic106—24no—
71C4×D212—24no—
72C4×D212—24no—
73#13133177—28no—
74#13133177—28no—
75#13133175—24no—
76#13133175—24no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53D412yes—
82C2×C4⋊Dic53C2312yes—
83C2×#11719919—22no—
84C4⋊C4×D53—24no—
85C4⋊C4×D53—24no—
86C2×#11719921—22no—
87C2×C4×Dic53—22no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—22no—
90C2×C4×Dic53—22no—
91C2×C4×Dic53C2312yes—
92#13133143—28no—
93#13133124—28no—
94C22×Q8×C53—21no—
95C22×Q8×C53—22no—
96C22×Q8×C53—21no—
97C22×Q8×C53—21no—
98C22×Q8×C53—22no—
99C22×Q8×C53—21no—
100C22×Q8×C53—21no—
101C22×Q8×C53—22no—
102C22×Q8×C53C2×C414yescomplemented
103C22×Q8×C53C2312yes—
104C23×C4×C53—21no—
105C23×C4×C53C2311yescharacteristic
106C2×M16×C53—21no—
107C2×M16×C53—22no—
108C2×M16×C53—22no—
109C2×M16×C53C2312yes—
110C22×C8×C53—21no—
111#117×C53—21no—
112#117×C53—21no—
113#117×C53D412yes—
114#117×C53C2312yes—
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C4⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C42×C53—21no—
119#103×C53—24no—
120#103×C53—24no—
121C24×C53C2×D411yescharacteristic
122C23×D53—21no—
123C2×Q8×C53—24no—
124C2×Q8×C53—22no—
125C2×Q8×C53—24no—
126C2×Q8×C53—24no—
127C2×Q8×C53—24no—
128C2×Q8×C53—24no—
129C2×Q8×C53—22no—
130C2×Q8×C53—22no—
131C2×Q8×C53—22no—
132C2×Q8×C53—22no—
133C2×Q8×C53—41no—
134C2×Q8×C53—41no—
135C2×Q8×C53—42no—
136C2×Q8×C53—42no—
137C2×Q8×C53—42no—
138C2×Q8×C53—42no—
139C2×Q8×C53—42no—
140C2×Q8×C53C22×C414yescomplemented
141C22×C4×C53—24no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—22no—
145C22×C4×C53—41no—
146C22×C4×C53—41no—
147C22×C4×C53—42no—
148C22×C4×C53C22×C412yes—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53#4112yes—
156C22×C4×C53#4112yes—
157C22×C4×C53C2411yescharacteristic
158C2×C22⋊D53—24no—
159C2×C22⋊D53—22no—
160C2×C22⋊D53—24no—
161C2×C22⋊D53—22no—
162C22×Dic53—21no—
163C22×Dic53—24no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—42no—
168C22×Dic53—42no—
169C22×Dic53—44no—
170C22×Dic53C2×D414yes—
171#11719945—24no—
172#11719945—24no—
173#11719945—24no—
174#11719945—24no—
175#11719945—24no—
176#11719945—22no—
177#11719945—22no—
178#11719945C22⋊C418yes—
179#11719945C22×C414yes—
180#11719945C2×D414yes—
181C2×D212—21no—
182C2×D212—22no—
183C2×D212C22⋊C414yes—
184C2×D212C2×D412yes—
185C2×C4×D53—24no—
186C2×C4×D53—22no—
187C2×C4×D53—24no—
188C2×C4×D53—22no—
189C2×C4×D53—28no—
190C2×C4×D53—84no—
191C2×Dic106—21no—
192C2×Dic106—22no—
193C2×Dic106C22⋊C414yes—
194C2×Dic106C2×D412yes—
195M16×C53—22no—
196M16×C53—22no—
197M16×C53—42no—
198C2×C8×C53—22no—
199C2×C8×C53—42no—
200C4⋊C4×C53—21no—
201C4⋊C4×C53—21no—
202C4⋊C4×C53C2×D412yes—
203C4⋊C4×C53C2×D412yes—
204C22⋊C4×C53—22no—
205C22⋊C4×C53—22no—
206C42×C53—22no—
207C42×C53—22no—
208#11719925—216no—
209#11719924—216no—
210#11719921—28no—
211C4⋊Dic53—22no—
212C4⋊Dic53—22no—
213C4⋊Dic53—22no—
214C4⋊Dic53—22no—
215#11719919—28no—
216C4×Dic53—28no—
217C4×Dic53—22no—
218C4×Dic53—22no—
219C4×Dic53—42no—
220#11719917—24no—
221#11719917—22no—
222#11719917C22×C414yescomplemented
223C2×#88322—24no—
224C2×#88322—22no—
225C23×C53—22no—
226C23×C53—22no—
227C23×C53C4×D412yes—
228C23×C53#12011yescharacteristic
229C23×C53#12011yescharacteristic
230C23×C53#12112yes—
231C23×C53C22×D411yescharacteristic
232C22×D53—22no—
233C22×D53—24no—
234C22×D53—42no—
235C22×D53—42no—
236Q8×C53—24no—
237Q8×C53—24no—
238Q8×C53—42no—
239Q8×C53—42no—
240Q8×C53—44no—
241Q8×C53—44no—
242C2×C4×C53—24no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—44no—
248C2×C4×C53—44no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53C2×C22⋊C412yes—
252C2×C4×C53C2×C22⋊C412yes—
253C2×C4×C53C4×D412yes—
254C2×C4×C53C4×D412yes—
255C2×C4×C53C4×D412yes—
256C2×C4×C53#12011yescharacteristic
257C2×C4×C53#12011yescharacteristic
258C2×C4×C53#12112yes—
259C2×C4×C53#12112yes—
260C2×C4×C53#12112yes—
261C2×C4×C53#12312yes—
262C2×C4×C53#12312yes—
263C2×C4×C53C23×C411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C22×D411yescharacteristic
266C2×C4×C53C22×D411yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C22⋊D53—24no—
270C22⋊D53—24no—
271C22⋊D53—28no—
272C2×Dic53—22no—
273C2×Dic53—24no—
274C2×Dic53—24no—
275C2×Dic53—24no—
276C2×Dic53—48no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—44no—
280C2×Dic53—44no—
281C2×Dic53—82no—
282D212—22no—
283D212—24no—
284D212C2×C22⋊C414yes—
285C4×D53—24no—
286C4×D53—24no—
287C4×D53—28no—
288C4×D53—88no—
289Dic106—22no—
290Dic106—24no—
291Dic106C2×C22⋊C414yes—
292C8×C53—42no—
293#88322—24no—
294C2×#1882—531nomaximal, Sylow
295C22×C53—24no—
296C22×C53—24no—
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—21no—
300C22×C53—21no—
301C22×C53#38612yes—
302C22×C53#38612yes—
303C22×C53#45512yes—
304C22×C53#45612yes—
305C22×C53C2×C4×D411yescharacteristic
306C22×C53C2×#12011yescharacteristic
307C22×C53C2×#12111yescharacteristic
308C2×D53—24no—
309C2×D53—48no—
310C2×D53—42no—
311C2×D53—42no—
312C4×C53—42no—
313C4×C53—42no—
314C4×C53—44no—
315C4×C53#38612yes—
316C4×C53#38612yes—
317C4×C53C22×C22⋊C411yescharacteristic
318C4×C53C2×C4×D411yescharacteristic
319C4×C53C2×#12111yescharacteristic
320C4×C53C2×#12311yescharacteristic
321Dic53—24no—
322Dic53—44no—
323Dic53—44no—
324C22×#137—531no—
325C2×#520—531no—
326C22×#104—531no—
327C2×#419D5311yescharacteristic, complemented
328C2×#391—531no—
329#1882—538no—
330C2×#343—531no—
331C2×#339—531no—
332C2×C53—21no—
333C2×C53—21no—
334C2×C53—22no—
335C2×C53#188214yescomplemented
336C2×C53C2×#38611yescharacteristic
337C2×C53C2×#45511yescharacteristic
338C2×C53C2×#45611yescharacteristic
339D53—44no—
340C22×#41—531no—
341C23×Q8C2×D5311yescharacteristic
342C2×#137—538no—
343C2×#137—534no—
344C2×#137—1062no—
345C2×#137—1064no—
346C22×Q16—1061no—
347C22×SD16—1061no—
348C22×M16—531no—
349C22×M16C2×D5311yescharacteristic
350#520—534no—
351#520—1062no—
352#520—1062no—
353#520—1062no—
354C2×C4×Q8—1061no—
355C2×C4×D4—1061no—
356C2×C4×D4—1062no—
357C2×#117—1061no—
358C2×#117C2×D5311yescharacteristic
359C22×C4⋊C4—531no—
360C22×C4⋊C4—1061no—
361C22×C42—531no—
362C2×#104—534no—
363C2×#104—1064no—
364#419—22no—
365#419C2×D5314yescomplemented
366C2×#103—21no—
367C2×#103—21no—
368#391—534no—
369#391—1062no—
370C2×#95—1062no—
371#343—534no—
372#339—534no—
373C53C2×#188211yescharacteristic, complemented, Sylow
374C2×#41—534no—
375C2×#41—532no—
376C2×#41—534no—
377C2×#41—532no—
378C2×#41—1062no—
379C2×#41—1061no—
380C2×#41—1062no—
381C2×#41—1061no—
382C22×Q8—21no—
383C22×Q8—22no—
384C22×Q8—22no—
385C22×Q8—21no—
386C22×Q8—21no—
387C22×Q8—22no—
388C22×Q8—21no—
389C22×Q8—21no—
390C22×Q8—531no—
391C22×Q8C4×D5314yescomplemented
392C22×Q8C22×D5312yes—
393C22×D4—531no—
394C22×D4—1061no—
395C23×C4—21no—
396C23×C4—532no—
397C23×C4—1062no—
398C23×C4—1061no—
399C23×C4—1061no—
400C23×C4C22×D5311yescharacteristic
401#137—5316no—
402#137—10616no—
403#137—1068no—
404C2×Q16—1068no—
405C2×Q16—1064no—
406C2×SD16—1068no—
407C2×SD16—1064no—
408C2×M16—21no—
409C2×M16—22no—
410C2×M16—22no—
411C2×M16—534no—
412C2×M16—532no—
413C2×M16—1062no—
414C2×M16—1061no—
415C2×M16C22×D5312yes—
416C22×C8—21no—
417C22×C8—1061no—
418C4×Q8—1064no—
419C4×Q8—1064no—
420C4×D4—1064no—
421C4×D4—1064no—
422C4×D4—1068no—
423C4×D4—1068no—
424#117—21no—
425#117—21no—
426#117—1064no—
427#117—1064no—
428#117C22⋊D5312yes—
429#117C22×D5312yes—
430C2×C4⋊C4—532no—
431C2×C4⋊C4—532no—
432C2×C4⋊C4—1064no—
433C2×C4⋊C4—1062no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1064no—
439C2×C4⋊C4C22⋊D5311yescharacteristic
440C2×C4⋊C4C22⋊D5311yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1062no—
443C2×C42—21no—
444C2×C42—532no—
445C2×C42—1062no—
446C2×C42—1062no—
447C2×C42—1062no—
448C2×C42—1061no—
449#104—1068no—
450#103—24no—
451#103—24no—
452#95—1068no—
453C24—1061no—
454C24D4×D5311yescharacteristic
455#41—538no—
456#41—534no—
457#41—534no—
458#41—1064no—
459#41—1064no—
460#41—1064no—
461#41—1064no—
462#41—1064no—
463#41—1062no—
464#41—1062no—
465C2×Q8—24no—
466C2×Q8—22no—
467C2×Q8—24no—
468C2×Q8—24no—
469C2×Q8—24no—
470C2×Q8—24no—
471C2×Q8—22no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—22no—
475C2×Q8—41no—
476C2×Q8—41no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—42no—
481C2×Q8—42no—
482C2×Q8—534no—
483C2×Q8—532no—
484C2×Q8—1061no—
485C2×Q8—1062no—
486C2×Q8C2×C4×D5314yescomplemented
487C2×D4—534no—
488C2×D4—532no—
489C2×D4—1064no—
490C2×D4—1062no—
491C2×D4—1064no—
492C2×D4—1062no—
493C2×D4—1061no—
494C2×D4—1062no—
495C22×C4—24no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—42no—
502C22×C4—534no—
503C22×C4—1068no—
504C22×C4—1061no—
505C22×C4—1064no—
506C22×C4—1062no—
507C22×C4—1064no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1064no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—4244no—
514C22×C4—2124no—
515C22×C4—2122no—
516C22×C4—2122no—
517C22×C4C2×C4×D5312yes—
518C22×C4#1171994512yes—
519C22×C4D4×D5311yescharacteristic
520C22×C4D4×D5311yescharacteristic
521C22×C4D4×D5312yes—
522C22×C4D4×D5311yescharacteristic
523C22×C4#1171994712yes—
524C22×C4C2×C22⋊D5311yescharacteristic
525C22×C4C2×C22⋊D5311yescharacteristic
526C22×C4C23×D5311yescharacteristic
527Q16—10616no—
528SD16—10616no—
529M16—22no—
530M16—22no—
531M16—42no—
532M16—534no—
533M16—1064no—
534M16—1062no—
535C2×C8—22no—
536C2×C8—42no—
537C2×C8—1064no—
538C2×C8—1062no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—1068no—
542C4⋊C4—1062no—
543C4⋊C4—1062no—
544C4⋊C4—1062no—
545C4⋊C4—1062no—
546C4⋊C4C2×C22⋊D5312yes—
547C4⋊C4C2×C22⋊D5312yes—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—1068no—
551C42—22no—
552C42—22no—
553C42—1068no—
554C42—1062no—
555C42—1062no—
556C42—2122no—
557C23—22no—
558C23—22no—
559C23—1064no—
560C23—1062no—
561C23—2122no—
562C23—2122no—
563C23#1313317712yes—
564C23#1313318511yescharacteristic
565C23#1313318511yescharacteristic
566C23#1313319012yes—
567C23C2×D4×D5311yescharacteristic
568Q8—24no—
569Q8—24no—
570Q8—44no—
571Q8—44no—
572Q8—42no—
573Q8—42no—
574Q8—534no—
575Q8—1064no—
576Q8—1062no—
577D4—534no—
578D4—1068no—
579D4—1064no—
580D4—1064no—
581D4—1064no—
582D4—1062no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—1064no—
593C2×C4—1062no—
594C2×C4—1068no—
595C2×C4—1064no—
596C2×C4—1064no—
597C2×C4—1064no—
598C2×C4—1064no—
599C2×C4—4248no—
600C2×C4—4242no—
601C2×C4—2128no—
602C2×C4—2122no—
603C2×C4—2124no—
604C2×C4—2122no—
605C2×C4—2124no—
606C2×C4C22⋊C4×D5312yes—
607C2×C4C22⋊C4×D5312yes—
608C2×C4#1313317712yes—
609C2×C4C4×C22⋊D5312yes—
610C2×C4C4×C22⋊D5312yes—
611C2×C4#1313318511yescharacteristic
612C2×C4#1313318511yescharacteristic
613C2×C4#1313318812yes—
614C2×C4#1313318812yes—
615C2×C4#1313319012yes—
616C2×C4#1313319912yes—
617C2×C4#1313319912yes—
618C2×C4C22×C4×D5311yescharacteristic
619C2×C4C2×D4×D5311yescharacteristic
620C2×C4C2×D4×D5311yescharacteristic
621C2×C4C22×C22⋊D5311yescharacteristic
622C2×C4C2×#1171994711yescharacteristic
623C2×C4C2×#1171994511yescharacteristic
624C8—42no—
625C8—1064no—
626C22—24no—
627C22—24no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—1064no—
633C22—2128no—
634C22—2122no—
635C22—2122no—
636C22#1681049912yes—
637C22#1681049912yes—
638C22#1681079912yes—
639C22#1681080412yes—
640C22C2×#1313317711yescharacteristic
641C22C2×#1313318511yescharacteristic
642C22C2×#1313319011yescharacteristic
643C4—44no—
644C4—42no—
645C4—42no—
646C4—1064no—
647C4—2124no—
648C4—2124no—
649C4#1681049912yes—
650C4#1681049912yes—
651C4C2×C22⋊C4×D5311yescharacteristic
652C4C2×C4×C22⋊D5311yescharacteristic
653C4C2×#1313318811yescharacteristic
654C4C2×#1313319911yescharacteristic
655C2—21no—
656C2—22no—
657C2—21no—
658C2—2124no—
659C2#2027508114yesdirect product factor
660C2C2×#1681049911yescharacteristic
661C2C2×#1681079911yescharacteristic
662C2C2×#1681080411yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#419C2
socleC23×C53#120
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#419D53
hypocenterC53C2×#1882
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
4. C2×#16810530C2semidirect
5. #20275081C2direct (N × H)
6. C2×#16810391C2semidirect
7. C2×#16810382C2semidirect
8. C2×C53×#419C2semidirect
12. C2×#13133248C4semidirect
31. #16810530C22semidirect
33. #16810391C22semidirect
34. #16810382C22semidirect
39. C53×#419C22semidirect
59. #13133248C2×C4semidirect
66. C2×#11719917C2×C4semidirect
102. C22×Q8×C53C2×C4semidirect
140. C2×Q8×C53C22×C4semidirect
222. #11719917C22×C4semidirect
327. C2×#419D53semidirect
335. C2×C53#1882semidirect
365. #419C2×D53semidirect
373. C53C2×#1882semidirect
391. C22×Q8C4×D53semidirect
486. C2×Q8C2×C4×D53semidirect
659. C2#20275081direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332482
C2×#168111032
C22×#131331432
C2×#168105302
#202750812
C2×#168103912
C2×#168103822
C2×C53×#4192
C2×#188253