Finite Group Databasebeta

← C2×#20275084 #27385582

Subgroups — C2×#20275084 #27385582

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×#13133247C211yesmaximal, characteristic
2C22×#13133144C211yesmaximal, characteristic
3C2×#16810631C211yesmaximal, characteristic, complemented
4C2×#16810528C211yesmaximal, characteristic, complemented
5#20275084C218yesmaximal, direct product factor
6C2×#16810392C211yesmaximal, characteristic, complemented
7C2×#16810382C211yesmaximal, characteristic, complemented
8C2×C53×#520C211yesmaximal, characteristic
9C23×Dic106C2211yescharacteristic
10C2×#13133247—24no—
11C2×#13133247—22no—
12C2×#13133247C418yescomplemented
13C2×#13133247C2214yes—
14C22×#11719925—21no—
15C22×#11719923—21no—
16C22×#11719917C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19C2×#13133144—24no—
20C2×#13133144C2214yes—
21#16810631—22no—
22#16810631C2214yescomplemented
23C2×#13133140—21no—
24C2×#13133140—21no—
25#16810528—22no—
26#16810528C2214yescomplemented
27C2×#13133124—22no—
28#16810392C2214yescomplemented
29#16810382C2214yescomplemented
30C22×#41×C53C2211yescharacteristic
31C53×#520—22no—
32C53×#520—22no—
33C53×#520—22no—
34C53×#520C2214yes—
35C2×C4×Q8×C53—21no—
36C2×C4×D4×C53—21no—
37C2×C4×D4×C53—22no—
38C2×#117×C53—21no—
39C2×#117×C53C2211yescharacteristic
40C22×C4⋊C4×C53—21no—
41C22×C4⋊C4×C53C2211yescharacteristic
42C22×Dic106—21no—
43C22×Dic106—21no—
44C22×Dic106—21no—
45C22×Dic106—22no—
46C22×Dic106—21no—
47C22×Dic106—22no—
48C22×Dic106—21no—
49C22×Dic106—22no—
50C22×Dic106C2×C414yescomplemented
51C22×Dic106C2312yes—
52C23×Dic53—21no—
53C23×Dic53—21no—
54#13133247—216no—
55#13133247—28no—
56#13133247C2×C4116yescomplemented
57C2×#11719925—28no—
58C2×#11719925—24no—
59C2×#11719923—28no—
60C2×#11719923—24no—
61C2×#11719917—21no—
62C2×#11719917—22no—
63C2×#11719917—21no—
64C2×#11719917—22no—
65C2×#11719917—22no—
66C2×#11719917C2×C414yescomplemented
67C2×#11719917C2312yes—
68C2×#11719917C2312yes—
69C22×#88322—21no—
70C22×#88322—21no—
71C2×C4×Dic53—21no—
72C2×C4×Dic53—22no—
73C2×C4×Dic53—22no—
74C2×C4×Dic53C2312yes—
75#13133144—28no—
76#13133140—24no—
77#13133140—24no—
78#13133124—28no—
79C2×#41×C53—21no—
80C2×#41×C53—21no—
81C2×#41×C53—22no—
82C2×#41×C53—22no—
83C2×#41×C53C2×C414yes—
84C2×#41×C53D414yes—
85C2×#41×C53D412yes—
86C2×#41×C53C2312yes—
87C22×Q8×C53D411yescharacteristic
88C22×D4×C53—21no—
89C22×D4×C53D411yescharacteristic
90C23×C4×C53—21no—
91C23×C4×C53—22no—
92C23×C4×C53D412yes—
93C23×C4×C53C2311yescharacteristic
94C4×Q8×C53—24no—
95C4×Q8×C53—24no—
96C4×D4×C53—24no—
97C4×D4×C53—28no—
98C4×D4×C53—24no—
99C4×D4×C53—28no—
100#117×C53—24no—
101#117×C53—24no—
102#117×C53—21no—
103#117×C53—21no—
104#117×C53D412yes—
105#117×C53C2312yes—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53—24no—
108C2×C4⋊C4×C53—24no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53D412yes—
116C2×C4⋊C4×C53C2312yes—
117C2×C22⋊C4×C53—22no—
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—22no—
120C2×C42×C53—21no—
121C24×C53—21no—
122C24×C53C2×D411yescharacteristic
123#41×C53—24no—
124#41×C53—24no—
125#41×C53—24no—
126#41×C53—24no—
127#41×C53—24no—
128#41×C53—22no—
129#41×C53—22no—
130#41×C53C22⋊C418yes—
131#41×C53C22×C414yes—
132#41×C53C2×D414yes—
133C2×Q8×C53—21no—
134C2×Q8×C53—22no—
135C2×Q8×C53C22⋊C414yes—
136C2×Q8×C53C2×D412yes—
137C2×D4×C53—24no—
138C2×D4×C53—22no—
139C2×D4×C53—24no—
140C2×D4×C53—22no—
141C2×D4×C53—21no—
142C2×D4×C53—22no—
143C2×D4×C53C22⋊C414yes—
144C2×D4×C53C2×D412yes—
145C22×C4×C53—21no—
146C22×C4×C53—24no—
147C22×C4×C53—22no—
148C22×C4×C53—24no—
149C22×C4×C53—22no—
150C22×C4×C53—21no—
151C22×C4×C53—22no—
152C22×C4×C53—21no—
153C22×C4×C53—28no—
154C22×C4×C53—22no—
155C22×C4×C53—44no—
156C22×C4×C53—84no—
157C22×C4×C53C22×C412yes—
158C22×C4×C53C2×D414yes—
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2×D412yes—
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53#4112yes—
166C22×C4×C53#4112yes—
167C22×C4×C53C2411yescharacteristic
168C22×Dic53—24no—
169C22×Dic53—24no—
170C22×Dic53—22no—
171C22×Dic53—22no—
172C22×Dic53—42no—
173C22×Dic53—41no—
174C22×Dic53—41no—
175C22×Dic53—42no—
176C22×Dic53—42no—
177C2×Dic106—24no—
178C2×Dic106—24no—
179C2×Dic106—24no—
180C2×Dic106—24no—
181C2×Dic106—22no—
182C2×Dic106—22no—
183C2×Dic106—22no—
184C2×Dic106—22no—
185C2×Dic106—24no—
186C2×Dic106—22no—
187C2×Dic106—41no—
188C2×Dic106—41no—
189C2×Dic106—42no—
190C2×Dic106—42no—
191C2×Dic106—42no—
192C2×Dic106—42no—
193C2×Dic106—42no—
194C2×Dic106C22×C414yescomplemented
195C4⋊C4×C53—28no—
196C4⋊C4×C53—21no—
197C4⋊C4×C53—22no—
198C4⋊C4×C53—22no—
199C4⋊C4×C53—21no—
200C4⋊C4×C53—22no—
201C4⋊C4×C53—22no—
202C4⋊C4×C53C2×D412yes—
203C4⋊C4×C53C2×D412yes—
204C22⋊C4×C53—28no—
205C22⋊C4×C53—22no—
206C22⋊C4×C53—22no—
207C42×C53—28no—
208C42×C53—22no—
209C42×C53—22no—
210#11719925—216no—
211#11719923—216no—
212C4×Dic53—22no—
213C4×Dic53—22no—
214C4×Dic53—42no—
215#11719917—24no—
216#11719917—22no—
217#11719917—22no—
218#11719917—22no—
219#11719917—42no—
220#11719917C22×C414yescomplemented
221C2×#88322—24no—
222C2×#88322—22no—
223C2×#88322—22no—
224C2×#88322—42no—
225C23×C53—22no—
226C23×C53—22no—
227C23×C53—22no—
228C23×C53—24no—
229C23×C53—42no—
230C23×C53—42no—
231C23×C53C4×D412yes—
232C23×C53#12011yescharacteristic
233C23×C53#12011yescharacteristic
234C23×C53#12112yes—
235C23×C53C22×D411yescharacteristic
236Q8×C53—22no—
237Q8×C53—24no—
238Q8×C53C2×C22⋊C414yes—
239D4×C53—24no—
240D4×C53—24no—
241D4×C53—28no—
242D4×C53—22no—
243D4×C53—24no—
244D4×C53C2×C22⋊C414yes—
245C2×C4×C53—22no—
246C2×C4×C53—24no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—28no—
250C2×C4×C53—22no—
251C2×C4×C53—24no—
252C2×C4×C53—22no—
253C2×C4×C53—44no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—44no—
257C2×C4×C53—82no—
258C2×C4×C53—88no—
259C2×C4×C53C2×C22⋊C412yes—
260C2×C4×C53C2×C22⋊C412yes—
261C2×C4×C53C4×D412yes—
262C2×C4×C53C4×D412yes—
263C2×C4×C53C4×D412yes—
264C2×C4×C53#12011yescharacteristic
265C2×C4×C53#12011yescharacteristic
266C2×C4×C53#12112yes—
267C2×C4×C53#12112yes—
268C2×C4×C53#12112yes—
269C2×C4×C53#12312yes—
270C2×C4×C53#12312yes—
271C2×C4×C53C23×C411yescharacteristic
272C2×C4×C53C22×D411yescharacteristic
273C2×C4×C53C22×D411yescharacteristic
274C2×C4×C53C22×D411yescharacteristic
275C2×C4×C53C2×#4111yescharacteristic
276C2×C4×C53C2×#4111yescharacteristic
277C2×Dic53—24no—
278C2×Dic53—24no—
279C2×Dic53—48no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—44no—
285C2×Dic53—44no—
286Dic106—24no—
287Dic106—24no—
288Dic106—42no—
289Dic106—42no—
290Dic106—44no—
291Dic106—44no—
292#88322—24no—
293#88322—42no—
294C2×#1882—531nomaximal, Sylow
295C22×C53—24no—
296C22×C53—24no—
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—21no—
300C22×C53—21no—
301C22×C53—24no—
302C22×C53—48no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53#38612yes—
306C22×C53#38612yes—
307C22×C53#45512yes—
308C22×C53#45612yes—
309C22×C53C2×C4×D411yescharacteristic
310C22×C53C2×#12011yescharacteristic
311C22×C53C2×#12111yescharacteristic
312C4×C53—24no—
313C4×C53—44no—
314C4×C53—42no—
315C4×C53—42no—
316C4×C53#38612yes—
317C4×C53#38612yes—
318C4×C53C22×C22⋊C411yescharacteristic
319C4×C53C2×C4×D411yescharacteristic
320C4×C53C2×#12111yescharacteristic
321C4×C53C2×#12311yescharacteristic
322Dic53—44no—
323Dic53—44no—
324C22×#137—531no—
325C2×#520D5311yescharacteristic
326C22×#104—531no—
327C2×#419—531no—
328C2×#391—531no—
329#1882—538no—
330C2×#343—531no—
331C2×#339—531no—
332C2×C53—21no—
333C2×C53—21no—
334C2×C53—22no—
335C2×C53—44no—
336C2×C53#188214yescomplemented
337C2×C53C2×#38611yescharacteristic
338C2×C53C2×#45511yescharacteristic
339C2×C53C2×#45611yescharacteristic
340C22×#41C2×D5311yescharacteristic
341C23×Q8—531no—
342C2×#137—538no—
343C2×#137—534no—
344C2×#137—1064no—
345C2×#137—1062no—
346C22×Q16—1061no—
347C22×SD16—1061no—
348C22×M16—531no—
349C22×M16—531no—
350#520—22no—
351#520—22no—
352#520—22no—
353#520C2×D5314yes—
354C2×C4×Q8—21no—
355C2×C4×D4—21no—
356C2×C4×D4—22no—
357C2×#117—21no—
358C2×#117C2×D5311yescharacteristic
359C22×C4⋊C4—21no—
360C22×C4⋊C4C2×D5311yescharacteristic
361C22×C42—531no—
362C2×#104—534no—
363C2×#104—1064no—
364#419—534no—
365#419—1062no—
366C2×#103—1061no—
367C2×#103—1061no—
368#391—534no—
369#391—1062no—
370C2×#95—1062no—
371#343—534no—
372#339—534no—
373C53C2×#188211yescharacteristic, complemented, Sylow
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—21no—
377C2×#41—21no—
378C2×#41C4×D5314yes—
379C2×#41D21214yes—
380C2×#41C22⋊D5312yes—
381C2×#41C22×D5312yes—
382C22×Q8—534no—
383C22×Q8—532no—
384C22×Q8—1062no—
385C22×Q8—1061no—
386C22×Q8—1062no—
387C22×Q8—1062no—
388C22×Q8—1061no—
389C22×Q8—1061no—
390C22×Q8—1061no—
391C22×Q8—1061no—
392C22×Q8C22⋊D5311yescharacteristic
393C22×D4—21no—
394C22×D4C22⋊D5311yescharacteristic
395C23×C4—21no—
396C23×C4—22no—
397C23×C4—1061no—
398C23×C4—1061no—
399C23×C4C22⋊D5312yes—
400C23×C4C22×D5311yescharacteristic
401#137—5316no—
402#137—1068no—
403#137—10616no—
404C2×Q16—1064no—
405C2×Q16—1068no—
406C2×SD16—1064no—
407C2×SD16—1068no—
408C2×M16—534no—
409C2×M16—532no—
410C2×M16—532no—
411C2×M16—1062no—
412C2×M16—1061no—
413C2×M16—1061no—
414C2×M16—1062no—
415C2×M16—1062no—
416C22×C8—1061no—
417C22×C8—1061no—
418C4×Q8—24no—
419C4×Q8—24no—
420C4×D4—28no—
421C4×D4—24no—
422C4×D4—24no—
423C4×D4—28no—
424#117—24no—
425#117—21no—
426#117—21no—
427#117—24no—
428#117C22⋊D5312yes—
429#117C22×D5312yes—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—24no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—24no—
437C2×C4⋊C4C22⋊D5312yes—
438C2×C4⋊C4C22⋊D5311yescharacteristic
439C2×C4⋊C4C22⋊D5311yescharacteristic
440C2×C4⋊C4C22×D5312yes—
441C2×C22⋊C4—22no—
442C2×C22⋊C4—21no—
443C2×C42—22no—
444C2×C42—21no—
445C2×C42—532no—
446C2×C42—1062no—
447C2×C42—1062no—
448C2×C42—1061no—
449#104—1068no—
450#103—1064no—
451#103—1064no—
452#95—1068no—
453C24—21no—
454C24C2×C22⋊D5311yescharacteristic
455#41—24no—
456#41—24no—
457#41—24no—
458#41—24no—
459#41—24no—
460#41—22no—
461#41—22no—
462#41#1171992118yes—
463#41C2×C4×D5314yes—
464#41C2×D21214yes—
465C2×Q8—21no—
466C2×Q8—22no—
467C2×Q8—534no—
468C2×Q8—1064no—
469C2×Q8—1062no—
470C2×Q8—1064no—
471C2×Q8—1064no—
472C2×Q8—1064no—
473C2×Q8—1064no—
474C2×Q8—1062no—
475C2×Q8—1062no—
476C2×Q8—1062no—
477C2×Q8—1062no—
478C2×Q8—2121no—
479C2×Q8—2121no—
480C2×Q8—2122no—
481C2×Q8—2122no—
482C2×Q8—2122no—
483C2×Q8—2122no—
484C2×Q8—2122no—
485C2×Q8#1171992114yes—
486C2×Q8C2×C22⋊D5312yes—
487C2×D4—24no—
488C2×D4—22no—
489C2×D4—24no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—22no—
493C2×D4#1171992114yes—
494C2×D4C2×C22⋊D5312yes—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—28no—
504C22×C4—22no—
505C22×C4—44no—
506C22×C4—84no—
507C22×C4—1064no—
508C22×C4—1064no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—2122no—
512C22×C4—2121no—
513C22×C4—2121no—
514C22×C4—2122no—
515C22×C4—2122no—
516C22×C4C2×C4×D5312yes—
517C22×C4C2×D21212yes—
518C22×C4#1171994512yes—
519C22×C4#1171994512yes—
520C22×C4C2×C22⋊D5314yes—
521C22×C4C2×C22⋊D5311yescharacteristic
522C22×C4C2×C22⋊D5311yescharacteristic
523C22×C4C2×C22⋊D5311yescharacteristic
524C22×C4C2×C22⋊D5311yescharacteristic
525C22×C4C2×C22⋊D5311yescharacteristic
526C22×C4C23×D5311yescharacteristic
527Q16—10616no—
528SD16—10616no—
529M16—534no—
530M16—1064no—
531M16—1062no—
532M16—1062no—
533M16—1062no—
534M16—2122no—
535C2×C8—1064no—
536C2×C8—1062no—
537C2×C8—1062no—
538C2×C8—2122no—
539C4⋊C4—28no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4C2×C22⋊D5312yes—
547C4⋊C4C2×C22⋊D5312yes—
548C22⋊C4—28no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C42—28no—
552C42—22no—
553C42—22no—
554C42—1062no—
555C42—1062no—
556C42—2122no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—42no—
562C23—42no—
563C23C4×C22⋊D5312yes—
564C23#1313318611yescharacteristic
565C23#1313318611yescharacteristic
566C23#1313319112yes—
567C23C22×C22⋊D5311yescharacteristic
568Q8—24no—
569Q8—22no—
570Q8—1064no—
571Q8—1064no—
572Q8—2124no—
573Q8—2124no—
574Q8—2122no—
575Q8—2122no—
576Q8C2×#1171992114yes—
577D4—28no—
578D4—24no—
579D4—24no—
580D4—24no—
581D4—22no—
582D4C2×#1171992114yes—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—82no—
596C2×C4—88no—
597C2×C4—1064no—
598C2×C4—1064no—
599C2×C4—2128no—
600C2×C4—2122no—
601C2×C4—2124no—
602C2×C4—2124no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4C2×#1171992112yes—
607C2×C4C2×#1171992112yes—
608C2×C4C4×C22⋊D5312yes—
609C2×C4C4×C22⋊D5312yes—
610C2×C4C4×C22⋊D5312yes—
611C2×C4#1313318611yescharacteristic
612C2×C4#1313318611yescharacteristic
613C2×C4#1313319112yes—
614C2×C4#1313319112yes—
615C2×C4#1313319112yes—
616C2×C4#1313320012yes—
617C2×C4#1313320012yes—
618C2×C4C22×C4×D5311yescharacteristic
619C2×C4C22×C22⋊D5311yescharacteristic
620C2×C4C22×C22⋊D5311yescharacteristic
621C2×C4C22×D21211yescharacteristic
622C2×C4C2×#1171994511yescharacteristic
623C2×C4C2×#1171994511yescharacteristic
624C8—1064no—
625C8—2122no—
626C22—24no—
627C22—24no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—24no—
633C22—48no—
634C22—42no—
635C22—42no—
636C22#1681050212yes—
637C22#1681050212yes—
638C22#1681080112yes—
639C22#1681080612yes—
640C22C2×C4×C22⋊D5311yescharacteristic
641C22C2×#1313318611yescharacteristic
642C22C2×#1313319111yescharacteristic
643C4—24no—
644C4—44no—
645C4—42no—
646C4—42no—
647C4—2124no—
648C4—2124no—
649C4#1681050212yes—
650C4#1681050212yes—
651C4C22×#1171992111yescharacteristic
652C4C2×C4×C22⋊D5311yescharacteristic
653C4C2×#1313319111yescharacteristic
654C4C2×#1313320011yescharacteristic
655C2—21no—
656C2—22no—
657C2—21no—
658C2—44no—
659C2#2027508414yesdirect product factor
660C2C2×#1681050211yescharacteristic
661C2C2×#1681080111yescharacteristic
662C2C2×#1681080611yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#520C2
socleC23×C53#120
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#520D53
hypocenterC53C2×#1882
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. C2×#16810631C2semidirect
4. C2×#16810528C2semidirect
5. #20275084C2direct (N × H)
6. C2×#16810392C2semidirect
7. C2×#16810382C2semidirect
12. C2×#13133247C4semidirect
22. #16810631C22semidirect
26. #16810528C22semidirect
28. #16810392C22semidirect
29. #16810382C22semidirect
50. C22×Dic106C2×C4semidirect
56. #13133247C2×C4semidirect
66. C2×#11719917C2×C4semidirect
194. C2×Dic106C22×C4semidirect
220. #11719917C22×C4semidirect
336. C2×C53#1882semidirect
373. C53C2×#1882semidirect
659. C2#20275084direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332472
C22×#131331442
C2×#168106312
C2×#168105282
#202750842
C2×#168103922
C2×#168103822
C2×C53×#5202
C2×#188253