Finite Group Databasebeta

← C2×#20275060 #27385413

Subgroups — C2×#20275060 #27385413

Order 13568.

2470 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719922C211yesmaximal, characteristic
2C22×#13133179C211yesmaximal, characteristic, complemented
3C22×#13133139C211yesmaximal, characteristic, complemented
4C2×#16810526C211yesmaximal, characteristic, complemented
5#20275060C218yesmaximal, direct product factor
6C2×#16810385C211yesmaximal, characteristic, complemented
7C2×#16810377C211yesmaximal, characteristic, complemented
8C22×#102×C53C211yesmaximal, characteristic, complemented
9C23×D212C2211yescharacteristic
10C23×C4×D53—21no—
11C22×#11719922—24no—
12C22×#11719922—24no—
13C22×#11719922C418yescomplemented
14C22×#11719922C2214yes—
15C23×#88322C2211yescharacteristic
16C2×#13133179—24no—
17C2×#13133179—22no—
18C2×#13133179—24no—
19C2×#13133179C2214yescomplemented
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C22⋊Dic53—21no—
22C22×C22⋊Dic53—21no—
23C22×#11719921—21no—
24C22×C4×Dic53C2211yescharacteristic
25C2×#13133139—24no—
26C2×#13133139C2214yescomplemented
27#16810526—22no—
28#16810526C2214yescomplemented
29#16810385C2214yescomplemented
30#16810377C2214yescomplemented
31C23×D4×C53C2211yescharacteristic
32C23×C8×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C2×#102×C53—24no—
35C2×#102×C53C2214yescomplemented
36C24×D53—21no—
37C22×D212—21no—
38C22×D212—22no—
39C22×D212—22no—
40C22×D212—22no—
41C22×D212C2×C414yes—
42C22×D212D414yes—
43C22×D212D414yes—
44C22×D212C2312yes—
45C23×Dic53—21no—
46C23×Dic53D411yescharacteristic
47C23×Dic53D411yescharacteristic
48C22×C4×D53—28no—
49C22×C4×D53—24no—
50C22×C4×D53—44no—
51C22×C4×D53—44no—
52C2×#11719922—232no—
53C2×#11719922—28no—
54C2×#11719922—28no—
55C2×#11719922C2×C4116yescomplemented
56C22×#88322—22no—
57C22×#88322—22no—
58C22×#88322C2×C414yescomplemented
59C22×#88322C2312yes—
60#13133179—216no—
61#13133179—28no—
62#13133179—28no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—22no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53C2312yes—
67C2×C22⋊Dic53—24no—
68C2×C22⋊Dic53—24no—
69C2×C22⋊Dic53—44no—
70C2×C22⋊Dic53—44no—
71C2×#11719921—28no—
72C2×#11719921—24no—
73C2×C4×Dic53—22no—
74C2×C4×Dic53—22no—
75C2×C4×Dic53D412yes—
76C2×C4×Dic53C2312yes—
77#13133139—28no—
78C25×C53—21no—
79C22×D4×C53—22no—
80C22×D4×C53—22no—
81C22×D4×C53—22no—
82C22×D4×C53—21no—
83C22×D4×C53—22no—
84C22×D4×C53—22no—
85C22×D4×C53C2×C414yescomplemented
86C22×D4×C53C2312yes—
87C23×C4×C53—21no—
88C23×C4×C53C2311yescharacteristic
89C22×C8×C53—22no—
90C22×C8×C53—22no—
91C22×C8×C53—22no—
92C22×C8×C53C2312yes—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53D414yes—
96C2×C4⋊C4×C53C2312yes—
97#102×C53—28no—
98C24×C53—24no—
99C24×C53—22no—
100C24×C53—42no—
101C24×C53—42no—
102C24×C53—42no—
103C24×C53—42no—
104C24×C53—42no—
105C24×C53—42no—
106C24×C53C2×D411yescharacteristic
107C23×D53—24no—
108C23×D53—24no—
109C23×D53—24no—
110C23×D53—22no—
111C23×D53—42no—
112C23×D53—42no—
113C23×D53—42no—
114C23×D53—42no—
115C2×D4×C53—28no—
116C2×D4×C53—28no—
117C2×D4×C53—24no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—42no—
124C2×D4×C53—44no—
125C2×D4×C53—42no—
126C2×D4×C53—44no—
127C2×D4×C53C22×C414yescomplemented
128C22×C4×C53—24no—
129C22×C4×C53—22no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D412yes—
137C22×C4×C53#4112yes—
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53D814yescomplemented
150C22×Dic53C2×D412yes—
151C22×Dic53C2×D412yes—
152C2×D212—28no—
153C2×D212—24no—
154C2×D212—24no—
155C2×D212—28no—
156C2×D212—24no—
157C2×D212—28no—
158C2×D212—22no—
159C2×D212—22no—
160C2×D212—22no—
161C2×D212C22⋊C4116yescomplemented
162C2×D212C22×C414yes—
163C2×D212C2×D414yes—
164C2×D212C2×D414yes—
165C2×C4×D53—216no—
166C2×C4×D53—48no—
167C2×C4×D53—432no—
168C2×C4×D53—48no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C2×C8×C53—22no—
172C2×C8×C53—44no—
173C4⋊C4×C53—24no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53C2×D414yes—
176C22⋊Dic53—48no—
177C22⋊Dic53—48no—
178#11719922—232no—
179#11719921—216no—
180C4⋊Dic53—22no—
181C4⋊Dic53—22no—
182C4⋊Dic53—42no—
183C4×Dic53—24no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C2×#88322—28no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322C22×C414yescomplemented
190C23×C53—24no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—48no—
198C23×C53—48no—
199C23×C53—48no—
200C23×C53—84no—
201C23×C53—84no—
202C23×C53—84no—
203C23×C53—84no—
204C23×C53C4×D412yes—
205C23×C53#12012yes—
206C23×C53#12112yes—
207C23×C53C2×D812yes—
208C23×C53#13512yes—
209C23×C53AGL(1,ℤ/8)12yes—
210C23×C53AGL(1,ℤ/8)12yes—
211C23×C53C22×D411yescharacteristic
212C22×D53—24no—
213C22×D53—24no—
214C22×D53—24no—
215C22×D53—216no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—48no—
221C22×D53—48no—
222C22×D53—48no—
223C22×D53—48no—
224C22×D53—48no—
225C22×D53—48no—
226D4×C53—24no—
227D4×C53—24no—
228D4×C53—42no—
229D4×C53—42no—
230D4×C53—48no—
231C2×C4×C53—24no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—48no—
235C2×C4×C53C2×C22⋊C414yes—
236C2×C4×C53C4×D412yes—
237C2×C4×C53C4×D414yes—
238C2×C4×C53#12012yes—
239C2×C4×C53#12114yes—
240C2×C4×C53#12112yes—
241C2×C4×C53#12314yes—
242C2×C4×C53C23×C411yescharacteristic
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—28no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—44no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—84no—
259C2×Dic53C2×D814yescomplemented
260D212—24no—
261D212—24no—
262D212—24no—
263D212—216no—
264D212C2×C22⋊C418yescomplemented
265C4×D53—432no—
266C8×C53—42no—
267#88322—24no—
268C2×#1879—531nomaximal, Sylow
269C22×C53—44no—
270C22×C53—44no—
271C22×C53—44no—
272C22×C53—816no—
273C22×C53—82no—
274C22×C53—82no—
275C22×C53—82no—
276C22×C53—82no—
277C22×C53#38614yes—
278C22×C53C4×D814yes—
279C22×C53#44214yes—
280C22×C53#44714yes—
281C22×C53#44914yes—
282C22×C53#45914yes—
283C22×C53#46314yes—
284C22×C53C2×C4×D411yescharacteristic
285C22×C53C2×#12011yescharacteristic
286C22×C53C2×#12111yescharacteristic
287C22×C53C22×D811yescharacteristic
288C22×C53C2×#13511yescharacteristic
289C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
291C2×D53—28no—
292C2×D53—432no—
293C2×D53—44no—
294C2×D53—44no—
295C2×D53—44no—
296C2×D53—44no—
297C2×D53—44no—
298C2×D53—44no—
299C4×C53—44no—
300C4×C53#38614yes—
301C4×C53C22×C22⋊C411yescharacteristic
302C4×C53C2×C4×D411yescharacteristic
303C4×C53C2×#12111yescharacteristic
304C4×C53C2×#12311yescharacteristic
305Dic53—24no—
306Dic53—42no—
307Dic53—82no—
308C23×D8—531no—
309C22×C4×D4—531no—
310C22×#102—531no—
311C22×#102D5311yescharacteristic, complemented
312C2×#390—531no—
313#1879—538no—
314C2×#340—531no—
315C2×#336—531no—
316C2×C53—84no—
317C2×C53#187918yescomplemented
318C2×C53C2×#38611yescharacteristic
319C2×C53C2×C4×D811yescharacteristic
320C2×C53C2×#44211yescharacteristic
321C2×C53C2×#44711yescharacteristic
322C2×C53C2×#44911yescharacteristic
323C2×C53C2×#45911yescharacteristic
324C2×C53C2×#46311yescharacteristic
325D53—48no—
326C23×D4—531no—
327C23×D4C2×D5311yescharacteristic
328C24×C4—1061no—
329C22×D8—538no—
330C22×D8—534no—
331C22×D8—1064no—
332C22×D8—1064no—
333C23×C8—531no—
334C23×C8C2×D5311yescharacteristic
335C2×C4×D4—534no—
336C2×C4×D4—1062no—
337C2×C4×D4—1064no—
338C2×C4×D4—1064no—
339C22×C4⋊C4—531no—
340C22×C4⋊C4C2×D5311yescharacteristic
341C22×C22⋊C4—1061no—
342C22×C22⋊C4—1061no—
343C22×C22⋊C4—1061no—
344C22×C42—531no—
345C2×#102—24no—
346C2×#102—534no—
347C2×#102—1064no—
348C2×#102C2×D5314yescomplemented
349#390—534no—
350#390—1062no—
351#340—534no—
352#336—534no—
353C53C2×#187911yescharacteristic, complemented, Sylow
354C25—21no—
355C25—1061no—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4—22no—
359C22×D4—21no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—534no—
363C22×D4—534no—
364C22×D4—534no—
365C22×D4—532no—
366C22×D4—1062no—
367C22×D4—1061no—
368C22×D4—1062no—
369C22×D4—1062no—
370C22×D4C4×D5314yescomplemented
371C22×D4C22×D5312yes—
372C23×C4—21no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—1064no—
376C23×C4—1068no—
377C23×C4—1061no—
378C23×C4—2124no—
379C23×C4—2124no—
380C23×C4C22×D5311yescharacteristic
381C2×D8—5316no—
382C2×D8—10632no—
383C2×D8—1068no—
384C2×D8—1068no—
385C22×C8—22no—
386C22×C8—22no—
387C22×C8—22no—
388C22×C8—534no—
389C22×C8—532no—
390C22×C8—1062no—
391C22×C8—1062no—
392C22×C8C22×D5312yes—
393C4×D4—1068no—
394C4×D4—10616no—
395C4×D4—1068no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—532no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4C22⋊D5314yes—
403C2×C4⋊C4C22×D5312yes—
404C2×C22⋊C4—1064no—
405C2×C22⋊C4—1068no—
406C2×C22⋊C4—1064no—
407C2×C22⋊C4—1064no—
408C2×C22⋊C4—2124no—
409C2×C22⋊C4—2124no—
410C2×C42—532no—
411C2×C42—532no—
412C2×C42—1062no—
413C2×C42—1062no—
414#102—28no—
415#102—1068no—
416C24—24no—
417C24—22no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—1064no—
425C24—1064no—
426C24—1064no—
427C24—1062no—
428C24—2122no—
429C24—2122no—
430C24—2122no—
431C24—2122no—
432C24D4×D5311yescharacteristic
433C2×D4—28no—
434C2×D4—24no—
435C2×D4—28no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—5316no—
446C2×D4—534no—
447C2×D4—534no—
448C2×D4—534no—
449C2×D4—1064no—
450C2×D4—1068no—
451C2×D4—1064no—
452C2×D4—1068no—
453C2×D4—1064no—
454C2×D4—1068no—
455C2×D4—1062no—
456C2×D4—1062no—
457C2×D4—1062no—
458C2×D4C2×C4×D5314yescomplemented
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—534no—
464C22×C4—532no—
465C22×C4—532no—
466C22×C4—10616no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—2128no—
474C22×C4—21232no—
475C22×C4—2128no—
476C22×C4—2122no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4C2×C4×D5312yes—
480C22×C4#1171994512yes—
481C22×C4D4×D5312yes—
482C22×C4D4×D5312yes—
483C22×C4D4×D5311yescharacteristic
484C22×C4#1171994712yes—
485C22×C4C2×C22⋊D5312yes—
486C22×C4C23×D5311yescharacteristic
487D8—10632no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—44no—
492C2×C8—534no—
493C2×C8—1068no—
494C2×C8—1062no—
495C2×C8—1062no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—1062no—
499C4⋊C4—1062no—
500C4⋊C4—2122no—
501C4⋊C4C2×C22⋊D5314yes—
502C22⋊C4—10616no—
503C22⋊C4—2128no—
504C22⋊C4—2128no—
505C42—1064no—
506C42—1062no—
507C42—1062no—
508C23—24no—
509C23—48no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—48no—
517C23—48no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—10616no—
523C23—1064no—
524C23—1064no—
525C23—1064no—
526C23—2128no—
527C23—2128no—
528C23—2128no—
529C23—2128no—
530C23—2122no—
531C23—2122no—
532C23—2122no—
533C23—2122no—
534C23—2128no—
535C23—2128no—
536C23#1313317712yes—
537C23#1313318512yes—
538C23#1313319012yes—
539C23D8×D5312yes—
540C23#1313323812yes—
541C23#1313324112yes—
542C23#1313324412yes—
543C23C2×D4×D5311yescharacteristic
544D4—24no—
545D4—24no—
546D4—48no—
547D4—42no—
548D4—42no—
549D4—538no—
550D4—10616no—
551D4—1064no—
552D4—1064no—
553D4—1064no—
554C2×C4—24no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—534no—
559C2×C4—1068no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—4244no—
566C2×C4—21232no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2124no—
571C2×C4C22⋊C4×D5314yes—
572C2×C4#1313317712yes—
573C2×C4C4×C22⋊D5314yes—
574C2×C4#1313318512yes—
575C2×C4#1313318814yes—
576C2×C4#1313319012yes—
577C2×C4#1313319914yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4C2×D4×D5311yescharacteristic
580C2×C4C2×D4×D5311yescharacteristic
581C2×C4C22×C22⋊D5311yescharacteristic
582C2×C4C2×#1171994711yescharacteristic
583C2×C4C2×#1171994511yescharacteristic
584C8—42no—
585C8—1064no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—82no—
590C22—816no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—1068no—
595C22—2124no—
596C22—2124no—
597C22—2124no—
598C22—21232no—
599C22—2124no—
600C22—2124no—
601C22—2124no—
602C22#1681049914yes—
603C22#1681070314yes—
604C22#1681073214yes—
605C22#1681075514yes—
606C22#1681076714yes—
607C22#1681081714yes—
608C22#1681083714yes—
609C22C2×#1313317711yescharacteristic
610C22C2×#1313318511yescharacteristic
611C22C2×#1313319011yescharacteristic
612C22C2×D8×D5311yescharacteristic
613C22C2×#1313323811yescharacteristic
614C22C2×#1313324111yescharacteristic
615C22C2×#1313324411yescharacteristic
616C4—44no—
617C4—1064no—
618C4—4242no—
619C4—2122no—
620C4#1681049914yes—
621C4C2×C22⋊C4×D5311yescharacteristic
622C4C2×C4×C22⋊D5311yescharacteristic
623C4C2×#1313318811yescharacteristic
624C4C2×#1313319911yescharacteristic
625C2—84no—
626C2—2128no—
627C2#2027506018yesdirect product factor
628C2C2×#1681049911yescharacteristic
629C2C2×#1681070311yescharacteristic
630C2C2×#1681073211yescharacteristic
631C2C2×#1681075511yescharacteristic
632C2C2×#1681076711yescharacteristic
633C2C2×#1681081711yescharacteristic
634C2C2×#1681083711yescharacteristic
subgroupG/H
centerC24D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×#102×C53C2
socleC24×C53C2×D4
prime radicalC23×#11719922C2
cosocleC22×C4×C53C24
hypercenterC22×#102D53
hypocenterC53C2×#1879
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×#13133179C2semidirect
3. C22×#13133139C2semidirect
4. C2×#16810526C2semidirect
5. #20275060C2direct (N × H)
6. C2×#16810385C2semidirect
7. C2×#16810377C2semidirect
8. C22×#102×C53C2semidirect
13. C22×#11719922C4semidirect
19. C2×#13133179C22semidirect
26. C2×#13133139C22semidirect
28. #16810526C22semidirect
29. #16810385C22semidirect
30. #16810377C22semidirect
35. C2×#102×C53C22semidirect
55. C2×#11719922C2×C4semidirect
58. C22×#88322C2×C4semidirect
85. C22×D4×C53C2×C4semidirect
127. C2×D4×C53C22×C4semidirect
149. C22×Dic53D8semidirect
161. C2×D212C22⋊C4semidirect
189. C2×#88322C22×C4semidirect
259. C2×Dic53C2×D8semidirect
264. D212C2×C22⋊C4semidirect
311. C22×#102D53semidirect
317. C2×C53#1879semidirect
348. C2×#102C2×D53semidirect
353. C53C2×#1879semidirect
370. C22×D4C4×D53semidirect
458. C2×D4C2×C4×D53semidirect
627. C2#20275060direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199222
C22×#131331792
C22×#131331392
C2×#168105262
#202750602
C2×#168103852
C2×#168103772
C22×#102×C532
C2×#187953