Finite Group Databasebeta

← C2×#20275162 #27385850

Subgroups — C2×#20275162 #27385850

Order 13568.

1954 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811135C211yesmaximal, characteristic
2C22×#13133174C211yesmaximal, characteristic, complemented
3C22×#13133145C211yesmaximal, characteristic, complemented
4C22×#13133143C211yesmaximal, characteristic, complemented
5C2×#16810586C211yesmaximal, characteristic, complemented
6#20275162C218yesmaximal, direct product factor
7C2×#16810382C211yesmaximal, characteristic, complemented
8C2×C53×#339C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C23×C4×D53C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12#16811135—22no—
13#16811135—22no—
14#16811135C414yescomplemented
15#16811135C2214yes—
16C2×#13133201—21no—
17C2×#13133197C412yescomplemented
18C2×#13133192C412yescomplemented
19C2×#13133187—21no—
20C2×C4×D212—22no—
21C2×#13133174—22no—
22C2×#13133174—24no—
23C2×#13133174—24no—
24C2×#13133174C2214yescomplemented
25C22×#11719919—21no—
26C22×#11719921—21no—
27C22×C4×Dic53C2211yescharacteristic
28C2×#13133145—24no—
29C2×#13133145C2214yescomplemented
30C2×#13133143—24no—
31C2×#13133143C2214yescomplemented
32#16810586—22no—
33#16810586C2214yescomplemented
34C2×#13133129—22no—
35#16810382C2214yescomplemented
36C22×M16×C53C2211yescharacteristic
37C2×#117×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C53×#339C2214yescomplemented
40C24×D53D411yescharacteristic
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—22no—
44C2×#11719945—22no—
45C2×#11719945—22no—
46C2×#11719945—22no—
47C2×#11719945C2×C412yes—
48C2×#11719945C2312yes—
49C22×Dic106C2×C411yescharacteristic
50C22×D212—22no—
51C22×D212C2×C411yescharacteristic
52C22×C22⋊D53—21no—
53C23×Dic53—21no—
54C23×Dic53D411yescharacteristic
55C22×C4×D53—22no—
56C22×C4×D53—22no—
57C22×C4×D53—22no—
58C22×C4×D53—22no—
59C22×C4×D53—21no—
60C22×C4×D53C2×C412yes—
61C22×C4×D53C2×C412yes—
62C22×C4×D53D412yes—
63C22×C4×D53D412yes—
64C22×C4×D53D412yes—
65C22×C4×D53C2312yes—
66C2×#11719917—21no—
67C2×#11719917—22no—
68C2×#11719917—22no—
69C2×#11719917C2312yes—
70C22×#88322—21no—
71#13133201—24no—
72#13133201—24no—
73#13133197—24no—
74#13133197C2×C418yescomplemented
75#13133192—24no—
76#13133192C2×C418yescomplemented
77#13133187—24no—
78#13133187—24no—
79C4×D212—28no—
80C4×D212—28no—
81#13133174—216no—
82#13133174—28no—
83#13133174—28no—
84C2×C4⋊Dic53—22no—
85C2×#11719919—28no—
86C2×#11719919—24no—
87C2×#11719921—28no—
88C2×#11719921—22no—
89C2×#11719921—24no—
90C2×#11719921—22no—
91C2×C4×Dic53—22no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53D414yescomplemented
94C2×C4×Dic53C2312yes—
95#13133145—28no—
96#13133143—28no—
97#13133129—28no—
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C2×M16×C53—21no—
101C2×M16×C53—22no—
102C2×M16×C53—22no—
103C2×M16×C53C2312yes—
104C22×C8×C53—21no—
105#117×C53—21no—
106#117×C53—21no—
107#117×C53C2×C412yes—
108#117×C53C2312yes—
109C2×C4⋊C4×C53C2×C412yes—
110C2×C22⋊C4×C53—21no—
111C2×C42×C53—22no—
112C2×C42×C53—21no—
113C2×C42×C53—21no—
114C2×C42×C53D414yescomplemented
115C2×C42×C53C2312yes—
116C24×C53C2×D411yescharacteristic
117C23×D53—22no—
118C23×D53—22no—
119C23×D53—21no—
120C23×D53—44no—
121C23×D53C22⋊C412yes—
122C23×D53C22⋊C412yes—
123C23×D53C2×D412yes—
124C22×C4×C53—22no—
125C22×C4×C53—24no—
126C22×C4×C53—22no—
127C22×C4×C53—44no—
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53#4112yes—
136C22×C4×C53#4112yes—
137C22×C4×C53C2411yescharacteristic
138C2×C22⋊D53—22no—
139C2×C22⋊D53—22no—
140C2×C22⋊D53—42no—
141C2×C22⋊D53—42no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—24no—
146C22×Dic53—22no—
147C22×Dic53—44no—
148C22×Dic53C22⋊C412yes—
149C22×Dic53C22⋊C412yes—
150C22×Dic53C2×D412yes—
151#11719945—24no—
152#11719945—24no—
153#11719945—22no—
154#11719945—22no—
155#11719945—22no—
156#11719945—22no—
157#11719945—42no—
158#11719945—42no—
159#11719945—42no—
160#11719945—42no—
161C2×D212—22no—
162C2×D212—22no—
163C2×D212—28no—
164C2×D212—21no—
165C2×D212—48no—
166C2×D212C22×C412yes—
167C2×C4×D53—24no—
168C2×C4×D53—24no—
169C2×C4×D53—24no—
170C2×C4×D53—24no—
171C2×C4×D53—24no—
172C2×C4×D53—24no—
173C2×C4×D53—24no—
174C2×C4×D53—24no—
175C2×C4×D53—24no—
176C2×C4×D53—24no—
177C2×C4×D53—22no—
178C2×C4×D53—22no—
179C2×C4×D53—22no—
180C2×C4×D53—22no—
181C2×C4×D53—22no—
182C2×C4×D53—22no—
183C2×C4×D53—42no—
184C2×C4×D53—42no—
185C2×C4×D53C22⋊C414yes—
186C2×C4×D53C22⋊C414yes—
187C2×C4×D53C22×C414yes—
188C2×C4×D53C2×D414yes—
189C2×Dic106—22no—
190C2×Dic106—22no—
191C2×Dic106—21no—
192C2×Dic106C22×C412yes—
193M16×C53—22no—
194M16×C53—22no—
195M16×C53—42no—
196C2×C8×C53—22no—
197C2×C8×C53—42no—
198C4⋊C4×C53—22no—
199C4⋊C4×C53C22×C414yes—
200C22⋊C4×C53—22no—
201C22⋊C4×C53—22no—
202C42×C53—22no—
203C42×C53—24no—
204C42×C53—22no—
205C42×C53—22no—
206C42×C53C2×D414yescomplemented
207#11719921—216no—
208#11719921—28no—
209#11719921—28no—
210C4⋊Dic53—28no—
211#11719919—216no—
212C4×Dic53—24no—
213C4×Dic53—22no—
214C4×Dic53C2×D414yescomplemented
215#11719917—22no—
216#11719917—22no—
217#11719917—42no—
218C2×#88322—22no—
219C2×#88322—42no—
220C23×C53—24no—
221C23×C53C2×C22⋊C411yescharacteristic
222C23×C53C2×C22⋊C411yescharacteristic
223C23×C53#12012yes—
224C23×C53#12312yes—
225C23×C53C22×D411yescharacteristic
226C22×D53—22no—
227C22×D53—22no—
228C22×D53—24no—
229C22×D53—24no—
230C22×D53—24no—
231C22×D53—24no—
232C22×D53—22no—
233C22×D53—44no—
234C22×D53—44no—
235C22×D53—48no—
236C22×D53—48no—
237C22×D53—42no—
238C22×D53—44no—
239C22×D53C2×C22⋊C414yes—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—44no—
243C2×C4×C53—48no—
244C2×C4×C53—44no—
245C2×C4×C53C2×C22⋊C411yescharacteristic
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53C4×D412yes—
248C2×C4×C53C4×D412yes—
249C2×C4×C53C4×D412yes—
250C2×C4×C53C4×D412yes—
251C2×C4×C53#12012yes—
252C2×C4×C53#12112yes—
253C2×C4×C53#12112yes—
254C2×C4×C53#12112yes—
255C2×C4×C53#12112yes—
256C2×C4×C53#12312yes—
257C2×C4×C53C23×C411yescharacteristic
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C22×D411yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C22⋊D53—42no—
264C22⋊D53—42no—
265C22⋊D53—44no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—22no—
271C2×Dic53—24no—
272C2×Dic53—44no—
273C2×Dic53—48no—
274C2×Dic53—42no—
275C2×Dic53C2×C22⋊C414yes—
276D212—22no—
277D212—22no—
278D212—416no—
279D212—42no—
280C4×D53—24no—
281C4×D53—24no—
282C4×D53—24no—
283C4×D53—24no—
284C4×D53—28no—
285C4×D53—28no—
286C4×D53—42no—
287C4×D53—42no—
288C4×D53—44no—
289Dic106—22no—
290Dic106—22no—
291Dic106—42no—
292C8×C53—42no—
293#88322—42no—
294C2×#1897—531nomaximal, Sylow
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—24no—
298C22×C53—24no—
299C22×C53#38612yes—
300C22×C53#38612yes—
301C22×C53#42114yes—
302C22×C53C22×C22⋊C411yescharacteristic
303C22×C53C2×#12011yescharacteristic
304C22×C53C2×#12311yescharacteristic
305C2×D53—24no—
306C2×D53—24no—
307C2×D53—24no—
308C2×D53—44no—
309C2×D53—44no—
310C2×D53—48no—
311C2×D53—48no—
312C2×D53—44no—
313C2×D53—42no—
314C2×D53—84no—
315C4×C53—44no—
316C4×C53—44no—
317C4×C53#38612yes—
318C4×C53#38612yes—
319C4×C53C2×C4×D411yescharacteristic
320C4×C53C2×C4×D411yescharacteristic
321C4×C53C2×#12111yescharacteristic
322C4×C53C2×#12111yescharacteristic
323Dic53—24no—
324Dic53—44no—
325Dic53—42no—
326C2×#525—531no—
327C22×#117—531no—
328C22×#104—531no—
329C22×#104—531no—
330C2×#407—531no—
331#1897—538no—
332C2×#339—531no—
333C2×#339D5311yescharacteristic, complemented
334C2×C53—21no—
335C2×C53—21no—
336C2×C53—22no—
337C2×C53#189714yescomplemented
338C2×C53C2×#38611yescharacteristic
339C2×C53C2×#42112yes—
340D53—44no—
341D53—82no—
342C22×#41—531no—
343C24×C4—531no—
344C22×M16—531no—
345C22×M16C2×D5311yescharacteristic
346#525—534no—
347#525—534no—
348#525—1062no—
349#525—1062no—
350C2×#123—1061no—
351C2×C22⋊Q8—532no—
352C2×#121—532no—
353C2×#120—1061no—
354C2×C4×D4—1062no—
355C2×#117—534no—
356C2×#117—1064no—
357C2×#117—1062no—
358C2×#117—1064no—
359C2×#117C2×D5311yescharacteristic
360C22×C4⋊C4—1061no—
361C22×C22⋊C4—1061no—
362C22×C42—531no—
363C22×C42C2×D5311yescharacteristic
364C2×#104—534no—
365C2×#104—534no—
366C2×#104—1064no—
367C2×#104—1064no—
368#407—534no—
369#407—1062no—
370C2×C22⋊C8—1062no—
371#339—534no—
372#339C2×D5314yescomplemented
373C53C2×#189711yescharacteristic, complemented, Sylow
374C25—531no—
375C2×#41—532no—
376C2×#41—532no—
377C2×#41—1061no—
378C2×#41—1061no—
379C2×#41—1062no—
380C2×#41—1062no—
381C2×#41—1062no—
382C2×#41—1062no—
383C22×Q8—531no—
384C22×D4—531no—
385C22×D4—1062no—
386C22×D4—1061no—
387C23×C4—21no—
388C23×C4—531no—
389C23×C4—532no—
390C23×C4—532no—
391C23×C4—532no—
392C23×C4—532no—
393C23×C4—532no—
394C23×C4—532no—
395C23×C4—1062no—
396C23×C4—1062no—
397C23×C4—1062no—
398C23×C4—1062no—
399C23×C4—1061no—
400C23×C4—1061no—
401C23×C4C22×D5311yescharacteristic
402C2×M16—21no—
403C2×M16—22no—
404C2×M16—22no—
405C2×M16—532no—
406C2×M16—1061no—
407C2×M16—1062no—
408C2×M16—1062no—
409C2×M16C22×D5312yes—
410C22×C8—21no—
411C22×C8—1061no—
412#123—1064no—
413#123—1064no—
414C22⋊Q8—538no—
415C22⋊Q8—1064no—
416#121—538no—
417#121—1064no—
418#120—1064no—
419#120—1064no—
420C4×D4—1068no—
421C4×D4—1068no—
422#117—21no—
423#117—21no—
424#117—1068no—
425#117—10616no—
426#117—1068no—
427#117C4×D5312yes—
428#117C22×D5312yes—
429C2×C4⋊C4—1064no—
430C2×C4⋊C4—1062no—
431C2×C4⋊C4—1068no—
432C2×C4⋊C4C4×D5312yes—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1062no—
435C2×C22⋊C4—1064no—
436C2×C22⋊C4—1062no—
437C2×C22⋊C4—1068no—
438C2×C42—21no—
439C2×C42—22no—
440C2×C42—21no—
441C2×C42—534no—
442C2×C42—532no—
443C2×C42—1062no—
444C2×C42—1061no—
445C2×C42D21214yescomplemented
446C2×C42C22×D5312yes—
447#104—1068no—
448#104—1068no—
449C22⋊C8—1068no—
450C24—532no—
451C24—532no—
452C24—532no—
453C24—1062no—
454C24—1062no—
455C24—1061no—
456C24—2124no—
457C24D4×D5311yescharacteristic
458#41—1064no—
459#41—1064no—
460#41—1062no—
461#41—1062no—
462#41—1062no—
463#41—1062no—
464#41—2122no—
465#41—2122no—
466#41—2122no—
467#41—2122no—
468C2×Q8—532no—
469C2×Q8—1061no—
470C2×Q8—1062no—
471C2×Q8—1062no—
472C2×D4—532no—
473C2×D4—1068no—
474C2×D4—1062no—
475C2×D4—1062no—
476C2×D4—1061no—
477C2×D4—1062no—
478C2×D4—1062no—
479C2×D4—2128no—
480C2×D4—2122no—
481C2×D4—2122no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—44no—
486C22×C4—532no—
487C22×C4—532no—
488C22×C4—532no—
489C22×C4—534no—
490C22×C4—534no—
491C22×C4—534no—
492C22×C4—534no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1064no—
496C22×C4—1064no—
497C22×C4—1064no—
498C22×C4—1064no—
499C22×C4—1064no—
500C22×C4—1064no—
501C22×C4—1064no—
502C22×C4—1064no—
503C22×C4—1064no—
504C22×C4—1064no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1061no—
510C22×C4—1064no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—2124no—
515C22×C4—2122no—
516C22×C4—2122no—
517C22×C4C2×C4×D5311yescharacteristic
518C22×C4C2×C4×D5311yescharacteristic
519C22×C4C2×D21212yes—
520C22×C4#1171994512yes—
521C22×C4D4×D5311yescharacteristic
522C22×C4D4×D5311yescharacteristic
523C22×C4D4×D5312yes—
524C22×C4D4×D5311yescharacteristic
525C22×C4#1171994912yes—
526C22×C4C23×D5311yescharacteristic
527M16—22no—
528M16—22no—
529M16—42no—
530M16—1062no—
531M16—1062no—
532M16—2122no—
533C2×C8—22no—
534C2×C8—42no—
535C2×C8—1062no—
536C2×C8—2122no—
537C4⋊C4—22no—
538C4⋊C4—1068no—
539C4⋊C4—10616no—
540C4⋊C4C2×C4×D5314yes—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—1068no—
544C22⋊C4—1068no—
545C22⋊C4—10616no—
546C42—22no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—534no—
551C42—1064no—
552C42—1062no—
553C42C2×D21214yescomplemented
554C23—24no—
555C23—534no—
556C23—1064no—
557C23—1064no—
558C23—1062no—
559C23—1062no—
560C23—1064no—
561C23—1064no—
562C23—1062no—
563C23—2124no—
564C23—2124no—
565C23—2128no—
566C23—2128no—
567C23—2122no—
568C23—2124no—
569C23C22⋊C4×D5311yescharacteristic
570C23C22⋊C4×D5311yescharacteristic
571C23#1313318712yes—
572C23#1313320212yes—
573C23C2×D4×D5311yescharacteristic
574Q8—1062no—
575Q8—1062no—
576Q8—2122no—
577D4—1062no—
578D4—1062no—
579D4—21216no—
580D4—2124no—
581D4—2122no—
582D4—2122no—
583D4—2122no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—534no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1068no—
595C2×C4—1068no—
596C2×C4—1064no—
597C2×C4—1064no—
598C2×C4—1064no—
599C2×C4—1064no—
600C2×C4—1062no—
601C2×C4—1064no—
602C2×C4—2124no—
603C2×C4—2128no—
604C2×C4—2122no—
605C2×C4—2124no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4C22⋊C4×D5311yescharacteristic
609C2×C4C22⋊C4×D5311yescharacteristic
610C2×C4#1313317912yes—
611C2×C4#1313317912yes—
612C2×C4C4×D21212yes—
613C2×C4C4×D21212yes—
614C2×C4#1313318712yes—
615C2×C4#1313319012yes—
616C2×C4#1313319012yes—
617C2×C4#1313319212yes—
618C2×C4#1313319212yes—
619C2×C4#1313320212yes—
620C2×C4C22×C4×D5311yescharacteristic
621C2×C4C2×D4×D5311yescharacteristic
622C2×C4C2×D4×D5311yescharacteristic
623C2×C4C22×D21211yescharacteristic
624C2×C4C2×#1171994511yescharacteristic
625C2×C4C2×#1171994911yescharacteristic
626C8—42no—
627C8—2122no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—1064no—
633C22—1064no—
634C22—1064no—
635C22—4244no—
636C22—2124no—
637C22—2128no—
638C22—2124no—
639C22—2124no—
640C22—2122no—
641C22—2128no—
642C22#1681050112yes—
643C22#1681050112yes—
644C22#1681063814yes—
645C22C2×C22⋊C4×D5311yescharacteristic
646C22C2×#1313318711yescharacteristic
647C22C2×#1313320211yescharacteristic
648C4—44no—
649C4—44no—
650C4—1064no—
651C4—2124no—
652C4—2122no—
653C4#1681050112yes—
654C4#1681050112yes—
655C4C2×#1313317911yescharacteristic
656C4C2×C4×D21211yescharacteristic
657C4C2×#1313319011yescharacteristic
658C4C2×#1313319211yescharacteristic
659C2—21no—
660C2—22no—
661C2—21no—
662C2—4242no—
663C2—2124no—
664C2#2027516214yesdirect product factor
665C2C2×#1681050111yescharacteristic
666C2C2×#1681063812yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#339C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#16811135C2
cosocleC22×C4×C53C24
hypercenterC2×#339D53
hypocenterC53C2×#1897
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
2. C22×#13133174C2semidirect
3. C22×#13133145C2semidirect
4. C22×#13133143C2semidirect
5. C2×#16810586C2semidirect
6. #20275162C2direct (N × H)
7. C2×#16810382C2semidirect
8. C2×C53×#339C2semidirect
14. #16811135C4semidirect
17. C2×#13133197C4semidirect
18. C2×#13133192C4semidirect
24. C2×#13133174C22semidirect
29. C2×#13133145C22semidirect
31. C2×#13133143C22semidirect
33. #16810586C22semidirect
35. #16810382C22semidirect
39. C53×#339C22semidirect
74. #13133197C2×C4semidirect
76. #13133192C2×C4semidirect
93. C2×C4×Dic53D4semidirect
114. C2×C42×C53D4semidirect
206. C42×C53C2×D4semidirect
214. C4×Dic53C2×D4semidirect
333. C2×#339D53semidirect
337. C2×C53#1897semidirect
372. #339C2×D53semidirect
373. C53C2×#1897semidirect
445. C2×C42D212semidirect
553. C42C2×D212semidirect
664. C2#20275162direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111352
C22×#131331742
C22×#131331452
C22×#131331432
C2×#168105862
#202751622
C2×#168103822
C2×C53×#3392
C2×#189753