Finite Group Databasebeta

← C2×#20275160 #27385848

Subgroups — C2×#20275160 #27385848

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×#16811130C211yesmaximal, characteristic
2C22×#13133174C211yesmaximal, characteristic, complemented
3C22×#13133144C211yesmaximal, characteristic, complemented
4C2×#16810586C211yesmaximal, characteristic, complemented
5#20275160C218yesmaximal, direct product factor
6C2×#16810384C211yesmaximal, characteristic, complemented
7C2×#16810383C211yesmaximal, characteristic, complemented
8C22×#104×C53C211yesmaximal, characteristic, complemented
9C23×C4×D53C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11#16811130—22no—
12#16811130—22no—
13#16811130C414yescomplemented
14#16811130C2214yes—
15C2×#13133198—21no—
16C2×#13133193C412yescomplemented
17C2×#13133188C412yescomplemented
18C2×#13133185—21no—
19C2×C4×C22⋊D53—22no—
20C2×#13133176C2211yescharacteristic
21C2×#13133174—22no—
22C2×#13133174—24no—
23C2×#13133174—24no—
24C2×#13133174C2214yescomplemented
25C22×#11719919—21no—
26C22×#11719921—21no—
27C22×C4×Dic53C2211yescharacteristic
28C2×#13133144—24no—
29C2×#13133144C2214yescomplemented
30#16810586—22no—
31#16810586C2214yescomplemented
32C2×#13133129—22no—
33#16810384C2214yescomplemented
34#16810383C2214yescomplemented
35C22×#41×C53C2211yescharacteristic
36C22×M16×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C2×#104×C53—24no—
39C2×#104×C53C2214yescomplemented
40C24×D53D411yescharacteristic
41C22×C22⋊D53—22no—
42C23×Dic53—21no—
43C23×Dic53D411yescharacteristic
44C22×C4×D53—22no—
45C22×C4×D53—22no—
46C22×C4×D53—22no—
47C22×C4×D53—22no—
48C22×C4×D53C2×C412yes—
49C22×C4×D53C2×C412yes—
50C22×C4×D53D412yes—
51C22×C4×D53D412yes—
52C22×C4×D53D412yes—
53C22×C4×D53C2312yes—
54C2×#11719917—21no—
55C2×#11719917—22no—
56C2×#11719917—22no—
57C2×#11719917C2312yes—
58C22×#88322—21no—
59#13133198—24no—
60#13133198—24no—
61#13133193—24no—
62#13133193C2×C418yescomplemented
63#13133188—24no—
64#13133188C2×C418yescomplemented
65#13133185—24no—
66#13133185—24no—
67C4×C22⋊D53—28no—
68C4×C22⋊D53—28no—
69#13133176—21no—
70#13133176—21no—
71#13133176C2×C412yes—
72#13133176C2312yes—
73#13133174—216no—
74#13133174—28no—
75#13133174—28no—
76C2×C4⋊Dic53C2×C412yes—
77C2×#11719919—28no—
78C2×#11719919—22no—
79C2×#11719919—24no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—22no—
82C2×#11719921—28no—
83C2×#11719921—22no—
84C2×#11719921—24no—
85C2×C4×Dic53—22no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53D414yescomplemented
89C2×C4×Dic53C2312yes—
90#13133144—28no—
91#13133129—28no—
92C2×#41×C53—21no—
93C2×#41×C53—21no—
94C2×#41×C53—22no—
95C2×#41×C53—22no—
96C2×#41×C53—22no—
97C2×#41×C53—22no—
98C2×#41×C53C2×C412yes—
99C2×#41×C53C2312yes—
100C22×Q8×C53C2×C411yescharacteristic
101C22×D4×C53—21no—
102C22×D4×C53C2×C411yescharacteristic
103C23×C4×C53—21no—
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—
111C2×C42×C53—22no—
112C2×C42×C53—21no—
113C2×C42×C53D414yescomplemented
114C2×C42×C53C2312yes—
115#104×C53—28no—
116C24×C53—21no—
117C24×C53C2×D411yescharacteristic
118C23×D53—22no—
119C23×D53—22no—
120C23×D53—44no—
121C23×D53C22⋊C412yes—
122C23×D53C22⋊C412yes—
123C23×D53C2×D412yes—
124#41×C53—24no—
125#41×C53—24no—
126#41×C53—22no—
127#41×C53—22no—
128#41×C53—22no—
129#41×C53—22no—
130#41×C53—42no—
131#41×C53—42no—
132#41×C53—42no—
133#41×C53—42no—
134C2×Q8×C53—21no—
135C2×Q8×C53—22no—
136C2×Q8×C53—22no—
137C2×Q8×C53C22×C412yes—
138C2×D4×C53—22no—
139C2×D4×C53—22no—
140C2×D4×C53—21no—
141C2×D4×C53—22no—
142C2×D4×C53—22no—
143C2×D4×C53—42no—
144C2×D4×C53—42no—
145C2×D4×C53C22×C412yes—
146C22×C4×C53—21no—
147C22×C4×C53—24no—
148C22×C4×C53—22no—
149C22×C4×C53—22no—
150C22×C4×C53—22no—
151C22×C4×C53—44no—
152C22×C4×C53—42no—
153C22×C4×C53—42no—
154C22×C4×C53C22×C411yescharacteristic
155C22×C4×C53C22×C411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D412yes—
160C22×C4×C53C2×D412yes—
161C22×C4×C53#4112yes—
162C22×C4×C53#4112yes—
163C22×C4×C53C2411yescharacteristic
164C2×C22⋊D53—28no—
165C2×C22⋊D53—48no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—24no—
170C22×Dic53—22no—
171C22×Dic53—44no—
172C22×Dic53C22⋊C412yes—
173C22×Dic53C22⋊C412yes—
174C22×Dic53C2×D412yes—
175C2×C4×D53—24no—
176C2×C4×D53—24no—
177C2×C4×D53—24no—
178C2×C4×D53—24no—
179C2×C4×D53—24no—
180C2×C4×D53—24no—
181C2×C4×D53—24no—
182C2×C4×D53—24no—
183C2×C4×D53—24no—
184C2×C4×D53—24no—
185C2×C4×D53—22no—
186C2×C4×D53—22no—
187C2×C4×D53—22no—
188C2×C4×D53—22no—
189C2×C4×D53C22⋊C414yes—
190C2×C4×D53C22⋊C414yes—
191C2×C4×D53C22×C414yes—
192C2×C4×D53C2×D414yes—
193M16×C53—22no—
194M16×C53—22no—
195M16×C53—42no—
196C2×C8×C53—22no—
197C2×C8×C53—42no—
198C42×C53—24no—
199C42×C53—22no—
200C42×C53C2×D414yescomplemented
201C22⋊Dic53—22no—
202C22⋊Dic53—22no—
203C22⋊Dic53—28no—
204#11719921—216no—
205#11719921—28no—
206C4⋊Dic53—22no—
207C4⋊Dic53C22×C414yes—
208#11719919—216no—
209#11719919—28no—
210C4×Dic53—22no—
211C4×Dic53—24no—
212C4×Dic53—22no—
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×C53—22no—
222C23×C53—42no—
223C23×C53—44no—
224C23×C53C2×C22⋊C411yescharacteristic
225C23×C53C2×C22⋊C411yescharacteristic
226C23×C53#12012yes—
227C23×C53#12312yes—
228C23×C53C22×D411yescharacteristic
229C22×D53—22no—
230C22×D53—22no—
231C22×D53—24no—
232C22×D53—24no—
233C22×D53—24no—
234C22×D53—24no—
235C22×D53—44no—
236C22×D53—44no—
237C22×D53—48no—
238C22×D53—48no—
239C22×D53C2×C22⋊C414yes—
240Q8×C53—22no—
241Q8×C53—22no—
242Q8×C53—42no—
243D4×C53—22no—
244D4×C53—22no—
245D4×C53—42no—
246D4×C53—42no—
247D4×C53—44no—
248D4×C53—42no—
249C2×C4×C53—22no—
250C2×C4×C53—24no—
251C2×C4×C53—44no—
252C2×C4×C53—48no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—44no—
257C2×C4×C53C2×C22⋊C411yescharacteristic
258C2×C4×C53C2×C22⋊C411yescharacteristic
259C2×C4×C53C4×D412yes—
260C2×C4×C53C4×D412yes—
261C2×C4×C53C4×D412yes—
262C2×C4×C53C4×D412yes—
263C2×C4×C53#12012yes—
264C2×C4×C53#12112yes—
265C2×C4×C53#12112yes—
266C2×C4×C53#12112yes—
267C2×C4×C53#12112yes—
268C2×C4×C53#12312yes—
269C2×C4×C53C23×C411yescharacteristic
270C2×C4×C53C22×D411yescharacteristic
271C2×C4×C53C22×D411yescharacteristic
272C2×C4×C53C22×D411yescharacteristic
273C2×C4×C53C2×#4111yescharacteristic
274C2×C4×C53C2×#4111yescharacteristic
275C22⋊D53—416no—
276C2×Dic53—22no—
277C2×Dic53—22no—
278C2×Dic53—24no—
279C2×Dic53—24no—
280C2×Dic53—24no—
281C2×Dic53—24no—
282C2×Dic53—44no—
283C2×Dic53—48no—
284C2×Dic53—44no—
285C2×Dic53C2×C22⋊C414yes—
286C4×D53—24no—
287C4×D53—24no—
288C4×D53—24no—
289C4×D53—24no—
290C4×D53—28no—
291C4×D53—28no—
292C8×C53—42no—
293#88322—42no—
294C2×#1897—531nomaximal, Sylow
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—24no—
298C22×C53—24no—
299C22×C53—44no—
300C22×C53—42no—
301C22×C53—84no—
302C22×C53#38612yes—
303C22×C53#38612yes—
304C22×C53#42114yes—
305C22×C53C22×C22⋊C411yescharacteristic
306C22×C53C2×#12011yescharacteristic
307C22×C53C2×#12311yescharacteristic
308C2×D53—24no—
309C2×D53—24no—
310C2×D53—24no—
311C2×D53—44no—
312C2×D53—44no—
313C2×D53—48no—
314C2×D53—48no—
315C4×C53—44no—
316C4×C53—42no—
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—44no—
326C2×#525—531no—
327C22×#117—531no—
328C22×#104—531no—
329C22×#104D5311yescharacteristic, complemented
330C2×#407—531no—
331#1897—538no—
332C2×#339—531no—
333C2×#339—531no—
334C2×C53—21no—
335C2×C53—21no—
336C2×C53—22no—
337C2×C53—82no—
338C2×C53#189714yescomplemented
339C2×C53C2×#38611yescharacteristic
340C2×C53C2×#42112yes—
341D53—44no—
342C22×#41C2×D5311yescharacteristic
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—531no—
357C2×#117—1064no—
358C2×#117—1062no—
359C2×#117—1064no—
360C22×C4⋊C4—1061no—
361C22×C22⋊C4—1061no—
362C22×C42—531no—
363C22×C42C2×D5311yescharacteristic
364C2×#104—24no—
365C2×#104—534no—
366C2×#104—1064no—
367C2×#104C2×D5314yescomplemented
368#407—534no—
369#407—1062no—
370C2×C22⋊C8—1062no—
371#339—534no—
372#339—534no—
373C53C2×#189711yescharacteristic, complemented, Sylow
374C25—531no—
375C2×#41—21no—
376C2×#41—21no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41—22no—
380C2×#41—22no—
381C2×#41C4×D5312yes—
382C2×#41C22×D5312yes—
383C22×Q8C4×D5311yescharacteristic
384C22×D4—21no—
385C22×D4—1062no—
386C22×D4C4×D5311yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4—531no—
390C23×C4—532no—
391C23×C4—532no—
392C23×C4—532no—
393C23×C4—532no—
394C23×C4—532no—
395C23×C4—532no—
396C23×C4—1062no—
397C23×C4—1062no—
398C23×C4—1062no—
399C23×C4—1062no—
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—532no—
423#117—532no—
424#117—1068no—
425#117—10616no—
426#117—1061no—
427#117—1061no—
428#117—1068no—
429C2×C4⋊C4—532no—
430C2×C4⋊C4—1064no—
431C2×C4⋊C4—1062no—
432C2×C4⋊C4—1068no—
433C2×C22⋊C4—1064no—
434C2×C22⋊C4—1062no—
435C2×C22⋊C4—1068no—
436C2×C22⋊C4—1062no—
437C2×C22⋊C4—1061no—
438C2×C42—21no—
439C2×C42—22no—
440C2×C42—534no—
441C2×C42—532no—
442C2×C42—1061no—
443C2×C42—1061no—
444C2×C42—1062no—
445C2×C42C22⋊D5314yescomplemented
446C2×C42C22×D5312yes—
447#104—28no—
448#104—1068no—
449C22⋊C8—1068no—
450C24—21no—
451C24—532no—
452C24—532no—
453C24—532no—
454C24—1062no—
455C24—1062no—
456C24—2124no—
457C24D4×D5311yescharacteristic
458#41—24no—
459#41—24no—
460#41—22no—
461#41—22no—
462#41—22no—
463#41—22no—
464#41—42no—
465#41—42no—
466#41—42no—
467#41—42no—
468C2×Q8—21no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8C2×C4×D5312yes—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—21no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—1068no—
480C2×D4—2128no—
481C2×D4C2×C4×D5312yes—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—532no—
491C22×C4—532no—
492C22×C4—532no—
493C22×C4—534no—
494C22×C4—534no—
495C22×C4—534no—
496C22×C4—534no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1064no—
500C22×C4—1064no—
501C22×C4—1064no—
502C22×C4—1064no—
503C22×C4—1064no—
504C22×C4—1064no—
505C22×C4—1064no—
506C22×C4—1064no—
507C22×C4—1064no—
508C22×C4—1064no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1064no—
515C22×C4—1062no—
516C22×C4—2124no—
517C22×C4C2×C4×D5311yescharacteristic
518C22×C4C2×C4×D5311yescharacteristic
519C22×C4#1171994512yes—
520C22×C4D4×D5311yescharacteristic
521C22×C4D4×D5311yescharacteristic
522C22×C4D4×D5312yes—
523C22×C4D4×D5311yescharacteristic
524C22×C4#1171994712yes—
525C22×C4C2×C22⋊D5312yes—
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—534no—
538C4⋊C4—1068no—
539C4⋊C4—10616no—
540C4⋊C4—1062no—
541C22⋊C4—1068no—
542C22⋊C4—10616no—
543C22⋊C4—1068no—
544C22⋊C4—1062no—
545C22⋊C4—1062no—
546C42—24no—
547C42—22no—
548C42—534no—
549C42—1062no—
550C42—1064no—
551C42—1062no—
552C42—1062no—
553C42C2×C22⋊D5314yescomplemented
554C23—24no—
555C23—22no—
556C23—42no—
557C23—44no—
558C23—534no—
559C23—1064no—
560C23—1064no—
561C23—1062no—
562C23—1062no—
563C23—1064no—
564C23—1064no—
565C23—2124no—
566C23—2124no—
567C23—2128no—
568C23—2128no—
569C23C22⋊C4×D5311yescharacteristic
570C23C22⋊C4×D5311yescharacteristic
571C23#1313318512yes—
572C23#1313319912yes—
573C23C2×D4×D5311yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—42no—
577D4—22no—
578D4—22no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—21216no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—534no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1068no—
598C2×C4—1068no—
599C2×C4—1064no—
600C2×C4—1064no—
601C2×C4—1064no—
602C2×C4—1064no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—2124no—
606C2×C4—2128no—
607C2×C4—2124no—
608C2×C4C22⋊C4×D5311yescharacteristic
609C2×C4C22⋊C4×D5311yescharacteristic
610C2×C4#1313317712yes—
611C2×C4#1313317712yes—
612C2×C4C4×C22⋊D5312yes—
613C2×C4C4×C22⋊D5312yes—
614C2×C4#1313318512yes—
615C2×C4#1313318812yes—
616C2×C4#1313318812yes—
617C2×C4#1313319012yes—
618C2×C4#1313319012yes—
619C2×C4#1313319912yes—
620C2×C4C22×C4×D5311yescharacteristic
621C2×C4C2×D4×D5311yescharacteristic
622C2×C4C2×D4×D5311yescharacteristic
623C2×C4C22×C22⋊D5311yescharacteristic
624C2×C4C2×#1171994711yescharacteristic
625C2×C4C2×#1171994511yescharacteristic
626C8—42no—
627C8—2122no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—44no—
633C22—42no—
634C22—84no—
635C22—1064no—
636C22—1064no—
637C22—1064no—
638C22—2124no—
639C22—2128no—
640C22—2124no—
641C22—2128no—
642C22#1681049912yes—
643C22#1681049912yes—
644C22#1681063814yes—
645C22C2×C22⋊C4×D5311yescharacteristic
646C22C2×#1313318511yescharacteristic
647C22C2×#1313319911yescharacteristic
648C4—44no—
649C4—42no—
650C4—1064no—
651C4—2124no—
652C4—2124no—
653C4#1681049912yes—
654C4#1681049912yes—
655C4C2×#1313317711yescharacteristic
656C4C2×C4×C22⋊D5311yescharacteristic
657C4C2×#1313318811yescharacteristic
658C4C2×#1313319011yescharacteristic
659C2—21no—
660C2—22no—
661C2—21no—
662C2—82no—
663C2—2124no—
664C2#2027516014yesdirect product factor
665C2C2×#1681049911yescharacteristic
666C2C2×#1681063812yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×#104×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#16811130C2
cosocleC22×C4×C53C24
hypercenterC22×#104D53
hypocenterC53C2×#1897
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C22×#13133174C2semidirect
3. C22×#13133144C2semidirect
4. C2×#16810586C2semidirect
5. #20275160C2direct (N × H)
6. C2×#16810384C2semidirect
7. C2×#16810383C2semidirect
8. C22×#104×C53C2semidirect
13. #16811130C4semidirect
16. C2×#13133193C4semidirect
17. C2×#13133188C4semidirect
24. C2×#13133174C22semidirect
29. C2×#13133144C22semidirect
31. #16810586C22semidirect
33. #16810384C22semidirect
34. #16810383C22semidirect
39. C2×#104×C53C22semidirect
62. #13133193C2×C4semidirect
64. #13133188C2×C4semidirect
88. C2×C4×Dic53D4semidirect
113. C2×C42×C53D4semidirect
200. C42×C53C2×D4semidirect
214. C4×Dic53C2×D4semidirect
329. C22×#104D53semidirect
338. C2×C53#1897semidirect
367. C2×#104C2×D53semidirect
373. C53C2×#1897semidirect
445. C2×C42C22⋊D53semidirect
553. C42C2×C22⋊D53semidirect
664. C2#20275160direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111302
C22×#131331742
C22×#131331442
C2×#168105862
#202751602
C2×#168103842
C2×#168103832
C22×#104×C532
C2×#189753