Finite Group Databasebeta

← C2×#20273218 #27351231

Subgroups — C2×#20273218 #27351231

Order 13568.

1834 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811136C211yesmaximal, characteristic
2C2×#16810585C211yesmaximal, characteristic, complemented
3C2×#95×D53C211yesmaximal, characteristic, complemented
4#20273218C218yesmaximal, direct product factor
5C2×#16810360C211yesmaximal, characteristic, complemented
6C2×#16810358C211yesmaximal, characteristic, complemented
7C2×#16810356C211yesmaximal, characteristic, complemented
8C2×C53×#328C211yesmaximal, characteristic, complemented
9C22×#11719949C411yescharacteristic, complemented
10C23×C4×D53—21no—
11C23×C4×D53C2211yescharacteristic
12C22×#11719912—21no—
13#16811136—22no—
14#16811136—22no—
15#16811136C414yescomplemented
16#16811136C2214yes—
17C2×#13133202—21no—
18C2×#13133193C411yescharacteristic, complemented
19C2×#13133192C411yescharacteristic, complemented
20C2×#13133191C2211yescharacteristic
21C2×#13133187—21no—
22C2×#13133179—21no—
23C2×#13133177—21no—
24C2×#13133175C411yescharacteristic, complemented
25#16810585—22no—
26#16810585C2214yescomplemented
27C2×C22⋊#88322C2211yescharacteristic
28C2×#13133129—21no—
29C2×#13133124—21no—
30C2×#13133124C2211yescharacteristic
31#95×D53—22no—
32#95×D53C2214yescomplemented
33#16810360C2214yescomplemented
34#16810358C2214yescomplemented
35#16810356C2214yescomplemented
36C2×C22⋊Q8×C53C2211yescharacteristic
37C2×C22⋊C8×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C53×#328C2214yescomplemented
40C24×D53D411yescharacteristic
41C2×#11719949—22no—
42C2×#11719949—44no—
43C2×#11719949C2×C414yescomplemented
44C22×D212—21no—
45C22×D212—21no—
46C22×D212C2×C411yescharacteristic, complemented
47C22×C22⋊D53—21no—
48C23×Dic53—21no—
49C23×Dic53D411yescharacteristic
50C22×C4×D53—28no—
51C22×C4×D53—24no—
52C22×C4×D53—21no—
53C22×C4×D53—44no—
54C22×C4×D53—44no—
55C22×C4×D53—44no—
56C22×C4×D53C2×C411yescharacteristic
57C22×C4×D53C2×C411yescharacteristic
58C22×C4×D53C2×C412yes—
59C22×C4×D53D411yescharacteristic
60C22×C4×D53D412yes—
61C22×C4×D53D411yescharacteristic
62C22×C4×D53D412yes—
63C22×C4×D53C2312yes—
64C2×#11719912—24no—
65C2×#11719912—44no—
66C22×#88322—21no—
67#13133202—24no—
68#13133202—24no—
69#13133193—22no—
70#13133193C2×C414yescomplemented
71#13133192—22no—
72#13133192C2×C414yescomplemented
73#13133191—21no—
74#13133191—21no—
75#13133191C2×C412yescomplemented
76#13133191C2312yes—
77#13133187—24no—
78#13133187—24no—
79#13133179—24no—
80#13133179—24no—
81#13133177—24no—
82#13133177—24no—
83#13133175—22no—
84#13133175C2×C414yescomplemented
85C2×C4⋊Dic53C2×C411yescharacteristic, complemented
86C2×#11719919—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×C4×Dic53—21no—
92C22⋊#88322—21no—
93C22⋊#88322C2312yes—
94#13133129—24no—
95#13133124—21no—
96#13133124—24no—
97#13133124C2312yes—
98C22×Q8×C53C2×C411yescharacteristic, complemented
99C23×C4×C53—21no—
100C23×C4×C53C2311yescharacteristic
101C22×C8×C53—21no—
102C22⋊Q8×C53—21no—
103C22⋊Q8×C53—21no—
104C22⋊Q8×C53C2×C412yescomplemented
105C22⋊Q8×C53C2312yes—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
108C2×C22⋊C4×C53—21no—
109C22⋊C8×C53—21no—
110C22⋊C8×C53C2312yes—
111#95×C53—21no—
112#95×C53C2312yes—
113C24×C53C2×D411yescharacteristic
114C23×D53—22no—
115C23×D53—22no—
116C23×D53—24no—
117C23×D53—24no—
118C23×D53—21no—
119C23×D53C22⋊C412yes—
120C23×D53C22⋊C411yescharacteristic
121C23×D53C22⋊C411yescharacteristic
122C23×D53C2×D412yes—
123C2×Q8×C53—21no—
124C2×Q8×C53—42no—
125C2×Q8×C53C22×C412yescomplemented
126C22×C4×C53—24no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C22×C411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C2×C22⋊D53—22no—
140C2×C22⋊D53—22no—
141C2×C22⋊D53—42no—
142C2×C22⋊D53—42no—
143C22×Dic53—24no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53C22⋊C411yescharacteristic
150C22×Dic53C22⋊C411yescharacteristic
151C22×Dic53C22⋊C412yes—
152C22×Dic53C2×D412yes—
153#11719949—48no—
154#11719949—48no—
155C2×D212—24no—
156C2×D212—24no—
157C2×D212—21no—
158C2×D212—42no—
159C2×D212—44no—
160C2×D212—44no—
161C2×D212C22×C412yescomplemented
162C2×C4×D53—24no—
163C2×C4×D53—24no—
164C2×C4×D53—216no—
165C2×C4×D53—24no—
166C2×C4×D53—44no—
167C2×C4×D53—44no—
168C2×C4×D53—44no—
169C2×C4×D53—44no—
170C2×C4×D53—42no—
171C2×C4×D53—42no—
172C2×C4×D53—44no—
173C2×C4×D53—432no—
174C2×C4×D53—48no—
175C2×C4×D53—48no—
176C2×C4×D53C22⋊C412yes—
177C2×C4×D53C22⋊C412yes—
178C2×C4×D53C22⋊C412yes—
179C2×C4×D53C22⋊C412yes—
180C2×C4×D53C22×C412yes—
181C2×C4×D53C22×C412yes—
182C2×C4×D53C2×D412yes—
183C2×C4×D53C2×D412yes—
184C2×C8×C53—22no—
185C2×C8×C53—42no—
186C4⋊C4×C53—22no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—22no—
189C4⋊C4×C53C22×C412yescomplemented
190C22⋊C4×C53—22no—
191C22⋊C4×C53—22no—
192#11719921—24no—
193#11719921—24no—
194#11719921—22no—
195#11719921—22no—
196#11719921—24no—
197C4⋊Dic53—21no—
198C4⋊Dic53C22×C412yescomplemented
199#11719919—24no—
200C4×Dic53—24no—
201C2×#88322—22no—
202C2×#88322—42no—
203#11719912—48no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53C2×C22⋊C411yescharacteristic
207C23×C53C2×C22⋊C411yescharacteristic
208C23×C53AGL(1,ℤ/8)12yes—
209C23×C53#13712yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—216no—
212C22×D53—216no—
213C22×D53—22no—
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—24no—
217C22×D53—22no—
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—24no—
221C22×D53—24no—
222C22×D53—24no—
223C22×D53—24no—
224C22×D53—22no—
225C22×D53—42no—
226C22×D53—42no—
227C22×D53—42no—
228C22×D53#9914yes—
229C22×D53#10414yes—
230C22×D53C2×C22⋊C412yes—
231C22×D53C2×C22⋊C412yes—
232Q8×C53—42no—
233Q8×C53—42no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—24no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—42no—
244C2×C4×C53—48no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53C2×C22⋊C411yescharacteristic
249C2×C4×C53C2×C22⋊C411yescharacteristic
250C2×C4×C53C23×C411yescharacteristic
251C2×C4×C53C22×D411yescharacteristic
252C22⋊D53—42no—
253C22⋊D53—42no—
254C22⋊D53—44no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—24no—
258C2×Dic53—22no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—48no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53C2×C22⋊C412yes—
266C2×Dic53C2×C22⋊C412yes—
267D212—42no—
268D212—42no—
269D212—48no—
270D212—48no—
271C4×D53—24no—
272C4×D53—24no—
273C4×D53—24no—
274C4×D53—24no—
275C4×D53—42no—
276C4×D53—42no—
277C4×D53—42no—
278C4×D53—42no—
279C4×D53—48no—
280C4×D53—432no—
281C8×C53—42no—
282#88322—42no—
283C2×#1504—531nomaximal, Sylow
284C22×C53—21no—
285C22×C53—21no—
286C22×C53—21no—
287C22×C53—21no—
288C22×C53—24no—
289C22×C53—24no—
290C22×C53C2×#9912yes—
291C22×C53#41712yes—
292C22×C53#41712yes—
293C22×C53C2×#10412yes—
294C22×C53C22×C22⋊C411yescharacteristic
295C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C53C2×#13711yescharacteristic
297C2×D53—28no—
298C2×D53—28no—
299C2×D53—216no—
300C2×D53—22no—
301C2×D53—22no—
302C2×D53—22no—
303C2×D53—22no—
304C2×D53—28no—
305C2×D53—28no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—42no—
309C2×D53—84no—
310C2×D53C2×#9914yes—
311C2×D53C2×#10414yes—
312C4×C53—21no—
313C4×C53—21no—
314C4×C53—21no—
315C4×C53—21no—
316C4×C53—44no—
317C4×C53—42no—
318C4×C53—42no—
319Dic53—22no—
320Dic53—22no—
321Dic53—44no—
322Dic53—42no—
323C2×#525—531no—
324C2×#407—531no—
325C22×#95—531no—
326#1504—538no—
327C2×#328—531no—
328C2×#328D5311yescharacteristic, complemented
329C2×#327—531no—
330C2×#327—531no—
331C2×C53—21no—
332C2×C53—21no—
333C2×C53—22no—
334C2×C53#150414yescomplemented
335C2×C53C22×#9911yescharacteristic
336C2×C53C2×#41711yescharacteristic
337C2×C53C22×#10411yescharacteristic
338D53—24no—
339D53—24no—
340D53—82no—
341C22×#41—531no—
342C24×C4—531no—
343C24×C4—1061no—
344C22×M16—1061no—
345#525—534no—
346#525—534no—
347#525—1062no—
348#525—1062no—
349C2×#123—1061no—
350C2×C22⋊Q8—531no—
351C2×C22⋊Q8C2×D5311yescharacteristic
352C2×#121—531no—
353C2×#121—531no—
354C2×#120—1061no—
355C2×C4×D4—1061no—
356C2×C4×D4—1061no—
357C2×#117—531no—
358#407—534no—
359#407—1062no—
360C2×C22⋊C8—531no—
361C2×C22⋊C8—1061no—
362C2×C22⋊C8C2×D5311yescharacteristic
363C2×#95—531no—
364C2×#95—534no—
365C2×#95—1062no—
366C2×#95—1061no—
367C2×#95C2×D5311yescharacteristic
368#328—534no—
369#328C2×D5314yescomplemented
370#327—534no—
371#327—534no—
372C53C2×#150411yescharacteristic, complemented, Sylow
373C25—531no—
374C2×#41—534no—
375C2×#41—1062no—
376C2×#41—2124no—
377C22×Q8C4×D5311yescharacteristic, complemented
378C22×D4—531no—
379C22×D4—1061no—
380C22×D4—1061no—
381C22×D4—1061no—
382C23×C4—21no—
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—531no—
387C23×C4—532no—
388C23×C4—531no—
389C23×C4—532no—
390C23×C4—532no—
391C23×C4—532no—
392C23×C4—1061no—
393C23×C4—1064no—
394C23×C4—1068no—
395C23×C4—1061no—
396C23×C4—2124no—
397C23×C4—2124no—
398C23×C4—2124no—
399C23×C4C22×D5311yescharacteristic
400C2×M16—1064no—
401C2×M16—2124no—
402C22×C8—21no—
403C22×C8—1061no—
404#123—1064no—
405#123—1064no—
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C22⋊Q8—534no—
409C22⋊Q8—1062no—
410C22⋊Q8C4×D5312yescomplemented
411C22⋊Q8C22×D5312yes—
412#121—532no—
413#121—532no—
414#121—534no—
415#121—1062no—
416#121—1061no—
417#121—1061no—
418#120—1064no—
419#120—1064no—
420C4×D4—1064no—
421C4×D4—1064no—
422C4×D4—1064no—
423C4×D4—1064no—
424#117—534no—
425#117—1062no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4C4×D5311yescharacteristic, complemented
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C42—1061no—
436C22⋊C8—21no—
437C22⋊C8—532no—
438C22⋊C8—1064no—
439C22⋊C8—1061no—
440C22⋊C8C22×D5312yes—
441#95—21no—
442#95—532no—
443#95—1064no—
444#95—1061no—
445#95C22×D5312yes—
446C24—531no—
447C24—531no—
448C24—532no—
449C24—532no—
450C24—1064no—
451C24—1064no—
452C24—1062no—
453C24—1062no—
454C24—1061no—
455C24D4×D5311yescharacteristic
456#41—2128no—
457#41—2128no—
458C2×Q8—21no—
459C2×Q8—42no—
460C2×Q8C2×C4×D5312yescomplemented
461C2×D4—532no—
462C2×D4—1064no—
463C2×D4—1064no—
464C2×D4—1062no—
465C2×D4—1062no—
466C2×D4—1061no—
467C2×D4—2122no—
468C2×D4—2124no—
469C2×D4—2124no—
470C2×D4—2122no—
471C2×D4—2122no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—532no—
482C22×C4—532no—
483C22×C4—532no—
484C22×C4—532no—
485C22×C4—532no—
486C22×C4—532no—
487C22×C4—532no—
488C22×C4—532no—
489C22×C4—532no—
490C22×C4—532no—
491C22×C4—1064no—
492C22×C4—1064no—
493C22×C4—1064no—
494C22×C4—10616no—
495C22×C4—1064no—
496C22×C4—1061no—
497C22×C4—1062no—
498C22×C4—2122no—
499C22×C4—2122no—
500C22×C4—2124no—
501C22×C4—2124no—
502C22×C4—2124no—
503C22×C4—2124no—
504C22×C4—2124no—
505C22×C4—2128no—
506C22×C4—21232no—
507C22×C4—2128no—
508C22×C4—2122no—
509C22×C4—2122no—
510C22×C4—2122no—
511C22×C4C2×C4×D5311yescharacteristic
512C22×C4C2×C4×D5311yescharacteristic
513C22×C4D4×D5311yescharacteristic
514C22×C4D4×D5311yescharacteristic
515C22×C4D4×D5311yescharacteristic
516C22×C4C23×D5311yescharacteristic
517M16—2128no—
518C2×C8—22no—
519C2×C8—42no—
520C2×C8—1062no—
521C2×C8—2122no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—532no—
526C4⋊C4—1064no—
527C4⋊C4—1061no—
528C4⋊C4C2×C4×D5312yescomplemented
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—1064no—
532C22⋊C4—1064no—
533C22⋊C4—1064no—
534C22⋊C4—1062no—
535C22⋊C4—1062no—
536C42—1064no—
537C23—22no—
538C23—22no—
539C23—532no—
540C23—532no—
541C23—534no—
542C23—534no—
543C23—1064no—
544C23—1064no—
545C23—1064no—
546C23—1064no—
547C23—10616no—
548C23—10616no—
549C23—1062no—
550C23—1064no—
551C23—1062no—
552C23—1062no—
553C23—1062no—
554C23—1062no—
555C23—1062no—
556C23—1062no—
557C23—2122no—
558C23—2122no—
559C23—2122no—
560C23C22⋊C4×D5311yescharacteristic
561C23C22⋊C4×D5311yescharacteristic
562C23#1313324412yes—
563C23#1313324912yes—
564C23C2×D4×D5311yescharacteristic
565Q8—42no—
566Q8—42no—
567D4—2122no—
568D4—2122no—
569D4—2128no—
570D4—2128no—
571D4—2124no—
572D4—2122no—
573D4—2122no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—532no—
589C2×C4—532no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1064no—
597C2×C4—1062no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2128no—
603C2×C4—21232no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2128no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4C22⋊C4×D5311yescharacteristic
611C2×C4C22⋊C4×D5311yescharacteristic
612C2×C4C22×C4×D5311yescharacteristic
613C2×C4C2×D4×D5311yescharacteristic
614C8—42no—
615C8—2122no—
616C22—21no—
617C22—21no—
618C22—24no—
619C22—24no—
620C22—21no—
621C22—21no—
622C22—534no—
623C22—534no—
624C22—1068no—
625C22—10616no—
626C22—1068no—
627C22—1062no—
628C22—1068no—
629C22—1068no—
630C22—1062no—
631C22—1062no—
632C22—1062no—
633C22—4244no—
634C22—2122no—
635C22—2122no—
636C22—2122no—
637C22#99×D5312yes—
638C22#1681062312yes—
639C22#1681062312yes—
640C22#104×D5312yes—
641C22C2×C22⋊C4×D5311yescharacteristic
642C22C2×#1313324411yescharacteristic
643C22C2×#1313324911yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—1062no—
652C4—1062no—
653C4—2124no—
654C4—2122no—
655C2—21no—
656C2—22no—
657C2—21no—
658C2—1064no—
659C2—1064no—
660C2—4242no—
661C2#2027321814yesdirect product factor
662C2C2×#99×D5311yescharacteristic
663C2C2×#1681062311yescharacteristic
664C2C2×#104×D5311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#328C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#16811136C2
cosocleC22×C4×C53C24
hypercenterC2×#328D53
hypocenterC53C2×#1504
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. C2×#16810585C2semidirect
3. C2×#95×D53C2semidirect
4. #20273218C2direct (N × H)
5. C2×#16810360C2semidirect
6. C2×#16810358C2semidirect
7. C2×#16810356C2semidirect
8. C2×C53×#328C2semidirect
9. C22×#11719949C4semidirect
15. #16811136C4semidirect
18. C2×#13133193C4semidirect
19. C2×#13133192C4semidirect
24. C2×#13133175C4semidirect
26. #16810585C22semidirect
32. #95×D53C22semidirect
33. #16810360C22semidirect
34. #16810358C22semidirect
35. #16810356C22semidirect
39. C53×#328C22semidirect
43. C2×#11719949C2×C4semidirect
46. C22×D212C2×C4semidirect
70. #13133193C2×C4semidirect
72. #13133192C2×C4semidirect
75. #13133191C2×C4semidirect
84. #13133175C2×C4semidirect
85. C2×C4⋊Dic53C2×C4semidirect
98. C22×Q8×C53C2×C4semidirect
104. C22⋊Q8×C53C2×C4semidirect
107. C2×C4⋊C4×C53C2×C4semidirect
125. C2×Q8×C53C22×C4semidirect
161. C2×D212C22×C4semidirect
189. C4⋊C4×C53C22×C4semidirect
198. C4⋊Dic53C22×C4semidirect
328. C2×#328D53semidirect
334. C2×C53#1504semidirect
369. #328C2×D53semidirect
372. C53C2×#1504semidirect
377. C22×Q8C4×D53semidirect
410. C22⋊Q8C4×D53semidirect
429. C2×C4⋊C4C4×D53semidirect
460. C2×Q8C2×C4×D53semidirect
528. C4⋊C4C2×C4×D53semidirect
661. C2#20273218direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111362
C2×#168105852
C2×#95×D532
#202732182
C2×#168103602
C2×#168103582
C2×#168103562
C2×C53×#3282
C2×#150453