Finite Group Databasebeta

← C2×#20275884 #27411513

Subgroups — C2×#20275884 #27411513

Order 13568.

1234 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810641C211yesmaximal, characteristic
2C22×#13133145C211yesmaximal, characteristic
3C2×#16810596C211yesmaximal, characteristic
4#20275884C218yesmaximal, direct product factor
5C2×#16810393C211yesmaximal, characteristic, complemented
6C2×#16810383C211yesmaximal, characteristic, complemented
7C2×#16810379C211yesmaximal, characteristic, complemented
8C2×C53×#533C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13#16810641—22no—
14#16810641C2214yes—
15C2×#13133145—21no—
16C2×#13133145—21no—
17C2×#13133145—22no—
18C2×#13133145—22no—
19C2×#13133145C2214yes—
20#16810596—22no—
21#16810596C2214yes—
22C2×#13133131—22no—
23#16810393C2214yescomplemented
24#16810383C2214yescomplemented
25#16810379C2214yescomplemented
26C53×#533—22no—
27C53×#533—22no—
28C53×#533—22no—
29C53×#533C2214yescomplemented
30C2×C4⋊Q8×C53—21no—
31C2×#125×C53—21no—
32C2×C22⋊Q8×C53—22no—
33C2×C4×Q8×C53—22no—
34C2×#117×C53C2211yescharacteristic
35C2×#117×C53C2211yescharacteristic
36C22×C42×C53C2211yescharacteristic
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—21no—
44C2×#11719945—21no—
45C2×#11719945—21no—
46C2×#11719945—21no—
47C2×#11719945Q812yescomplemented
48C2×#11719945C2312yes—
49C22×Dic106Q811yescharacteristic, complemented
50C22×D212Q811yescharacteristic, complemented
51C22×C22⋊D53—21no—
52C22×C4×D53—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917—21no—
60C2×#11719917—21no—
61C2×#11719917—21no—
62C2×#11719917—21no—
63C2×#11719917C2312yes—
64C2×#11719917C2312yes—
65C22×#88322—21no—
66C22×#88322—21no—
67#13133176—21no—
68#13133176—21no—
69#13133176—21no—
70#13133176C2312yes—
71C2×C4⋊Dic53—21no—
72C2×C22⋊Dic53—21no—
73C2×C4×Dic53—21no—
74#13133145—24no—
75#13133145—24no—
76#13133145—28no—
77#13133131—28no—
78C22×Q8×C53—22no—
79C23×C4×C53—21no—
80C23×C4×C53C2311yescharacteristic
81C4⋊Q8×C53—24no—
82C4⋊Q8×C53—24no—
83#125×C53—24no—
84#125×C53—24no—
85C22⋊Q8×C53—28no—
86C22⋊Q8×C53—28no—
87C4×Q8×C53—28no—
88C4×Q8×C53—28no—
89#117×C53—21no—
90#117×C53—21no—
91#117×C53—21no—
92#117×C53—21no—
93#117×C53D412yes—
94#117×C53D412yes—
95#117×C53C2312yes—
96#117×C53C2312yes—
97C2×C4⋊C4×C53—24no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53D412yes—
103C2×C42×C53—21no—
104C2×C42×C53—21no—
105C2×C42×C53—21no—
106C2×C42×C53—21no—
107C2×C42×C53D411yescharacteristic
108C2×C42×C53D411yescharacteristic
109C2×C42×C53D412yes—
110C2×C42×C53D412yes—
111C2×C42×C53C2312yes—
112C24×C53C2×D411yescharacteristic
113C23×D53—21no—
114C2×Q8×C53—28no—
115C2×Q8×C53—48no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—44no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×Q811yescharacteristic
129C22×C4×C53C2×Q811yescharacteristic
130C22×C4×C53#4111yescharacteristic
131C22×C4×C53#4111yescharacteristic
132C22×C4×C53#4111yescharacteristic
133C22×C4×C53#4111yescharacteristic
134C22×C4×C53#4111yescharacteristic
135C22×C4×C53#4111yescharacteristic
136C22×C4×C53C2411yescharacteristic
137C2×C22⋊D53—22no—
138C2×C22⋊D53—22no—
139C2×C22⋊D53—41no—
140C2×C22⋊D53—41no—
141C2×C22⋊D53—41no—
142C2×C22⋊D53—41no—
143C22×Dic53—21no—
144C22×Dic53—41no—
145#11719945—22no—
146#11719945—22no—
147#11719945—22no—
148#11719945—22no—
149#11719945—22no—
150#11719945—22no—
151#11719945—22no—
152#11719945—22no—
153#11719945—41no—
154#11719945—41no—
155#11719945—41no—
156#11719945—41no—
157#11719945—41no—
158#11719945—41no—
159#11719945—41no—
160#11719945—41no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212—21no—
164C2×D212—21no—
165C2×D212—21no—
166C2×D212C2×Q812yescomplemented
167C2×C4×D53—22no—
168C2×C4×D53—22no—
169C2×C4×D53—41no—
170C2×C4×D53—41no—
171C2×C4×D53—41no—
172C2×C4×D53—41no—
173C2×Dic106—21no—
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106—21no—
178C2×Dic106C2×Q812yescomplemented
179C4⋊C4×C53—216no—
180C4⋊C4×C53—28no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—22no—
188C22⋊C4×C53C2×D414yes—
189C42×C53—22no—
190C42×C53—21no—
191C42×C53—21no—
192C42×C53—22no—
193C42×C53—22no—
194C42×C53—22no—
195C42×C53—22no—
196C42×C53C2×D412yes—
197C42×C53C2×D412yes—
198C42×C53C2×D414yes—
199C22⋊Dic53—22no—
200C22⋊Dic53—41no—
201C4⋊Dic53—22no—
202C4⋊Dic53—41no—
203C4×Dic53—22no—
204C4×Dic53—41no—
205#11719917—22no—
206#11719917—22no—
207#11719917—22no—
208#11719917—22no—
209#11719917—41no—
210#11719917—41no—
211#11719917—41no—
212#11719917—41no—
213C2×#88322—22no—
214C2×#88322—22no—
215C2×#88322—41no—
216C2×#88322—41no—
217C2×#88322—41no—
218C2×#88322—41no—
219C23×C53—42no—
220C23×C53#12011yescharacteristic
221C23×C53#12011yescharacteristic
222C23×C53C22⋊Q811yescharacteristic
223C23×C53C22⋊Q811yescharacteristic
224C23×C53#12411yescharacteristic
225C23×C53#12411yescharacteristic
226C23×C53C22×D411yescharacteristic
227C22×D53—22no—
228C22×D53—41no—
229C22×D53—41no—
230C22×D53—82no—
231Q8×C53—416no—
232C2×C4×C53—24no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—44no—
238C2×C4×C53—44no—
239C2×C4×C53—42no—
240C2×C4×C53—44no—
241C2×C4×C53—44no—
242C2×C4×C53#12011yescharacteristic
243C2×C4×C53#12011yescharacteristic
244C2×C4×C53C22⋊Q811yescharacteristic
245C2×C4×C53C22⋊Q811yescharacteristic
246C2×C4×C53C22⋊Q811yescharacteristic
247C2×C4×C53C22⋊Q811yescharacteristic
248C2×C4×C53C22⋊Q811yescharacteristic
249C2×C4×C53C22⋊Q811yescharacteristic
250C2×C4×C53C22⋊Q811yescharacteristic
251C2×C4×C53C22⋊Q811yescharacteristic
252C2×C4×C53C22⋊Q811yescharacteristic
253C2×C4×C53C22⋊Q811yescharacteristic
254C2×C4×C53#12411yescharacteristic
255C2×C4×C53#12411yescharacteristic
256C2×C4×C53#12411yescharacteristic
257C2×C4×C53#12411yescharacteristic
258C2×C4×C53#12411yescharacteristic
259C2×C4×C53#12411yescharacteristic
260C2×C4×C53#12411yescharacteristic
261C2×C4×C53#12411yescharacteristic
262C2×C4×C53#12411yescharacteristic
263C2×C4×C53#12411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C22×D411yescharacteristic
266C2×C4×C53C22×Q811yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53C2×#4111yescharacteristic
270C22⋊D53—42no—
271C22⋊D53—42no—
272C22⋊D53—42no—
273C22⋊D53—42no—
274C2×Dic53—22no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—42no—
278C2×Dic53—82no—
279D212—22no—
280D212—22no—
281D212—41no—
282D212—41no—
283C4×D53—42no—
284C4×D53—42no—
285C4×D53—42no—
286C4×D53—42no—
287Dic106—22no—
288Dic106—22no—
289Dic106—41no—
290Dic106—41no—
291#88322—42no—
292#88322—42no—
293C2×#2031—531nomaximal, Sylow
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53#39311yescharacteristic
302C22×C53#39311yescharacteristic
303C22×C53#39311yescharacteristic
304C22×C53#39311yescharacteristic
305C22×C53C2×#12011yescharacteristic
306C22×C53C2×C22⋊Q811yescharacteristic
307C22×C53C2×#12411yescharacteristic
308C2×D53—42no—
309C2×D53—82no—
310C2×D53—82no—
311C2×D53—82no—
312C4×C53—24no—
313C4×C53—42no—
314C4×C53—44no—
315C4×C53—44no—
316C4×C53#39311yescharacteristic
317C4×C53#39311yescharacteristic
318C4×C53#39311yescharacteristic
319C4×C53#39311yescharacteristic
320C4×C53C2×C22⋊Q811yescharacteristic
321C4×C53C2×C22⋊Q811yescharacteristic
322C4×C53C2×#12411yescharacteristic
323C4×C53C2×#12411yescharacteristic
324Dic53—42no—
325Dic53—82no—
326C2×#533D5311yescharacteristic, complemented
327C2×#421—531no—
328C22×#104—531no—
329C2×#410—531no—
330#2031—538no—
331C2×#344—531no—
332C2×#339—531no—
333C2×H(3,ℤ/4)—531no—
334C2×C53—21no—
335C2×C53—21no—
336C2×C53—21no—
337C2×C53—21no—
338C2×C53—21no—
339C2×C53—21no—
340C2×C53#203112yescomplemented
341C2×C53C2×#39311yescharacteristic
342D53—82no—
343C22×#41—531no—
344C22×M16—531no—
345C22×M16—531no—
346#533—22no—
347#533—22no—
348#533—22no—
349#533C2×D5314yescomplemented
350C2×C4⋊Q8—21no—
351C2×#125—21no—
352C2×C22⋊Q8—22no—
353C2×C4×Q8—22no—
354C2×#117—531no—
355C2×#117C2×D5311yescharacteristic
356C2×#117C2×D5311yescharacteristic
357C22×C42C2×D5311yescharacteristic
358#421—534no—
359#421—1062no—
360C2×#104—534no—
361C2×#104—1061no—
362C2×#104—1061no—
363C2×#104—1062no—
364C2×#104—1062no—
365#410—534no—
366#410—1062no—
367C2×#99—1062no—
368#344—534no—
369#339—534no—
370H(3,ℤ/4)—534no—
371C53C2×#203111yescharacteristic, complemented, Sylow
372C2×#41—532no—
373C2×#41—532no—
374C2×#41—1061no—
375C2×#41—1061no—
376C2×#41—1061no—
377C2×#41—1061no—
378C2×#41—1061no—
379C2×#41—1061no—
380C2×#41—1061no—
381C2×#41—1061no—
382C2×#41—1061no—
383C2×#41—1061no—
384C22×Q8—22no—
385C22×Q8—531no—
386C22×D4—531no—
387C22×D4—1061no—
388C23×C4—21no—
389C23×C4—1061no—
390C23×C4C22×D5311yescharacteristic
391C2×M16—532no—
392C2×M16—532no—
393C2×M16—1061no—
394C2×M16—1061no—
395C2×M16—1061no—
396C2×M16—1061no—
397C2×M16—1061no—
398C2×M16—1061no—
399C2×M16—1061no—
400C2×M16—1061no—
401C2×M16—1061no—
402C2×M16—1061no—
403C22×C8—1061no—
404C22×C8—1061no—
405C4⋊Q8—24no—
406C4⋊Q8—24no—
407#125—24no—
408#125—24no—
409C22⋊Q8—28no—
410C22⋊Q8—28no—
411C4×Q8—28no—
412C4×Q8—28no—
413#117—21no—
414#117—21no—
415#117—21no—
416#117—21no—
417#117—532no—
418#117—1061no—
419#117—1061no—
420#117—1061no—
421#117C22⋊D5312yes—
422#117C22⋊D5312yes—
423#117C22×D5312yes—
424#117C22×D5312yes—
425C2×C4⋊C4—24no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—1061no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4C22⋊D5312yes—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—1061no—
438C2×C42C22⋊D5312yes—
439C2×C42C22⋊D5311yescharacteristic
440C2×C42C22⋊D5311yescharacteristic
441C2×C42C22⋊D5312yes—
442C2×C42C22×D5312yes—
443#104—1064no—
444#104—1064no—
445#104—1068no—
446#99—1068no—
447C24—1061no—
448C24C2×C22⋊D5311yescharacteristic
449#41—1062no—
450#41—1062no—
451#41—1062no—
452#41—1062no—
453#41—1062no—
454#41—1062no—
455#41—1062no—
456#41—1062no—
457#41—2121no—
458#41—2121no—
459#41—2121no—
460#41—2121no—
461#41—2121no—
462#41—2121no—
463#41—2121no—
464#41—2121no—
465C2×Q8—28no—
466C2×Q8—48no—
467C2×Q8—532no—
468C2×Q8—1061no—
469C2×Q8—1061no—
470C2×Q8—1061no—
471C2×Q8—1061no—
472C2×Q8—1061no—
473C2×D4—532no—
474C2×D4—1062no—
475C2×D4—1062no—
476C2×D4—1061no—
477C2×D4—1061no—
478C2×D4—1061no—
479C2×D4—1061no—
480C2×D4—1061no—
481C2×D4—2121no—
482C2×D4—2121no—
483C2×D4—2121no—
484C2×D4—2121no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—44no—
492C22×C4—1061no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—2121no—
496C22×C4—2121no—
497C22×C4—2121no—
498C22×C4—2121no—
499C22×C4—2121no—
500C22×C4Q8×D5311yescharacteristic
501C22×C4Q8×D5311yescharacteristic
502C22×C4#1171994911yescharacteristic
503C22×C4#1171994911yescharacteristic
504C22×C4#1171994911yescharacteristic
505C22×C4#1171994911yescharacteristic
506C22×C4#1171994911yescharacteristic
507C22×C4#1171994911yescharacteristic
508C22×C4C2×C22⋊D5311yescharacteristic
509C22×C4C2×C22⋊D5311yescharacteristic
510C22×C4C2×C22⋊D5311yescharacteristic
511C22×C4C2×C22⋊D5311yescharacteristic
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C23×D5311yescharacteristic
514M16—1062no—
515M16—1062no—
516M16—1062no—
517M16—1062no—
518M16—2121no—
519M16—2121no—
520M16—2121no—
521M16—2121no—
522C2×C8—1062no—
523C2×C8—1062no—
524C2×C8—2121no—
525C2×C8—2121no—
526C2×C8—2121no—
527C2×C8—2121no—
528C4⋊C4—216no—
529C4⋊C4—28no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—1062no—
535C4⋊C4—2121no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—1062no—
540C22⋊C4—2121no—
541C22⋊C4C2×C22⋊D5314yes—
542C42—22no—
543C42—21no—
544C42—21no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—1062no—
550C42—2121no—
551C42C2×C22⋊D5312yes—
552C42C2×C22⋊D5312yes—
553C42C2×C22⋊D5314yes—
554C23—42no—
555C23—1062no—
556C23—4242no—
557C23—2121no—
558C23—2121no—
559C23#1313318611yescharacteristic
560C23#1313318611yescharacteristic
561C23#1313319311yescharacteristic
562C23#1313319311yescharacteristic
563C23#1313320611yescharacteristic
564C23#1313320611yescharacteristic
565C23C22×C22⋊D5311yescharacteristic
566Q8—416no—
567Q8—1062no—
568Q8—1062no—
569Q8—2121no—
570Q8—2121no—
571D4—1062no—
572D4—1062no—
573D4—2122no—
574D4—2122no—
575D4—2122no—
576D4—2122no—
577D4—2121no—
578D4—2121no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—1062no—
590C2×C4—4242no—
591C2×C4—2122no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4#1313318611yescharacteristic
599C2×C4#1313318611yescharacteristic
600C2×C4#1313319311yescharacteristic
601C2×C4#1313319311yescharacteristic
602C2×C4#1313319311yescharacteristic
603C2×C4#1313319311yescharacteristic
604C2×C4#1313319311yescharacteristic
605C2×C4#1313319311yescharacteristic
606C2×C4#1313319311yescharacteristic
607C2×C4#1313319311yescharacteristic
608C2×C4#1313319311yescharacteristic
609C2×C4#1313319311yescharacteristic
610C2×C4#1313320611yescharacteristic
611C2×C4#1313320611yescharacteristic
612C2×C4#1313320611yescharacteristic
613C2×C4#1313320611yescharacteristic
614C2×C4#1313320611yescharacteristic
615C2×C4#1313320611yescharacteristic
616C2×C4#1313320611yescharacteristic
617C2×C4#1313320611yescharacteristic
618C2×C4#1313320611yescharacteristic
619C2×C4#1313320611yescharacteristic
620C2×C4C22×C22⋊D5311yescharacteristic
621C2×C4C22×C22⋊D5311yescharacteristic
622C2×C4C2×Q8×D5311yescharacteristic
623C2×C4C2×#1171994911yescharacteristic
624C2×C4C2×#1171994911yescharacteristic
625C2×C4C2×#1171994911yescharacteristic
626C8—2122no—
627C8—2122no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—4242no—
636C22—4242no—
637C22—4242no—
638C22—2122no—
639C22#1681053411yescharacteristic
640C22#1681053411yescharacteristic
641C22#1681053411yescharacteristic
642C22#1681053411yescharacteristic
643C22C2×#1313318611yescharacteristic
644C22C2×#1313319311yescharacteristic
645C22C2×#1313320611yescharacteristic
646C4—24no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—4242no—
651C4—2122no—
652C4#1681053411yescharacteristic
653C4#1681053411yescharacteristic
654C4#1681053411yescharacteristic
655C4#1681053411yescharacteristic
656C4C2×#1313319311yescharacteristic
657C4C2×#1313319311yescharacteristic
658C4C2×#1313320611yescharacteristic
659C4C2×#1313320611yescharacteristic
660C2—21no—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—4242no—
667C2#2027588412yesdirect product factor
668C2C2×#1681053411yescharacteristic
subgroupG/H
centerC2×C4#13133206
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#533C2
socleC22×C53#393
prime radicalC22×#11719945C22
hypercenterC2×#533D53
hypocenterC53C2×#2031
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
4. #20275884C2direct (N × H)
5. C2×#16810393C2semidirect
6. C2×#16810383C2semidirect
7. C2×#16810379C2semidirect
8. C2×C53×#533C2semidirect
23. #16810393C22semidirect
24. #16810383C22semidirect
25. #16810379C22semidirect
29. C53×#533C22semidirect
47. C2×#11719945Q8semidirect
49. C22×Dic106Q8semidirect
50. C22×D212Q8semidirect
166. C2×D212C2×Q8semidirect
178. C2×Dic106C2×Q8semidirect
326. C2×#533D53semidirect
340. C2×C53#2031semidirect
349. #533C2×D53semidirect
371. C53C2×#2031semidirect
667. C2#20275884direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168106412
C22×#131331452
C2×#168105962
#202758842
C2×#168103932
C2×#168103832
C2×#168103792
C2×C53×#5332
C2×#203153