Finite Group Databasebeta

← C2×#20277665 #27543442

Subgroups — C2×#20277665 #27543442

Order 13568.

3678 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D53C211yesmaximal, characteristic
2C22×#13133175C211yesmaximal, characteristic, complemented
3#20277665C2116yesmaximal, direct product factor
4C2×#16810523C212yesmaximal
5C2×#16810514C212yesmaximal, complemented
6C2×#16810511C212yesmaximal, complemented
7C2×#16810492C211yesmaximal, characteristic, complemented
8C2×#16810483C211yesmaximal, characteristic, complemented
9C2×#16810480C211yesmaximal, characteristic, complemented
10C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
11C2×C4×#11719919C212yesmaximal, complemented
12C22×#117×C53C211yesmaximal, characteristic, complemented
13C23×C22⋊D53C2211yescharacteristic
14C23×C4×D53C2211yescharacteristic
15C2×C4×C22⋊D53C4116yescomplemented
16C2×C4×C22⋊D53C2218yes—
17C2×#13133175—28no—
18C2×#13133175C2218yescomplemented
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C22×#11719919C2212yes—
22C22×#11719919C2212yes—
23C22×#11719919C2211yescharacteristic
24C22×C22⋊Dic53C2212yes—
25C22×C22⋊Dic53C2211yescharacteristic
26C22×#11719921C2212yes—
27C22×#11719921C2212yes—
28C22×#11719921C2211yescharacteristic
29C22×C4×Dic53C2212yes—
30C22×C4×Dic53C2212yes—
31C22×C4×Dic53C2211yescharacteristic
32#16810523C22116yes—
33#16810514C22116yescomplemented
34#16810511C22116yescomplemented
35#16810492C2218yescomplemented
36#16810483C2218yescomplemented
37#16810480C2218yescomplemented
38C4⋊C4×Dic53C2218yescomplemented
39C4×#11719919C22116yescomplemented
40C2×#13133124C2212yes—
41C2×#13133124C2212yes—
42C2×#13133124C2212yes—
43C24×C4×C53C2211yescharacteristic
44C2×#117×C53—28no—
45C2×#117×C53C2218yescomplemented
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×C4⋊C4×C53C2211yescharacteristic
48C22×C22⋊C4×C53C2212yes—
49C22×C42×C53C2212yes—
50C24×D53C2311yescharacteristic
51C22×C22⋊D53—24no—
52C22×C22⋊D53—24no—
53C22×C22⋊D53C2×C418yes—
54C22×C22⋊D53C2314yes—
55C23×Dic53C2312yes—
56C23×Dic53C2312yes—
57C23×Dic53C2311yescharacteristic
58C23×Dic53C2311yescharacteristic
59C23×Dic53C2311yescharacteristic
60C22×C4×D53—24no—
61C22×C4×D53—24no—
62C22×C4×D53C2×C418yes—
63C22×C4×D53C2314yes—
64C4×C22⋊D53C2×C4164yescomplemented
65#13133175—232no—
66C2×C4⋊Dic53—24no—
67C2×C4⋊Dic53—24no—
68C2×C4⋊Dic53C2314yes—
69C2×C4⋊Dic53C2314yes—
70C2×#11719919—28no—
71C2×#11719919—44no—
72C2×#11719919C2×C418yes—
73C2×#11719919C2318yes—
74C2×#11719919C2318yes—
75C2×#11719919C2314yes—
76C2×C22⋊Dic53—44no—
77C2×C22⋊Dic53C2×C418yes—
78C2×C22⋊Dic53C2318yes—
79C2×C22⋊Dic53C2314yes—
80C2×#11719921—28no—
81C2×#11719921—44no—
82C2×#11719921C2×C418yes—
83C2×#11719921C2318yes—
84C2×#11719921C2318yes—
85C2×#11719921C2314yes—
86C2×C4×Dic53—28no—
87C2×C4×Dic53—28no—
88C2×C4×Dic53C2×C418yes—
89C2×C4×Dic53C2318yes—
90C2×C4×Dic53C2318yes—
91C2×C4×Dic53C2314yes—
92#13133124C2318yes—
93#13133124C2318yes—
94#13133124C2318yes—
95C25×C53C2311yescharacteristic
96C23×C4×C53—24no—
97C23×C4×C53—24no—
98C23×C4×C53C2×C418yes—
99C23×C4×C53C2312yes—
100C23×C4×C53C2311yescharacteristic
101C23×C4×C53C2314yes—
102C23×C4×C53C2312yes—
103C23×C4×C53C2311yescharacteristic
104#117×C53—232no—
105C2×C4⋊C4×C53—24no—
106C2×C4⋊C4×C53D418yescomplemented
107C2×C4⋊C4×C53C2314yes—
108C2×C4⋊C4×C53C2314yes—
109C2×C22⋊C4×C53—28no—
110C2×C22⋊C4×C53C2318yes—
111C2×C42×C53—28no—
112C2×C42×C53C2318yes—
113C24×C53—22no—
114C24×C53—22no—
115C24×C53—22no—
116C24×C53—22no—
117C24×C53—22no—
118C24×C53—22no—
119C24×C53C22×C414yes—
120C24×C53C2412yes—
121C24×C53C2411yescharacteristic
122C23×D53—22no—
123C23×D53—22no—
124C23×D53—22no—
125C23×D53—22no—
126C23×D53—22no—
127C23×D53—22no—
128C23×D53C22×C414yes—
129C23×D53C2412yes—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—24no—
133C22×C4×C53—24no—
134C22×C4×C53—232no—
135C22×C4×C53—28no—
136C22×C4×C53—28no—
137C22×C4×C53—24no—
138C22×C4×C53—24no—
139C22×C4×C53—24no—
140C22×C4×C53C22×C414yes—
141C22×C4×C53C22×C4116yes—
142C22×C4×C53C22×C414yes—
143C22×C4×C53C2×D418yes—
144C22×C4×C53C2×D414yes—
145C22×C4×C53#4114yes—
146C22×C4×C53C2414yes—
147C22×C4×C53C2412yes—
148C22×C4×C53C2414yes—
149C22×C4×C53C2412yes—
150C2×C22⋊D53—232no—
151C2×C22⋊D53—28no—
152C2×C22⋊D53—28no—
153C2×C22⋊D53C22×C4116yes—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—24no—
159C22×Dic53—24no—
160C22×Dic53—24no—
161C22×Dic53—24no—
162C22×Dic53—24no—
163C22×Dic53—24no—
164C22×Dic53C22×C414yes—
165C22×Dic53C22×C414yes—
166C22×Dic53C22×C414yes—
167C22×Dic53#4114yes—
168C22×Dic53#4114yes—
169C22×Dic53C2414yes—
170C22×Dic53C2414yes—
171C22×Dic53C2412yes—
172C22×Dic53C2412yes—
173C22×Dic53C2412yes—
174C2×C4×D53—232no—
175C2×C4×D53—28no—
176C2×C4×D53—28no—
177C2×C4×D53C22×C4116yes—
178C4⋊C4×C53—28no—
179C4⋊C4×C53C2×D4116yescomplemented
180C22⋊C4×C53—216no—
181C42×C53—216no—
182C22⋊Dic53—48no—
183C22⋊Dic53C22×C4116yes—
184#11719921—216no—
185#11719921—48no—
186#11719921C22×C4116yes—
187C4⋊Dic53—28no—
188C4⋊Dic53—28no—
189#11719919—216no—
190#11719919—48no—
191#11719919C22×C4116yes—
192C4×Dic53—216no—
193C4×Dic53—216no—
194C4×Dic53C22×C4116yes—
195C23×C53—28no—
196C23×C53—28no—
197C23×C53—28no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—22no—
204C23×C53—44no—
205C23×C53—44no—
206C23×C53—44no—
207C23×C53—44no—
208C23×C53C23×C414yes—
209C23×C53C23×C412yes—
210C23×C53C22×D412yes—
211C23×C53C2×#4112yes—
212C23×C53C2×#4112yes—
213C23×C53C2×#4112yes—
214C23×C53H(5,𝔽2)12yes—
215C23×C532−1+412yes—
216C23×C53C2511yescharacteristic
217C22×D53—28no—
218C22×D53—28no—
219C22×D53—28no—
220C22×D53—22no—
221C22×D53—22no—
222C22×D53—22no—
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53—44no—
227C22×D53—44no—
228C22×D53—44no—
229C22×D53—44no—
230C22×D53C23×C414yes—
231C2×C4×C53—28no—
232C2×C4×C53—216no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—232no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—48no—
242C2×C4×C53C4×D4116yes—
243C2×C4×C53C23×C414yes—
244C2×C4×C53C23×C414yes—
245C2×C4×C53C22×D418yes—
246C2×C4×C53C22×D414yes—
247C2×C4×C53C2×#4114yes—
248C22⋊D53—232no—
249C2×Dic53—28no—
250C2×Dic53—28no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—48no—
262C2×Dic53—48no—
263C2×Dic53#117116yescomplemented
264C2×Dic53C23×C414yes—
265C2×Dic53C23×C414yes—
266C2×Dic53C23×C414yes—
267C2×Dic53C2×#4114yes—
268C2×Dic53C2×#4114yes—
269C4×D53—232no—
270C2×#2354—531nomaximal, Sylow
271C22×C53—24no—
272C22×C53—24no—
273C22×C53—24no—
274C22×C53—416no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53C2×#11714yes—
280C22×C53C2×C4×D414yes—
281C22×C53#52014yes—
282C22×C53#54714yes—
283C22×C53#55014yes—
284C22×C53#55514yes—
285C22×C53#55614yes—
286C22×C53C24×C411yescharacteristic
287C22×C53C23×D411yescharacteristic
288C22×C53C22×#4111yescharacteristic
289C22×C53C22×#4111yescharacteristic
290C22×C53C22×#4111yescharacteristic
291C22×C53C2×H(5,𝔽2)11yescharacteristic
292C22×C53C2×2−1+411yescharacteristic
293C2×D53—24no—
294C2×D53—24no—
295C2×D53—24no—
296C2×D53—416no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—42no—
301C4×C53—24no—
302C4×C53—28no—
303C4×C53—44no—
304C4×C53C2×C4×D418yes—
305Dic53—24no—
306Dic53—24no—
307Dic53—44no—
308Dic53—44no—
309Dic53C2×#11718yescomplemented
310C22×C4×D4—531no—
311C22×#117—531no—
312C22×#117D5311yescharacteristic, complemented
313#2354—5316no—
314C2×#390—532no—
315C2×#388—532no—
316C2×#388—532no—
317C2×#384—531no—
318C2×#380—531no—
319C2×#380—531no—
320C2×C4×C4⋊C4—532no—
321C2×C4×C4⋊C4—531no—
322C2×C53—44no—
323C2×C53#235418yescomplemented
324C2×C53C22×#11711yescharacteristic
325C2×C53C22×C4×D411yescharacteristic
326C2×C53C2×#52011yescharacteristic
327C2×C53C2×#54711yescharacteristic
328C2×C53C2×#55011yescharacteristic
329C2×C53C2×#55511yescharacteristic
330C2×C53C2×#55611yescharacteristic
331D53—44no—
332C23×D4—531no—
333C24×C4—531no—
334C24×C4C2×D5311yescharacteristic
335C2×C4×D4—5316no—
336C2×C4×D4—538no—
337C2×#117—28no—
338C2×#117—538no—
339C2×#117—1068no—
340C2×#117C2×D5318yescomplemented
341C22×C4⋊C4—531no—
342C22×C4⋊C4—532no—
343C22×C4⋊C4—531no—
344C22×C4⋊C4—531no—
345C22×C4⋊C4—532no—
346C22×C4⋊C4C2×D5311yescharacteristic
347C22×C4⋊C4C2×D5311yescharacteristic
348C22×C22⋊C4—531no—
349C22×C22⋊C4—531no—
350C22×C22⋊C4—532no—
351C22×C22⋊C4—532no—
352C22×C22⋊C4—532no—
353C22×C22⋊C4C2×D5312yes—
354C22×C42—531no—
355C22×C42—532no—
356C22×C42—532no—
357C22×C42C2×D5312yes—
358#390—5316no—
359#388—5316no—
360#388—5316no—
361#384—538no—
362#380—538no—
363#380—538no—
364C4×C4⋊C4—538no—
365C4×C4⋊C4—5316no—
366C2×#95—532no—
367C2×#95—532no—
368C2×#95—532no—
369C53C2×#235411yescharacteristic, complemented, Sylow
370C25—531no—
371C25C22×D5311yescharacteristic
372C22×D4—538no—
373C22×D4—534no—
374C22×D4—1064no—
375C22×D4—1064no—
376C23×C4—24no—
377C23×C4—24no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—531no—
381C23×C4—538no—
382C23×C4—534no—
383C23×C4—532no—
384C23×C4—532no—
385C23×C4—1064no—
386C23×C4—1064no—
387C23×C4C4×D5318yes—
388C23×C4C22×D5312yes—
389C23×C4C22×D5314yes—
390C23×C4C22×D5311yescharacteristic
391C23×C4C22×D5312yes—
392C23×C4C22×D5311yescharacteristic
393C4×D4—5364no—
394#117—232no—
395#117—10632no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—538no—
398C2×C4⋊C4—534no—
399C2×C4⋊C4—534no—
400C2×C4⋊C4—538no—
401C2×C4⋊C4—534no—
402C2×C4⋊C4—538no—
403C2×C4⋊C4—1068no—
404C2×C4⋊C4—1064no—
405C2×C4⋊C4—1064no—
406C2×C4⋊C4—2124no—
407C2×C4⋊C4C22⋊D5318yescomplemented
408C2×C4⋊C4C22×D5314yes—
409C2×C4⋊C4C22×D5314yes—
410C2×C22⋊C4—28no—
411C2×C22⋊C4—538no—
412C2×C22⋊C4—534no—
413C2×C22⋊C4—538no—
414C2×C22⋊C4—534no—
415C2×C22⋊C4—538no—
416C2×C22⋊C4—538no—
417C2×C22⋊C4—538no—
418C2×C22⋊C4—1068no—
419C2×C22⋊C4—2124no—
420C2×C22⋊C4—2124no—
421C2×C22⋊C4C22×D5318yes—
422C2×C42—28no—
423C2×C42—538no—
424C2×C42—534no—
425C2×C42—538no—
426C2×C42—538no—
427C2×C42—1068no—
428C2×C42—1068no—
429C2×C42C22×D5318yes—
430#95—538no—
431#95—538no—
432#95—538no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—534no—
440C24—532no—
441C24—1062no—
442C24—1062no—
443C24—1062no—
444C24—1062no—
445C24—1062no—
446C24—1062no—
447C24C2×C4×D5314yes—
448C24C23×D5312yes—
449C24C23×D5311yescharacteristic
450C2×D4—5316no—
451C2×D4—10632no—
452C2×D4—1068no—
453C2×D4—1068no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—232no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—534no—
465C22×C4—534no—
466C22×C4—534no—
467C22×C4—532no—
468C22×C4—534no—
469C22×C4—534no—
470C22×C4—532no—
471C22×C4—532no—
472C22×C4—5316no—
473C22×C4—534no—
474C22×C4—534no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—10632no—
480C22×C4—1068no—
481C22×C4—1068no—
482C22×C4—1064no—
483C22×C4—1064no—
484C22×C4—1064no—
485C22×C4—1064no—
486C22×C4—1064no—
487C22×C4—1064no—
488C22×C4C2×C4×D5314yes—
489C22×C4C2×C4×D53116yes—
490C22×C4C2×C4×D5314yes—
491C22×C4#1171994514yes—
492C22×C4C2×C22⋊D5318yes—
493C22×C4C2×C22⋊D5314yes—
494C22×C4C23×D5314yes—
495C22×C4C23×D5312yes—
496C22×C4C23×D5314yes—
497C22×C4C23×D5312yes—
498C4⋊C4—28no—
499C4⋊C4—5316no—
500C4⋊C4—10616no—
501C4⋊C4—1068no—
502C4⋊C4—1068no—
503C4⋊C4—2128no—
504C4⋊C4C2×C22⋊D53116yescomplemented
505C22⋊C4—216no—
506C22⋊C4—5316no—
507C22⋊C4—5316no—
508C22⋊C4—10616no—
509C22⋊C4—2128no—
510C22⋊C4—2128no—
511C42—216no—
512C42—5316no—
513C42—10616no—
514C42—10616no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—534no—
529C23—1068no—
530C23—1068no—
531C23—1068no—
532C23—1062no—
533C23—1062no—
534C23—1062no—
535C23—1062no—
536C23—1062no—
537C23—1062no—
538C23—2124no—
539C23—2124no—
540C23—2124no—
541C23—2124no—
542C23C22×C4×D5314yes—
543C23C22×C4×D5312yes—
544C23C22×C22⋊D5312yes—
545C23#41×D5312yes—
546C23#41×D5312yes—
547C23C2×#1171994512yes—
548C23#1313326312yes—
549C23#1313326412yes—
550C23C24×D5311yescharacteristic
551D4—10632no—
552C2×C4—216no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—232no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—48no—
563C2×C4—5316no—
564C2×C4—534no—
565C2×C4—534no—
566C2×C4—534no—
567C2×C4—534no—
568C2×C4—534no—
569C2×C4—1068no—
570C2×C4—1068no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—10632no—
576C2×C4—1064no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—2128no—
583C2×C4—2128no—
584C2×C4C4×C22⋊D53116yes—
585C2×C4C22×C4×D5314yes—
586C2×C4C22×C4×D5314yes—
587C2×C4C22×C22⋊D5318yes—
588C2×C4C22×C22⋊D5314yes—
589C2×C4C2×#1171994514yes—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—416no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—1064no—
599C22—1064no—
600C22—1064no—
601C22—2122no—
602C22—21216no—
603C22—2122no—
604C22—2122no—
605C22—2122no—
606C22#117×D5314yes—
607C22C2×C4×C22⋊D5314yes—
608C22#1681110314yes—
609C22#1681127214yes—
610C22#1681129214yes—
611C22#1681133614yes—
612C22#1681134114yes—
613C22C23×C4×D5311yescharacteristic
614C22C23×C22⋊D5311yescharacteristic
615C22C2×#41×D5311yescharacteristic
616C22C2×#41×D5311yescharacteristic
617C22C22×#1171994511yescharacteristic
618C22C2×#1313326311yescharacteristic
619C22C2×#1313326411yescharacteristic
620C4—24no—
621C4—28no—
622C4—44no—
623C4—538no—
624C4—1064no—
625C4—1064no—
626C4—2124no—
627C4—2124no—
628C4C2×C4×C22⋊D5318yes—
629C2—44no—
630C2—2124no—
631C2#2027766518yesdirect product factor
632C2C2×#117×D5311yescharacteristic
633C2C22×C4×C22⋊D5311yescharacteristic
634C2C2×#1681110311yescharacteristic
635C2C2×#1681127211yescharacteristic
636C2C2×#1681129211yescharacteristic
637C2C2×#1681133611yescharacteristic
638C2C2×#1681134111yescharacteristic
subgroupG/H
centerC24C23×D53
commutatorC22×C53C24×C4
FrattiniC23C24×D53
FittingC22×#117×C53C2
socleC24×C53C24
prime radicalC23×C22⋊D53C22
cosocleC23×C53C25
hypercenterC22×#117D53
hypocenterC53C2×#2354
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C22×#13133175C2semidirect
3. #20277665C2direct (N × H)
5. C2×#16810514C2semidirect
6. C2×#16810511C2semidirect
7. C2×#16810492C2semidirect
8. C2×#16810483C2semidirect
9. C2×#16810480C2semidirect
10. C2×C4⋊C4×Dic53C2semidirect
11. C2×C4×#11719919C2semidirect
12. C22×#117×C53C2semidirect
15. C2×C4×C22⋊D53C4semidirect
18. C2×#13133175C22semidirect
33. #16810514C22semidirect
34. #16810511C22semidirect
35. #16810492C22semidirect
36. #16810483C22semidirect
37. #16810480C22semidirect
38. C4⋊C4×Dic53C22semidirect
39. C4×#11719919C22semidirect
45. C2×#117×C53C22semidirect
64. C4×C22⋊D53C2×C4semidirect
106. C2×C4⋊C4×C53D4semidirect
179. C4⋊C4×C53C2×D4semidirect
263. C2×Dic53#117semidirect
309. Dic53C2×#117semidirect
312. C22×#117D53semidirect
323. C2×C53#2354semidirect
340. C2×#117C2×D53semidirect
369. C53C2×#2354semidirect
407. C2×C4⋊C4C22⋊D53semidirect
504. C4⋊C4C2×C22⋊D53semidirect
631. C2#20277665direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D532
C22×#131331752
#202776652
C2×#168105232
C2×#168105142
C2×#168105112
C2×#168104922
C2×#168104832
C2×#168104802
C2×C4⋊C4×Dic532
C2×C4×#117199192
C22×#117×C532
C2×#235453