Finite Group Databasebeta

← C2×#20279626 #27555355

Subgroups — C2×#20279626 #27555355

Order 13568.

3526 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133197C211yesmaximal, characteristic, complemented
2C22×#13133196C211yesmaximal, characteristic, complemented
3C22×#13133182C211yesmaximal, characteristic, complemented
4C22×C4×C22⋊D53C211yesmaximal, characteristic
5#20279626C2116yesmaximal, direct product factor
6C2×#16810557C212yesmaximal, complemented
7C2×#16810538C212yesmaximal
8C2×#16810516C212yesmaximal, complemented
9C2×#16810512C212yesmaximal, complemented
10C2×#16810488C211yesmaximal, characteristic, complemented
11C2×#16810480C211yesmaximal, characteristic, complemented
12C2×C53×#380C211yesmaximal, characteristic, complemented
13C23×Dic106C2211yescharacteristic
14C23×C22⋊D53C2211yescharacteristic
15C23×C4×D53C2211yescharacteristic
16C2×#13133197—28no—
17C2×#13133197C2218yescomplemented
18C2×#13133196—28no—
19C2×#13133196C2218yescomplemented
20C2×#13133182—28no—
21C2×#13133182C2218yescomplemented
22C2×C4×C22⋊D53—28no—
23C2×C4×C22⋊D53C2218yes—
24C22×C4⋊Dic53C2211yescharacteristic
25C22×C4⋊Dic53C2212yes—
26C22×#11719919C2211yescharacteristic
27C22×#11719919C2212yes—
28C22×#11719919C2212yes—
29C22×C22⋊Dic53C2211yescharacteristic
30C22×C22⋊Dic53C2212yes—
31C22×#11719921C2212yes—
32C22×#11719921C2211yescharacteristic
33C22×#11719921C2212yes—
34C22×C4×Dic53C2212yes—
35C22×C4×Dic53C2211yescharacteristic
36#16810557C22116yescomplemented
37#16810538C22116yes—
38#16810516C22116yescomplemented
39#16810512C22116yescomplemented
40#16810488C2218yescomplemented
41#16810480C2218yescomplemented
42C2×#13133124C2212yes—
43C2×#13133124C2212yes—
44C2×#13133124C2212yes—
45C24×C4×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×C4⋊C4×C53C2211yescharacteristic
48C22×C22⋊C4×C53C2212yes—
49C53×#380C2218yescomplemented
50C2×#95×C53C2212yes—
51C24×D53C2311yescharacteristic
52C22×Dic106—24no—
53C22×Dic106—24no—
54C22×Dic106D418yescomplemented
55C22×Dic106C2314yes—
56C22×C22⋊D53—24no—
57C22×C22⋊D53—24no—
58C22×C22⋊D53—24no—
59C22×C22⋊D53C2314yes—
60C23×Dic53C2312yes—
61C23×Dic53C2311yescharacteristic
62C23×Dic53C2311yescharacteristic
63C23×Dic53C2312yes—
64C23×Dic53C2311yescharacteristic
65C22×C4×D53—24no—
66C22×C4×D53—24no—
67C22×C4×D53—24no—
68C22×C4×D53C2314yes—
69#13133197—232no—
70#13133196—232no—
71#13133182—232no—
72C4×C22⋊D53—232no—
73C2×C4⋊Dic53—24no—
74C2×C4⋊Dic53—28no—
75C2×C4⋊Dic53C2318yes—
76C2×C4⋊Dic53C2314yes—
77C2×#11719919—24no—
78C2×#11719919—28no—
79C2×#11719919—28no—
80C2×#11719919C2318yes—
81C2×#11719919C2318yes—
82C2×#11719919C2314yes—
83C2×C22⋊Dic53—24no—
84C2×C22⋊Dic53—28no—
85C2×C22⋊Dic53C2314yes—
86C2×C22⋊Dic53C2318yes—
87C2×#11719921—28no—
88C2×#11719921—24no—
89C2×#11719921—28no—
90C2×#11719921C2314yes—
91C2×#11719921C2318yes—
92C2×#11719921C2318yes—
93C2×C4×Dic53—28no—
94C2×C4×Dic53D418yes—
95C2×C4×Dic53C2318yes—
96C2×C4×Dic53C2314yes—
97#13133124C2318yes—
98#13133124C2318yes—
99#13133124C2318yes—
100C25×C53C2311yescharacteristic
101C23×C4×C53—24no—
102C23×C4×C53—24no—
103C23×C4×C53—24no—
104C23×C4×C53C2312yes—
105C23×C4×C53C2312yes—
106C23×C4×C53C2311yescharacteristic
107C23×C4×C53C2314yes—
108C23×C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53—24no—
110C2×C4⋊C4×C53—24no—
111C2×C4⋊C4×C53C2314yes—
112C2×C4⋊C4×C53C2314yes—
113C2×C22⋊C4×C53—28no—
114C2×C22⋊C4×C53C2318yes—
115#95×C53C2318yes—
116C24×C53—22no—
117C24×C53—22no—
118C24×C53—22no—
119C24×C53—22no—
120C24×C53—22no—
121C24×C53—22no—
122C24×C53#4114yes—
123C24×C53C2412yes—
124C24×C53C2411yescharacteristic
125C23×D53—22no—
126C23×D53—22no—
127C23×D53—22no—
128C23×D53—22no—
129C23×D53—22no—
130C23×D53—22no—
131C23×D53#4114yes—
132C23×D53C2412yes—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—24no—
138C22×C4×C53—24no—
139C22×C4×C53—22no—
140C22×C4×C53—24no—
141C22×C4×C53—24no—
142C22×C4×C53—28no—
143C22×C4×C53—28no—
144C22×C4×C53—28no—
145C22×C4×C53—416no—
146C22×C4×C53C2×D414yes—
147C22×C4×C53C2×D414yes—
148C22×C4×C53#4114yes—
149C22×C4×C53C2412yes—
150C22×C4×C53C2414yes—
151C22×C4×C53C2414yes—
152C22×C4×C53C2412yes—
153C2×C22⋊D53—232no—
154C2×C22⋊D53—28no—
155C2×C22⋊D53—28no—
156C2×C22⋊D53—28no—
157C22×Dic53—24no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—24no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—24no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—24no—
167C22×Dic53—22no—
168C22×Dic53—24no—
169C22×Dic53C2×D414yes—
170C22×Dic53C2×D414yes—
171C22×Dic53C2×D418yes—
172C22×Dic53C2414yes—
173C22×Dic53C2412yes—
174C22×Dic53C2412yes—
175C22×Dic53C2412yes—
176C22×Dic53C2414yes—
177C2×C4×D53—28no—
178C2×C4×D53—28no—
179C2×C4×D53—28no—
180C2×C4×D53—416no—
181C2×Dic106—232no—
182C2×Dic106—28no—
183C2×Dic106—28no—
184C2×Dic106C2×D4116yescomplemented
185C4⋊C4×C53—28no—
186C4⋊C4×C53—28no—
187C22⋊C4×C53—216no—
188C22⋊Dic53—28no—
189C22⋊Dic53—216no—
190#11719921—216no—
191#11719921—28no—
192#11719921—216no—
193C4⋊Dic53—28no—
194C4⋊Dic53—216no—
195#11719919—28no—
196#11719919—216no—
197#11719919—216no—
198C4×Dic53—216no—
199C4×Dic53C2×D4116yes—
200C23×C53—28no—
201C23×C53—28no—
202C23×C53—28no—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—44no—
210C23×C53—44no—
211C23×C53—44no—
212C23×C53—44no—
213C23×C53C22×D412yes—
214C23×C53C22×D412yes—
215C23×C53C2×#4112yes—
216C23×C53C2×#4114yes—
217C23×C53C2×#4112yes—
218C23×C53C2×#4112yes—
219C23×C532−1+412yes—
220C23×C532−1+412yes—
221C23×C53C2511yescharacteristic
222C22×D53—28no—
223C22×D53—28no—
224C22×D53—28no—
225C22×D53—22no—
226C22×D53—22no—
227C22×D53—22no—
228C22×D53—22no—
229C22×D53—22no—
230C22×D53—22no—
231C22×D53—44no—
232C22×D53—44no—
233C22×D53—44no—
234C22×D53—44no—
235C22×D53C2×#4114yes—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—22no—
241C2×C4×C53—24no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—24no—
245C2×C4×C53—44no—
246C2×C4×C53—48no—
247C2×C4×C53—48no—
248C2×C4×C53—416no—
249C2×C4×C53#121116yes—
250C2×C4×C53C22×D414yes—
251C2×C4×C53C22×D414yes—
252C2×C4×C53C2×#4114yes—
253C22⋊D53—232no—
254C2×Dic53—216no—
255C2×Dic53—28no—
256C2×Dic53—28no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—24no—
266C2×Dic53—22no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—44no—
270C2×Dic53—48no—
271C2×Dic53C22×D418yes—
272C2×Dic53C22×D414yes—
273C2×Dic53C22×D414yes—
274C4×D53—416no—
275Dic106—232no—
276C2×#2536—531nomaximal, Sylow
277C22×C53—24no—
278C22×C53—24no—
279C22×C53—24no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—416no—
285C22×C53C2×#12114yes—
286C22×C53#52514yes—
287C22×C53#53614yes—
288C22×C53#54814yes—
289C22×C53#54814yes—
290C22×C53#55214yes—
291C22×C53#55214yes—
292C22×C53C23×D411yescharacteristic
293C22×C53C23×D411yescharacteristic
294C22×C53C22×#4111yescharacteristic
295C22×C53C22×#4111yescharacteristic
296C22×C53C22×#4111yescharacteristic
297C22×C53C2×2−1+411yescharacteristic
298C22×C53C2×2−1+411yescharacteristic
299C2×D53—24no—
300C2×D53—24no—
301C2×D53—24no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—416no—
307C4×C53—42no—
308C4×C53—44no—
309C4×C53—44no—
310C4×C53C2×#12118yes—
311Dic53—28no—
312Dic53—24no—
313Dic53—24no—
314Dic53—42no—
315Dic53—44no—
316C22×C22⋊Q8—531no—
317C22×C22⋊Q8—531no—
318C22×C4×Q8—531no—
319C22×C4×D4—531no—
320#2536—5316no—
321C2×#398—532no—
322C2×#394—532no—
323C2×#388—532no—
324C2×#388—532no—
325C2×#382—531no—
326C2×#380—531no—
327C2×#380D5311yescharacteristic, complemented
328C2×C53—44no—
329C2×C53#253618yescomplemented
330C2×C53C22×#12111yescharacteristic
331C2×C53C2×#52511yescharacteristic
332C2×C53C2×#53611yescharacteristic
333C2×C53C2×#54811yescharacteristic
334C2×C53C2×#54811yescharacteristic
335C2×C53C2×#55211yescharacteristic
336C2×C53C2×#55211yescharacteristic
337D53—44no—
338C23×Q8—531no—
339C23×D4—531no—
340C24×C4—531no—
341C24×C4C2×D5311yescharacteristic
342C2×C22⋊Q8—538no—
343C2×C22⋊Q8—538no—
344C2×C22⋊Q8—1068no—
345C2×C22⋊Q8—1068no—
346C2×C4×Q8—538no—
347C2×C4×Q8—1068no—
348C2×C4×D4—538no—
349C2×C4×D4—1068no—
350C22×C4⋊C4—532no—
351C22×C4⋊C4—531no—
352C22×C4⋊C4—532no—
353C22×C4⋊C4—532no—
354C22×C4⋊C4—531no—
355C22×C4⋊C4C2×D5311yescharacteristic
356C22×C4⋊C4C2×D5311yescharacteristic
357C22×C22⋊C4—532no—
358C22×C22⋊C4—531no—
359C22×C22⋊C4—532no—
360C22×C22⋊C4—532no—
361C22×C22⋊C4—531no—
362C22×C22⋊C4C2×D5312yes—
363C22×C42—531no—
364C22×C42—532no—
365#398—5316no—
366#394—5316no—
367#388—5316no—
368#388—5316no—
369#382—538no—
370#380—538no—
371#380C2×D5318yescomplemented
372C2×#95—532no—
373C2×#95—532no—
374C2×#95—532no—
375C2×#95C2×D5312yes—
376C53C2×#253611yescharacteristic, complemented, Sylow
377C25—531no—
378C25C22×D5311yescharacteristic
379C22×Q8—538no—
380C22×Q8—534no—
381C22×Q8—1064no—
382C22×Q8—1064no—
383C22×D4—534no—
384C22×D4—1064no—
385C22×D4—1064no—
386C22×D4—1064no—
387C23×C4—24no—
388C23×C4—24no—
389C23×C4—24no—
390C23×C4—532no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4—532no—
394C23×C4—531no—
395C23×C4—534no—
396C23×C4—1064no—
397C23×C4—1064no—
398C23×C4—1064no—
399C23×C4C22×D5314yes—
400C23×C4C22×D5312yes—
401C23×C4C22×D5312yes—
402C23×C4C22×D5311yescharacteristic
403C23×C4C22×D5311yescharacteristic
404C22⋊Q8—10632no—
405C22⋊Q8—10632no—
406C4×Q8—10632no—
407C4×D4—10632no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4—24no—
410C2×C4⋊C4—534no—
411C2×C4⋊C4—538no—
412C2×C4⋊C4—538no—
413C2×C4⋊C4—534no—
414C2×C4⋊C4—538no—
415C2×C4⋊C4—1064no—
416C2×C4⋊C4—1068no—
417C2×C4⋊C4—1068no—
418C2×C4⋊C4—1068no—
419C2×C4⋊C4—1064no—
420C2×C4⋊C4C22×D5314yes—
421C2×C4⋊C4C22×D5314yes—
422C2×C22⋊C4—28no—
423C2×C22⋊C4—534no—
424C2×C22⋊C4—538no—
425C2×C22⋊C4—534no—
426C2×C22⋊C4—538no—
427C2×C22⋊C4—538no—
428C2×C22⋊C4—1064no—
429C2×C22⋊C4—1068no—
430C2×C22⋊C4—1068no—
431C2×C22⋊C4—1064no—
432C2×C22⋊C4—1068no—
433C2×C22⋊C4C22×D5318yes—
434C2×C42—538no—
435C2×C42—534no—
436C2×C42—538no—
437C2×C42—1068no—
438#95—538no—
439#95—538no—
440#95—538no—
441#95C22×D5318yes—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—534no—
449C24—532no—
450C24—1062no—
451C24—1062no—
452C24—1062no—
453C24—1062no—
454C24—1062no—
455C24—1062no—
456C24#1171994714yes—
457C24C23×D5312yes—
458C24C23×D5311yescharacteristic
459C2×Q8—5316no—
460C2×Q8—10632no—
461C2×Q8—1068no—
462C2×Q8—1068no—
463C2×D4—1068no—
464C2×D4—1068no—
465C2×D4—10632no—
466C2×D4—1068no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—28no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—416no—
480C22×C4—534no—
481C22×C4—534no—
482C22×C4—538no—
483C22×C4—532no—
484C22×C4—532no—
485C22×C4—534no—
486C22×C4—534no—
487C22×C4—532no—
488C22×C4—1064no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1064no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1064no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1064no—
498C22×C4—1062no—
499C22×C4—1064no—
500C22×C4—1068no—
501C22×C4—1068no—
502C22×C4—1068no—
503C22×C4—21216no—
504C22×C4#1171994514yes—
505C22×C4D4×D5314yes—
506C22×C4D4×D5314yes—
507C22×C4C23×D5314yes—
508C22×C4C23×D5314yes—
509C22×C4C23×D5312yes—
510C22×C4C23×D5312yes—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—1068no—
514C4⋊C4—10616no—
515C4⋊C4—10616no—
516C4⋊C4—10616no—
517C4⋊C4—1068no—
518C22⋊C4—216no—
519C22⋊C4—1068no—
520C22⋊C4—10616no—
521C22⋊C4—10616no—
522C22⋊C4—1068no—
523C22⋊C4—10616no—
524C42—5316no—
525C42—10616no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—534no—
540C23—1068no—
541C23—1068no—
542C23—1068no—
543C23—1062no—
544C23—1062no—
545C23—1062no—
546C23—1062no—
547C23—1062no—
548C23—1062no—
549C23—2124no—
550C23—2124no—
551C23—2124no—
552C23—2124no—
553C23C2×D4×D5312yes—
554C23C2×D4×D5312yes—
555C23#41×D5312yes—
556C23C2×#1171994714yes—
557C23C2×#1171994712yes—
558C23C2×#1171994512yes—
559C23#1313326412yes—
560C23#1313326512yes—
561C23C24×D5311yescharacteristic
562Q8—10632no—
563D4—10632no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—416no—
577C2×C4—534no—
578C2×C4—534no—
579C2×C4—538no—
580C2×C4—10616no—
581C2×C4—1068no—
582C2×C4—1064no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1064no—
586C2×C4—1068no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1064no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1064no—
593C2×C4—1062no—
594C2×C4—1064no—
595C2×C4—2128no—
596C2×C4—2124no—
597C2×C4—21216no—
598C2×C4#13133190116yes—
599C2×C4C2×D4×D5314yes—
600C2×C4C2×D4×D5314yes—
601C2×C4C2×#1171994514yes—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—42no—
606C22—416no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—1064no—
611C22—1064no—
612C22—1064no—
613C22—2122no—
614C22—21216no—
615C22—2122no—
616C22—2122no—
617C22—2122no—
618C22C2×#1313319014yes—
619C22#1681113314yes—
620C22#1681119614yes—
621C22#1681127914yes—
622C22#1681128314yes—
623C22#1681130814yes—
624C22#1681131214yes—
625C22C22×D4×D5311yescharacteristic
626C22C22×D4×D5311yescharacteristic
627C22C2×#41×D5311yescharacteristic
628C22C22×#1171994711yescharacteristic
629C22C22×#1171994511yescharacteristic
630C22C2×#1313326411yescharacteristic
631C22C2×#1313326511yescharacteristic
632C4—44no—
633C4—42no—
634C4—44no—
635C4—1064no—
636C4—1068no—
637C4—1064no—
638C4—2124no—
639C4—2122no—
640C4C2×#1313319018yes—
641C2—44no—
642C2—2124no—
643C2#2027962618yesdirect product factor
644C2C22×#1313319011yescharacteristic
645C2C2×#1681113311yescharacteristic
646C2C2×#1681119611yescharacteristic
647C2C2×#1681127911yescharacteristic
648C2C2×#1681128311yescharacteristic
649C2C2×#1681130811yescharacteristic
650C2C2×#1681131211yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#380C2
socleC24×C53C24
prime radicalC23×C22⋊D53C22
hypercenterC2×#380D53
hypocenterC53C2×#2536
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#13133197C2semidirect
2. C22×#13133196C2semidirect
3. C22×#13133182C2semidirect
5. #20279626C2direct (N × H)
6. C2×#16810557C2semidirect
8. C2×#16810516C2semidirect
9. C2×#16810512C2semidirect
10. C2×#16810488C2semidirect
11. C2×#16810480C2semidirect
12. C2×C53×#380C2semidirect
17. C2×#13133197C22semidirect
19. C2×#13133196C22semidirect
21. C2×#13133182C22semidirect
36. #16810557C22semidirect
38. #16810516C22semidirect
39. #16810512C22semidirect
40. #16810488C22semidirect
41. #16810480C22semidirect
49. C53×#380C22semidirect
54. C22×Dic106D4semidirect
184. C2×Dic106C2×D4semidirect
327. C2×#380D53semidirect
329. C2×C53#2536semidirect
371. #380C2×D53semidirect
376. C53C2×#2536semidirect
643. C2#20279626direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331972
C22×#131331962
C22×#131331822
C22×C4×C22⋊D532
#202796262
C2×#168105572
C2×#168105382
C2×#168105162
C2×#168105122
C2×#168104882
C2×#168104802
C2×C53×#3802
C2×#253653