Finite Group Databasebeta

← C2×#20279181 #27553148

Subgroups — C2×#20279181 #27553148

Order 13568.

4222 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133202C212yesmaximal, complemented
2C22×#13133199C212yesmaximal, complemented
3C22×#13133190C211yesmaximal, characteristic
4C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
5#20279181C2116yesmaximal, direct product factor
6C2×#16810560C211yesmaximal, characteristic, complemented
7C2×#16810559C211yesmaximal, characteristic, complemented
8C2×#16810558C211yesmaximal, characteristic, complemented
9C2×#16810523C212yesmaximal, complemented
10C2×#16810492C212yesmaximal, complemented
11C2×#16810470C211yesmaximal, characteristic, complemented
12C2×C53×#399C211yesmaximal, characteristic, complemented
13C23×D212C2211yescharacteristic
14C23×C22⋊D53C2211yescharacteristic
15C23×C22⋊D53C2211yescharacteristic
16C23×C4×D53C2212yes—
17C23×C4×D53C2211yescharacteristic
18C2×#13133202—216no—
19C2×#13133202C22116yescomplemented
20C2×#13133199—216no—
21C2×#13133199C22116yescomplemented
22C2×#13133190—28no—
23C2×#13133190C2218yes—
24C22×C4⋊Dic53C2212yes—
25C22×C4⋊Dic53C2211yescharacteristic
26C22×#11719919C2211yescharacteristic
27C22×#11719919C2212yes—
28C22×#11719919C2212yes—
29C22×#11719919C2212yes—
30C2×C4⋊C4×D53—28no—
31C2×C4⋊C4×D53C2218yescomplemented
32C22×C22⋊Dic53C2212yes—
33C22×#11719921C2212yes—
34C22×#11719921C2212yes—
35C22×#11719921C2211yescharacteristic
36C22×#11719921C2212yes—
37C22×C4×Dic53C2212yes—
38#16810560C2218yescomplemented
39#16810559C2218yescomplemented
40#16810558C2218yescomplemented
41#16810523C22116yescomplemented
42#16810492C22116yescomplemented
43#16810470C2218yescomplemented
44C2×#13133124C2211yescharacteristic
45C22×C4⋊C4×C53C2212yes—
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×C22⋊C4×C53C2212yes—
48C22×C22⋊C4×C53C2211yescharacteristic
49C53×#399C2218yescomplemented
50C2×#95×C53C2211yescharacteristic
51C24×D53C2311yescharacteristic
52C24×D53C2311yescharacteristic
53C22×D212—24no—
54C22×D212—28no—
55C22×D212C2314yes—
56C22×C22⋊D53—24no—
57C22×C22⋊D53—28no—
58C22×C22⋊D53—24no—
59C22×C22⋊D53—28no—
60C22×C22⋊D53C2314yes—
61C22×C22⋊D53C2314yes—
62C23×Dic53C2312yes—
63C23×Dic53C2311yescharacteristic
64C23×Dic53C2312yes—
65C23×Dic53C2311yescharacteristic
66C22×C4×D53—28no—
67C22×C4×D53—28no—
68C22×C4×D53—28no—
69C22×C4×D53—28no—
70C22×C4×D53D418yes—
71C22×C4×D53C2318yes—
72C22×C4×D53C2314yes—
73#13133202—264no—
74#13133199—264no—
75#13133190—232no—
76C2×C4⋊Dic53—24no—
77C2×C4⋊Dic53—28no—
78C2×C4⋊Dic53C2314yes—
79C2×C4⋊Dic53C2318yes—
80C2×#11719919—28no—
81C2×#11719919—24no—
82C2×#11719919—28no—
83C2×#11719919—28no—
84C2×#11719919C2318yes—
85C2×#11719919C2318yes—
86C2×#11719919C2318yes—
87C2×#11719919C2314yes—
88C4⋊C4×D53—232no—
89C2×C22⋊Dic53—28no—
90C2×C22⋊Dic53C2318yes—
91C2×#11719921—24no—
92C2×#11719921—28no—
93C2×#11719921—28no—
94C2×#11719921D4116yescomplemented
95C2×#11719921C2318yes—
96C2×#11719921C2318yes—
97C2×#11719921C2314yes—
98C2×#11719921C2318yes—
99C2×C4×Dic53—28no—
100C2×C4×Dic53C2318yes—
101#13133124C2314yes—
102C25×C53C2311yescharacteristic
103C23×C4×C53C2312yes—
104C23×C4×C53C2311yescharacteristic
105C23×C4×C53C2312yes—
106C23×C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53—24no—
108C2×C4⋊C4×C53—28no—
109C2×C4⋊C4×C53C2314yes—
110C2×C4⋊C4×C53C2318yes—
111C2×C22⋊C4×C53—24no—
112C2×C22⋊C4×C53—28no—
113C2×C22⋊C4×C53C2318yes—
114C2×C22⋊C4×C53C2314yes—
115#95×C53C2314yes—
116C24×C53—24no—
117C24×C53—22no—
118C24×C53—24no—
119C24×C53—22no—
120C24×C53—22no—
121C24×C53C2412yes—
122C24×C53C2411yescharacteristic
123C23×D53—24no—
124C23×D53—22no—
125C23×D53—22no—
126C23×D53—24no—
127C23×D53—22no—
128C23×D53—24no—
129C23×D53—22no—
130C23×D53—22no—
131C23×D53C2×D418yes—
132C23×D53C2×D414yes—
133C23×D53C2412yes—
134C23×D53C2412yes—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—24no—
138C22×C4×C53—24no—
139C22×C4×C53—22no—
140C22×C4×C53—24no—
141C22×C4×C53—24no—
142C22×C4×C53—24no—
143C22×C4×C53C2×D418yes—
144C22×C4×C53C2×D414yes—
145C22×C4×C53C2414yes—
146C22×C4×C53C2412yes—
147C22×C4×C53C2412yes—
148C22×C4×C53C2414yes—
149C2×C22⋊D53—28no—
150C2×C22⋊D53—216no—
151C2×C22⋊D53—28no—
152C2×C22⋊D53—216no—
153C2×C22⋊D53—416no—
154C2×C22⋊D53—416no—
155C22×Dic53—24no—
156C22×Dic53—24no—
157C22×Dic53—24no—
158C22×Dic53—22no—
159C22×Dic53—24no—
160C22×Dic53—24no—
161C22×Dic53—24no—
162C22×Dic53C2×D418yes—
163C22×Dic53C2×D414yes—
164C22×Dic53#4118yes—
165C22×Dic53C2414yes—
166C22×Dic53C2412yes—
167C22×Dic53C2414yes—
168C22×Dic53C2412yes—
169C2×D212—28no—
170C2×D212—216no—
171C2×D212—416no—
172C2×C4×D53—232no—
173C2×C4×D53—216no—
174C2×C4×D53—216no—
175C2×C4×D53—216no—
176C2×C4×D53—216no—
177C2×C4×D53—432no—
178C2×C4×D53C2×D4116yes—
179C4⋊C4×C53—28no—
180C4⋊C4×C53—216no—
181C22⋊C4×C53—28no—
182C22⋊C4×C53—216no—
183C22⋊Dic53—216no—
184#11719921—28no—
185#11719921—216no—
186#11719921—216no—
187#11719921C2×D4132yescomplemented
188C4⋊Dic53—28no—
189C4⋊Dic53—216no—
190#11719919—216no—
191#11719919—28no—
192#11719919—216no—
193#11719919—216no—
194C4×Dic53—216no—
195C23×C53—22no—
196C23×C53—24no—
197C23×C53—24no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—44no—
201C23×C53—44no—
202C23×C53—44no—
203C23×C53—48no—
204C23×C53—48no—
205C23×C53C22×D412yes—
206C23×C53C22×D414yes—
207C23×C53C2×#4114yes—
208C23×C532−1+412yes—
209C23×C532−1+412yes—
210C23×C53C2511yescharacteristic
211C22×D53—28no—
212C22×D53—28no—
213C22×D53—216no—
214C22×D53—216no—
215C22×D53—24no—
216C22×D53—22no—
217C22×D53—22no—
218C22×D53—22no—
219C22×D53—24no—
220C22×D53—24no—
221C22×D53—22no—
222C22×D53—22no—
223C22×D53—44no—
224C22×D53—44no—
225C22×D53—44no—
226C22×D53—48no—
227C22×D53—48no—
228C22×D53#127116yes—
229C22×D53C22×D414yes—
230C22×D53C22×D418yes—
231C2×C4×C53—28no—
232C2×C4×C53—216no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—22no—
240C2×C4×C53—24no—
241C2×C4×C53—48no—
242C2×C4×C53—44no—
243C2×C4×C53C22×D414yes—
244C2×C4×C53C22×D418yes—
245C22⋊D53—416no—
246C22⋊D53—416no—
247C2×Dic53—216no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—22no—
255C2×Dic53—48no—
256C2×Dic53—44no—
257C2×Dic53#123116yescomplemented
258C2×Dic53C22×D418yes—
259C2×Dic53C22×D414yes—
260C2×Dic53C2×#4118yes—
261D212—416no—
262C4×D53—232no—
263C4×D53—432no—
264C2×#2501—531nomaximal, Sylow
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—44no—
269C22×C53—44no—
270C22×C53—88no—
271C22×C53C2×#12314yes—
272C22×C53C2×#12714yes—
273C22×C53#53614yes—
274C22×C53#54718yes—
275C22×C53#54718yes—
276C22×C53C23×D411yescharacteristic
277C22×C53C23×D412yes—
278C22×C53C22×#4112yes—
279C22×C53C2×2−1+411yescharacteristic
280C22×C53C2×2−1+411yescharacteristic
281C2×D53—24no—
282C2×D53—24no—
283C2×D53—28no—
284C2×D53—28no—
285C2×D53—232no—
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—44no—
290C2×D53—44no—
291C2×D53—88no—
292C2×D53C2×#12718yes—
293C4×C53—24no—
294C4×C53—28no—
295C4×C53—44no—
296C4×C53—42no—
297Dic53—28no—
298Dic53—44no—
299Dic53—42no—
300Dic53C2×#12318yescomplemented
301C22×#123—532no—
302C22×#123—532no—
303C22×#121—531no—
304C23×C4⋊C4—531no—
305#2501—5316no—
306C2×#399—531no—
307C2×#399—531no—
308C2×#399—531no—
309C2×#399D5311yescharacteristic, complemented
310C2×#390—532no—
311C2×#384—532no—
312C2×C4×C22⋊C4—531no—
313C2×C53—82no—
314C2×C53#250118yescomplemented
315C2×C53C22×#12311yescharacteristic
316C2×C53C22×#12711yescharacteristic
317C2×C53C2×#53611yescharacteristic
318C2×C53C2×#54712yes—
319C2×C53C2×#54712yes—
320D53—28no—
321D53—82no—
322C23×D4—531no—
323C23×D4—531no—
324C23×D4—531no—
325C24×C4—531no—
326C24×C4—532no—
327C2×#123—5316no—
328C2×#123—5316no—
329C2×#123—10616no—
330C2×#123—10616no—
331C2×#121—538no—
332C2×#121—1068no—
333C22×C4⋊C4—532no—
334C22×C4⋊C4—532no—
335C22×C4⋊C4—532no—
336C22×C4⋊C4—531no—
337C22×C4⋊C4—531no—
338C22×C4⋊C4—532no—
339C22×C4⋊C4—538no—
340C22×C4⋊C4—1068no—
341C22×C4⋊C4C2×D5311yescharacteristic
342C22×C4⋊C4C2×D5312yes—
343C22×C22⋊C4—532no—
344C22×C22⋊C4—532no—
345C22×C22⋊C4—531no—
346C22×C22⋊C4—532no—
347C22×C22⋊C4—532no—
348C22×C22⋊C4C2×D5311yescharacteristic
349C22×C22⋊C4C2×D5312yes—
350C22×C42—532no—
351#399—538no—
352#399—538no—
353#399—538no—
354#399C2×D5318yescomplemented
355#390—5316no—
356#384—5316no—
357C4×C22⋊C4—538no—
358C2×#95—531no—
359C2×#95C2×D5311yescharacteristic
360C53C2×#250111yescharacteristic, complemented, Sylow
361C25—531no—
362C25—531no—
363C25C22×D5311yescharacteristic
364C22×D4—534no—
365C22×D4—534no—
366C22×D4—534no—
367C22×D4—1064no—
368C22×D4—1068no—
369C22×D4—1064no—
370C22×D4—1068no—
371C22×D4—1064no—
372C22×D4—1068no—
373C23×C4—531no—
374C23×C4—532no—
375C23×C4—538no—
376C23×C4—534no—
377C23×C4—532no—
378C23×C4—531no—
379C23×C4—538no—
380C23×C4—1068no—
381C23×C4—1068no—
382C23×C4—1068no—
383C23×C4—1068no—
384C23×C4C22×D5312yes—
385C23×C4C22×D5311yescharacteristic
386C23×C4C22×D5312yes—
387C23×C4C22×D5311yescharacteristic
388#123—10664no—
389#123—10664no—
390#121—10632no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—28no—
393C2×C4⋊C4—538no—
394C2×C4⋊C4—534no—
395C2×C4⋊C4—534no—
396C2×C4⋊C4—538no—
397C2×C4⋊C4—538no—
398C2×C4⋊C4—538no—
399C2×C4⋊C4—10632no—
400C2×C4⋊C4—1068no—
401C2×C4⋊C4—1064no—
402C2×C4⋊C4—1068no—
403C2×C4⋊C4—1068no—
404C2×C4⋊C4—1064no—
405C2×C4⋊C4—1068no—
406C2×C4⋊C4C22×D5314yes—
407C2×C4⋊C4C22×D5318yes—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—28no—
410C2×C22⋊C4—5316no—
411C2×C22⋊C4—538no—
412C2×C22⋊C4—538no—
413C2×C22⋊C4—534no—
414C2×C22⋊C4—538no—
415C2×C22⋊C4—538no—
416C2×C22⋊C4—1068no—
417C2×C22⋊C4—1068no—
418C2×C22⋊C4—1064no—
419C2×C22⋊C4—1068no—
420C2×C22⋊C4C22×D5314yes—
421C2×C22⋊C4C22×D5318yes—
422C2×C42—538no—
423C2×C42—1068no—
424#95—534no—
425#95C22×D5314yes—
426C24—24no—
427C24—22no—
428C24—24no—
429C24—22no—
430C24—22no—
431C24—538no—
432C24—534no—
433C24—532no—
434C24—532no—
435C24—1064no—
436C24—1062no—
437C24—1062no—
438C24—1064no—
439C24—1062no—
440C24—1064no—
441C24—1062no—
442C24—1062no—
443C24C23×D5312yes—
444C24C23×D5311yescharacteristic
445C2×D4—1068no—
446C2×D4—10616no—
447C2×D4—1068no—
448C2×D4—10616no—
449C2×D4—1068no—
450C2×D4—10616no—
451C2×D4—21216no—
452C2×D4—21216no—
453C2×D4—21216no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—534no—
463C22×C4—538no—
464C22×C4—532no—
465C22×C4—538no—
466C22×C4—534no—
467C22×C4—5316no—
468C22×C4—534no—
469C22×C4—532no—
470C22×C4—10632no—
471C22×C4—10616no—
472C22×C4—10616no—
473C22×C4—10616no—
474C22×C4—10616no—
475C22×C4—1064no—
476C22×C4—1064no—
477C22×C4—1064no—
478C22×C4—1064no—
479C22×C4—1062no—
480C22×C4—1064no—
481C22×C4—1064no—
482C22×C4—21232no—
483C22×C4D4×D5318yes—
484C22×C4D4×D5314yes—
485C22×C4C23×D5314yes—
486C22×C4C23×D5312yes—
487C22×C4C23×D5312yes—
488C22×C4C23×D5314yes—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—10616no—
492C4⋊C4—1068no—
493C4⋊C4—10616no—
494C4⋊C4—10616no—
495C4⋊C4—1068no—
496C4⋊C4—10616no—
497C22⋊C4—28no—
498C22⋊C4—216no—
499C22⋊C4—5332no—
500C22⋊C4—10616no—
501C22⋊C4—10616no—
502C22⋊C4—1068no—
503C22⋊C4—10616no—
504C42—10616no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—48no—
511C23—48no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—5316no—
516C23—538no—
517C23—534no—
518C23—10616no—
519C23—10616no—
520C23—1068no—
521C23—1064no—
522C23—1068no—
523C23—1062no—
524C23—1062no—
525C23—1064no—
526C23—1062no—
527C23—1064no—
528C23—1062no—
529C23—1062no—
530C23—2128no—
531C23—2128no—
532C23—2124no—
533C23—2124no—
534C23—2124no—
535C23C2×D4×D5314yes—
536C23C2×D4×D5312yes—
537C23#41×D5314yes—
538C23#1313326412yes—
539C23#1313326512yes—
540C23C24×D5311yescharacteristic
541D4—21216no—
542D4—21216no—
543D4—21216no—
544C2×C4—216no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—5316no—
557C2×C4—534no—
558C2×C4—538no—
559C2×C4—538no—
560C2×C4—10632no—
561C2×C4—10616no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—1064no—
565C2×C4—1064no—
566C2×C4—1062no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—21232no—
570C2×C4—2128no—
571C2×C4—2124no—
572C2×C4C2×D4×D5318yes—
573C2×C4C2×D4×D5314yes—
574C22—42no—
575C22—44no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—88no—
580C22—538no—
581C22—1064no—
582C22—10632no—
583C22—1068no—
584C22—1068no—
585C22—1064no—
586C22—4248no—
587C22—2122no—
588C22—2124no—
589C22—2124no—
590C22—2122no—
591C22—2122no—
592C22#123×D5314yes—
593C22#127×D5314yes—
594C22#1681119614yes—
595C22#1681127118yes—
596C22#1681127518yes—
597C22C22×D4×D5311yescharacteristic
598C22C22×D4×D5312yes—
599C22C2×#41×D5312yes—
600C22C2×#1313326411yescharacteristic
601C22C2×#1313326511yescharacteristic
602C4—28no—
603C4—24no—
604C4—44no—
605C4—42no—
606C4—538no—
607C4—1068no—
608C4—2124no—
609C4—2122no—
610C2—82no—
611C2—1068no—
612C2—4242no—
613C2#2027918118yesdirect product factor
614C2C2×#123×D5311yescharacteristic
615C2C2×#127×D5311yescharacteristic
616C2C2×#1681119611yescharacteristic
617C2C2×#1681127112yes—
618C2C2×#1681127512yes—
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#399C2
socleC24×C53C24
prime radicalC22×#13133190C2
hypercenterC2×#399D53
hypocenterC53C2×#2501
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#13133202C2semidirect
2. C22×#13133199C2semidirect
4. C22×C4⋊C4×D53C2semidirect
5. #20279181C2direct (N × H)
6. C2×#16810560C2semidirect
7. C2×#16810559C2semidirect
8. C2×#16810558C2semidirect
9. C2×#16810523C2semidirect
10. C2×#16810492C2semidirect
11. C2×#16810470C2semidirect
12. C2×C53×#399C2semidirect
19. C2×#13133202C22semidirect
21. C2×#13133199C22semidirect
31. C2×C4⋊C4×D53C22semidirect
38. #16810560C22semidirect
39. #16810559C22semidirect
40. #16810558C22semidirect
41. #16810523C22semidirect
42. #16810492C22semidirect
43. #16810470C22semidirect
49. C53×#399C22semidirect
94. C2×#11719921D4semidirect
187. #11719921C2×D4semidirect
257. C2×Dic53#123semidirect
300. Dic53C2×#123semidirect
309. C2×#399D53semidirect
314. C2×C53#2501semidirect
354. #399C2×D53semidirect
360. C53C2×#2501semidirect
613. C2#20279181direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332022
C22×#131331992
C22×#131331902
C22×C4⋊C4×D532
#202791812
C2×#168105602
C2×#168105592
C2×#168105582
C2×#168105232
C2×#168104922
C2×#168104702
C2×C53×#3992
C2×#250153