Finite Group Databasebeta

← C2×#20275124 #27385706

Subgroups — C2×#20275124 #27385706

Order 13568.

3566 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811098C211yesmaximal, characteristic, complemented
2C22×#13133132C211yesmaximal, characteristic, complemented
3C2×#16810479C212yesmaximal, complemented
4#20275124C218yesmaximal, direct product factor
5C2×#16810389C212yesmaximal, complemented
6C22×#120×C53C211yesmaximal, characteristic
7#16811098—24no—
8#16811098—22no—
9#16811098C2214yescomplemented
10C2×D4×Dic53—24no—
11C2×#13133176—21no—
12C22×C22⋊Dic53—24no—
13C22×C22⋊Dic53—21no—
14C22×C22⋊Dic53C2212yes—
15C22×C22⋊Dic53C2212yes—
16C2×#13133132—24no—
17C2×#13133132C2214yescomplemented
18#16810479—24no—
19#16810479C2218yescomplemented
20#16810389C2218yescomplemented
21C26×C53C2211yescharacteristic
22C23×D4×C53—21no—
23C23×D4×C53C2211yescharacteristic
24C2×#120×C53—24no—
25C2×#120×C53—22no—
26C2×#120×C53C418yescomplemented
27C2×#120×C53C2214yes—
28C22×C22⋊C4×C53—21no—
29C22×C22⋊C4×C53C2211yescharacteristic
30C23×Dic53—22no—
31C23×Dic53—22no—
32C23×Dic53D414yescomplemented
33D4×Dic53—216no—
34D4×Dic53—216no—
35#13133176—24no—
36#13133176—24no—
37C2×C4⋊Dic53—22no—
38C2×C22⋊Dic53—22no—
39C2×C22⋊Dic53—24no—
40C2×C22⋊Dic53—22no—
41C2×C22⋊Dic53—24no—
42C2×C22⋊Dic53—22no—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53—22no—
45C2×C22⋊Dic53—22no—
46C2×C22⋊Dic53—216no—
47C2×C22⋊Dic53—24no—
48C2×C22⋊Dic53—24no—
49C2×C22⋊Dic53—416no—
50C2×C22⋊Dic53D414yescomplemented
51C2×C22⋊Dic53C2314yes—
52C2×C22⋊Dic53C2314yes—
53C2×C4×Dic53—22no—
54#13133132—28no—
55C25×C53—21no—
56C25×C53—22no—
57C25×C53—22no—
58C25×C53—21no—
59C25×C53—22no—
60C25×C53—24no—
61C25×C53—48no—
62C25×C53C2×C414yescomplemented
63C25×C53D411yescharacteristic
64C25×C53C2312yes—
65C25×C53C2311yescharacteristic
66C22×D4×C53—22no—
67C22×D4×C53—22no—
68C22×D4×C53—24no—
69C22×D4×C53—28no—
70C22×D4×C53—24no—
71C22×D4×C53—48no—
72C22×D4×C53C2×C414yes—
73C22×D4×C53D414yes—
74C22×D4×C53D412yes—
75C22×D4×C53C2312yes—
76C23×C4×C53—21no—
77C23×C4×C53D411yescharacteristic
78#120×C53—216no—
79#120×C53—28no—
80#120×C53C2×C4116yescomplemented
81C2×C22⋊C4×C53—22no—
82C2×C22⋊C4×C53—21no—
83C2×C22⋊C4×C53—28no—
84C2×C22⋊C4×C53—24no—
85C2×C22⋊C4×C53C2×C414yescomplemented
86C2×C22⋊C4×C53C2312yes—
87C24×C53—28no—
88C24×C53—216no—
89C24×C53—24no—
90C24×C53—28no—
91C24×C53—21no—
92C24×C53—22no—
93C24×C53—24no—
94C24×C53—21no—
95C24×C53—22no—
96C24×C53—24no—
97C24×C53—28no—
98C24×C53—22no—
99C24×C53—24no—
100C24×C53—416no—
101C24×C53—44no—
102C24×C53—48no—
103C24×C53—48no—
104C24×C53—44no—
105C24×C53—416no—
106C24×C53—48no—
107C24×C53—432no—
108C24×C53—416no—
109C24×C53—44no—
110C24×C53—42no—
111C24×C53—41no—
112C24×C53—41no—
113C24×C53—44no—
114C24×C53—44no—
115C24×C53—42no—
116C24×C53—42no—
117C24×C53C22⋊C414yescomplemented
118C24×C53C22×C412yes—
119C24×C53C22×C414yescomplemented
120C24×C53C2×D411yescharacteristic
121C24×C53C2×D412yes—
122C24×C53C2×D412yes—
123C24×C53C2×D411yescharacteristic
124C24×C53C2×D412yes—
125C24×C53C2×D412yes—
126C24×C53#4114yes—
127C24×C53C2411yescharacteristic
128C2×D4×C53—28no—
129C2×D4×C53—28no—
130C2×D4×C53—24no—
131C2×D4×C53—28no—
132C2×D4×C53—28no—
133C2×D4×C53—24no—
134C2×D4×C53—24no—
135C2×D4×C53—24no—
136C2×D4×C53—216no—
137C2×D4×C53—432no—
138C2×D4×C53—416no—
139C2×D4×C53C22⋊C418yescomplemented
140C2×D4×C53C22×C414yes—
141C2×D4×C53C2×D414yes—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—22no—
145C22×C4×C53—24no—
146C22×C4×C53—22no—
147C22×C4×C53—44no—
148C22×C4×C53C22⋊C414yescomplemented
149C22×C4×C53C2×D412yes—
150C22×Dic53—24no—
151C22×Dic53—28no—
152C22×Dic53—24no—
153C22×Dic53—24no—
154C22×Dic53—48no—
155C22×Dic53—48no—
156C22×Dic53—44no—
157C22×Dic53—84no—
158C22×Dic53C2×D418yescomplemented
159C22⋊C4×C53—24no—
160C22⋊C4×C53—22no—
161C22⋊C4×C53—216no—
162C22⋊C4×C53C22×C414yescomplemented
163C22⋊Dic53—28no—
164C22⋊Dic53—24no—
165C22⋊Dic53—24no—
166C22⋊Dic53—24no—
167C22⋊Dic53—24no—
168C22⋊Dic53—24no—
169C22⋊Dic53—24no—
170C22⋊Dic53—432no—
171C22⋊Dic53—44no—
172C4⋊Dic53—28no—
173C4×Dic53—28no—
174C23×C53—28no—
175C23×C53—24no—
176C23×C53—216no—
177C23×C53—28no—
178C23×C53—232no—
179C23×C53—216no—
180C23×C53—24no—
181C23×C53—28no—
182C23×C53—22no—
183C23×C53—24no—
184C23×C53—22no—
185C23×C53—24no—
186C23×C53—44no—
187C23×C53—48no—
188C23×C53—44no—
189C23×C53—44no—
190C23×C53—42no—
191C23×C53—48no—
192C23×C53—42no—
193C23×C53—432no—
194C23×C53—48no—
195C23×C53—416no—
196C23×C53—416no—
197C23×C53—48no—
198C23×C53—416no—
199C23×C53—44no—
200C23×C53—48no—
201C23×C53—416no—
202C23×C53—48no—
203C23×C53—432no—
204C23×C53—416no—
205C23×C53—48no—
206C23×C53—44no—
207C23×C53—432no—
208C23×C53—416no—
209C23×C53—42no—
210C23×C53—44no—
211C23×C53—42no—
212C23×C53—44no—
213C23×C53—48no—
214C23×C53—44no—
215C23×C53—44no—
216C23×C53—48no—
217C23×C53—84no—
218C23×C53—82no—
219C23×C53C2×C22⋊C412yes—
220C23×C53C2×C22⋊C412yes—
221C23×C53C2×C22⋊C414yescomplemented
222C23×C53C4×D414yes—
223C23×C53C4×D414yes—
224C23×C53#12012yes—
225C23×C53#12011yescharacteristic
226C23×C53#12011yescharacteristic
227C23×C53#12114yes—
228C23×C53#12114yes—
229C23×C53#12312yes—
230C23×C53#12312yes—
231C23×C53C23×C411yescharacteristic
232C23×C53C22×D412yes—
233C23×C53C22×D411yescharacteristic
234C23×C53C22×D411yescharacteristic
235C23×C53C2×#4112yes—
236D4×C53—28no—
237D4×C53—28no—
238D4×C53—216no—
239D4×C53—432no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—24no—
245C2×C4×C53—48no—
246C2×C4×C53—44no—
247C2×C4×C53C2×C22⋊C414yescomplemented
248C2×Dic53—28no—
249C2×Dic53—48no—
250C2×Dic53—48no—
251C2×Dic53—48no—
252C2×Dic53—44no—
253C2×Dic53—84no—
254C2×Dic53—88no—
255C2×Dic53—88no—
256C2×#1889—531nomaximal, Sylow
257C22×C53—28no—
258C22×C53—24no—
259C22×C53—28no—
260C22×C53—24no—
261C22×C53—24no—
262C22×C53—24no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—28no—
266C22×C53—216no—
267C22×C53—24no—
268C22×C53—42no—
269C22×C53—44no—
270C22×C53—44no—
271C22×C53—42no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—42no—
275C22×C53—44no—
276C22×C53—416no—
277C22×C53—44no—
278C22×C53—48no—
279C22×C53—48no—
280C22×C53—416no—
281C22×C53—432no—
282C22×C53—416no—
283C22×C53—48no—
284C22×C53—44no—
285C22×C53—48no—
286C22×C53—44no—
287C22×C53—82no—
288C22×C53—84no—
289C22×C53#38612yes—
290C22×C53#38614yes—
291C22×C53#38612yes—
292C22×C53B(4,𝔽2)14yes—
293C22×C53C22×C22⋊C411yescharacteristic
294C22×C53C2×C4×D412yes—
295C22×C53C2×#12011yescharacteristic
296C22×C53C2×#12112yes—
297C22×C53C2×#12311yescharacteristic
298C4×C53—24no—
299C4×C53—44no—
300Dic53—48no—
301Dic53—84no—
302C22×#120D5311yescharacteristic
303C2×#518—531no—
304C22×#99—531no—
305C2×#379—532no—
306#1889—538no—
307C2×#342—532no—
308C2×C53—22no—
309C2×C53—21no—
310C2×C53—22no—
311C2×C53—21no—
312C2×C53—22no—
313C2×C53—24no—
314C2×C53—48no—
315C2×C53—44no—
316C2×C53#188914yescomplemented
317C2×C53C2×#38611yescharacteristic
318C2×C53C2×B(4,𝔽2)12yes—
319C26C2×D5311yescharacteristic
320C23×D4—21no—
321C23×D4C2×D5311yescharacteristic
322C2×#120—22no—
323C2×#120—24no—
324C2×#120Dic5318yescomplemented
325C2×#120C2×D5314yes—
326#518—534no—
327#518—1064no—
328#518—1062no—
329C2×C4×D4—1064no—
330C2×#117—1061no—
331C22×C22⋊C4—21no—
332C22×C22⋊C4—532no—
333C22×C22⋊C4—532no—
334C22×C22⋊C4—1061no—
335C22×C22⋊C4—1064no—
336C22×C22⋊C4C2×D5311yescharacteristic
337C2×#99—534no—
338C2×#99—1064no—
339#379—538no—
340#379—1064no—
341#342—538no—
342C53C2×#188911yescharacteristic, complemented, Sylow
343C25—22no—
344C25—21no—
345C25—22no—
346C25—24no—
347C25—22no—
348C25—21no—
349C25—48no—
350C25C2×Dic5314yescomplemented
351C25C22⋊D5311yescharacteristic
352C25C22×D5312yes—
353C25C22×D5311yescharacteristic
354C22×D4—22no—
355C22×D4—22no—
356C22×D4—24no—
357C22×D4—24no—
358C22×D4—28no—
359C22×D4—48no—
360C22×D4C2×Dic5314yes—
361C22×D4C22⋊D5312yes—
362C22×D4C22⋊D5314yes—
363C22×D4C22×D5312yes—
364C23×C4—21no—
365C23×C4—534no—
366C23×C4—1062no—
367C23×C4—1062no—
368C23×C4C22⋊D5311yescharacteristic
369#120—28no—
370#120—216no—
371#120C2×Dic53116yescomplemented
372C4×D4—10616no—
373C4×D4—10616no—
374#117—1064no—
375#117—1064no—
376C2×C4⋊C4—1062no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—28no—
381C2×C22⋊C4—534no—
382C2×C22⋊C4—534no—
383C2×C22⋊C4—534no—
384C2×C22⋊C4—1062no—
385C2×C22⋊C4—1064no—
386C2×C22⋊C4—1062no—
387C2×C22⋊C4—1062no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4—1062no—
390C2×C22⋊C4—1064no—
391C2×C22⋊C4—1064no—
392C2×C22⋊C4—1064no—
393C2×C22⋊C4—10616no—
394C2×C22⋊C4—1062no—
395C2×C22⋊C4—21216no—
396C2×C22⋊C4C2×Dic5314yescomplemented
397C2×C22⋊C4C22×D5312yes—
398C2×C42—1062no—
399#99—1068no—
400C24—216no—
401C24—28no—
402C24—24no—
403C24—28no—
404C24—21no—
405C24—24no—
406C24—22no—
407C24—21no—
408C24—22no—
409C24—24no—
410C24—28no—
411C24—22no—
412C24—24no—
413C24—416no—
414C24—48no—
415C24—44no—
416C24—44no—
417C24—48no—
418C24—416no—
419C24—416no—
420C24—48no—
421C24—432no—
422C24—44no—
423C24—41no—
424C24—42no—
425C24—41no—
426C24—44no—
427C24—44no—
428C24—42no—
429C24—42no—
430C24C22⋊Dic5314yescomplemented
431C24D4×D5312yes—
432C24D4×D5312yes—
433C24#1171994714yes—
434C24C22×Dic5312yes—
435C24C22×Dic5314yescomplemented
436C24C2×C22⋊D5311yescharacteristic
437C24C2×C22⋊D5312yes—
438C24C2×C22⋊D5311yescharacteristic
439C24C2×C22⋊D5312yes—
440C24C23×D5311yescharacteristic
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—28no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—216no—
450C2×D4—416no—
451C2×D4—432no—
452C2×D4C22⋊Dic5318yescomplemented
453C2×D4C22×Dic5314yes—
454C2×D4C2×C22⋊D5314yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—44no—
461C22×C4—538no—
462C22×C4—1068no—
463C22×C4—1064no—
464C22×C4—1064no—
465C22×C4—1064no—
466C22×C4—4244no—
467C22×C4—2128no—
468C22×C4—2124no—
469C22×C4—2128no—
470C22×C4C22⋊Dic5314yescomplemented
471C22×C4C2×C22⋊D5312yes—
472C4⋊C4—1068no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—216no—
476C22⋊C4—1068no—
477C22⋊C4—1064no—
478C22⋊C4—1064no—
479C22⋊C4—1064no—
480C22⋊C4—1064no—
481C22⋊C4—1064no—
482C22⋊C4—1064no—
483C22⋊C4—21232no—
484C22⋊C4—2124no—
485C22⋊C4C22×Dic5314yescomplemented
486C42—1068no—
487C23—24no—
488C23—28no—
489C23—216no—
490C23—216no—
491C23—28no—
492C23—232no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—44no—
500C23—48no—
501C23—44no—
502C23—48no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—48no—
507C23—432no—
508C23—416no—
509C23—416no—
510C23—44no—
511C23—416no—
512C23—48no—
513C23—48no—
514C23—416no—
515C23—48no—
516C23—432no—
517C23—416no—
518C23—44no—
519C23—48no—
520C23—432no—
521C23—416no—
522C23—44no—
523C23—48no—
524C23—44no—
525C23—44no—
526C23—48no—
527C23—42no—
528C23—44no—
529C23—42no—
530C23—84no—
531C23—82no—
532C23C2×C22⋊Dic5312yes—
533C23C2×C22⋊Dic5312yes—
534C23C2×C22⋊Dic5314yescomplemented
535C23D4×Dic5314yes—
536C23D4×Dic5314yes—
537C23#1313318511yescharacteristic
538C23#1313318512yes—
539C23#1313318511yescharacteristic
540C23#1313318914yes—
541C23#1313318914yes—
542C23#1313319812yes—
543C23#1313319812yes—
544C23C23×Dic5311yescharacteristic
545C23C2×D4×D5312yes—
546C23C22×C22⋊D5311yescharacteristic
547C23C22×C22⋊D5311yescharacteristic
548C23C2×#1171994712yes—
549D4—216no—
550D4—28no—
551D4—28no—
552D4—432no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—1068no—
561C2×C4—4248no—
562C2×C4—4244no—
563C2×C4—4248no—
564C2×C4—2128no—
565C2×C4—2124no—
566C2×C4—2128no—
567C2×C4—2128no—
568C2×C4C2×C22⋊Dic5314yescomplemented
569C22—24no—
570C22—28no—
571C22—28no—
572C22—24no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—22no—
577C22—28no—
578C22—216no—
579C22—24no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—416no—
590C22—48no—
591C22—416no—
592C22—48no—
593C22—432no—
594C22—44no—
595C22—416no—
596C22—48no—
597C22—44no—
598C22—48no—
599C22—84no—
600C22—82no—
601C22#1681050312yes—
602C22#1681050314yes—
603C22#1681050312yes—
604C22#1681080914yes—
605C22C22×C22⋊Dic5311yescharacteristic
606C22C2×D4×Dic5312yes—
607C22C2×#1313318511yescharacteristic
608C22C2×#1313318912yes—
609C22C2×#1313319811yescharacteristic
610C4—24no—
611C4—44no—
612C4—4244no—
613C4—2128no—
614C2—21no—
615C2—22no—
616C2—22no—
617C2—22no—
618C2—21no—
619C2—24no—
620C2—48no—
621C2—44no—
622C2#2027512414yesdirect product factor
623C2C2×#1681050311yescharacteristic
624C2C2×#1681080912yes—
subgroupG/H
centerC23#13133185
commutatorC23×C53C23×C4
FrattiniC24C23×D53
Fitting, prime radicalC22×#120×C53C2
socleC23×C53#120
cosocleC24×C53C24
hypercenterC22×#120D53
hypocenterC53C2×#1889
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×#120

Complements

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

NH ≅ G/Ntype
1. C2×#16811098C2semidirect
2. C22×#13133132C2semidirect
3. C2×#16810479C2semidirect
4. #20275124C2direct (N × H)
5. C2×#16810389C2semidirect
9. #16811098C22semidirect
17. C2×#13133132C22semidirect
19. #16810479C22semidirect
20. #16810389C22semidirect
26. C2×#120×C53C4semidirect
32. C23×Dic53D4semidirect
50. C2×C22⋊Dic53D4semidirect
62. C25×C53C2×C4semidirect
80. #120×C53C2×C4semidirect
85. C2×C22⋊C4×C53C2×C4semidirect
117. C24×C53C22⋊C4semidirect
119. C24×C53C22×C4semidirect
139. C2×D4×C53C22⋊C4semidirect
148. C22×C4×C53C22⋊C4semidirect
158. C22×Dic53C2×D4semidirect
162. C22⋊C4×C53C22×C4semidirect
221. C23×C53C2×C22⋊C4semidirect
247. C2×C4×C53C2×C22⋊C4semidirect
316. C2×C53#1889semidirect
324. C2×#120Dic53semidirect
342. C53C2×#1889semidirect
350. C25C2×Dic53semidirect
371. #120C2×Dic53semidirect
396. C2×C22⋊C4C2×Dic53semidirect
430. C24C22⋊Dic53semidirect
435. C24C22×Dic53semidirect
452. C2×D4C22⋊Dic53semidirect
470. C22×C4C22⋊Dic53semidirect
485. C22⋊C4C22×Dic53semidirect
534. C23C2×C22⋊Dic53semidirect
568. C2×C4C2×C22⋊Dic53semidirect
622. C2#20275124direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110982
C22×#131331322
C2×#168104792
#202751242
C2×#168103892
C22×#120×C532
C2×#188953