Finite Group Databasebeta

← C2×#20273065 #27350039

Subgroups — C2×#20273065 #27350039

Order 13568.

1346 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133203C211yesmaximal, characteristic
2C2×#16810677C211yesmaximal, characteristic, complemented
3C2×#16810673C211yesmaximal, characteristic, complemented
4#20273065C218yesmaximal, direct product factor
5C2×#16810347C211yesmaximal, characteristic, complemented
6C2×#16810346C211yesmaximal, characteristic, complemented
7C2×#16810346C211yesmaximal, characteristic, complemented
8C2×C53×#323C211yesmaximal, characteristic, complemented
9C23×Dic106C411yescharacteristic
10C23×C22⋊D53C411yescharacteristic
11C2×#13133203—21no—
12C2×#13133203—21no—
13C2×#13133203—21no—
14C2×#13133203—21no—
15C2×#13133203—22no—
16C2×#13133203—22no—
17C2×#13133203C414yes—
18C2×#13133203C2214yes—
19C22×C22⋊Dic53C411yescharacteristic
20C22×#11719921C411yescharacteristic
21C22×#11719921C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23#16810677—22no—
24#16810677C2214yescomplemented
25#16810673—22no—
26#16810673C2214yescomplemented
27C2×#13133146—21no—
28C2×#13133146—21no—
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×#13133127—21no—
32C2×C8×Dic53—21no—
33#16810347C2214yescomplemented
34#16810346C2214yescomplemented
35#16810346C2214yescomplemented
36C22×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C8×C53C2211yescharacteristic
38C2×C22⋊C8×C53C2211yescharacteristic
39C53×#323C2214yescomplemented
40C24×D53C2×C411yescharacteristic
41C22×Dic106—21no—
42C22×Dic106—22no—
43C22×Dic106—21no—
44C22×Dic106—44no—
45C22×Dic106C2×C414yes—
46C22×C22⋊D53—21no—
47C22×C22⋊D53—22no—
48C22×C22⋊D53—21no—
49C22×C22⋊D53—44no—
50C22×C22⋊D53C2×C414yes—
51C23×Dic53C2×C411yescharacteristic
52C23×Dic53C2×C411yescharacteristic
53C23×Dic53C2×C411yescharacteristic
54C22×#88322—21no—
55C22×#88322—21no—
56#13133203—24no—
57#13133203—24no—
58#13133203—24no—
59#13133203—24no—
60#13133203—24no—
61#13133203—24no—
62#13133203—24no—
63#13133203—24no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—21no—
67C2×C22⋊Dic53C2×C414yes—
68C2×#11719921—21no—
69C2×#11719921—21no—
70C2×#11719921—21no—
71C2×#11719921—21no—
72C2×#11719921—22no—
73C2×#11719921—21no—
74C2×#11719921—21no—
75C2×#11719921C2×C414yes—
76C2×#11719921C2×C412yes—
77C2×#11719921C2312yes—
78C2×C4×Dic53C2×C412yes—
79C2×C4×Dic53D411yescharacteristic
80C2×C4×Dic53D411yescharacteristic
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D412yes—
84C2×C4×Dic53D412yes—
85C2×C4×Dic53C2312yes—
86#13133146—24no—
87#13133146—24no—
88C22⋊#88322—21no—
89C22⋊#88322—21no—
90C22⋊#88322C2312yes—
91C22⋊#88322C2312yes—
92#13133127—24no—
93C8×Dic53—24no—
94C25×C53C2×C411yescharacteristic
95C23×C4×C53C2×C411yescharacteristic
96C23×C4×C53C2311yescharacteristic
97C22×C8×C53—21no—
98C22×C8×C53—21no—
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53C2×C412yes—
104C2×C22⋊C4×C53C2312yes—
105C22⋊C8×C53—21no—
106C22⋊C8×C53—21no—
107C22⋊C8×C53C2312yes—
108C22⋊C8×C53C2312yes—
109C24×C53—21no—
110C24×C53—21no—
111C24×C53—22no—
112C24×C53—22no—
113C24×C53—42no—
114C24×C53—42no—
115C24×C53C22×C411yescharacteristic
116C24×C53C22×C412yes—
117C23×D53—21no—
118C23×D53—21no—
119C23×D53—22no—
120C23×D53—22no—
121C23×D53—42no—
122C23×D53—42no—
123C23×D53C22×C412yes—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2411yescharacteristic
135C2×C22⋊D53—24no—
136C2×C22⋊D53—24no—
137C2×C22⋊D53—44no—
138C2×C22⋊D53—48no—
139C2×C22⋊D53—48no—
140C2×C22⋊D53—44no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—24no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—42no—
147C22×Dic53C22⋊C412yes—
148C22×Dic53C22⋊C411yescharacteristic
149C22×Dic53C22⋊C411yescharacteristic
150C22×Dic53C22⋊C411yescharacteristic
151C22×Dic53C22⋊C412yes—
152C22×Dic53C22⋊C411yescharacteristic
153C22×Dic53C22×C412yes—
154C22×Dic53C22×C412yes—
155C22×Dic53C22×C412yes—
156C2×Dic106—24no—
157C2×Dic106—24no—
158C2×Dic106—44no—
159C2×Dic106—48no—
160C2×Dic106—44no—
161C2×Dic106—48no—
162C2×C8×C53—22no—
163C2×C8×C53—22no—
164C2×C8×C53—42no—
165C2×C8×C53—42no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊Dic53—24no—
171C22⋊Dic53—24no—
172#11719921—22no—
173#11719921—22no—
174#11719921—22no—
175#11719921—22no—
176#11719921—24no—
177#11719921—24no—
178C4×Dic53C22⋊C412yes—
179C4×Dic53C22⋊C412yes—
180C4×Dic53C22⋊C412yes—
181C4×Dic53C22⋊C412yes—
182C4×Dic53C2×D412yes—
183C4×Dic53C2×D412yes—
184C4×Dic53C2×D412yes—
185C4×Dic53C2×D412yes—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—42no—
189C2×#88322—42no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—84no—
204C23×C53—84no—
205C23×C53C2×C22⋊C411yescharacteristic
206C23×C53C2×C22⋊C411yescharacteristic
207C23×C53#13112yes—
208C23×C53#13112yes—
209C23×C53C23×C411yescharacteristic
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—22no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—84no—
222C22×D53—84no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53C2×C22⋊C411yescharacteristic
234C2×C4×C53C2×C22⋊C411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C22⋊D53—48no—
238C22⋊D53—48no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—24no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—22no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53#10014yes—
252C2×Dic53C2×C22⋊C412yes—
253C2×Dic53C2×C22⋊C412yes—
254C2×Dic53C2×C22⋊C412yes—
255C2×Dic53C2×C22⋊C412yes—
256Dic106—48no—
257Dic106—48no—
258C8×C53—42no—
259C8×C53—42no—
260#88322—42no—
261#88322—42no—
262C2×#1469—531nomaximal, Sylow
263C22×C53—21no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—24no—
268C22×C53—24no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53—82no—
273C22×C53—82no—
274C22×C53—84no—
275C22×C53—84no—
276C22×C53#40812yes—
277C22×C53#40812yes—
278C22×C53#41012yes—
279C22×C53C2×#10012yes—
280C22×C53C22×C22⋊C411yescharacteristic
281C22×C53C2×#13111yescharacteristic
282C22×C53C2×#13111yescharacteristic
283C2×D53—42no—
284C2×D53—42no—
285C2×D53—42no—
286C2×D53—84no—
287C2×D53—84no—
288C2×D53—82no—
289C2×D53—82no—
290C4×C53—21no—
291C4×C53—21no—
292C4×C53—21no—
293C4×C53—21no—
294C4×C53—42no—
295Dic53—22no—
296Dic53—22no—
297Dic53—22no—
298Dic53—42no—
299Dic53C2×#10014yes—
300C22×#124—531no—
301C2×#432—531no—
302C2×#431—531no—
303#1469—538no—
304C2×#323—531no—
305C2×#323—531no—
306C2×#323—531no—
307C2×#323D5311yescharacteristic, complemented
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—22no—
311C2×C53—82no—
312C2×C53#146914yescomplemented
313C2×C53C2×#40811yescharacteristic
314C2×C53C2×#41011yescharacteristic
315C2×C53C22×#10011yescharacteristic
316D53—82no—
317C23×Q8—531no—
318C23×D4—531no—
319C2×#124—534no—
320C2×#124—534no—
321C2×#124—1061no—
322C2×#124—1061no—
323C2×#124—1061no—
324C2×#124—1061no—
325C2×#124—1062no—
326C2×#124—1062no—
327C22×C22⋊C4—531no—
328C22×C22⋊C4—531no—
329C22×C22⋊C4—531no—
330C22×C22⋊C4C2×D5311yescharacteristic
331C22×C42—531no—
332#432—534no—
333#432—1062no—
334#431—534no—
335#431—1062no—
336C2×C4⋊C8—1061no—
337C2×C4⋊C8—1061no—
338C2×C22⋊C8—531no—
339C2×C22⋊C8—531no—
340C2×C22⋊C8C2×D5311yescharacteristic
341C2×C22⋊C8C2×D5311yescharacteristic
342C2×#97—1061no—
343C2×C4×C8—1061no—
344#323—534no—
345#323—534no—
346#323—534no—
347#323C2×D5314yescomplemented
348C53C2×#146911yescharacteristic, complemented, Sylow
349C25—531no—
350C25C4×D5311yescharacteristic
351C22×Q8—534no—
352C22×Q8—1062no—
353C22×Q8—1061no—
354C22×Q8—1061no—
355C22×Q8—2124no—
356C22×D4—534no—
357C22×D4—1061no—
358C22×D4—1062no—
359C22×D4—1061no—
360C22×D4—2124no—
361C23×C4—531no—
362C23×C4—531no—
363C23×C4—531no—
364C23×C4C4×D5311yescharacteristic
365C23×C4C22×D5311yescharacteristic
366C22×C8—21no—
367C22×C8—21no—
368C22×C8—1061no—
369C22×C8—1061no—
370#124—1064no—
371#124—1064no—
372#124—1064no—
373#124—1064no—
374#124—1064no—
375#124—1064no—
376#124—1064no—
377#124—1064no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—534no—
383C2×C22⋊C4—534no—
384C2×C22⋊C4—532no—
385C2×C22⋊C4—532no—
386C2×C22⋊C4—1062no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1062no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4C4×D5312yes—
397C2×C22⋊C4C22×D5312yes—
398C2×C42—532no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—532no—
404C2×C42—532no—
405C2×C42—532no—
406C4⋊C8—1064no—
407C4⋊C8—1064no—
408C22⋊C8—21no—
409C22⋊C8—21no—
410C22⋊C8—532no—
411C22⋊C8—532no—
412C22⋊C8—1061no—
413C22⋊C8—1061no—
414C22⋊C8C22×D5312yes—
415C22⋊C8C22×D5312yes—
416#97—1064no—
417C4×C8—1064no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24—42no—
423C24—42no—
424C24—532no—
425C24—1062no—
426C24—1062no—
427C24—1061no—
428C24—1061no—
429C24—2122no—
430C24—2122no—
431C24C2×C4×D5311yescharacteristic
432C24C2×C4×D5312yes—
433C2×Q8—1064no—
434C2×Q8—1064no—
435C2×Q8—2124no—
436C2×Q8—2128no—
437C2×Q8—2128no—
438C2×Q8—2124no—
439C2×D4—1064no—
440C2×D4—1064no—
441C2×D4—2124no—
442C2×D4—2128no—
443C2×D4—2124no—
444C2×D4—2128no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—531no—
450C22×C4—532no—
451C22×C4—532no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—1064no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—2122no—
464C22×C4C2×C4×D5312yes—
465C22×C4C2×C4×D5311yescharacteristic
466C22×C4D4×D5311yescharacteristic
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4D4×D5311yescharacteristic
470C22×C4C23×D5311yescharacteristic
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—1062no—
476C2×C8—1062no—
477C2×C8—2122no—
478C2×C8—2122no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—1064no—
484C22⋊C4—1064no—
485C22⋊C4—1064no—
486C22⋊C4—1064no—
487C22⋊C4—1062no—
488C22⋊C4—1062no—
489C22⋊C4—1062no—
490C22⋊C4—1062no—
491C42—532no—
492C42—532no—
493C42—532no—
494C42—532no—
495C42—532no—
496C42—532no—
497C42—532no—
498C42—532no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—84no—
513C23—84no—
514C23—1062no—
515C23—1062no—
516C23—1062no—
517C23—4244no—
518C23—4244no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23—2122no—
524C23—2122no—
525C23—2122no—
526C23—2122no—
527C23C22⋊C4×D5311yescharacteristic
528C23C22⋊C4×D5311yescharacteristic
529C23#1313322312yes—
530C23#1313322512yes—
531C23C22×C4×D5311yescharacteristic
532Q8—2128no—
533Q8—2128no—
534D4—2128no—
535D4—2128no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—532no—
547C2×C4—534no—
548C2×C4—532no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—1064no—
552C2×C4—1064no—
553C2×C4—1064no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4C22⋊C4×D5311yescharacteristic
564C2×C4C22⋊C4×D5311yescharacteristic
565C2×C4C2×D4×D5311yescharacteristic
566C2×C4C2×D4×D5311yescharacteristic
567C8—42no—
568C8—42no—
569C8—2122no—
570C8—2122no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—82no—
581C22—82no—
582C22—84no—
583C22—84no—
584C22—4242no—
585C22—4242no—
586C22—4244no—
587C22—4244no—
588C22—2122no—
589C22—2122no—
590C22—2122no—
591C22#1681058812yes—
592C22#1681058812yes—
593C22#1681059512yes—
594C22#100×D5312yes—
595C22C2×C22⋊C4×D5311yescharacteristic
596C22C2×#1313322311yescharacteristic
597C22C2×#1313322511yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—42no—
603C4—534no—
604C4—1062no—
605C4—1062no—
606C4—1062no—
607C4—2122no—
608C2—22no—
609C2—21no—
610C2—21no—
611C2—82no—
612C2—4242no—
613C2#2027306514yesdirect product factor
614C2C2×#1681058811yescharacteristic
615C2C2×#1681059511yescharacteristic
616C2C2×#100×D5311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#323C2
socleC23×C53C2×C22⋊C4
prime radicalC23×C22⋊D53C4
cosocleC22×C4×C53C24
hypercenterC2×#323D53
hypocenterC53C2×#1469
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. C2×#16810677C2semidirect
3. C2×#16810673C2semidirect
4. #20273065C2direct (N × H)
5. C2×#16810347C2semidirect
6. C2×#16810346C2semidirect
7. C2×#16810346C2semidirect
8. C2×C53×#323C2semidirect
24. #16810677C22semidirect
26. #16810673C22semidirect
33. #16810347C22semidirect
34. #16810346C22semidirect
35. #16810346C22semidirect
39. C53×#323C22semidirect
307. C2×#323D53semidirect
312. C2×C53#1469semidirect
347. #323C2×D53semidirect
348. C53C2×#1469semidirect
613. C2#20273065direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332032
C2×#168106772
C2×#168106732
#202730652
C2×#168103472
C2×#168103462
C2×C53×#3232
C2×#146953