Finite Group Databasebeta

← C2×#20276270 #27413976

Subgroups — C2×#20276270 #27413976

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810671C211yesmaximal, characteristic, complemented
2C2×#16810651C211yesmaximal, characteristic, complemented
3C2×#16810650C211yesmaximal, characteristic, complemented
4C2×#16810640C211yesmaximal, characteristic
5C2×#16810590C211yesmaximal, characteristic
6#20276270C218yesmaximal, direct product factor
7C2×#16810379C211yesmaximal, characteristic, complemented
8C2×C53×#421C211yesmaximal, characteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C23×#88322C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13#16810671—22no—
14#16810671C2214yescomplemented
15C2×#13133152—22no—
16#16810651—22no—
17#16810651C2214yescomplemented
18#16810650—22no—
19#16810650C2214yescomplemented
20C2×C4⋊#88322—22no—
21C2×#13133146—22no—
22#16810640—22no—
23#16810640C2214yes—
24C2×#13133144—22no—
25#16810590—22no—
26#16810590C2214yes—
27C2×C22⋊#88322—22no—
28#16810379C2214yescomplemented
29C22×#41×C53C2211yescharacteristic
30C22×M16×C53C2211yescharacteristic
31C2×#117×C53C2211yescharacteristic
32C53×#421—22no—
33C53×#421C2214yes—
34C2×#104×C53—22no—
35C2×#11719917—21no—
36C2×#11719917—21no—
37C2×#11719917—21no—
38C2×#11719917—21no—
39C2×#11719917—21no—
40C2×#11719917—21no—
41C2×#11719917—21no—
42C2×#11719917—21no—
43C2×#11719917—21no—
44C2×#11719917D412yes—
45C2×#11719917C2312yes—
46C2×#11719917C2312yes—
47C22×#88322—21no—
48C22×#88322—21no—
49C22×#88322—21no—
50C22×#88322—21no—
51C22×#88322—21no—
52C22×#88322—21no—
53C22×#88322—21no—
54C22×#88322D412yes—
55C22×#88322C2312yes—
56#13133176—21no—
57#13133176—21no—
58#13133176D412yes—
59#13133176C2312yes—
60C2×C4⋊Dic53—21no—
61C2×C22⋊Dic53—21no—
62C2×C4×Dic53D412yes—
63#13133152—28no—
64C4⋊#88322—28no—
65#13133146—28no—
66#13133144—28no—
67C22⋊#88322—28no—
68C2×#41×C53—21no—
69C2×#41×C53—21no—
70C2×#41×C53—21no—
71C2×#41×C53—21no—
72C2×#41×C53—21no—
73C2×#41×C53—21no—
74C2×#41×C53—21no—
75C2×#41×C53—21no—
76C2×#41×C53—21no—
77C2×#41×C53—21no—
78C2×#41×C53Q812yes—
79C2×#41×C53C2312yes—
80C22×Q8×C53Q811yescharacteristic
81C22×D4×C53—21no—
82C22×D4×C53Q811yescharacteristic
83C23×C4×C53—21no—
84C23×C4×C53C2311yescharacteristic
85C2×M16×C53—21no—
86C2×M16×C53—21no—
87C2×M16×C53—21no—
88C2×M16×C53—21no—
89C2×M16×C53—21no—
90C2×M16×C53C2312yes—
91C22×C8×C53—21no—
92#117×C53—21no—
93#117×C53—21no—
94#117×C53D412yes—
95#117×C53C2312yes—
96C2×C4⋊C4×C53—21no—
97C2×C22⋊C4×C53—21no—
98C2×C42×C53D412yes—
99#104×C53—28no—
100C24×C53—21no—
101C24×C53#4111yescharacteristic
102#41×C53—22no—
103#41×C53—22no—
104#41×C53—22no—
105#41×C53—22no—
106#41×C53—22no—
107#41×C53—22no—
108#41×C53—22no—
109#41×C53—22no—
110#41×C53—41no—
111#41×C53—41no—
112#41×C53—41no—
113#41×C53—41no—
114#41×C53—41no—
115#41×C53—41no—
116#41×C53—41no—
117#41×C53—41no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53C2×Q812yes—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—41no—
132C2×D4×C53—41no—
133C2×D4×C53—41no—
134C2×D4×C53—41no—
135C2×D4×C53C2×Q812yes—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—41no—
141C22×C4×C53—41no—
142C22×C4×C53—41no—
143C22×C4×C53—41no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×Q811yescharacteristic
151C22×C4×C53C2×Q811yescharacteristic
152C22×C4×C53#4111yescharacteristic
153C22×C4×C53#4111yescharacteristic
154C22×C4×C53#4111yescharacteristic
155C22×C4×C53#4111yescharacteristic
156C22×C4×C53#4111yescharacteristic
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—22no—
159M16×C53—22no—
160M16×C53—22no—
161M16×C53—41no—
162M16×C53—41no—
163C2×C8×C53—22no—
164C2×C8×C53—41no—
165C2×C8×C53—41no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C42×C53—22no—
171C42×C53C2×D414yes—
172C22⋊Dic53—22no—
173C22⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—22no—
176C4×Dic53—22no—
177C4×Dic53C2×D414yes—
178#11719917—22no—
179#11719917—22no—
180#11719917—22no—
181#11719917—22no—
182#11719917—22no—
183#11719917—22no—
184#11719917—41no—
185#11719917—41no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—41no—
193C2×#88322—41no—
194C2×#88322—41no—
195C2×#88322—41no—
196C2×#88322—41no—
197C2×#88322—41no—
198C2×#88322C2×D414yes—
199C23×C53—22no—
200C23×C53—42no—
201C23×C53—44no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53C22⋊Q811yescharacteristic
205C23×C53C22⋊Q811yescharacteristic
206C23×C53#12311yescharacteristic
207C23×C53#12311yescharacteristic
208C23×C53#12311yescharacteristic
209C23×C53#12311yescharacteristic
210C23×C53C2×#4111yescharacteristic
211Q8×C53—22no—
212Q8×C53—22no—
213Q8×C53—41no—
214Q8×C53—41no—
215D4×C53—22no—
216D4×C53—22no—
217D4×C53—42no—
218D4×C53—42no—
219D4×C53—42no—
220D4×C53—42no—
221D4×C53—41no—
222D4×C53—41no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—44no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53C22⋊Q811yescharacteristic
233C2×C4×C53C22⋊Q811yescharacteristic
234C2×C4×C53C22⋊Q811yescharacteristic
235C2×C4×C53C22⋊Q811yescharacteristic
236C2×C4×C53C22⋊Q811yescharacteristic
237C2×C4×C53C22⋊Q811yescharacteristic
238C2×C4×C53C22⋊Q811yescharacteristic
239C2×C4×C53C22⋊Q811yescharacteristic
240C2×C4×C53C22⋊Q811yescharacteristic
241C2×C4×C53C22⋊Q811yescharacteristic
242C2×C4×C53#12311yescharacteristic
243C2×C4×C53#12311yescharacteristic
244C2×C4×C53#12311yescharacteristic
245C2×C4×C53#12311yescharacteristic
246C2×C4×C53#12311yescharacteristic
247C2×C4×C53#12311yescharacteristic
248C2×C4×C53#12311yescharacteristic
249C2×C4×C53#12311yescharacteristic
250C2×C4×C53#12711yescharacteristic
251C2×C4×C53#12711yescharacteristic
252C2×C4×C53#12711yescharacteristic
253C2×C4×C53#12711yescharacteristic
254C2×C4×C53C22×D411yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C22×Q811yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×Dic53—24no—
261C2×Dic53—44no—
262C8×C53—42no—
263#88322—24no—
264#88322—42no—
265#88322—42no—
266C2×#2098—531nomaximal, Sylow
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—44no—
275C22×C53—42no—
276C22×C53—82no—
277C22×C53—82no—
278C22×C53#39911yescharacteristic
279C22×C53#39911yescharacteristic
280C22×C53#39911yescharacteristic
281C22×C53#39911yescharacteristic
282C22×C53C2×C22⋊Q811yescharacteristic
283C22×C53C2×#12311yescharacteristic
284C22×C53C2×#12311yescharacteristic
285C4×C53—44no—
286C4×C53—42no—
287C4×C53#39911yescharacteristic
288C4×C53#39911yescharacteristic
289C4×C53#39911yescharacteristic
290C4×C53#39911yescharacteristic
291C4×C53C2×C22⋊Q811yescharacteristic
292C4×C53C2×C22⋊Q811yescharacteristic
293C4×C53C2×#12311yescharacteristic
294C4×C53C2×#12711yescharacteristic
295Dic53—44no—
296C2×#430—531no—
297C2×#424—531no—
298C2×#424—531no—
299C2×#421—531no—
300C2×#421D5311yescharacteristic
301C2×#408—531no—
302#2098—538no—
303C2×H(3,ℤ/4)—531no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53—21no—
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—82no—
311C2×C53#209812yescomplemented
312C2×C53C2×#39911yescharacteristic
313C22×#41C2×D5311yescharacteristic
314C22×M16—531no—
315C22×M16—531no—
316C22×M16C2×D5311yescharacteristic
317C23×C8—531no—
318C2×#117—531no—
319C2×#117C2×D5311yescharacteristic
320#430—534no—
321#430—1062no—
322C2×#108—1062no—
323#424—534no—
324#424—534no—
325#424—1062no—
326#424—1062no—
327C2×C4⋊C8—1062no—
328C2×C4⋊C8—1062no—
329#421—22no—
330#421—534no—
331#421—1062no—
332#421C2×D5314yes—
333C2×#104—22no—
334C2×#104—1062no—
335#408—534no—
336#408—1062no—
337C2×C22⋊C8—1062no—
338H(3,ℤ/4)—534no—
339C53C2×#209811yescharacteristic, complemented, Sylow
340C2×#41—21no—
341C2×#41—21no—
342C2×#41—21no—
343C2×#41—21no—
344C2×#41—21no—
345C2×#41—21no—
346C2×#41—21no—
347C2×#41—21no—
348C2×#41—21no—
349C2×#41—21no—
350C2×#41Dic10612yes—
351C2×#41C22×D5312yes—
352C22×Q8Dic10611yescharacteristic
353C22×D4—21no—
354C22×D4Dic10611yescharacteristic
355C23×C4—21no—
356C23×C4C22×D5311yescharacteristic
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—532no—
363C2×M16—532no—
364C2×M16—532no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16—1061no—
368C2×M16—1061no—
369C2×M16—1061no—
370C2×M16—1061no—
371C2×M16—1061no—
372C2×M16—1061no—
373C2×M16—1061no—
374C2×M16C22×D5312yes—
375C22×C8—21no—
376C22×C8—532no—
377C22×C8—532no—
378C22×C8—1061no—
379C22×C8—1061no—
380C22×C8—1061no—
381C22×C8—1061no—
382C22×C8—1061no—
383C22×C8—1061no—
384C22×C8—1061no—
385#117—21no—
386#117—21no—
387#117—532no—
388#117—532no—
389#117—1061no—
390#117—1061no—
391#117C22⋊D5312yes—
392#117C22×D5312yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—1061no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1061no—
397C2×C42—532no—
398C2×C42C22⋊D5312yes—
399#108—1068no—
400C4⋊C8—1068no—
401C4⋊C8—1068no—
402#104—28no—
403#104—1068no—
404C22⋊C8—1068no—
405C24—21no—
406C24#1171994711yescharacteristic
407#41—22no—
408#41—22no—
409#41—22no—
410#41—22no—
411#41—22no—
412#41—22no—
413#41—22no—
414#41—22no—
415#41—41no—
416#41—41no—
417#41—41no—
418#41—41no—
419#41—41no—
420#41—41no—
421#41—41no—
422#41—41no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8C2×Dic10612yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4C2×Dic10612yes—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—1062no—
450C22×C4C2×Dic10611yescharacteristic
451C22×C4C2×Dic10611yescharacteristic
452C22×C4#1171994511yescharacteristic
453C22×C4#1171994511yescharacteristic
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5311yescharacteristic
456C22×C4D4×D5311yescharacteristic
457C22×C4D4×D5311yescharacteristic
458C22×C4#1171994711yescharacteristic
459C22×C4#1171994711yescharacteristic
460C22×C4#1171994711yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C23×D5311yescharacteristic
464M16—22no—
465M16—22no—
466M16—41no—
467M16—41no—
468M16—1062no—
469M16—1062no—
470M16—1062no—
471M16—1062no—
472M16—1062no—
473M16—1062no—
474M16—2121no—
475M16—2121no—
476C2×C8—22no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—534no—
480C2×C8—1062no—
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C2×C8—2121no—
487C2×C8—2121no—
488C2×C8—2121no—
489C2×C8—2121no—
490C2×C8—2121no—
491C2×C8—2121no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—1062no—
495C4⋊C4—1062no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—1062no—
499C22⋊C4—1062no—
500C42—22no—
501C42—534no—
502C42—1062no—
503C42C2×C22⋊D5314yes—
504C23—22no—
505C23—42no—
506C23—44no—
507C23—41no—
508C23—41no—
509C23#1313319411yescharacteristic
510C23#1313319411yescharacteristic
511C23#1313319811yescharacteristic
512C23#1313319811yescharacteristic
513C23#1313319911yescharacteristic
514C23#1313319911yescharacteristic
515C23C2×#1171994711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520D4—22no—
521D4—22no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—41no—
527D4—41no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—1064no—
538C2×C4—2124no—
539C2×C4#1313319411yescharacteristic
540C2×C4#1313319411yescharacteristic
541C2×C4#1313319411yescharacteristic
542C2×C4#1313319411yescharacteristic
543C2×C4#1313319411yescharacteristic
544C2×C4#1313319411yescharacteristic
545C2×C4#1313319611yescharacteristic
546C2×C4#1313319611yescharacteristic
547C2×C4#1313319611yescharacteristic
548C2×C4#1313319611yescharacteristic
549C2×C4#1313319811yescharacteristic
550C2×C4#1313319811yescharacteristic
551C2×C4#1313319911yescharacteristic
552C2×C4#1313319911yescharacteristic
553C2×C4#1313319911yescharacteristic
554C2×C4#1313319911yescharacteristic
555C2×C4#1313319911yescharacteristic
556C2×C4#1313319911yescharacteristic
557C2×C4#1313321311yescharacteristic
558C2×C4#1313321311yescharacteristic
559C2×C4#1313321311yescharacteristic
560C2×C4#1313321311yescharacteristic
561C2×C4C2×D4×D5311yescharacteristic
562C2×C4C2×D4×D5311yescharacteristic
563C2×C4C22×C22⋊D5311yescharacteristic
564C2×C4C22×Dic10611yescharacteristic
565C2×C4C2×#1171994711yescharacteristic
566C2×C4C2×#1171994511yescharacteristic
567C8—42no—
568C8—1064no—
569C8—2122no—
570C8—2122no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—44no—
579C22—42no—
580C22—82no—
581C22—82no—
582C22#1681055911yescharacteristic
583C22#1681055911yescharacteristic
584C22#1681055911yescharacteristic
585C22#1681055911yescharacteristic
586C22C2×#1313319411yescharacteristic
587C22C2×#1313319811yescharacteristic
588C22C2×#1313319911yescharacteristic
589C4—44no—
590C4—42no—
591C4—2124no—
592C4#1681055911yescharacteristic
593C4#1681055911yescharacteristic
594C4#1681055911yescharacteristic
595C4#1681055911yescharacteristic
596C4C2×#1313319411yescharacteristic
597C4C2×#1313319611yescharacteristic
598C4C2×#1313319911yescharacteristic
599C4C2×#1313321311yescharacteristic
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—82no—
607C2#2027627012yesdirect product factor
608C2C2×#1681055911yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#421C2
socleC22×C53#399
prime radicalC22×#41×C53C22
hypercenterC2×#421D53
hypocenterC53C2×#2098
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
1. C2×#16810671C2semidirect
2. C2×#16810651C2semidirect
3. C2×#16810650C2semidirect
6. #20276270C2direct (N × H)
7. C2×#16810379C2semidirect
14. #16810671C22semidirect
17. #16810651C22semidirect
19. #16810650C22semidirect
28. #16810379C22semidirect
311. C2×C53#2098semidirect
339. C53C2×#2098semidirect
607. C2#20276270direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168106712
C2×#168106512
C2×#168106502
C2×#168106402
C2×#168105902
#202762702
C2×#168103792
C2×C53×#4212
C2×#209853