Finite Group Databasebeta

← C2×#20276386 #27536880

Subgroups — C2×#20276386 #27536880

Order 13568.

1422 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811187C211yesmaximal, characteristic
2C2×#16810597C211yesmaximal, characteristic, complemented
3#20276386C218yesmaximal, direct product factor
4C2×#16810415C214yesmaximal, complemented
5C2×C53×#410C211yesmaximal, characteristic, complemented
6C23×D212C411yescharacteristic, complemented
7#16811187—22no—
8#16811187—22no—
9#16811187C414yescomplemented
10#16811187C2214yes—
11C2×#13133213C2212yes—
12C2×#13133206C412yescomplemented
13C2×#13133191—21no—
14C2×#13133191—21no—
15C2×#13133185—21no—
16C2×#13133185—21no—
17C2×#13133176—21no—
18C2×#13133176C411yescharacteristic, complemented
19#16810597—22no—
20#16810597C2214yescomplemented
21C2×#13133132—21no—
22C2×#13133132C2212yescomplemented
23#16810415C22116yescomplemented
24C22×#41×C53C2211yescharacteristic
25C2×#117×C53—21no—
26C53×#410—22no—
27C53×#410C2214yescomplemented
28C2×#99×C53—21no—
29C2×#99×C53C2212yescomplemented
30C24×D53—21no—
31C22×D212—21no—
32C22×D212—22no—
33C22×D212—42no—
34C22×D212—41no—
35C22×D212—41no—
36C22×D212C2×C412yes—
37C22×D212C2×C414yescomplemented
38C22×C22⋊D53—22no—
39C22×C22⋊D53—22no—
40#13133213—22no—
41#13133213—22no—
42#13133213C2×C414yes—
43#13133213C2314yes—
44#13133206—24no—
45#13133206C2×C418yescomplemented
46#13133191—24no—
47#13133191—24no—
48#13133191—24no—
49#13133191—24no—
50#13133185—24no—
51#13133185—24no—
52#13133185—24no—
53#13133185—24no—
54#13133176—22no—
55#13133176—24no—
56#13133176—42no—
57#13133176C2×C414yescomplemented
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—41no—
60C2×C22⋊Dic53—21no—
61C2×C22⋊Dic53—22no—
62C2×#11719921—22no—
63C2×#11719921—22no—
64C2×C4×Dic53—41no—
65C2×C4×Dic53C2×C412yes—
66#13133132—24no—
67#13133132—22no—
68#13133132C2314yescomplemented
69C2×#41×C53—41no—
70C2×#41×C53—41no—
71C2×#41×C53—42no—
72C2×#41×C53C2×C412yes—
73C2×#41×C53D412yes—
74C2×#41×C53D412yes—
75C2×#41×C53C2312yes—
76C22×Q8×C53C2×C411yescharacteristic
77C22×D4×C53D411yescharacteristic
78C22×D4×C53D411yescharacteristic
79C22×D4×C53C2311yescharacteristic
80C23×C4×C53C2×C411yescharacteristic
81C23×C4×C53D411yescharacteristic
82C23×C4×C53D411yescharacteristic
83#117×C53—24no—
84#117×C53—42no—
85C2×C4⋊C4×C53—41no—
86C2×C22⋊C4×C53—22no—
87C2×C42×C53—41no—
88#99×C53—24no—
89#99×C53—22no—
90#99×C53C2314yescomplemented
91C24×C53C22⋊C411yescharacteristic
92C24×C53C2×D411yescharacteristic
93C24×C53C2×D411yescharacteristic
94C23×D53—22no—
95C23×D53—24no—
96C23×D53—42no—
97C23×D53—42no—
98C23×D53—84no—
99#41×C53—42no—
100#41×C53—42no—
101#41×C53—42no—
102#41×C53—42no—
103#41×C53—42no—
104#41×C53—42no—
105#41×C53—42no—
106#41×C53—42no—
107C2×Q8×C53—41no—
108C2×Q8×C53—41no—
109C2×Q8×C53C22⋊C412yes—
110C2×Q8×C53C22×C412yes—
111C2×D4×C53—41no—
112C2×D4×C53—41no—
113C2×D4×C53—41no—
114C2×D4×C53—41no—
115C2×D4×C53—41no—
116C2×D4×C53—41no—
117C2×D4×C53C22⋊C412yes—
118C2×D4×C53C22⋊C412yes—
119C2×D4×C53C22×C411yescharacteristic
120C2×D4×C53C2×D412yes—
121C2×D4×C53C2×D411yescharacteristic
122C2×D4×C53C2×D412yes—
123C2×D4×C53C2×D411yescharacteristic
124C2×D4×C53C2411yescharacteristic
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—22no—
134C22×C4×C53—42no—
135C22×C4×C53C22⋊C412yes—
136C22×C4×C53C22⋊C411yescharacteristic
137C22×C4×C53C22⋊C411yescharacteristic
138C22×C4×C53C22⋊C412yes—
139C22×C4×C53C22⋊C412yes—
140C22×C4×C53C22⋊C411yescharacteristic
141C22×C4×C53C22×C411yescharacteristic
142C22×C4×C53C22×C412yes—
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D412yes—
145C2×C22⋊D53—24no—
146C2×C22⋊D53—24no—
147C2×C22⋊D53—24no—
148C2×C22⋊D53—24no—
149C2×C22⋊D53—84no—
150C2×C22⋊D53—84no—
151C22×Dic53—22no—
152C22×Dic53—42no—
153C2×D212—24no—
154C2×D212—22no—
155C2×D212—44no—
156C2×D212—44no—
157C2×D212—42no—
158C2×D212—42no—
159C2×D212—42no—
160C2×D212—41no—
161C2×D212—41no—
162C2×D212—44no—
163C2×D212—44no—
164C2×D212C22×C414yes—
165C4⋊C4×C53—44no—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—42no—
168C42×C53—44no—
169C22⋊Dic53—24no—
170C22⋊Dic53—24no—
171C22⋊Dic53—42no—
172#11719921—28no—
173#11719921—28no—
174C4⋊Dic53—24no—
175C4⋊Dic53—44no—
176C4×Dic53—22no—
177C4×Dic53—44no—
178C4×Dic53C22×C414yes—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53C2×C22⋊C411yescharacteristic
189C23×C53C2×C22⋊C411yescharacteristic
190C23×C53C2×C22⋊C411yescharacteristic
191C23×C53C2×C22⋊C412yes—
192C23×C53C22×D411yescharacteristic
193C23×C53C22×D411yescharacteristic
194C22×D53—24no—
195C22×D53—48no—
196C22×D53—44no—
197C22×D53—44no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—88no—
201C22×D53—88no—
202C22×D53—88no—
203Q8×C53—42no—
204Q8×C53—42no—
205D4×C53—42no—
206D4×C53—42no—
207D4×C53—41no—
208D4×C53—41no—
209D4×C53—42no—
210D4×C53—41no—
211D4×C53—42no—
212D4×C53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—44no—
232C2×C4×C53—84no—
233C2×C4×C53#9912yes—
234C2×C4×C53#9911yescharacteristic
235C2×C4×C53#9911yescharacteristic
236C2×C4×C53C2×C22⋊C411yescharacteristic
237C2×C4×C53C2×C22⋊C412yes—
238C2×C4×C53C2×C22⋊C412yes—
239C2×C4×C53C2×C22⋊C412yes—
240C2×C4×C53C23×C411yescharacteristic
241C22⋊D53—84no—
242C22⋊D53—84no—
243C22⋊D53—84no—
244C22⋊D53—84no—
245C2×Dic53—24no—
246C2×Dic53—44no—
247C2×Dic53—44no—
248C2×Dic53—84no—
249D212—44no—
250D212—44no—
251D212—44no—
252D212—44no—
253C2×#2128—531nomaximal, Sylow
254C22×C53—22no—
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—22no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53C2×#9911yescharacteristic
272C22×C53C2×#9911yescharacteristic
273C22×C53C22×C22⋊C411yescharacteristic
274C2×D53—44no—
275C2×D53—44no—
276C2×D53—44no—
277C2×D53—84no—
278C2×D53—816no—
279C2×D53—84no—
280C4×C53—21no—
281C4×C53—21no—
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—84no—
285C4×C53C2×#9912yes—
286C4×C53C2×#9912yes—
287Dic53—44no—
288Dic53—84no—
289C2×#535—531no—
290C2×#410—531no—
291C2×#410D5311yescharacteristic, complemented
292#2128—538no—
293C2×#353—534no—
294C2×C53—21no—
295C2×C53—21no—
296C2×C53—41no—
297C2×C53—41no—
298C2×C53—42no—
299C2×C53—41no—
300C2×C53—41no—
301C2×C53#212812yescomplemented
302C2×C53C22×#9911yescharacteristic
303D53—84no—
304C22×#41C2×D5311yescharacteristic
305C23×D4—531no—
306#535—534no—
307#535—534no—
308#535—1062no—
309#535—1062no—
310C2×#127—532no—
311C2×#124—532no—
312C2×#121—1061no—
313C2×#121—1061no—
314C2×#120—1061no—
315C2×#120—1061no—
316C2×#117—21no—
317C2×#117—531no—
318C2×#117—1061no—
319#410—22no—
320#410—534no—
321#410—1062no—
322#410C2×D5314yescomplemented
323C2×#99—21no—
324C2×#99—532no—
325C2×#99—1061no—
326C2×#99C2×D5312yescomplemented
327#353—5316no—
328C53C2×#212811yescharacteristic, complemented, Sylow
329C25—1061no—
330C2×#41—41no—
331C2×#41—41no—
332C2×#41—42no—
333C2×#41C4×D5312yes—
334C2×#41D21212yes—
335C2×#41C22⋊D5312yes—
336C2×#41C22×D5312yes—
337C22×Q8C4×D5311yescharacteristic
338C22×D4—532no—
339C22×D4—534no—
340C22×D4—1062no—
341C22×D4—1062no—
342C22×D4—1062no—
343C22×D4—1061no—
344C22×D4—2122no—
345C22×D4—2121no—
346C22×D4—2121no—
347C22×D4D21211yescharacteristic
348C22×D4C22⋊D5311yescharacteristic
349C22×D4C22×D5311yescharacteristic
350C23×C4C4×D5311yescharacteristic
351C23×C4D21211yescharacteristic
352C23×C4C22⋊D5311yescharacteristic
353#127—534no—
354#127—534no—
355#127—1062no—
356#127—1062no—
357#124—538no—
358#124—1064no—
359#121—1064no—
360#121—1064no—
361#121—1064no—
362#121—1064no—
363#120—1064no—
364#120—1064no—
365#120—1064no—
366#120—1064no—
367#117—24no—
368#117—42no—
369#117—534no—
370#117—1062no—
371#117—1064no—
372#117—2122no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—2121no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—1061no—
378C2×C22⋊C4—1062no—
379C2×C22⋊C4—1062no—
380C2×C22⋊C4—1062no—
381C2×C42—41no—
382C2×C42—532no—
383C2×C42—2121no—
384#99—24no—
385#99—22no—
386#99—534no—
387#99—1062no—
388#99—1064no—
389#99C22×D5314yescomplemented
390C24—1064no—
391C24—1062no—
392C24—4244no—
393C24—2122no—
394C24—2122no—
395C24#1171992111yescharacteristic
396C24C2×D21211yescharacteristic
397C24C2×C22⋊D5311yescharacteristic
398#41—42no—
399#41—42no—
400#41—42no—
401#41—42no—
402#41—42no—
403#41—42no—
404#41—42no—
405#41—42no—
406C2×Q8—41no—
407C2×Q8—41no—
408C2×Q8#1171992112yes—
409C2×Q8C2×C4×D5312yes—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—534no—
417C2×D4—1064no—
418C2×D4—1064no—
419C2×D4—1064no—
420C2×D4—1064no—
421C2×D4—1064no—
422C2×D4—1062no—
423C2×D4—4244no—
424C2×D4—4244no—
425C2×D4—2122no—
426C2×D4—2122no—
427C2×D4—2124no—
428C2×D4—2124no—
429C2×D4—2124no—
430C2×D4—2124no—
431C2×D4—2122no—
432C2×D4—2121no—
433C2×D4—2121no—
434C2×D4#1171992112yes—
435C2×D4#1171992112yes—
436C2×D4C2×C4×D5311yescharacteristic
437C2×D4C2×D21212yes—
438C2×D4C2×D21211yescharacteristic
439C2×D4C2×C22⋊D5312yes—
440C2×D4C2×C22⋊D5311yescharacteristic
441C2×D4C23×D5311yescharacteristic
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—1062no—
453C22×C4—2122no—
454C22×C4#1171992111yescharacteristic
455C22×C4#1171992111yescharacteristic
456C22×C4#1171992112yes—
457C22×C4#1171992112yes—
458C22×C4#1171992112yes—
459C22×C4#1171992111yescharacteristic
460C22×C4C2×C4×D5311yescharacteristic
461C22×C4C2×C4×D5312yes—
462C22×C4C2×D21212yes—
463C22×C4C2×C22⋊D5312yes—
464C4⋊C4—44no—
465C4⋊C4—1064no—
466C4⋊C4—2124no—
467C22⋊C4—24no—
468C22⋊C4—42no—
469C22⋊C4—1064no—
470C22⋊C4—1068no—
471C22⋊C4—1068no—
472C22⋊C4—1064no—
473C22⋊C4—2122no—
474C42—44no—
475C42—534no—
476C42—1062no—
477C42—2124no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—1064no—
488C23—4248no—
489C23—4248no—
490C23—4248no—
491C23—2124no—
492C23—2124no—
493C23—2128no—
494C23—2122no—
495C23—2122no—
496C23C2×#1171992111yescharacteristic
497C23C2×#1171992112yes—
498C23C2×#1171992111yescharacteristic
499C23C2×#1171992111yescharacteristic
500C23C22×C22⋊D5311yescharacteristic
501C23C22×D21211yescharacteristic
502Q8—42no—
503Q8—42no—
504D4—41no—
505D4—41no—
506D4—42no—
507D4—41no—
508D4—42no—
509D4—41no—
510D4—42no—
511D4—42no—
512D4—4244no—
513D4—4244no—
514D4—4244no—
515D4—4244no—
516D4—2124no—
517D4—2124no—
518D4—2124no—
519D4—2124no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—44no—
539C2×C4—84no—
540C2×C4—1064no—
541C2×C4—4244no—
542C2×C4—2124no—
543C2×C4—2124no—
544C2×C4#1313313112yes—
545C2×C4#1313313111yescharacteristic
546C2×C4#1313313111yescharacteristic
547C2×C4C2×#1171992111yescharacteristic
548C2×C4C2×#1171992112yes—
549C2×C4C2×#1171992112yes—
550C2×C4C2×#1171992112yes—
551C2×C4C22×C4×D5311yescharacteristic
552C22—22no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—21no—
557C22—21no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—81no—
568C22—81no—
569C22—4244no—
570C22—42416no—
571C22—4244no—
572C22—2124no—
573C22—2124no—
574C22—2124no—
575C22C2×#1313313111yescharacteristic
576C22C2×#1313313111yescharacteristic
577C22C22×#1171992111yescharacteristic
578C4—21no—
579C4—22no—
580C4—22no—
581C4—21no—
582C4—84no—
583C4—4244no—
584C4—2124no—
585C4C2×#1313313112yes—
586C4C2×#1313313112yes—
587C2—21no—
588C2—21no—
589C2—41no—
590C2—41no—
591C2—42no—
592C2—41no—
593C2—41no—
594C2—4244no—
595C2#2027638612yesdirect product factor
596C2C22×#1313313111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC2×C4×C53C23×C4
FrattiniC2×D4C23×D53
FittingC2×C53×#410C2
socleC22×C53C2×#99
prime radicalC2×#16811187C2
cosocleC2×D4×C53C24
hypercenterC2×#410D53
hypocenterC53C2×#2128
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#16810597C2semidirect
3. #20276386C2direct (N × H)
4. C2×#16810415C2semidirect
5. C2×C53×#410C2semidirect
6. C23×D212C4semidirect
9. #16811187C4semidirect
12. C2×#13133206C4semidirect
18. C2×#13133176C4semidirect
20. #16810597C22semidirect
22. C2×#13133132C22semidirect
23. #16810415C22semidirect
27. C53×#410C22semidirect
29. C2×#99×C53C22semidirect
37. C22×D212C2×C4semidirect
45. #13133206C2×C4semidirect
57. #13133176C2×C4semidirect
68. #13133132C23semidirect
90. #99×C53C23semidirect
291. C2×#410D53semidirect
301. C2×C53#2128semidirect
322. #410C2×D53semidirect
326. C2×#99C2×D53semidirect
328. C53C2×#2128semidirect
389. #99C22×D53semidirect
595. C2#20276386direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111872
C2×#168105972
#202763862
C2×#168104152
C2×C53×#4102
C2×#212853