Finite Group Databasebeta

← C2×#20285906 #28256198

Subgroups — C2×#20285906 #28256198

Order 13568.

2050 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811174C211yesmaximal, characteristic
2C2×#125×D53C211yesmaximal, characteristic, complemented
3#20285906C2116yesmaximal, direct product factor
4C2×#16811029C211yesmaximal, characteristic, complemented
5C2×#16810989C211yesmaximal, characteristic, complemented
6C2×#16810988C211yesmaximal, characteristic, complemented
7C2×#16810943C212yesmaximal, complemented
8C2×#16810938C212yesmaximal, complemented
9C2×#16810926C212yesmaximal, complemented
10C2×#16810922C212yesmaximal, complemented
11C2×#16810678C211yesmaximal, characteristic, complemented
12C2×C53×#501C211yesmaximal, characteristic, complemented
13C22×#11719949—21no—
14#16811174—24no—
15#16811174C2218yes—
16C2×#13133214C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133208C2211yescharacteristic
19C2×#13133208C2211yescharacteristic
20C2×#13133207—21no—
21C2×#13133207—21no—
22#125×D53—24no—
23#125×D53C2218yescomplemented
24C2×#13133206—21no—
25C2×#13133192—21no—
26C2×#13133192C2212yes—
27C2×#13133179—21no—
28C2×#13133179C2212yes—
29C2×C4⋊C4×D53—21no—
30C2×C4⋊C4×D53—21no—
31C2×C4⋊C4×D53C2211yescharacteristic
32C2×C4⋊C4×D53C2211yescharacteristic
33C2×C42×D53C2211yescharacteristic
34#16811029C2218yescomplemented
35#16810989C2218yescomplemented
36#16810988C2218yescomplemented
37#16810943C22116yescomplemented
38#16810938C22116yescomplemented
39#16810926C22116yescomplemented
40#16810922C22116yescomplemented
41#16810678C2218yescomplemented
42C2×#13133150C2212yes—
43C2×#13133148C2212yes—
44C2×#13133146C2212yes—
45C2×#13133139C2212yes—
46C2×#13133139C2212yes—
47C2×#13133137C2212yes—
48C2×#13133137C2212yes—
49C2×#13133129C2212yes—
50C2×#13133128C2211yescharacteristic
51C2×C4⋊Q8×C53C2211yescharacteristic
52C2×#125×C53C2211yescharacteristic
53C53×#501C2218yescomplemented
54C2×#107×C53C2212yes—
55C2×#106×C53C2212yes—
56C2×#97×C53C2211yescharacteristic
57C2×#11719949—28no—
58C2×#11719949—48no—
59C22×D212—22no—
60C22×D212—21no—
61C22×D212—21no—
62C22×D212C2312yes—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C22×C4×D53—22no—
66C22×C4×D53D411yescharacteristic
67C22×C4×D53D411yescharacteristic
68C22×C4×D53C2311yescharacteristic
69C22×#88322C2312yes—
70#13133214—22no—
71#13133214C2314yes—
72#13133209—22no—
73#13133209C2314yes—
74#13133208—22no—
75#13133208—22no—
76#13133208C2314yes—
77#13133208C2314yes—
78#13133207—28no—
79#13133207—28no—
80#13133206—28no—
81#13133192—28no—
82#13133192—24no—
83#13133192C2318yes—
84#13133179—28no—
85#13133179—24no—
86#13133179C2318yes—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53—21no—
89C2×C4⋊Dic53C2311yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C4⋊C4×D53—28no—
98C4⋊C4×D53—28no—
99C4⋊C4×D53—22no—
100C4⋊C4×D53—22no—
101C4⋊C4×D53C2314yes—
102C4⋊C4×D53C2314yes—
103C2×#11719921—22no—
104C2×#11719921—22no—
105C2×C4×Dic53D411yescharacteristic
106C2×C4×Dic53D411yescharacteristic
107C2×C4×Dic53C2311yescharacteristic
108C42×D53D414yes—
109C42×D53C2314yes—
110#13133150C2318yes—
111#13133148C2318yes—
112#13133146C2318yes—
113#13133139C2318yes—
114#13133139C2318yes—
115#13133137C2318yes—
116#13133137C2318yes—
117#13133129C2318yes—
118#13133128C2314yes—
119C22×Q8×C53—21no—
120C22×C8×C53C2312yes—
121C4⋊Q8×C53—22no—
122C4⋊Q8×C53C2314yes—
123#125×C53—22no—
124#125×C53C2314yes—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—21no—
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2312yes—
131C2×C42×C53C2311yescharacteristic
132#107×C53C2318yes—
133#106×C53C2318yes—
134#97×C53C2314yes—
135C23×D53—22no—
136C23×D53C2×D411yescharacteristic
137C2×Q8×C53—24no—
138C2×Q8×C53—44no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—22no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150#11719949—432no—
151C2×D212—28no—
152C2×D212—28no—
153C2×D212—22no—
154C2×D212—24no—
155C2×D212—24no—
156C2×D212—44no—
157C2×D212—44no—
158C2×D212—48no—
159C2×D212C2414yes—
160C2×C4×D53—24no—
161C2×C4×D53—24no—
162C2×C4×D53—24no—
163C2×C4×D53—24no—
164C2×C4×D53—28no—
165C2×C4×D53—42no—
166C2×C4×D53—48no—
167C2×C4×D53—44no—
168C2×C4×D53—44no—
169C2×C4×D53C2×D412yes—
170C2×C4×D53C2×D414yes—
171C2×C4×D53C2×D414yes—
172C2×C4×D53C2412yes—
173C2×C8×C53—24no—
174C2×C8×C53C2414yes—
175C4⋊C4×C53—24no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—24no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53C2412yes—
182C4⋊C4×C53C2414yes—
183C4⋊C4×C53C2412yes—
184C42×C53C2×D412yes—
185C42×C53C2412yes—
186#11719921—28no—
187#11719921—28no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—24no—
191C4⋊Dic53—24no—
192C4⋊Dic53C2412yes—
193C4⋊Dic53C2412yes—
194#11719919—24no—
195#11719919—24no—
196#11719919—24no—
197#11719919—24no—
198#11719919—24no—
199#11719919—24no—
200C4×Dic53C2×D412yes—
201C4×Dic53C2×D414yes—
202C4×Dic53C2×D414yes—
203C4×Dic53C2412yes—
204C2×#88322—24no—
205C2×#88322C2414yes—
206C23×C53C22×D411yescharacteristic
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—24no—
210C22×D53—44no—
211C22×D53—44no—
212C22×D53—44no—
213C22×D53AGL(1,ℤ/8)14yes—
214C22×D53C22×D412yes—
215Q8×C53—48no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—24no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—44no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53C22×D412yes—
225C2×C4×C53C22×D412yes—
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C532−1+414yes—
228C2×C4×C53C2511yescharacteristic
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53C22×D412yes—
237C2×Dic53C22×D412yes—
238C2×Dic53C22×D412yes—
239D212—28no—
240D212—48no—
241D212—48no—
242D212—416no—
243D212—44no—
244C4×D53—28no—
245C4×D53—24no—
246C4×D53—24no—
247C4×D53—44no—
248C4×D53—416no—
249C4×D53—48no—
250C4×D53—48no—
251C8×C53—24no—
252#88322—24no—
253C2×#3234—531nomaximal, Sylow
254C22×C53#53612yes—
255C22×C53C2×AGL(1,ℤ/8)12yes—
256C22×C53#57512yes—
257C22×C53C23×D411yescharacteristic
258C2×D53—22no—
259C2×D53—28no—
260C2×D53—22no—
261C2×D53—44no—
262C2×D53—44no—
263C2×D53—44no—
264C2×D53—88no—
265C2×D53C2×AGL(1,ℤ/8)14yes—
266C4×C53—22no—
267C4×C53—22no—
268C4×C53—44no—
269C4×C53—42no—
270C4×C53—42no—
271C4×C53#53612yes—
272C4×C53C2×2−1+412yes—
273Dic53—22no—
274Dic53—22no—
275Dic53—22no—
276Dic53—42no—
277Dic53—42no—
278C2×#532—531no—
279C22×#125—531no—
280#3234—5316no—
281C2×#501—531no—
282C2×#501D5311yescharacteristic, complemented
283C2×#490—531no—
284C2×#490—531no—
285C2×#481—532no—
286C2×#480—532no—
287C2×#478—532no—
288C2×#478—532no—
289C2×#432—531no—
290C2×C53#323414yescomplemented
291C2×C53C2×#53611yescharacteristic
292C2×C53C22×AGL(1,ℤ/8)11yescharacteristic
293C2×C53C2×#57511yescharacteristic
294D53—24no—
295D53—84no—
296C22×#41—1061no—
297#532—538no—
298#532—1064no—
299C2×C4⋊Q8C2×D5311yescharacteristic
300C2×#127—531no—
301C2×#125—538no—
302C2×#125—531no—
303C2×#125—531no—
304C2×#125—531no—
305C2×#125—1061no—
306C2×#125—1061no—
307C2×#125—1064no—
308C2×#125C2×D5311yescharacteristic
309C2×#124—1061no—
310C2×#121—532no—
311C2×#121—1061no—
312C2×C4×D4—532no—
313C2×C4×D4—1061no—
314C22×C4⋊C4—531no—
315C22×C4⋊C4—531no—
316C22×C4⋊C4—1061no—
317C22×C4⋊C4—1061no—
318C22×C42—531no—
319#501—538no—
320#501C2×D5318yescomplemented
321#490—538no—
322#490—538no—
323#481—5316no—
324#480—5316no—
325#478—5316no—
326#478—5316no—
327#432—538no—
328C2×#107—532no—
329C2×#107C2×D5312yes—
330C2×#106—532no—
331C2×#106C2×D5312yes—
332C2×C4⋊C8—532no—
333C2×#102—532no—
334C2×#102—532no—
335C2×#102—532no—
336C2×#102—532no—
337C2×C22⋊C8—532no—
338C2×#97—531no—
339C2×#97C2×D5311yescharacteristic
340C53C2×#323411yescharacteristic, complemented, Sylow
341C2×#41—1068no—
342C2×#41—2128no—
343C22×Q8—21no—
344C22×D4—532no—
345C22×D4—1062no—
346C22×D4—1061no—
347C22×D4—1061no—
348C23×C4—531no—
349C23×C4—531no—
350C23×C4—531no—
351C23×C4—1062no—
352C23×C4—1061no—
353C23×C4—1061no—
354C22×C8—532no—
355C22×C8C22×D5312yes—
356C4⋊Q8—22no—
357C4⋊Q8C22×D5314yes—
358#127—534no—
359#127—1062no—
360#125—22no—
361#125—534no—
362#125—534no—
363#125—534no—
364#125—1068no—
365#125—1068no—
366#125—1062no—
367#125—1062no—
368#125—1062no—
369#125C22×D5314yes—
370#124—1068no—
371#121—538no—
372#121—1064no—
373#121—1068no—
374C4×D4—538no—
375C4×D4—1064no—
376C4×D4—1068no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—534no—
383C2×C4⋊C4—534no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1068no—
387C2×C4⋊C4—1068no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5312yes—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1062no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—534no—
405C2×C42—534no—
406C2×C42C22×D5311yescharacteristic
407#107—538no—
408#107C22×D5318yes—
409#106—538no—
410#106C22×D5318yes—
411C4⋊C8—538no—
412#102—538no—
413#102—538no—
414#102—538no—
415#102—538no—
416C22⋊C8—538no—
417#97—534no—
418#97C22×D5314yes—
419C24—531no—
420C24—1062no—
421#41—21232no—
422C2×Q8—24no—
423C2×Q8—44no—
424C2×D4—534no—
425C2×D4—1068no—
426C2×D4—1064no—
427C2×D4—1062no—
428C2×D4—1064no—
429C2×D4—1068no—
430C2×D4—2124no—
431C2×D4—2124no—
432C2×D4—2128no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—532no—
440C22×C4—534no—
441C22×C4—532no—
442C22×C4—534no—
443C22×C4—1068no—
444C22×C4—1064no—
445C22×C4—1064no—
446C22×C4—1064no—
447C22×C4—1064no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—2122no—
451C22×C4—2128no—
452C22×C4—2124no—
453C22×C4—2124no—
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5311yescharacteristic
456C22×C4C23×D5311yescharacteristic
457C2×C8—24no—
458C2×C8—534no—
459C2×C8—1064no—
460C2×C8C23×D5314yes—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—532no—
468C4⋊C4—532no—
469C4⋊C4—1064no—
470C4⋊C4—1064no—
471C4⋊C4—1064no—
472C4⋊C4—1064no—
473C4⋊C4—1064no—
474C4⋊C4—1064no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1064no—
478C4⋊C4—1064no—
479C4⋊C4C23×D5312yes—
480C4⋊C4C23×D5312yes—
481C4⋊C4C23×D5314yes—
482C22⋊C4—1068no—
483C22⋊C4—1068no—
484C42—534no—
485C42—532no—
486C42—534no—
487C42—532no—
488C42D4×D5312yes—
489C42C23×D5312yes—
490C23—534no—
491C23—532no—
492C23—1062no—
493C23—1062no—
494C23—1064no—
495C23—2124no—
496C23—2124no—
497C23—2124no—
498C23C2×D4×D5311yescharacteristic
499Q8—48no—
500D4—1068no—
501D4—2128no—
502D4—2128no—
503D4—21216no—
504D4—2124no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—532no—
514C2×C4—532no—
515C2×C4—532no—
516C2×C4—1068no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1064no—
521C2×C4—1064no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—2124no—
525C2×C4—21216no—
526C2×C4—2128no—
527C2×C4—2128no—
528C2×C4—2122no—
529C2×C4—2122no—
530C2×C4C2×D4×D5312yes—
531C2×C4C2×D4×D5312yes—
532C2×C4C2×D4×D5311yescharacteristic
533C2×C4#1313326514yes—
534C2×C4C24×D5311yescharacteristic
535C8—24no—
536C8—1064no—
537C22—534no—
538C22—1062no—
539C22—1062no—
540C22—1068no—
541C22—4248no—
542C22—2124no—
543C22—2124no—
544C22—2124no—
545C22#1681119812yes—
546C22AGL(1,ℤ/8)×D5312yes—
547C22#1681142212yes—
548C22C22×D4×D5311yescharacteristic
549C4—22no—
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—1062no—
555C4—1062no—
556C4—1062no—
557C4—2122no—
558C4—2122no—
559C4#1681119812yes—
560C4C2×#1313326512yes—
561C2—1064no—
562C2—4244no—
563C2#2028590614yesdirect product factor
564C2C2×#1681119811yescharacteristic
565C2C2×AGL(1,ℤ/8)×D5311yescharacteristic
566C2C2×#1681142211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#501C2
socleC23×C53C22×D4
prime radicalC2×#16811174C2
hypercenterC2×#501D53
hypocenterC53C2×#3234
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
2. C2×#125×D53C2semidirect
3. #20285906C2direct (N × H)
4. C2×#16811029C2semidirect
5. C2×#16810989C2semidirect
6. C2×#16810988C2semidirect
7. C2×#16810943C2semidirect
8. C2×#16810938C2semidirect
9. C2×#16810926C2semidirect
10. C2×#16810922C2semidirect
11. C2×#16810678C2semidirect
12. C2×C53×#501C2semidirect
23. #125×D53C22semidirect
34. #16811029C22semidirect
35. #16810989C22semidirect
36. #16810988C22semidirect
37. #16810943C22semidirect
38. #16810938C22semidirect
39. #16810926C22semidirect
40. #16810922C22semidirect
41. #16810678C22semidirect
53. C53×#501C22semidirect
282. C2×#501D53semidirect
290. C2×C53#3234semidirect
320. #501C2×D53semidirect
340. C53C2×#3234semidirect
563. C2#20285906direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111742
C2×#125×D532
#202859062
C2×#168110292
C2×#168109892
C2×#168109882
C2×#168109432
C2×#168109382
C2×#168109262
C2×#168109222
C2×#168106782
C2×C53×#5012
C2×#323453