Finite Group Databasebeta

← C2×#20285930 #28256245

Subgroups — C2×#20285930 #28256245

Order 13568.

1758 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811179C211yesmaximal, characteristic
2C2×#16811178C211yesmaximal, characteristic
3#20285930C2116yesmaximal, direct product factor
4C2×#16811029C211yesmaximal, characteristic, complemented
5C2×#16810993C211yesmaximal, characteristic, complemented
6C2×#16810992C211yesmaximal, characteristic, complemented
7C2×#16810967C212yesmaximal
8C2×#16810966C212yesmaximal
9C2×#16810917C212yesmaximal, complemented
10C2×#16810907C212yesmaximal, complemented
11C2×#16810678C211yesmaximal, characteristic
12C2×C53×#501C211yesmaximal, characteristic, complemented
13#16811179—24no—
14#16811179C2218yes—
15#16811178—24no—
16#16811178C2218yes—
17C2×#13133217C2211yescharacteristic
18C2×#13133216C2211yescharacteristic
19C2×#13133216C2211yescharacteristic
20C2×#13133209C2211yescharacteristic
21C2×#13133208—21no—
22C2×#13133197C2212yes—
23C2×#13133195—21no—
24C2×#13133195—21no—
25C2×#13133193—21no—
26C2×Q8×Dic53—21no—
27C2×#13133182—21no—
28C2×#13133182—21no—
29C2×#13133182C2212yes—
30C2×#13133175—21no—
31C2×#13133175C2211yescharacteristic
32C2×#13133175C2211yescharacteristic
33C2×C42×D53C2211yescharacteristic
34#16811029C2218yescomplemented
35#16810993C2218yescomplemented
36#16810992C2218yescomplemented
37#16810967C22116yes—
38#16810966C22116yes—
39#16810917C22116yescomplemented
40#16810907C22116yescomplemented
41#16810678C2218yes—
42C2×#13133150C2212yes—
43C2×#13133148C2212yes—
44C2×#13133146C2212yes—
45C2×#13133142C2212yes—
46C2×#13133142C2212yes—
47C2×#13133140C2212yes—
48C2×#13133140C2212yes—
49C2×#13133129C2212yes—
50C2×#13133128C2211yescharacteristic
51C2×C4⋊Q8×C53C2211yescharacteristic
52C2×#125×C53C2211yescharacteristic
53C53×#501C2218yescomplemented
54C2×#107×C53C2212yes—
55C2×#106×C53C2212yes—
56C2×#97×C53C2211yescharacteristic
57C22×Dic106—21no—
58C22×Dic106—21no—
59C22×Dic106C2312yes—
60C22×C4×D53D411yescharacteristic
61C22×C4×D53D411yescharacteristic
62C22×C4×D53C2311yescharacteristic
63C22×#88322C2312yes—
64#13133217—22no—
65#13133217C2314yes—
66#13133216—22no—
67#13133216—22no—
68#13133216C2314yes—
69#13133216C2314yes—
70#13133209—22no—
71#13133209C2314yes—
72#13133208—28no—
73#13133197—24no—
74#13133197C2318yes—
75#13133195—28no—
76#13133195—28no—
77#13133193—28no—
78Q8×Dic53—28no—
79#13133182—28no—
80#13133182—28no—
81#13133182—24no—
82#13133182C2318yes—
83#13133175—28no—
84#13133175—22no—
85#13133175—22no—
86#13133175C2314yes—
87#13133175C2314yes—
88C2×C4⋊Dic53—21no—
89C2×C4⋊Dic53—21no—
90C2×C4⋊Dic53—22no—
91C2×C4⋊Dic53C2311yescharacteristic
92C2×C4⋊Dic53C2311yescharacteristic
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—22no—
98C2×#11719921—22no—
99C2×#11719921—21no—
100C2×#11719921—21no—
101C2×C4×Dic53—22no—
102C2×C4×Dic53—21no—
103C2×C4×Dic53—21no—
104C2×C4×Dic53D411yescharacteristic
105C2×C4×Dic53D411yescharacteristic
106C2×C4×Dic53C2311yescharacteristic
107C42×D53D414yes—
108C42×D53C2314yes—
109#13133150C2318yes—
110#13133148C2318yes—
111#13133146C2318yes—
112#13133142C2318yes—
113#13133142C2318yes—
114#13133140C2318yes—
115#13133140C2318yes—
116#13133129C2318yes—
117#13133128C2314yes—
118C22×Q8×C53—21no—
119C22×C8×C53C2312yes—
120C4⋊Q8×C53—22no—
121C4⋊Q8×C53C2314yes—
122#125×C53—22no—
123#125×C53C2314yes—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2312yes—
130C2×C42×C53C2311yescharacteristic
131#107×C53C2318yes—
132#106×C53C2318yes—
133#97×C53C2314yes—
134C23×D53C2×D411yescharacteristic
135C2×Q8×C53—24no—
136C2×Q8×C53—44no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C2×C4×D53—24no—
150C2×C4×D53—24no—
151C2×C4×D53—42no—
152C2×C4×D53C2×D412yes—
153C2×C4×D53C2×D414yes—
154C2×C4×D53C2×D414yes—
155C2×C4×D53C2412yes—
156C2×Dic106—28no—
157C2×Dic106—22no—
158C2×Dic106—24no—
159C2×Dic106—24no—
160C2×Dic106—44no—
161C2×Dic106—44no—
162C2×Dic106C2414yes—
163C2×C8×C53—24no—
164C2×C8×C53C2414yes—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—24no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53C2412yes—
172C4⋊C4×C53C2414yes—
173C4⋊C4×C53C2412yes—
174C42×C53C2×D412yes—
175C42×C53C2412yes—
176#11719921—28no—
177#11719921—24no—
178#11719921—24no—
179C4⋊Dic53—24no—
180C4⋊Dic53—24no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—28no—
184C4⋊Dic53C2412yes—
185C4⋊Dic53C2412yes—
186#11719919—24no—
187#11719919—24no—
188#11719919—24no—
189#11719919—24no—
190#11719919—28no—
191C4×Dic53—28no—
192C4×Dic53—24no—
193C4×Dic53—24no—
194C4×Dic53C2×D412yes—
195C4×Dic53C2×D414yes—
196C4×Dic53C2×D414yes—
197C4×Dic53C2412yes—
198C2×#88322—24no—
199C2×#88322C2414yes—
200C23×C53C22×D411yescharacteristic
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—22no—
204C22×D53C22×D412yes—
205Q8×C53—48no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—24no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—44no—
214C2×C4×C53C22×D412yes—
215C2×C4×C53C22×D412yes—
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C532−1+414yes—
218C2×C4×C53C2511yescharacteristic
219C2×Dic53—22no—
220C2×Dic53—22no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—24no—
224C2×Dic53—44no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53#13714yes—
228C2×Dic53C22×D412yes—
229C2×Dic53C22×D412yes—
230C2×Dic53C22×D412yes—
231C4×D53—24no—
232C4×D53—24no—
233C4×D53—28no—
234C4×D53—44no—
235Dic106—28no—
236Dic106—48no—
237Dic106—44no—
238Dic106—48no—
239C8×C53—24no—
240#88322—24no—
241C2×#3237—531nomaximal, Sylow
242C22×C53#53612yes—
243C22×C53C2×#13712yes—
244C22×C53#57512yes—
245C22×C53C23×D411yescharacteristic
246C2×D53—22no—
247C2×D53—22no—
248C2×D53—22no—
249C2×D53—44no—
250C4×C53—22no—
251C4×C53—22no—
252C4×C53—42no—
253C4×C53—42no—
254C4×C53—44no—
255C4×C53#53612yes—
256C4×C53C2×2−1+412yes—
257Dic53—22no—
258Dic53—22no—
259Dic53—44no—
260Dic53—42no—
261Dic53—42no—
262Dic53C2×#13714yes—
263C2×#533—531no—
264C2×#533—531no—
265#3237—5316no—
266C2×#501—531no—
267C2×#501D5311yescharacteristic, complemented
268C2×#491—531no—
269C2×#491—531no—
270C2×#485—532no—
271C2×#485—532no—
272C2×#477—532no—
273C2×#475—532no—
274C2×#432—531no—
275C2×C53#323714yescomplemented
276C2×C53C2×#53611yescharacteristic
277C2×C53C22×#13711yescharacteristic
278C2×C53C2×#57511yescharacteristic
279D53—42no—
280#533—538no—
281#533—538no—
282#533—1064no—
283#533—1064no—
284C2×C4⋊Q8—531no—
285C2×C4⋊Q8—531no—
286C2×C4⋊Q8—531no—
287C2×C4⋊Q8C2×D5311yescharacteristic
288C2×#125—531no—
289C2×#125—1061no—
290C2×#125C2×D5311yescharacteristic
291C2×C22⋊Q8—532no—
292C2×C22⋊Q8—1061no—
293C2×C22⋊Q8—1061no—
294C2×C22⋊Q8—1061no—
295C2×C4×Q8—532no—
296C2×C4×Q8—1061no—
297C2×C4×Q8—1061no—
298C2×C4×Q8—1061no—
299C2×#117—531no—
300C2×#117—531no—
301C2×#117—1061no—
302C22×C42—531no—
303#501—538no—
304#501C2×D5318yescomplemented
305#491—538no—
306#491—538no—
307#485—5316no—
308#485—5316no—
309#477—5316no—
310#475—5316no—
311#432—538no—
312C2×#107—532no—
313C2×#107C2×D5312yes—
314C2×#106—532no—
315C2×#106C2×D5312yes—
316C2×C4⋊C8—532no—
317C2×#103—532no—
318C2×#103—532no—
319C2×#103—532no—
320C2×#103—532no—
321C2×C22⋊C8—532no—
322C2×#97—531no—
323C2×#97C2×D5311yescharacteristic
324C53C2×#323711yescharacteristic, complemented, Sylow
325C22×Q8—21no—
326C22×Q8—532no—
327C22×Q8—1061no—
328C22×Q8—1061no—
329C23×C4—531no—
330C23×C4—531no—
331C23×C4—531no—
332C22×C8—532no—
333C22×C8C22×D5312yes—
334C4⋊Q8—22no—
335C4⋊Q8—534no—
336C4⋊Q8—534no—
337C4⋊Q8—534no—
338C4⋊Q8—1062no—
339C4⋊Q8—1062no—
340C4⋊Q8—1062no—
341C4⋊Q8C22×D5314yes—
342#125—22no—
343#125—534no—
344#125—1068no—
345#125—1062no—
346#125C22×D5314yes—
347C22⋊Q8—538no—
348C22⋊Q8—1064no—
349C22⋊Q8—1068no—
350C22⋊Q8—1068no—
351C22⋊Q8—1068no—
352C4×Q8—538no—
353C4×Q8—1064no—
354C4×Q8—1068no—
355C4×Q8—1068no—
356C4×Q8—1068no—
357#117—534no—
358#117—534no—
359#117—1062no—
360#117—1062no—
361#117—1068no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1062no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4C22×D5311yescharacteristic
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C4⋊C4C22×D5312yes—
378C2×C22⋊C4—1061no—
379C2×C22⋊C4—1061no—
380C2×C22⋊C4—1062no—
381C2×C42—531no—
382C2×C42—531no—
383C2×C42—531no—
384C2×C42—534no—
385C2×C42—534no—
386C2×C42—1061no—
387C2×C42—1061no—
388C2×C42—1062no—
389C2×C42C22×D5311yescharacteristic
390#107—538no—
391#107C22×D5318yes—
392#106—538no—
393#106C22×D5318yes—
394C4⋊C8—538no—
395#103—538no—
396#103—538no—
397#103—538no—
398#103—538no—
399C22⋊C8—538no—
400#97—534no—
401#97C22×D5314yes—
402C24—531no—
403C2×Q8—24no—
404C2×Q8—44no—
405C2×Q8—534no—
406C2×Q8—1064no—
407C2×Q8—1064no—
408C2×Q8—1062no—
409C2×Q8—1068no—
410C2×Q8—2124no—
411C2×Q8—2124no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—531no—
416C22×C4—531no—
417C22×C4—531no—
418C22×C4—534no—
419C22×C4—532no—
420C22×C4—532no—
421C22×C4—534no—
422C22×C4—1064no—
423C22×C4—1064no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—1062no—
427C22×C4—2122no—
428C22×C4D4×D5311yescharacteristic
429C22×C4D4×D5311yescharacteristic
430C22×C4C23×D5311yescharacteristic
431C2×C8—24no—
432C2×C8—534no—
433C2×C8—1064no—
434C2×C8C23×D5314yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—532no—
442C4⋊C4—532no—
443C4⋊C4—1068no—
444C4⋊C4—1068no—
445C4⋊C4—1064no—
446C4⋊C4—1064no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C4⋊C4—1064no—
450C4⋊C4—1064no—
451C4⋊C4—1064no—
452C4⋊C4—1064no—
453C4⋊C4C23×D5312yes—
454C4⋊C4C23×D5312yes—
455C4⋊C4C23×D5314yes—
456C22⋊C4—1064no—
457C22⋊C4—1064no—
458C22⋊C4—1068no—
459C42—534no—
460C42—534no—
461C42—532no—
462C42—532no—
463C42—1064no—
464C42—1064no—
465C42—1068no—
466C42D4×D5312yes—
467C42C23×D5312yes—
468C23—532no—
469C23—1062no—
470C23—1062no—
471C23—1062no—
472C23C2×D4×D5311yescharacteristic
473Q8—48no—
474Q8—1068no—
475Q8—2128no—
476Q8—2128no—
477Q8—2124no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—534no—
487C2×C4—532no—
488C2×C4—532no—
489C2×C4—532no—
490C2×C4—1068no—
491C2×C4—1062no—
492C2×C4—1062no—
493C2×C4—1064no—
494C2×C4—1064no—
495C2×C4—1062no—
496C2×C4—1062no—
497C2×C4—1064no—
498C2×C4—2124no—
499C2×C4—2122no—
500C2×C4—2122no—
501C2×C4—2124no—
502C2×C4C2×D4×D5312yes—
503C2×C4C2×D4×D5312yes—
504C2×C4C2×D4×D5311yescharacteristic
505C2×C4#1313326514yes—
506C2×C4C24×D5311yescharacteristic
507C8—24no—
508C8—1064no—
509C22—1062no—
510C22—1062no—
511C22—1062no—
512C22—2124no—
513C22#1681119812yes—
514C22#137×D5312yes—
515C22#1681141912yes—
516C22C22×D4×D5311yescharacteristic
517C4—22no—
518C4—22no—
519C4—42no—
520C4—44no—
521C4—42no—
522C4—534no—
523C4—1062no—
524C4—1062no—
525C4—2124no—
526C4—2122no—
527C4—2122no—
528C4#1681119812yes—
529C4C2×#1313326512yes—
530C2—2122no—
531C2#2028593014yesdirect product factor
532C2C2×#1681119811yescharacteristic
533C2C2×#137×D5311yescharacteristic
534C2C2×#1681141911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#501C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenterC2×#501D53
hypocenterC53C2×#3237
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
3. #20285930C2direct (N × H)
4. C2×#16811029C2semidirect
5. C2×#16810993C2semidirect
6. C2×#16810992C2semidirect
9. C2×#16810917C2semidirect
10. C2×#16810907C2semidirect
12. C2×C53×#501C2semidirect
34. #16811029C22semidirect
35. #16810993C22semidirect
36. #16810992C22semidirect
39. #16810917C22semidirect
40. #16810907C22semidirect
53. C53×#501C22semidirect
267. C2×#501D53semidirect
275. C2×C53#3237semidirect
304. #501C2×D53semidirect
324. C53C2×#3237semidirect
531. C2#20285930direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111792
C2×#168111782
#202859302
C2×#168110292
C2×#168109932
C2×#168109922
C2×#168109672
C2×#168109662
C2×#168109172
C2×#168109072
C2×#168106782
C2×C53×#5012
C2×#323753