Finite Group Databasebeta

← C2×#20287578 #28264518

Subgroups — C2×#20287578 #28264518

Order 13568.

1918 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811333C211yesmaximal, characteristic, complemented
2#20287578C2116yesmaximal, direct product factor
3C2×#16811026C211yesmaximal, characteristic, complemented
4C2×#16811022C212yesmaximal, complemented
5C2×#16810923C212yesmaximal, complemented
6C2×#16810920C212yesmaximal, complemented
7C2×#16810913C211yesmaximal, characteristic, complemented
8C2×#16810910C211yesmaximal, characteristic, complemented
9C2×Q8×#88322C211yesmaximal, characteristic, complemented
10C2×C4×#11719922C211yesmaximal, characteristic
11C2×C4×#11719922C212yesmaximal
12C2×C53×#554C211yesmaximal, characteristic, complemented
13C22×#11719922C2211yescharacteristic
14#16811333—24no—
15#16811333C2218yescomplemented
16C2×#13133217C2211yescharacteristic
17C2×#13133209C2212yes—
18C2×#13133195—22no—
19C2×#13133193—21no—
20C2×C4×D212C2211yescharacteristic
21C2×C4×D212C2212yes—
22C2×C4⋊C4×D53—21no—
23#16811026C2218yescomplemented
24#16811022C22116yescomplemented
25#16810923C22116yescomplemented
26#16810920C22116yescomplemented
27#16810913C2218yescomplemented
28#16810910C2218yescomplemented
29Q8×#88322C2218yescomplemented
30C4×#11719922C2218yes—
31C4×#11719922C22116yes—
32C2×#13133150C2211yescharacteristic
33C2×#13133150C2212yes—
34C2×#13133150C2212yes—
35C2×#13133148C2212yes—
36C2×#13133148C2212yes—
37C2×C4⋊#88322C2211yescharacteristic
38C2×C4⋊#88322C2212yes—
39C2×#13133138C2212yes—
40C2×#13133138C2211yescharacteristic
41C2×#13133137C2212yes—
42C2×#13133137C2211yescharacteristic
43C2×C4×#88322C2212yes—
44C2×C4×#88322C2211yescharacteristic
45C53×#554—24no—
46C53×#554C2218yescomplemented
47C2×C4⋊Q8×C53C2211yescharacteristic
48C2×#125×C53C2212yes—
49C2×C22⋊Q8×C53—21no—
50C2×C22⋊Q8×C53—22no—
51C2×C4×Q8×C53C2211yescharacteristic
52C2×C4×D4×C53C2211yescharacteristic
53C2×C4×D4×C53C2212yes—
54C22×C4⋊C4×C53—21no—
55C22×Dic106—21no—
56C22×D212C2311yescharacteristic
57C22×C4×D53—21no—
58C22×C4×D53—22no—
59C2×#11719922Q818yescomplemented
60C2×#11719922C2314yes—
61C22×#88322C2312yes—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133217—22no—
65#13133217C2314yes—
66#13133209—24no—
67#13133209C2318yes—
68#13133195—216no—
69#13133193—28no—
70C4×D212—22no—
71C4×D212—24no—
72C4×D212C2314yes—
73C4×D212C2318yes—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2312yes—
77C2×#11719919—22no—
78C2×#11719919—22no—
79C4⋊C4×D53—28no—
80C2×#11719921—22no—
81C2×#11719921—21no—
82#13133150C2314yes—
83#13133150C2318yes—
84#13133150C2318yes—
85#13133148C2318yes—
86#13133148C2318yes—
87C4⋊#88322C2314yes—
88C4⋊#88322C2318yes—
89#13133138C2318yes—
90#13133138C2314yes—
91#13133137C2318yes—
92#13133137C2314yes—
93C4×#88322C2318yes—
94C4×#88322C2314yes—
95C22×Q8×C53—21no—
96C22×Q8×C53D411yescharacteristic
97C22×D4×C53C2311yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53—22no—
100C4⋊Q8×C53—22no—
101C4⋊Q8×C53C2314yes—
102#125×C53—24no—
103#125×C53C2318yes—
104C22⋊Q8×C53—216no—
105C22⋊Q8×C53—28no—
106C4×Q8×C53D414yes—
107C4×Q8×C53C2314yes—
108C4×D4×C53—22no—
109C4×D4×C53—24no—
110C4×D4×C53C2314yes—
111C4×D4×C53C2318yes—
112C2×C4⋊C4×C53—28no—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—22no—
116C2×C4⋊C4×C53D411yescharacteristic
117C2×C4⋊C4×C53D412yes—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2312yes—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—22no—
122C2×C42×C53C2312yes—
123C2×C42×C53C2311yescharacteristic
124C24×C53—21no—
125C23×D53—21no—
126C2×Q8×C53—24no—
127C2×Q8×C53—24no—
128C2×Q8×C53—82no—
129C2×Q8×C53C2×D414yes—
130C2×D4×C53—21no—
131C2×D4×C53—22no—
132C2×D4×C53C2×Q814yes—
133C2×D4×C53C2412yes—
134C22×C4×C53—21no—
135C22×C4×C53—22no—
136C22×C4×C53—24no—
137C22×C4×C53—28no—
138C22×C4×C53—44no—
139C22×C4×C53—84no—
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—22no—
147C2×D212—21no—
148C2×D212—22no—
149C2×D212C2×Q814yes—
150C2×D212C2412yes—
151C2×C4×D53—24no—
152C2×C4×D53—28no—
153C2×C4×D53—44no—
154C2×C4×D53—84no—
155C2×Dic106—24no—
156C2×Dic106—82no—
157C4⋊C4×C53—28no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—28no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53C2×D414yes—
163C4⋊C4×C53C2×D418yes—
164C4⋊C4×C53C2412yes—
165C4⋊C4×C53C2414yes—
166C22⋊C4×C53—28no—
167C22⋊C4×C53—24no—
168C42×C53C2×D414yes—
169C42×C53C2×D412yes—
170C42×C53C2414yes—
171C42×C53C2412yes—
172#11719922C2×Q8116yescomplemented
173#11719921—28no—
174#11719921—24no—
175C4⋊Dic53—24no—
176C4⋊Dic53—21no—
177C4⋊Dic53—22no—
178C4⋊Dic53C2412yes—
179C4⋊Dic53C2414yes—
180#11719919—28no—
181#11719919—28no—
182C2×#88322—24no—
183C2×#88322—22no—
184C2×#88322C2×Q814yes—
185C2×#88322C2414yes—
186C2×#88322C2412yes—
187C2×#88322C2412yes—
188C23×C53—24no—
189C23×C53—22no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53C22×D411yescharacteristic
193C22×D53—24no—
194C22×D53—22no—
195C22×D53—42no—
196C22×D53—42no—
197Q8×C53—28no—
198Q8×C53—84no—
199D4×C53—24no—
200D4×C53—22no—
201D4×C53C22×Q814yes—
202C2×C4×C53—24no—
203C2×C4×C53—22no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—22no—
207C2×C4×C53—42no—
208C2×C4×C53—44no—
209C2×C4×C53—48no—
210C2×C4×C53—88no—
211C2×C4×C53#13514yes—
212C2×C4×C53C22×D412yes—
213C2×C4×C53C22×D414yes—
214C2×C4×C53C22×D414yes—
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C22×D412yes—
217C2×C4×C53C22×Q812yes—
218C2×C4×C532−1+412yes—
219C2×C4×C53C2511yescharacteristic
220C2×Dic53—24no—
221C2×Dic53—22no—
222C2×Dic53—42no—
223C2×Dic53—44no—
224D212—24no—
225D212—22no—
226D212C22×Q814yes—
227C4×D53—48no—
228C4×D53—88no—
229Dic106—84no—
230#88322—24no—
231#88322—22no—
232#88322C22×Q814yes—
233C2×#3380—531nomaximal, Sylow
234C22×C53—24no—
235C22×C53—48no—
236C22×C53—42no—
237C22×C53—42no—
238C22×C53D4×Q812yes—
239C22×C53C2×#13512yes—
240C22×C53#57612yes—
241C22×C53C23×D411yescharacteristic
242C2×D53—24no—
243C2×D53—48no—
244C2×D53—42no—
245C2×D53—42no—
246C4×C53—24no—
247C4×C53—22no—
248C4×C53—24no—
249C4×C53—42no—
250C4×C53—44no—
251C4×C53D4×Q812yes—
252C4×C53C2×#13514yes—
253C4×C53C23×Q811yescharacteristic
254C4×C53C2×2−1+411yescharacteristic
255Dic53—42no—
256Dic53—44no—
257C2×#554—531no—
258C2×#554D5311yescharacteristic, complemented
259#3380—5316no—
260C2×#500—531no—
261C2×#499—532no—
262C2×#478—532no—
263C2×#478—532no—
264C2×#476—531no—
265C2×#476—531no—
266C2×C8×Q8—531no—
267C2×C4×D8—532no—
268C2×C4×D8—531no—
269C2×C53—44no—
270C2×C53#338014yescomplemented
271C2×C53C2×D4×Q811yescharacteristic
272C2×C53C22×#13511yescharacteristic
273C2×C53C2×#57611yescharacteristic
274D53—44no—
275C22×D8—531no—
276#554—24no—
277#554—538no—
278#554—1064no—
279#554C2×D5318yescomplemented
280C2×C4⋊Q8—531no—
281C2×C4⋊Q8C2×D5311yescharacteristic
282C2×#125—532no—
283C2×#125C2×D5312yes—
284C2×C22⋊Q8—21no—
285C2×C22⋊Q8—22no—
286C2×C22⋊Q8—1061no—
287C2×C22⋊Q8—1062no—
288C2×C4×Q8C2×D5311yescharacteristic
289C2×C4×D4—532no—
290C2×C4×D4—531no—
291C2×C4×D4C2×D5312yes—
292C2×C4×D4C2×D5311yescharacteristic
293C22×C4⋊C4—21no—
294C22×C4⋊C4—1061no—
295#500—538no—
296#499—5316no—
297#478—5316no—
298#478—5316no—
299#476—538no—
300#476—538no—
301C8×Q8—538no—
302C4×D8—5316no—
303C4×D8—538no—
304C2×#107—532no—
305C2×#107—531no—
306C2×#107—532no—
307C2×#106—532no—
308C2×#106—532no—
309C2×C4⋊C8—532no—
310C2×C4⋊C8—531no—
311C2×#102—531no—
312C2×#102—532no—
313C2×#102—531no—
314C2×#102—532no—
315C2×C4×C8—532no—
316C2×C4×C8—531no—
317C53C2×#338011yescharacteristic, complemented, Sylow
318C22×Q8—21no—
319C22×Q8—1061no—
320C22×Q8C22⋊D5311yescharacteristic
321C22×D4—531no—
322C22×D4C22×D5311yescharacteristic
323C23×C4—21no—
324C23×C4—22no—
325C23×C4—1062no—
326C23×C4—1061no—
327C2×D8—538no—
328C2×D8—534no—
329C22×C8—531no—
330C22×C8—532no—
331C22×C8—531no—
332C4⋊Q8—22no—
333C4⋊Q8—534no—
334C4⋊Q8—1062no—
335C4⋊Q8C22×D5314yes—
336#125—24no—
337#125—538no—
338#125—1064no—
339#125C22×D5318yes—
340C22⋊Q8—28no—
341C22⋊Q8—216no—
342C22⋊Q8—1068no—
343C22⋊Q8—10616no—
344C4×Q8C22⋊D5314yes—
345C4×Q8C22×D5314yes—
346C4×D4—24no—
347C4×D4—22no—
348C4×D4—534no—
349C4×D4—538no—
350C4×D4—1064no—
351C4×D4—1062no—
352C4×D4C22×D5314yes—
353C4×D4C22×D5318yes—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—28no—
358C2×C4⋊C4—531no—
359C2×C4⋊C4—532no—
360C2×C4⋊C4—1062no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1062no—
363C2×C4⋊C4—1068no—
364C2×C4⋊C4C22⋊D5311yescharacteristic
365C2×C4⋊C4C22⋊D5312yes—
366C2×C4⋊C4C22×D5311yescharacteristic
367C2×C4⋊C4C22×D5312yes—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1062no—
372C2×C42C22×D5312yes—
373C2×C42C22×D5311yescharacteristic
374#107—534no—
375#107—538no—
376#107—538no—
377#106—538no—
378#106—538no—
379C4⋊C8—534no—
380C4⋊C8—538no—
381#102—538no—
382#102—534no—
383#102—538no—
384#102—534no—
385C4×C8—538no—
386C4×C8—534no—
387C24—21no—
388C24—1061no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—82no—
392C2×Q8—1064no—
393C2×Q8—4242no—
394C2×Q8C2×C22⋊D5314yes—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—534no—
398C2×D4—532no—
399C2×D4—1061no—
400C2×D4—1062no—
401C2×D4Q8×D5314yes—
402C2×D4C23×D5312yes—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—44no—
408C22×C4—84no—
409C22×C4—1061no—
410C22×C4—1062no—
411C22×C4—1068no—
412C22×C4—1064no—
413C22×C4—4244no—
414C22×C4—2124no—
415C22×C4C2×C22⋊D5312yes—
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C2×C22⋊D5312yes—
418C22×C4C2×C22⋊D5311yescharacteristic
419C22×C4C23×D5311yescharacteristic
420D8—5316no—
421C2×C8—534no—
422C2×C8—532no—
423C2×C8—534no—
424C2×C8—532no—
425C2×C8—1064no—
426C2×C8—1062no—
427C4⋊C4—28no—
428C4⋊C4—24no—
429C4⋊C4—28no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—532no—
433C4⋊C4—534no—
434C4⋊C4—1061no—
435C4⋊C4—1062no—
436C4⋊C4—1068no—
437C4⋊C4—1064no—
438C4⋊C4—1068no—
439C4⋊C4C2×C22⋊D5314yes—
440C4⋊C4C2×C22⋊D5318yes—
441C4⋊C4C23×D5312yes—
442C4⋊C4C23×D5314yes—
443C22⋊C4—24no—
444C22⋊C4—28no—
445C22⋊C4—1064no—
446C22⋊C4—1068no—
447C42C2×C22⋊D5314yes—
448C42C2×C22⋊D5312yes—
449C42C23×D5314yes—
450C42C23×D5312yes—
451C23—24no—
452C23—22no—
453C23—42no—
454C23—42no—
455C23—1064no—
456C23—1062no—
457C23—2122no—
458C23—2122no—
459C23C22×C22⋊D5311yescharacteristic
460Q8—28no—
461Q8—84no—
462Q8—4244no—
463D4—24no—
464D4—22no—
465D4—534no—
466D4—1064no—
467D4—1062no—
468D4C2×Q8×D5314yes—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—48no—
477C2×C4—88no—
478C2×C4—1062no—
479C2×C4—1064no—
480C2×C4—4248no—
481C2×C4—2122no—
482C2×C4—2124no—
483C2×C4—2128no—
484C2×C4#1313323714yes—
485C2×C4C22×C22⋊D5314yes—
486C2×C4C22×C22⋊D5312yes—
487C2×C4C22×C22⋊D5314yes—
488C2×C4C22×C22⋊D5311yescharacteristic
489C2×C4C22×C22⋊D5312yes—
490C2×C4C2×Q8×D5312yes—
491C2×C4#1313326512yes—
492C2×C4C24×D5311yescharacteristic
493C8—534no—
494C8—1064no—
495C8—1062no—
496C22—24no—
497C22—48no—
498C22—42no—
499C22—42no—
500C22—1064no—
501C22—2128no—
502C22—2122no—
503C22—2122no—
504C22Q8×C22⋊D5312yes—
505C22C2×#1313323712yes—
506C22#1681142612yes—
507C22C23×C22⋊D5311yescharacteristic
508C4—24no—
509C4—24no—
510C4—22no—
511C4—42no—
512C4—44no—
513C4—2122no—
514C4—2124no—
515C4Q8×C22⋊D5312yes—
516C4C2×#1313323714yes—
517C4C22×Q8×D5311yescharacteristic
518C4C2×#1313326511yescharacteristic
519C2—44no—
520C2—2124no—
521C2#2028757814yesdirect product factor
522C2C2×Q8×C22⋊D5311yescharacteristic
523C2C22×#1313323711yescharacteristic
524C2C2×#1681142611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#554C2
socleC23×C53C22×D4
prime radicalC22×#11719922C22
hypercenterC2×#554D53
hypocenterC53C2×#3380
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16811333C2semidirect
2. #20287578C2direct (N × H)
3. C2×#16811026C2semidirect
4. C2×#16811022C2semidirect
5. C2×#16810923C2semidirect
6. C2×#16810920C2semidirect
7. C2×#16810913C2semidirect
8. C2×#16810910C2semidirect
9. C2×Q8×#88322C2semidirect
12. C2×C53×#554C2semidirect
15. #16811333C22semidirect
23. #16811026C22semidirect
24. #16811022C22semidirect
25. #16810923C22semidirect
26. #16810920C22semidirect
27. #16810913C22semidirect
28. #16810910C22semidirect
29. Q8×#88322C22semidirect
46. C53×#554C22semidirect
59. C2×#11719922Q8semidirect
172. #11719922C2×Q8semidirect
258. C2×#554D53semidirect
270. C2×C53#3380semidirect
279. #554C2×D53semidirect
317. C53C2×#3380semidirect
521. C2#20287578direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168113332
#202875782
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