Finite Group Databasebeta

← C2×#20284571 #28251678

Subgroups — C2×#20284571 #28251678

Order 13568.

2094 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811254C211yesmaximal, characteristic, complemented
2C2×#16811104C211yesmaximal, characteristic
3#20284571C2116yesmaximal, direct product factor
4C2×#16810929C212yesmaximal, complemented
5C2×#16810917C212yesmaximal, complemented
6C2×#16810926C212yesmaximal, complemented
7C2×#16810902C212yesmaximal, complemented
8C2×#16810663C211yesmaximal, characteristic, complemented
9C2×#16810649C211yesmaximal, characteristic, complemented
10C2×#16810622C211yesmaximal, characteristic
11C2×#16810620C211yesmaximal, characteristic
12C22×#107×C53C211yesmaximal, characteristic, complemented
13C22×#11719945C2211yescharacteristic
14C22×#11719917C2211yescharacteristic
15#16811254—24no—
16#16811254C2218yescomplemented
17C2×#13133216—21no—
18C2×#13133216C2212yes—
19C2×#13133208—21no—
20C2×#13133208C2212yes—
21C2×#13133196—22no—
22#16811104—24no—
23#16811104C2218yes—
24C2×#13133182C2212yes—
25C2×#13133179C2212yes—
26C2×C4×C22⋊D53—22no—
27C2×#13133176C2211yescharacteristic
28C2×#13133176C2211yescharacteristic
29C2×#13133175—21no—
30C2×C4⋊C4×D53—21no—
31#16810929C22116yescomplemented
32#16810926C22116yescomplemented
33#16810917C22116yescomplemented
34#16810902C22116yescomplemented
35#16810663C2218yescomplemented
36C2×#13133150C2212yes—
37C2×#13133148C2212yes—
38#16810649C2218yescomplemented
39C2×#13133146C2212yes—
40C2×#13133146C2212yes—
41#16810622C2218yes—
42#16810620C2218yes—
43C2×#13133142C2211yescharacteristic
44C2×#13133142C2211yescharacteristic
45C2×#13133140C2212yes—
46C2×#13133139C2211yescharacteristic
47C2×#13133139C2211yescharacteristic
48C2×#13133137C2212yes—
49C23×C8×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C2×#107×C53C2212yes—
53C2×#107×C53C2212yes—
54C2×#107×C53C2218yescomplemented
55C2×#11719945—22no—
56C2×#11719945—24no—
57C2×#11719945Q818yescomplemented
58C2×#11719945C2314yes—
59C22×Dic106—22no—
60C22×Dic106C2311yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C22⋊D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—22no—
65C2×#11719917—24no—
66C2×#11719917C2314yes—
67C22×#88322C2312yes—
68#13133216—28no—
69#13133216—24no—
70#13133216C2318yes—
71#13133208—28no—
72#13133208—24no—
73#13133208C2318yes—
74#13133196—216no—
75#13133182—24no—
76#13133182C2318yes—
77#13133179—24no—
78#13133179C2318yes—
79C4×C22⋊D53—216no—
80#13133176—22no—
81#13133176D414yes—
82#13133176C2314yes—
83#13133176C2314yes—
84#13133175—28no—
85C2×C4⋊Dic53—22no—
86C2×C4⋊Dic53D411yescharacteristic
87C2×C4⋊Dic53D411yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×#11719919—22no—
91C2×#11719919—22no—
92C2×#11719919—22no—
93C4⋊C4×D53—28no—
94C2×C22⋊Dic53—21no—
95C2×C22⋊Dic53D412yes—
96C2×#11719921—22no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53—22no—
99C2×C4×Dic53C2312yes—
100#13133150C2318yes—
101#13133148C2318yes—
102#13133146C2318yes—
103#13133146C2318yes—
104#13133142C2314yes—
105#13133142C2314yes—
106#13133140C2318yes—
107#13133139C2314yes—
108#13133139C2314yes—
109#13133137C2318yes—
110C23×C4×C53—21no—
111C23×C4×C53—21no—
112C23×C4×C53C2311yescharacteristic
113C22×C8×C53—24no—
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2314yes—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—22no—
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53—22no—
121C2×C4⋊C4×C53C2312yes—
122C2×C4⋊C4×C53C2314yes—
123C2×C4⋊C4×C53C2312yes—
124C2×C4⋊C4×C53C2314yes—
125#107×C53C2318yes—
126#107×C53C2318yes—
127C24×C53C2×D411yescharacteristic
128C23×D53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—24no—
132C22×C4×C53—24no—
133C22×C4×C53—82no—
134C22×C4×C53—82no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×Q814yes—
138C22×C4×C53#4114yes—
139C22×C4×C53C2411yescharacteristic
140C22×C4×C53C2412yes—
141C2×C22⋊D53—28no—
142C2×C22⋊D53—24no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53C2×D412yes—
147C22×Dic53C2×D412yes—
148#11719945—216no—
149#11719945—28no—
150#11719945C2×Q8116yescomplemented
151C2×D212—21no—
152C2×D212—22no—
153C2×D212C2×Q814yes—
154C2×D212C2412yes—
155C2×C4×D53—28no—
156C2×C4×D53—24no—
157C2×C4×D53—28no—
158C2×C4×D53—84no—
159C2×Dic106—21no—
160C2×Dic106—22no—
161C2×Dic106—28no—
162C2×Dic106—84no—
163C2×Dic106C2×Q814yes—
164C2×Dic106C2412yes—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—28no—
168C2×C8×C53C2412yes—
169C2×C8×C53C2412yes—
170C4⋊C4×C53—24no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—24no—
174C4⋊C4×C53C2414yes—
175C4⋊C4×C53C2414yes—
176C22⋊Dic53—24no—
177C22⋊Dic53C2×D418yes—
178#11719921—28no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—28no—
182C4⋊Dic53C2×D414yes—
183C4⋊Dic53C2×D414yes—
184C4⋊Dic53C2412yes—
185C4⋊Dic53C2412yes—
186#11719919—28no—
187#11719919—28no—
188#11719919—28no—
189C4×Dic53—24no—
190C4×Dic53—28no—
191C4×Dic53C2×D414yes—
192C4×Dic53C2414yes—
193#11719917—28no—
194C2×#88322—24no—
195C2×#88322C2414yes—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53C22⋊Q814yes—
199C23×C53C22×D412yes—
200C23×C53C22×D411yescharacteristic
201C22×D53—24no—
202C22×D53—22no—
203C22×D53—42no—
204C22×D53—42no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—44no—
208C2×C4×C53—44no—
209C2×C4×C53—84no—
210C2×C4×C53—84no—
211C2×C4×C53C22⋊Q814yes—
212C2×C4×C53C22⋊Q818yes—
213C2×C4×C53C22×D411yescharacteristic
214C2×C4×C53C22×D412yes—
215C2×C4×C53C22×Q812yes—
216C2×C4×C53C22×Q814yes—
217C2×C4×C53C2×#4112yes—
218C2×C4×C53C2×#4114yes—
219C2×C4×C53C2511yescharacteristic
220C22⋊D53—216no—
221C2×Dic53—24no—
222C2×Dic53—24no—
223C2×Dic53—24no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53C22×D414yes—
230C2×Dic53C22×D414yes—
231D212—24no—
232D212—22no—
233D212C22×Q814yes—
234C4×D53—216no—
235C4×D53—88no—
236Dic106—24no—
237Dic106—22no—
238Dic106—88no—
239Dic106C22×Q814yes—
240C8×C53—22no—
241C8×C53—22no—
242#88322—24no—
243C2×#3083—531nomaximal, Sylow
244C22×C53—28no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53C2×C22⋊Q814yes—
248C22×C53C2×C22⋊Q812yes—
249C22×C53#57612yes—
250C22×C53#57812yes—
251C22×C53C23×D411yescharacteristic
252C2×D53—24no—
253C2×D53—48no—
254C2×D53—42no—
255C2×D53—42no—
256C4×C53—44no—
257C4×C53—44no—
258C4×C53C2×C22⋊Q812yes—
259C4×C53C2×C22⋊Q814yes—
260C4×C53C23×Q811yescharacteristic
261C4×C53C22×#4111yescharacteristic
262Dic53—24no—
263Dic53—24no—
264Dic53—24no—
265Dic53—42no—
266Dic53—42no—
267C2×#544—531no—
268C2×#520—531no—
269#3083—5316no—
270C2×#479—532no—
271C2×#477—532no—
272C2×#478—532no—
273C2×#474—532no—
274C2×#428—531no—
275C22×#107D5311yescharacteristic, complemented
276C2×#424—531no—
277C2×#417—531no—
278C2×#416—531no—
279C2×C53—24no—
280C2×C53#308314yescomplemented
281C2×C53C22×C22⋊Q811yescharacteristic
282C2×C53C2×#57611yescharacteristic
283C2×C53C2×#57811yescharacteristic
284D53—44no—
285C22×#41—531no—
286C22×M16—531no—
287C23×C8C2×D5311yescharacteristic
288#544—538no—
289#544—1064no—
290C2×C4⋊Q8—532no—
291C2×C4⋊Q8—1061no—
292C2×#125—532no—
293C2×#125—1061no—
294C2×C22⋊Q8—1062no—
295#520—538no—
296#520—1064no—
297C2×C4×Q8—532no—
298C2×C4×D4—532no—
299C2×C4×D4—1062no—
300C2×#117—531no—
301C2×#117—531no—
302C2×#117—1061no—
303C22×C4⋊C4—1061no—
304C22×C4⋊C4C2×D5311yescharacteristic
305C22×C4⋊C4C2×D5311yescharacteristic
306#479—5316no—
307#478—5316no—
308#477—5316no—
309#474—5316no—
310#428—538no—
311C2×#107—532no—
312C2×#107C2×D5312yes—
313C2×#107C2×D5312yes—
314C2×#107C2×D5318yescomplemented
315C2×#106—532no—
316#424—538no—
317C2×C4⋊C8—532no—
318C2×C4⋊C8—532no—
319#417—538no—
320#416—538no—
321C2×#103—531no—
322C2×#103—531no—
323C2×#103—532no—
324C2×#102—531no—
325C2×#102—531no—
326C2×#102—532no—
327C53C2×#308311yescharacteristic, complemented, Sylow
328C2×#41—538no—
329C2×#41—534no—
330C2×#41—1064no—
331C2×#41—1062no—
332C22×Q8—531no—
333C22×Q8—1062no—
334C22×D4—531no—
335C22×D4—1061no—
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—1062no—
339C23×C4—1061no—
340C23×C4C22×D5311yescharacteristic
341C2×M16—534no—
342C2×M16—1064no—
343C22×C8—24no—
344C22×C8—532no—
345C22×C8C22×D5311yescharacteristic
346C22×C8C22×D5311yescharacteristic
347C22×C8C22×D5314yes—
348C4⋊Q8—538no—
349C4⋊Q8—1064no—
350C4⋊Q8—1068no—
351#125—538no—
352#125—1064no—
353#125—1068no—
354C22⋊Q8—10616no—
355C4×Q8—538no—
356C4×Q8—1064no—
357C4×D4—538no—
358C4×D4—1064no—
359C4×D4—10616no—
360#117—534no—
361#117—534no—
362#117—534no—
363#117—1068no—
364#117—1062no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—1068no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4C22×D5312yes—
379C2×C4⋊C4C22×D5314yes—
380C2×C4⋊C4C22×D5312yes—
381C2×C4⋊C4C22×D5314yes—
382C2×C22⋊C4—532no—
383C2×C22⋊C4—1062no—
384C2×C22⋊C4—1061no—
385C2×C42—532no—
386C2×C42—1062no—
387C2×C42—1061no—
388#107—538no—
389#107C22×D5318yes—
390#107C22×D5318yes—
391#106—538no—
392C4⋊C8—538no—
393C4⋊C8—538no—
394#103—534no—
395#103—534no—
396#103—538no—
397#102—534no—
398#102—534no—
399#102—538no—
400C24—1061no—
401C24D4×D5311yescharacteristic
402#41—5316no—
403#41—10616no—
404#41—1068no—
405C2×Q8—534no—
406C2×Q8—532no—
407C2×Q8—1068no—
408C2×Q8—1061no—
409C2×Q8—1062no—
410C2×Q8—4244no—
411C2×D4—534no—
412C2×D4—532no—
413C2×D4—1068no—
414C2×D4—1064no—
415C2×D4—1061no—
416C2×D4—1062no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—532no—
424C22×C4—532no—
425C22×C4—1068no—
426C22×C4—1061no—
427C22×C4—1068no—
428C22×C4—1064no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—4244no—
432C22×C4#1171994514yes—
433C22×C4D4×D5312yes—
434C22×C4D4×D5311yescharacteristic
435C22×C4Q8×D5314yes—
436C22×C4C23×D5312yes—
437C22×C4C23×D5311yescharacteristic
438M16—1068no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—28no—
442C2×C8—534no—
443C2×C8—1064no—
444C2×C8C23×D5312yes—
445C2×C8C23×D5312yes—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—534no—
451C4⋊C4—534no—
452C4⋊C4—532no—
453C4⋊C4—532no—
454C4⋊C4—1068no—
455C4⋊C4—1068no—
456C4⋊C4—1068no—
457C4⋊C4—1068no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4C23×D5314yes—
461C4⋊C4C23×D5314yes—
462C22⋊C4—538no—
463C22⋊C4—1068no—
464C22⋊C4—1064no—
465C42—534no—
466C42—534no—
467C42—1068no—
468C42—1064no—
469C23—22no—
470C23—22no—
471C23—1064no—
472C23—1062no—
473C23—2122no—
474C23—2122no—
475C23#1313319514yes—
476C23C2×D4×D5312yes—
477C23C2×D4×D5311yescharacteristic
478Q8—534no—
479Q8—1064no—
480Q8—1062no—
481Q8—4248no—
482D4—534no—
483D4—10616no—
484D4—1064no—
485D4—1062no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—534no—
493C2×C4—534no—
494C2×C4—1064no—
495C2×C4—1064no—
496C2×C4—1064no—
497C2×C4—1062no—
498C2×C4—10616no—
499C2×C4—1062no—
500C2×C4—1062no—
501C2×C4—4248no—
502C2×C4—2122no—
503C2×C4—2122no—
504C2×C4#1313319514yes—
505C2×C4#1313319518yes—
506C2×C4C2×D4×D5311yescharacteristic
507C2×C4C2×D4×D5312yes—
508C2×C4C2×Q8×D5312yes—
509C2×C4C2×Q8×D5314yes—
510C2×C4C2×#1171994512yes—
511C2×C4C2×#1171994514yes—
512C2×C4C24×D5311yescharacteristic
513C8—22no—
514C8—22no—
515C8—1064no—
516C22—28no—
517C22—22no—
518C22—22no—
519C22—1064no—
520C22—2128no—
521C22—2122no—
522C22—2122no—
523C22C2×#1313319514yes—
524C22C2×#1313319512yes—
525C22#1681142412yes—
526C22#1681143612yes—
527C22C22×D4×D5311yescharacteristic
528C4—44no—
529C4—44no—
530C4—1064no—
531C4—1064no—
532C4—1064no—
533C4—2122no—
534C4—2122no—
535C4C2×#1313319514yes—
536C4C2×#1313319512yes—
537C4C22×Q8×D5311yescharacteristic
538C4C22×#1171994511yescharacteristic
539C2—24no—
540C2—2124no—
541C2#2028457114yesdirect product factor
542C2C22×#1313319511yescharacteristic
543C2C2×#1681142411yescharacteristic
544C2C2×#1681143611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC22×#107×C53C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
hypercenterC22×#107D53
hypocenterC53C2×#3083
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
1. C2×#16811254C2semidirect
3. #20284571C2direct (N × H)
4. C2×#16810929C2semidirect
5. C2×#16810917C2semidirect
6. C2×#16810926C2semidirect
7. C2×#16810902C2semidirect
8. C2×#16810663C2semidirect
9. C2×#16810649C2semidirect
12. C22×#107×C53C2semidirect
16. #16811254C22semidirect
31. #16810929C22semidirect
32. #16810926C22semidirect
33. #16810917C22semidirect
34. #16810902C22semidirect
35. #16810663C22semidirect
38. #16810649C22semidirect
54. C2×#107×C53C22semidirect
57. C2×#11719945Q8semidirect
150. #11719945C2×Q8semidirect
275. C22×#107D53semidirect
280. C2×C53#3083semidirect
314. C2×#107C2×D53semidirect
327. C53C2×#3083semidirect
541. C2#20284571direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168112542
C2×#168111042
#202845712
C2×#168109292
C2×#168109172
C2×#168109262
C2×#168109022
C2×#168106632
C2×#168106492
C2×#168106222
C2×#168106202
C22×#107×C532
C2×#308353