Finite Group Databasebeta

← C2×#20254723 #27089081

Subgroups — C2×#20254723 #27089081

Order 12032.

1930 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689678C211yesmaximal, characteristic, complemented
2C2×#15689660C211yesmaximal, characteristic
3#20254723C2116yesmaximal, direct product factor
4C2×#15689432C212yesmaximal, complemented
5C2×#15689408C211yesmaximal, characteristic, complemented
6C2×#15689404C211yesmaximal, characteristic, complemented
7C2×#15689361C212yesmaximal, complemented
8C2×#15689355C212yesmaximal
9C2×#15689347C212yesmaximal, complemented
10C2×#15689325C211yesmaximal, characteristic, complemented
11C2×C4⋊C8×D47C211yesmaximal, characteristic, complemented
12C2×C47×#462C211yesmaximal, characteristic, complemented
13C22×#10625336C2212yes—
14C22×C8×D47C2212yes—
15#15689678—24no—
16#15689678C2218yescomplemented
17C2×#13094024C2211yescharacteristic
18#15689660—24no—
19#15689660C2218yes—
20C2×#13094019—21no—
21C2×#13094015—21no—
22C2×#13094015C2211yescharacteristic
23C2×#13094013C2211yescharacteristic
24C2×#13094009—21no—
25C2×#13094004—21no—
26C2×#13094000C2212yes—
27C2×#13094000C2211yescharacteristic
28C2×#13093999C2211yescharacteristic
29C2×Q8×Dic47C2212yes—
30C2×#13093989—21no—
31C2×C4×D188—21no—
32C2×C4×D188C2211yescharacteristic
33C2×#13093982—21no—
34C2×#13093981—21no—
35C2×C42×D47C2211yescharacteristic
36#15689432C22116yescomplemented
37#15689408C2218yescomplemented
38#15689404C2218yescomplemented
39#15689361C22116yescomplemented
40#15689355C22116yes—
41#15689347C22116yescomplemented
42#15689325C2218yescomplemented
43C2×#13093958C2212yes—
44C2×#13093955C2212yes—
45C2×C4⋊(C47⋊C8)C2211yescharacteristic
46C2×Dic47⋊C8C2212yes—
47C4⋊C8×D47C2218yescomplemented
48C2×#13093948C2212yes—
49C2×#13093948C2212yes—
50C2×#13093946C2212yes—
51C22×Q16×C47C2212yes—
52C2×C4⋊Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C47×#462C2218yescomplemented
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×#103×C47C2212yes—
57C22×Dic94—21no—
58C22×D188—21no—
59C22×D188C2311yescharacteristic
60C22×C4×D47—21no—
61C22×C4×D47D411yescharacteristic
62C22×C4×D47D411yescharacteristic
63C22×C4×D47C2311yescharacteristic
64C2×#10625336—44no—
65C2×#10625336C2318yes—
66C22×C47⋊C8C2312yes—
67C2×C8×D47—44no—
68C2×C8×D47C2318yes—
69#13094024—22no—
70#13094024C2314yes—
71#13094019—28no—
72#13094015—28no—
73#13094015—22no—
74#13094015C2314yes—
75#13094013—22no—
76#13094013C2314yes—
77#13094009—28no—
78#13094004—28no—
79#13094000—22no—
80#13094000—24no—
81#13094000C2318yes—
82#13094000C2314yes—
83#13093999—22no—
84#13093999C2314yes—
85Q8×Dic47—24no—
86Q8×Dic47C2318yes—
87#13093989—28no—
88C4×D188—28no—
89C4×D188—22no—
90C4×D188C2314yes—
91#13093982—28no—
92#13093981—28no—
93C2×C4⋊Dic47—21no—
94C2×C4⋊Dic47—21no—
95C2×C4⋊Dic47—22no—
96C2×C4⋊Dic47C2311yescharacteristic
97C2×C4⋊Dic47C2312yes—
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4—21no—
100C2×Dic47⋊C4—22no—
101C2×(C2×D47)⋊C4—22no—
102C2×(C2×D47)⋊C4—21no—
103C2×(C2×D47)⋊C4—21no—
104C2×(C2×D47)⋊C4—21no—
105C2×(C2×D47)⋊C4—21no—
106C2×(C2×D47)⋊C4—21no—
107C2×C4×Dic47—22no—
108C2×C4×Dic47D411yescharacteristic
109C2×C4×Dic47D411yescharacteristic
110C2×C4×Dic47C2311yescharacteristic
111C42×D47D414yes—
112C42×D47C2314yes—
113#13093958C2318yes—
114#13093955C2318yes—
115C4⋊(C47⋊C8)C2314yes—
116Dic47⋊C8C2318yes—
117#13093948C2318yes—
118#13093948C2318yes—
119#13093946C2318yes—
120C22×Q8×C47C2311yescharacteristic
121C22×Q8×C47C2312yes—
122C2×Q16×C47—44no—
123C2×Q16×C47C2318yes—
124C22×C8×C47C2312yes—
125C4⋊Q8×C47—22no—
126C4⋊Q8×C47C2314yes—
127C4×Q8×C47—22no—
128C4×Q8×C47C2314yes—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47—21no—
131C2×C4⋊C4×C47—21no—
132C2×C4⋊C4×C47C2311yescharacteristic
133C2×C42×C47—21no—
134C2×C42×C47C2311yescharacteristic
135C4⋊C8×C47C2314yes—
136#103×C47C2318yes—
137C23×D47—21no—
138C23×D47C2×D411yescharacteristic
139C2×Q8×C47—21no—
140C2×Q8×C47—22no—
141C2×Q8×C47—22no—
142C2×Q8×C47—24no—
143C2×Q8×C47—22no—
144C2×Q8×C47—24no—
145C2×Q8×C47C2414yes—
146C2×Q8×C47C2412yes—
147C22×C4×C47—21no—
148C22×C4×C47—21no—
149C22×C4×C47—22no—
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2411yescharacteristic
153C22×Dic47—22no—
154C22×Dic47—21no—
155C22×Dic47C2×D411yescharacteristic
156C22×Dic47C2×D411yescharacteristic
157C22×Dic47C2×D411yescharacteristic
158C2×D188—22no—
159C2×D188—22no—
160C2×D188—24no—
161C2×D188—21no—
162C2×D188—44no—
163C2×D188C2412yes—
164C2×C4×D47—24no—
165C2×C4×D47—24no—
166C2×C4×D47—22no—
167C2×C4×D47—22no—
168C2×C4×D47—24no—
169C2×C4×D47—44no—
170C2×C4×D47C2×D412yes—
171C2×C4×D47C2×D414yes—
172C2×C4×D47C2×D414yes—
173C2×C4×D47C2412yes—
174C2×Dic94—24no—
175C2×Dic94—44no—
176Q16×C47—48no—
177C2×C8×C47—24no—
178C2×C8×C47C2414yes—
179C4⋊C4×C47—24no—
180C4⋊C4×C47—24no—
181C4⋊C4×C47—24no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47C2412yes—
184C42×C47—24no—
185C42×C47C2×D412yes—
186C42×C47C2412yes—
187#10625336—48no—
188(C2×D47)⋊C4—28no—
189(C2×D47)⋊C4—24no—
190(C2×D47)⋊C4—24no—
191(C2×D47)⋊C4—24no—
192(C2×D47)⋊C4—24no—
193(C2×D47)⋊C4—24no—
194C4⋊Dic47—24no—
195C4⋊Dic47—21no—
196C4⋊Dic47—22no—
197C4⋊Dic47—24no—
198C4⋊Dic47—28no—
199C4⋊Dic47C2414yes—
200C4⋊Dic47C2412yes—
201Dic47⋊C4—24no—
202Dic47⋊C4—24no—
203Dic47⋊C4—28no—
204C4×Dic47—28no—
205C4×Dic47C2×D412yes—
206C4×Dic47C2×D414yes—
207C4×Dic47C2×D414yes—
208C4×Dic47C2412yes—
209C2×C47⋊C8—24no—
210C2×C47⋊C8C2414yes—
211C8×D47—48no—
212C23×C47C22×D411yescharacteristic
213C22×D47—22no—
214C22×D47—22no—
215C22×D47—22no—
216C22×D47—42no—
217C22×D47—42no—
218C22×D47—42no—
219C22×D47#13714yes—
220C22×D47C22×D412yes—
221Q8×C47—24no—
222Q8×C47—22no—
223Q8×C47—24no—
224Q8×C47—22no—
225Q8×C47—42no—
226Q8×C47—44no—
227C2×C4×C47—22no—
228C2×C4×C47—24no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—42no—
232C2×C4×C47—44no—
233C2×C4×C47—42no—
234C2×C4×C47#13514yes—
235C2×C4×C47C22×D412yes—
236C2×C4×C47C22×D411yescharacteristic
237C2×C4×C47C22×D412yes—
238C2×C4×C47H(5,𝔽2)12yes—
239C2×C4×C472−1+412yes—
240C2×C4×C47C2511yescharacteristic
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—24no—
245C2×Dic47—22no—
246C2×Dic47—44no—
247C2×Dic47—42no—
248C2×Dic47C22×D412yes—
249C2×Dic47C22×D412yes—
250C2×Dic47C22×D412yes—
251D188—22no—
252D188—22no—
253D188—48no—
254D188—42no—
255C4×D47—28no—
256C4×D47—24no—
257C4×D47—28no—
258C4×D47—22no—
259C4×D47—22no—
260C4×D47—48no—
261Dic94—48no—
262C8×C47—24no—
263C47⋊C8—24no—
264C2×#3248—471nomaximal, Sylow
265C22×C47#53712yes—
266C22×C47C2×#13512yes—
267C22×C47C2×#13712yes—
268C22×C47C23×D411yescharacteristic
269C2×D47—22no—
270C2×D47—28no—
271C2×D47—22no—
272C2×D47—42no—
273C2×D47—42no—
274C2×D47—42no—
275C2×D47—84no—
276C2×D47C2×#13714yes—
277C4×C47—22no—
278C4×C47—44no—
279C4×C47—42no—
280C4×C47—42no—
281C4×C47#53712yes—
282C4×C47C2×#13514yes—
283C4×C47C2×H(5,𝔽2)11yescharacteristic
284C4×C47C2×2−1+411yescharacteristic
285Dic47—22no—
286Dic47—22no—
287Dic47—22no—
288Dic47—44no—
289Dic47—42no—
290C2×#533—471no—
291C2×#529—471no—
292#3248—4716no—
293C2×#479—472no—
294C2×#475—471no—
295C2×#474—471no—
296C2×#467—472no—
297C2×#465—472no—
298C2×#464—472no—
299C2×#462D4711yescharacteristic, complemented
300C2×#461—471no—
301C22×C4⋊C8—471no—
302C2×C47#324814yescomplemented
303C2×C47C2×#53711yescharacteristic
304C2×C47C22×#13511yescharacteristic
305C2×C47C22×#13711yescharacteristic
306D47—24no—
307D47—82no—
308C22×Q16C2×D4712yes—
309C22×SD16—472no—
310C23×C8—472no—
311#533—478no—
312#533—944no—
313C2×C4⋊Q8—471no—
314C2×C4⋊Q8C2×D4711yescharacteristic
315#529—478no—
316#529—944no—
317C2×#126—941no—
318C2×#125—471no—
319C2×#125—941no—
320C2×#124—471no—
321C2×#123—941no—
322C2×C22⋊Q8—471no—
323C2×C22⋊Q8—472no—
324C2×C22⋊Q8—941no—
325C2×#121—471no—
326C2×C4×Q8—472no—
327C2×C4×Q8—941no—
328C2×C4×Q8C2×D4711yescharacteristic
329C2×C4×D4—471no—
330C2×C4×D4—941no—
331C2×#117—941no—
332C2×#117—941no—
333C22×C42—471no—
334#479—4716no—
335#475—478no—
336#474—478no—
337#467—4716no—
338#465—4716no—
339#464—4716no—
340#462C2×D4718yescomplemented
341#461—478no—
342C2×#107—472no—
343C2×#106—472no—
344C2×C4⋊C8—472no—
345C2×C4⋊C8—471no—
346C2×C4⋊C8—478no—
347C2×C4⋊C8C2×D4711yescharacteristic
348C2×#103—472no—
349C2×#103—472no—
350C2×#103C2×D4712yes—
351C2×#102—472no—
352C47C2×#324811yescharacteristic, complemented, Sylow
353C22×Q8—941no—
354C22×Q8C22×D4712yes—
355C22×Q8C22×D4711yescharacteristic
356C22×D4—471no—
357C22×D4—941no—
358C23×C4—471no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—941no—
362C2×Q16—44no—
363C2×Q16C22×D4718yes—
364C2×SD16—478no—
365C2×SD16—1884no—
366C22×C8—472no—
367C22×C8—478no—
368C22×C8—1884no—
369C22×C8C22×D4712yes—
370C4⋊Q8—22no—
371C4⋊Q8—474no—
372C4⋊Q8—942no—
373C4⋊Q8C22×D4714yes—
374#126—948no—
375#125—474no—
376#125—942no—
377#125—948no—
378#124—474no—
379#124—942no—
380#123—948no—
381C22⋊Q8—474no—
382C22⋊Q8—478no—
383C22⋊Q8—948no—
384C22⋊Q8—942no—
385C22⋊Q8—944no—
386#121—474no—
387#121—942no—
388C4×Q8—22no—
389C4×Q8—478no—
390C4×Q8—948no—
391C4×Q8—944no—
392C4×Q8C22×D4714yes—
393C4×D4—474no—
394C4×D4—948no—
395C4×D4—942no—
396#117—948no—
397#117—948no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4—472no—
403C2×C4⋊C4—941no—
404C2×C4⋊C4—941no—
405C2×C4⋊C4—941no—
406C2×C4⋊C4—942no—
407C2×C4⋊C4—942no—
408C2×C4⋊C4—941no—
409C2×C4⋊C4C22×D4711yescharacteristic
410C2×C22⋊C4—941no—
411C2×C22⋊C4—941no—
412C2×C22⋊C4—941no—
413C2×C22⋊C4—942no—
414C2×C22⋊C4—941no—
415C2×C22⋊C4—941no—
416C2×C42—21no—
417C2×C42—471no—
418C2×C42—471no—
419C2×C42—471no—
420C2×C42—474no—
421C2×C42—474no—
422C2×C42—942no—
423C2×C42C22×D4711yescharacteristic
424#107—478no—
425#106—478no—
426C4⋊C8—478no—
427C4⋊C8—474no—
428C4⋊C8C22×D4714yes—
429#103—478no—
430#103—478no—
431#103C22×D4718yes—
432#102—478no—
433C24—471no—
434C24—941no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—944no—
442C2×Q8—1884no—
443C2×Q8C23×D4712yes—
444C2×Q8C23×D4714yes—
445C2×D4—472no—
446C2×D4—944no—
447C2×D4—941no—
448C2×D4—942no—
449C2×D4—942no—
450C2×D4—1884no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—474no—
458C22×C4—474no—
459C22×C4—472no—
460C22×C4—472no—
461C22×C4—944no—
462C22×C4—944no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—941no—
467C22×C4—944no—
468C22×C4—1884no—
469C22×C4D4×D4711yescharacteristic
470C22×C4D4×D4711yescharacteristic
471C22×C4C23×D4711yescharacteristic
472Q16—48no—
473SD16—1888no—
474C2×C8—24no—
475C2×C8—474no—
476C2×C8—944no—
477C2×C8—1888no—
478C2×C8C23×D4714yes—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—474no—
484C4⋊C4—472no—
485C4⋊C4—944no—
486C4⋊C4—944no—
487C4⋊C4—948no—
488C4⋊C4—944no—
489C4⋊C4—948no—
490C4⋊C4—941no—
491C4⋊C4—942no—
492C4⋊C4—944no—
493C4⋊C4C23×D4712yes—
494C22⋊C4—944no—
495C22⋊C4—944no—
496C22⋊C4—944no—
497C22⋊C4—948no—
498C22⋊C4—944no—
499C22⋊C4—944no—
500C42—24no—
501C42—474no—
502C42—472no—
503C42—474no—
504C42—472no—
505C42—948no—
506C42D4×D4712yes—
507C42C23×D4712yes—
508C23—474no—
509C23—472no—
510C23—942no—
511C23—942no—
512C23—942no—
513C23—1882no—
514C23—1882no—
515C23—1882no—
516C23C2×D4×D4711yescharacteristic
517Q8—24no—
518Q8—22no—
519Q8—22no—
520Q8—24no—
521Q8—42no—
522Q8—44no—
523Q8—1888no—
524D4—942no—
525D4—942no—
526D4—1888no—
527D4—1882no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—472no—
536C2×C4—472no—
537C2×C4—472no—
538C2×C4—948no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—942no—
542C2×C4—948no—
543C2×C4—942no—
544C2×C4—942no—
545C2×C4—944no—
546C2×C4—944no—
547C2×C4—942no—
548C2×C4—1882no—
549C2×C4—1884no—
550C2×C4—1888no—
551C2×C4#1309404714yes—
552C2×C4C2×D4×D4712yes—
553C2×C4C2×D4×D4712yes—
554C2×C4C2×D4×D4711yescharacteristic
555C2×C4#1309406912yes—
556C2×C4#1309407112yes—
557C2×C4C24×D4711yescharacteristic
558C8—24no—
559C8—944no—
560C22—474no—
561C22—948no—
562C22—942no—
563C22—942no—
564C22—3764no—
565C22—1882no—
566C22—1882no—
567C22—1882no—
568C22#1568970212yes—
569C22C2×#1309404712yes—
570C22#137×D4712yes—
571C22C22×D4×D4711yescharacteristic
572C4—22no—
573C4—42no—
574C4—42no—
575C4—44no—
576C4—942no—
577C4—942no—
578C4—942no—
579C4—1882no—
580C4—1884no—
581C4#1568970212yes—
582C4C2×#1309404714yes—
583C4C2×#1309406911yescharacteristic
584C4C2×#1309407111yescharacteristic
585C2—944no—
586C2—3762no—
587C2#2025472314yesdirect product factor
588C2C2×#1568970211yescharacteristic
589C2C22×#1309404711yescharacteristic
590C2C2×#137×D4711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#462C2
socleC23×C47C22×D4
prime radicalC2×#13093999C22
hypercenterC2×#462D47
hypocenterC47C2×#3248
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
1. C2×#15689678C2semidirect
3. #20254723C2direct (N × H)
4. C2×#15689432C2semidirect
5. C2×#15689408C2semidirect
6. C2×#15689404C2semidirect
7. C2×#15689361C2semidirect
9. C2×#15689347C2semidirect
10. C2×#15689325C2semidirect
11. C2×C4⋊C8×D47C2semidirect
12. C2×C47×#462C2semidirect
16. #15689678C22semidirect
36. #15689432C22semidirect
37. #15689408C22semidirect
38. #15689404C22semidirect
39. #15689361C22semidirect
41. #15689347C22semidirect
42. #15689325C22semidirect
47. C4⋊C8×D47C22semidirect
54. C47×#462C22semidirect
299. C2×#462D47semidirect
302. C2×C47#3248semidirect
340. #462C2×D47semidirect
352. C47C2×#3248semidirect
587. C2#20254723direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896782
C2×#156896602
#202547232
C2×#156894322
C2×#156894082
C2×#156894042
C2×#156893612
C2×#156893552
C2×#156893472
C2×#156893252
C2×C4⋊C8×D472
C2×C47×#4622
C2×#324847