Finite Group Databasebeta

← C2×#11748166 #14423547

Subgroups — C2×#11748166 #14423547

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6769C211yesmaximal, characteristic
2C22×#6706C211yesmaximal, characteristic, complemented
3C2×#91365C211yesmaximal, characteristic, complemented
4C22×#6662C211yesmaximal, characteristic, complemented
5C2×C5×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6C2×C3×#3948C211yesmaximal, characteristic, complemented
7C2×#90968C211yesmaximal, characteristic, complemented
8#11748166C218yesmaximal, direct product factor
9C2×#65022—31nomaximal, Hall
10C22×D60C2211yescharacteristic
11C22×Dic30C2211yescharacteristic
12C22×C4×C5×D3C2211yescharacteristic
13C22×C3×Dic10C2211yescharacteristic
14C22×#1203—21no—
15C22×#1200—21no—
16C2×#6769—24no—
17C2×#6769—22no—
18C2×#6769—24no—
19C2×#6769C2214yes—
20C22×C3×C5×C8C2211yescharacteristic
21C2×#6706—24no—
22C2×#6706C2214yescomplemented
23C22×C5×C3⋊C8C2211yescharacteristic
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×#6668—21no—
26C2×(C2×D3)⋊Dic5—21no—
27#91365—22no—
28#91365C2214yescomplemented
29C2×#6662—24no—
30C2×#6662C2214yescomplemented
31C5×(C2×D3)⋊C8C2214yescomplemented
32C3×#3948C2214yescomplemented
33#90968C2214yescomplemented
34C2×#4808—51nomaximal, Hall
35C22×#709—31no—
36C2×#4070—31no—
37C22×Dic20—31no—
38C22×#677—31no—
39C2×C5×C22⋊C8—31no—
40C2×#3953—31no—
41C2×#3948D311yescharacteristic, complemented
42#65022—38no—
43C23×D15—21no—
44C23×C5×D3D411yescharacteristic
45C22×C3×C4×C5C2311yescharacteristic
46C22×Dic15—21no—
47C2×D60—21no—
48C2×D60—22no—
49C2×D60D414yes—
50C2×D60C2312yes—
51C2×Dic30—22no—
52C2×Dic30—22no—
53C2×Dic30—21no—
54C2×Dic30C2312yes—
55C22×C5×Dic3D411yescharacteristic
56C2×C4×C5×D3—22no—
57C2×C4×C5×D3—22no—
58C2×C4×C5×D3D412yes—
59C2×C4×C5×D3C2312yes—
60C22×C3×Dic5—21no—
61C22×C3×Dic5—21no—
62C2×C3×Dic10—21no—
63C2×C3×Dic10—22no—
64C2×C3×Dic10D414yes—
65C2×C3×Dic10C2312yes—
66C2×#1203—28no—
67C2×#1203—24no—
68C2×#1200—28no—
69C2×#1200—24no—
70#6769—216no—
71#6769—28no—
72#6769—28no—
73C2×C3×C5×C8—22no—
74C2×C3×C5×C8C2312yes—
75#6706—28no—
76C2×C5×C3⋊C8—22no—
77C2×C5×C3⋊C8C2312yes—
78C3×C4⋊Dic5—21no—
79C3×C4⋊Dic5C2312yes—
80#6668—24no—
81(C2×D3)⋊Dic5—24no—
82#6662—28no—
83C22×#239—51no—
84C22×#216—51no—
85C22×#204—51no—
86C2×#897—51no—
87C2×C3×#103—51no—
88C2×(C2×D3)⋊C8D511yescharacteristic, complemented
89C2×#809—51no—
90#4808—58no—
91C23×C4×C5—31no—
92C22×C22⋊D5—61no—
93C2×#709—34no—
94C2×#709—64no—
95C2×#709—62no—
96C2×#709—64no—
97C22×D20—31no—
98C22×C4×D5—61no—
99C22×Dic10—31no—
100C22×Dic10C2×D311yescharacteristic
101C22×C5×C8—31no—
102C22×C5×C8C2×D311yescharacteristic
103C2×C22⋊Dic5—61no—
104C2×C4⋊Dic5C2×D311yescharacteristic
105#4070—34no—
106#4070—62no—
107C2×#683—61no—
108C2×Dic20—34no—
109C2×Dic20—64no—
110C2×#677—34no—
111C2×#677—64no—
112C5×C22⋊C8—34no—
113#3953—34no—
114#3948C2×D314yescomplemented
115C2×#452—151noSylow
116C23×C3×C5C2×D411yescharacteristic
117C22×D15—24no—
118C22×D15—22no—
119C22×D15—42no—
120C22×D15—42no—
121C22×C5×D3—22no—
122C22×C5×D3—22no—
123C22×C5×D3—22no—
124C22×C5×D3C2×D412yes—
125C2×C3×C4×C5C2×D411yescharacteristic
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5C2×D412yes—
128C2×C3×C4×C5C2411yescharacteristic
129C2×Dic15—22no—
130C2×Dic15—42no—
131D60—24no—
132D60—22no—
133D60C2×D414yes—
134Dic30—22no—
135Dic30—22no—
136Dic30—42no—
137C2×C5×Dic3—22no—
138C2×C5×Dic3C2×D412yes—
139C4×C5×D3—24no—
140C4×C5×D3—22no—
141C4×C5×D3—22no—
142C2×C3×Dic5—24no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—22no—
145C2×C3×Dic5—42no—
146C3×Dic10—24no—
147C3×Dic10—22no—
148C3×Dic10C2×D414yes—
149#1203—216no—
150#1200—216no—
151C3×C5×C8—22no—
152C5×C3⋊C8—22no—
153C22×C3×Q8—51no—
154C2×#239—54no—
155C2×#239—104no—
156C2×#239—104no—
157C2×#239—102no—
158C22×D12—51no—
159C22×D12—101no—
160C22×C4×D3—101no—
161C22×C4×D3C2×D511yescharacteristic
162C22×Dic6—51no—
163C22×C3×C8C2×D511yescharacteristic
164C2×C3×C4⋊C4—51no—
165C2×#216—54no—
166C2×#216—104no—
167C2×(C2×D3)⋊C4—101no—
168C2×C4⋊Dic3—101no—
169C22×C3⋊C8C2×D511yescharacteristic
170C2×#204—54no—
171C2×#204—104no—
172#897—54no—
173#897—102no—
174C3×#103—54no—
175(C2×D3)⋊C8C2×D514yescomplemented
176#809—54no—
177C24×C5—31no—
178C23×D5—61no—
179C22×C4×C5—31no—
180C22×C4×C5—32no—
181C22×C4×C5—32no—
182C22×C4×C5—62no—
183C22×C4×C5—62no—
184C22×C4×C5C22×D311yescharacteristic
185C2×C22⋊D5—64no—
186C2×C22⋊D5—68no—
187C22×Dic5—21no—
188C22×Dic5—21no—
189C22×Dic5—61no—
190#709—68no—
191#709—616no—
192#709—68no—
193C2×D20—34no—
194C2×D20—32no—
195C2×D20—62no—
196C2×D20—61no—
197C2×C4×D5—64no—
198C2×C4×D5—68no—
199C2×Dic10—21no—
200C2×Dic10—22no—
201C2×Dic10—32no—
202C2×Dic10—62no—
203C2×Dic10—62no—
204C2×Dic10—61no—
205C2×Dic10D1214yes—
206C2×Dic10C22×D312yes—
207C2×C5×C8—22no—
208C2×C5×C8—32no—
209C2×C5×C8—62no—
210C2×C5×C8C22×D312yes—
211C22⋊Dic5—64no—
212C4⋊Dic5—21no—
213C4⋊Dic5C22×D312yes—
214#683—64no—
215Dic20—68no—
216#677—68no—
217C22×#41—151no—
218C22×Q16—151no—
219C22×SD16—151no—
220C2×C22⋊Q8—151no—
221#452—158no—
222C2×#103—51no—
223C2×#102—151no—
224C2×C22⋊C8—31no—
225C22×C3×C5#12012yes—
226C22×C3×C5#13512yes—
227C22×C3×C5#13712yes—
228C22×C3×C5C22×D411yescharacteristic
229C2×D15—24no—
230C2×D15—42no—
231C2×D15—42no—
232C2×D15—48no—
233C2×C5×D3—22no—
234C2×C5×D3—22no—
235C2×C5×D3—22no—
236C2×C5×D3—44no—
237C3×C4×C5#12012yes—
238C3×C4×C5C22×D411yescharacteristic
239C3×C4×C5C22×D411yescharacteristic
240Dic15—42no—
241C3×Dic5—24no—
242C3×Dic5—42no—
243C5×Dic3—22no—
244C23×D3—101no—
245C23×D3D2011yescharacteristic
246C2×C3×Q8—54no—
247C2×C3×Q8—52no—
248C2×C3×Q8—101no—
249C2×C3×Q8—102no—
250C22×C3×C4—101no—
251C22×C3×C4—101no—
252C22×C3×C4C22×D511yescharacteristic
253C22×Dic3—101no—
254C22×Dic3D2011yescharacteristic
255#239—108no—
256#239—108no—
257#239—1016no—
258C2×D12—54no—
259C2×D12—52no—
260C2×D12—108no—
261C2×D12—104no—
262C2×D12—101no—
263C2×D12—102no—
264C2×C4×D3—22no—
265C2×C4×D3—22no—
266C2×C4×D3—108no—
267C2×C4×D3—104no—
268C2×C4×D3D2012yes—
269C2×C4×D3C22×D512yes—
270C2×Dic6—52no—
271C2×Dic6—102no—
272C2×Dic6—102no—
273C2×Dic6—101no—
274C2×C3×C8—22no—
275C2×C3×C8C22×D512yes—
276C3×C4⋊C4—52no—
277C3×C4⋊C4—101no—
278#216—108no—
279(C2×D3)⋊C4—104no—
280C4⋊Dic3—104no—
281C2×C3⋊C8—22no—
282C2×C3⋊C8C22×D512yes—
283#204—108no—
284C23×C5—32no—
285C23×C5—62no—
286C23×C5—62no—
287C23×C5—62no—
288C23×C5D3×D411yescharacteristic
289C22×D5—64no—
290C22×D5—62no—
291C22×D5—122no—
292C22×D5—122no—
293C2×C4×C5—32no—
294C2×C4×C5—62no—
295C2×C4×C5—64no—
296C2×C4×C5—62no—
297C2×C4×C5—62no—
298C2×C4×C5C2×D1212yes—
299C2×C4×C5D3×D411yescharacteristic
300C2×C4×C5D3×D412yes—
301C2×C4×C5C23×D311yescharacteristic
302C22⋊D5—616no—
303C2×Dic5—24no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—42no—
307C2×Dic5—62no—
308C2×Dic5—122no—
309D20—34no—
310D20—64no—
311D20—62no—
312C4×D5—616no—
313Dic10—24no—
314Dic10—22no—
315Dic10—62no—
316Dic10—62no—
317Dic10—122no—
318Dic10C2×D1214yes—
319C5×C8—22no—
320C5×C8—62no—
321C2×#41—154no—
322C2×#41—302no—
323C2×#41—304no—
324C2×#41—304no—
325C22×Q8—51no—
326C22×Q8—151no—
327C22×D4—151no—
328C22×D4—301no—
329C23×C4—31no—
330C23×C4—301no—
331C2×Q16—154no—
332C2×Q16—304no—
333C2×SD16—154no—
334C2×SD16—304no—
335C22×C8—31no—
336C22×C8D3×D511yescharacteristic
337C22⋊Q8—154no—
338C22⋊Q8—302no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—301no—
341C2×C22⋊C4—301no—
342#103—54no—
343#102—154no—
344C22⋊C8—34no—
345C2×C3×C5#45214yescomplemented
346C2×C3×C5C2×#12011yescharacteristic
347C2×C3×C5C2×#13511yescharacteristic
348C2×C3×C5C2×#13711yescharacteristic
349D15—44no—
350C5×D3—42no—
351C23×C3C2×D2011yescharacteristic
352C22×D3—22no—
353C22×D3—22no—
354C22×D3—22no—
355C22×D3—102no—
356C22×D3—104no—
357C22×D3—202no—
358C22×D3—202no—
359C22×D3C2×D2012yes—
360C3×Q8—54no—
361C3×Q8—102no—
362C3×Q8—104no—
363C2×C3×C4—102no—
364C2×C3×C4—104no—
365C2×C3×C4—102no—
366C2×C3×C4—202no—
367C2×C3×C4C2×D2011yescharacteristic
368C2×C3×C4D4×D512yes—
369C2×C3×C4D4×D512yes—
370C2×C3×C4C23×D511yescharacteristic
371C2×Dic3—22no—
372C2×Dic3—102no—
373C2×Dic3—202no—
374C2×Dic3C2×D2012yes—
375D12—54no—
376D12—1016no—
377D12—102no—
378D12—104no—
379C4×D3—22no—
380C4×D3—22no—
381C4×D3—24no—
382C4×D3—1016no—
383Dic6—102no—
384Dic6—102no—
385Dic6—202no—
386C3×C8—22no—
387C3⋊C8—22no—
388C22×C5—62no—
389C22×C5—62no—
390C22×C5—62no—
391C22×C5—124no—
392C22×C5#88212yes—
393C22×C5#91612yes—
394C22×C5#91812yes—
395C22×C5C2×D3×D411yescharacteristic
396C2×D5—64no—
397C2×D5—122no—
398C2×D5—122no—
399C2×D5—128no—
400C4×C5—62no—
401C4×C5#88212yes—
402C4×C5C22×D1211yescharacteristic
403C4×C5C2×D3×D411yescharacteristic
404Dic5—24no—
405Dic5—42no—
406Dic5—122no—
407C24—31no—
408C24—301no—
409#41—308no—
410#41—308no—
411#41—3016no—
412C2×Q8—54no—
413C2×Q8—52no—
414C2×Q8—101no—
415C2×Q8—102no—
416C2×Q8—152no—
417C2×Q8—301no—
418C2×Q8—302no—
419C2×Q8—302no—
420C2×D4—154no—
421C2×D4—152no—
422C2×D4—304no—
423C2×D4—308no—
424C2×D4—302no—
425C2×D4—301no—
426C22×C4—31no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—101no—
432C22×C4—101no—
433C22×C4—304no—
434C22×C4—308no—
435C22×C4—301no—
436C22×C4C2×D3×D511yescharacteristic
437Q16—308no—
438SD16—308no—
439C2×C8—22no—
440C2×C8—32no—
441C2×C8—62no—
442C2×C8C2×D3×D512yes—
443C4⋊C4—52no—
444C4⋊C4—101no—
445C4⋊C4—304no—
446C22⋊C4—304no—
447C3×C5C2×#45211yescharacteristic, complemented, Hall
448C22×C3#402812yes—
449C22×C3#405012yes—
450C22×C3#405512yes—
451C22×C3C22×D2011yescharacteristic
452C2×D3—22no—
453C2×D3—22no—
454C2×D3—22no—
455C2×D3—44no—
456C2×D3—104no—
457C2×D3—202no—
458C2×D3—202no—
459C2×D3—208no—
460C3×C4—104no—
461C3×C4—202no—
462C3×C4#402812yes—
463C3×C4C2×D4×D511yescharacteristic
464C3×C4C2×D4×D511yescharacteristic
465Dic3—22no—
466Dic3—202no—
467C2×C5—122no—
468C2×C5#480814yescomplemented
469C2×C5C2×#88211yescharacteristic
470C2×C5C2×#91611yescharacteristic
471C2×C5C2×#91811yescharacteristic
472D5—124no—
473C23—32no—
474C23—62no—
475C23—62no—
476C23—62no—
477C23—304no—
478C23—302no—
479C23—602no—
480C23—602no—
481C23D3×D2011yescharacteristic
482Q8—54no—
483Q8—104no—
484Q8—102no—
485Q8—302no—
486Q8—302no—
487Q8—602no—
488D4—154no—
489D4—3016no—
490D4—304no—
491D4—302no—
492C2×C4—32no—
493C2×C4—62no—
494C2×C4—64no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—104no—
498C2×C4—102no—
499C2×C4—102no—
500C2×C4—202no—
501C2×C4—3016no—
502C2×C4—302no—
503C2×C4—602no—
504C2×C4D5×D1212yes—
505C2×C4D3×D2011yescharacteristic
506C2×C4#679012yes—
507C2×C4C22×D3×D511yescharacteristic
508C8—22no—
509C8—62no—
510C2×C3#6502214yescomplemented
511C2×C3C2×#402811yescharacteristic
512C2×C3C2×#405011yescharacteristic
513C2×C3C2×#405511yescharacteristic
514D3—42no—
515D3—204no—
516C5C2×#480811yescharacteristic, complemented, Sylow
517C22—62no—
518C22—62no—
519C22—62no—
520C22—124no—
521C22—304no—
522C22—602no—
523C22—602no—
524C22—608no—
525C22#9126312yes—
526C22#9127212yes—
527C22#9147612yes—
528C22C2×D3×D2011yescharacteristic
529C4—62no—
530C4—104no—
531C4—202no—
532C4—602no—
533C4#9147612yes—
534C4C2×D5×D1211yescharacteristic
535C4C2×#679011yescharacteristic
536C3C2×#6502211yescharacteristic, complemented, Sylow
537C2—122no—
538C2—604no—
539C2#1174816614yesdirect product factor
540C2C2×#9126311yescharacteristic
541C2C2×#9127211yescharacteristic
542C2C2×#9147611yescharacteristic
subgroupG/H
centerC23D3×D20
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×C8C22
socleC23×C3×C5C2×D4
prime radicalC22×#6769C2
hypercenterC22×C8D3×D5
hypocenterC3×C5C2×#452
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C22×#6706C2semidirect
3. C2×#91365C2semidirect
4. C22×#6662C2semidirect
5. C2×C5×(C2×D3)⋊C8C2semidirect
6. C2×C3×#3948C2semidirect
7. C2×#90968C2semidirect
8. #11748166C2direct (N × H)
22. C2×#6706C22semidirect
28. #91365C22semidirect
30. C2×#6662C22semidirect
31. C5×(C2×D3)⋊C8C22semidirect
32. C3×#3948C22semidirect
33. #90968C22semidirect
41. C2×#3948D3semidirect
88. C2×(C2×D3)⋊C8D5semidirect
114. #3948C2×D3semidirect
175. (C2×D3)⋊C8C2×D5semidirect
345. C2×C3×C5#452semidirect
447. C3×C5C2×#452semidirect
468. C2×C5#4808semidirect
510. C2×C3#65022semidirect
516. C5C2×#4808semidirect
536. C3C2×#65022semidirect
539. C2#11748166direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#67692
C22×#67062
C2×#913652
C22×#66622
C2×C5×(C2×D3)⋊C82
C2×C3×#39482
C2×#909682
#117481662
C2×#650223
C2×#48085