Finite Group Databasebeta

← C2×#11749406 #14423531

Subgroups — C2×#11749406 #14423531

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×#6773C211yesmaximal, characteristic
2C22×#6717C211yesmaximal, characteristic, complemented
3C22×C5×#216C211yesmaximal, characteristic, complemented
4C2×C3×#4092C211yesmaximal, characteristic, complemented
5#11749406C218yesmaximal, direct product factor
6C2×#90972C211yesmaximal, characteristic, complemented
7C2×#90968C211yesmaximal, characteristic, complemented
8C2×#90955C211yesmaximal, characteristic, complemented
9C2×#65476—31nomaximal, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×D60C2211yescharacteristic
12C22×C5×Dic6C2211yescharacteristic
13C22×C3×C4×D5C2211yescharacteristic
14C22×#1202—21no—
15C22×#1200—21no—
16C2×#6773—24no—
17C2×#6773—24no—
18C2×#6773—22no—
19C2×#6773C2214yes—
20C2×#6717—24no—
21C2×#6717C2214yescomplemented
22C22×C3⋊#168C2211yescharacteristic
23C2×C5×#216—24no—
24C2×C5×#216C2214yescomplemented
25C22×C5×C3⋊C8C2211yescharacteristic
26C3×#4092—22no—
27C3×#4092C2214yescomplemented
28C2×C3×#685—21no—
29C2×C3×C4⋊Dic5C2211yescharacteristic
30C2×C3×#683—21no—
31#90972C2214yescomplemented
32#90968C2214yescomplemented
33#90955C2214yescomplemented
34C2×#5047—51nomaximal, Hall
35C22×#713—31no—
36C22×C5×Q16—31no—
37C2×#4092D311yescharacteristic, complemented
38C22×#688—31no—
39C2×#3967—31no—
40#65476—38no—
41C2×#3953—31no—
42C2×#3952—31no—
43C23×D15—21no—
44C23×C3×D5D411yescharacteristic
45C2×C3×C5×Q8—21no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8—22no—
48C2×C3×C5×Q8C2312yes—
49C22×C3×C4×C5—21no—
50C22×C3×C4×C5C2311yescharacteristic
51C2×D60—21no—
52C2×D60—22no—
53C2×D60D414yes—
54C2×D60C2312yes—
55C22×C5×Dic3—21no—
56C2×C5×Dic6—21no—
57C2×C5×Dic6—22no—
58C2×C5×Dic6D414yes—
59C2×C5×Dic6C2312yes—
60C22×C3×Dic5—21no—
61C22×C3×Dic5D411yescharacteristic
62C2×C3×C4×D5—22no—
63C2×C3×C4×D5—22no—
64C2×C3×C4×D5D412yes—
65C2×C3×C4×D5C2312yes—
66C2×#1202—28no—
67C2×#1202—24no—
68C2×#1200—28no—
69C2×#1200—24no—
70#6773—216no—
71#6773—28no—
72#6773—28no—
73#6717—28no—
74C2×C3⋊#168—22no—
75C2×C3⋊#168C2312yes—
76C5×#216—28no—
77C2×C5×C3⋊C8—22no—
78C2×C5×C3⋊C8C2312yes—
79C3×#685—24no—
80C3×C4⋊Dic5—21no—
81C3×C4⋊Dic5C2312yes—
82C3×#683—24no—
83C22×#235—51no—
84C2×C3×C22⋊Q8—51no—
85C22×#216D511yescharacteristic, complemented
86C22×#215—51no—
87C2×#830—51no—
88#5047—58no—
89C2×#810—51no—
90C2×#809—51no—
91C22×C5×Q8—31no—
92C22×C5×Q8C2×D311yescharacteristic
93C2×#713—34no—
94C2×#713—64no—
95C2×#713—64no—
96C2×#713—62no—
97C22×D20—31no—
98C22×D20—61no—
99C22×C4×D5—61no—
100C22×C4×D5C2×D311yescharacteristic
101C2×C5×Q16—34no—
102C2×C5×Q16—64no—
103C22×C5×C8—31no—
104#4092—22no—
105#4092C2×D314yescomplemented
106C2×#688—34no—
107C2×#688—64no—
108C2×#685—21no—
109C2×C4⋊Dic5C2×D311yescharacteristic
110C2×#683—21no—
111C22×#168—31no—
112#3967—34no—
113#3953—34no—
114#3952—34no—
115C2×#452—151noSylow
116C23×C3×C5C2×D411yescharacteristic
117C22×D15—24no—
118C22×D15—22no—
119C22×D15—42no—
120C22×D15—42no—
121C22×C3×D5—22no—
122C22×C3×D5—22no—
123C22×C3×D5—22no—
124C22×C3×D5C2×D412yes—
125C3×C5×Q8—22no—
126C3×C5×Q8—22no—
127C3×C5×Q8—42no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—42no—
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D412yes—
132C2×C3×C4×C5C2×D412yes—
133C2×C3×C4×C5C2411yescharacteristic
134D60—24no—
135D60—22no—
136D60C2×D414yes—
137C2×C5×Dic3—24no—
138C2×C5×Dic3—22no—
139C5×Dic6—24no—
140C5×Dic6—22no—
141C5×Dic6C2×D414yes—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—42no—
145C2×C3×Dic5C2×D412yes—
146C3×C4×D5—24no—
147C3×C4×D5—22no—
148C3×C4×D5—22no—
149#1202—216no—
150#1200—216no—
151C3⋊#168—22no—
152C5×C3⋊C8—22no—
153C22×C3×Q8C2×D511yescharacteristic
154C23×C3×C4—51no—
155C22×#67—101no—
156C2×#235—54no—
157C2×#235—104no—
158C2×#235—102no—
159C2×#235—104no—
160C22×D12—51no—
161C22×C4×D3—101no—
162C22×Dic6C2×D511yescharacteristic
163C3×C22⋊Q8—54no—
164C3×C22⋊Q8—102no—
165C2×C3×C4⋊C4—51no—
166C2×C3×C4⋊C4—101no—
167C2×C3×C22⋊C4—101no—
168C2×#216—24no—
169C2×#216C2×D514yescomplemented
170C2×#215—54no—
171C2×#215—104no—
172C22×C3⋊C8—51no—
173C22×C3⋊C8C2×D511yescharacteristic
174#830—54no—
175#810—54no—
176#809—54no—
177C23×D5—61no—
178C23×D5#6711yescharacteristic
179C2×C5×Q8—21no—
180C2×C5×Q8—22no—
181C2×C5×Q8—22no—
182C2×C5×Q8—34no—
183C2×C5×Q8—32no—
184C2×C5×Q8—62no—
185C2×C5×Q8—61no—
186C2×C5×Q8C22×D312yes—
187C22×C4×C5—21no—
188C22×C4×C5—61no—
189C22×C4×C5C22×D311yescharacteristic
190C22×Dic5—21no—
191C22×Dic5#6711yescharacteristic
192#713—68no—
193#713—68no—
194#713—616no—
195C2×D20—34no—
196C2×D20—32no—
197C2×D20—68no—
198C2×D20—64no—
199C2×D20—62no—
200C2×D20—61no—
201C2×C4×D5—22no—
202C2×C4×D5—22no—
203C2×C4×D5—68no—
204C2×C4×D5—64no—
205C2×C4×D5#6712yes—
206C2×C4×D5C22×D312yes—
207C5×Q16—68no—
208C2×C5×C8—32no—
209C2×C5×C8—62no—
210#688—68no—
211#685—24no—
212C4⋊Dic5—21no—
213C4⋊Dic5C22×D312yes—
214#683—24no—
215C2×#168—32no—
216C2×#168—62no—
217C22×#41—151no—
218C22×Q16—31no—
219C22×SD16—151no—
220C2×C22⋊Q8—51no—
221#452—158no—
222C2×#103—151no—
223C2×#102—151no—
224C2×C22⋊C8—151no—
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×C3×D5—22no—
234C2×C3×D5—22no—
235C2×C3×D5—22no—
236C2×C3×D5—44no—
237C3×C4×C5—42no—
238C3×C4×C5#12012yes—
239C3×C4×C5C22×D411yescharacteristic
240C3×C4×C5C22×D411yescharacteristic
241C3×Dic5—22no—
242C3×Dic5—42no—
243C5×Dic3—24no—
244C24×C3—51no—
245C23×D3—101no—
246C2×C3×Q8—21no—
247C2×C3×Q8—22no—
248C2×C3×Q8—22no—
249C2×C3×Q8C22×D512yes—
250C22×C3×C4—21no—
251C22×C3×C4—51no—
252C22×C3×C4—52no—
253C22×C3×C4—52no—
254C22×C3×C4—102no—
255C22×C3×C4—102no—
256C22×C3×C4—101no—
257C22×C3×C4C22×D511yescharacteristic
258C2×#67—104no—
259C2×#67—108no—
260C22×Dic3—21no—
261#235—108no—
262#235—1016no—
263#235—108no—
264C2×D12—54no—
265C2×D12—52no—
266C2×D12—101no—
267C2×D12—102no—
268C2×C4×D3—104no—
269C2×C4×D3—108no—
270C2×Dic6—21no—
271C2×Dic6—22no—
272C2×Dic6C22⋊D514yes—
273C2×Dic6C22×D512yes—
274C3×C4⋊C4—52no—
275C3×C4⋊C4—104no—
276C3×C4⋊C4—101no—
277C3×C22⋊C4—104no—
278#216—28no—
279#215—108no—
280C2×C3⋊C8—22no—
281C2×C3⋊C8—52no—
282C2×C3⋊C8—102no—
283C2×C3⋊C8C22×D512yes—
284C23×C5C2×#6711yescharacteristic
285C22×D5—22no—
286C22×D5—22no—
287C22×D5—22no—
288C22×D5—64no—
289C22×D5—62no—
290C22×D5—122no—
291C22×D5—122no—
292C22×D5C2×#6712yes—
293C5×Q8—22no—
294C5×Q8—22no—
295C5×Q8—34no—
296C5×Q8—42no—
297C5×Q8—64no—
298C5×Q8—62no—
299C2×C4×C5—22no—
300C2×C4×C5—42no—
301C2×C4×C5—64no—
302C2×C4×C5—62no—
303C2×C4×C5D3×D412yes—
304C2×C4×C5D3×D412yes—
305C2×C4×C5C2×#6711yescharacteristic
306C2×C4×C5C23×D311yescharacteristic
307C2×Dic5—22no—
308C2×Dic5—22no—
309C2×Dic5—42no—
310C2×Dic5C2×#6712yes—
311D20—34no—
312D20—616no—
313D20—64no—
314D20—62no—
315C4×D5—24no—
316C4×D5—22no—
317C4×D5—22no—
318C4×D5—616no—
319C5×C8—62no—
320#168—62no—
321C2×#41—154no—
322C2×#41—302no—
323C2×#41—304no—
324C2×#41—304no—
325C22×Q8—31no—
326C22×Q8D3×D511yescharacteristic
327C22×D4—151no—
328C22×D4—301no—
329C23×C4—51no—
330C23×C4—301no—
331C2×Q16—34no—
332C2×Q16—64no—
333C2×SD16—154no—
334C2×SD16—304no—
335C22×C8—31no—
336C22×C8—151no—
337C22⋊Q8—54no—
338C22⋊Q8—102no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—101no—
341C2×C22⋊C4—101no—
342#103—154no—
343#102—154no—
344C22⋊C8—154no—
345C2×C3×C5#45214yescomplemented
346C2×C3×C5C2×#12011yescharacteristic
347C2×C3×C5C2×#13511yescharacteristic
348C2×C3×C5C2×#13711yescharacteristic
349D15—44no—
350C3×D5—42no—
351C23×C3—52no—
352C23×C3—102no—
353C23×C3—102no—
354C23×C3—102no—
355C23×C3D4×D511yescharacteristic
356C22×D3—102no—
357C22×D3—104no—
358C22×D3—202no—
359C22×D3—202no—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—42no—
363C2×C3×C4—22no—
364C2×C3×C4—42no—
365C2×C3×C4—52no—
366C2×C3×C4—102no—
367C2×C3×C4—102no—
368C2×C3×C4—102no—
369C2×C3×C4—104no—
370C2×C3×C4—102no—
371C2×C3×C4—202no—
372C2×C3×C4D4×D511yescharacteristic
373C2×C3×C4D4×D512yes—
374C2×C3×C4C2×C22⋊D512yes—
375C2×C3×C4C23×D511yescharacteristic
376#67—1016no—
377C2×Dic3—22no—
378C2×Dic3—24no—
379D12—54no—
380D12—102no—
381D12—104no—
382C4×D3—1016no—
383Dic6—22no—
384Dic6—24no—
385Dic6C2×C22⋊D514yes—
386C3⋊C8—22no—
387C3⋊C8—102no—
388C22×C5#93712yes—
389C22×C5#94212yes—
390C22×C5#95012yes—
391C22×C5C22×#6711yescharacteristic
392C2×D5—22no—
393C2×D5—22no—
394C2×D5—22no—
395C2×D5—44no—
396C2×D5—64no—
397C2×D5—122no—
398C2×D5—128no—
399C2×D5—122no—
400C4×C5—42no—
401C4×C5—64no—
402C4×C5#93712yes—
403C4×C5C2×D3×D411yescharacteristic
404C4×C5C2×D3×D411yescharacteristic
405Dic5—22no—
406Dic5—42no—
407C24—51no—
408C24—301no—
409#41—308no—
410#41—308no—
411#41—3016no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—34no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—62no—
419C2×Q8C2×D3×D512yes—
420C2×D4—154no—
421C2×D4—152no—
422C2×D4—304no—
423C2×D4—308no—
424C2×D4—301no—
425C2×D4—302no—
426C22×C4—21no—
427C22×C4—51no—
428C22×C4—52no—
429C22×C4—52no—
430C22×C4—61no—
431C22×C4—102no—
432C22×C4—102no—
433C22×C4—101no—
434C22×C4—304no—
435C22×C4—308no—
436C22×C4C2×D3×D511yescharacteristic
437Q16—68no—
438SD16—308no—
439C2×C8—32no—
440C2×C8—62no—
441C2×C8—152no—
442C2×C8—302no—
443C4⋊C4—52no—
444C4⋊C4—104no—
445C4⋊C4—101no—
446C22⋊C4—104no—
447C3×C5C2×#45211yescharacteristic, complemented, Hall
448C22×C3—102no—
449C22×C3—102no—
450C22×C3—102no—
451C22×C3—204no—
452C22×C3#406412yes—
453C22×C3#406512yes—
454C22×C3#408312yes—
455C22×C3C2×D4×D511yescharacteristic
456C2×D3—104no—
457C2×D3—202no—
458C2×D3—202no—
459C2×D3—208no—
460C3×C4—42no—
461C3×C4—102no—
462C3×C4—202no—
463C3×C4#408312yes—
464C3×C4C2×D4×D511yescharacteristic
465C3×C4C22×C22⋊D511yescharacteristic
466Dic3—24no—
467C2×C5#504714yescomplemented
468C2×C5C2×#93711yescharacteristic
469C2×C5C2×#94211yescharacteristic
470C2×C5C2×#95011yescharacteristic
471D5—42no—
472D5—124no—
473C23—52no—
474C23—102no—
475C23—102no—
476C23—102no—
477C23—302no—
478C23—304no—
479C23—602no—
480C23—602no—
481C23D5×#6711yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—34no—
485Q8—42no—
486Q8—62no—
487Q8—64no—
488D4—154no—
489D4—3016no—
490D4—302no—
491D4—304no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—52no—
495C2×C4—62no—
496C2×C4—64no—
497C2×C4—102no—
498C2×C4—102no—
499C2×C4—102no—
500C2×C4—104no—
501C2×C4—102no—
502C2×C4—202no—
503C2×C4—3016no—
504C2×C4D5×#6711yescharacteristic
505C2×C4D3×C22⋊D512yes—
506C2×C4#679012yes—
507C2×C4C22×D3×D511yescharacteristic
508C8—62no—
509C8—302no—
510C2×C3—202no—
511C2×C3#6547614yescomplemented
512C2×C3C2×#406411yescharacteristic
513C2×C3C2×#406511yescharacteristic
514C2×C3C2×#408311yescharacteristic
515D3—204no—
516C5C2×#504711yescharacteristic, complemented, Sylow
517C22—102no—
518C22—102no—
519C22—102no—
520C22—204no—
521C22—304no—
522C22—608no—
523C22—602no—
524C22—602no—
525C22#9151512yes—
526C22#9151612yes—
527C22#9157712yes—
528C22C2×D5×#6711yescharacteristic
529C4—42no—
530C4—64no—
531C4—102no—
532C4—202no—
533C4#9157712yes—
534C4C2×D3×C22⋊D511yescharacteristic
535C4C2×#679011yescharacteristic
536C3C2×#6547611yescharacteristic, complemented, Sylow
537C2—202no—
538C2—604no—
539C2#1174940614yesdirect product factor
540C2C2×#9151511yescharacteristic
541C2C2×#9151611yescharacteristic
542C2C2×#9157711yescharacteristic
subgroupG/H
centerC23D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C2×D4
prime radicalC22×#6773C2
hypercenterC22×Q8D3×D5
hypocenterC3×C5C2×#452
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C22×#6717C2semidirect
3. C22×C5×#216C2semidirect
4. C2×C3×#4092C2semidirect
5. #11749406C2direct (N × H)
6. C2×#90972C2semidirect
7. C2×#90968C2semidirect
8. C2×#90955C2semidirect
21. C2×#6717C22semidirect
24. C2×C5×#216C22semidirect
27. C3×#4092C22semidirect
31. #90972C22semidirect
32. #90968C22semidirect
33. #90955C22semidirect
37. C2×#4092D3semidirect
85. C22×#216D5semidirect
105. #4092C2×D3semidirect
169. C2×#216C2×D5semidirect
345. C2×C3×C5#452semidirect
447. C3×C5C2×#452semidirect
467. C2×C5#5047semidirect
511. C2×C3#65476semidirect
516. C5C2×#5047semidirect
536. C3C2×#65476semidirect
539. C2#11749406direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#67732
C22×#67172
C22×C5×#2162
C2×C3×#40922
#117494062
C2×#909722
C2×#909682
C2×#909552
C2×#654763
C2×#50475