Finite Group Databasebeta

← C2×#11754586 #14480079

Subgroups — C2×#11754586 #14480079

Order 1920.

3456 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×C5×#235C211yesmaximal, characteristic, complemented
2C22×C3×#709C211yesmaximal, characteristic, complemented
3C22×C4×Dic15C211yesmaximal, characteristic, complemented
4C2×#91571C214yesmaximal, complemented
5#11754586C2116yesmaximal, direct product factor
6C2×#91463C211yesmaximal, characteristic, complemented
7C2×C4×#1201C214yesmaximal, complemented
8C2×#91385C211yesmaximal, characteristic, complemented
9C2×#91370C211yesmaximal, characteristic, complemented
10C2×#91368C211yesmaximal, characteristic, complemented
11C2×#66166—31nomaximal, Hall
12C23×C3×C4×C5C2211yescharacteristic
13C23×Dic15C2212yes—
14C22×C5×#67C2212yes—
15C2×C5×#235—28no—
16C2×C5×#235C2218yescomplemented
17C22×C5×D12C2211yescharacteristic
18C22×C4×C5×D3C2212yes—
19C22×C5×Dic6C2211yescharacteristic
20C22×C3×C22⋊D5C2212yes—
21C2×C3×#709—28no—
22C2×C3×#709C2218yescomplemented
23C22×C3×D20C2211yescharacteristic
24C22×C3×C4×D5C2212yes—
25C22×C3×Dic10C2211yescharacteristic
26C22×#1201C2214yes—
27C2×C4×Dic15C2211yescharacteristic
28C2×C4×Dic15C2212yes—
29C2×C4×Dic15C2211yescharacteristic
30C2×C4×Dic15C2218yescomplemented
31#91571C22132yescomplemented
32C2×#6669C2214yes—
33C2×(C2×D3)⋊Dic5C2214yes—
34C2×#6665C2214yes—
35#91463C2218yescomplemented
36C4×#1201C22132yescomplemented
37#91385C2218yescomplemented
38#91370C2218yescomplemented
39#91368C2218yescomplemented
40C2×#5832—51nomaximal, Hall
41C22×C5×#41—31no—
42C22×#709D311yescharacteristic, complemented
43#66166—316no—
44C2×#4093—31no—
45C2×#4090—31no—
46C2×#4086—31no—
47C2×#4085—34no—
48C2×#4082—31no—
49C2×C4×C22⋊D5—34no—
50C22×C4×Dic5—31no—
51C24×C3×C5C2311yescharacteristic
52C23×C5×D3C2312yes—
53C23×C3×D5C2312yes—
54C22×C3×C4×C5D418yescomplemented
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2314yes—
58C22×Dic15—28no—
59C22×Dic15C2312yes—
60C22×Dic15C2318yes—
61C22×Dic15C2312yes—
62C2×C5×#67—28no—
63C2×C5×#67C2318yes—
64C22×C5×Dic3C2312yes—
65C5×#235—232no—
66C2×C5×D12—24no—
67C2×C5×D12C2314yes—
68C2×C4×C5×D3—28no—
69C2×C4×C5×D3C2318yes—
70C2×C5×Dic6—24no—
71C2×C5×Dic6C2314yes—
72C2×C3×C22⋊D5—28no—
73C2×C3×C22⋊D5C2318yes—
74C22×C3×Dic5C2312yes—
75C3×#709—232no—
76C2×C3×D20—24no—
77C2×C3×D20C2314yes—
78C2×C3×C4×D5—28no—
79C2×C3×C4×D5C2318yes—
80C2×C3×Dic10—24no—
81C2×C3×Dic10C2314yes—
82C2×#1201—216no—
83C2×#1201C23116yes—
84C4×Dic15C2318yes—
85C4×Dic15C2314yes—
86C4×Dic15C2314yes—
87#6669C23116yes—
88(C2×D3)⋊Dic5C23116yes—
89#6665C23116yes—
90C22×C3×#41—51no—
91C22×#235D511yescharacteristic, complemented
92#5832—516no—
93C2×#947—51no—
94C2×#944—51no—
95C2×#940—51no—
96C2×#939—54no—
97C2×#936—51no—
98C2×C4×#67—54no—
99C22×C4×Dic3—51no—
100C2×C5×#41—38no—
101C2×C5×#41—68no—
102C22×C5×Q8—31no—
103C22×C5×D4—32no—
104C22×C5×D4—31no—
105C23×C4×C5—32no—
106C23×C4×C5C2×D311yescharacteristic
107C22×C22⋊D5—34no—
108C22×C22⋊D5C2×D312yes—
109C23×Dic5—32no—
110C2×#709—28no—
111C2×#709C2×D318yescomplemented
112C22×D20C2×D311yescharacteristic
113C22×C4×D5C2×D312yes—
114C22×Dic10C2×D311yescharacteristic
115C2×C22⋊Dic5—34no—
116#4093—38no—
117#4090—38no—
118#4086—38no—
119#4085—332no—
120#4082—38no—
121C4×C22⋊D5—332no—
122C2×#685—34no—
123C2×#683—34no—
124C2×C4×Dic5—32no—
125C2×C4×Dic5—31no—
126C2×C4×Dic5—31no—
127C2×C4×Dic5—38no—
128C2×#532—151noSylow
129C23×C3×C5—24no—
130C23×C3×C5C2×D414yes—
131C23×C3×C5C2412yes—
132C23×C3×C5C2411yescharacteristic
133C22×C5×D3—24no—
134C22×C5×D3—28no—
135C22×C5×D3C2414yes—
136C22×C3×D5—24no—
137C22×C3×D5—28no—
138C22×C3×D5C2414yes—
139C2×C3×C4×C5C2×D414yes—
140C2×C3×C4×C5C2×D414yes—
141C2×C3×C4×C5C2×D4116yescomplemented
142C2×C3×C4×C5C2412yes—
143C2×C3×C4×C5C2412yes—
144C2×Dic15—24no—
145C2×Dic15—216no—
146C2×Dic15#4118yescomplemented
147C2×Dic15C2414yes—
148C2×Dic15C2414yes—
149C5×#67—216no—
150C2×C5×Dic3—24no—
151C2×C5×Dic3C2414yes—
152C5×D12—28no—
153C4×C5×D3—216no—
154C5×Dic6—28no—
155C3×C22⋊D5—216no—
156C2×C3×Dic5—24no—
157C2×C3×Dic5C2414yes—
158C3×D20—28no—
159C3×C4×D5—216no—
160C3×Dic10—28no—
161#1201—232no—
162C2×C3×#41—58no—
163C2×C3×#41—108no—
164C22×C3×Q8—51no—
165C22×C3×D4—52no—
166C22×C3×D4—51no—
167C23×C3×C4—52no—
168C23×C3×C4C2×D511yescharacteristic
169C22×#67—54no—
170C22×#67C2×D512yes—
171C23×Dic3—52no—
172C2×#235—28no—
173C2×#235C2×D518yescomplemented
174C22×D12C2×D511yescharacteristic
175C22×C4×D3C2×D512yes—
176C22×Dic6C2×D511yescharacteristic
177C2×#217—54no—
178#947—58no—
179#944—58no—
180#940—58no—
181#939—532no—
182#936—58no—
183C4×#67—532no—
184C2×(C2×D3)⋊C4—54no—
185C2×Dic3⋊C4—54no—
186C2×C4×Dic3—52no—
187C2×C4×Dic3—51no—
188C2×C4×Dic3—51no—
189C2×C4×Dic3—58no—
190C24×C5—32no—
191C24×C5C22×D311yescharacteristic
192C23×D5C22×D312yes—
193C5×#41—632no—
194C2×C5×Q8—34no—
195C2×C5×Q8—64no—
196C2×C5×D4—38no—
197C2×C5×D4—34no—
198C2×C5×D4—68no—
199C2×C5×D4—64no—
200C22×C4×C5—32no—
201C22×C4×C5—38no—
202C22×C4×C5—68no—
203C22×C4×C5#6718yescomplemented
204C22×C4×C5C22×D311yescharacteristic
205C22×C4×C5C22×D314yes—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—28no—
208C2×C22⋊D5—316no—
209C2×C22⋊D5—616no—
210C2×C22⋊D5C22×D318yes—
211C22×Dic5—32no—
212C22×Dic5—32no—
213C22×Dic5—38no—
214C22×Dic5—68no—
215C22×Dic5C22×D312yes—
216#709—232no—
217C2×D20—24no—
218C2×D20C22×D314yes—
219C2×C4×D5—28no—
220C2×C4×D5C22×D318yes—
221C2×Dic10—24no—
222C2×Dic10C22×D314yes—
223C22⋊Dic5—316no—
224#685—316no—
225#683—316no—
226C4×Dic5—38no—
227C4×Dic5—34no—
228C4×Dic5—34no—
229C22×#41—31no—
230C22×#41—51no—
231#532—1516no—
232C2×C4⋊Q8—151no—
233C2×#127—151no—
234C2×#124—151no—
235C2×#124—151no—
236C2×#121—154no—
237C2×C4×D4—154no—
238C22×C42—151no—
239C22×C3×C5—28no—
240C22×C3×C5—24no—
241C22×C3×C5C22×D414yes—
242C22×C3×C5C22×D412yes—
243C22×C3×C5C2×#4114yes—
244C22×C3×C5C2511yescharacteristic
245C2×C5×D3—24no—
246C2×C5×D3—28no—
247C2×C5×D3—48no—
248C2×C3×D5—24no—
249C2×C3×D5—28no—
250C2×C3×D5—48no—
251C3×C4×C5C22×D414yes—
252C3×C4×C5C22×D414yes—
253Dic15—24no—
254Dic15C2×#4118yescomplemented
255C3×Dic5—24no—
256C5×Dic3—24no—
257C24×C3—52no—
258C24×C3C22×D511yescharacteristic
259C23×D3C22×D512yes—
260C3×#41—1032no—
261C2×C3×Q8—54no—
262C2×C3×Q8—104no—
263C2×C3×D4—58no—
264C2×C3×D4—54no—
265C2×C3×D4—108no—
266C2×C3×D4—104no—
267C22×C3×C4—52no—
268C22×C3×C4—58no—
269C22×C3×C4—108no—
270C22×C3×C4C22⋊D518yescomplemented
271C22×C3×C4C22×D511yescharacteristic
272C22×C3×C4C22×D514yes—
273C22×C3×C4C22×D511yescharacteristic
274C2×#67—28no—
275C2×#67—516no—
276C2×#67—1016no—
277C2×#67C22×D518yes—
278C22×Dic3—52no—
279C22×Dic3—52no—
280C22×Dic3—58no—
281C22×Dic3—108no—
282C22×Dic3C22×D512yes—
283#235—232no—
284C2×D12—24no—
285C2×D12C22×D514yes—
286C2×C4×D3—28no—
287C2×C4×D3C22×D518yes—
288C2×Dic6—24no—
289C2×Dic6C22×D514yes—
290#217—516no—
291(C2×D3)⋊C4—516no—
292Dic3⋊C4—516no—
293C4×Dic3—58no—
294C4×Dic3—54no—
295C4×Dic3—54no—
296C23×C5—24no—
297C23×C5—34no—
298C23×C5—68no—
299C23×C5—64no—
300C23×C5C2×#6714yes—
301C23×C5C23×D312yes—
302C23×C5C23×D311yescharacteristic
303C22×D5—28no—
304C22×D5—24no—
305C22×D5C23×D314yes—
306C5×Q8—68no—
307C5×D4—616no—
308C5×D4—68no—
309C2×C4×C5—34no—
310C2×C4×C5—64no—
311C2×C4×C5—616no—
312C2×C4×C5C2×#6714yes—
313C2×C4×C5C2×#6714yes—
314C2×C4×C5C2×#67116yescomplemented
315C2×C4×C5C23×D312yes—
316C2×C4×C5C23×D312yes—
317C22⋊D5—216no—
318C22⋊D5—632no—
319C2×Dic5—24no—
320C2×Dic5—38no—
321C2×Dic5—34no—
322C2×Dic5—34no—
323C2×Dic5—616no—
324C2×Dic5—64no—
325C2×Dic5C23×D314yes—
326D20—28no—
327C4×D5—216no—
328Dic10—28no—
329C2×#41—38no—
330C2×#41—58no—
331C2×#41—68no—
332C2×#41—108no—
333C22×Q8—31no—
334C22×Q8—51no—
335C22×D4—32no—
336C22×D4—31no—
337C22×D4—52no—
338C22×D4—51no—
339C22×D4—154no—
340C23×C4—32no—
341C23×C4—52no—
342C23×C4—152no—
343C23×C4D3×D511yescharacteristic
344C4⋊Q8—158no—
345#127—158no—
346#124—158no—
347#124—158no—
348#121—1532no—
349C4×D4—1532no—
350C2×C4⋊C4—154no—
351C2×C22⋊C4—154no—
352C2×C22⋊C4—154no—
353C2×C42—151no—
354C2×C42—151no—
355C2×C42—158no—
356C2×C42—152no—
357C2×C3×C5—24no—
358C2×C3×C5#53214yescomplemented
359C2×C3×C5C23×D411yescharacteristic
360C2×C3×C5C22×#4112yes—
361C3×D5—44no—
362C5×D3—44no—
363C23×C3—24no—
364C23×C3—54no—
365C23×C3—108no—
366C23×C3—104no—
367C23×C3C2×C22⋊D514yes—
368C23×C3C23×D512yes—
369C23×C3C23×D511yescharacteristic
370C22×D3—28no—
371C22×D3—24no—
372C22×D3C23×D514yes—
373C3×Q8—108no—
374C3×D4—1016no—
375C3×D4—108no—
376C2×C3×C4—54no—
377C2×C3×C4—104no—
378C2×C3×C4—1016no—
379C2×C3×C4C2×C22⋊D514yes—
380C2×C3×C4C2×C22⋊D514yes—
381C2×C3×C4C2×C22⋊D5116yescomplemented
382C2×C3×C4C23×D512yes—
383C2×C3×C4C23×D512yes—
384#67—216no—
385#67—1032no—
386C2×Dic3—24no—
387C2×Dic3—58no—
388C2×Dic3—54no—
389C2×Dic3—54no—
390C2×Dic3—1016no—
391C2×Dic3—104no—
392C2×Dic3C23×D514yes—
393D12—28no—
394C4×D3—216no—
395Dic6—28no—
396C22×C5—24no—
397C22×C5—28no—
398C22×C5—68no—
399C22×C5—64no—
400C22×C5—128no—
401C22×C5D3×#4114yes—
402C22×C5C22×#6714yes—
403C22×C5C22×#6712yes—
404C22×C5C24×D311yescharacteristic
405C2×D5—28no—
406C2×D5—24no—
407C2×D5—48no—
408C4×C5—64no—
409C4×C5C22×#6714yes—
410C4×C5C22×#6714yes—
411Dic5—24no—
412Dic5—38no—
413Dic5—64no—
414C24—32no—
415C24—52no—
416C24C2×D3×D511yescharacteristic
417#41—632no—
418#41—1032no—
419C2×Q8—34no—
420C2×Q8—54no—
421C2×Q8—64no—
422C2×Q8—104no—
423C2×D4—38no—
424C2×D4—34no—
425C2×D4—58no—
426C2×D4—54no—
427C2×D4—68no—
428C2×D4—64no—
429C2×D4—108no—
430C2×D4—104no—
431C2×D4—1516no—
432C2×D4—3016no—
433C22×C4—32no—
434C22×C4—38no—
435C22×C4—52no—
436C22×C4—58no—
437C22×C4—68no—
438C22×C4—108no—
439C22×C4—152no—
440C22×C4—152no—
441C22×C4—158no—
442C22×C4—308no—
443C22×C4#120118yescomplemented
444C22×C4C2×D3×D511yescharacteristic
445C22×C4C2×D3×D514yes—
446C22×C4C2×D3×D511yescharacteristic
447C4⋊C4—1516no—
448C22⋊C4—1516no—
449C22⋊C4—1516no—
450C42—158no—
451C42—154no—
452C42—154no—
453C3×C5C2×#53211yescharacteristic, complemented, Hall
454C22×C3—28no—
455C22×C3—24no—
456C22×C3—108no—
457C22×C3—104no—
458C22×C3—208no—
459C22×C3D5×#4114yes—
460C22×C3C22×C22⋊D514yes—
461C22×C3C22×C22⋊D512yes—
462C22×C3C24×D511yescharacteristic
463C2×D3—28no—
464C2×D3—24no—
465C2×D3—48no—
466C3×C4—104no—
467C3×C4C22×C22⋊D514yes—
468C3×C4C22×C22⋊D514yes—
469Dic3—24no—
470Dic3—58no—
471Dic3—104no—
472C2×C5—24no—
473C2×C5—124no—
474C2×C5#583214yescomplemented
475C2×C5C2×D3×#4112yes—
476C2×C5C23×#6711yescharacteristic
477D5—44no—
478C23—24no—
479C23—34no—
480C23—54no—
481C23—68no—
482C23—64no—
483C23—108no—
484C23—104no—
485C23C2×#120114yes—
486C23C22×D3×D512yes—
487C23C22×D3×D511yescharacteristic
488Q8—68no—
489Q8—108no—
490D4—616no—
491D4—68no—
492D4—1016no—
493D4—108no—
494D4—3032no—
495C2×C4—34no—
496C2×C4—54no—
497C2×C4—64no—
498C2×C4—616no—
499C2×C4—104no—
500C2×C4—1016no—
501C2×C4—158no—
502C2×C4—154no—
503C2×C4—154no—
504C2×C4—3016no—
505C2×C4—304no—
506C2×C4C2×#120114yes—
507C2×C4C2×#120114yes—
508C2×C4C2×#1201116yescomplemented
509C2×C4C22×D3×D512yes—
510C2×C4C22×D3×D512yes—
511C2×C3—24no—
512C2×C3—204no—
513C2×C3#6616614yescomplemented
514C2×C3C2×D5×#4112yes—
515C2×C3C23×C22⋊D511yescharacteristic
516D3—44no—
517C5C2×#583211yescharacteristic, complemented, Sylow
518C22—28no—
519C22—24no—
520C22—64no—
521C22—68no—
522C22—104no—
523C22—108no—
524C22—128no—
525C22—208no—
526C22#9201614yes—
527C22C22×#120114yes—
528C22C22×#120112yes—
529C22C23×D3×D511yescharacteristic
530C4—64no—
531C4—104no—
532C4—158no—
533C4—304no—
534C4C22×#120114yes—
535C4C22×#120114yes—
536C3C2×#6616611yescharacteristic, complemented, Sylow
537C2—24no—
538C2—124no—
539C2—204no—
540C2#1175458614yesdirect product factor
541C2C2×#9201612yes—
542C2C23×#120111yescharacteristic
subgroupG/H
centerC22×C4C2×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC23×C3×C4×C5C22
socleC23×C3×C5C24
hypercenterC23×C4D3×D5
hypocenterC3×C5C2×#532
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×C5×#235C2semidirect
2. C22×C3×#709C2semidirect
3. C22×C4×Dic15C2semidirect
4. C2×#91571C2semidirect
5. #11754586C2direct (N × H)
6. C2×#91463C2semidirect
7. C2×C4×#1201C2semidirect
8. C2×#91385C2semidirect
9. C2×#91370C2semidirect
10. C2×#91368C2semidirect
16. C2×C5×#235C22semidirect
22. C2×C3×#709C22semidirect
30. C2×C4×Dic15C22semidirect
31. #91571C22semidirect
35. #91463C22semidirect
36. C4×#1201C22semidirect
37. #91385C22semidirect
38. #91370C22semidirect
39. #91368C22semidirect
42. C22×#709D3semidirect
54. C22×C3×C4×C5D4semidirect
91. C22×#235D5semidirect
111. C2×#709C2×D3semidirect
141. C2×C3×C4×C5C2×D4semidirect
146. C2×Dic15#41semidirect
173. C2×#235C2×D5semidirect
203. C22×C4×C5#67semidirect
254. Dic15C2×#41semidirect
270. C22×C3×C4C22⋊D5semidirect
314. C2×C4×C5C2×#67semidirect
358. C2×C3×C5#532semidirect
381. C2×C3×C4C2×C22⋊D5semidirect
443. C22×C4#1201semidirect
453. C3×C5C2×#532semidirect
474. C2×C5#5832semidirect
508. C2×C4C2×#1201semidirect
513. C2×C3#66166semidirect
517. C5C2×#5832semidirect
536. C3C2×#66166semidirect
540. C2#11754586direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#2352
C22×C3×#7092
C22×C4×Dic152
C2×#915712
#117545862
C2×#914632
C2×C4×#12012
C2×#913852
C2×#913702
C2×#913682
C2×#661663
C2×#58325