Finite Group Databasebeta

← C2×#11752601 #14484048

Subgroups — C2×#11752601 #14484048

Order 1920.

2224 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic30C211yesmaximal, characteristic, complemented
2C2×C4×C5×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#4021C211yesmaximal, characteristic
4C2×Dic3×D20C211yesmaximal, characteristic
5C2×#91421C212yesmaximal, complemented
6C2×#91394C212yesmaximal, complemented
7C2×#91389C211yesmaximal, characteristic, complemented
8C2×#91366C212yesmaximal, complemented
9C2×#91364C211yesmaximal, characteristic
10C2×#91361C212yesmaximal, complemented
11C2×Dic3×Dic10C211yesmaximal, characteristic, complemented
12#11752601C2116yesmaximal, direct product factor
13C2×#65918—31nomaximal, Hall
14C22×Dic30C2211yescharacteristic
15C22×C5×Dic6C2211yescharacteristic
16C22×C3×D20C2211yescharacteristic
17C22×C3×Dic10C2211yescharacteristic
18C22×D5×Dic3C2212yes—
19C2×C3×C42×C5C2211yescharacteristic
20C2×C4⋊Dic15C2211yescharacteristic
21C2×#6712C2212yes—
22C2×C4×Dic15C2212yes—
23C4×Dic30C2218yescomplemented
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×Dic3⋊C4C2211yescharacteristic
26C2×C5×Dic3⋊C4C2211yescharacteristic
27C2×C4×C5×Dic3C2211yescharacteristic
28C2×C4×C5×Dic3C2211yescharacteristic
29C4×C5×Dic6C2218yescomplemented
30C2×C3×#685C2212yes—
31C2×C3×#685C2212yes—
32C3×#4021C2218yes—
33C2×Dic3⋊Dic5C2212yes—
34C2×Dic3⋊Dic5C2212yes—
35C2×#6668C2212yes—
36C2×#6665C2212yes—
37C2×#6665C2212yes—
38C2×#6665C2212yes—
39C2×Dic3×Dic5C2212yes—
40Dic3×D20C2218yes—
41#91421C22116yescomplemented
42#91394C22116yescomplemented
43#91389C2218yescomplemented
44#91366C22116yescomplemented
45#91364C2218yes—
46#91361C22116yescomplemented
47Dic3×Dic10C2218yescomplemented
48C2×#5671—51nomaximal, Hall
49C2×C4×C5×Q8—31no—
50C2×#4092—32no—
51#65918—316no—
52C2×#4044—32no—
53C2×#4044—32no—
54C2×#4038—32no—
55C2×#4021—31no—
56C2×#4021D311yescharacteristic
57C2×C4×D20—31no—
58C2×#4015—31no—
59C2×C4×Dic10—31no—
60C2×C4×Dic10—31no—
61C23×C3×D5C2312yes—
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2311yescharacteristic
64C22×C3×C4×C5C2311yescharacteristic
65C22×Dic15C2312yes—
66C22×Dic15C2312yes—
67C2×Dic30—24no—
68C2×Dic30C2314yes—
69C22×C5×Dic3C2311yescharacteristic
70C22×C5×Dic3C2311yescharacteristic
71C22×C5×Dic3C2311yescharacteristic
72C22×C5×Dic3C2311yescharacteristic
73C2×C5×Dic6—24no—
74C2×C5×Dic6C2314yes—
75C22×C3×Dic5C2312yes—
76C2×C3×D20—24no—
77C2×C3×D20C2314yes—
78C2×C3×Dic10—24no—
79C2×C3×Dic10C2314yes—
80C2×D5×Dic3—28no—
81C2×D5×Dic3C2318yes—
82C3×C42×C5C2314yes—
83C4⋊Dic15C2314yes—
84#6712C2318yes—
85C4×Dic15C2318yes—
86C5×C4⋊Dic3C2314yes—
87C5×Dic3⋊C4C2314yes—
88C5×Dic3⋊C4C2314yes—
89C4×C5×Dic3C2314yes—
90C4×C5×Dic3C2314yes—
91C3×#685C2318yes—
92C3×#685C2318yes—
93Dic3⋊Dic5C2318yes—
94Dic3⋊Dic5C2318yes—
95#6668C2318yes—
96#6665C2318yes—
97#6665C2318yes—
98#6665C2318yes—
99Dic3×Dic5C2318yes—
100C2×C3×#124—51no—
101C2×Q8×Dic3—51no—
102C2×#944—51no—
103C2×#936—51no—
104C2×D4×Dic3—51no—
105#5671—516no—
106C2×#879—52no—
107C2×#879—52no—
108C2×#878—52no—
109C2×#877—52no—
110C2×C4×Dic6—51no—
111C2×C4×Dic6D511yescharacteristic, complemented
112C22×C5×Q8—31no—
113C22×D20C2×D311yescharacteristic
114C22×C4×D5—32no—
115C22×Dic10—31no—
116C22×Dic10C2×D311yescharacteristic
117C4×C5×Q8—38no—
118C2×C5×C4⋊C4—31no—
119C2×C5×C4⋊C4—31no—
120C2×C5×C4⋊C4—31no—
121C2×C42×C5—31no—
122C2×C42×C5—31no—
123C2×C42×C5C2×D311yescharacteristic
124#4092—316no—
125C2×#685—32no—
126C2×#685—32no—
127C2×#685—32no—
128C2×#685C2×D312yes—
129C2×#685C2×D312yes—
130C2×C4⋊Dic5—31no—
131C2×C4⋊Dic5—32no—
132C2×C4⋊Dic5—32no—
133C2×#683—32no—
134C2×#683—32no—
135C2×C4×Dic5—32no—
136C2×C4×Dic5—32no—
137#4044—316no—
138#4044—316no—
139#4038—316no—
140#4021—38no—
141#4021C2×D318yes—
142C4×D20—38no—
143#4015—38no—
144C4×Dic10—38no—
145C4×Dic10—38no—
146C2×#556—151noSylow
147C23×C3×C5C2411yescharacteristic
148C22×C3×D5—24no—
149C22×C3×D5—24no—
150C22×C3×D5—24no—
151C22×C3×D5C2414yes—
152C2×C3×C4×C5—22no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5#4114yes—
155C2×C3×C4×C5C2412yes—
156C2×C3×C4×C5C2412yes—
157C2×C3×C4×C5C2412yes—
158C2×Dic15—24no—
159C2×Dic15—24no—
160C2×Dic15C2414yes—
161C2×Dic15C2414yes—
162Dic30—28no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3#4114yes—
167C2×C5×Dic3C2412yes—
168C2×C5×Dic3C2412yes—
169C2×C5×Dic3C2412yes—
170C2×C5×Dic3C2412yes—
171C5×Dic6—28no—
172C2×C3×Dic5—24no—
173C2×C3×Dic5C2414yes—
174C3×D20—28no—
175C3×Dic10—28no—
176D5×Dic3—216no—
177C22×C3×Q8—51no—
178C22×C3×D4—51no—
179C23×Dic3—52no—
180C22×Dic6—51no—
181C22×Dic6C2×D511yescharacteristic
182C3×#124—58no—
183C2×C3×C22⋊C4—52no—
184C2×C3×C22⋊C4—52no—
185C2×C3×C42C2×D511yescharacteristic
186C2×#217—52no—
187C2×#217—52no—
188C2×#217—52no—
189Q8×Dic3—58no—
190#944—58no—
191#936—58no—
192D4×Dic3—58no—
193C2×C4⋊Dic3—51no—
194C2×C4⋊Dic3—52no—
195C2×C4⋊Dic3C2×D511yescharacteristic
196C2×Dic3⋊C4—52no—
197C2×Dic3⋊C4—52no—
198C2×Dic3⋊C4—52no—
199C2×Dic3⋊C4C2×D511yescharacteristic
200C2×Dic3⋊C4C2×D511yescharacteristic
201C2×C4×Dic3—52no—
202C2×C4×Dic3—52no—
203C2×C4×Dic3C2×D511yescharacteristic
204C2×C4×Dic3C2×D511yescharacteristic
205#879—516no—
206#879—516no—
207#878—516no—
208#877—516no—
209C4×Dic6—58no—
210C4×Dic6C2×D518yescomplemented
211C23×D5C22×D312yes—
212C2×C5×Q8—34no—
213C2×C5×Q8—64no—
214C22×C4×C5—31no—
215C22×C4×C5—31no—
216C22×C4×C5—31no—
217C22×C4×C5—31no—
218C22×C4×C5C22×D311yescharacteristic
219C22×C4×C5C22×D311yescharacteristic
220C22×C4×C5C22×D311yescharacteristic
221C22×Dic5—32no—
222C22×Dic5—32no—
223C22×Dic5C22×D312yes—
224C2×D20—24no—
225C2×D20C22×D314yes—
226C2×C4×D5—38no—
227C2×C4×D5—68no—
228C2×Dic10—24no—
229C2×Dic10—34no—
230C2×Dic10—64no—
231C2×Dic10C22×D314yes—
232C5×C4⋊C4—34no—
233C5×C4⋊C4—34no—
234C5×C4⋊C4—34no—
235C42×C5—34no—
236C42×C5—34no—
237C42×C5C22×D314yes—
238#685—38no—
239#685—38no—
240#685—38no—
241#685C22×D318yes—
242#685C22×D318yes—
243C4⋊Dic5—34no—
244C4⋊Dic5—38no—
245C4⋊Dic5—38no—
246#683—38no—
247#683—38no—
248C4×Dic5—38no—
249C4×Dic5—38no—
250#556—1516no—
251C2×C4⋊Q8—151no—
252C2×#126—152no—
253C2×#126—152no—
254C2×#124—51no—
255C2×#124—151no—
256C2×C22⋊Q8—152no—
257C2×C4×Q8—31no—
258C2×C4×Q8—151no—
259C2×C4×Q8—151no—
260C2×C4×D4—151no—
261C2×#117—152no—
262C22×C3×C5C2×#4112yes—
263C22×C3×C5C2×#4112yes—
264C22×C3×C52−1+412yes—
265C22×C3×C5C2511yescharacteristic
266C2×C3×D5—24no—
267C2×C3×D5—24no—
268C2×C3×D5—24no—
269C2×C3×D5—48no—
270C3×C4×C5—22no—
271C3×C4×C5—22no—
272C3×C4×C5C2×#4114yes—
273Dic15—24no—
274Dic15—24no—
275C3×Dic5—24no—
276C5×Dic3—22no—
277C5×Dic3—22no—
278C5×Dic3—22no—
279C5×Dic3C2×#4114yes—
280C24×C3—52no—
281C2×C3×Q8—54no—
282C2×C3×Q8—104no—
283C2×C3×D4—54no—
284C2×C3×D4—104no—
285C22×C3×C4—52no—
286C22×C3×C4C22×D511yescharacteristic
287C22×C3×C4C22×D511yescharacteristic
288C22×C3×C4C22×D511yescharacteristic
289C22×Dic3—52no—
290C22×Dic3—52no—
291C22×Dic3—58no—
292C22×Dic3—108no—
293C22×Dic3C22×D511yescharacteristic
294C22×Dic3C22×D511yescharacteristic
295C22×Dic3C22×D511yescharacteristic
296C22×Dic3C22×D511yescharacteristic
297C2×Dic6—24no—
298C2×Dic6—54no—
299C2×Dic6—104no—
300C2×Dic6C22×D514yes—
301C3×C22⋊C4—58no—
302C3×C22⋊C4—58no—
303C3×C42C22×D514yes—
304#217—58no—
305#217—58no—
306#217—58no—
307C4⋊Dic3—54no—
308C4⋊Dic3—58no—
309C4⋊Dic3C22×D514yes—
310Dic3⋊C4—58no—
311Dic3⋊C4—58no—
312Dic3⋊C4—58no—
313Dic3⋊C4C22×D514yes—
314Dic3⋊C4C22×D514yes—
315C4×Dic3—58no—
316C4×Dic3—58no—
317C4×Dic3C22×D514yes—
318C4×Dic3C22×D514yes—
319C23×C5C23×D311yescharacteristic
320C22×D5—24no—
321C22×D5—24no—
322C22×D5—24no—
323C22×D5C23×D314yes—
324C5×Q8—68no—
325C2×C4×C5—22no—
326C2×C4×C5—22no—
327C2×C4×C5—34no—
328C2×C4×C5—32no—
329C2×C4×C5—32no—
330C2×C4×C5—32no—
331C2×C4×C5—32no—
332C2×C4×C5—62no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5#23714yes—
336C2×C4×C5C23×D312yes—
337C2×C4×C5C23×D312yes—
338C2×C4×C5C23×D312yes—
339C2×Dic5—24no—
340C2×Dic5—34no—
341C2×Dic5—34no—
342C2×Dic5—64no—
343C2×Dic5—64no—
344C2×Dic5C23×D314yes—
345D20—28no—
346C4×D5—616no—
347Dic10—28no—
348Dic10—68no—
349C22×Q8—31no—
350C22×Q8—51no—
351C22×Q8—151no—
352C22×D4—51no—
353C23×C4—152no—
354C4⋊Q8—158no—
355#126—1516no—
356#126—1516no—
357#124—58no—
358#124—158no—
359C22⋊Q8—1516no—
360C4×Q8—38no—
361C4×Q8—158no—
362C4×Q8—158no—
363C4×D4—158no—
364#117—1516no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—31no—
367C2×C4⋊C4—31no—
368C2×C4⋊C4—152no—
369C2×C4⋊C4—152no—
370C2×C4⋊C4—152no—
371C2×C4⋊C4—151no—
372C2×C4⋊C4—152no—
373C2×C22⋊C4—52no—
374C2×C22⋊C4—52no—
375C2×C22⋊C4—152no—
376C2×C22⋊C4—152no—
377C2×C22⋊C4—152no—
378C2×C42—31no—
379C2×C42—31no—
380C2×C42—152no—
381C2×C42—152no—
382C2×C42D3×D511yescharacteristic
383C2×C3×C5#55614yescomplemented
384C2×C3×C5C22×#4111yescharacteristic
385C2×C3×C5C22×#4111yescharacteristic
386C2×C3×C5C2×2−1+411yescharacteristic
387C3×D5—44no—
388C23×C3—54no—
389C23×C3—104no—
390C23×C3—104no—
391C23×C3—104no—
392C23×C3C23×D511yescharacteristic
393C3×Q8—108no—
394C3×D4—108no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—54no—
398C2×C3×C4—104no—
399C2×C3×C4#71314yes—
400C2×C3×C4C23×D512yes—
401C2×C3×C4C23×D512yes—
402C2×C3×C4C23×D512yes—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—54no—
407C2×Dic3—54no—
408C2×Dic3—104no—
409C2×Dic3—104no—
410C2×Dic3—1016no—
411C2×Dic3#70914yes—
412C2×Dic3C23×D512yes—
413C2×Dic3C23×D512yes—
414C2×Dic3C23×D512yes—
415C2×Dic3C23×D512yes—
416Dic6—28no—
417Dic6—108no—
418C22×C5C2×#23712yes—
419C22×C5D3×#4112yes—
420C22×C5#101012yes—
421C22×C5C24×D311yescharacteristic
422C2×D5—24no—
423C2×D5—24no—
424C2×D5—24no—
425C2×D5—48no—
426C4×C5—22no—
427C4×C5—22no—
428C4×C5—34no—
429C4×C5—62no—
430C4×C5—62no—
431C4×C5—62no—
432C4×C5C2×#23714yes—
433Dic5—24no—
434Dic5—64no—
435Dic5—64no—
436C24—52no—
437C2×Q8—34no—
438C2×Q8—54no—
439C2×Q8—64no—
440C2×Q8—104no—
441C2×Q8—154no—
442C2×Q8—304no—
443C2×D4—54no—
444C2×D4—104no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—52no—
450C22×C4—158no—
451C22×C4—152no—
452C22×C4—152no—
453C22×C4—308no—
454C22×C4C2×D3×D511yescharacteristic
455C22×C4C2×D3×D511yescharacteristic
456C22×C4C2×D3×D511yescharacteristic
457C4⋊C4—34no—
458C4⋊C4—34no—
459C4⋊C4—34no—
460C4⋊C4—158no—
461C4⋊C4—158no—
462C4⋊C4—158no—
463C4⋊C4—154no—
464C4⋊C4—158no—
465C22⋊C4—58no—
466C22⋊C4—58no—
467C22⋊C4—158no—
468C22⋊C4—158no—
469C22⋊C4—158no—
470C42—34no—
471C42—34no—
472C42—158no—
473C42—158no—
474C42C2×D3×D514yes—
475C3×C5C2×#55611yescharacteristic, complemented, Hall
476C22×C3—104no—
477C22×C3—104no—
478C22×C3—104no—
479C22×C3—208no—
480C22×C3C2×#70912yes—
481C22×C3C2×#71312yes—
482C22×C3#415012yes—
483C22×C3C24×D511yescharacteristic
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—104no—
487C3×C4C2×#71314yes—
488Dic3—22no—
489Dic3—22no—
490Dic3—22no—
491Dic3—104no—
492Dic3—104no—
493Dic3C2×#70914yes—
494C2×C5#567114yescomplemented
495C2×C5C22×#23711yescharacteristic
496C2×C5C2×D3×#4111yescharacteristic
497C2×C5C2×#101011yescharacteristic
498D5—44no—
499C23—54no—
500C23—104no—
501C23—104no—
502C23—104no—
503C23C22×D3×D511yescharacteristic
504Q8—68no—
505Q8—108no—
506Q8—308no—
507D4—108no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—34no—
511C2×C4—32no—
512C2×C4—32no—
513C2×C4—32no—
514C2×C4—32no—
515C2×C4—54no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—104no—
520C2×C4—154no—
521C2×C4—154no—
522C2×C4—3016no—
523C2×C4—304no—
524C2×C4—304no—
525C2×C4#676614yes—
526C2×C4C22×D3×D512yes—
527C2×C4C22×D3×D512yes—
528C2×C4C22×D3×D512yes—
529C2×C3—204no—
530C2×C3#6591814yescomplemented
531C2×C3C22×#70911yescharacteristic
532C2×C3C22×#71311yescharacteristic
533C2×C3C2×#415011yescharacteristic
534C5C2×#567111yescharacteristic, complemented, Sylow
535C22—104no—
536C22—104no—
537C22—104no—
538C22—208no—
539C22C2×#676612yes—
540C22#9200112yes—
541C22D3×#70912yes—
542C22C23×D3×D511yescharacteristic
543C4—22no—
544C4—22no—
545C4—34no—
546C4—62no—
547C4—62no—
548C4—62no—
549C4—104no—
550C4—304no—
551C4—304no—
552C4C2×#676614yes—
553C3C2×#6591811yescharacteristic, complemented, Sylow
554C2—204no—
555C2#1175260114yesdirect product factor
556C2C22×#676611yescharacteristic
557C2C2×#9200111yescharacteristic
558C2C2×D3×#70911yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C42×C5C22
socleC23×C3×C5C24
prime radicalC22×C3×D20C22
hypercenterC2×C42D3×D5
hypocenterC3×C5C2×#556
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic30C2semidirect
2. C2×C4×C5×Dic6C2semidirect
5. C2×#91421C2semidirect
6. C2×#91394C2semidirect
7. C2×#91389C2semidirect
8. C2×#91366C2semidirect
10. C2×#91361C2semidirect
11. C2×Dic3×Dic10C2semidirect
12. #11752601C2direct (N × H)
23. C4×Dic30C22semidirect
29. C4×C5×Dic6C22semidirect
41. #91421C22semidirect
42. #91394C22semidirect
43. #91389C22semidirect
44. #91366C22semidirect
46. #91361C22semidirect
47. Dic3×Dic10C22semidirect
111. C2×C4×Dic6D5semidirect
210. C4×Dic6C2×D5semidirect
383. C2×C3×C5#556semidirect
475. C3×C5C2×#556semidirect
494. C2×C5#5671semidirect
530. C2×C3#65918semidirect
534. C5C2×#5671semidirect
553. C3C2×#65918semidirect
555. C2#11752601direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×Dic302
C2×C4×C5×Dic62
C2×C3×#40212
C2×Dic3×D202
C2×#914212
C2×#913942
C2×#913892
C2×#913662
C2×#913642
C2×#913612
C2×Dic3×Dic102
#117526012
C2×#659183
C2×#56715