Finite Group Databasebeta

← #14321116

Subgroups — #14321116

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4018C211yesmaximal, characteristic
2C2×C4×D5×Dic3C211yesmaximal, characteristic
3C2×#91357C211yesmaximal, characteristic
4C4×#6712C211yesmaximal, characteristic, complemented
5C4×C5×Dic3⋊C4C211yesmaximal, characteristic, complemented
6#11748891C212yesmaximal
7#11748890C212yesmaximal
8C4×Dic3⋊Dic5C211yesmaximal, characteristic, complemented
9C4×#6669C211yesmaximal, characteristic, complemented
10#11748629C212yesmaximal, complemented
11#11748627C212yesmaximal, complemented
12#10605486—31nomaximal, Hall
13C22×C3×C4×D5C2211yescharacteristic
14C22×D5×Dic3C2212yes—
15C2×C3×C42×C5C2211yescharacteristic
16C2×#6712C2212yes—
17C2×C4×Dic15C2211yescharacteristic
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×Dic3⋊C4C2212yes—
20C2×C4×C5×Dic3C2211yescharacteristic
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C3×#685C2212yes—
23C2×C3×#683C2212yes—
24C2×C3×C4×Dic5C2211yescharacteristic
25C3×#4018—24no—
26C2×Dic3⋊Dic5C2212yes—
27C2×#6669C2212yes—
28C2×#6668C2211yescharacteristic
29C2×#6668C2211yescharacteristic
30C2×#6665C2211yescharacteristic
31C2×#6665C2211yescharacteristic
32C2×Dic3×Dic5C2212yes—
33C4×D5×Dic3—24no—
34#91357C418yescomplemented
35#91115C2212yes—
36C5×C3⋊#95C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82226—51nomaximal, Hall
40C2×#4018—31no—
41C2×#4018D311yescharacteristic
42C2×C42×D5—31no—
43C4×C5×C4⋊C4—31no—
44#65278—32no—
45#65261—32no—
46C4×C4⋊Dic5—31no—
47C4×#683—31no—
48C4×#683—31no—
49#64957—32no—
50#64936—32no—
51C23×C3×D5C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2312yes—
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2311yescharacteristic
59C22×C5×Dic3C2312yes—
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2311yescharacteristic
62C2×C3×C4×D5C2×C414yes—
63C2×C3×C4×D5D414yes—
64C2×C3×C4×D5Q814yes—
65C2×D5×Dic3—24no—
66C2×D5×Dic3—24no—
67C2×D5×Dic3—24no—
68C3×C42×C5—22no—
69#6712—24no—
70C4×Dic15—22no—
71C4×Dic15C2×C414yescomplemented
72C5×Dic3⋊C4—24no—
73C4×C5×Dic3—22no—
74C4×C5×Dic3C2×C414yescomplemented
75C3×#685—24no—
76C3×#683—24no—
77C3×C4×Dic5—22no—
78Dic3⋊Dic5—24no—
79#6669—24no—
80#6668C2×C414yescomplemented
81#6668C2×C414yescomplemented
82#6665C2×C414yes—
83#6665C2×C414yes—
84Dic3×Dic5—24no—
85C2×C3×#117—51no—
86C2×#926—51no—
87C22×C4×Dic3—51no—
88#4972—52no—
89#4972—52no—
90#4946—52no—
91#4946—52no—
92C4×Dic3⋊C4—51no—
93C4×Dic3⋊C4—51no—
94C4×Dic3⋊C4—51no—
95C4×Dic3⋊C4D511yescharacteristic, complemented
96C22×C4×D5—32no—
97C22×C4×D5C2×D311yescharacteristic
98C2×C5×C4⋊C4—32no—
99C2×C42×C5—31no—
100C2×C42×C5—31no—
101C2×C42×C5C2×D311yescharacteristic
102C2×#685—31no—
103C2×#685—31no—
104C2×#685C2×D312yes—
105C2×C4⋊Dic5—32no—
106C2×#683—31no—
107C2×#683—31no—
108C2×#683—32no—
109C2×#683—32no—
110C2×#683C2×D312yes—
111C2×C4×Dic5—31no—
112C2×C4×Dic5—31no—
113C2×C4×Dic5—32no—
114C2×C4×Dic5C2×D311yescharacteristic
115#4018—24no—
116#4018—38no—
117C42×D5—64no—
118C5×#95—32no—
119#3963—32no—
120#3963—32no—
121#3963—32no—
122#2296—151noSylow
123C23×C3×C5C2411yescharacteristic
124C22×C3×D5—22no—
125C22×C3×D5—22no—
126C22×C3×D5C22×C412yes—
127C22×C3×D5C2×D412yes—
128C22×C3×D5C2×Q812yes—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5C22×C412yes—
133C2×C3×C4×C5C2×D412yes—
134C2×C3×C4×C5C2×Q812yes—
135C2×Dic15—22no—
136C2×Dic15—22no—
137C2×Dic15—22no—
138C2×Dic15—22no—
139C2×Dic15C22×C412yes—
140C2×Dic15C22×C412yes—
141C2×Dic15#4114yes—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3C22×C412yes—
147C2×C5×Dic3C22×C412yes—
148C2×C5×Dic3#4114yes—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—22no—
152C2×C3×Dic5C22×C412yes—
153C2×C3×Dic5C2×D412yes—
154C2×C3×Dic5C2×Q812yes—
155C3×C4×D5C4⋊C4116yescomplemented
156D5×Dic3—216no—
157C23×C3×C4—51no—
158C23×Dic3—52no—
159C3×#117—104no—
160C2×C3×C4⋊C4—52no—
161C2×C3×C22⋊C4—52no—
162C2×C3×C42—51no—
163C2×C3×C42C2×D511yescharacteristic
164C2×#217—51no—
165C2×#217—51no—
166#926—58no—
167C2×C4⋊Dic3—51no—
168C2×C4⋊Dic3—51no—
169C2×Dic3⋊C4—52no—
170C2×Dic3⋊C4—52no—
171C2×Dic3⋊C4—52no—
172C2×Dic3⋊C4C2×D512yes—
173C2×C4×Dic3—51no—
174C2×C4×Dic3—51no—
175C2×C4×Dic3—52no—
176C2×C4×Dic3—104no—
177C2×C4×Dic3C2×D511yescharacteristic
178C2×C4×Dic3C2×D511yescharacteristic
179C3⋊#95—52no—
180C3⋊#95—52no—
181C3⋊#95—52no—
182C3⋊#95C2×D512yes—
183C23×D5C22×D311yescharacteristic
184C22×C4×C5—31no—
185C22×C4×C5—32no—
186C22×C4×C5—31no—
187C22×C4×C5C22×D312yes—
188C22×C4×C5C22×D311yescharacteristic
189C22×Dic5—32no—
190C22×Dic5—31no—
191C22×Dic5—31no—
192C22×Dic5C22×D312yes—
193C22×Dic5C22×D311yescharacteristic
194C2×C4×D5—64no—
195C2×C4×D5—64no—
196C2×C4×D5—64no—
197C2×C4×D5Dic614yes—
198C2×C4×D5C4×D314yes—
199C2×C4×D5#6714yes—
200C5×C4⋊C4—64no—
201C42×C5—22no—
202C42×C5—34no—
203C42×C5—62no—
204#685—24no—
205#685—34no—
206#685—34no—
207C4⋊Dic5—64no—
208#683—24no—
209#683—34no—
210#683—34no—
211#683—64no—
212#683—64no—
213C4×Dic5—22no—
214C4×Dic5—34no—
215C4×Dic5—62no—
216C4×Dic5—64no—
217C2×#117—51no—
218C2×#117—151no—
219C22×C42—151no—
220#389—152no—
221#389—152no—
222#385—152no—
223#385—152no—
224C4×C4⋊C4—31no—
225C4×C4⋊C4—151no—
226C4×C4⋊C4—151no—
227C4×C4⋊C4—151no—
228C22×C3×C5C23×C411yescharacteristic
229C22×C3×C5C22×D411yescharacteristic
230C22×C3×C5C22×Q811yescharacteristic
231C22×C3×C5C2×#4112yes—
232C22×C3×C5C2×#4112yes—
233C2×C3×D5—22no—
234C2×C3×D5—22no—
235C2×C3×D5—24no—
236C2×C3×D5—24no—
237C2×C3×D5C2×C4⋊C414yes—
238C3×C4×C5—42no—
239C3×C4×C5C2×C4⋊C414yes—
240Dic15—22no—
241Dic15—22no—
242Dic15—24no—
243C3×Dic5—42no—
244C3×Dic5C2×C4⋊C414yes—
245C5×Dic3—22no—
246C5×Dic3—22no—
247C5×Dic3—24no—
248C24×C3—51no—
249C22×C3×C4—51no—
250C22×C3×C4—54no—
251C22×C3×C4—54no—
252C22×C3×C4—54no—
253C22×C3×C4—52no—
254C22×C3×C4C22×D512yes—
255C22×C3×C4C22×D511yescharacteristic
256C22×Dic3—52no—
257C22×Dic3—51no—
258C22×Dic3—51no—
259C22×Dic3—104no—
260C22×Dic3—104no—
261C22×Dic3—104no—
262C22×Dic3C22×D511yescharacteristic
263C22×Dic3C22×D511yescharacteristic
264C22×Dic3C22×D512yes—
265C3×C4⋊C4—104no—
266C3×C22⋊C4—104no—
267C3×C42—22no—
268C3×C42—102no—
269#217—54no—
270#217—54no—
271C4⋊Dic3—54no—
272C4⋊Dic3—54no—
273Dic3⋊C4—24no—
274Dic3⋊C4—104no—
275Dic3⋊C4—104no—
276Dic3⋊C4—104no—
277C4×Dic3—22no—
278C4×Dic3—54no—
279C4×Dic3—104no—
280C4×Dic3—102no—
281C4×Dic3C4×D514yescomplemented
282C23×C5C23×D311yescharacteristic
283C22×D5—22no—
284C22×D5—22no—
285C22×D5C2×Dic612yes—
286C22×D5C2×C4×D312yes—
287C22×D5C2×#6712yes—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—32no—
292C2×C4×C5—32no—
293C2×C4×C5—34no—
294C2×C4×C5—62no—
295C2×C4×C5—62no—
296C2×C4×C5—62no—
297C2×C4×C5—62no—
298C2×C4×C5C2×Dic612yes—
299C2×C4×C5C2×C4×D312yes—
300C2×C4×C5C2×#6712yes—
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—32no—
305C2×Dic5—32no—
306C2×Dic5—34no—
307C2×Dic5—62no—
308C2×Dic5—62no—
309C2×Dic5—62no—
310C2×Dic5—62no—
311C2×Dic5C2×Dic612yes—
312C2×Dic5C2×C4×D312yes—
313C2×Dic5C2×#6712yes—
314C4×D5—616no—
315C4×D5Dic3⋊C4116yescomplemented
316C23×C4—51no—
317C23×C4—152no—
318#117—104no—
319#117—158no—
320C2×C4⋊C4—32no—
321C2×C4⋊C4—52no—
322C2×C4⋊C4—151no—
323C2×C4⋊C4—151no—
324C2×C4⋊C4—152no—
325C2×C4⋊C4—152no—
326C2×C4⋊C4—152no—
327C2×C22⋊C4—52no—
328C2×C22⋊C4—151no—
329C2×C22⋊C4—151no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—51no—
333C2×C42—151no—
334C2×C42—152no—
335C2×C42—151no—
336C2×C42—304no—
337C2×C42D3×D511yescharacteristic
338#95—32no—
339#95—152no—
340#95—152no—
341#95—152no—
342C2×C3×C5C22×C4⋊C411yescharacteristic
343C2×C3×C5C4×#4111yescharacteristic
344C2×C3×C5C4×#4111yescharacteristic
345C2×C3×C5#52512yes—
346C2×C3×C5#53312yes—
347C3×D5—24no—
348C23×C3—52no—
349C23×C3—52no—
350C23×C3—52no—
351C23×C3—102no—
352C23×C3—102no—
353C23×C3C23×D511yescharacteristic
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—52no—
358C2×C3×C4—52no—
359C2×C3×C4—52no—
360C2×C3×C4—516no—
361C2×C3×C4—102no—
362C2×C3×C4—102no—
363C2×C3×C4—102no—
364C2×C3×C4C2×C4×D512yes—
365C2×C3×C4D4×D512yes—
366C2×C3×C4Q8×D512yes—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—52no—
372C2×Dic3—52no—
373C2×Dic3—54no—
374C2×Dic3—1016no—
375C2×Dic3—102no—
376C2×Dic3—102no—
377C2×Dic3—102no—
378C2×Dic3—102no—
379C2×Dic3C2×C4×D512yes—
380C2×Dic3C2×C4×D512yes—
381C2×Dic3#70914yes—
382C22×C5C22×Dic611yescharacteristic
383C22×C5C22×C4×D311yescharacteristic
384C22×C5D3×#4112yes—
385C22×C5D3×#4112yes—
386C22×C5C22×#6711yescharacteristic
387C2×D5—24no—
388C2×D5—22no—
389C2×D5—22no—
390C2×D5—24no—
391C2×D5C2×Dic3⋊C414yes—
392C4×C5—42no—
393C4×C5—62no—
394C4×C5—62no—
395C4×C5—64no—
396C4×C5C2×Dic3⋊C414yes—
397Dic5—42no—
398Dic5—62no—
399Dic5—62no—
400Dic5—64no—
401Dic5C2×Dic3⋊C414yes—
402C24—51no—
403C22×C4—31no—
404C22×C4—32no—
405C22×C4—31no—
406C22×C4—51no—
407C22×C4—54no—
408C22×C4—54no—
409C22×C4—54no—
410C22×C4—52no—
411C22×C4—152no—
412C22×C4—151no—
413C22×C4—151no—
414C22×C4—304no—
415C22×C4—304no—
416C22×C4—304no—
417C22×C4C2×D3×D512yes—
418C22×C4C2×D3×D511yescharacteristic
419C4⋊C4—64no—
420C4⋊C4—104no—
421C4⋊C4—154no—
422C4⋊C4—154no—
423C4⋊C4—304no—
424C4⋊C4—304no—
425C4⋊C4—304no—
426C22⋊C4—104no—
427C22⋊C4—154no—
428C22⋊C4—154no—
429C42—22no—
430C42—34no—
431C42—62no—
432C42—102no—
433C42—154no—
434C42—304no—
435C42—302no—
436C3×C5#229611yescharacteristic, complemented, Hall
437C22×C3—54no—
438C22×C3—102no—
439C22×C3—102no—
440C22×C3—104no—
441C22×C3—104no—
442C22×C3C22×C4×D511yescharacteristic
443C22×C3C2×#70912yes—
444C22×C3C2×D4×D511yescharacteristic
445C22×C3C2×Q8×D511yescharacteristic
446C22×C3D5×#4112yes—
447C3×C4—42no—
448C3×C4—54no—
449C3×C4—202no—
450C3×C4D5×C4⋊C414yes—
451Dic3—22no—
452Dic3—22no—
453Dic3—24no—
454Dic3—102no—
455Dic3—102no—
456Dic3—104no—
457C2×C5#551712yes—
458C2×C5#552211yescharacteristic
459C2×C5#552211yescharacteristic
460C2×C5C22×Dic3⋊C411yescharacteristic
461C2×C5#582812yes—
462D5—24no—
463C23—52no—
464C23—52no—
465C23—52no—
466C23—102no—
467C23—102no—
468C23C22×D3×D511yescharacteristic
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—32no—
473C2×C4—32no—
474C2×C4—34no—
475C2×C4—52no—
476C2×C4—52no—
477C2×C4—52no—
478C2×C4—516no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—102no—
484C2×C4—102no—
485C2×C4—102no—
486C2×C4—152no—
487C2×C4—152no—
488C2×C4—154no—
489C2×C4—3016no—
490C2×C4—302no—
491C2×C4—302no—
492C2×C4—302no—
493C2×C4—302no—
494C2×C4D5×Dic612yes—
495C2×C4C4×D3×D512yes—
496C2×C4D5×#6712yes—
497C2×C3—104no—
498C2×C3C4×#70911yescharacteristic
499C2×C3C2×D5×C4⋊C411yescharacteristic
500C2×C3#6585611yescharacteristic
501C2×C3#6588112yes—
502C2×C3#6591212yes—
503C5#8222611yescharacteristic, complemented, Sylow
504C22—54no—
505C22—104no—
506C22—104no—
507C22—102no—
508C22—102no—
509C22C2×D5×Dic611yescharacteristic
510C22C2×C4×D3×D511yescharacteristic
511C22D3×#70912yes—
512C22#9201612yes—
513C22C2×D5×#6711yescharacteristic
514C4—42no—
515C4—54no—
516C4—64no—
517C4—62no—
518C4—62no—
519C4—202no—
520C4—304no—
521C4—302no—
522C4—302no—
523C4D5×Dic3⋊C414yes—
524C3#1060548611yescharacteristic, complemented, Sylow
525C2—104no—
526C2#1175259612yes—
527C2#1175266611yescharacteristic
528C2#1175266711yescharacteristic
529C2C2×D5×Dic3⋊C411yescharacteristic
530C2#1175463112yes—
subgroupG/H
centerC22×C4C2×D3×D5
commutatorC22×C3×C5C23×C4
FrattiniC23C22×D3×D5
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C3×D5C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#2296
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
4. C4×#6712C2semidirect
5. C4×C5×Dic3⋊C4C2semidirect
8. C4×Dic3⋊Dic5C2semidirect
9. C4×#6669C2semidirect
10. #11748629C2semidirect
11. #11748627C2semidirect
34. #91357C4semidirect
71. C4×Dic15C2×C4semidirect
74. C4×C5×Dic3C2×C4semidirect
80. #6668C2×C4semidirect
81. #6668C2×C4semidirect
95. C4×Dic3⋊C4D5semidirect
155. C3×C4×D5C4⋊C4semidirect
281. C4×Dic3C4×D5semidirect
315. C4×D5Dic3⋊C4semidirect
436. C3×C5#2296semidirect
503. C5#82226semidirect
524. C3#10605486semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40182
C2×C4×D5×Dic32
C2×#913572
C4×#67122
C4×C5×Dic3⋊C42
#117488912
#117488902
C4×Dic3⋊Dic52
C4×#66692
#117486292
#117486272
#106054863
#822265