Finite Group Databasebeta

← #14430714

Subgroups — #14430714

Order 1920.

768 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
1C2×C5×#890C211yesmaximal, characteristic, complemented
2Dic3×#709C211yesmaximal, characteristic
3#11751065C211yesmaximal, characteristic, complemented
4C3×#65255C211yesmaximal, characteristic
5#11748459C211yesmaximal, characteristic, complemented
6#11748440C211yesmaximal, characteristic, complemented
7#11748368C211yesmaximal, characteristic
8#11747296C212yesmaximal, complemented
9#11747295C212yesmaximal, complemented
10#11747251C212yesmaximal, complemented
11#11747249C212yesmaximal, complemented
12#10614748—31nomaximal, Hall
13C2×C3×#709C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15#91815C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17C2×C5×C4⋊Dic3—21no—
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C2×C5×Dic3⋊C4—21no—
20C2×C4×C5×Dic3C2211yescharacteristic
21C5×#890—21no—
22C5×#890—21no—
23C5×#890C2212yes—
24C5×#890C2212yes—
25C2×C3×#681C2211yescharacteristic
26Dic3×C22⋊D5—21no—
27Dic3×C22⋊D5—21no—
28Dic3×D20C2211yescharacteristic
29Dic3×D20C2211yescharacteristic
30C4×D5×Dic3—21no—
31#91357—21no—
32Dic3×Dic10C2211yescharacteristic
33Dic3×Dic10C2211yescharacteristic
34#91092C2212yes—
35#91091C2212yes—
36C3×#3942C2212yes—
37C3×#3941C2212yes—
38#90988C2212yes—
39#90985C2212yes—
40Dic3⋊#168C2212yes—
41Dic3⋊#168C2212yes—
42#90970C2212yes—
43#90964C2212yes—
44#78871—51nomaximal, Hall
45C2×C5×#125—31no—
46C4×#709—31no—
47#65355—32no—
48#65355—32no—
49#65352—32no—
50#65352—32no—
51#65255—31no—
52#65255D311yescharacteristic
53#65253—31no—
54#65253—31no—
55#65213—31no—
56C22×C3×C4×C5—21no—
57C22×C3×C4×C5C2311yescharacteristic
58C22×C5×Dic3—21no—
59C22×C5×Dic3D411yescharacteristic
60C22×C5×Dic3D411yescharacteristic
61C2×C3×C22⋊D5—21no—
62C3×#709—22no—
63C3×#709Q814yes—
64C2×C3×D20C2311yescharacteristic
65C2×C3×C4×D5—21no—
66C2×C3×Dic10C2311yescharacteristic
67C2×D5×Dic3—21no—
68C2×D5×Dic3—21no—
69C3×C5×C4⋊C4—21no—
70C3×C5×C4⋊C4C2312yes—
71#6719—21no—
72C4⋊Dic15C2311yescharacteristic
73C4⋊Dic15C2311yescharacteristic
74C4×Dic15—21no—
75#6710—22no—
76C2×C3⋊#168C2312yes—
77C5×C4⋊Dic3—21no—
78C5×C4⋊Dic3—22no—
79C5×C4⋊Dic3—22no—
80C5×C4⋊Dic3C2312yes—
81C5×Dic3⋊C4—22no—
82C5×Dic3⋊C4—22no—
83C4×C5×Dic3D411yescharacteristic
84C4×C5×Dic3D411yescharacteristic
85C4×C5×Dic3C2311yescharacteristic
86C4×C5×Dic3C2311yescharacteristic
87C3×#681—22no—
88C2×C3×#168C2312yes—
89#6668—21no—
90#6668—21no—
91#6665—21no—
92#6665—21no—
93Dic3×Dic5—21no—
94Dic3×Dic5—21no—
95Dic3×#41—51no—
96C2×#890D511yescharacteristic, complemented
97C3×#417—51no—
98#5232—51no—
99#5121—51no—
100#5118—51no—
101#4964—51no—
102#4799—52no—
103#4796—52no—
104#4766—52no—
105#4763—52no—
106C2×#709C2×D311yescharacteristic
107C5×#125—32no—
108C5×#125—32no—
109C5×#125—61no—
110C5×#125—61no—
111C2×C5×C4⋊C4—31no—
112C2×C5×C4⋊C4—61no—
113C2×C5×C4⋊C4—61no—
114C2×C5×C4⋊C4C2×D311yescharacteristic
115C2×C42×C5—31no—
116C4×C22⋊D5—61no—
117C4×C22⋊D5—61no—
118#4072—31no—
119C2×#681—31no—
120C2×#681C2×D311yescharacteristic
121C4×D20—31no—
122C4×D20—31no—
123#4018—61no—
124C42×D5—61no—
125C4×Dic10—31no—
126C4×Dic10—31no—
127#3942—32no—
128#3942C2×D312yes—
129#3941—32no—
130#3941C2×D312yes—
131#3940—32no—
132#3940—32no—
133#3939—32no—
134#3939—32no—
135C4⋊#168—32no—
136C4⋊#168—32no—
137#3080—151noSylow
138C23×C3×C5C2×D411yescharacteristic
139C22×C3×D5—21no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—42no—
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2×Q812yes—
144C2×C3×C4×C5#4112yes—
145C2×C3×C4×C5C2411yescharacteristic
146C2×Dic15—21no—
147C2×Dic15—21no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—41no—
150C2×C5×Dic3—41no—
151C2×C5×Dic3—42no—
152C2×C5×Dic3C2×D411yescharacteristic
153C2×C5×Dic3C2×D411yescharacteristic
154C2×C5×Dic3C2×D411yescharacteristic
155C2×C5×Dic3C2×D411yescharacteristic
156C3×C22⋊D5—24no—
157C2×C3×Dic5—21no—
158C3×D20—21no—
159C3×D20C2×Q812yes—
160C3×C4×D5—24no—
161C3×Dic10—21no—
162C3×Dic10C2×Q812yes—
163D5×Dic3—42no—
164D5×Dic3—42no—
165C3⋊#168—22no—
166C3×#168—22no—
167C2×C3×#41—51no—
168C2×C3×M16—51no—
169C2×C3×C4⋊C4C2×D511yescharacteristic
170Q8×Dic3—51no—
171Q8×Dic3—51no—
172D4×Dic3—51no—
173D4×Dic3—51no—
174D4×Dic3—101no—
175D4×Dic3—101no—
176#926—51no—
177#926—101no—
178C2×C4⋊Dic3—21no—
179C2×C4⋊Dic3C2×D511yescharacteristic
180C2×Dic3⋊C4—21no—
181C2×C4×Dic3—101no—
182C2×C4×Dic3C2×D511yescharacteristic
183C2×#208—51no—
184#890—21no—
185#890—21no—
186#890C2×D512yes—
187#890C2×D512yes—
188C3×#103—52no—
189C3×#102—52no—
190#835—52no—
191#832—52no—
192#818—52no—
193#817—52no—
194Dic3⋊C8—52no—
195Dic3⋊C8—52no—
196#808—52no—
197#807—52no—
198C22×C4×C5—21no—
199C22×C4×C5—31no—
200C22×C4×C5—31no—
201C22×C4×C5—61no—
202C22×C4×C5C22×D311yescharacteristic
203C2×C22⋊D5—21no—
204#709—22no—
205#709Dic614yes—
206C2×D20C22×D311yescharacteristic
207C2×C4×D5—21no—
208C2×C4×D5—61no—
209C2×C4×D5—61no—
210C2×Dic10C22×D311yescharacteristic
211C5×C4⋊C4—21no—
212C5×C4⋊C4—32no—
213C5×C4⋊C4—62no—
214C5×C4⋊C4—62no—
215C5×C4⋊C4—62no—
216C5×C4⋊C4—62no—
217C5×C4⋊C4—61no—
218C5×C4⋊C4C22×D312yes—
219C42×C5—31no—
220C42×C5—31no—
221C42×C5—31no—
222C42×C5—31no—
223C22⋊Dic5—61no—
224#685—61no—
225#685—61no—
226C4⋊Dic5—31no—
227C4⋊Dic5—31no—
228#683—61no—
229#683—61no—
230C4×Dic5—61no—
231C4×Dic5—61no—
232C4×Dic5—61no—
233#681—22no—
234#681—62no—
235C2×#168—32no—
236C2×#168C22×D312yes—
237C2×#125—31no—
238C4×#41—151no—
239#479—152no—
240#479—152no—
241#478—152no—
242#478—152no—
243#428—151no—
244#428—151no—
245#423—151no—
246#417—51no—
247#417—151no—
248C22×C3×C5—21no—
249C22×C3×C5—21no—
250C22×C3×C5C22⋊Q812yes—
251C22×C3×C5C22×D411yescharacteristic
252C2×C3×D5—22no—
253C2×C3×D5—41no—
254C2×C3×D5—41no—
255C3×C4×C5—42no—
256C3×C4×C5C22⋊Q812yes—
257C3×C4×C5C22×Q811yescharacteristic
258C3×C4×C5C2×#4111yescharacteristic
259Dic15—41no—
260Dic15—41no—
261C3×Dic5—22no—
262C5×Dic3—21no—
263C5×Dic3—21no—
264C5×Dic3—21no—
265C5×Dic3—21no—
266C5×Dic3—42no—
267C3×#41—54no—
268C3×#41—102no—
269C2×C3×Q8—51no—
270C2×C3×D4—51no—
271C2×C3×D4—101no—
272C22×C3×C4—21no—
273C22×C3×C4—101no—
274C22×C3×C4C22×D511yescharacteristic
275C22×Dic3—21no—
276C22×Dic3—101no—
277C22×Dic3—101no—
278C22×Dic3C22⋊D511yescharacteristic
279C22×Dic3C22⋊D511yescharacteristic
280C3×M16—102no—
281C2×C3×C8—52no—
282C3×C4⋊C4—21no—
283C3×C4⋊C4C22×D512yes—
284#217—101no—
285#217—101no—
286#217—101no—
287C4⋊Dic3—21no—
288C4⋊Dic3—22no—
289C4⋊Dic3—22no—
290C4⋊Dic3—51no—
291C4⋊Dic3—51no—
292C4⋊Dic3—101no—
293C4⋊Dic3—101no—
294C4⋊Dic3C22×D512yes—
295Dic3⋊C4—22no—
296Dic3⋊C4—22no—
297C4×Dic3—101no—
298C4×Dic3—101no—
299C4×Dic3—101no—
300C4×Dic3C22⋊D511yescharacteristic
301C4×Dic3C22⋊D511yescharacteristic
302C4×Dic3C22×D511yescharacteristic
303C4×Dic3C22×D511yescharacteristic
304#208—102no—
305C2×C3⋊C8—52no—
306C23×C5D3×D411yescharacteristic
307C22×D5—21no—
308C2×C4×C5—22no—
309C2×C4×C5—31no—
310C2×C4×C5—31no—
311C2×C4×C5—31no—
312C2×C4×C5—31no—
313C2×C4×C5—42no—
314C2×C4×C5—62no—
315C2×C4×C5—121no—
316C2×C4×C5—121no—
317C2×C4×C5—122no—
318C2×C4×C5C2×Dic612yes—
319C2×C4×C5D3×D411yescharacteristic
320C2×C4×C5#23712yes—
321C2×C4×C5C23×D311yescharacteristic
322C22⋊D5—24no—
323C2×Dic5—21no—
324C2×Dic5—61no—
325C2×Dic5—61no—
326D20—21no—
327D20C2×Dic612yes—
328C4×D5—24no—
329C4×D5—122no—
330C4×D5—122no—
331Dic10—21no—
332Dic10C2×Dic612yes—
333#168—22no—
334#168—62no—
335C2×#41—51no—
336C2×M16—51no—
337C2×M16—151no—
338#125—32no—
339#125—32no—
340#125—61no—
341#125—61no—
342C4×Q8—151no—
343C4×Q8—151no—
344C4×D4—151no—
345C4×D4—151no—
346C4×D4—301no—
347C4×D4—301no—
348#117—151no—
349#117—301no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4D3×D511yescharacteristic
354C2×C42—31no—
355C2×C42—301no—
356#107—152no—
357#107—152no—
358#106—152no—
359#106—152no—
360C4⋊C8—152no—
361C4⋊C8—152no—
362#103—52no—
363#103—152no—
364#102—52no—
365#102—152no—
366C2×C3×C5—22no—
367C2×C3×C5C2×C22⋊Q811yescharacteristic
368C2×C3×C5#57511yescharacteristic
369C2×C3×C5#57511yescharacteristic
370C3×D5—42no—
371C23×C3—101no—
372C23×C3C2×C22⋊D511yescharacteristic
373C3×Q8—52no—
374C3×Q8—101no—
375C3×D4—52no—
376C3×D4—104no—
377C3×D4—101no—
378C2×C3×C4—22no—
379C2×C3×C4—42no—
380C2×C3×C4—101no—
381C2×C3×C4—104no—
382C2×C3×C4Q8×D512yes—
383C2×C3×C4#71312yes—
384C2×C3×C4C2×C22⋊D511yescharacteristic
385C2×C3×C4C23×D511yescharacteristic
386C2×Dic3—22no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—101no—
391C2×Dic3—101no—
392C2×Dic3—202no—
393C2×Dic3—202no—
394C2×Dic3C2×C22⋊D511yescharacteristic
395C2×Dic3C2×C22⋊D511yescharacteristic
396C2×Dic3C2×C22⋊D511yescharacteristic
397C2×Dic3C2×C22⋊D511yescharacteristic
398C3×C8—102no—
399C3⋊C8—102no—
400C22×C5—21no—
401C22×C5—21no—
402C22×C5#87812yes—
403C22×C5C2×D3×D411yescharacteristic
404C2×D5—22no—
405C2×D5—41no—
406C2×D5—41no—
407C4×C5—42no—
408C4×C5—61no—
409C4×C5—61no—
410C4×C5—61no—
411C4×C5—61no—
412C4×C5—122no—
413C4×C5#87812yes—
414C4×C5C22×Dic611yescharacteristic
415C4×C5C2×#23711yescharacteristic
416Dic5—22no—
417Dic5—121no—
418Dic5—121no—
419#41—54no—
420#41—102no—
421C2×Q8—51no—
422C2×D4—51no—
423C2×D4—101no—
424C22×C4—21no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—61no—
428C22×C4—101no—
429C22×C4—301no—
430C22×C4—301no—
431C22×C4C2×D3×D511yescharacteristic
432M16—102no—
433M16—302no—
434C2×C8—52no—
435C2×C8—152no—
436C4⋊C4—21no—
437C4⋊C4—32no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—62no—
442C4⋊C4—61no—
443C4⋊C4—151no—
444C4⋊C4—151no—
445C4⋊C4—301no—
446C4⋊C4—301no—
447C4⋊C4C2×D3×D512yes—
448C22⋊C4—301no—
449C22⋊C4—301no—
450C22⋊C4—301no—
451C42—31no—
452C42—31no—
453C42—31no—
454C42—31no—
455C42—301no—
456C42—301no—
457C42—301no—
458C3×C5#308011yescharacteristic, complemented, Hall
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—102no—
462C22×C3—201no—
463C22×C3—201no—
464C22×C3#409212yes—
465C22×C3C22×C22⋊D511yescharacteristic
466C3×C4—42no—
467C3×C4—102no—
468C3×C4#409212yes—
469C3×C4C2×Q8×D511yescharacteristic
470C3×C4C2×#71311yescharacteristic
471Dic3—21no—
472Dic3—21no—
473Dic3—21no—
474Dic3—21no—
475Dic3—42no—
476Dic3—201no—
477Dic3—201no—
478C2×C5—22no—
479C2×C5C2×#87811yescharacteristic
480C2×C5#577311yescharacteristic
481C2×C5#577711yescharacteristic
482D5—42no—
483C23—101no—
484C23D3×C22⋊D511yescharacteristic
485Q8—52no—
486Q8—101no—
487D4—52no—
488D4—104no—
489D4—101no—
490C2×C4—22no—
491C2×C4—31no—
492C2×C4—31no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—42no—
496C2×C4—62no—
497C2×C4—101no—
498C2×C4—104no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—301no—
503C2×C4—301no—
504C2×C4—602no—
505C2×C4—602no—
506C2×C4D5×Dic612yes—
507C2×C4#676612yes—
508C2×C4D3×C22⋊D511yescharacteristic
509C2×C4C22×D3×D511yescharacteristic
510C8—102no—
511C8—302no—
512C2×C3—22no—
513C2×C3—202no—
514C2×C3C2×#409211yescharacteristic
515C2×C3#6615611yescharacteristic
516C2×C3#6615611yescharacteristic
517C5#7887111yescharacteristic, complemented, Sylow
518C22—21no—
519C22—21no—
520C22—102no—
521C22—201no—
522C22—201no—
523C22#9142112yes—
524C22C2×D3×C22⋊D511yescharacteristic
525C4—42no—
526C4—61no—
527C4—61no—
528C4—61no—
529C4—61no—
530C4—102no—
531C4—122no—
532C4—601no—
533C4—601no—
534C4#9142112yes—
535C4C2×D5×Dic611yescharacteristic
536C4C2×#676611yescharacteristic
537C3#1061474811yescharacteristic, complemented, Sylow
538C2—22no—
539C2—202no—
540C2C2×#9142111yescharacteristic
541C2#1175469611yescharacteristic
542C2#1175509011yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#709C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#3080
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#890C2semidirect
3. #11751065C2semidirect
5. #11748459C2semidirect
6. #11748440C2semidirect
8. #11747296C2semidirect
9. #11747295C2semidirect
10. #11747251C2semidirect
11. #11747249C2semidirect
96. C2×#890D5semidirect
458. C3×C5#3080semidirect
517. C5#78871semidirect
537. C3#10614748semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#8902
Dic3×#7092
#117510652
C3×#652552
#117484592
#117484402
#117483682
#117472962
#117472952
#117472512
#117472492
#106147483
#788715