Finite Group Databasebeta

← #14430732

Subgroups — #14430732

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
1#11754455C211yesmaximal, characteristic
2C2×#91384C211yesmaximal, characteristic, complemented
3#11751219C211yesmaximal, characteristic, complemented
4C5×#5133C211yesmaximal, characteristic
5C3×#65410C211yesmaximal, characteristic, complemented
6#11748472C211yesmaximal, characteristic, complemented
7#11748384C211yesmaximal, characteristic
8#11747887C212yesmaximal, complemented
9#11747885C212yesmaximal, complemented
10#11747817C212yesmaximal, complemented
11#11747814C212yesmaximal, complemented
12#10610602—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C2×C3×C5×M16C2211yescharacteristic
15C2×C4×Dic15C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17C2×C5×C4⋊Dic3C2211yescharacteristic
18C3×#4072C2211yescharacteristic
19C2×C3×C4⋊Dic5C2211yescharacteristic
20#67⋊Dic5—21no—
21#67⋊Dic5—21no—
22C2×Dic3⋊Dic5—21no—
23C2×#6668—21no—
24#91435C2211yescharacteristic
25#91435C2211yescharacteristic
26C4×D3×Dic5—21no—
27#91384—21no—
28#91384—21no—
29#91384C2212yes—
30#91384C2212yes—
31#91371—21no—
32#91344C2211yescharacteristic
33#91344C2211yescharacteristic
34#91098C2212yes—
35#91098C2212yes—
36C5×#821C2212yes—
37C5×#816C2212yes—
38C3×#3950C2212yes—
39C3×#3949C2212yes—
40#90989C2212yes—
41#90986C2212yes—
42#90972C2212yes—
43#90966C2212yes—
44#79241—51nomaximal, Hall
45#41×Dic5—31no—
46C2×#4036—31no—
47C5×#417—31no—
48#65521—31no—
49#65410D311yescharacteristic, complemented
50#65407—31no—
51#65253—31no—
52#65088—32no—
53#65085—32no—
54#65055—32no—
55#65052—32no—
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15D411yescharacteristic
58C22×Dic15D411yescharacteristic
59C2×C5×#67—21no—
60C22×C5×Dic3—21no—
61C5×#235—22no—
62C5×#235Q814yes—
63C2×C5×D12C2311yescharacteristic
64C2×C4×C5×D3—21no—
65C2×C5×Dic6C2311yescharacteristic
66C22×C3×Dic5—21no—
67C2×D3×Dic5—21no—
68C2×D3×Dic5—21no—
69C3×C5×M16—22no—
70C2×C3×C5×C8C2312yes—
71C4×Dic15D411yescharacteristic
72C4×Dic15D411yescharacteristic
73C4×Dic15C2311yescharacteristic
74C4×Dic15C2311yescharacteristic
75#6710—22no—
76C2×C3⋊#168C2312yes—
77C5×C4⋊Dic3—21no—
78C5×C4⋊Dic3C2312yes—
79C3×C22⋊Dic5—21no—
80C3×C4⋊Dic5—21no—
81C3×C4⋊Dic5C2311yescharacteristic
82C3×C4⋊Dic5C2311yescharacteristic
83C3×C4⋊Dic5C2312yes—
84C3×C4×Dic5—21no—
85Dic3⋊Dic5—22no—
86Dic3⋊Dic5—22no—
87Dic3⋊Dic5—21no—
88Dic3⋊Dic5—21no—
89#6668—22no—
90#6668—22no—
91(C2×D3)⋊Dic5—21no—
92(C2×D3)⋊Dic5—21no—
93Dic3×Dic5—21no—
94Dic3×Dic5—21no—
95#5522—51no—
96C2×#890—51no—
97C3×#428—51no—
98#5133D511yescharacteristic
99#5118—51no—
100#4995—51no—
101#4966—51no—
102#4891—52no—
103#4877—52no—
104#4871—52no—
105#4857—52no—
106C2×C5×#41—31no—
107C2×C5×M16C2×D311yescharacteristic
108C2×C5×C4⋊C4—31no—
109Q8×Dic5—31no—
110Q8×Dic5—31no—
111D4×Dic5—31no—
112D4×Dic5—31no—
113D4×Dic5—61no—
114D4×Dic5—61no—
115#4072—61no—
116#4072C2×D311yescharacteristic
117C2×C4⋊Dic5—61no—
118C2×C4⋊Dic5C2×D311yescharacteristic
119C2×#683—61no—
120C2×C4×Dic5—31no—
121C2×C4×Dic5—61no—
122C2×#681—31no—
123#4036—32no—
124#4036—32no—
125#4036—61no—
126#4036—61no—
127C5×#103—32no—
128C5×#102—32no—
129#3967—32no—
130#3964—32no—
131#3950C2×D312yes—
132#3949C2×D312yes—
133#3946—32no—
134#3946—32no—
135#3940—32no—
136#3939—32no—
137#3080—151noSylow
138C23×C3×C5C2×D411yescharacteristic
139C22×C5×D3—21no—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×Q812yes—
142C2×C3×C4×C5#4112yes—
143C2×C3×C4×C5C2411yescharacteristic
144C2×Dic15—41no—
145C2×Dic15—41no—
146C2×Dic15C2×D411yescharacteristic
147C2×Dic15C2×D411yescharacteristic
148C2×Dic15C2×D411yescharacteristic
149C2×Dic15C2×D411yescharacteristic
150C5×#67—24no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—42no—
154C5×D12—21no—
155C5×D12C2×Q812yes—
156C4×C5×D3—24no—
157C5×Dic6—21no—
158C5×Dic6C2×Q812yes—
159C2×C3×Dic5—21no—
160C2×C3×Dic5—21no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5—42no—
163D3×Dic5—42no—
164D3×Dic5—42no—
165C3×C5×C8—22no—
166C3⋊#168—22no—
167C2×#235C2×D511yescharacteristic
168C2×C3×M16C2×D511yescharacteristic
169C3×#117—51no—
170C2×C3×C4⋊C4—51no—
171C2×C4⋊Dic3—101no—
172C2×C4⋊Dic3C2×D511yescharacteristic
173C2×Dic3⋊C4—101no—
174C2×C4×Dic3—51no—
175C2×#208—51no—
176#893—51no—
177#893—51no—
178#890—52no—
179#890—52no—
180#890—101no—
181#890—101no—
182#887—51no—
183#887—51no—
184#67⋊C4—101no—
185#67⋊C4—101no—
186#872—101no—
187C42×D3—101no—
188C3×#107—52no—
189C3×#106—52no—
190#821C2×D512yes—
191#816C2×D512yes—
192Dic3⋊C8—52no—
193Dic3⋊C8—52no—
194#810—52no—
195#809—52no—
196#808—52no—
197#807—52no—
198C5×#41—34no—
199C5×#41—62no—
200C2×C5×Q8—31no—
201C2×C5×D4—31no—
202C2×C5×D4—61no—
203C22×C4×C5—61no—
204C22×C4×C5—61no—
205C22×C4×C5C22×D311yescharacteristic
206C22×Dic5—21no—
207C22×Dic5—31no—
208C22×Dic5—31no—
209C22×Dic5—61no—
210C22×Dic5—61no—
211C5×M16—22no—
212C2×C5×C8C22×D312yes—
213C5×C4⋊C4—32no—
214C5×C4⋊C4—61no—
215C22⋊Dic5—21no—
216C22⋊Dic5—61no—
217C22⋊Dic5—61no—
218C4⋊Dic5—21no—
219C4⋊Dic5—62no—
220C4⋊Dic5—62no—
221C4⋊Dic5—61no—
222C4⋊Dic5—61no—
223C4⋊Dic5C22×D311yescharacteristic
224C4⋊Dic5C22×D311yescharacteristic
225C4⋊Dic5C22×D312yes—
226#683—62no—
227#683—62no—
228C4×Dic5—21no—
229C4×Dic5—31no—
230C4×Dic5—31no—
231C4×Dic5—31no—
232C4×Dic5—31no—
233C4×Dic5—61no—
234C4×Dic5—61no—
235#681—62no—
236C2×#168—32no—
237C2×#125—151no—
238C4×#41—151no—
239#479—152no—
240#479—152no—
241#478—152no—
242#478—152no—
243#428—51no—
244#428—151no—
245#423—151no—
246#417—31no—
247#417—151no—
248C22×C3×C5—21no—
249C22×C3×C5—21no—
250C22×C3×C5C22⋊Q812yes—
251C22×C3×C5C22×D411yescharacteristic
252C2×C5×D3—22no—
253C2×C5×D3—41no—
254C2×C5×D3—41no—
255C3×C4×C5C22⋊Q812yes—
256C3×C4×C5C22×Q811yescharacteristic
257C3×C4×C5C2×#4111yescharacteristic
258Dic15—21no—
259Dic15—21no—
260Dic15—21no—
261Dic15—21no—
262C3×Dic5—41no—
263C3×Dic5—41no—
264C3×Dic5—42no—
265C5×Dic3—22no—
266C5×Dic3—42no—
267C22×C3×C4—101no—
268C22×C3×C4C22×D511yescharacteristic
269C2×#67—21no—
270C22×Dic3—21no—
271C22×Dic3—51no—
272C22×Dic3—51no—
273#235—22no—
274#235Dic1014yes—
275C2×D12C22×D511yescharacteristic
276C2×C4×D3—21no—
277C2×C4×D3—101no—
278C2×C4×D3—101no—
279C2×Dic6C22×D511yescharacteristic
280C3×M16—22no—
281C2×C3×C8C22×D512yes—
282C3×C4⋊C4—51no—
283C3×C4⋊C4—51no—
284C3×C4⋊C4—52no—
285C3×C4⋊C4—101no—
286C3×C22⋊C4—101no—
287C3×C42—101no—
288(C2×D3)⋊C4—101no—
289(C2×D3)⋊C4—101no—
290C4⋊Dic3—21no—
291C4⋊Dic3—102no—
292C4⋊Dic3—102no—
293C4⋊Dic3C22×D512yes—
294Dic3⋊C4—101no—
295Dic3⋊C4—101no—
296Dic3⋊C4—102no—
297Dic3⋊C4—102no—
298C4×Dic3—51no—
299C4×Dic3—51no—
300C4×Dic3—51no—
301C4×Dic3—51no—
302C4×Dic3—101no—
303C4×Dic3—101no—
304#208—102no—
305C2×C3⋊C8—52no—
306C23×C5—61no—
307C23×C5D3×D411yescharacteristic
308C5×Q8—32no—
309C5×Q8—61no—
310C5×D4—32no—
311C5×D4—64no—
312C5×D4—61no—
313C2×C4×C5—61no—
314C2×C4×C5—64no—
315C2×C4×C5—62no—
316C2×C4×C5—122no—
317C2×C4×C5#23512yes—
318C2×C4×C5D3×D411yescharacteristic
319C2×C4×C5D3×Q812yes—
320C2×C4×C5C23×D311yescharacteristic
321C2×Dic5—21no—
322C2×Dic5—21no—
323C2×Dic5—22no—
324C2×Dic5—31no—
325C2×Dic5—31no—
326C2×Dic5—31no—
327C2×Dic5—31no—
328C2×Dic5—42no—
329C2×Dic5—122no—
330C2×Dic5—122no—
331C2×Dic5—121no—
332C2×Dic5—121no—
333C5×C8—22no—
334#168—62no—
335C2×#41—31no—
336C2×M16—151no—
337C2×M16D3×D511yescharacteristic
338#125—152no—
339#125—152no—
340#125—301no—
341#125—301no—
342C4×Q8—151no—
343C4×Q8—151no—
344C4×D4—151no—
345C4×D4—151no—
346C4×D4—301no—
347C4×D4—301no—
348#117—51no—
349#117—301no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—51no—
352C2×C4⋊C4—301no—
353C2×C4⋊C4—301no—
354C2×C42—151no—
355C2×C42—301no—
356#107—52no—
357#107—152no—
358#106—52no—
359#106—152no—
360C4⋊C8—152no—
361C4⋊C8—152no—
362#103—32no—
363#103—152no—
364#102—32no—
365#102—152no—
366C2×C3×C5—22no—
367C2×C3×C5C2×C22⋊Q811yescharacteristic
368C2×C3×C5#57511yescharacteristic
369C2×C3×C5#57511yescharacteristic
370C5×D3—42no—
371C23×C3D4×D511yescharacteristic
372C22×D3—21no—
373C2×C3×C4—101no—
374C2×C3×C4—101no—
375C2×C3×C4—102no—
376C2×C3×C4—202no—
377C2×C3×C4C2×Dic1012yes—
378C2×C3×C4D4×D511yescharacteristic
379C2×C3×C4#71112yes—
380C2×C3×C4C23×D511yescharacteristic
381#67—24no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—42no—
385C2×Dic3—51no—
386C2×Dic3—51no—
387C2×Dic3—51no—
388C2×Dic3—51no—
389C2×Dic3—201no—
390C2×Dic3—201no—
391D12—21no—
392D12C2×Dic1012yes—
393C4×D3—24no—
394C4×D3—202no—
395C4×D3—202no—
396Dic6—21no—
397Dic6C2×Dic1012yes—
398C3×C8—22no—
399C3⋊C8—102no—
400C22×C5—21no—
401C22×C5—21no—
402C22×C5—62no—
403C22×C5—121no—
404C22×C5—121no—
405C22×C5#89612yes—
406C22×C5C2×D3×D411yescharacteristic
407C4×C5—62no—
408C4×C5—122no—
409C4×C5#89612yes—
410C4×C5C2×#23511yescharacteristic
411C4×C5C2×D3×Q811yescharacteristic
412Dic5—42no—
413Dic5—41no—
414Dic5—41no—
415Dic5—61no—
416Dic5—61no—
417Dic5—61no—
418Dic5—61no—
419#41—34no—
420#41—62no—
421C2×Q8—31no—
422C2×D4—31no—
423C2×D4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—101no—
427C22×C4—151no—
428C22×C4—151no—
429C22×C4—301no—
430C22×C4—301no—
431C22×C4C2×D3×D511yescharacteristic
432M16—22no—
433M16—302no—
434C2×C8—152no—
435C2×C8C2×D3×D512yes—
436C4⋊C4—32no—
437C4⋊C4—52no—
438C4⋊C4—51no—
439C4⋊C4—51no—
440C4⋊C4—61no—
441C4⋊C4—101no—
442C4⋊C4—302no—
443C4⋊C4—302no—
444C4⋊C4—302no—
445C4⋊C4—302no—
446C4⋊C4—301no—
447C4⋊C4—301no—
448C22⋊C4—101no—
449C22⋊C4—301no—
450C22⋊C4—301no—
451C42—101no—
452C42—151no—
453C42—151no—
454C42—151no—
455C42—151no—
456C42—301no—
457C42—301no—
458C3×C5#308011yescharacteristic, complemented, Hall
459C22×C3—21no—
460C22×C3—21no—
461C22×C3#402412yes—
462C22×C3C2×D4×D511yescharacteristic
463C2×D3—22no—
464C2×D3—41no—
465C2×D3—41no—
466C3×C4—201no—
467C3×C4—201no—
468C3×C4—202no—
469C3×C4#402412yes—
470C3×C4C22×Dic1011yescharacteristic
471C3×C4C2×#71111yescharacteristic
472Dic3—22no—
473Dic3—42no—
474Dic3—101no—
475Dic3—101no—
476Dic3—101no—
477Dic3—101no—
478C2×C5—22no—
479C2×C5—122no—
480C2×C5C2×#89611yescharacteristic
481C2×C5#577311yescharacteristic
482C2×C5#577711yescharacteristic
483C23—61no—
484C23#677711yescharacteristic
485Q8—32no—
486Q8—61no—
487D4—32no—
488D4—64no—
489D4—61no—
490C2×C4—61no—
491C2×C4—64no—
492C2×C4—62no—
493C2×C4—101no—
494C2×C4—101no—
495C2×C4—102no—
496C2×C4—122no—
497C2×C4—151no—
498C2×C4—151no—
499C2×C4—151no—
500C2×C4—151no—
501C2×C4—202no—
502C2×C4—602no—
503C2×C4—602no—
504C2×C4—601no—
505C2×C4—601no—
506C2×C4D3×Dic1012yes—
507C2×C4#677711yescharacteristic
508C2×C4#678212yes—
509C2×C4C22×D3×D511yescharacteristic
510C8—22no—
511C8—302no—
512C2×C3—22no—
513C2×C3C2×#402411yescharacteristic
514C2×C3#6610711yescharacteristic
515C2×C3#6611111yescharacteristic
516D3—42no—
517C5#7924111yescharacteristic, complemented, Sylow
518C22—21no—
519C22—21no—
520C22—62no—
521C22—121no—
522C22—121no—
523C22#9143612yes—
524C22C2×#677711yescharacteristic
525C4—62no—
526C4—122no—
527C4—201no—
528C4—202no—
529C4—201no—
530C4—301no—
531C4—301no—
532C4—301no—
533C4—301no—
534C4#9143612yes—
535C4C2×D3×Dic1011yescharacteristic
536C4C2×#678211yescharacteristic
537C3#1061060211yescharacteristic, complemented, Sylow
538C2—22no—
539C2—122no—
540C2#1175403511yescharacteristic
541C2#1175403611yescharacteristic
542C2C2×#9143611yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×#235C22
hypercenterC2×M16D3×D5
hypocenterC3×C5#3080
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×#91384C2semidirect
3. #11751219C2semidirect
5. C3×#65410C2semidirect
6. #11748472C2semidirect
8. #11747887C2semidirect
9. #11747885C2semidirect
10. #11747817C2semidirect
11. #11747814C2semidirect
49. #65410D3semidirect
458. C3×C5#3080semidirect
517. C5#79241semidirect
537. C3#10610602semidirect

Maximal subgroups

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

subgroupindex
#117544552
C2×#913842
#117512192
C5×#51332
C3×#654102
#117484722
#117483842
#117478872
#117478852
#117478172
#117478142
#106106023
#792415