Finite Group Databasebeta

← #14417179

Subgroups — #14417179

Order 1920.

940 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#137C211yesmaximal, characteristic
2#11756586C211yesmaximal, characteristic
3#11756571C211yesmaximal, characteristic, complemented
4#11751334C211yesmaximal, characteristic
5C2×#91119C211yesmaximal, characteristic, complemented
6#11751285C212yesmaximal, complemented
7Q16×Dic15C212yesmaximal, complemented
8#11751265C212yesmaximal, complemented
9SD16×Dic15C212yesmaximal, complemented
10#11751223C211yesmaximal, characteristic, complemented
11#11751222C211yesmaximal, characteristic, complemented
12#10615696—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×C3×C5×Q8C2211yescharacteristic
15C3×C5×#137C418yescomplemented
16C2×C3×C5×Q16C2212yes—
17C2×C3×C5×SD16C2212yes—
18C2×C3×C5×M16C2211yescharacteristic
19Q8×Dic15—21no—
20Q8×Dic15—21no—
21Q8×Dic15C2212yes—
22Q8×Dic15C2212yes—
23Q8×Dic15C2212yes—
24D4×Dic15—22no—
25D4×Dic15C2212yes—
26#91815—22no—
27#91815—21no—
28#91815C2211yescharacteristic
29#91815C2211yescharacteristic
30C2×C4⋊Dic15—21no—
31C2×C4⋊Dic15C2211yescharacteristic
32C22×C3⋊#168C2211yescharacteristic
33#91119C2212yes—
34#91119C2212yes—
35#91119C2211yescharacteristic
36#91119C2211yescharacteristic
37#91116C2211yescharacteristic
38#91116C2211yescharacteristic
39#91102C2212yes—
40#91101C2212yes—
41#91099C2212yes—
42C8×Dic15C2212yes—
43#83965—51nomaximal, Hall
44C2×C5×#137D311yescharacteristic
45#66161—31no—
46#66146—31no—
47#65522—31no—
48C2×#3967—31no—
49#65473—32no—
50Q16×Dic5—32no—
51#65453—32no—
52SD16×Dic5—32no—
53#65411—31no—
54#65410—31no—
55C3×C5×#41—22no—
56C3×C5×#41C2×C414yes—
57C2×C3×C5×Q8—21no—
58C2×C3×C5×Q8—22no—
59C2×C3×C5×Q8C2×C414yescomplemented
60C2×C3×C5×Q8C2312yes—
61C2×C3×C5×Q8C2311yescharacteristic
62C2×C3×C5×D4—21no—
63C2×C3×C5×D4C2311yescharacteristic
64C22×C3×C4×C5—21no—
65C22×C3×C4×C5—21no—
66C22×C3×C4×C5C2311yescharacteristic
67C22×Dic15—21no—
68C22×Dic15—22no—
69C3×C5×Q16C2×C418yescomplemented
70C3×C5×SD16C2×C418yescomplemented
71C3×C5×M16C2×C414yescomplemented
72C2×C3×C5×C8C2312yes—
73#6719—21no—
74#6719—22no—
75#6719D412yes—
76C4⋊Dic15—21no—
77C4⋊Dic15—22no—
78C4⋊Dic15—22no—
79C4⋊Dic15—22no—
80C4⋊Dic15D411yescharacteristic
81C4⋊Dic15D411yescharacteristic
82C4⋊Dic15C2312yes—
83C4⋊Dic15C2311yescharacteristic
84C4⋊Dic15C2311yescharacteristic
85C4×Dic15—21no—
86C4×Dic15—22no—
87C4×Dic15—22no—
88C4×Dic15—22no—
89C4×Dic15C2312yes—
90C2×C3⋊#168—22no—
91C2×C3⋊#168C2311yescharacteristic
92C2×C3⋊#168C2311yescharacteristic
93C2×C3×#137D511yescharacteristic
94#5827—51no—
95#5812—51no—
96#5233—51no—
97C2×#835—51no—
98#5184—52no—
99Dic3×Q16—52no—
100#5164—52no—
101Dic3×SD16—52no—
102#5122—51no—
103#5121—51no—
104C2×C5×#41C2×D311yescharacteristic
105C22×C5×Q8C2×D311yescharacteristic
106C5×#137Dic318yescomplemented
107C2×C5×Q16C2×D312yes—
108C2×C5×SD16C2×D312yes—
109C2×C5×M16C2×D311yescharacteristic
110Q8×Dic5—32no—
111Q8×Dic5—32no—
112Q8×Dic5—32no—
113Q8×Dic5—61no—
114Q8×Dic5—61no—
115D4×Dic5—32no—
116D4×Dic5—62no—
117#4072—31no—
118#4072—31no—
119#4072—61no—
120#4072—62no—
121C2×C4⋊Dic5—31no—
122C2×C4⋊Dic5—61no—
123C22×#168—31no—
124#3967—32no—
125#3967—32no—
126#3967—31no—
127#3967—31no—
128#3964—31no—
129#3964—31no—
130#3950—32no—
131#3949—32no—
132#3947—32no—
133C8×Dic5—32no—
134#2947—151noSylow
135C23×C3×C5—21no—
136C23×C3×C5C2×D411yescharacteristic
137C3×C5×Q8—24no—
138C3×C5×Q8—22no—
139C3×C5×Q8—21no—
140C3×C5×Q8C22×C414yes—
141C3×C5×Q8C22×C412yes—
142C3×C5×D4—24no—
143C3×C5×D4—21no—
144C3×C5×D4C22×C412yes—
145C2×C3×C4×C5—24no—
146C2×C3×C4×C5—21no—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5—24no—
149C2×C3×C4×C5C22×C412yes—
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5#4112yes—
152C2×C3×C4×C5C2411yescharacteristic
153C2×Dic15—21no—
154C2×Dic15—21no—
155C2×Dic15—22no—
156C2×Dic15—82no—
157C2×Dic15—81no—
158C2×Dic15C2×D412yes—
159C2×Dic15C2×D412yes—
160C3×C5×C8C22×C414yes—
161C3⋊#168—21no—
162C3⋊#168—21no—
163C2×C3×#41C2×D511yescharacteristic
164C22×C3×Q8C2×D511yescharacteristic
165C3×#137Dic518yescomplemented
166C2×C3×Q16C2×D512yes—
167C2×C3×SD16C2×D512yes—
168C2×C3×M16C2×D511yescharacteristic
169Q8×Dic3—52no—
170Q8×Dic3—52no—
171Q8×Dic3—52no—
172Q8×Dic3—101no—
173Q8×Dic3—101no—
174D4×Dic3—52no—
175D4×Dic3—102no—
176#926—51no—
177#926—51no—
178#926—101no—
179#926—102no—
180C2×C4⋊Dic3—51no—
181C2×C4⋊Dic3—101no—
182C22×C3⋊C8—51no—
183#835—52no—
184#835—52no—
185#835—51no—
186#835—51no—
187#832—51no—
188#832—51no—
189#818—52no—
190#817—52no—
191#815—52no—
192C8×Dic3—52no—
193C5×#41—22no—
194C5×#41C2×Dic314yes—
195C2×C5×Q8—21no—
196C2×C5×Q8—22no—
197C2×C5×Q8C2×Dic314yescomplemented
198C2×C5×Q8C22×D312yes—
199C2×C5×Q8C22×D311yescharacteristic
200C2×C5×D4—21no—
201C2×C5×D4C22×D311yescharacteristic
202C22×C4×C5—21no—
203C22×C4×C5—21no—
204C22×C4×C5C22×D311yescharacteristic
205C22×Dic5—62no—
206C22×Dic5—61no—
207C5×Q16C2×Dic318yescomplemented
208C5×SD16C2×Dic318yescomplemented
209C5×M16C2×Dic314yescomplemented
210C2×C5×C8C22×D312yes—
211C22⋊Dic5—32no—
212C22⋊Dic5—62no—
213C22⋊Dic5—61no—
214C4⋊Dic5—31no—
215C4⋊Dic5—31no—
216C4⋊Dic5—31no—
217C4⋊Dic5—32no—
218C4⋊Dic5—31no—
219C4⋊Dic5—62no—
220C4⋊Dic5—62no—
221C4⋊Dic5—62no—
222C4⋊Dic5—61no—
223C4×Dic5—32no—
224C4×Dic5—62no—
225C4×Dic5—62no—
226C4×Dic5—62no—
227C4×Dic5—61no—
228C2×#168—31no—
229C2×#168—31no—
230C2×#168—62no—
231C2×#137D1511yescharacteristic
232#520—151no—
233#519—151no—
234#441—152no—
235#440—152no—
236C4×Q16—152no—
237C4×SD16—152no—
238#428—151no—
239#416—151no—
240C2×#103—151no—
241#405—151no—
242C22×C3×C5—21no—
243C22×C3×C5—21no—
244C22×C3×C5—22no—
245C22×C3×C5—41no—
246C22×C3×C5—41no—
247C22×C3×C5C4×D412yes—
248C22×C3×C5C22×D411yescharacteristic
249C3×C4×C5—22no—
250C3×C4×C5—24no—
251C3×C4×C5C4×D412yes—
252C3×C4×C5C23×C411yescharacteristic
253C3×C4×C5C2×#4111yescharacteristic
254Dic15—22no—
255Dic15—22no—
256Dic15—41no—
257Dic15—42no—
258Dic15—41no—
259C3×#41—22no—
260C3×#41C2×Dic514yes—
261C2×C3×Q8—21no—
262C2×C3×Q8—22no—
263C2×C3×Q8C2×Dic514yescomplemented
264C2×C3×Q8C22×D512yes—
265C2×C3×Q8C22×D511yescharacteristic
266C2×C3×D4—21no—
267C2×C3×D4C22×D511yescharacteristic
268C22×C3×C4—21no—
269C22×C3×C4—21no—
270C22×C3×C4C22×D511yescharacteristic
271C22×Dic3—102no—
272C22×Dic3—101no—
273C3×Q16C2×Dic518yescomplemented
274C3×SD16C2×Dic518yescomplemented
275C3×M16C2×Dic514yescomplemented
276C2×C3×C8C22×D512yes—
277#217—52no—
278#217—102no—
279#217—101no—
280C4⋊Dic3—51no—
281C4⋊Dic3—51no—
282C4⋊Dic3—51no—
283C4⋊Dic3—52no—
284C4⋊Dic3—51no—
285C4⋊Dic3—102no—
286C4⋊Dic3—102no—
287C4⋊Dic3—102no—
288C4⋊Dic3—101no—
289C4×Dic3—52no—
290C4×Dic3—102no—
291C4×Dic3—102no—
292C4×Dic3—102no—
293C4×Dic3—101no—
294C2×C3⋊C8—51no—
295C2×C3⋊C8—51no—
296C2×C3⋊C8—102no—
297C23×C5—21no—
298C23×C5D3×D411yescharacteristic
299C5×Q8—24no—
300C5×Q8—22no—
301C5×Q8—21no—
302C5×Q8C22×Dic314yes—
303C5×Q8C22×Dic312yes—
304C5×D4—24no—
305C5×D4—21no—
306C5×D4C22×Dic312yes—
307C2×C4×C5—24no—
308C2×C4×C5—21no—
309C2×C4×C5—22no—
310C2×C4×C5—24no—
311C2×C4×C5D3×D411yescharacteristic
312C2×C4×C5#23712yes—
313C2×C4×C5C22×Dic312yes—
314C2×C4×C5C23×D311yescharacteristic
315C2×Dic5—32no—
316C2×Dic5—32no—
317C2×Dic5—61no—
318C2×Dic5—61no—
319C2×Dic5—62no—
320C2×Dic5—242no—
321C2×Dic5—241no—
322C5×C8C22×Dic314yes—
323#168—61no—
324#168—61no—
325C2×#41C2×D1511yescharacteristic
326C22×Q8C2×D1511yescharacteristic
327#137Dic1518yescomplemented
328C2×Q16C2×D1512yes—
329C2×SD16C2×D1512yes—
330C2×M16C2×D1511yescharacteristic
331C22×C8—151no—
332C4×Q8—152no—
333C4×Q8—152no—
334C4×Q8—152no—
335C4×Q8—301no—
336C4×Q8—301no—
337C4×D4—152no—
338C4×D4—302no—
339#117—151no—
340#117—151no—
341#117—302no—
342#117—301no—
343C2×C4⋊C4—151no—
344C2×C4⋊C4—301no—
345#107—152no—
346#106—152no—
347#103—152no—
348#103—152no—
349#103—151no—
350#103—151no—
351#102—151no—
352#102—151no—
353#97—152no—
354C4×C8—152no—
355C2×C3×C5—22no—
356C2×C3×C5—42no—
357C2×C3×C5C2×C4×D411yescharacteristic
358C2×C3×C5#57711yescharacteristic
359C2×C3×C5#57811yescharacteristic
360C23×C3—21no—
361C23×C3D4×D511yescharacteristic
362C3×Q8—22no—
363C3×Q8—21no—
364C3×Q8—24no—
365C3×Q8C22×Dic514yes—
366C3×Q8C22×Dic512yes—
367C3×D4—24no—
368C3×D4—21no—
369C3×D4C22×Dic512yes—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4D4×D511yescharacteristic
375C2×C3×C4#71112yes—
376C2×C3×C4C22×Dic512yes—
377C2×C3×C4C23×D511yescharacteristic
378C2×Dic3—52no—
379C2×Dic3—52no—
380C2×Dic3—101no—
381C2×Dic3—101no—
382C2×Dic3—102no—
383C2×Dic3—402no—
384C2×Dic3—401no—
385C3×C8C22×Dic514yes—
386C3⋊C8—101no—
387C3⋊C8—101no—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5—22no—
391C22×C5—41no—
392C22×C5—41no—
393C22×C5D4×Dic312yes—
394C22×C5C2×D3×D411yescharacteristic
395C4×C5—22no—
396C4×C5—24no—
397C4×C5D4×Dic312yes—
398C4×C5C2×#23711yescharacteristic
399C4×C5C23×Dic311yescharacteristic
400Dic5—62no—
401Dic5—62no—
402Dic5—121no—
403Dic5—122no—
404Dic5—121no—
405#41—22no—
406#41C2×Dic1514yes—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8C2×Dic1514yescomplemented
410C2×Q8C22×D1512yes—
411C2×Q8C22×D1511yescharacteristic
412C2×D4—21no—
413C2×D4C22×D1511yescharacteristic
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—302no—
417C22×C4—301no—
418C22×C4C22×D1511yescharacteristic
419Q16C2×Dic1518yescomplemented
420SD16C2×Dic1518yescomplemented
421M16C2×Dic1514yescomplemented
422C2×C8—151no—
423C2×C8—151no—
424C2×C8—302no—
425C2×C8C22×D1512yes—
426C4⋊C4—151no—
427C4⋊C4—151no—
428C4⋊C4—151no—
429C4⋊C4—151no—
430C4⋊C4—152no—
431C4⋊C4—302no—
432C4⋊C4—302no—
433C4⋊C4—302no—
434C4⋊C4—301no—
435C22⋊C4—152no—
436C22⋊C4—302no—
437C22⋊C4—301no—
438C42—152no—
439C42—302no—
440C42—302no—
441C42—302no—
442C42—301no—
443C3×C5#294711yescharacteristic, complemented, Hall
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—22no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3D4×Dic512yes—
450C22×C3C2×D4×D511yescharacteristic
451C3×C4—22no—
452C3×C4—24no—
453C3×C4D4×Dic512yes—
454C3×C4C2×#71111yescharacteristic
455C3×C4C23×Dic511yescharacteristic
456Dic3—102no—
457Dic3—102no—
458Dic3—201no—
459Dic3—202no—
460Dic3—201no—
461C2×C5—22no—
462C2×C5—42no—
463C2×C5#577811yescharacteristic
464C2×C5#577911yescharacteristic
465C2×C5C2×D4×Dic311yescharacteristic
466C23—21no—
467C23D4×D1511yescharacteristic
468Q8—24no—
469Q8—22no—
470Q8—21no—
471Q8C22×Dic1514yes—
472Q8C22×Dic1512yes—
473D4—24no—
474D4—21no—
475D4C22×Dic1512yes—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—152no—
481C2×C4—152no—
482C2×C4—301no—
483C2×C4—301no—
484C2×C4—302no—
485C2×C4—1202no—
486C2×C4—1201no—
487C2×C4D4×D1511yescharacteristic
488C2×C4#681912yes—
489C2×C4C22×Dic1512yes—
490C2×C4C23×D1511yescharacteristic
491C8—301no—
492C8—301no—
493C8C22×Dic1514yes—
494C2×C3—22no—
495C2×C3—42no—
496C2×C3#6611211yescharacteristic
497C2×C3#6611311yescharacteristic
498C2×C3C2×D4×Dic511yescharacteristic
499C5#8396511yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22D4×Dic1512yes—
506C22C2×D4×D1511yescharacteristic
507C4—22no—
508C4—24no—
509C4—302no—
510C4—302no—
511C4—602no—
512C4—601no—
513C4—601no—
514C4D4×Dic1512yes—
515C4C2×#681911yescharacteristic
516C4C23×Dic1511yescharacteristic
517C3#1061569611yescharacteristic, complemented, Sylow
518C2—22no—
519C2—42no—
520C2#1175653711yescharacteristic
521C2#1175653811yescharacteristic
522C2C2×D4×Dic1511yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
FittingC2×C3×C5×#137C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×C5×#41C22
cosocleC2×C3×C4×C5C24
hypercenterC2×#137D15
hypocenterC3×C5#2947
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
3. #11756571C2semidirect
5. C2×#91119C2semidirect
6. #11751285C2semidirect
7. Q16×Dic15C2semidirect
8. #11751265C2semidirect
9. SD16×Dic15C2semidirect
10. #11751223C2semidirect
11. #11751222C2semidirect
15. C3×C5×#137C4semidirect
59. C2×C3×C5×Q8C2×C4semidirect
69. C3×C5×Q16C2×C4semidirect
70. C3×C5×SD16C2×C4semidirect
71. C3×C5×M16C2×C4semidirect
106. C5×#137Dic3semidirect
165. C3×#137Dic5semidirect
197. C2×C5×Q8C2×Dic3semidirect
207. C5×Q16C2×Dic3semidirect
208. C5×SD16C2×Dic3semidirect
209. C5×M16C2×Dic3semidirect
263. C2×C3×Q8C2×Dic5semidirect
273. C3×Q16C2×Dic5semidirect
274. C3×SD16C2×Dic5semidirect
275. C3×M16C2×Dic5semidirect
327. #137Dic15semidirect
409. C2×Q8C2×Dic15semidirect
419. Q16C2×Dic15semidirect
420. SD16C2×Dic15semidirect
421. M16C2×Dic15semidirect
443. C3×C5#2947semidirect
499. C5#83965semidirect
517. C3#10615696semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1372
#117565862
#117565712
#117513342
C2×#911192
#117512852
Q16×Dic152
#117512652
SD16×Dic152
#117512232
#117512222
#106156963
#839655