Finite Group Databasebeta

← #14417114

Subgroups — #14417114

Order 1920.

1012 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#908C211yesmaximal, characteristic
2#11754503C211yesmaximal, characteristic, complemented
3#11754492C211yesmaximal, characteristic, complemented
4#11751223C211yesmaximal, characteristic, complemented
5C3×#65410C211yesmaximal, characteristic, complemented
6#11748382C211yesmaximal, characteristic, complemented
7C2×#90966C211yesmaximal, characteristic, complemented
8#11747816C212yesmaximal, complemented
9#11747813C212yesmaximal, complemented
10#11747810C212yesmaximal, complemented
11#11747807C212yesmaximal, complemented
12#10615661—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C22×C5×D12C2211yescharacteristic
15C2×C3×C5×M16C2211yescharacteristic
16#91815C2211yescharacteristic
17C22×C3⋊#168C2211yescharacteristic
18C5×#908C418yescomplemented
19C2×C5×D24C2212yes—
20C2×C5×#204C2212yes—
21C3×#4072C2211yescharacteristic
22C2×C3×C4⋊Dic5C2211yescharacteristic
23D3×C22⋊Dic5—22no—
24Dic5×#67—22no—
25#91435C2212yes—
26#91435C2212yes—
27#91435C2212yescomplemented
28D3×C4⋊Dic5—21no—
29Dic5×D12—21no—
30Dic5×D12—21no—
31#91344C2212yescomplemented
32#91338—21no—
33#91099C2212yescomplemented
34C8×Dic15C2212yescomplemented
35C3×#3950C2212yescomplemented
36C3×#3949C2212yescomplemented
37#90972C2211yescharacteristic, complemented
38#90972C2211yescharacteristic, complemented
39#90966C2212yes—
40#90966C2212yes—
41#90966C2211yescharacteristic, complemented
42#90966C2211yescharacteristic, complemented
43#79201—51nomaximal, Hall
44C2×C5×AGL(1,ℤ/8)—31no—
45#66161—31no—
46#66132—31no—
47#65522—31no—
48C2×#3964—31no—
49#65453—32no—
50SD16×Dic5—32no—
51#65441—32no—
52D8×Dic5—32no—
53#65411—31no—
54#65410D311yescharacteristic, complemented
55C23×C5×D3—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C2×C5×#67—21no—
58C5×#235—22no—
59C5×#235C2×C414yescomplemented
60C2×C5×D12—21no—
61C2×C5×D12—22no—
62C2×C5×D12C2×C414yescomplemented
63C2×C5×D12C2312yes—
64C2×C5×D12C2311yescharacteristic
65C2×C4×C5×D3—21no—
66C2×C5×Dic6C2311yescharacteristic
67C22×C3×Dic5—21no—
68C2×D3×Dic5—22no—
69C2×D3×Dic5—22no—
70C2×D3×Dic5—22no—
71C3×C5×M16C2×C414yescomplemented
72C2×C3×C5×C8C2312yes—
73#6719D412yes—
74C4⋊Dic15D411yescharacteristic
75C4⋊Dic15D411yescharacteristic
76C4×Dic15C2312yes—
77C2×C3⋊#168—22no—
78C2×C3⋊#168C2311yescharacteristic
79C2×C3⋊#168C2311yescharacteristic
80C5×D24C2×C418yescomplemented
81C5×#204C2×C418yescomplemented
82C3×C22⋊Dic5—21no—
83C3×C4⋊Dic5—21no—
84C3×C4⋊Dic5C2311yescharacteristic
85C3×C4⋊Dic5C2311yescharacteristic
86C3×C4⋊Dic5C2312yes—
87C3×C4×Dic5—21no—
88Dic3⋊Dic5—22no—
89(C2×D3)⋊Dic5—22no—
90(C2×D3)⋊Dic5—22no—
91(C2×D3)⋊Dic5—22no—
92Dic3×Dic5—22no—
93C2×#908D511yescharacteristic
94#5521—51no—
95#5504—51no—
96C3×#428—51no—
97#5122—51no—
98#5002—51no—
99C2×#809—51no—
100#4889—52no—
101#4872—52no—
102#4869—52no—
103#4852—52no—
104C2×C5×#41—31no—
105C22×C5×D4—31no—
106C5×AGL(1,ℤ/8)—38no—
107C2×C5×SD16—32no—
108C2×C5×D8—32no—
109C2×C5×M16C2×D311yescharacteristic
110C2×C22⋊Dic5—62no—
111Q8×Dic5—32no—
112D4×Dic5—32no—
113D4×Dic5—32no—
114D4×Dic5—32no—
115D4×Dic5—61no—
116D4×Dic5—61no—
117D4×Dic5—62no—
118#4072—31no—
119#4072—61no—
120#4072C2×D311yescharacteristic
121C2×C4⋊Dic5—61no—
122C2×C4⋊Dic5C2×D311yescharacteristic
123C22×#168—31no—
124#3967—31no—
125#3967—31no—
126#3964—32no—
127#3964—32no—
128#3964—31no—
129#3964—31no—
130#3950C2×D312yescomplemented
131#3949C2×D312yescomplemented
132#3947—32no—
133C8×Dic5—32no—
134#2946—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×C5×D3—21no—
137C22×C5×D3—22no—
138C22×C5×D3—24no—
139C22×C5×D3—81no—
140C22×C5×D3—81no—
141C2×C3×C4×C5C22×C412yes—
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5#4112yes—
144C2×C3×C4×C5C2411yescharacteristic
145C2×Dic15C2×D412yes—
146C2×Dic15C2×D412yes—
147C5×#67—24no—
148C2×C5×Dic3—21no—
149C5×D12—24no—
150C5×D12—22no—
151C5×D12—21no—
152C5×D12C22×C414yes—
153C5×D12C22×C412yes—
154C4×C5×D3—24no—
155C5×Dic6—21no—
156C5×Dic6C22×C412yes—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5—81no—
161D3×Dic5—82no—
162D3×Dic5—82no—
163D3×Dic5—82no—
164C3×C5×C8C22×C414yes—
165C3⋊#168—21no—
166C3⋊#168—21no—
167C2×#235C2×D511yescharacteristic
168C22×D12C2×D511yescharacteristic
169C2×C3×M16C2×D511yescharacteristic
170C3×#117—51no—
171C2×C3×C4⋊C4—51no—
172C4×#67—102no—
173#926—51no—
174C22×C3⋊C8—51no—
175#908Dic518yescomplemented
176C2×D24C2×D512yes—
177C2×#204C2×D512yes—
178#893—52no—
179#893—52no—
180#893—52no—
181#892—101no—
182D3×C4⋊C4—101no—
183#887—52no—
184D3×C22⋊C4—102no—
185C4×D12—101no—
186C4×D12—101no—
187C3×#107—52no—
188C3×#106—52no—
189#815—52no—
190C8×Dic3—52no—
191#810—51no—
192#810—51no—
193#809—51no—
194#809—52no—
195#809—51no—
196#809—52no—
197C24×C5—61no—
198C5×#41—34no—
199C5×#41—62no—
200C2×C5×Q8—31no—
201C2×C5×D4—34no—
202C2×C5×D4—31no—
203C2×C5×D4—32no—
204C2×C5×D4—61no—
205C2×C5×D4—61no—
206C2×C5×D4—62no—
207C22×C4×C5—61no—
208C22×C4×C5C22×D311yescharacteristic
209C22×Dic5—21no—
210C22×Dic5—62no—
211C22×Dic5—62no—
212C22×Dic5—62no—
213C5×SD16—38no—
214C5×D8—38no—
215C5×M16C4×D314yescomplemented
216C2×C5×C8C22×D312yes—
217C22⋊Dic5—21no—
218C22⋊Dic5—32no—
219C22⋊Dic5—62no—
220C22⋊Dic5—62no—
221C22⋊Dic5—62no—
222C4⋊Dic5—21no—
223C4⋊Dic5—31no—
224C4⋊Dic5—31no—
225C4⋊Dic5—62no—
226C4⋊Dic5C22×D311yescharacteristic
227C4⋊Dic5C22×D312yes—
228C4⋊Dic5C22×D311yescharacteristic
229C4×Dic5—21no—
230C4×Dic5—32no—
231C4×Dic5—62no—
232C2×#168—31no—
233C2×#168—31no—
234C2×#168—62no—
235C2×AGL(1,ℤ/8)—31no—
236#520—151no—
237#518—151no—
238#442—152no—
239#440—152no—
240C4×SD16—152no—
241C4×D8—152no—
242#428—51no—
243#416—151no—
244C2×#102—151no—
245#405—151no—
246C22×C3×C5—21no—
247C22×C3×C5—21no—
248C22×C3×C5C4×D412yes—
249C22×C3×C5C22×D411yescharacteristic
250C2×C5×D3—22no—
251C2×C5×D3—24no—
252C2×C5×D3—41no—
253C2×C5×D3—42no—
254C2×C5×D3—42no—
255C2×C5×D3—41no—
256C2×C5×D3—84no—
257C3×C4×C5C4×D412yes—
258C3×C4×C5C23×C411yescharacteristic
259C3×C4×C5C2×#4111yescharacteristic
260Dic15—22no—
261Dic15—22no—
262C3×Dic5—41no—
263C3×Dic5—42no—
264C3×Dic5—41no—
265C5×Dic3—22no—
266C23×D3—21no—
267C22×C3×C4—101no—
268C22×C3×C4C22×D511yescharacteristic
269C2×#67—21no—
270#235—22no—
271#235C2×Dic514yescomplemented
272C2×D12—21no—
273C2×D12—22no—
274C2×D12C2×Dic514yescomplemented
275C2×D12C22×D512yes—
276C2×D12C22×D511yescharacteristic
277C2×C4×D3—21no—
278C2×C4×D3—102no—
279C2×C4×D3—102no—
280C2×C4×D3—102no—
281C2×Dic6C22×D511yescharacteristic
282C3×M16C2×Dic514yescomplemented
283C2×C3×C8C22×D512yes—
284C3×C4⋊C4—52no—
285C3×C4⋊C4—51no—
286C3×C4⋊C4—51no—
287C3×C4⋊C4—101no—
288C3×C22⋊C4—101no—
289C3×C42—101no—
290#217—52no—
291(C2×D3)⋊C4—102no—
292(C2×D3)⋊C4—102no—
293(C2×D3)⋊C4—102no—
294C4⋊Dic3—51no—
295C4⋊Dic3—51no—
296Dic3⋊C4—102no—
297C4×Dic3—52no—
298C4×Dic3—102no—
299C2×C3⋊C8—51no—
300C2×C3⋊C8—51no—
301C2×C3⋊C8—102no—
302D24C2×Dic518yescomplemented
303#204C2×Dic518yescomplemented
304C23×C5—61no—
305C23×C5—62no—
306C23×C5—64no—
307C23×C5—241no—
308C23×C5—241no—
309C23×C5D3×D411yescharacteristic
310C5×Q8—32no—
311C5×Q8—61no—
312C5×D4—32no—
313C5×D4—34no—
314C5×D4—64no—
315C5×D4—64no—
316C5×D4—62no—
317C5×D4—61no—
318C2×C4×C5—61no—
319C2×C4×C5—64no—
320C2×C4×C5C2×C4×D312yes—
321C2×C4×C5D3×D411yescharacteristic
322C2×C4×C5#23912yes—
323C2×C4×C5C23×D311yescharacteristic
324C2×Dic5—21no—
325C2×Dic5—21no—
326C2×Dic5—22no—
327C2×Dic5—32no—
328C2×Dic5—32no—
329C2×Dic5—81no—
330C2×Dic5—242no—
331C2×Dic5—242no—
332C2×Dic5—242no—
333C5×C8C2×C4×D314yes—
334#168—61no—
335#168—61no—
336C2×#41—31no—
337C22×D4—31no—
338AGL(1,ℤ/8)—38no—
339C2×SD16—32no—
340C2×D8—32no—
341C2×M16D3×D511yescharacteristic
342C22×C8—151no—
343C4×Q8—152no—
344C4×D4—152no—
345C4×D4—152no—
346C4×D4—152no—
347C4×D4—301no—
348C4×D4—301no—
349C4×D4—302no—
350#117—51no—
351#117—151no—
352#117—301no—
353C2×C4⋊C4—51no—
354C2×C4⋊C4—301no—
355C2×C22⋊C4—302no—
356#107—52no—
357#106—52no—
358#103—151no—
359#103—151no—
360#102—152no—
361#102—152no—
362#102—151no—
363#102—151no—
364#97—152no—
365C4×C8—152no—
366C2×C3×C5—22no—
367C2×C3×C5C2×C4×D411yescharacteristic
368C2×C3×C5#57611yescharacteristic
369C2×C3×C5#57711yescharacteristic
370C5×D3—42no—
371C5×D3—44no—
372C23×C3D4×D511yescharacteristic
373C22×D3—21no—
374C22×D3—22no—
375C22×D3—24no—
376C22×D3—81no—
377C22×D3—81no—
378C2×C3×C4—102no—
379C2×C3×C4—101no—
380C2×C3×C4—101no—
381C2×C3×C4—401no—
382C2×C3×C4D4×D511yescharacteristic
383C2×C3×C4#71112yes—
384C2×C3×C4C22×Dic512yes—
385C2×C3×C4C23×D511yescharacteristic
386#67—24no—
387C2×Dic3—21no—
388C2×Dic3—52no—
389C2×Dic3—52no—
390D12—22no—
391D12—21no—
392D12—24no—
393D12C22×Dic514yes—
394D12C22×Dic512yes—
395C4×D3—24no—
396C4×D3—402no—
397C4×D3—402no—
398C4×D3—402no—
399Dic6—21no—
400Dic6C22×Dic512yes—
401C3×C8C22×Dic514yes—
402C3⋊C8—101no—
403C3⋊C8—101no—
404C22×C5—21no—
405C22×C5—21no—
406C22×C5—62no—
407C22×C5—64no—
408C22×C5—121no—
409C22×C5—122no—
410C22×C5—122no—
411C22×C5—121no—
412C22×C5—244no—
413C22×C5#89312yes—
414C22×C5C2×D3×D411yescharacteristic
415C4×C5—62no—
416C4×C5#89312yes—
417C4×C5C22×C4×D311yescharacteristic
418C4×C5C2×#23911yescharacteristic
419Dic5—42no—
420Dic5—41no—
421Dic5—41no—
422Dic5—62no—
423Dic5—62no—
424C24—61no—
425#41—34no—
426#41—62no—
427C2×Q8—31no—
428C2×D4—34no—
429C2×D4—31no—
430C2×D4—32no—
431C2×D4—61no—
432C2×D4—62no—
433C2×D4—61no—
434C22×C4—61no—
435C22×C4—101no—
436C22×C4—302no—
437C22×C4—302no—
438C22×C4—302no—
439C22×C4C2×D3×D511yescharacteristic
440SD16—38no—
441D8—38no—
442M16D3×Dic514yescomplemented
443C2×C8—151no—
444C2×C8—151no—
445C2×C8—302no—
446C2×C8C2×D3×D512yes—
447C4⋊C4—52no—
448C4⋊C4—51no—
449C4⋊C4—51no—
450C4⋊C4—101no—
451C4⋊C4—151no—
452C4⋊C4—151no—
453C4⋊C4—302no—
454C22⋊C4—101no—
455C22⋊C4—152no—
456C22⋊C4—302no—
457C22⋊C4—302no—
458C22⋊C4—302no—
459C42—101no—
460C42—152no—
461C42—302no—
462C3×C5#294611yescharacteristic, complemented, Hall
463C22×C3—21no—
464C22×C3—21no—
465C22×C3D4×Dic512yes—
466C22×C3C2×D4×D511yescharacteristic
467C2×D3—22no—
468C2×D3—24no—
469C2×D3—41no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—41no—
473C2×D3—84no—
474C3×C4—201no—
475C3×C4—202no—
476C3×C4—201no—
477C3×C4D4×Dic512yes—
478C3×C4C2×#71111yescharacteristic
479C3×C4C23×Dic511yescharacteristic
480Dic3—22no—
481Dic3—102no—
482Dic3—102no—
483C2×C5—22no—
484C2×C5—124no—
485C2×C5—122no—
486C2×C5C2×#89311yescharacteristic
487C2×C5#577411yescharacteristic
488C2×C5#577811yescharacteristic
489C23—61no—
490C23—62no—
491C23—64no—
492C23—241no—
493C23—241no—
494C23#677711yescharacteristic
495Q8—32no—
496Q8—61no—
497D4—32no—
498D4—34no—
499D4—64no—
500D4—64no—
501D4—62no—
502D4—61no—
503C2×C4—61no—
504C2×C4—64no—
505C2×C4—101no—
506C2×C4—101no—
507C2×C4—102no—
508C2×C4—152no—
509C2×C4—152no—
510C2×C4—401no—
511C2×C4—1202no—
512C2×C4—1202no—
513C2×C4—1202no—
514C2×C4#676812yes—
515C2×C4#677711yescharacteristic
516C2×C4C2×D3×Dic512yes—
517C2×C4C22×D3×D511yescharacteristic
518C8—301no—
519C8—301no—
520C8C2×D3×Dic514yes—
521C2×C3—22no—
522C2×C3#6610811yescharacteristic
523C2×C3#6610911yescharacteristic
524C2×C3C2×D4×Dic511yescharacteristic
525D3—44no—
526D3—42no—
527C5#7920111yescharacteristic, complemented, Sylow
528C22—21no—
529C22—21no—
530C22—62no—
531C22—64no—
532C22—121no—
533C22—122no—
534C22—122no—
535C22—121no—
536C22—244no—
537C22#9143512yes—
538C22C2×#677711yescharacteristic
539C4—62no—
540C4—202no—
541C4—201no—
542C4—201no—
543C4—302no—
544C4—302no—
545C4#9143512yes—
546C4C2×#676811yescharacteristic
547C4C22×D3×Dic511yescharacteristic
548C3#1061566111yescharacteristic, complemented, Sylow
549C2—22no—
550C2—124no—
551C2—122no—
552C2#1175403811yescharacteristic
553C2#1175403911yescharacteristic
554C2C2×#9143511yescharacteristic
subgroupG/H
centerC22C2×#6777
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×#908C2
cosocleC2×C3×C4×C5C24
hypercenterC2×M16D3×D5
hypocenterC3×C5#2946
LCSG ▷ 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. #11754503C2semidirect
3. #11754492C2semidirect
4. #11751223C2semidirect
5. C3×#65410C2semidirect
6. #11748382C2semidirect
7. C2×#90966C2semidirect
8. #11747816C2semidirect
9. #11747813C2semidirect
10. #11747810C2semidirect
11. #11747807C2semidirect
18. C5×#908C4semidirect
27. #91435C22semidirect
31. #91344C22semidirect
33. #91099C22semidirect
34. C8×Dic15C22semidirect
35. C3×#3950C22semidirect
36. C3×#3949C22semidirect
37. #90972C22semidirect
38. #90972C22semidirect
41. #90966C22semidirect
42. #90966C22semidirect
54. #65410D3semidirect
59. C5×#235C2×C4semidirect
62. C2×C5×D12C2×C4semidirect
71. C3×C5×M16C2×C4semidirect
80. C5×D24C2×C4semidirect
81. C5×#204C2×C4semidirect
130. #3950C2×D3semidirect
131. #3949C2×D3semidirect
175. #908Dic5semidirect
215. C5×M16C4×D3semidirect
271. #235C2×Dic5semidirect
274. C2×D12C2×Dic5semidirect
282. C3×M16C2×Dic5semidirect
302. D24C2×Dic5semidirect
303. #204C2×Dic5semidirect
442. M16D3×Dic5semidirect
462. C3×C5#2946semidirect
527. C5#79201semidirect
548. C3#10615661semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9082
#117545032
#117544922
#117512232
C3×#654102
#117483822
C2×#909662
#117478162
#117478132
#117478102
#117478072
#106156613
#792015