Finite Group Databasebeta

← C5×#69538 #14301654

Subgroups — C5×#69538 #14301654

Order 1920.

772 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#929C211yesmaximal, characteristic
2C5×#5486C211yesmaximal, characteristic
3C3×C5×#381C211yesmaximal, characteristic, complemented
4C5×#5211C211yesmaximal, characteristic, complemented
5C2×C5×#806C211yesmaximal, characteristic
6C5×#4527C211yesmaximal, characteristic, complemented
7C5×#4526C211yesmaximal, characteristic, complemented
8C5×#1890—31nomaximal, Hall
9C23×C3×C4×C5C2211yescharacteristic
10C22×C5×#67C2211yescharacteristic
11C2×C3×C5×C22⋊C4C2211yescharacteristic
12C2×C3×C5×C22⋊C4C2211yescharacteristic
13C2×C5×#217C2211yescharacteristic
14C2×C5×#217C2211yescharacteristic
15C2×C5×#217C2211yescharacteristic
16C5×#929—22no—
17C5×#929C414yescomplemented
18C4×C5×#67—22no—
19C2×C5×(C2×D3)⋊C4—21no—
20C2×C5×Dic3⋊C4—21no—
21C5×#67⋊C4—22no—
22C5×D3×C22⋊C4—21no—
23C5×#877—21no—
24C3×C5×#95—22no—
25C5×C3⋊#95—22no—
26C5×#806—22no—
27#69538C511yesmaximal, characteristic, direct product factor, Hall
28C2×C5×#123—31no—
29C5×#518—31no—
30C2×C5×#99—31no—
31C5×#381—31no—
32C5×#381D311yescharacteristic, complemented
33C5×#342—31no—
34C5×#342—31no—
35C24×C3×C5C2311yescharacteristic
36C23×C5×D3D411yescharacteristic
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5C2×C412yescomplemented
45C22×C3×C4×C5D412yes—
46C2×C5×#67—21no—
47C2×C5×#67—21no—
48C2×C5×#67—21no—
49C2×C5×#67—21no—
50C2×C5×#67C2×C412yes—
51C2×C5×#67D412yes—
52C22×C5×Dic3—21no—
53C22×C5×Dic3—21no—
54C22×C5×Dic3D411yescharacteristic
55C22×C5×Dic3D412yes—
56C2×C4×C5×D3—22no—
57C3×C5×C22⋊C4—21no—
58C3×C5×C22⋊C4—21no—
59C3×C5×C22⋊C4—21no—
60C3×C5×C22⋊C4—21no—
61C5×#217—21no—
62C5×#217—21no—
63C5×#217—21no—
64C5×#217—21no—
65C5×#217—21no—
66C5×#217C2×C412yescomplemented
67C5×(C2×D3)⋊C4—22no—
68C5×(C2×D3)⋊C4—24no—
69C5×Dic3⋊C4—22no—
70C5×Dic3⋊C4—24no—
71C4×C5×Dic3—22no—
72C2×#929C2×C511yescharacteristic
73#5486C2×C511yescharacteristic
74C3×#381C2×C511yescharacteristic, complemented
75#5211C2×C511yescharacteristic, complemented
76C2×#806C2×C511yescharacteristic
77#4527C2×C511yescharacteristic, complemented
78#4526C2×C511yescharacteristic, complemented
79C22×C5×D4—31no—
80C23×C4×C5C2×D311yescharacteristic
81C5×#123—34no—
82C5×#123—62no—
83C4×C5×D4—62no—
84C4×C5×D4—62no—
85C5×#117—61no—
86C2×C5×C4⋊C4—61no—
87C2×C5×C22⋊C4—31no—
88C2×C5×C22⋊C4—31no—
89C2×C5×C22⋊C4—31no—
90C2×C5×C22⋊C4—61no—
91C2×C5×C22⋊C4—61no—
92C2×C5×C22⋊C4C2×D311yescharacteristic
93C2×C5×C22⋊C4C2×D311yescharacteristic
94C5×#99—62no—
95C5×#95—22no—
96C5×#95—62no—
97#1890—31noSylow
98C23×C3×C5—42no—
99C23×C3×C5C22×C411yescharacteristic
100C23×C3×C5C2×D411yescharacteristic
101C23×C3×C5C2×D411yescharacteristic
102C23×C3×C5C2×D411yescharacteristic
103C23×C3×C5C2×D411yescharacteristic
104C23×C3×C5#4111yescharacteristic
105C23×C3×C5#4111yescharacteristic
106C22×C5×D3—21no—
107C22×C5×D3—21no—
108C22×C5×D3—42no—
109C22×C5×D3C22⋊C412yes—
110C2×C3×C4×C5—22no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5—41no—
115C2×C3×C4×C5—41no—
116C2×C3×C4×C5—41no—
117C2×C3×C4×C5—41no—
118C2×C3×C4×C5—42no—
119C2×C3×C4×C5—42no—
120C2×C3×C4×C5—41no—
121C5×#67—22no—
122C5×#67—22no—
123C5×#67—22no—
124C5×#67—22no—
125C2×C5×Dic3—21no—
126C2×C5×Dic3—21no—
127C2×C5×Dic3—22no—
128C2×C5×Dic3—22no—
129C2×C5×Dic3—42no—
130C2×C5×Dic3—42no—
131C2×C5×Dic3—42no—
132C2×C5×Dic3—41no—
133C2×C5×Dic3C22⋊C412yes—
134C4×C5×D3—82no—
135C23×C3×C4C22×C511yescharacteristic
136C22×#67C22×C511yescharacteristic
137C2×C3×C22⋊C4C22×C511yescharacteristic
138C2×C3×C22⋊C4C22×C511yescharacteristic
139C2×#217C22×C511yescharacteristic
140C2×#217C22×C511yescharacteristic
141C2×#217C22×C511yescharacteristic
142#929—22no—
143#929C4×C514yescomplemented
144C4×#67—22no—
145C2×(C2×D3)⋊C4—21no—
146C2×Dic3⋊C4—21no—
147#67⋊C4—22no—
148D3×C22⋊C4—21no—
149#877—21no—
150C3×#95—22no—
151C3⋊#95—22no—
152#806—22no—
153C24×C5—31no—
154C24×C5C22×D311yescharacteristic
155C2×C5×D4—32no—
156C2×C5×D4—32no—
157C2×C5×D4—61no—
158C2×C5×D4—61no—
159C2×C5×D4—61no—
160C2×C5×D4—61no—
161C22×C4×C5—21no—
162C22×C4×C5—21no—
163C22×C4×C5—21no—
164C22×C4×C5—21no—
165C22×C4×C5—21no—
166C22×C4×C5—21no—
167C22×C4×C5—21no—
168C22×C4×C5—31no—
169C22×C4×C5—32no—
170C22×C4×C5—62no—
171C22×C4×C5—61no—
172C22×C4×C5—61no—
173C22×C4×C5C4×D312yescomplemented
174C22×C4×C5#6712yes—
175C5×C4⋊C4—64no—
176C5×C4⋊C4—62no—
177C5×C22⋊C4—21no—
178C5×C22⋊C4—21no—
179C5×C22⋊C4—21no—
180C5×C22⋊C4—21no—
181C5×C22⋊C4—32no—
182C5×C22⋊C4—64no—
183C5×C22⋊C4—62no—
184C5×C22⋊C4—61no—
185C5×C22⋊C4—61no—
186C5×C22⋊C4—61no—
187C5×C22⋊C4—61no—
188C5×C22⋊C4—61no—
189C42×C5—62no—
190C2×#123—31no—
191#518—31no—
192C2×#99—31no—
193#381—31no—
194#381C5×D311yescharacteristic, complemented
195#342—31no—
196#342—31no—
197C22×C3×C5—22no—
198C22×C3×C5—22no—
199C22×C3×C5—22no—
200C22×C3×C5—41no—
201C22×C3×C5—41no—
202C22×C3×C5—41no—
203C22×C3×C5—41no—
204C22×C3×C5C2×C22⋊C411yescharacteristic
205C22×C3×C5C4×D411yescharacteristic
206C22×C3×C5C4×D411yescharacteristic
207C22×C3×C5#12011yescharacteristic
208C22×C3×C5#12111yescharacteristic
209C22×C3×C5#12111yescharacteristic
210C22×C3×C5#12311yescharacteristic
211C2×C5×D3—22no—
212C2×C5×D3—42no—
213C2×C5×D3—42no—
214C2×C5×D3—81no—
215C3×C4×C5—42no—
216C3×C4×C5—42no—
217C3×C4×C5—81no—
218C5×Dic3—22no—
219C5×Dic3—42no—
220C5×Dic3—42no—
221C5×Dic3—81no—
222C24×C3C23×C511yescharacteristic
223C23×D3C5×D411yescharacteristic
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4C2×C4×C512yescomplemented
232C22×C3×C4C5×D412yes—
233C2×#67—21no—
234C2×#67—21no—
235C2×#67—21no—
236C2×#67—21no—
237C2×#67C2×C4×C512yes—
238C2×#67C5×D412yes—
239C22×Dic3—21no—
240C22×Dic3—21no—
241C22×Dic3C5×D411yescharacteristic
242C22×Dic3C5×D412yes—
243C2×C4×D3—22no—
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248#217—21no—
249#217—21no—
250#217—21no—
251#217—21no—
252#217—21no—
253#217C2×C4×C512yescomplemented
254(C2×D3)⋊C4—22no—
255(C2×D3)⋊C4—24no—
256Dic3⋊C4—22no—
257Dic3⋊C4—24no—
258C4×Dic3—22no—
259C23×C5—32no—
260C23×C5—42no—
261C23×C5—61no—
262C23×C5—61no—
263C23×C5—122no—
264C23×C5C2×C4×D311yescharacteristic
265C23×C5#23511yescharacteristic
266C23×C5D3×D411yescharacteristic
267C23×C5D3×D411yescharacteristic
268C23×C5D3×D411yescharacteristic
269C23×C5#23711yescharacteristic
270C23×C5C2×#6711yescharacteristic
271C5×D4—62no—
272C5×D4—62no—
273C5×D4—62no—
274C5×D4—62no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—32no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—42no—
285C2×C4×C5—42no—
286C2×C4×C5—41no—
287C2×C4×C5—61no—
288C2×C4×C5—61no—
289C2×C4×C5—62no—
290C2×C4×C5—62no—
291C2×C4×C5—122no—
292C2×C4×C5—122no—
293C2×C4×C5—122no—
294C2×C4×C5—121no—
295C2×C4×C5—242no—
296C22×D4—31no—
297C23×C4C2×C5×D311yescharacteristic
298#123—34no—
299#123—62no—
300C4×D4—62no—
301C4×D4—62no—
302#117—61no—
303C2×C4⋊C4—61no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4C2×C5×D311yescharacteristic
310C2×C22⋊C4C2×C5×D311yescharacteristic
311#99—62no—
312#95—22no—
313#95—62no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5—21no—
317C2×C3×C5—21no—
318C2×C3×C5—21no—
319C2×C3×C5—21no—
320C2×C3×C5#38611yescharacteristic
321C2×C3×C5#45812yes—
322C5×D3—42no—
323C23×C3—42no—
324C23×C3C22×C4×C511yescharacteristic
325C23×C3C2×C5×D411yescharacteristic
326C23×C3C2×C5×D411yescharacteristic
327C23×C3C2×C5×D411yescharacteristic
328C23×C3C2×C5×D411yescharacteristic
329C23×C3C5×#4111yescharacteristic
330C23×C3C5×#4111yescharacteristic
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—42no—
334C22×D3C5×C22⋊C412yes—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346#67—22no—
347#67—22no—
348#67—22no—
349#67—22no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3C5×C22⋊C412yes—
359C4×D3—82no—
360C22×C5—22no—
361C22×C5—22no—
362C22×C5—22no—
363C22×C5—41no—
364C22×C5—41no—
365C22×C5—41no—
366C22×C5—41no—
367C22×C5—62no—
368C22×C5—122no—
369C22×C5—122no—
370C22×C5—241no—
371C22×C5D3×C22⋊C411yescharacteristic
372C22×C5#67⋊C411yescharacteristic
373C22×C5#88311yescharacteristic
374C22×C5#88411yescharacteristic
375C22×C5C4×#6711yescharacteristic
376C22×C5#93711yescharacteristic
377C22×C5#93811yescharacteristic
378C4×C5—42no—
379C4×C5—42no—
380C4×C5—62no—
381C4×C5—81no—
382C4×C5—122no—
383C4×C5—122no—
384C4×C5—241no—
385C24—31no—
386C24C22×C5×D311yescharacteristic
387C2×D4—32no—
388C2×D4—32no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—31no—
401C22×C4—32no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4C4×C5×D312yescomplemented
406C22×C4C5×#6712yes—
407C4⋊C4—64no—
408C4⋊C4—62no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—32no—
414C22⋊C4—64no—
415C22⋊C4—62no—
416C22⋊C4—61no—
417C22⋊C4—61no—
418C22⋊C4—61no—
419C22⋊C4—61no—
420C22⋊C4—61no—
421C42—62no—
422C3×C5#189011yescharacteristic, complemented, Hall
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3C2×C5×C22⋊C411yescharacteristic
431C22×C3C4×C5×D411yescharacteristic
432C22×C3C4×C5×D411yescharacteristic
433C22×C3C5×#12011yescharacteristic
434C22×C3C5×#12111yescharacteristic
435C22×C3C5×#12111yescharacteristic
436C22×C3C5×#12311yescharacteristic
437C2×D3—22no—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—81no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—81no—
444Dic3—22no—
445Dic3—42no—
446Dic3—42no—
447Dic3—81no—
448C2×C5—21no—
449C2×C5—21no—
450C2×C5—21no—
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—21no—
454C2×C5—122no—
455C2×C5#495611yescharacteristic
456C2×C5#505312yes—
457C23—32no—
458C23—42no—
459C23—61no—
460C23—61no—
461C23—122no—
462C23C2×C4×C5×D311yescharacteristic
463C23C5×#23511yescharacteristic
464C23C5×D3×D411yescharacteristic
465C23C5×D3×D411yescharacteristic
466C23C5×D3×D411yescharacteristic
467C23C5×#23711yescharacteristic
468C23C2×C5×#6711yescharacteristic
469D4—62no—
470D4—62no—
471D4—62no—
472D4—62no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—32no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—121no—
493C2×C4—242no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3C5×#38611yescharacteristic
501C2×C3C5×#45812yes—
502D3—42no—
503C5#6953811yescharacteristic, direct product factor, Sylow
504C22—22no—
505C22—22no—
506C22—22no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—62no—
512C22—122no—
513C22—122no—
514C22—241no—
515C22C5×D3×C22⋊C411yescharacteristic
516C22C5×#67⋊C411yescharacteristic
517C22C5×#88311yescharacteristic
518C22C5×#88411yescharacteristic
519C22C4×C5×#6711yescharacteristic
520C22C5×#93711yescharacteristic
521C22C5×#93811yescharacteristic
522C4—42no—
523C4—42no—
524C4—62no—
525C4—81no—
526C4—122no—
527C4—122no—
528C4—241no—
529C3C5×#189011yescharacteristic, complemented, Sylow
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—21no—
535C2—21no—
536C2—122no—
537C2C5×#495611yescharacteristic
538C2C5×#505312yes—
subgroupG/H
centerC22×C5#937
commutatorC23×C3C22×C4×C5
FrattiniC24C22×C5×D3
FittingC3×C5×#381C2
socleC22×C3×C5#120
prime radicalC22×C5×#67C22
cosocleC24×C3C23×C5
hypercenterC5×#381D3
hypocenterC3C5×#1890
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C24×C5 ◁ C5×#381

Complements

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

NH ≅ G/Ntype
3. C3×C5×#381C2semidirect
4. C5×#5211C2semidirect
6. C5×#4527C2semidirect
7. C5×#4526C2semidirect
17. C5×#929C4semidirect
27. #69538C5direct (N × H)
32. C5×#381D3semidirect
44. C22×C3×C4×C5C2×C4semidirect
66. C5×#217C2×C4semidirect
74. C3×#381C2×C5semidirect
75. #5211C2×C5semidirect
77. #4527C2×C5semidirect
78. #4526C2×C5semidirect
143. #929C4×C5semidirect
173. C22×C4×C5C4×D3semidirect
194. #381C5×D3semidirect
231. C22×C3×C4C2×C4×C5semidirect
253. #217C2×C4×C5semidirect
405. C22×C4C4×C5×D3semidirect
422. C3×C5#1890semidirect
503. C5#69538direct (N × H)
529. C3C5×#1890semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9292
C5×#54862
C3×C5×#3812
C5×#52112
C2×C5×#8062
C5×#45272
C5×#45262
C5×#18903
#695385