Finite Group Databasebeta

← C5×#71713 #14300535

Subgroups — C5×#71713 #14300535

Order 1920.

752 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#525C211yesmaximal, characteristic
2C22×C5×#208C211yesmaximal, characteristic, complemented
3C2×C5×#837C212yesmaximal, complemented
4C4×C5×#217C211yesmaximal, characteristic, complemented
5C5×#4532C212yesmaximal, complemented
6C5×#1871—31nomaximal, Hall
7C2×C3×C5×#41C2211yescharacteristic
8C23×C3×C4×C5C2211yescharacteristic
9C3×C5×#123—21no—
10C3×C5×C22⋊Q8C411yescharacteristic, complemented
11C3×C5×C22⋊Q8C411yescharacteristic, complemented
12C3×C5×#121C411yescharacteristic, complemented
13C3×C5×#121C411yescharacteristic, complemented
14C3×C5×#120—21no—
15C3×C4×C5×D4—21no—
16C3×C4×C5×D4—21no—
17C3×C5×#117C2211yescharacteristic
18C2×C5×#217—21no—
19C2×C4×C5×Dic3C2212yes—
20C2×C5×#208—21no—
21C2×C5×#208—24no—
22C2×C5×#208C2212yes—
23C22×C5×C3⋊C8—21no—
24C5×#837—24no—
25C5×C3⋊#95—21no—
26#71713C511yesmaximal, characteristic, direct product factor, Hall
27C22×C5×M16—31no—
28C5×#525D311yescharacteristic
29C2×C5×#104—32no—
30C4×C5×C22⋊C4—31no—
31C5×#339—32no—
32C24×C3×C5D411yescharacteristic
33C3×C5×#41—22no—
34C3×C5×#41—22no—
35C2×C3×C5×Q8C2×C411yescharacteristic
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4C2×C411yescharacteristic
40C22×C3×C4×C5—22no—
41C22×C3×C4×C5—22no—
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5C2×C411yescharacteristic
44C22×C3×C4×C5C2×C411yescharacteristic
45C22×C3×C4×C5D411yescharacteristic
46C22×C3×C4×C5D411yescharacteristic
47C22×C3×C4×C5D411yescharacteristic
48C22×C3×C4×C5C2311yescharacteristic
49C22×C5×Dic3—22no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4C2×C411yescharacteristic
53C3×C5×C4⋊C4C2×C411yescharacteristic
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4—21no—
57C3×C5×C22⋊C4—21no—
58C3×C5×C22⋊C4—21no—
59C3×C42×C5—21no—
60C5×#217—24no—
61C4×C5×Dic3—22no—
62C4×C5×Dic3—22no—
63C5×#208—24no—
64C5×#208—22no—
65C5×#208D414yescomplemented
66C2×C5×C3⋊C8—24no—
67C2×C5×C3⋊C8—22no—
68C3×#525C2×C511yescharacteristic
69C22×#208C2×C511yescharacteristic, complemented
70C2×#837C2×C512yescomplemented
71C4×#217C2×C511yescharacteristic, complemented
72#4532C2×C512yescomplemented
73C2×C5×#41C2×D311yescharacteristic
74C23×C4×C5C2×D311yescharacteristic
75C2×C5×M16—32no—
76C2×C5×M16—64no—
77C2×C5×M16—61no—
78C22×C5×C8—61no—
79C5×#123—21no—
80C5×C22⋊Q8Dic311yescharacteristic, complemented
81C5×C22⋊Q8Dic311yescharacteristic, complemented
82C5×#121Dic311yescharacteristic, complemented
83C5×#121Dic311yescharacteristic, complemented
84C5×#120—21no—
85C4×C5×D4—21no—
86C4×C5×D4—21no—
87C5×#117C2×D311yescharacteristic
88C2×C5×C22⋊C4—61no—
89C2×C42×C5—32no—
90C5×#104—64no—
91C5×#95—61no—
92#1871—31noSylow
93C23×C3×C5—22no—
94C23×C3×C5—22no—
95C23×C3×C5—21no—
96C23×C3×C5—41no—
97C23×C3×C5C22⋊C411yescharacteristic
98C23×C3×C5C22⋊C411yescharacteristic
99C23×C3×C5C2×D411yescharacteristic
100C3×C5×Q8—22no—
101C3×C5×D4—22no—
102C3×C5×D4—42no—
103C3×C5×D4—42no—
104C3×C5×D4—42no—
105C2×C3×C4×C5—22no—
106C2×C3×C4×C5—22no—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5—21no—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—41no—
116C2×C3×C4×C5—41no—
117C2×C3×C4×C5C22⋊C411yescharacteristic
118C2×C3×C4×C5C22⋊C411yescharacteristic
119C2×C3×C4×C5C22×C411yescharacteristic
120C2×C3×C4×C5C2×D412yes—
121C2×C3×C4×C5C2×D411yescharacteristic
122C2×C3×C4×C5#4112yes—
123C2×C5×Dic3—24no—
124C2×C5×Dic3—42no—
125C2×C5×Dic3—42no—
126C5×C3⋊C8—24no—
127C2×C3×#41C22×C511yescharacteristic
128C23×C3×C4C22×C511yescharacteristic
129C3×#123—21no—
130C3×C22⋊Q8C4×C511yescharacteristic, complemented
131C3×C22⋊Q8C4×C511yescharacteristic, complemented
132C3×#121C4×C511yescharacteristic, complemented
133C3×#121C4×C511yescharacteristic, complemented
134C3×#120—21no—
135C3×C4×D4—21no—
136C3×C4×D4—21no—
137C3×#117C22×C511yescharacteristic
138C2×#217—21no—
139C2×C4×Dic3C22×C512yes—
140C2×#208—21no—
141C2×#208—24no—
142C2×#208C22×C512yes—
143C22×C3⋊C8—21no—
144#837—24no—
145C3⋊#95—21no—
146C24×C5#6711yescharacteristic
147C5×#41—22no—
148C5×#41—22no—
149C2×C5×Q8C2×Dic311yescharacteristic
150C2×C5×D4—21no—
151C2×C5×D4—21no—
152C2×C5×D4—21no—
153C2×C5×D4C2×Dic311yescharacteristic
154C22×C4×C5—22no—
155C22×C4×C5—22no—
156C22×C4×C5—21no—
157C22×C4×C5—62no—
158C22×C4×C5C2×Dic311yescharacteristic
159C22×C4×C5C2×Dic311yescharacteristic
160C22×C4×C5#6711yescharacteristic
161C22×C4×C5#6711yescharacteristic
162C22×C4×C5#6711yescharacteristic
163C22×C4×C5C22×D311yescharacteristic
164C5×M16—34no—
165C5×M16—64no—
166C5×M16—62no—
167C2×C5×C8—64no—
168C2×C5×C8—62no—
169C5×C4⋊C4—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4C2×Dic311yescharacteristic
172C5×C4⋊C4C2×Dic311yescharacteristic
173C5×C22⋊C4—21no—
174C5×C22⋊C4—21no—
175C5×C22⋊C4—21no—
176C5×C22⋊C4—21no—
177C5×C22⋊C4—21no—
178C5×C22⋊C4—64no—
179C42×C5—21no—
180C42×C5—62no—
181C42×C5—62no—
182C22×M16—31no—
183#525C5×D311yescharacteristic
184C2×#104—32no—
185C4×C22⋊C4—31no—
186#339—32no—
187C22×C3×C5—21no—
188C22×C3×C5—21no—
189C22×C3×C5—21no—
190C22×C3×C5—21no—
191C22×C3×C5—21no—
192C22×C3×C5—21no—
193C22×C3×C5—21no—
194C22×C3×C5—44no—
195C22×C3×C5—41no—
196C22×C3×C5—41no—
197C22×C3×C5—41no—
198C22×C3×C5#10412yescomplemented
199C22×C3×C5C2×C22⋊C411yescharacteristic
200C22×C3×C5#12112yes—
201C3×C4×C5—21no—
202C3×C4×C5—21no—
203C3×C4×C5—41no—
204C3×C4×C5—41no—
205C3×C4×C5—41no—
206C3×C4×C5C4×D412yes—
207C3×C4×C5#12011yescharacteristic
208C3×C4×C5#12311yescharacteristic
209C5×Dic3—44no—
210C24×C3C5×D411yescharacteristic
211C3×#41—22no—
212C3×#41—22no—
213C2×C3×Q8C2×C4×C511yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4C2×C4×C511yescharacteristic
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4—21no—
221C22×C3×C4C2×C4×C511yescharacteristic
222C22×C3×C4C2×C4×C511yescharacteristic
223C22×C3×C4C5×D411yescharacteristic
224C22×C3×C4C5×D411yescharacteristic
225C22×C3×C4C5×D411yescharacteristic
226C22×C3×C4C23×C511yescharacteristic
227C22×Dic3—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4C2×C4×C511yescharacteristic
231C3×C4⋊C4C2×C4×C511yescharacteristic
232C3×C22⋊C4—21no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—21no—
237C3×C42—21no—
238#217—24no—
239C4×Dic3—22no—
240C4×Dic3—22no—
241#208—24no—
242#208—22no—
243#208C5×D414yescomplemented
244C2×C3⋊C8—24no—
245C2×C3⋊C8—22no—
246C23×C5—22no—
247C23×C5—22no—
248C23×C5—21no—
249C23×C5—41no—
250C23×C5#21711yescharacteristic
251C23×C5#21711yescharacteristic
252C23×C5C2×#6711yescharacteristic
253C5×Q8—22no—
254C5×D4—22no—
255C5×D4—42no—
256C5×D4—42no—
257C5×D4—42no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—64no—
271C2×C4×C5—122no—
272C2×C4×C5—122no—
273C2×C4×C5#21711yescharacteristic
274C2×C4×C5#21711yescharacteristic
275C2×C4×C5D3×D412yes—
276C2×C4×C5#23712yes—
277C2×C4×C5C22×Dic311yescharacteristic
278C2×C4×C5C2×#6711yescharacteristic
279C5×C8—64no—
280C2×#41C2×C5×D311yescharacteristic
281C23×C4C2×C5×D311yescharacteristic
282C2×M16—32no—
283C2×M16—64no—
284C2×M16—61no—
285C22×C8—61no—
286#123—21no—
287C22⋊Q8C5×Dic311yescharacteristic, complemented
288C22⋊Q8C5×Dic311yescharacteristic, complemented
289#121C5×Dic311yescharacteristic, complemented
290#121C5×Dic311yescharacteristic, complemented
291#120—21no—
292C4×D4—21no—
293C4×D4—21no—
294#117C2×C5×D311yescharacteristic
295C2×C22⋊C4—61no—
296C2×C42—32no—
297#104—64no—
298#95—61no—
299C2×C3×C5—22no—
300C2×C3×C5—22no—
301C2×C3×C5—41no—
302C2×C3×C5—81no—
303C2×C3×C5#38611yescharacteristic
304C2×C3×C5C2×#10411yescharacteristic
305C2×C3×C5#42111yescharacteristic
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—21no—
309C23×C3—41no—
310C23×C3C5×C22⋊C411yescharacteristic
311C23×C3C5×C22⋊C411yescharacteristic
312C23×C3C2×C5×D411yescharacteristic
313C3×Q8—22no—
314C3×D4—22no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—42no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4C5×C22⋊C411yescharacteristic
331C2×C3×C4C5×C22⋊C411yescharacteristic
332C2×C3×C4C22×C4×C511yescharacteristic
333C2×C3×C4C2×C5×D412yes—
334C2×C3×C4C2×C5×D411yescharacteristic
335C2×C3×C4C5×#4112yes—
336C2×Dic3—24no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C3⋊C8—24no—
340C22×C5—21no—
341C22×C5—21no—
342C22×C5—21no—
343C22×C5—21no—
344C22×C5—21no—
345C22×C5—21no—
346C22×C5—21no—
347C22×C5—44no—
348C22×C5—41no—
349C22×C5—41no—
350C22×C5—41no—
351C22×C5#83712yescomplemented
352C22×C5#93912yes—
353C22×C5C2×#21711yescharacteristic
354C4×C5—21no—
355C4×C5—21no—
356C4×C5—41no—
357C4×C5—41no—
358C4×C5—41no—
359C4×C5—124no—
360C4×C5D4×Dic312yes—
361C4×C5#93511yescharacteristic
362C4×C5#93711yescharacteristic
363C24C5×#6711yescharacteristic
364#41—22no—
365#41—22no—
366C2×Q8C2×C5×Dic311yescharacteristic
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4C2×C5×Dic311yescharacteristic
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—62no—
375C22×C4C2×C5×Dic311yescharacteristic
376C22×C4C2×C5×Dic311yescharacteristic
377C22×C4C5×#6711yescharacteristic
378C22×C4C5×#6711yescharacteristic
379C22×C4C5×#6711yescharacteristic
380C22×C4C22×C5×D311yescharacteristic
381M16—34no—
382M16—64no—
383M16—62no—
384C2×C8—64no—
385C2×C8—62no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4C2×C5×Dic311yescharacteristic
389C4⋊C4C2×C5×Dic311yescharacteristic
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—21no—
394C22⋊C4—21no—
395C22⋊C4—64no—
396C42—21no—
397C42—62no—
398C42—62no—
399C3×C5#187111yescharacteristic, complemented, Hall
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—44no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3C5×#10412yescomplemented
412C22×C3C2×C5×C22⋊C411yescharacteristic
413C22×C3C5×#12112yes—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—41no—
419C3×C4C4×C5×D412yes—
420C3×C4C5×#12011yescharacteristic
421C3×C4C5×#12311yescharacteristic
422Dic3—44no—
423C2×C5—22no—
424C2×C5—22no—
425C2×C5—41no—
426C2×C5—81no—
427C2×C5#522011yescharacteristic
428C2×C5C2×#83711yescharacteristic
429C2×C5#523611yescharacteristic
430C23—22no—
431C23—22no—
432C23—21no—
433C23—41no—
434C23C5×#21711yescharacteristic
435C23C5×#21711yescharacteristic
436C23C2×C5×#6711yescharacteristic
437Q8—22no—
438D4—22no—
439D4—42no—
440D4—42no—
441D4—42no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—64no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4C5×#21711yescharacteristic
458C2×C4C5×#21711yescharacteristic
459C2×C4C5×D3×D412yes—
460C2×C4C5×#23712yes—
461C2×C4C22×C5×Dic311yescharacteristic
462C2×C4C2×C5×#6711yescharacteristic
463C8—64no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—41no—
467C2×C3—81no—
468C2×C3C5×#38611yescharacteristic
469C2×C3C2×C5×#10411yescharacteristic
470C2×C3C5×#42111yescharacteristic
471C5#7171311yescharacteristic, direct product factor, Sylow
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—44no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22C5×#83712yescomplemented
484C22C5×#93912yes—
485C22C2×C5×#21711yescharacteristic
486C4—21no—
487C4—21no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—124no—
492C4C5×D4×Dic312yes—
493C4C5×#93511yescharacteristic
494C4C5×#93711yescharacteristic
495C3C5×#187111yescharacteristic, complemented, Sylow
496C2—22no—
497C2—22no—
498C2—41no—
499C2—81no—
500C2C5×#522011yescharacteristic
501C2C2×C5×#83711yescharacteristic
502C2C5×#523611yescharacteristic
subgroupG/H
centerC2×C4×C5C2×#67
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
Fitting, prime radicalC3×C5×#525C2
socleC22×C3×C5C2×C22⋊C4
cosocleC22×C3×C4C23×C5
hypercenterC5×#525D3
hypocenterC3C5×#1871
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ C5×#525

Complements

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

NH ≅ G/Ntype
2. C22×C5×#208C2semidirect
3. C2×C5×#837C2semidirect
4. C4×C5×#217C2semidirect
5. C5×#4532C2semidirect
10. C3×C5×C22⋊Q8C4semidirect
11. C3×C5×C22⋊Q8C4semidirect
12. C3×C5×#121C4semidirect
13. C3×C5×#121C4semidirect
26. #71713C5direct (N × H)
65. C5×#208D4semidirect
69. C22×#208C2×C5semidirect
70. C2×#837C2×C5semidirect
71. C4×#217C2×C5semidirect
72. #4532C2×C5semidirect
80. C5×C22⋊Q8Dic3semidirect
81. C5×C22⋊Q8Dic3semidirect
82. C5×#121Dic3semidirect
83. C5×#121Dic3semidirect
130. C3×C22⋊Q8C4×C5semidirect
131. C3×C22⋊Q8C4×C5semidirect
132. C3×#121C4×C5semidirect
133. C3×#121C4×C5semidirect
198. C22×C3×C5#104semidirect
243. #208C5×D4semidirect
287. C22⋊Q8C5×Dic3semidirect
288. C22⋊Q8C5×Dic3semidirect
289. #121C5×Dic3semidirect
290. #121C5×Dic3semidirect
351. C22×C5#837semidirect
399. C3×C5#1871semidirect
411. C22×C3C5×#104semidirect
471. C5#71713direct (N × H)
483. C22C5×#837semidirect
495. C3C5×#1871semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5252
C22×C5×#2082
C2×C5×#8372
C4×C5×#2172
C5×#45322
C5×#18713
#717135