Finite Group Databasebeta

← C5×#69506 #14299439

Subgroups — C5×#69506 #14299439

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#926C211yesmaximal, characteristic
2C3×C5×#380C211yesmaximal, characteristic
3C2×C5×#830C211yesmaximal, characteristic
4C2×C5×#808C211yesmaximal, characteristic, complemented
5C2×C5×#807C211yesmaximal, characteristic, complemented
6C5×#4529C211yesmaximal, characteristic, complemented
7C5×#4529C211yesmaximal, characteristic, complemented
8C5×#1854—31nomaximal, Hall
9C23×C3×C4×C5C2211yescharacteristic
10C2×C3×C5×C4⋊C4C2211yescharacteristic
11C2×C3×C5×C4⋊C4C2211yescharacteristic
12C2×C3×C5×C22⋊C4—21no—
13C2×C5×#217—21no—
14C5×#926—22no—
15C5×#926—22no—
16C2×C5×C4⋊Dic3C2211yescharacteristic
17C2×C5×C4⋊Dic3C2211yescharacteristic
18C2×C4×C5×Dic3—21no—
19C22×C5×C3⋊C8C2211yescharacteristic
20C22×C5×C3⋊C8C2211yescharacteristic
21C3×C5×#95—21no—
22C5×#830C414yescomplemented
23C5×#808—22no—
24C5×#807—22no—
25#69506C511yesmaximal, characteristic, direct product factor, Hall
26C2×C5×#117—31no—
27C2×C5×#107—31no—
28C2×C5×#106—31no—
29C2×C5×C22⋊C8—31no—
30C5×#380D311yescharacteristic
31C5×#340—31no—
32C5×#340—31no—
33C24×C3×C5D411yescharacteristic
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5C2×C412yes—
41C22×C3×C4×C5D411yescharacteristic
42C22×C3×C4×C5Q812yes—
43C22×C3×C4×C5C2311yescharacteristic
44C22×C5×Dic3—21no—
45C22×C5×Dic3—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C22⋊C4—42no—
51C5×#217—24no—
52C5×C4⋊Dic3—21no—
53C5×C4⋊Dic3—21no—
54C5×C4⋊Dic3D412yes—
55C5×C4⋊Dic3D412yes—
56C4×C5×Dic3—24no—
57C2×C5×C3⋊C8—21no—
58C2×C5×C3⋊C8—21no—
59C2×C5×C3⋊C8—21no—
60C2×C5×C3⋊C8—21no—
61C2×C5×C3⋊C8C2×C412yes—
62C2×C5×C3⋊C8C2×C412yes—
63C2×#926C2×C511yescharacteristic
64C3×#380C2×C511yescharacteristic
65C2×#830C2×C511yescharacteristic
66C2×#808C2×C511yescharacteristic, complemented
67C2×#807C2×C511yescharacteristic, complemented
68#4529C2×C511yescharacteristic, complemented
69#4529C2×C511yescharacteristic, complemented
70C23×C4×C5C2×D311yescharacteristic
71C22×C5×C8—31no—
72C22×C5×C8—31no—
73C5×#117—62no—
74C5×#117—62no—
75C2×C5×C4⋊C4—31no—
76C2×C5×C4⋊C4—31no—
77C2×C5×C4⋊C4C2×D311yescharacteristic
78C2×C5×C4⋊C4C2×D311yescharacteristic
79C2×C5×C22⋊C4—21no—
80C2×C5×C22⋊C4—61no—
81C2×C42×C5—61no—
82C5×#107—62no—
83C5×#106—62no—
84C5×C22⋊C8—34no—
85C5×#95—21no—
86#1854—31noSylow
87C23×C3×C5—21no—
88C23×C3×C5—21no—
89C23×C3×C5—21no—
90C23×C3×C5—21no—
91C23×C3×C5—21no—
92C23×C3×C5—21no—
93C23×C3×C5C4⋊C412yes—
94C23×C3×C5C2×D411yescharacteristic
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—22no—
98C2×C3×C4×C5—22no—
99C2×C3×C4×C5—22no—
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5C4⋊C412yes—
108C2×C3×C4×C5C22×C411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5C2×D411yescharacteristic
111C2×C3×C4×C5C2×Q811yescharacteristic
112C2×C3×C4×C5#4111yescharacteristic
113C2×C3×C4×C5#4111yescharacteristic
114C2×C5×Dic3—22no—
115C2×C5×Dic3—22no—
116C2×C5×Dic3—41no—
117C2×C5×Dic3—41no—
118C2×C5×Dic3—41no—
119C2×C5×Dic3—41no—
120C5×C3⋊C8—22no—
121C5×C3⋊C8—22no—
122C23×C3×C4C22×C511yescharacteristic
123C2×C3×C4⋊C4C22×C511yescharacteristic
124C2×C3×C4⋊C4C22×C511yescharacteristic
125C2×C3×C22⋊C4—21no—
126C2×#217—21no—
127#926—22no—
128#926—22no—
129C2×C4⋊Dic3C22×C511yescharacteristic
130C2×C4⋊Dic3C22×C511yescharacteristic
131C2×C4×Dic3—21no—
132C22×C3⋊C8C22×C511yescharacteristic
133C22×C3⋊C8C22×C511yescharacteristic
134C3×#95—21no—
135#830C4×C514yescomplemented
136#808—22no—
137#807—22no—
138C24×C5#6711yescharacteristic
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—61no—
146C22×C4×C5—61no—
147C22×C4×C5Dic612yes—
148C22×C4×C5C4×D312yes—
149C22×C4×C5#6711yescharacteristic
150C22×C4×C5C22×D311yescharacteristic
151C2×C5×C8—32no—
152C2×C5×C8—32no—
153C2×C5×C8—61no—
154C2×C5×C8—61no—
155C2×C5×C8—61no—
156C2×C5×C8—61no—
157C5×C4⋊C4—21no—
158C5×C4⋊C4—21no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—32no—
162C5×C4⋊C4—32no—
163C5×C4⋊C4—61no—
164C5×C4⋊C4—61no—
165C5×C22⋊C4—42no—
166C5×C22⋊C4—64no—
167C42×C5—64no—
168C2×#117—31no—
169C2×#107—31no—
170C2×#106—31no—
171C2×C22⋊C8—31no—
172#380C5×D311yescharacteristic
173#340—31no—
174#340—31no—
175C22×C3×C5—22no—
176C22×C3×C5—22no—
177C22×C3×C5—22no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5—41no—
181C22×C3×C5—41no—
182C22×C3×C5C2×C4⋊C411yescharacteristic
183C22×C3×C5#12011yescharacteristic
184C22×C3×C5#12311yescharacteristic
185C22×C3×C5#13511yescharacteristic
186C22×C3×C5#13511yescharacteristic
187C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
188C22×C3×C5#13711yescharacteristic
189C3×C4×C5—22no—
190C3×C4×C5—81no—
191C3×C4×C5—81no—
192C3×C4×C5C4×D411yescharacteristic
193C3×C4×C5C4×D411yescharacteristic
194C3×C4×C5C22⋊Q811yescharacteristic
195C3×C4×C5C22⋊Q811yescharacteristic
196C5×Dic3—42no—
197C5×Dic3—42no—
198C24×C3C5×D411yescharacteristic
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×C4×C512yes—
206C22×C3×C4C5×D411yescharacteristic
207C22×C3×C4C5×Q812yes—
208C22×C3×C4C23×C511yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C22⋊C4—42no—
216#217—24no—
217C4⋊Dic3—21no—
218C4⋊Dic3—21no—
219C4⋊Dic3C5×D412yes—
220C4⋊Dic3C5×D412yes—
221C4×Dic3—24no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8C2×C4×C512yes—
227C2×C3⋊C8C2×C4×C512yes—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5—21no—
231C23×C5—21no—
232C23×C5—21no—
233C23×C5—21no—
234C23×C5Dic3⋊C412yes—
235C23×C5C2×#6711yescharacteristic
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—41no—
243C2×C4×C5—41no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—41no—
248C2×C4×C5—62no—
249C2×C4×C5—62no—
250C2×C4×C5—121no—
251C2×C4×C5—121no—
252C2×C4×C5—121no—
253C2×C4×C5—121no—
254C2×C4×C5Dic3⋊C412yes—
255C2×C4×C5C2×Dic611yescharacteristic
256C2×C4×C5C2×C4×D311yescharacteristic
257C2×C4×C5D3×D411yescharacteristic
258C2×C4×C5D3×D411yescharacteristic
259C2×C4×C5#23711yescharacteristic
260C2×C4×C5#23711yescharacteristic
261C5×C8—62no—
262C5×C8—62no—
263C23×C4C2×C5×D311yescharacteristic
264C22×C8—31no—
265C22×C8—31no—
266#117—62no—
267#117—62no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4C2×C5×D311yescharacteristic
271C2×C4⋊C4C2×C5×D311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—61no—
274C2×C42—61no—
275#107—62no—
276#106—62no—
277C22⋊C8—34no—
278#95—21no—
279C2×C3×C5—42no—
280C2×C3×C5#38511yescharacteristic
281C2×C3×C5#42711yescharacteristic
282C2×C3×C5#42811yescharacteristic
283C2×C3×C5#44911yescharacteristic
284C2×C3×C5#45211yescharacteristic
285C2×C3×C5#48211yescharacteristic
286C2×C3×C5#48511yescharacteristic
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3C5×C4⋊C412yes—
294C23×C3C2×C5×D411yescharacteristic
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4C5×C4⋊C412yes—
308C2×C3×C4C22×C4×C511yescharacteristic
309C2×C3×C4C2×C5×D411yescharacteristic
310C2×C3×C4C2×C5×D411yescharacteristic
311C2×C3×C4C2×C5×Q811yescharacteristic
312C2×C3×C4C5×#4111yescharacteristic
313C2×C3×C4C5×#4111yescharacteristic
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C3⋊C8—22no—
321C3⋊C8—22no—
322C22×C5—22no—
323C22×C5—22no—
324C22×C5—22no—
325C22×C5—41no—
326C22×C5—41no—
327C22×C5—41no—
328C22×C5—41no—
329C22×C5C2×Dic3⋊C411yescharacteristic
330C22×C5#93211yescharacteristic
331C22×C5#93511yescharacteristic
332C22×C5#93711yescharacteristic
333C22×C5#94211yescharacteristic
334C22×C5#95011yescharacteristic
335C22×C5#95011yescharacteristic
336C4×C5—22no—
337C4×C5—81no—
338C4×C5—81no—
339C4×C5—122no—
340C4×C5—122no—
341C4×C5#87811yescharacteristic
342C4×C5#87811yescharacteristic
343C4×C5#67⋊C411yescharacteristic
344C4×C5#67⋊C411yescharacteristic
345C24C5×#6711yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4C5×Dic612yes—
355C22×C4C4×C5×D312yes—
356C22×C4C5×#6711yescharacteristic
357C22×C4C22×C5×D311yescharacteristic
358C2×C8—32no—
359C2×C8—32no—
360C2×C8—61no—
361C2×C8—61no—
362C2×C8—61no—
363C2×C8—61no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—32no—
369C4⋊C4—32no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C22⋊C4—42no—
373C22⋊C4—64no—
374C42—64no—
375C3×C5#185411yescharacteristic, complemented, Hall
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3C2×C5×C4⋊C411yescharacteristic
384C22×C3C5×#12011yescharacteristic
385C22×C3C5×#12311yescharacteristic
386C22×C3C5×#13511yescharacteristic
387C22×C3C5×#13511yescharacteristic
388C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
389C22×C3C5×#13711yescharacteristic
390C3×C4—22no—
391C3×C4—81no—
392C3×C4—81no—
393C3×C4C4×C5×D411yescharacteristic
394C3×C4C4×C5×D411yescharacteristic
395C3×C4C5×C22⋊Q811yescharacteristic
396C3×C4C5×C22⋊Q811yescharacteristic
397Dic3—42no—
398Dic3—42no—
399C2×C5—42no—
400C2×C5#494611yescharacteristic
401C2×C5#496411yescharacteristic
402C2×C5#499811yescharacteristic
403C2×C5#503511yescharacteristic
404C2×C5#503711yescharacteristic
405C2×C5#504511yescharacteristic
406C2×C5#504711yescharacteristic
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23C5×Dic3⋊C412yes—
414C23C2×C5×#6711yescharacteristic
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4C5×Dic3⋊C412yes—
434C2×C4C2×C5×Dic611yescharacteristic
435C2×C4C2×C4×C5×D311yescharacteristic
436C2×C4C5×D3×D411yescharacteristic
437C2×C4C5×D3×D411yescharacteristic
438C2×C4C5×#23711yescharacteristic
439C2×C4C5×#23711yescharacteristic
440C8—62no—
441C8—62no—
442C2×C3—42no—
443C2×C3C5×#38511yescharacteristic
444C2×C3C5×#42711yescharacteristic
445C2×C3C5×#42811yescharacteristic
446C2×C3C5×#44911yescharacteristic
447C2×C3C5×#45211yescharacteristic
448C2×C3C5×#48211yescharacteristic
449C2×C3C5×#48511yescharacteristic
450C5#6950611yescharacteristic, direct product factor, Sylow
451C22—22no—
452C22—22no—
453C22—22no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22C2×C5×Dic3⋊C411yescharacteristic
459C22C5×#93211yescharacteristic
460C22C5×#93511yescharacteristic
461C22C5×#93711yescharacteristic
462C22C5×#94211yescharacteristic
463C22C5×#95011yescharacteristic
464C22C5×#95011yescharacteristic
465C4—22no—
466C4—81no—
467C4—81no—
468C4—122no—
469C4—122no—
470C4C5×#87811yescharacteristic
471C4C5×#87811yescharacteristic
472C4C5×#67⋊C411yescharacteristic
473C4C5×#67⋊C411yescharacteristic
474C3C5×#185411yescharacteristic, complemented, Sylow
475C2—42no—
476C2C5×#494611yescharacteristic
477C2C5×#496411yescharacteristic
478C2C5×#499811yescharacteristic
479C2C5×#503511yescharacteristic
480C2C5×#503711yescharacteristic
481C2C5×#504511yescharacteristic
482C2C5×#504711yescharacteristic
subgroupG/H
centerC23×C5C2×#67
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#380C2
socleC23×C3×C5C2×D4
prime radicalC24×C3×C5D4
cosocleC22×C3×C4C23×C5
hypercenterC5×#380D3
hypocenterC3C5×#1854
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C23×C4×C5 ◁ C5×#380

Complements

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

NH ≅ G/Ntype
4. C2×C5×#808C2semidirect
5. C2×C5×#807C2semidirect
6. C5×#4529C2semidirect
7. C5×#4529C2semidirect
22. C5×#830C4semidirect
25. #69506C5direct (N × H)
66. C2×#808C2×C5semidirect
67. C2×#807C2×C5semidirect
68. #4529C2×C5semidirect
69. #4529C2×C5semidirect
135. #830C4×C5semidirect
375. C3×C5#1854semidirect
450. C5#69506direct (N × H)
474. C3C5×#1854semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9262
C3×C5×#3802
C2×C5×#8302
C2×C5×#8082
C2×C5×#8072
C5×#45292
C5×#18543
#695065