Finite Group Databasebeta

← C5×#68168 #14281766

Subgroups — C5×#68168 #14281766

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5548C211yesmaximal, characteristic
2C5×D3×C4⋊C8C211yesmaximal, characteristic, complemented
3C5×#4685C211yesmaximal, characteristic, complemented
4C3×C5×D4⋊C8C211yesmaximal, characteristic, complemented
5C5×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C5×#4483C211yesmaximal, characteristic, complemented
7C5×#4459C211yesmaximal, characteristic, complemented
8C5×#1480—31nomaximal, Hall
9C2×C5×#235—21no—
10C3×C4×C5×D4C2211yescharacteristic
11C5×#940C411yescharacteristic
12C5×#938C411yescharacteristic
13C4×C5×#67—21no—
14C2×C5×D3×C8—21no—
15C5×#895C411yescharacteristic
16C5×#888C411yescharacteristic
17C5×#885—21no—
18C5×#884—21no—
19C4×C5×D12C2211yescharacteristic
20C42×C5×D3C2211yescharacteristic
21C3×C5×C4⋊C8C2211yescharacteristic
22C3×C4×C5×C8C2211yescharacteristic
23C5×(C2×D3)⋊C8—21no—
24C5×Dic3⋊C8—21no—
25C5×Dic3⋊C8—21no—
26C5×C4⋊(C3⋊C8)C2211yescharacteristic
27C4×C5×C3⋊C8C2211yescharacteristic
28#68168C511yesmaximal, characteristic, direct product factor, Hall
29C5×#532—31no—
30C5×#431—31no—
31C2×C5×C4⋊C8—31no—
32C5×D4⋊C8—31no—
33C5×D4⋊C8—31no—
34C5×D4⋊C8—31no—
35C5×D4⋊C8D311yescharacteristic, complemented
36C2×C3×C5×D4C2×C411yescharacteristic
37C22×C3×C4×C5—21no—
38C2×C5×#67—21no—
39C2×C5×#67—21no—
40C2×C5×#67—21no—
41C5×#235—42no—
42C5×#235—42no—
43C2×C5×D12—21no—
44C2×C5×D12C2×C411yescharacteristic
45C2×C4×C5×D3—21no—
46C2×C4×C5×D3C2×C411yescharacteristic
47C2×C4×C5×D3D411yescharacteristic
48C2×C4×C5×D3D411yescharacteristic
49C2×C5×Dic6—21no—
50C2×C3×C5×C8—21no—
51C2×C3×C5×C8—21no—
52C3×C5×C4⋊C4C2×C411yescharacteristic
53C3×C5×C22⋊C4—21no—
54C3×C42×C5C2311yescharacteristic
55C5×#217—21no—
56C5×(C2×D3)⋊C4—21no—
57C5×(C2×D3)⋊C4—21no—
58C5×C4⋊Dic3C2×C411yescharacteristic
59C5×Dic3⋊C4—21no—
60C4×C5×Dic3C2×C411yescharacteristic
61C4×C5×Dic3D411yescharacteristic
62C4×C5×Dic3D411yescharacteristic
63C2×C5×C3⋊C8—21no—
64C2×C5×C3⋊C8—21no—
65C5×D3×C8—42no—
66#5548C2×C511yescharacteristic
67D3×C4⋊C8C2×C511yescharacteristic, complemented
68#4685C2×C511yescharacteristic, complemented
69C3×D4⋊C8C2×C511yescharacteristic, complemented
70D4⋊(C3⋊C8)C2×C511yescharacteristic, complemented
71#4483C2×C511yescharacteristic, complemented
72#4459C2×C511yescharacteristic, complemented
73C2×C5×#41—61no—
74C22×C5×C8—61no—
75C5×C4⋊Q8—31no—
76C5×#127—31no—
77C5×#124—61no—
78C5×#121—31no—
79C5×#121—31no—
80C5×#121—61no—
81C4×C5×D4—31no—
82C4×C5×D4—61no—
83C4×C5×D4C2×D311yescharacteristic
84C2×C42×C5—31no—
85C5×C4⋊C8—31no—
86C5×C4⋊C8—61no—
87C5×C4⋊C8—61no—
88C5×C4⋊C8C2×D311yescharacteristic
89C5×C22⋊C8—61no—
90C4×C5×C8—31no—
91C4×C5×C8C2×D311yescharacteristic
92#1480—31noSylow
93C23×C3×C5—21no—
94C22×C5×D3—21no—
95C22×C5×D3C22⋊C411yescharacteristic
96C3×C5×D4—21no—
97C3×C5×D4—21no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5C22×C411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C5×#67—42no—
105C5×#67—42no—
106C5×#67—42no—
107C2×C5×Dic3—21no—
108C2×C5×Dic3C22⋊C411yescharacteristic
109C2×C5×Dic3C22⋊C411yescharacteristic
110C2×C5×Dic3C22⋊C411yescharacteristic
111C5×D12—21no—
112C5×D12—21no—
113C5×D12—42no—
114C4×C5×D3—21no—
115C4×C5×D3—21no—
116C4×C5×D3—41no—
117C4×C5×D3—41no—
118C4×C5×D3—41no—
119C4×C5×D3D812yes—
120C4×C5×D3SD1612yes—
121C5×Dic6—42no—
122C3×C5×C8—41no—
123C3×C5×C8—41no—
124C5×C3⋊C8—41no—
125C5×C3⋊C8—41no—
126C2×#235—21no—
127C3×C4×D4C22×C511yescharacteristic
128#940C4×C511yescharacteristic
129#938C4×C511yescharacteristic
130C4×#67—21no—
131C2×D3×C8—21no—
132#895C4×C511yescharacteristic
133#888C4×C511yescharacteristic
134#885—21no—
135#884—21no—
136C4×D12C22×C511yescharacteristic
137C42×D3C22×C511yescharacteristic
138C3×C4⋊C8C22×C511yescharacteristic
139C3×C4×C8C22×C511yescharacteristic
140(C2×D3)⋊C8—21no—
141Dic3⋊C8—21no—
142Dic3⋊C8—21no—
143C4⋊(C3⋊C8)C22×C511yescharacteristic
144C4×C3⋊C8C22×C511yescharacteristic
145C5×#41—122no—
146C5×#41—122no—
147C2×C5×Q8—61no—
148C2×C5×D4—31no—
149C2×C5×D4—61no—
150C2×C5×D4—61no—
151C2×C5×D4—61no—
152C2×C5×D4—61no—
153C2×C5×D4C4×D311yescharacteristic
154C22×C4×C5—21no—
155C22×C4×C5—31no—
156C22×C4×C5—31no—
157C22×C4×C5—31no—
158C22×C4×C5—61no—
159C2×C5×C8—21no—
160C2×C5×C8—21no—
161C2×C5×C8—61no—
162C2×C5×C8—61no—
163C2×C5×C8—122no—
164C5×C4⋊C4—31no—
165C5×C4⋊C4—61no—
166C5×C4⋊C4C4×D311yescharacteristic
167C5×C22⋊C4—21no—
168C5×C22⋊C4—61no—
169C5×C22⋊C4—61no—
170C5×C22⋊C4—61no—
171C42×C5—31no—
172C42×C5—31no—
173C42×C5—31no—
174C42×C5C22×D311yescharacteristic
175#532—31no—
176#431—31no—
177C2×C4⋊C8—31no—
178D4⋊C8—31no—
179D4⋊C8—31no—
180D4⋊C8—31no—
181D4⋊C8C5×D311yescharacteristic, complemented
182C22×C3×C5—41no—
183C22×C3×C5—41no—
184C22×C3×C5—41no—
185C22×C3×C5C2×C22⋊C411yescharacteristic
186C2×C5×D3—21no—
187C2×C5×D3—21no—
188C2×C5×D3—41no—
189C2×C5×D3—41no—
190C2×C5×D3—41no—
191C2×C5×D3#10212yes—
192C3×C4×C5—21no—
193C3×C4×C5—41no—
194C3×C4×C5#13111yescharacteristic
195C3×C4×C5#13111yescharacteristic
196C3×C4×C5C2×D811yescharacteristic
197C3×C4×C5C2×SD1611yescharacteristic
198C5×Dic3—21no—
199C5×Dic3—21no—
200C5×Dic3—41no—
201C5×Dic3#10212yes—
202C2×C3×D4C2×C4×C511yescharacteristic
203C22×C3×C4—21no—
204C2×#67—21no—
205C2×#67—21no—
206C2×#67—21no—
207#235—42no—
208#235—42no—
209C2×D12—21no—
210C2×D12C2×C4×C511yescharacteristic
211C2×C4×D3—21no—
212C2×C4×D3C2×C4×C511yescharacteristic
213C2×C4×D3C5×D411yescharacteristic
214C2×C4×D3C5×D411yescharacteristic
215C2×Dic6—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C3×C4⋊C4C2×C4×C511yescharacteristic
219C3×C22⋊C4—21no—
220C3×C42C23×C511yescharacteristic
221#217—21no—
222(C2×D3)⋊C4—21no—
223(C2×D3)⋊C4—21no—
224C4⋊Dic3C2×C4×C511yescharacteristic
225Dic3⋊C4—21no—
226C4×Dic3C2×C4×C511yescharacteristic
227C4×Dic3C5×D411yescharacteristic
228C4×Dic3C5×D411yescharacteristic
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231D3×C8—42no—
232C23×C5—21no—
233C23×C5—31no—
234C23×C5—61no—
235C5×Q8—122no—
236C5×D4—21no—
237C5×D4—21no—
238C5×D4—61no—
239C5×D4—61no—
240C5×D4—122no—
241C5×D4—122no—
242C5×D4—122no—
243C5×D4—122no—
244C2×C4×C5—21no—
245C2×C4×C5—31no—
246C2×C4×C5—32no—
247C2×C4×C5—32no—
248C2×C4×C5—31no—
249C2×C4×C5—31no—
250C2×C4×C5—41no—
251C2×C4×C5—41no—
252C2×C4×C5—61no—
253C2×C4×C5—61no—
254C2×C4×C5—61no—
255C2×C4×C5—121no—
256C2×C4×C5—121no—
257C2×C4×C5—121no—
258C2×C4×C5C2×C4×D311yescharacteristic
259C2×C4×C5D3×D411yescharacteristic
260C2×C4×C5D3×D411yescharacteristic
261C5×C8—41no—
262C5×C8—41no—
263C5×C8—121no—
264C5×C8—121no—
265C2×#41—61no—
266C22×C8—61no—
267C4⋊Q8—31no—
268#127—31no—
269#124—61no—
270#121—31no—
271#121—31no—
272#121—61no—
273C4×D4—31no—
274C4×D4—61no—
275C4×D4C2×C5×D311yescharacteristic
276C2×C42—31no—
277C4⋊C8—31no—
278C4⋊C8—61no—
279C4⋊C8—61no—
280C4⋊C8C2×C5×D311yescharacteristic
281C22⋊C8—61no—
282C4×C8—31no—
283C4×C8C2×C5×D311yescharacteristic
284C2×C3×C5—81no—
285C2×C3×C5#40811yescharacteristic
286C2×C3×C5C2×#10211yescharacteristic
287C2×C3×C5#42111yescharacteristic
288C5×D3—22no—
289C5×D3—81no—
290C23×C3—21no—
291C22×D3—21no—
292C22×D3C5×C22⋊C411yescharacteristic
293C3×D4—21no—
294C3×D4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4C22×C4×C511yescharacteristic
299C2×C3×C4C2×C5×D411yescharacteristic
300C2×C3×C4C2×C5×D411yescharacteristic
301#67—42no—
302#67—42no—
303#67—42no—
304C2×Dic3—21no—
305C2×Dic3C5×C22⋊C411yescharacteristic
306C2×Dic3C5×C22⋊C411yescharacteristic
307C2×Dic3C5×C22⋊C411yescharacteristic
308D12—21no—
309D12—21no—
310D12—42no—
311C4×D3—21no—
312C4×D3—21no—
313C4×D3—41no—
314C4×D3—41no—
315C4×D3—41no—
316C4×D3C5×D812yes—
317C4×D3C5×SD1612yes—
318Dic6—42no—
319C3×C8—41no—
320C3×C8—41no—
321C3⋊C8—41no—
322C3⋊C8—41no—
323C22×C5—32no—
324C22×C5—41no—
325C22×C5—41no—
326C22×C5—41no—
327C22×C5—61no—
328C22×C5—61no—
329C22×C5—121no—
330C22×C5—121no—
331C22×C5—121no—
332C22×C5D3×C22⋊C411yescharacteristic
333C4×C5—21no—
334C4×C5—32no—
335C4×C5—41no—
336C4×C5—61no—
337C4×C5—61no—
338C4×C5—121no—
339C4×C5#90111yescharacteristic
340C4×C5#90711yescharacteristic
341C4×C5D3×D811yescharacteristic
342C4×C5D3×SD1611yescharacteristic
343#41—122no—
344#41—122no—
345C2×Q8—61no—
346C2×D4—31no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4C4×C5×D311yescharacteristic
352C22×C4—21no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—61no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—61no—
360C2×C8—61no—
361C2×C8—122no—
362C4⋊C4—31no—
363C4⋊C4—61no—
364C4⋊C4C4×C5×D311yescharacteristic
365C22⋊C4—21no—
366C22⋊C4—61no—
367C22⋊C4—61no—
368C22⋊C4—61no—
369C42—31no—
370C42—31no—
371C42—31no—
372C42C22×C5×D311yescharacteristic
373C3×C5#148011yescharacteristic, complemented, Hall
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3C2×C5×C22⋊C411yescharacteristic
378C2×D3—21no—
379C2×D3—21no—
380C2×D3—41no—
381C2×D3—41no—
382C2×D3—41no—
383C2×D3C5×#10212yes—
384C3×C4—21no—
385C3×C4—41no—
386C3×C4C5×#13111yescharacteristic
387C3×C4C5×#13111yescharacteristic
388C3×C4C2×C5×D811yescharacteristic
389C3×C4C2×C5×SD1611yescharacteristic
390Dic3—21no—
391Dic3—21no—
392Dic3—41no—
393Dic3C5×#10212yes—
394C2×C5—62no—
395C2×C5—81no—
396C2×C5—241no—
397C2×C5#472711yescharacteristic
398C2×C5D3×#10211yescharacteristic
399C2×C5#482111yescharacteristic
400C23—21no—
401C23—31no—
402C23—61no—
403Q8—122no—
404D4—21no—
405D4—21no—
406D4—61no—
407D4—61no—
408D4—122no—
409D4—122no—
410D4—122no—
411D4—122no—
412C2×C4—21no—
413C2×C4—31no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—31no—
417C2×C4—31no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4C2×C4×C5×D311yescharacteristic
427C2×C4C5×D3×D411yescharacteristic
428C2×C4C5×D3×D411yescharacteristic
429C8—41no—
430C8—41no—
431C8—121no—
432C8—121no—
433C2×C3—81no—
434C2×C3C5×#40811yescharacteristic
435C2×C3C2×C5×#10211yescharacteristic
436C2×C3C5×#42111yescharacteristic
437D3—22no—
438D3—81no—
439C5#6816811yescharacteristic, direct product factor, Sylow
440C22—32no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—61no—
445C22—61no—
446C22—121no—
447C22—121no—
448C22—121no—
449C22C5×D3×C22⋊C411yescharacteristic
450C4—21no—
451C4—32no—
452C4—41no—
453C4—61no—
454C4—61no—
455C4—121no—
456C4C5×#90111yescharacteristic
457C4C5×#90711yescharacteristic
458C4C5×D3×D811yescharacteristic
459C4C5×D3×SD1611yescharacteristic
460C3C5×#148011yescharacteristic, complemented, Sylow
461C2—62no—
462C2—81no—
463C2—241no—
464C2C5×#472711yescharacteristic
465C2C5×D3×#10211yescharacteristic
466C2C5×#482111yescharacteristic
subgroupG/H
centerC2×C4×C5D3×D4
commutatorC2×C3×C4C22×C4×C5
FrattiniC42C22×C5×D3
FittingC3×C5×D4⋊C8C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC5×#5548C2
cosocleC3×C42C23×C5
hypercenterC5×D4⋊C8D3
hypocenterC3C5×#1480
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4×C5 ◁ C42×C5 ◁ C5×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C5×D3×C4⋊C8C2semidirect
3. C5×#4685C2semidirect
4. C3×C5×D4⋊C8C2semidirect
5. C5×D4⋊(C3⋊C8)C2semidirect
6. C5×#4483C2semidirect
7. C5×#4459C2semidirect
28. #68168C5direct (N × H)
35. C5×D4⋊C8D3semidirect
67. D3×C4⋊C8C2×C5semidirect
68. #4685C2×C5semidirect
69. C3×D4⋊C8C2×C5semidirect
70. D4⋊(C3⋊C8)C2×C5semidirect
71. #4483C2×C5semidirect
72. #4459C2×C5semidirect
181. D4⋊C8C5×D3semidirect
373. C3×C5#1480semidirect
439. C5#68168direct (N × H)
460. C3C5×#1480semidirect

Maximal subgroups

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

subgroupindex
C5×#55482
C5×D3×C4⋊C82
C5×#46852
C3×C5×D4⋊C82
C5×D4⋊(C3⋊C8)2
C5×#44832
C5×#44592
C5×#14803
#681685