Finite Group Databasebeta

← C5×#71197 #14289274

Subgroups — C5×#71197 #14289274

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5732C211yesmaximal, characteristic, complemented
2C3×C5×#486C211yesmaximal, characteristic, complemented
3C5×#5075C211yesmaximal, characteristic, complemented
4C5×#5072C211yesmaximal, characteristic
5C5×#4547C211yesmaximal, characteristic, complemented
6C5×#4459C211yesmaximal, characteristic, complemented
7C5×#4456C211yesmaximal, characteristic, complemented
8C5×#1628—31nomaximal, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C5×#127C2211yescharacteristic
11C5×#947—21no—
12C5×Q8×Dic3—21no—
13C2×C5×#214—21no—
14C2×C5×#213—21no—
15C5×#898—22no—
16C5×#897—21no—
17C5×#892—21no—
18C4×C5×D12C2211yescharacteristic
19C4×C5×Dic6C2211yescharacteristic
20C3×C5×#102—22no—
21C3×C4×C5×C8C2211yescharacteristic
22C5×#832—21no—
23C5×#832—21no—
24C5×C4⋊(C3⋊C8)C2211yescharacteristic
25C5×C4⋊(C3⋊C8)C2211yescharacteristic
26#71197C511yesmaximal, characteristic, direct product factor, Hall
27C5×#556—31no—
28C5×#486D311yescharacteristic, complemented
29C5×#460—31no—
30C5×#459—31no—
31C5×#332—31no—
32C5×Q8⋊C8—31no—
33C5×D4⋊C8—31no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—21no—
38C2×C5×D12D411yescharacteristic
39C2×C4×C5×D3—21no—
40C2×C5×Dic6D411yescharacteristic
41C2×C3×C5×C8—21no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4D412yes—
44C3×C42×C5C2311yescharacteristic
45C5×#214—42no—
46C5×#213—42no—
47C5×(C2×D3)⋊C4—22no—
48C5×(C2×D3)⋊C4—22no—
49C5×(C2×D3)⋊C4—21no—
50C5×C4⋊Dic3—22no—
51C5×C4⋊Dic3D411yescharacteristic
52C5×C4⋊Dic3D411yescharacteristic
53C5×Dic3⋊C4—21no—
54C4×C5×Dic3—22no—
55C4×C5×Dic3—21no—
56C2×C5×C3⋊C8—21no—
57C2×C5×C3⋊C8—21no—
58#5732C2×C511yescharacteristic, complemented
59C3×#486C2×C511yescharacteristic, complemented
60#5075C2×C511yescharacteristic, complemented
61#5072C2×C511yescharacteristic
62#4547C2×C511yescharacteristic, complemented
63#4459C2×C511yescharacteristic, complemented
64#4456C2×C511yescharacteristic, complemented
65C2×C5×SD16—61no—
66C2×C5×D8—61no—
67C5×C4⋊Q8C2×D311yescharacteristic
68C5×#127C2×D311yescharacteristic
69C5×#126—62no—
70C5×#124—61no—
71C5×C22⋊Q8—61no—
72C4×C5×Q8—31no—
73C4×C5×Q8—61no—
74C4×C5×D4—31no—
75C5×#117—61no—
76C5×C4⋊C8—31no—
77C5×C4⋊C8—31no—
78C5×#102—22no—
79C5×#102—61no—
80C5×#102—61no—
81C4×C5×C8C2×D311yescharacteristic
82#1628—31noSylow
83C23×C3×C5—41no—
84C22×C5×D3—21no—
85C3×C5×Q8—41no—
86C3×C5×Q8—41no—
87C3×C5×D4—42no—
88C3×C5×D4—41no—
89C3×C5×D4—41no—
90C3×C5×D4—41no—
91C3×C5×D4—41no—
92C2×C3×C4×C5—22no—
93C2×C3×C4×C5C2×D411yescharacteristic
94C2×C3×C4×C5C2×D411yescharacteristic
95C2×C3×C4×C5C2×D411yescharacteristic
96C2×C5×Dic3—21no—
97C2×C5×Dic3—21no—
98C2×C5×Dic3—21no—
99C5×D12—21no—
100C5×D12—21no—
101C4×C5×D3—41no—
102C4×C5×D3—41no—
103C5×Dic6—21no—
104C5×Dic6—21no—
105C3×C5×C8—41no—
106C5×C3⋊C8—41no—
107C5×C3⋊C8—41no—
108C3×C4⋊Q8C22×C511yescharacteristic
109C3×#127C22×C511yescharacteristic
110#947—21no—
111Q8×Dic3—21no—
112C2×#214—21no—
113C2×#213—21no—
114#898—22no—
115#897—21no—
116#892—21no—
117C4×D12C22×C511yescharacteristic
118C4×Dic6C22×C511yescharacteristic
119C3×#102—22no—
120C3×C4×C8C22×C511yescharacteristic
121#832—21no—
122#832—21no—
123C4⋊(C3⋊C8)C22×C511yescharacteristic
124C4⋊(C3⋊C8)C22×C511yescharacteristic
125C2×C5×Q8—21no—
126C2×C5×Q8—31no—
127C2×C5×D4—21no—
128C2×C5×D4—21no—
129C2×C5×D4—21no—
130C2×C5×D4—31no—
131C22×C4×C5—61no—
132C5×SD16—122no—
133C5×D8—122no—
134C2×C5×C8—21no—
135C2×C5×C8—61no—
136C2×C5×C8—61no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4—31no—
139C5×C4⋊C4—31no—
140C5×C4⋊C4—62no—
141C5×C4⋊C4—61no—
142C5×C4⋊C4#6712yes—
143C5×C22⋊C4—62no—
144C5×C22⋊C4—62no—
145C5×C22⋊C4—61no—
146C42×C5—62no—
147C42×C5—61no—
148C42×C5C22×D311yescharacteristic
149#556—31no—
150#486C5×D311yescharacteristic, complemented
151#460—31no—
152#459—31no—
153#332—31no—
154Q8⋊C8—31no—
155D4⋊C8—31no—
156C22×C3×C5—81no—
157C22×C3×C5—81no—
158C22×C3×C5—81no—
159C22×C3×C5#12011yescharacteristic
160C2×C5×D3—41no—
161C2×C5×D3—41no—
162C2×C5×D3—41no—
163C3×C4×C5—21no—
164C3×C4×C5—42no—
165C3×C4×C5#13511yescharacteristic
166C3×C4×C5#13511yescharacteristic
167C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
168C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
169C5×Dic3—41no—
170C5×Dic3—41no—
171C5×Dic3—41no—
172C2×C3×Q8—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×D12C5×D411yescharacteristic
177C2×C4×D3—21no—
178C2×Dic6C5×D411yescharacteristic
179C2×C3×C8—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4C5×D412yes—
182C3×C42C23×C511yescharacteristic
183#214—42no—
184#213—42no—
185(C2×D3)⋊C4—22no—
186(C2×D3)⋊C4—22no—
187(C2×D3)⋊C4—21no—
188C4⋊Dic3—22no—
189C4⋊Dic3C5×D411yescharacteristic
190C4⋊Dic3C5×D411yescharacteristic
191Dic3⋊C4—21no—
192C4×Dic3—22no—
193C4×Dic3—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C23×C5—41no—
197C23×C5—61no—
198C5×Q8—41no—
199C5×Q8—41no—
200C5×Q8—61no—
201C5×Q8—61no—
202C5×D4—42no—
203C5×D4—41no—
204C5×D4—41no—
205C5×D4—41no—
206C5×D4—41no—
207C5×D4—61no—
208C5×D4—61no—
209C2×C4×C5—22no—
210C2×C4×C5—61no—
211C2×C4×C5—61no—
212C2×C4×C5—61no—
213C2×C4×C5—121no—
214C2×C4×C5—121no—
215C2×C4×C5D3×D411yescharacteristic
216C2×C4×C5D3×D411yescharacteristic
217C2×C4×C5C2×#6711yescharacteristic
218C5×C8—41no—
219C5×C8—121no—
220C5×C8—121no—
221C2×SD16—61no—
222C2×D8—61no—
223C4⋊Q8C2×C5×D311yescharacteristic
224#127C2×C5×D311yescharacteristic
225#126—62no—
226#124—61no—
227C22⋊Q8—61no—
228C4×Q8—31no—
229C4×Q8—61no—
230C4×D4—31no—
231#117—61no—
232C4⋊C8—31no—
233C4⋊C8—31no—
234#102—22no—
235#102—61no—
236#102—61no—
237C4×C8C2×C5×D311yescharacteristic
238C2×C3×C5—161no—
239C2×C3×C5#44911yescharacteristic
240C2×C3×C5#44911yescharacteristic
241C2×C3×C5#45411yescharacteristic
242C5×D3—81no—
243C23×C3—41no—
244C22×D3—21no—
245C3×Q8—41no—
246C3×Q8—41no—
247C3×D4—42no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—41no—
251C3×D4—41no—
252C2×C3×C4—22no—
253C2×C3×C4C2×C5×D411yescharacteristic
254C2×C3×C4C2×C5×D411yescharacteristic
255C2×C3×C4C2×C5×D411yescharacteristic
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259D12—21no—
260D12—21no—
261C4×D3—41no—
262C4×D3—41no—
263Dic6—21no—
264Dic6—21no—
265C3×C8—41no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C22×C5—81no—
269C22×C5—81no—
270C22×C5—81no—
271C22×C5—121no—
272C22×C5—121no—
273C22×C5—121no—
274C22×C5#93711yescharacteristic
275C4×C5—21no—
276C4×C5—42no—
277C4×C5—121no—
278C4×C5—121no—
279C4×C5—121no—
280C4×C5#91111yescharacteristic
281C4×C5#91211yescharacteristic
282C4×C5#91411yescharacteristic
283C4×C5#91611yescharacteristic
284C2×Q8—21no—
285C2×Q8—31no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—31no—
290C22×C4—61no—
291SD16—122no—
292D8—122no—
293C2×C8—21no—
294C2×C8—61no—
295C2×C8—61no—
296C4⋊C4—21no—
297C4⋊C4—31no—
298C4⋊C4—31no—
299C4⋊C4—62no—
300C4⋊C4—61no—
301C4⋊C4C5×#6712yes—
302C22⋊C4—62no—
303C22⋊C4—62no—
304C22⋊C4—61no—
305C42—62no—
306C42—61no—
307C42C22×C5×D311yescharacteristic
308C3×C5#162811yescharacteristic, complemented, Hall
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3C5×#12011yescharacteristic
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C3×C4—21no—
317C3×C4—42no—
318C3×C4C5×#13511yescharacteristic
319C3×C4C5×#13511yescharacteristic
320C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
321C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
322Dic3—41no—
323Dic3—41no—
324Dic3—41no—
325C2×C5—161no—
326C2×C5—241no—
327C2×C5#506211yescharacteristic
328C2×C5#515811yescharacteristic
329C2×C5#517211yescharacteristic
330C23—41no—
331C23—61no—
332Q8—41no—
333Q8—41no—
334Q8—61no—
335Q8—61no—
336D4—42no—
337D4—41no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—61no—
342D4—61no—
343C2×C4—22no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4C5×D3×D411yescharacteristic
350C2×C4C5×D3×D411yescharacteristic
351C2×C4C2×C5×#6711yescharacteristic
352C8—41no—
353C8—121no—
354C8—121no—
355C2×C3—161no—
356C2×C3C5×#44911yescharacteristic
357C2×C3C5×#44911yescharacteristic
358C2×C3C5×#45411yescharacteristic
359D3—81no—
360C5#7119711yescharacteristic, direct product factor, Sylow
361C22—81no—
362C22—81no—
363C22—81no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22C5×#93711yescharacteristic
368C4—21no—
369C4—42no—
370C4—121no—
371C4—121no—
372C4—121no—
373C4C5×#91111yescharacteristic
374C4C5×#91211yescharacteristic
375C4C5×#91411yescharacteristic
376C4C5×#91611yescharacteristic
377C3C5×#162811yescharacteristic, complemented, Sylow
378C2—161no—
379C2—241no—
380C2C5×#506211yescharacteristic
381C2C5×#515811yescharacteristic
382C2C5×#517211yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC3×C42C23×C5
FrattiniC42C22×C5×D3
FittingC3×C5×#486C2
socleC22×C3×C5#120
prime radicalC5×#5072C2
hypercenterC5×#486D3
hypocenterC3C5×#1628
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C42×C5 ◁ C5×#486

Complements

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

NH ≅ G/Ntype
1. C5×#5732C2semidirect
2. C3×C5×#486C2semidirect
3. C5×#5075C2semidirect
5. C5×#4547C2semidirect
6. C5×#4459C2semidirect
7. C5×#4456C2semidirect
26. #71197C5direct (N × H)
28. C5×#486D3semidirect
58. #5732C2×C5semidirect
59. C3×#486C2×C5semidirect
60. #5075C2×C5semidirect
62. #4547C2×C5semidirect
63. #4459C2×C5semidirect
64. #4456C2×C5semidirect
150. #486C5×D3semidirect
308. C3×C5#1628semidirect
360. C5#71197direct (N × H)
377. C3C5×#1628semidirect

Maximal subgroups

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

subgroupindex
C5×#57322
C3×C5×#4862
C5×#50752
C5×#50722
C5×#45472
C5×#44592
C5×#44562
C5×#16283
#711975