Finite Group Databasebeta

← C5×#71157 #14289814

Subgroups — C5×#71157 #14289814

Order 1920.

388 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5584C211yesmaximal, characteristic
2C3×C5×#477C211yesmaximal, characteristic, complemented
3C5×#5084C211yesmaximal, characteristic
4C5×#5079C211yesmaximal, characteristic, complemented
5C5×#4535C211yesmaximal, characteristic, complemented
6C5×#4486C211yesmaximal, characteristic, complemented
7C5×#4483C211yesmaximal, characteristic
8C5×#1638—31nomaximal, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C4×C5×Q8C2211yescharacteristic
11C5×#946—21no—
12C5×#898—22no—
13C5×#892—21no—
14C5×#875—21no—
15C4×C5×D12C2211yescharacteristic
16C5×#869C2211yescharacteristic
17C4×C5×Dic6—21no—
18C3×C5×#107—21no—
19C3×C5×C4⋊C8C2211yescharacteristic
20C3×C5×#103—21no—
21C5×#835—22no—
22C5×#809—21no—
23C5×#808—21no—
24C5×C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C5×C3⋊C8C2211yescharacteristic
26#71157C511yesmaximal, characteristic, direct product factor, Hall
27C5×#556—31no—
28C5×#487—31no—
29C5×#477D311yescharacteristic, complemented
30C5×#476—31no—
31C5×#332—31no—
32C5×Q8⋊C8—31no—
33C5×D4⋊C8—31no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C5×D12D411yescharacteristic
37C2×C4×C5×D3—21no—
38C2×C5×Dic6—21no—
39C2×C3×C5×C8—21no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4D411yescharacteristic
44C3×C42×C5—21no—
45C3×C42×C5C2311yescharacteristic
46C5×(C2×D3)⋊C4—22no—
47C5×(C2×D3)⋊C4—22no—
48C5×(C2×D3)⋊C4—21no—
49C5×C4⋊Dic3—21no—
50C5×C4⋊Dic3D411yescharacteristic
51C5×C4⋊Dic3D412yes—
52C5×Dic3⋊C4—22no—
53C4×C5×Dic3—22no—
54C2×C5×C3⋊C8—21no—
55C2×C5×C3⋊C8—21no—
56#5584C2×C511yescharacteristic
57C3×#477C2×C511yescharacteristic, complemented
58#5084C2×C511yescharacteristic
59#5079C2×C511yescharacteristic, complemented
60#4535C2×C511yescharacteristic, complemented
61#4486C2×C511yescharacteristic, complemented
62#4483C2×C511yescharacteristic
63C5×C4⋊Q8—31no—
64C5×C4⋊Q8C2×D311yescharacteristic
65C5×#126—62no—
66C5×#124—61no—
67C5×C22⋊Q8—61no—
68C4×C5×Q8—61no—
69C4×C5×Q8C2×D311yescharacteristic
70C4×C5×D4—31no—
71C5×#117—61no—
72C5×#107—21no—
73C5×#106—61no—
74C5×C4⋊C8—31no—
75C5×C4⋊C8C2×D311yescharacteristic
76C5×#103—21no—
77C5×#103—62no—
78C5×#102—61no—
79C4×C5×C8—31no—
80#1638—31noSylow
81C22×C5×D3—21no—
82C3×C5×Q8—21no—
83C3×C5×Q8—21no—
84C3×C5×Q8—42no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5C2×D411yescharacteristic
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C5×Dic3—21no—
92C2×C5×Dic3—22no—
93C5×D12—21no—
94C5×D12—21no—
95C4×C5×D3—41no—
96C4×C5×D3—41no—
97C5×Dic6—41no—
98C5×Dic6—41no—
99C3×C5×C8—41no—
100C5×C3⋊C8—41no—
101C5×C3⋊C8—41no—
102C3×C4⋊Q8C22×C511yescharacteristic
103C3×C4×Q8C22×C511yescharacteristic
104#946—21no—
105#898—22no—
106#892—21no—
107#875—21no—
108C4×D12C22×C511yescharacteristic
109#869C22×C511yescharacteristic
110C4×Dic6—21no—
111C3×#107—21no—
112C3×C4⋊C8C22×C511yescharacteristic
113C3×#103—21no—
114#835—22no—
115#809—21no—
116#808—21no—
117C4⋊(C3⋊C8)C22×C511yescharacteristic
118C4×C3⋊C8C22×C511yescharacteristic
119C2×C5×Q8—21no—
120C2×C5×Q8—61no—
121C2×C5×Q8#6711yescharacteristic
122C2×C5×D4—31no—
123C22×C4×C5—61no—
124C2×C5×C8—21no—
125C2×C5×C8—61no—
126C2×C5×C8—61no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4—31no—
131C5×C4⋊C4—32no—
132C5×C4⋊C4—62no—
133C5×C4⋊C4—61no—
134C5×C4⋊C4#6711yescharacteristic
135C5×C22⋊C4—62no—
136C5×C22⋊C4—62no—
137C5×C22⋊C4—61no—
138C42×C5—21no—
139C42×C5—62no—
140C42×C5C22×D311yescharacteristic
141#556—31no—
142#487—31no—
143#477C5×D311yescharacteristic, complemented
144#476—31no—
145#332—31no—
146Q8⋊C8—31no—
147D4⋊C8—31no—
148C22×C3×C5#12011yescharacteristic
149C2×C5×D3—41no—
150C2×C5×D3—41no—
151C2×C5×D3—41no—
152C3×C4×C5—21no—
153C3×C4×C5—41no—
154C3×C4×C5—41no—
155C3×C4×C5—81no—
156C3×C4×C5#13511yescharacteristic
157C3×C4×C5#13511yescharacteristic
158C3×C4×C5#13711yescharacteristic
159C3×C4×C5#13711yescharacteristic
160C5×Dic3—42no—
161C5×Dic3—41no—
162C2×C3×Q8—21no—
163C2×C3×Q8C5×D411yescharacteristic
164C2×D12C5×D411yescharacteristic
165C2×C4×D3—21no—
166C2×Dic6—21no—
167C2×C3×C8—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4C5×D411yescharacteristic
172C3×C42—21no—
173C3×C42C23×C511yescharacteristic
174(C2×D3)⋊C4—22no—
175(C2×D3)⋊C4—22no—
176(C2×D3)⋊C4—21no—
177C4⋊Dic3—21no—
178C4⋊Dic3C5×D411yescharacteristic
179C4⋊Dic3C5×D412yes—
180Dic3⋊C4—22no—
181C4×Dic3—22no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C23×C5—61no—
185C5×Q8—21no—
186C5×Q8—21no—
187C5×Q8—42no—
188C5×Q8—121no—
189C5×Q8—121no—
190C5×D4—61no—
191C5×D4—61no—
192C2×C4×C5—21no—
193C2×C4×C5—21no—
194C2×C4×C5—41no—
195C2×C4×C5—61no—
196C2×C4×C5—62no—
197C2×C4×C5—121no—
198C2×C4×C5—121no—
199C2×C4×C5D3×D411yescharacteristic
200C2×C4×C5D3×D411yescharacteristic
201C2×C4×C5C2×#6711yescharacteristic
202C5×C8—41no—
203C5×C8—121no—
204C5×C8—121no—
205C4⋊Q8—31no—
206C4⋊Q8C2×C5×D311yescharacteristic
207#126—62no—
208#124—61no—
209C22⋊Q8—61no—
210C4×Q8—61no—
211C4×Q8C2×C5×D311yescharacteristic
212C4×D4—31no—
213#117—61no—
214#107—21no—
215#106—61no—
216C4⋊C8—31no—
217C4⋊C8C2×C5×D311yescharacteristic
218#103—21no—
219#103—62no—
220#102—61no—
221C4×C8—31no—
222C2×C3×C5#45211yescharacteristic
223C2×C3×C5#45211yescharacteristic
224C2×C3×C5#45411yescharacteristic
225C5×D3—81no—
226C22×D3—21no—
227C3×Q8—21no—
228C3×Q8—21no—
229C3×Q8—42no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4C2×C5×D411yescharacteristic
234C2×C3×C4C2×C5×D411yescharacteristic
235C2×C3×C4C2×C5×D411yescharacteristic
236C2×Dic3—21no—
237C2×Dic3—22no—
238D12—21no—
239D12—21no—
240C4×D3—41no—
241C4×D3—41no—
242Dic6—41no—
243Dic6—41no—
244C3×C8—41no—
245C3⋊C8—41no—
246C3⋊C8—41no—
247C22×C5—121no—
248C22×C5—121no—
249C22×C5—121no—
250C22×C5#93711yescharacteristic
251C4×C5—21no—
252C4×C5—41no—
253C4×C5—41no—
254C4×C5—81no—
255C4×C5—122no—
256C4×C5—121no—
257C4×C5#91811yescharacteristic
258C4×C5#91911yescharacteristic
259C4×C5#94211yescharacteristic
260C4×C5#95011yescharacteristic
261C2×Q8—21no—
262C2×Q8—61no—
263C2×Q8C5×#6711yescharacteristic
264C2×D4—31no—
265C22×C4—61no—
266C2×C8—21no—
267C2×C8—61no—
268C2×C8—61no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—32no—
273C4⋊C4—31no—
274C4⋊C4—62no—
275C4⋊C4—61no—
276C4⋊C4C5×#6711yescharacteristic
277C22⋊C4—62no—
278C22⋊C4—62no—
279C22⋊C4—61no—
280C42—21no—
281C42—62no—
282C42C22×C5×D311yescharacteristic
283C3×C5#163811yescharacteristic, complemented, Hall
284C22×C3C5×#12011yescharacteristic
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C3×C4—21no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—81no—
292C3×C4C5×#13511yescharacteristic
293C3×C4C5×#13511yescharacteristic
294C3×C4C5×#13711yescharacteristic
295C3×C4C5×#13711yescharacteristic
296Dic3—42no—
297Dic3—41no—
298C2×C5—241no—
299C2×C5#504711yescharacteristic
300C2×C5#518711yescharacteristic
301C2×C5#520111yescharacteristic
302C23—61no—
303Q8—21no—
304Q8—21no—
305Q8—42no—
306Q8—121no—
307Q8—121no—
308D4—61no—
309D4—61no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—61no—
314C2×C4—62no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4C5×D3×D411yescharacteristic
318C2×C4C5×D3×D411yescharacteristic
319C2×C4C2×C5×#6711yescharacteristic
320C8—41no—
321C8—121no—
322C8—121no—
323C2×C3C5×#45211yescharacteristic
324C2×C3C5×#45211yescharacteristic
325C2×C3C5×#45411yescharacteristic
326D3—81no—
327C5#7115711yescharacteristic, direct product factor, Sylow
328C22—121no—
329C22—121no—
330C22—121no—
331C22C5×#93711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—122no—
337C4—121no—
338C4C5×#91811yescharacteristic
339C4C5×#91911yescharacteristic
340C4C5×#94211yescharacteristic
341C4C5×#95011yescharacteristic
342C3C5×#163811yescharacteristic, complemented, Sylow
343C2—241no—
344C2C5×#504711yescharacteristic
345C2C5×#518711yescharacteristic
346C2C5×#520111yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC3×C42C23×C5
FrattiniC42C22×C5×D3
FittingC3×C5×#477C2
socleC22×C3×C5#120
prime radicalC2×C5×D12D4
hypercenterC5×#477D3
hypocenterC3C5×#1638
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C42×C5 ◁ C5×#477

Complements

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

NH ≅ G/Ntype
2. C3×C5×#477C2semidirect
4. C5×#5079C2semidirect
5. C5×#4535C2semidirect
6. C5×#4486C2semidirect
26. #71157C5direct (N × H)
29. C5×#477D3semidirect
57. C3×#477C2×C5semidirect
59. #5079C2×C5semidirect
60. #4535C2×C5semidirect
61. #4486C2×C5semidirect
143. #477C5×D3semidirect
283. C3×C5#1638semidirect
327. C5#71157direct (N × H)
342. C3C5×#1638semidirect

Maximal subgroups

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

subgroupindex
C5×#55842
C3×C5×#4772
C5×#50842
C5×#50792
C5×#45352
C5×#44862
C5×#44832
C5×#16383
#711575