Finite Group Databasebeta

← C5×#71229 #14289808

Subgroups — C5×#71229 #14289808

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×#5732C211yesmaximal, characteristic
2C3×C5×#487C211yesmaximal, characteristic, complemented
3C5×#5091C211yesmaximal, characteristic, complemented
4C5×#5084C211yesmaximal, characteristic
5C5×#4547C211yesmaximal, characteristic, complemented
6C5×#4459C211yesmaximal, characteristic
7C5×#4456C211yesmaximal, characteristic, complemented
8C5×#1638—31nomaximal, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C5×C4⋊Q8C2211yescharacteristic
11C5×#947—21no—
12C5×Q8×Dic3—21no—
13C5×#898—22no—
14C5×#897—21no—
15C5×#892—21no—
16C4×C5×D12C2211yescharacteristic
17C4×C5×Dic6C2211yescharacteristic
18C3×C5×#103—22no—
19C3×C4×C5×C8C2211yescharacteristic
20C5×#810—21no—
21C5×#809—21no—
22C5×#808—21no—
23C5×#807—21no—
24C5×C4⋊(C3⋊C8)C2211yescharacteristic
25C5×C4⋊(C3⋊C8)C2211yescharacteristic
26#71229C511yesmaximal, characteristic, direct product factor, Hall
27C5×#556—31no—
28C5×#487D311yescharacteristic, complemented
29C5×#477—31no—
30C5×#476—31no—
31C5×#332—31no—
32C5×Q8⋊C8—31no—
33C5×D4⋊C8—31no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8—21no—
36C2×C5×D12D411yescharacteristic
37C2×C4×C5×D3—21no—
38C2×C5×Dic6D411yescharacteristic
39C2×C3×C5×C8—21no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4D412yes—
44C3×C42×C5C2311yescharacteristic
45C5×(C2×D3)⋊C4—22no—
46C5×(C2×D3)⋊C4—22no—
47C5×(C2×D3)⋊C4—21no—
48C5×C4⋊Dic3—22no—
49C5×C4⋊Dic3D411yescharacteristic
50C5×C4⋊Dic3D411yescharacteristic
51C5×Dic3⋊C4—21no—
52C4×C5×Dic3—22no—
53C4×C5×Dic3—21no—
54C2×C5×C3⋊C8—21no—
55C2×C5×C3⋊C8—21no—
56#5732C2×C511yescharacteristic
57C3×#487C2×C511yescharacteristic, complemented
58#5091C2×C511yescharacteristic, complemented
59#5084C2×C511yescharacteristic
60#4547C2×C511yescharacteristic, complemented
61#4459C2×C511yescharacteristic
62#4456C2×C511yescharacteristic, complemented
63C5×C4⋊Q8C2×D311yescharacteristic
64C5×C4⋊Q8C2×D311yescharacteristic
65C5×#126—62no—
66C5×#124—61no—
67C5×C22⋊Q8—61no—
68C4×C5×Q8—31no—
69C4×C5×Q8—61no—
70C4×C5×D4—31no—
71C5×#117—61no—
72C5×#107—61no—
73C5×#106—61no—
74C5×C4⋊C8—31no—
75C5×C4⋊C8—31no—
76C5×#103—22no—
77C5×#103—61no—
78C5×#102—61no—
79C4×C5×C8C2×D311yescharacteristic
80#1638—31noSylow
81C22×C5×D3—21no—
82C3×C5×Q8—42no—
83C3×C5×Q8—41no—
84C3×C5×Q8—41no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5C2×D411yescharacteristic
88C2×C3×C4×C5C2×D411yescharacteristic
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C5×Dic3—21no—
91C2×C5×Dic3—21no—
92C2×C5×Dic3—21no—
93C5×D12—21no—
94C5×D12—21no—
95C4×C5×D3—41no—
96C4×C5×D3—41no—
97C5×Dic6—21no—
98C5×Dic6—21no—
99C3×C5×C8—41no—
100C5×C3⋊C8—41no—
101C5×C3⋊C8—41no—
102C3×C4⋊Q8C22×C511yescharacteristic
103C3×C4⋊Q8C22×C511yescharacteristic
104#947—21no—
105Q8×Dic3—21no—
106#898—22no—
107#897—21no—
108#892—21no—
109C4×D12C22×C511yescharacteristic
110C4×Dic6C22×C511yescharacteristic
111C3×#103—22no—
112C3×C4×C8C22×C511yescharacteristic
113#810—21no—
114#809—21no—
115#808—21no—
116#807—21no—
117C4⋊(C3⋊C8)C22×C511yescharacteristic
118C4⋊(C3⋊C8)C22×C511yescharacteristic
119C2×C5×Q8—21no—
120C2×C5×Q8—21no—
121C2×C5×Q8—31no—
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—31no—
132C5×C4⋊C4—62no—
133C5×C4⋊C4—61no—
134C5×C4⋊C4#6712yes—
135C5×C22⋊C4—62no—
136C5×C22⋊C4—62no—
137C5×C22⋊C4—61no—
138C42×C5—62no—
139C42×C5—61no—
140C42×C5C22×D311yescharacteristic
141#556—31no—
142#487C5×D311yescharacteristic, complemented
143#477—31no—
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—42no—
154C3×C4×C5—81no—
155C3×C4×C5#13511yescharacteristic
156C3×C4×C5#13511yescharacteristic
157C3×C4×C5#13711yescharacteristic
158C3×C4×C5#13711yescharacteristic
159C5×Dic3—41no—
160C5×Dic3—41no—
161C5×Dic3—41no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×D12C5×D411yescharacteristic
165C2×C4×D3—21no—
166C2×Dic6C5×D411yescharacteristic
167C2×C3×C8—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4C5×D412yes—
172C3×C42C23×C511yescharacteristic
173(C2×D3)⋊C4—22no—
174(C2×D3)⋊C4—22no—
175(C2×D3)⋊C4—21no—
176C4⋊Dic3—22no—
177C4⋊Dic3C5×D411yescharacteristic
178C4⋊Dic3C5×D411yescharacteristic
179Dic3⋊C4—21no—
180C4×Dic3—22no—
181C4×Dic3—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C23×C5—61no—
185C5×Q8—42no—
186C5×Q8—41no—
187C5×Q8—41no—
188C5×Q8—61no—
189C5×Q8—61no—
190C5×D4—61no—
191C5×D4—61no—
192C2×C4×C5—22no—
193C2×C4×C5—41no—
194C2×C4×C5—61no—
195C2×C4×C5—61no—
196C2×C4×C5—61no—
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⋊Q8C2×C5×D311yescharacteristic
206C4⋊Q8C2×C5×D311yescharacteristic
207#126—62no—
208#124—61no—
209C22⋊Q8—61no—
210C4×Q8—31no—
211C4×Q8—61no—
212C4×D4—31no—
213#117—61no—
214#107—61no—
215#106—61no—
216C4⋊C8—31no—
217C4⋊C8—31no—
218#103—22no—
219#103—61no—
220#102—61no—
221C4×C8C2×C5×D311yescharacteristic
222C2×C3×C5#45211yescharacteristic
223C2×C3×C5#45211yescharacteristic
224C2×C3×C5#45411yescharacteristic
225C5×D3—81no—
226C22×D3—21no—
227C3×Q8—42no—
228C3×Q8—41no—
229C3×Q8—41no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4C2×C5×D411yescharacteristic
233C2×C3×C4C2×C5×D411yescharacteristic
234C2×C3×C4C2×C5×D411yescharacteristic
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238D12—21no—
239D12—21no—
240C4×D3—41no—
241C4×D3—41no—
242Dic6—21no—
243Dic6—21no—
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—42no—
253C4×C5—81no—
254C4×C5—121no—
255C4×C5—121no—
256C4×C5—121no—
257C4×C5#91511yescharacteristic
258C4×C5#91611yescharacteristic
259C4×C5#91811yescharacteristic
260C4×C5#91911yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—31no—
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—31no—
273C4⋊C4—31no—
274C4⋊C4—62no—
275C4⋊C4—61no—
276C4⋊C4C5×#6712yes—
277C22⋊C4—62no—
278C22⋊C4—62no—
279C22⋊C4—61no—
280C42—62no—
281C42—61no—
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—42no—
290C3×C4—81no—
291C3×C4C5×#13511yescharacteristic
292C3×C4C5×#13511yescharacteristic
293C3×C4C5×#13711yescharacteristic
294C3×C4C5×#13711yescharacteristic
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298C2×C5—241no—
299C2×C5#506211yescharacteristic
300C2×C5#517311yescharacteristic
301C2×C5#518711yescharacteristic
302C23—61no—
303Q8—42no—
304Q8—41no—
305Q8—41no—
306Q8—61no—
307Q8—61no—
308D4—61no—
309D4—61no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
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#7122911yescharacteristic, direct product factor, Sylow
328C22—121no—
329C22—121no—
330C22—121no—
331C22C5×#93711yescharacteristic
332C4—21no—
333C4—42no—
334C4—81no—
335C4—121no—
336C4—121no—
337C4—121no—
338C4C5×#91511yescharacteristic
339C4C5×#91611yescharacteristic
340C4C5×#91811yescharacteristic
341C4C5×#91911yescharacteristic
342C3C5×#163811yescharacteristic, complemented, Sylow
343C2—241no—
344C2C5×#506211yescharacteristic
345C2C5×#517311yescharacteristic
346C2C5×#518711yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC3×C42C23×C5
FrattiniC42C22×C5×D3
FittingC3×C5×#487C2
socleC22×C3×C5#120
prime radicalC2×C5×D12D4
hypercenterC5×#487D3
hypocenterC3C5×#1638
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C42×C5 ◁ C5×#487

Complements

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

NH ≅ G/Ntype
2. C3×C5×#487C2semidirect
3. C5×#5091C2semidirect
5. C5×#4547C2semidirect
7. C5×#4456C2semidirect
26. #71229C5direct (N × H)
28. C5×#487D3semidirect
57. C3×#487C2×C5semidirect
58. #5091C2×C5semidirect
60. #4547C2×C5semidirect
62. #4456C2×C5semidirect
142. #487C5×D3semidirect
283. C3×C5#1638semidirect
327. C5#71229direct (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×#57322
C3×C5×#4872
C5×#50912
C5×#50842
C5×#45472
C5×#44592
C5×#44562
C5×#16383
#712295