Finite Group Databasebeta

← C5×Q8×ΓL(1,𝔽25) #14418805

Subgroups — C5×Q8×ΓL(1,𝔽25) #14418805

Order 1920.

776 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C5×D3×Q8C211yesmaximal, characteristic
2C3×C5×C8×Q8C211yesmaximal, characteristic, complemented
3C5×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4C5×#4837C213yesmaximal
5C5×#4831C213yesmaximal, complemented
6C4×C5×ΓL(1,𝔽25)C213yesmaximal
7C5×#4679C213yesmaximal, complemented
8C5×Q8×M16—31nomaximal, Hall
9C2×C5×D3×Q8C411yescharacteristic
10C3×C4×C5×Q8C2211yescharacteristic
11C5×Q8×Dic3C411yescharacteristic
12C2×C5×ΓL(1,𝔽25)C2213yes—
13C2×C5×ΓL(1,𝔽25)C2211yescharacteristic
14C5×D3×C4⋊C4C413yes—
15C5×#887C413yes—
16C42×C5×D3C2213yes—
17C4×C5×Dic6C2213yes—
18C3×C5×C4⋊C8C2213yes—
19C3×C4×C5×C8C2213yes—
20C5×#815C2213yes—
21C5×#815C2213yes—
22C5×Dic3⋊C8C2216yes—
23C5×C4⋊(C3⋊C8)C2213yes—
24C4×C5×C3⋊C8C2213yes—
25Q8×ΓL(1,𝔽25)C511yesmaximal, characteristic, direct product factor, Hall
26C2×C4×C5×Q8—31no—
27C5×#446—33no—
28C5×#446—33no—
29C5×C8×Q8—31no—
30C5×C8×Q8D311yescharacteristic, complemented
31C5×#423—33no—
32C4×C5×M16—33no—
33C2×C3×C5×Q8C2×C411yescharacteristic
34C5×D3×Q8—24no—
35C2×C4×C5×D3C2×C413yes—
36C2×C4×C5×D3C2×C413yes—
37C2×C4×C5×D3C2311yescharacteristic
38C2×C5×Dic6C2×C413yes—
39C2×C3×C5×C8C2313yes—
40C2×C3×C5×C8C2311yescharacteristic
41C3×C5×C4⋊C4C2×C413yes—
42C3×C42×C5C2313yes—
43C5×C4⋊Dic3C2×C413yes—
44C5×Dic3⋊C4C2×C416yes—
45C4×C5×Dic3C2×C413yes—
46C4×C5×Dic3C2×C413yes—
47C4×C5×Dic3C2313yes—
48C2×C5×C3⋊C8C2313yes—
49C2×C5×C3⋊C8C2311yescharacteristic
50C5×ΓL(1,𝔽25)—26no—
51C5×ΓL(1,𝔽25)Q814yesdirect product factor
52C4×D3×Q8C2×C511yescharacteristic
53C3×C8×Q8C2×C511yescharacteristic, complemented
54Q8×C3⋊C8C2×C511yescharacteristic, complemented
55#4837C2×C513yes—
56#4831C2×C513yescomplemented
57C4×ΓL(1,𝔽25)C2×C513yes—
58#4679C2×C513yescomplemented
59C22×C5×Q8—31no—
60C2×C5×M16—31no—
61C2×C5×M16—33no—
62C4×C5×Q8—31no—
63C4×C5×Q8—33no—
64C4×C5×Q8—33no—
65C4×C5×Q8C2×D311yescharacteristic
66C2×C5×C4⋊C4—33no—
67C2×C42×C5—33no—
68C5×C4⋊C8—36no—
69C5×C4⋊C8—33no—
70C5×C4⋊C8C2×D313yes—
71C5×#97—33no—
72C5×#97—33no—
73C4×C5×C8—33no—
74C4×C5×C8C2×D313yes—
75Q8×M16—31noSylow
76C22×C5×D3C22×C411yescharacteristic
77C3×C5×Q8M1614yescomplemented
78C2×C3×C4×C5C22×C413yes—
79C2×C3×C4×C5C22×C413yes—
80C2×C3×C4×C5C2411yescharacteristic
81C2×C5×Dic3C22×C413yes—
82C2×C5×Dic3C22×C413yes—
83C2×C5×Dic3C22×C411yescharacteristic
84C4×C5×D3—26no—
85C4×C5×D3—43no—
86C4×C5×D3C2×Q812yes—
87C4×C5×D3#4112yes—
88C5×Dic6—26no—
89C3×C5×C8—23no—
90C3×C5×C8C2×Q812yescomplemented
91C5×C3⋊C8—23no—
92C5×C3⋊C8C2×Q812yescomplemented
93C2×D3×Q8C4×C511yescharacteristic
94C3×C4×Q8C22×C511yescharacteristic
95Q8×Dic3C4×C511yescharacteristic
96C2×ΓL(1,𝔽25)C22×C513yes—
97C2×ΓL(1,𝔽25)C22×C511yescharacteristic
98D3×C4⋊C4C4×C513yes—
99#887C4×C513yes—
100C42×D3C22×C513yes—
101C4×Dic6C22×C513yes—
102C3×C4⋊C8C22×C513yes—
103C3×C4×C8C22×C513yes—
104#815C22×C513yes—
105#815C22×C513yes—
106Dic3⋊C8C22×C516yes—
107C4⋊(C3⋊C8)C22×C513yes—
108C4×C3⋊C8C22×C513yes—
109C2×C5×Q8—33no—
110C2×C5×Q8—64no—
111C2×C5×Q8C4×D311yescharacteristic
112C22×C4×C5—31no—
113C22×C4×C5—33no—
114C22×C4×C5—33no—
115C5×M16—34no—
116C5×M16—66no—
117C2×C5×C8—31no—
118C2×C5×C8—33no—
119C2×C5×C8C22×D313yes—
120C2×C5×C8C22×D311yescharacteristic
121C5×C4⋊C4—36no—
122C5×C4⋊C4—33no—
123C5×C4⋊C4C4×D313yes—
124C42×C5—33no—
125C42×C5—33no—
126C42×C5—33no—
127C42×C5C22×D313yes—
128C2×C4×Q8—31no—
129#446—33no—
130#446—33no—
131C8×Q8—31no—
132C8×Q8C5×D311yescharacteristic, complemented
133#423—33no—
134C4×M16—33no—
135C22×C3×C5C23×C411yescharacteristic
136C2×C5×D3—21no—
137C2×C5×D3—21no—
138C2×C5×D3C4×Q812yes—
139C3×C4×C5—23no—
140C3×C4×C5C2×M1616yes—
141C3×C4×C5C22×Q811yescharacteristic
142C3×C4×C5C2×#4111yescharacteristic
143C5×Dic3—23no—
144C5×Dic3—23no—
145C5×Dic3C4×Q812yes—
146C2×C3×Q8C2×C4×C511yescharacteristic
147D3×Q8—24no—
148C2×C4×D3C2×C4×C513yes—
149C2×C4×D3C2×C4×C513yes—
150C2×C4×D3C23×C511yescharacteristic
151C2×Dic6C2×C4×C513yes—
152C2×C3×C8C23×C513yes—
153C2×C3×C8C23×C511yescharacteristic
154C3×C4⋊C4C2×C4×C513yes—
155C3×C42C23×C513yes—
156C4⋊Dic3C2×C4×C513yes—
157Dic3⋊C4C2×C4×C516yes—
158C4×Dic3C2×C4×C513yes—
159C4×Dic3C2×C4×C513yes—
160C4×Dic3C23×C513yes—
161C2×C3⋊C8C23×C513yes—
162C2×C3⋊C8C23×C511yescharacteristic
163ΓL(1,𝔽25)—26no—
164ΓL(1,𝔽25)C5×Q814yesdirect product factor
165C23×C5—31no—
166C5×Q8—66no—
167C5×Q8ΓL(1,𝔽25)14yesdirect product factor
168C2×C4×C5—31no—
169C2×C4×C5—32no—
170C2×C4×C5—32no—
171C2×C4×C5—33no—
172C2×C4×C5—33no—
173C2×C4×C5—66no—
174C2×C4×C5—123no—
175C2×C4×C5C2×C4×D313yes—
176C2×C4×C5C2×C4×D313yes—
177C2×C4×C5C23×D311yescharacteristic
178C5×C8—23no—
179C5×C8—32no—
180C5×C8—63no—
181C5×C8D3×Q812yescomplemented
182C22×Q8—31no—
183C2×M16—31no—
184C2×M16—33no—
185C4×Q8—31no—
186C4×Q8—33no—
187C4×Q8—33no—
188C4×Q8C2×C5×D311yescharacteristic
189C2×C4⋊C4—33no—
190C2×C42—33no—
191C4⋊C8—36no—
192C4⋊C8—33no—
193C4⋊C8C2×C5×D313yes—
194#97—33no—
195#97—33no—
196C4×C8—33no—
197C4×C8C2×C5×D313yes—
198C2×C3×C5C2×C4×Q811yescharacteristic
199C2×C3×C5C22×M1611yescharacteristic
200C2×C3×C5#56811yescharacteristic
201C5×D3—22no—
202C22×D3C22×C4×C511yescharacteristic
203C3×Q8C5×M1614yescomplemented
204C2×C3×C4C22×C4×C513yes—
205C2×C3×C4C22×C4×C513yes—
206C2×C3×C4C24×C511yescharacteristic
207C2×Dic3C22×C4×C513yes—
208C2×Dic3C22×C4×C513yes—
209C2×Dic3C22×C4×C511yescharacteristic
210C4×D3—26no—
211C4×D3—43no—
212C4×D3C2×C5×Q812yes—
213C4×D3C5×#4112yes—
214Dic6—26no—
215C3×C8—23no—
216C3×C8C2×C5×Q812yescomplemented
217C3⋊C8—23no—
218C3⋊C8C2×C5×Q812yescomplemented
219C22×C5—32no—
220C22×C5—61no—
221C22×C5—61no—
222C22×C5C22×C4×D311yescharacteristic
223C4×C5—23no—
224C4×C5—32no—
225C4×C5—63no—
226C4×C5—63no—
227C4×C5C2×ΓL(1,𝔽25)16yes—
228C4×C5C2×D3×Q811yescharacteristic
229C4×C5D3×#4111yescharacteristic
230C2×Q8—33no—
231C2×Q8—64no—
232C2×Q8C4×C5×D311yescharacteristic
233C22×C4—31no—
234C22×C4—33no—
235C22×C4—33no—
236M16—34no—
237M16—66no—
238C2×C8—31no—
239C2×C8—33no—
240C2×C8C22×C5×D313yes—
241C2×C8C22×C5×D311yescharacteristic
242C4⋊C4—36no—
243C4⋊C4—33no—
244C4⋊C4C4×C5×D313yes—
245C42—33no—
246C42—33no—
247C42—33no—
248C42C22×C5×D313yes—
249C3×C5Q8×M1611yescharacteristic, complemented, Hall
250C22×C3C23×C4×C511yescharacteristic
251C2×D3—21no—
252C2×D3—21no—
253C2×D3C4×C5×Q812yes—
254C3×C4—23no—
255C3×C4C2×C5×M1616yes—
256C3×C4C22×C5×Q811yescharacteristic
257C3×C4C2×C5×#4111yescharacteristic
258Dic3—23no—
259Dic3—23no—
260Dic3C4×C5×Q812yes—
261C2×C5—62no—
262C2×C5C4×D3×Q811yescharacteristic
263C2×C5C22×ΓL(1,𝔽25)11yescharacteristic
264C2×C5#575011yescharacteristic
265C23—31no—
266Q8—66no—
267Q8C5×ΓL(1,𝔽25)14yesdirect product factor
268C2×C4—31no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—33no—
272C2×C4—33no—
273C2×C4—66no—
274C2×C4—123no—
275C2×C4C2×C4×C5×D313yes—
276C2×C4C2×C4×C5×D313yes—
277C2×C4C23×C5×D311yescharacteristic
278C8—23no—
279C8—32no—
280C8—63no—
281C8C5×D3×Q812yescomplemented
282C2×C3C2×C4×C5×Q811yescharacteristic
283C2×C3C22×C5×M1611yescharacteristic
284C2×C3C5×#56811yescharacteristic
285D3—22no—
286C5Q8×ΓL(1,𝔽25)11yescharacteristic, direct product factor, Sylow
287C22—32no—
288C22—61no—
289C22—61no—
290C22C22×C4×C5×D311yescharacteristic
291C4—23no—
292C4—32no—
293C4—63no—
294C4—63no—
295C4C2×C5×ΓL(1,𝔽25)16yes—
296C4C2×C5×D3×Q811yescharacteristic
297C4C5×D3×#4111yescharacteristic
298C3C5×Q8×M1611yescharacteristic, complemented, Sylow
299C2—62no—
300C2C4×C5×D3×Q811yescharacteristic
301C2C22×C5×ΓL(1,𝔽25)11yescharacteristic
302C2C5×#575011yescharacteristic
subgroupG/H
centerC2×C4×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC2×C4C23×C5×D3
FittingC3×C5×C8×Q8C2
socleC22×C3×C5C23×C4
prime radicalC22×C5×D3C22×C4
cosocleC2×C3×C4C24×C5
hypercenterC5×C8×Q8D3
hypocenterC3C5×Q8×M16
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4×C5 ◁ C5×C8×Q8

Complements

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

NH ≅ G/Ntype
2. C3×C5×C8×Q8C2semidirect
3. C5×Q8×C3⋊C8C2semidirect
5. C5×#4831C2semidirect
7. C5×#4679C2semidirect
25. Q8×ΓL(1,𝔽25)C5direct (N × H)
30. C5×C8×Q8D3semidirect
51. C5×ΓL(1,𝔽25)Q8direct (N × H)
53. C3×C8×Q8C2×C5semidirect
54. Q8×C3⋊C8C2×C5semidirect
56. #4831C2×C5semidirect
58. #4679C2×C5semidirect
77. C3×C5×Q8M16semidirect
90. C3×C5×C8C2×Q8semidirect
92. C5×C3⋊C8C2×Q8semidirect
132. C8×Q8C5×D3semidirect
164. ΓL(1,𝔽25)C5×Q8direct (N × H)
167. C5×Q8ΓL(1,𝔽25)direct (N × H)
181. C5×C8D3×Q8semidirect
203. C3×Q8C5×M16semidirect
216. C3×C8C2×C5×Q8semidirect
218. C3⋊C8C2×C5×Q8semidirect
249. C3×C5Q8×M16semidirect
267. Q8C5×ΓL(1,𝔽25)direct (N × H)
281. C8C5×D3×Q8semidirect
286. C5Q8×ΓL(1,𝔽25)direct (N × H)
298. C3C5×Q8×M16semidirect

Maximal subgroups

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

subgroupindex
C4×C5×D3×Q82
C3×C5×C8×Q82
C5×Q8×C3⋊C82
C5×#48372
C5×#48312
C4×C5×ΓL(1,𝔽25)2
C5×#46792
C5×Q8×M163
Q8×ΓL(1,𝔽25)5