Finite Group Databasebeta

← C5×#85403 #14511141

Subgroups — C5×#85403 #14511141

Order 1920.

416 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×C4⋊(#41⋊C3)C211yesmaximal, characteristic
2C5×#5425C212yesmaximal, complemented
3C5×#5424C212yesmaximal, complemented
4C4×C5×#226C211yesmaximal, characteristic, complemented
5C5×#5386C211yesmaximal, characteristic, complemented
6C5×#3304—31nomaximal, Hall
7C2×C5×#41⋊C3C2212yes—
8C2×C5×#226C2211yescharacteristic
9C5×#887—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C5×#859C2212yes—
12C5×#859C2211yescharacteristic
13#85403C511yesmaximal, characteristic, direct product factor, Hall
14C5×#556—31no—
15C5×#550D311yescharacteristic
16C5×#486—31no—
17C5×#482—32no—
18C5×#475—31no—
19C5×#465—32no—
20C5×#460—31no—
21C5×#452—32no—
22C4×C5×Q16—31no—
23C5×C8×D4—31no—
24C5×#416—32no—
25C2×C5×Dic6—41no—
26C5×#41⋊C3—22no—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C5×#226—21no—
29C3×C5×C4⋊C4—41no—
30C5×Dic3⋊C4—42no—
31C4×C5×Dic3—41no—
32C4×C5×Dic3—42no—
33C4⋊(#41⋊C3)C2×C511yescharacteristic
34#5425C2×C512yescomplemented
35#5424C2×C512yescomplemented
36C4×#226C2×C511yescharacteristic, complemented
37#5386C2×C511yescharacteristic, complemented
38C2×C5×#41C2×D312yes—
39C2×C5×Q16—31no—
40C2×C5×SD16—32no—
41C22×C5×C8—32no—
42C5×C4⋊Q8—31no—
43C5×#127—31no—
44C5×#126—62no—
45C5×#124—61no—
46C5×C22⋊Q8—32no—
47C5×#121—32no—
48C4×C5×Q8—31no—
49C4×C5×Q8—61no—
50C4×C5×Q8C2×D311yescharacteristic
51C4×C5×D4—31no—
52C5×#117—32no—
53C5×#107—31no—
54C5×#106—32no—
55C5×C4⋊C8—31no—
56C5×#103—31no—
57C5×#103—31no—
58C5×#103—32no—
59C5×#102—32no—
60C5×#102—32no—
61C5×#102—32no—
62C5×C22⋊C8—32no—
63C4×C5×C8—31no—
64#3304—31noSylow
65C2×C3×C4×C5—41no—
66C2×C3×C4×C5—42no—
67C2×C5×Dic3—41no—
68C2×C5×Dic3—42no—
69C2×C5×Dic3—41no—
70C5×Dic6—44no—
71C5×SL(2,𝔽3)#4111yescharacteristic
72C2×#41⋊C3C22×C512yes—
73C2×#226C22×C511yescharacteristic
74#887—41no—
75C4×SL(2,𝔽3)C22×C511yescharacteristic
76#859C22×C512yes—
77#859C22×C511yescharacteristic
78C5×#41—22no—
79C5×#41—62no—
80C2×C5×Q8—31no—
81C2×C5×Q8—61no—
82C2×C5×Q8C22×D311yescharacteristic
83C2×C5×D4—32no—
84C2×C5×D4—31no—
85C2×C5×D4—61no—
86C22×C4×C5—32no—
87C5×Q16—61no—
88C5×Q16—61no—
89C5×SD16—122no—
90C2×C5×C8—31no—
91C2×C5×C8—32no—
92C2×C5×C8—31no—
93C2×C5×C8—122no—
94C5×C4⋊C4—31no—
95C5×C4⋊C4—31no—
96C5×C4⋊C4—32no—
97C5×C4⋊C4—61no—
98C5×C4⋊C4—61no—
99C5×C4⋊C4—62no—
100C5×C4⋊C4S411yescharacteristic
101C5×C22⋊C4—32no—
102C5×C22⋊C4—62no—
103C5×C22⋊C4—62no—
104C42×C5—31no—
105C42×C5—62no—
106C42×C5—61no—
107#556—31no—
108#550C5×D311yescharacteristic
109#486—31no—
110#482—32no—
111#475—31no—
112#465—32no—
113#460—31no—
114#452—32no—
115C4×Q16—31no—
116C8×D4—31no—
117#416—32no—
118C22×C3×C5—41no—
119C3×C4×C5—42no—
120C3×C4×C5—82no—
121C5×Dic3—44no—
122C5×Dic3—42no—
123C5×Dic3—81no—
124C2×Dic6—41no—
125#41⋊C3—22no—
126C2×SL(2,𝔽3)C23×C511yescharacteristic
127#226—21no—
128C3×C4⋊C4—41no—
129Dic3⋊C4—42no—
130C4×Dic3—41no—
131C4×Dic3—42no—
132C23×C5—32no—
133C5×Q8—31no—
134C5×Q8—61no—
135C5×Q8—61no—
136C5×Q8—122no—
137C5×Q8#23711yescharacteristic
138C5×D4—62no—
139C5×D4—62no—
140C5×D4—62no—
141C5×D4—241no—
142C2×C4×C5—31no—
143C2×C4×C5—31no—
144C2×C4×C5—62no—
145C2×C4×C5—62no—
146C2×C4×C5—61no—
147C2×C4×C5—62no—
148C2×C4×C5—61no—
149C2×C4×C5C2×S412yes—
150C2×C4×C5C2×S411yescharacteristic
151C5×C8—61no—
152C5×C8—62no—
153C5×C8—61no—
154C2×#41C2×C5×D312yes—
155C2×Q16—31no—
156C2×SD16—32no—
157C22×C8—32no—
158C4⋊Q8—31no—
159#127—31no—
160#126—62no—
161#124—61no—
162C22⋊Q8—32no—
163#121—32no—
164C4×Q8—31no—
165C4×Q8—61no—
166C4×Q8C2×C5×D311yescharacteristic
167C4×D4—31no—
168#117—32no—
169#107—31no—
170#106—32no—
171C4⋊C8—31no—
172#103—31no—
173#103—31no—
174#103—32no—
175#102—32no—
176#102—32no—
177#102—32no—
178C22⋊C8—32no—
179C4×C8—31no—
180C2×C3×C5—41no—
181C2×C3×C5—41no—
182C2×C3×C5—41no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×Dic3—41no—
186C2×Dic3—42no—
187C2×Dic3—41no—
188Dic6—44no—
189SL(2,𝔽3)C5×#4111yescharacteristic
190C22×C5—62no—
191C22×C5—62no—
192C22×C5—62no—
193C22×C5C22×S411yescharacteristic
194C4×C5—22no—
195C4×C5—31no—
196C4×C5—31no—
197C4×C5—61no—
198C4×C5—121no—
199C4×C5—122no—
200C4×C5—121no—
201C4×C5#98512yes—
202#41—22no—
203#41—62no—
204C2×Q8—31no—
205C2×Q8—61no—
206C2×Q8C22×C5×D311yescharacteristic
207C2×D4—32no—
208C2×D4—31no—
209C2×D4—61no—
210C22×C4—32no—
211Q16—61no—
212Q16—61no—
213SD16—122no—
214C2×C8—31no—
215C2×C8—31no—
216C2×C8—32no—
217C2×C8—122no—
218C4⋊C4—31no—
219C4⋊C4—32no—
220C4⋊C4—31no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—62no—
224C4⋊C4C5×S411yescharacteristic
225C22⋊C4—32no—
226C22⋊C4—62no—
227C22⋊C4—62no—
228C42—31no—
229C42—62no—
230C42—61no—
231C3×C5—41noHall
232C22×C3—41no—
233C3×C4—42no—
234C3×C4—82no—
235Dic3—44no—
236Dic3—42no—
237Dic3—81no—
238C2×C5—122no—
239C2×C5#591411yescharacteristic
240C2×C5C2×#98511yescharacteristic
241C2×C5#592411yescharacteristic
242C23—32no—
243Q8—31no—
244Q8—61no—
245Q8—61no—
246Q8—122no—
247Q8C5×#23711yescharacteristic
248D4—62no—
249D4—62no—
250D4—62no—
251D4—241no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—61no—
257C2×C4—62no—
258C2×C4—61no—
259C2×C4C2×C5×S412yes—
260C2×C4C2×C5×S411yescharacteristic
261C8—61no—
262C8—62no—
263C8—61no—
264C2×C3—41no—
265C2×C3—41no—
266C2×C3—41no—
267C5#8540311yescharacteristic, direct product factor, Sylow
268C22—62no—
269C22—62no—
270C22—62no—
271C22C22×C5×S411yescharacteristic
272C4—22no—
273C4—31no—
274C4—31no—
275C4—61no—
276C4—121no—
277C4—122no—
278C4—121no—
279C4C5×#98512yes—
280C3—41noSylow
281C2—122no—
282C2C5×#591411yescharacteristic
283C2C2×C5×#98511yescharacteristic
284C2C5×#592411yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×#550D3
prime radicalC5×C4⋊(#41⋊C3)C2
hypercenterC5×C4⋊C4S4
hypocenterSL(2,𝔽3)C5×#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#5425C2semidirect
3. C5×#5424C2semidirect
4. C4×C5×#226C2semidirect
5. C5×#5386C2semidirect
13. #85403C5direct (N × H)
34. #5425C2×C5semidirect
35. #5424C2×C5semidirect
36. C4×#226C2×C5semidirect
37. #5386C2×C5semidirect
267. C5#85403direct (N × H)

Maximal subgroups

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

subgroupindex
C5×C4⋊(#41⋊C3)2
C5×#54252
C5×#54242
C4×C5×#2262
C5×#53862
C5×#33043
C5×#8874
#854035