Finite Group Databasebeta

← C5×#85562 #14511300

Subgroups — C5×#85562 #14511300

Order 1920.

376 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C2×C5×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C5×#1787—31nomaximal, Hall
3C5×C22⋊SL(2,𝔽3)C411yescharacteristic, complemented
4C2×C5×SL(2,ℤ/4)—41nomaximal
5C5×#859—42nomaximal
6#85562C511yesmaximal, characteristic, direct product factor, Hall
7C23×C5×Q8D311yescharacteristic
8C2×C5×#103—32no—
9C2×C5×#103—32no—
10C2×C5×C22⋊C8—31no—
11C5×#380—31no—
12C22×C5×A4—41no—
13C2×C5×SL(2,𝔽3)—42no—
14C2×C5×SL(2,𝔽3)—81no—
15C5×SL(2,ℤ/4)—42no—
16C2×C22⋊SL(2,𝔽3)C2×C511yescharacteristic
17C22×C5×Q8—31no—
18C22×C5×Q8—32no—
19C22×C5×Q8—32no—
20C22×C5×Q8—61no—
21C22×C5×Q8—61no—
22C22×C5×Q8Dic311yescharacteristic, complemented
23C23×C4×C5—31no—
24C22×C5×C8—32no—
25C2×C5×C4⋊C4—31no—
26C2×C5×C4⋊C4—31no—
27C2×C5×C22⋊C4—61no—
28C5×#103—62no—
29C5×#103—62no—
30C5×#103—62no—
31C5×#103—62no—
32C5×C22⋊C8—61no—
33C5×C22⋊C8—61no—
34C5×#95—61no—
35#1787—31noSylow
36C2×C5×A4—41no—
37C2×C5×A4—41no—
38C2×C5×A4—41no—
39C2×C5×Dic3—161no—
40C5×SL(2,𝔽3)—42no—
41C5×SL(2,𝔽3)—81no—
42C22⋊SL(2,𝔽3)C4×C511yescharacteristic, complemented
43C2×SL(2,ℤ/4)—41no—
44#859—42no—
45C24×C5S411yescharacteristic
46C2×C5×Q8—21no—
47C2×C5×Q8—32no—
48C2×C5×Q8—32no—
49C2×C5×Q8—32no—
50C2×C5×Q8—32no—
51C2×C5×Q8—32no—
52C2×C5×Q8—61no—
53C2×C5×Q8—61no—
54C2×C5×Q8—62no—
55C2×C5×Q8—62no—
56C2×C5×Q8—62no—
57C2×C5×Q8—61no—
58C2×C5×Q8—61no—
59C2×C5×Q8—61no—
60C2×C5×Q8—62no—
61C2×C5×Q8S412yes—
62C22×C4×C5—31no—
63C22×C4×C5—31no—
64C22×C4×C5—31no—
65C22×C4×C5—31no—
66C22×C4×C5—31no—
67C22×C4×C5—31no—
68C22×C4×C5—62no—
69C22×C4×C5—62no—
70C22×C4×C5—61no—
71C22×C4×C5—61no—
72C2×C5×C8—62no—
73C2×C5×C8—62no—
74C2×C5×C8—62no—
75C5×C4⋊C4—62no—
76C5×C4⋊C4—62no—
77C5×C22⋊C4—121no—
78C5×C22⋊C4—121no—
79C23×Q8C5×D311yescharacteristic
80C2×#103—32no—
81C2×#103—32no—
82C2×C22⋊C8—31no—
83#380—31no—
84C22×C3×C5—161no—
85C5×A4—41no—
86C5×Dic3—162no—
87C22×A4—41no—
88C2×SL(2,𝔽3)—42no—
89C2×SL(2,𝔽3)—81no—
90SL(2,ℤ/4)—42no—
91C23×C5—31no—
92C23×C5—31no—
93C23×C5—31no—
94C23×C5—31no—
95C23×C5#22611yescharacteristic
96C23×C5GL(2,𝔽3)11yescharacteristic
97C23×C5SL(2,ℤ/4)11yescharacteristic
98C5×Q8—21no—
99C5×Q8—32no—
100C5×Q8—32no—
101C5×Q8—32no—
102C5×Q8—61no—
103C5×Q8—62no—
104C5×Q8—61no—
105C5×Q8—62no—
106C5×Q8—61no—
107C5×Q8SL(2,ℤ/4)12yescomplemented
108C2×C4×C5—32no—
109C2×C4×C5—31no—
110C2×C4×C5—31no—
111C2×C4×C5—32no—
112C2×C4×C5—31no—
113C2×C4×C5—31no—
114C2×C4×C5—62no—
115C2×C4×C5—62no—
116C2×C4×C5—62no—
117C2×C4×C5—62no—
118C2×C4×C5—61no—
119C2×C4×C5—61no—
120C2×C4×C5—121no—
121C2×C4×C5—121no—
122C2×C4×C5—121no—
123C2×C4×C5—121no—
124C2×C4×C5—121no—
125C2×C4×C5—121no—
126C5×C8—122no—
127C22×Q8—31no—
128C22×Q8—32no—
129C22×Q8—32no—
130C22×Q8—61no—
131C22×Q8—61no—
132C22×Q8C5×Dic311yescharacteristic, complemented
133C23×C4—31no—
134C22×C8—32no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C22⋊C4—61no—
138#103—62no—
139#103—62no—
140#103—62no—
141#103—62no—
142C22⋊C8—61no—
143C22⋊C8—61no—
144#95—61no—
145C2×C3×C5—161no—
146C2×C3×C5—161no—
147C2×C3×C5—161no—
148C2×A4—41no—
149C2×A4—41no—
150C2×A4—41no—
151C2×Dic3—161no—
152SL(2,𝔽3)—42no—
153SL(2,𝔽3)—81no—
154C22×C5—31no—
155C22×C5—31no—
156C22×C5—31no—
157C22×C5—31no—
158C22×C5—31no—
159C22×C5—31no—
160C22×C5—31no—
161C22×C5—31no—
162C22×C5—31no—
163C22×C5—61no—
164C22×C5#85911yescharacteristic, complemented
165C22×C5#102011yescharacteristic
166C4×C5—32no—
167C4×C5—32no—
168C4×C5—61no—
169C4×C5—61no—
170C4×C5—241no—
171C4×C5—241no—
172C24C5×S411yescharacteristic
173C2×Q8—21no—
174C2×Q8—32no—
175C2×Q8—32no—
176C2×Q8—32no—
177C2×Q8—32no—
178C2×Q8—32no—
179C2×Q8—61no—
180C2×Q8—61no—
181C2×Q8—62no—
182C2×Q8—62no—
183C2×Q8—62no—
184C2×Q8—61no—
185C2×Q8—61no—
186C2×Q8—61no—
187C2×Q8—62no—
188C2×Q8C5×S412yes—
189C22×C4—31no—
190C22×C4—31no—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—62no—
196C22×C4—62no—
197C22×C4—61no—
198C22×C4—61no—
199C2×C8—62no—
200C2×C8—62no—
201C2×C8—62no—
202C4⋊C4—62no—
203C4⋊C4—62no—
204C22⋊C4—121no—
205C22⋊C4—121no—
206C3×C5—161noHall
207C22×C3—161no—
208A4—41no—
209Dic3—162no—
210C2×C5—31no—
211C2×C5—31no—
212C2×C5—31no—
213C2×C5—31no—
214C2×C5#593011yescharacteristic
215C2×C5#593111yescharacteristic
216C2×C5#593611yescharacteristic
217C23—31no—
218C23—31no—
219C23—31no—
220C23—31no—
221C23C5×#22611yescharacteristic
222C23C5×GL(2,𝔽3)11yescharacteristic
223C23C5×SL(2,ℤ/4)11yescharacteristic
224Q8—21no—
225Q8—32no—
226Q8—32no—
227Q8—32no—
228Q8—61no—
229Q8—62no—
230Q8—61no—
231Q8—62no—
232Q8—61no—
233Q8C5×SL(2,ℤ/4)12yescomplemented
234C2×C4—32no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—32no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—121no—
247C2×C4—121no—
248C2×C4—121no—
249C2×C4—121no—
250C2×C4—121no—
251C2×C4—121no—
252C8—122no—
253C2×C3—161no—
254C2×C3—161no—
255C2×C3—161no—
256C5#8556211yescharacteristic, direct product factor, Sylow
257C22—31no—
258C22—31no—
259C22—31no—
260C22—31no—
261C22—31no—
262C22—31no—
263C22—31no—
264C22—31no—
265C22—31no—
266C22—61no—
267C22C5×#85911yescharacteristic, complemented
268C22C5×#102011yescharacteristic
269C4—32no—
270C4—32no—
271C4—61no—
272C4—61no—
273C4—241no—
274C4—241no—
275C3—161noSylow
276C2—31no—
277C2—31no—
278C2—31no—
279C2—31no—
280C2C5×#593011yescharacteristic
281C2C5×#593111yescharacteristic
282C2C5×#593611yescharacteristic
subgroupG/H
center, hypercenterC22×C5#1020
commutator, hypocenterC22⋊SL(2,𝔽3)C4×C5
FrattiniC22C5×#1020
FittingC23×C5×Q8D3
socleC24×C5S4
prime radicalC2×C5×C22⋊SL(2,𝔽3)C2
cosocleC2×C22⋊SL(2,𝔽3)C2×C5
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C22×C5

Complements

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

NH ≅ G/Ntype
3. C5×C22⋊SL(2,𝔽3)C4semidirect
6. #85562C5direct (N × H)
22. C22×C5×Q8Dic3semidirect
42. C22⋊SL(2,𝔽3)C4×C5semidirect
107. C5×Q8SL(2,ℤ/4)semidirect
132. C22×Q8C5×Dic3semidirect
164. C22×C5#859semidirect
233. Q8C5×SL(2,ℤ/4)semidirect
256. C5#85562direct (N × H)
267. C22C5×#859semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊SL(2,𝔽3)2
C5×#17873
C2×C5×SL(2,ℤ/4)4
C5×#8594
#855625