Finite Group Databasebeta

← #10617790⋊C3 #14514052

Subgroups — #10617790⋊C3 #14514052

Order 1920.

530 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 91 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×#520⋊C3C211yesmaximal, characteristic
2#66161⋊C3C211yesmaximal, characteristic
3Dic5×#41⋊C3C211yesmaximal, characteristic
4#65903⋊C3C212yesmaximal, complemented
5SL(2,𝔽3)×Dic10C212yesmaximal, complemented
6#10617790C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2211yescharacteristic
8C3×#4034—41nomaximal
9C4×C5×SL(2,𝔽3)C2212yes—
10Dic5×SL(2,𝔽3)C2211yescharacteristic
11Dic5×SL(2,𝔽3)C2211yescharacteristic
12Dic5×SL(2,𝔽3)C2212yes—
13#3503⋊C3—51nomaximal, Hall
14C5×#520C2×C311yescharacteristic
15#66161C2×C311yescharacteristic
16#41×Dic5C2×C311yescharacteristic
17#65903C2×C312yes—
18Q8×Dic10C2×C312yes—
19#65873—32no—
20D4×Dic10—32no—
21#65853—31no—
22#65851—31no—
23#65834—31no—
24C2×#4034—31no—
25C2×C3×Dic10—42no—
26C5×#41⋊C3Q812yescomplemented
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C3×C5×C4⋊C4—41no—
29C3×C4⋊Dic5—41no—
30C3×#683—42no—
31C3×C4×Dic5—41no—
32Q8×SL(2,𝔽3)—52no—
33#557⋊C3—52no—
34#520⋊C3—51no—
35#520⋊C3D511yescharacteristic
36C4×#41⋊C3—51no—
37C2×C5×#41C22×C311yescharacteristic
38C22×Dic10—32no—
39C4×C5×Q8C22×C312yes—
40C4×C5×D4—32no—
41C5×#117—31no—
42C2×C5×C4⋊C4—31no—
43Q8×Dic5C22×C311yescharacteristic
44Q8×Dic5C22×C312yes—
45Q8×Dic5C22×C311yescharacteristic
46D4×Dic5—31no—
47D4×Dic5—31no—
48D4×Dic5—32no—
49#4072—31no—
50#4072—31no—
51C2×C4⋊Dic5—31no—
52#4070—34no—
53C2×#683—32no—
54C2×C4×Dic5—31no—
55#4036—31no—
56#4036—31no—
57#4036—32no—
58#4034—32no—
59#4034—32no—
60#4034—32no—
61#4034—32no—
62#4034—31no—
63#4034A411yescharacteristic
64#4024—34no—
65#4024—32no—
66#4024—32no—
67#4016—32no—
68#4015—32no—
69C4×Dic10—32no—
70C4×Dic10—32no—
71#3503—51noSylow
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5—42no—
74C2×C3×Dic5—41no—
75C2×C3×Dic5—41no—
76C2×C3×Dic5—42no—
77C3×Dic10—84no—
78C5×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
79C2×#41⋊C3C2×D511yescharacteristic
80C3×C4⋊Q8—201no—
81C4×SL(2,𝔽3)—51no—
82C4×SL(2,𝔽3)—51no—
83C4×SL(2,𝔽3)—52no—
84C4×SL(2,𝔽3)C2×D512yes—
85C5×#41—32no—
86C5×#41C3×Q812yescomplemented
87C2×C5×Q8C23×C311yescharacteristic
88C2×C5×D4—31no—
89C22×C4×C5—31no—
90C22×C4×C5—32no—
91C22×Dic5—32no—
92C22×Dic5—31no—
93C22×Dic5—31no—
94C2×Dic10—32no—
95C2×Dic10—34no—
96C2×Dic10—124no—
97C2×Dic10C2×A412yes—
98C5×C4⋊C4—32no—
99C5×C4⋊C4—31no—
100C5×C4⋊C4C2×A411yescharacteristic
101C5×C22⋊C4—32no—
102C42×C5—32no—
103C22⋊Dic5—32no—
104C22⋊Dic5—32no—
105C4⋊Dic5—32no—
106C4⋊Dic5—31no—
107C4⋊Dic5—32no—
108C4⋊Dic5—31no—
109C4⋊Dic5—32no—
110C4⋊Dic5—31no—
111C4⋊Dic5C2×A411yescharacteristic
112#683—34no—
113#683—32no—
114#683—32no—
115#683—32no—
116#683C2×A412yes—
117C4×Dic5—31no—
118C4×Dic5—31no—
119C4×Dic5—31no—
120C4×Dic5—32no—
121C4×Dic5C2×A411yescharacteristic
122Q82—52no—
123#557—52no—
124#554—152no—
125D4×Q8—152no—
126#544—151no—
127#544—151no—
128#533—151no—
129C2×C4⋊Q8—151no—
130#520—51no—
131#520C3×D511yescharacteristic
132C4×#41—51no—
133C22×C3×C5—41no—
134C3×C4×C5—42no—
135C3×C4×C5—82no—
136C3×Dic5—42no—
137C3×Dic5—42no—
138C3×Dic5—82no—
139C2×C3×Q8—202no—
140#41⋊C3Dic1012yescomplemented
141C2×SL(2,𝔽3)C22×D511yescharacteristic
142C3×C4⋊C4—41no—
143C3×C4⋊C4—202no—
144C3×C4⋊C4—201no—
145C3×C42—201no—
146C23×C5—31no—
147C5×Q8—31no—
148C5×Q8C2×C3×Q811yescharacteristic
149C5×D4—32no—
150C5×D4—32no—
151C2×C4×C5—31no—
152C2×C4×C5—32no—
153C2×C4×C5—32no—
154C2×C4×C5—32no—
155C2×C4×C5—122no—
156C2×C4×C5C22×A412yes—
157C2×C4×C5C22×A411yescharacteristic
158C2×Dic5—31no—
159C2×Dic5—31no—
160C2×Dic5—32no—
161C2×Dic5—62no—
162C2×Dic5—62no—
163C2×Dic5—122no—
164C2×Dic5C22×A412yes—
165C2×Dic5C22×A411yescharacteristic
166C2×Dic5C22×A411yescharacteristic
167Dic10—24no—
168Dic10—64no—
169Dic10—124no—
170C2×#41C2×C3×D511yescharacteristic
171C22×Q8—152no—
172C4⋊Q8—51no—
173C4⋊Q8—152no—
174C4⋊Q8—151no—
175C4⋊Q8—152no—
176C4⋊Q8—152no—
177C4⋊Q8—152no—
178C4⋊Q8—152no—
179#125—152no—
180#125—151no—
181#125—152no—
182#125—151no—
183C22⋊Q8—152no—
184C22⋊Q8—152no—
185C22⋊Q8—154no—
186C22⋊Q8—154no—
187C4×Q8—51no—
188C4×Q8—51no—
189C4×Q8—52no—
190C4×Q8—152no—
191C4×Q8—152no—
192C4×Q8C2×C3×D512yes—
193C4×D4—32no—
194C4×D4—151no—
195C4×D4—151no—
196C4×D4—152no—
197#117—31no—
198#117—151no—
199#117—151no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—152no—
202C2×C4⋊C4—151no—
203C2×C42—151no—
204C2×C3×C5—41no—
205C2×C3×C5—41no—
206C2×C3×C5—41no—
207C3×Q8—404no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—201no—
211C2×C3×C4—201no—
212C2×C3×C4—202no—
213SL(2,𝔽3)C2×Dic1011yescharacteristic, complemented
214C22×C5—32no—
215C22×C5—61no—
216C22×C5—61no—
217C22×C5C23×A411yescharacteristic
218C4×C5—22no—
219C4×C5—31no—
220C4×C5—31no—
221C4×C5—62no—
222C4×C5Q8×A412yes—
223Dic5—22no—
224Dic5—62no—
225Dic5—61no—
226Dic5—61no—
227Dic5#99412yes—
228Dic52−1+4⋊C312yes—
229#41—32no—
230#41C3×Dic1012yescomplemented
231C2×Q8—52no—
232C2×Q8—152no—
233C2×Q8—154no—
234C2×Q8—604no—
235C2×Q8C22×C3×D511yescharacteristic
236C2×D4—31no—
237C22×C4—31no—
238C22×C4—32no—
239C22×C4—151no—
240C22×C4—151no—
241C22×C4—152no—
242C4⋊C4—32no—
243C4⋊C4—31no—
244C4⋊C4—51no—
245C4⋊C4—52no—
246C4⋊C4—152no—
247C4⋊C4—152no—
248C4⋊C4—152no—
249C4⋊C4—152no—
250C4⋊C4—151no—
251C4⋊C4—151no—
252C4⋊C4—154no—
253C4⋊C4—151no—
254C4⋊C4—152no—
255C4⋊C4—152no—
256C4⋊C4D5×A411yescharacteristic
257C22⋊C4—32no—
258C22⋊C4—152no—
259C22⋊C4—152no—
260C42—32no—
261C42—51no—
262C42—151no—
263C42—151no—
264C42—152no—
265C42—151no—
266C3×C5—41noHall
267C22×C3—41no—
268C3×C4—42no—
269C3×C4—82no—
270C3×C4—202no—
271C3×C4—202no—
272C3×C4—402no—
273C2×C5—62no—
274C2×C5C2×Q8×A411yescharacteristic
275C2×C5C2×#99411yescharacteristic
276C2×C5C2×2−1+4⋊C311yescharacteristic
277C23—31no—
278Q8—31no—
279Q8—104no—
280Q8—304no—
281Q8—604no—
282Q8C2×C3×Dic1011yescharacteristic
283D4—32no—
284D4—32no—
285C2×C4—31no—
286C2×C4—32no—
287C2×C4—32no—
288C2×C4—32no—
289C2×C4—51no—
290C2×C4—51no—
291C2×C4—52no—
292C2×C4—122no—
293C2×C4—151no—
294C2×C4—151no—
295C2×C4—152no—
296C2×C4—302no—
297C2×C4—302no—
298C2×C4—602no—
299C2×C4C2×D5×A412yes—
300C2×C4C2×D5×A411yescharacteristic
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—41no—
304C5#3503⋊C311yescharacteristic, complemented, Sylow
305C22—32no—
306C22—61no—
307C22—61no—
308C22C22×D5×A411yescharacteristic
309C4—22no—
310C4—31no—
311C4—31no—
312C4—52no—
313C4—52no—
314C4—62no—
315C4—102no—
316C4—301no—
317C4—302no—
318C4—301no—
319C4A4×Dic1012yes—
320C3—41noSylow
321C2—62no—
322C2C2×A4×Dic1011yescharacteristic
323C2D5×#99411yescharacteristic
324C2D5×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC5×#520C2×C3
socleC22×C5C23×A4
prime radicalC2×C5×#41⋊C3C22
hypercenterC4⋊C4D5×A4
hypocenterC5×Q8C2×C3×Q8
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #65903⋊C3C2semidirect
5. SL(2,𝔽3)×Dic10C2semidirect
6. #10617790C3semidirect
26. C5×#41⋊C3Q8semidirect
78. C5×SL(2,𝔽3)C2×Q8semidirect
86. C5×#41C3×Q8semidirect
140. #41⋊C3Dic10semidirect
213. SL(2,𝔽3)C2×Dic10semidirect
230. #41C3×Dic10semidirect
304. C5#3503⋊C3semidirect

Maximal subgroups

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

subgroupindex
C5×#520⋊C32
#66161⋊C32
Dic5×#41⋊C32
#65903⋊C32
SL(2,𝔽3)×Dic102
#106177903
C3×#40344
#3503⋊C35