Finite Group Databasebeta

← #10617791⋊C3 #14514054

Subgroups — #10617791⋊C3 #14514054

Order 1920.

486 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 87 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#65904⋊C3C212yesmaximal, complemented
5#65903⋊C3C212yesmaximal, complemented
6#10617791C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2211yescharacteristic
8C3×#4036—41nomaximal
9C4×C5×SL(2,𝔽3)C2212yes—
10Dic5×SL(2,𝔽3)C2211yescharacteristic
11Dic5×SL(2,𝔽3)C2211yescharacteristic
12Dic5×SL(2,𝔽3)C2212yes—
13#3504⋊C3—51nomaximal, Hall
14C5×#520C2×C311yescharacteristic
15#66161C2×C311yescharacteristic
16#41×Dic5C2×C311yescharacteristic
17#65904C2×C312yes—
18#65903C2×C312yes—
19#65877—32no—
20#65873—32no—
21#65853—31no—
22#65851—31no—
23#65834—31no—
24C2×#4036—31no—
25C5×#41⋊C3Q812yescomplemented
26C2×C5×SL(2,𝔽3)C2311yescharacteristic
27C3×C5×C4⋊C4—41no—
28C3×C4⋊Dic5—41no—
29C3×C4⋊Dic5—42no—
30C3×#683—42no—
31C3×C4×Dic5—41no—
32#557⋊C3—52no—
33#557⋊C3—52no—
34#520⋊C3—51no—
35#520⋊C3D511yescharacteristic
36C4×#41⋊C3—51no—
37C2×C5×#41C22×C311yescharacteristic
38C4×C5×Q8C22×C312yes—
39C4×C5×D4—32no—
40C5×#117—31no—
41C2×C5×C4⋊C4—31no—
42Q8×Dic5C22×C311yescharacteristic
43Q8×Dic5C22×C312yes—
44Q8×Dic5C22×C311yescharacteristic
45D4×Dic5—31no—
46D4×Dic5—31no—
47D4×Dic5—32no—
48#4072—31no—
49#4072—31no—
50C2×C4⋊Dic5—32no—
51C2×C4⋊Dic5—31no—
52#4070—34no—
53C2×#683—32no—
54C2×C4×Dic5—31no—
55#4036—32no—
56#4036—32no—
57#4036—32no—
58#4036—32no—
59#4036—31no—
60#4036A411yescharacteristic
61#4034—32no—
62#4034—31no—
63#4034—31no—
64#4024—34no—
65#4024—32no—
66#4024—32no—
67#4016—32no—
68#4015—32no—
69C4×Dic10—32no—
70C4×Dic10—32no—
71#3504—51noSylow
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5—42no—
74C2×C3×Dic5—41no—
75C2×C3×Dic5—41no—
76C2×C3×Dic5—42no—
77C5×SL(2,𝔽3)C2×Q811yescharacteristic
78C2×#41⋊C3C2×D511yescharacteristic
79C3×#125—201no—
80C4×SL(2,𝔽3)—51no—
81C4×SL(2,𝔽3)—51no—
82C4×SL(2,𝔽3)—52no—
83C4×SL(2,𝔽3)C2×D512yes—
84C5×#41—32no—
85C5×#41C3×Q812yes—
86C2×C5×Q8C23×C311yescharacteristic
87C2×C5×D4—31no—
88C22×C4×C5—31no—
89C22×C4×C5—32no—
90C22×Dic5—32no—
91C22×Dic5—31no—
92C22×Dic5—31no—
93C2×Dic10—32no—
94C2×Dic10—32no—
95C5×C4⋊C4—32no—
96C5×C4⋊C4—31no—
97C5×C4⋊C4C2×A411yescharacteristic
98C5×C22⋊C4—32no—
99C42×C5—32no—
100C22⋊Dic5—32no—
101C22⋊Dic5—32no—
102C4⋊Dic5—34no—
103C4⋊Dic5—31no—
104C4⋊Dic5—32no—
105C4⋊Dic5—31no—
106C4⋊Dic5—32no—
107C4⋊Dic5—31no—
108C4⋊Dic5C2×A412yes—
109C4⋊Dic5C2×A411yescharacteristic
110#683—34no—
111#683—32no—
112#683—32no—
113#683—32no—
114#683C2×A412yes—
115C4×Dic5—31no—
116C4×Dic5—32no—
117C4×Dic5—31no—
118C4×Dic5—31no—
119C4×Dic5C2×A411yescharacteristic
120#557—52no—
121#557—52no—
122#554—152no—
123#554—152no—
124#544—151no—
125#544—151no—
126#533—151no—
127C2×#125—151no—
128#520—51no—
129#520C3×D511yescharacteristic
130C4×#41—51no—
131C22×C3×C5—41no—
132C3×C4×C5—42no—
133C3×C4×C5—82no—
134C3×Dic5—82no—
135C3×Dic5—81no—
136C3×Dic5—81no—
137#41⋊C3Dic1012yescomplemented
138C2×SL(2,𝔽3)C22×D511yescharacteristic
139C3×C4⋊C4—41no—
140C3×C4⋊C4—202no—
141C3×C4⋊C4—201no—
142C3×C4⋊C4—202no—
143C3×C42—201no—
144C23×C5—31no—
145C5×Q8—31no—
146C5×Q8C2×C3×Q811yescharacteristic
147C5×D4—32no—
148C5×D4—32no—
149C2×C4×C5—31no—
150C2×C4×C5—32no—
151C2×C4×C5—32no—
152C2×C4×C5—32no—
153C2×C4×C5—122no—
154C2×C4×C5C22×A412yes—
155C2×C4×C5C22×A411yescharacteristic
156C2×Dic5—31no—
157C2×Dic5—31no—
158C2×Dic5—32no—
159C2×Dic5—121no—
160C2×Dic5—121no—
161C2×Dic5—122no—
162C2×Dic5C22×A411yescharacteristic
163C2×Dic5C22×A411yescharacteristic
164C2×Dic5C22×A412yes—
165Dic10—122no—
166Dic10—122no—
167C2×#41C2×C3×D511yescharacteristic
168C4⋊Q8—152no—
169C4⋊Q8—151no—
170C4⋊Q8—152no—
171C4⋊Q8—151no—
172#125—51no—
173#125—152no—
174#125—151no—
175#125—152no—
176#125—152no—
177#125—152no—
178#125—152no—
179C22⋊Q8—152no—
180C22⋊Q8—152no—
181C22⋊Q8—154no—
182C22⋊Q8—154no—
183C4×Q8—51no—
184C4×Q8—51no—
185C4×Q8—52no—
186C4×Q8—152no—
187C4×Q8—152no—
188C4×Q8C2×C3×D512yes—
189C4×D4—32no—
190C4×D4—151no—
191C4×D4—151no—
192C4×D4—152no—
193#117—31no—
194#117—151no—
195#117—151no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—152no—
198C2×C4⋊C4—152no—
199C2×C4⋊C4—151no—
200C2×C42—151no—
201C2×C3×C5—41no—
202C2×C3×C5—41no—
203C2×C3×C5—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—201no—
207C2×C3×C4—201no—
208C2×C3×C4—202no—
209SL(2,𝔽3)C2×Dic1011yescharacteristic
210C22×C5—32no—
211C22×C5—61no—
212C22×C5—61no—
213C22×C5C23×A411yescharacteristic
214C4×C5—22no—
215C4×C5—31no—
216C4×C5—31no—
217C4×C5—62no—
218C4×C5Q8×A412yes—
219Dic5—22no—
220Dic5—21no—
221Dic5—21no—
222Dic5—61no—
223Dic5—61no—
224Dic5—62no—
225#41—32no—
226#41C3×Dic1012yes—
227C2×Q8—152no—
228C2×Q8—152no—
229C2×Q8C22×C3×D511yescharacteristic
230C2×D4—31no—
231C22×C4—31no—
232C22×C4—32no—
233C22×C4—151no—
234C22×C4—151no—
235C22×C4—152no—
236C4⋊C4—32no—
237C4⋊C4—31no—
238C4⋊C4—51no—
239C4⋊C4—52no—
240C4⋊C4—52no—
241C4⋊C4—152no—
242C4⋊C4—152no—
243C4⋊C4—152no—
244C4⋊C4—152no—
245C4⋊C4—151no—
246C4⋊C4—151no—
247C4⋊C4—154no—
248C4⋊C4—154no—
249C4⋊C4—151no—
250C4⋊C4—152no—
251C4⋊C4D5×A411yescharacteristic
252C22⋊C4—32no—
253C22⋊C4—152no—
254C22⋊C4—152no—
255C42—32no—
256C42—51no—
257C42—151no—
258C42—151no—
259C42—152no—
260C42—151no—
261C3×C5—41noHall
262C22×C3—41no—
263C3×C4—42no—
264C3×C4—82no—
265C3×C4—401no—
266C3×C4—402no—
267C3×C4—401no—
268C2×C5—62no—
269C2×C5C2×Q8×A411yescharacteristic
270C2×C5#594511yescharacteristic
271C2×C5#594511yescharacteristic
272C23—31no—
273Q8—31no—
274Q8—602no—
275Q8—602no—
276Q8C2×C3×Dic1011yescharacteristic
277D4—32no—
278D4—32no—
279C2×C4—31no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—52no—
286C2×C4—122no—
287C2×C4—151no—
288C2×C4—151no—
289C2×C4—152no—
290C2×C4—601no—
291C2×C4—601no—
292C2×C4—602no—
293C2×C4C2×D5×A412yes—
294C2×C4C2×D5×A411yescharacteristic
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C5#3504⋊C311yescharacteristic, complemented, Sylow
299C22—32no—
300C22—61no—
301C22—61no—
302C22C22×D5×A411yescharacteristic
303C4—22no—
304C4—31no—
305C4—31no—
306C4—62no—
307C4—101no—
308C4—102no—
309C4—101no—
310C4—301no—
311C4—302no—
312C4—301no—
313C4A4×Dic1012yes—
314C3—41noSylow
315C2—62no—
316C2C2×A4×Dic1011yescharacteristic
317C2#1175694411yescharacteristic
318C2#66287⋊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. #65904⋊C3C2semidirect
5. #65903⋊C3C2semidirect
6. #10617791C3semidirect
25. C5×#41⋊C3Q8semidirect
137. #41⋊C3Dic10semidirect
298. C5#3504⋊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
#65904⋊C32
#65903⋊C32
#106177913
C3×#40364
#3504⋊C35