Finite Group Databasebeta

← #14306044

Subgroups — #14306044

Order 1920.

640 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C3×#66059C211yesmaximal, characteristic
2C2×#91239C214yesmaximal, complemented
3#11752345C211yesmaximal, characteristic, complemented
4#11752341C211yesmaximal, characteristic, complemented
5#10604664—31nomaximal, Hall
6C2×C3×#713C2212yes—
7C2×C4×ΓL(1,𝔽16)C2212yes—
8C2×#6758C2212yes—
9C3×C5×C4⋊Q8C411yescharacteristic, complemented
10C3×#4093C412yescomplemented
11C3×#4041—22no—
12C3×#4039—22no—
13C3×#4020C411yescharacteristic, complemented
14C3×C42×D5C2211yescharacteristic
15#91239—28no—
16#91235—22no—
17#91233—22no—
18#71816—51nomaximal, Hall
19#66059D311yescharacteristic
20C2×#4010—34no—
21#65686—31no—
22#65682—31no—
23C22×ΓL(1,𝔽16)—22no—
24C2×C3×C5×Q8C2×C412yes—
25C3×#713—24no—
26C3×#713—24no—
27C2×C3×D20—22no—
28C2×C3×D20—22no—
29C2×C3×D20C2×C412yes—
30C2×C3×C4×D5—22no—
31C2×C3×C4×D5D412yes—
32C2×C3×C4×D5C2311yescharacteristic
33C2×C3⋊#170—22no—
34C4×ΓL(1,𝔽16)—22no—
35C4×ΓL(1,𝔽16)D414yescomplemented
36#6758—24no—
37C3×C5×C4⋊C4—22no—
38C3×C42×C5C2×C411yescharacteristic
39C3×#685—24no—
40C3×C4×Dic5C2×C411yescharacteristic
41C3×C4×Dic5D412yes—
42C3×#532—51no—
43C2×#837—54no—
44#4932—51no—
45#4924—51no—
46C2×#713C2×D312yes—
47C2×C4×AGL(1,𝔽5)—32no—
48C2×#700—32no—
49C5×C4⋊Q8Dic311yescharacteristic, complemented
50#4093Dic312yescomplemented
51#4041—22no—
52#4039—22no—
53#4020Dic311yescharacteristic, complemented
54C42×D5C2×D311yescharacteristic
55#4010—68no—
56(C2×AGL(1,𝔽5))⋊C4—62no—
57#4004—62no—
58#1963—151noSylow
59C22×C3×D5—22no—
60C22×C3×D5C2×D411yescharacteristic
61C2×ΓL(1,𝔽16)—24no—
62C2×ΓL(1,𝔽16)—44no—
63C3×C5×Q8—24no—
64C2×C3×C4×C5—22no—
65C2×C3×C4×C5C22⋊C412yes—
66C2×C3×C4×C5C22×C411yescharacteristic
67C2×C3×Dic5C22⋊C412yes—
68C2×C3×Dic5C2×D411yescharacteristic
69C3×D20—24no—
70C3×D20—44no—
71C3×D20—44no—
72C3×C4×D5—42no—
73C3×C4×D5—42no—
74C3×C4×D5—42no—
75C3×C4×D5C2×D412yes—
76C3×C4×D5#4112yes—
77C3⋊#170—42no—
78C2×C3×#41—52no—
79C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
80C3×#127—51no—
81C3×#124—52no—
82C3×#121—102no—
83C3×C4×D4—102no—
84C2×C3×C42—51no—
85C2×C4×Dic3—52no—
86C2×#208—52no—
87#837—108no—
88C3⋊#95—102no—
89C4⋊(C3⋊C8)—102no—
90C22×AGL(1,𝔽5)—62no—
91C2×C5×Q8C2×Dic312yes—
92#713—24no—
93#713—24no—
94C2×D20—22no—
95C2×D20—22no—
96C2×D20C2×Dic312yes—
97C2×C4×D5—22no—
98C2×C4×D5#6712yes—
99C2×C4×D5C22×D311yescharacteristic
100C2×#170—62no—
101C4×AGL(1,𝔽5)—34no—
102C4×AGL(1,𝔽5)—62no—
103#700—64no—
104C5×C4⋊C4—22no—
105C42×C5C2×Dic311yescharacteristic
106#685—24no—
107C4×Dic5C2×Dic311yescharacteristic
108C4×Dic5#6712yes—
109#532—51no—
110#423—151no—
111C2×#104—154no—
112#376—151no—
113C22×C3×C5C2×C22⋊C411yescharacteristic
114C2×C3×D5—22no—
115C2×C3×D5—42no—
116C2×C3×D5—44no—
117C2×C3×D5#12112yes—
118ΓL(1,𝔽16)—44no—
119C3×C4×C5—22no—
120C3×C4×C5—42no—
121C3×C4×C5C4×D412yes—
122C3×Dic5—22no—
123C3×Dic5#12411yescharacteristic
124C3×Dic5#12711yescharacteristic
125C3×#41—104no—
126C3×#41—104no—
127C2×C3×Q8C2×AGL(1,𝔽5)12yes—
128C2×C3×D4—52no—
129C2×C3×D4—102no—
130C2×C3×D4—102no—
131C22×C3×C4—51no—
132C22×C3×C4—52no—
133C22×C3×C4—102no—
134C22×Dic3—102no—
135C3×C4⋊C4—22no—
136C3×C22⋊C4—104no—
137C3×C42—52no—
138C3×C42—51no—
139C3×C42C2×AGL(1,𝔽5)11yescharacteristic
140C4×Dic3—54no—
141C4×Dic3—102no—
142#208—104no—
143C2×C3⋊C8—102no—
144C22×D5—22no—
145C22×D5C2×#6711yescharacteristic
146C2×AGL(1,𝔽5)—64no—
147C2×AGL(1,𝔽5)—124no—
148C5×Q8—24no—
149C2×C4×C5—22no—
150C2×C4×C5#21712yes—
151C2×C4×C5C22×Dic311yescharacteristic
152C2×Dic5#21712yes—
153C2×Dic5C2×#6711yescharacteristic
154D20—24no—
155D20—44no—
156D20—44no—
157C4×D5—42no—
158C4×D5—42no—
159C4×D5—42no—
160C4×D5D3×D412yes—
161C4×D5#23712yes—
162#170—122no—
163C2×#41—52no—
164C2×M16—152no—
165C4⋊Q8ΓL(1,𝔽16)11yescharacteristic, complemented
166#127—51no—
167#124—52no—
168#121—102no—
169C4×D4—102no—
170C2×C42—51no—
171C2×C42—152no—
172C4⋊C8—302no—
173#104—308no—
174#95—302no—
175C2×C3×C5#39011yescharacteristic
176C2×C3×C5#42112yes—
177C3×D5—22no—
178C3×D5—82no—
179C23×C3—51no—
180C23×C3—102no—
181C3×Q8—24no—
182C3×D4—104no—
183C3×D4—204no—
184C3×D4—204no—
185C2×C3×C4—22no—
186C2×C3×C4—51no—
187C2×C3×C4—52no—
188C2×C3×C4—52no—
189C2×C3×C4—52no—
190C2×C3×C4—202no—
191C2×C3×C4—202no—
192C2×C3×C4—202no—
193C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
194C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
195C2×Dic3—104no—
196C2×Dic3—204no—
197C3⋊C8—202no—
198C22×C5C2×#21711yescharacteristic
199C2×D5—22no—
200C2×D5—42no—
201C2×D5—44no—
202C2×D5#93812yes—
203AGL(1,𝔽5)—124no—
204C4×C5—22no—
205C4×C5—42no—
206C4×C5D4×Dic312yes—
207Dic5—22no—
208Dic5#93611yescharacteristic
209Dic5#94011yescharacteristic
210#41—104no—
211#41—104no—
212C2×Q8C2×ΓL(1,𝔽16)12yes—
213C2×D4—52no—
214C2×D4—102no—
215C2×D4—102no—
216C22×C4—51no—
217C22×C4—52no—
218C22×C4—102no—
219C22×C4—302no—
220M16—304no—
221C2×C8—302no—
222C4⋊C4—22no—
223C22⋊C4—104no—
224C42—52no—
225C42—51no—
226C42—154no—
227C42—302no—
228C42C2×ΓL(1,𝔽16)11yescharacteristic
229C3×C5#196311yescharacteristic, complemented, Hall
230C22×C3—52no—
231C22×C3—102no—
232C22×C3—202no—
233C22×C3—204no—
234C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
235C3×C4—22no—
236C3×C4—42no—
237C3×C4—51no—
238C3×C4—51no—
239C3×C4—102no—
240C3×C4D4×AGL(1,𝔽5)12yes—
241Dic3—204no—
242C2×C5#522111yescharacteristic
243C2×C5#523612yes—
244D5—22no—
245D5—82no—
246C23—51no—
247C23—102no—
248Q8—24no—
249D4—104no—
250D4—204no—
251D4—204no—
252C2×C4—22no—
253C2×C4—51no—
254C2×C4—52no—
255C2×C4—52no—
256C2×C4—52no—
257C2×C4—202no—
258C2×C4—202no—
259C2×C4—202no—
260C2×C4—304no—
261C2×C4—604no—
262C2×C4#676312yes—
263C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
264C8—602no—
265C2×C3—102no—
266C2×C3—402no—
267C2×C3#6578011yescharacteristic
268C2×C3#6578412yes—
269C5#7181611yescharacteristic, complemented, Sylow
270C22—52no—
271C22—102no—
272C22—202no—
273C22—204no—
274C22C2×#676311yescharacteristic
275C4—22no—
276C4—42no—
277C4—51no—
278C4—51no—
279C4—102no—
280C4—604no—
281C4D4×ΓL(1,𝔽16)12yes—
282C3#1060466411yescharacteristic, complemented, Sylow
283C2—102no—
284C2—402no—
285C2#1175243911yescharacteristic
286C2#1175244312yes—
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×C4⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#66059C2
cosocleC2×C3×C4×D5C23
hypercenterC4⋊Q8ΓL(1,𝔽16)
hypocenterC3×C5#1963
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#91239C2semidirect
3. #11752345C2semidirect
4. #11752341C2semidirect
9. C3×C5×C4⋊Q8C4semidirect
10. C3×#4093C4semidirect
13. C3×#4020C4semidirect
35. C4×ΓL(1,𝔽16)D4semidirect
49. C5×C4⋊Q8Dic3semidirect
50. #4093Dic3semidirect
53. #4020Dic3semidirect
79. C3×C4⋊Q8AGL(1,𝔽5)semidirect
165. C4⋊Q8ΓL(1,𝔽16)semidirect
229. C3×C5#1963semidirect
269. C5#71816semidirect
282. C3#10604664semidirect

Maximal subgroups

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

subgroupindex
C3×#660592
C2×#912392
#117523452
#117523412
#106046643
#718165