Finite Group Databasebeta

← #14277044

Subgroups — #14277044

Order 1920.

456 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 105 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
1#11752424C212yesmaximal
2C2×C3×#3952C211yesmaximal, characteristic
3#10598764—31nomaximal, Hall
4C22×C3×C4×D5C2211yescharacteristic
5C2×#6763—22no—
6C2×C4⋊ΓL(1,𝔽16)—22no—
7C22×C3×C5×C8C411yescharacteristic, complemented
8C22×C3×#168C411yescharacteristic, complemented
9#91235C414yescomplemented
10C3×#3952—22no—
11#67676—51nomaximal, Hall
12#65765—32no—
13C2×#3952D311yescharacteristic
14C23×C3×D5D411yescharacteristic
15C22×ΓL(1,𝔽16)—24no—
16C22×C3×C4×C5C2×C411yescharacteristic
17C22×C3×Dic5Q811yescharacteristic
18C2×C3×C4×D5—41no—
19C2×C3×C4×D5—41no—
20C2×C3×C4×D5C2×C412yes—
21C2×C3×C4×D5D412yes—
22#6763—42no—
23#6763—42no—
24C4⋊ΓL(1,𝔽16)—42no—
25C4⋊ΓL(1,𝔽16)—42no—
26C2×C3×C5×C8—21no—
27C2×C3×C5×C8—41no—
28C2×C3×#168—21no—
29C2×C3×#168—41no—
30C2×C3×C22⋊C8—51no—
31#5212—52no—
32C22×C4×D5C2×D311yescharacteristic
33C2×C22⋊AGL(1,𝔽5)—62no—
34C2×C4⋊AGL(1,𝔽5)—62no—
35C22×C5×C8Dic311yescharacteristic, complemented
36C22×#168Dic311yescharacteristic, complemented
37(C2×AGL(1,𝔽5))⋊C4—34no—
38#3952—22no—
39#1289—151noSylow
40C23×C3×C5C4⋊C411yescharacteristic
41C22×C3×D5—41no—
42C22×C3×D5—41no—
43C22×C3×D5C22⋊C412yes—
44C22×C3×D5D811yescharacteristic
45C22×C3×D5SD1612yes—
46C22×C3×D5Q1611yescharacteristic
47C2×ΓL(1,𝔽16)—44no—
48C2×ΓL(1,𝔽16)—44no—
49C2×ΓL(1,𝔽16)—44no—
50C2×C3×C4×C5—21no—
51C2×C3×C4×C5—21no—
52C2×C3×C4×C5C4211yescharacteristic
53C2×C3×C4×C5C22⋊C411yescharacteristic
54C2×C3×Dic5—41no—
55C2×C3×Dic5C4⋊C412yes—
56C3×C4×D5—22no—
57C3×C4×D5—22no—
58C3×C4×D5—42no—
59C3×C5×C8—41no—
60C3×#168—41no—
61C23×C3×C4—51no—
62C22×C3×C8—51no—
63C22×C3×C8AGL(1,𝔽5)11yescharacteristic, complemented
64C2×#217—102no—
65C2×C4⋊Dic3—102no—
66C3×C22⋊C8—102no—
67C3⋊#95—54no—
68C23×D5D1211yescharacteristic
69C22×AGL(1,𝔽5)—64no—
70C22×C4×C5C2×Dic311yescharacteristic
71C22×Dic5Dic611yescharacteristic
72C2×C4×D5—41no—
73C2×C4×D5—41no—
74C2×C4×D5C4×D312yes—
75C2×C4×D5#6712yes—
76C22⋊AGL(1,𝔽5)—122no—
77C22⋊AGL(1,𝔽5)—122no—
78C4⋊AGL(1,𝔽5)—122no—
79C4⋊AGL(1,𝔽5)—122no—
80C2×C5×C8—21no—
81C2×C5×C8—41no—
82C2×#168—21no—
83C2×#168—41no—
84C2×C22⋊C8—51no—
85#380—152no—
86C22×C3×C5—21no—
87C22×C3×C5—21no—
88C22×C3×C5#9511yescharacteristic
89C22×C3×C5#10611yescharacteristic
90C22×C3×C5#10711yescharacteristic
91C2×C3×D5—22no—
92C2×C3×D5—22no—
93C2×C3×D5—41no—
94C2×C3×D5—42no—
95C2×C3×D5—41no—
96C2×C3×D5#10212yes—
97C2×C3×D5#10312yes—
98ΓL(1,𝔽16)—84no—
99C3×C4×C5—21no—
100C3×C4×C5#9911yescharacteristic
101C3×C4×C5#9911yescharacteristic
102C3×Dic5—22no—
103C24×C3—51no—
104C22×C3×C4—51no—
105C22×C3×C4—52no—
106C22×C3×C4—52no—
107C22×C3×C4—201no—
108C22×C3×C4—201no—
109C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
110C22×Dic3—104no—
111C2×C3×C8—21no—
112C2×C3×C8—41no—
113C2×C3×C8—101no—
114C2×C3×C8—201no—
115#217—202no—
116#217—202no—
117C4⋊Dic3—202no—
118C4⋊Dic3—202no—
119C23×C5C4⋊Dic311yescharacteristic
120C22×D5—41no—
121C22×D5—41no—
122C22×D5#20412yes—
123C22×D5D2411yescharacteristic
124C22×D5Dic1211yescharacteristic
125C22×D5(C2×D3)⋊C412yes—
126C2×AGL(1,𝔽5)—124no—
127C2×AGL(1,𝔽5)—124no—
128C2×AGL(1,𝔽5)—124no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5C4×Dic311yescharacteristic
132C2×C4×C5#21711yescharacteristic
133C2×Dic5—41no—
134C2×Dic5Dic3⋊C412yes—
135C4×D5—22no—
136C4×D5—22no—
137C4×D5—42no—
138C5×C8—41no—
139#168—41no—
140C23×C4—51no—
141C22×C8—51no—
142C22×C8ΓL(1,𝔽16)11yescharacteristic, complemented
143C2×C4⋊C4—302no—
144C2×C22⋊C4—302no—
145C22⋊C8—102no—
146#95—154no—
147C2×C3×C5—21no—
148C2×C3×C5—21no—
149C2×C3×C5H(3,ℤ/4)11yescharacteristic
150C2×C3×C5#34011yescharacteristic
151C2×C3×C5#34211yescharacteristic
152C3×D5—22no—
153C3×D5—22no—
154C23×C3—52no—
155C23×C3—52no—
156C23×C3—51no—
157C23×C3—51no—
158C23×C3—201no—
159C23×C3—201no—
160C23×C3C4⋊AGL(1,𝔽5)11yescharacteristic
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—52no—
164C2×C3×C4—102no—
165C2×C3×C4—102no—
166C2×C3×C4—202no—
167C2×C3×C4—201no—
168C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
169C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
170C2×Dic3—204no—
171C2×Dic3—204no—
172C2×Dic3—204no—
173C3×C8—41no—
174C3×C8—201no—
175C22×C5—21no—
176C22×C5—21no—
177C22×C5#81711yescharacteristic
178C22×C5#81811yescharacteristic
179C22×C5C3⋊#9511yescharacteristic
180C2×D5—22no—
181C2×D5—22no—
182C2×D5—42no—
183C2×D5—41no—
184C2×D5—41no—
185C2×D5#81612yes—
186C2×D5#82112yes—
187AGL(1,𝔽5)—244no—
188C4×C5—21no—
189C4×C5#83411yescharacteristic
190C4×C5#83411yescharacteristic
191Dic5—22no—
192C24—51no—
193C22×C4—51no—
194C22×C4—52no—
195C22×C4—52no—
196C22×C4—201no—
197C22×C4—201no—
198C22×C4—304no—
199C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
200C2×C8—21no—
201C2×C8—41no—
202C2×C8—101no—
203C2×C8—201no—
204C4⋊C4—602no—
205C4⋊C4—602no—
206C22⋊C4—602no—
207C22⋊C4—602no—
208C3×C5#128911yescharacteristic, complemented, Hall
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—52no—
212C22×C3—52no—
213C22×C3—102no—
214C22×C3—102no—
215C22×C3—202no—
216C22×C3—201no—
217C22×C3—201no—
218C22×C3#399311yescharacteristic
219C22×C3#399411yescharacteristic
220C22×C3(C2×AGL(1,𝔽5))⋊C411yescharacteristic
221C3×C4—21no—
222C3×C4—102no—
223C3×C4#399911yescharacteristic
224C3×C4#401111yescharacteristic
225Dic3—404no—
226C2×C5—21no—
227C2×C5—21no—
228C2×C5#452611yescharacteristic
229C2×C5#455111yescharacteristic
230C2×C5#455711yescharacteristic
231D5—22no—
232D5—22no—
233C23—52no—
234C23—52no—
235C23—51no—
236C23—51no—
237C23—201no—
238C23—201no—
239C23C4⋊ΓL(1,𝔽16)11yescharacteristic
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—52no—
243C2×C4—102no—
244C2×C4—102no—
245C2×C4—202no—
246C2×C4—201no—
247C2×C4—604no—
248C2×C4—604no—
249C2×C4—604no—
250C2×C4C4×ΓL(1,𝔽16)11yescharacteristic
251C2×C4#676311yescharacteristic
252C8—41no—
253C8—201no—
254C2×C3—21no—
255C2×C3—21no—
256C2×C3—102no—
257C2×C3—102no—
258C2×C3#6489311yescharacteristic
259C2×C3#6489511yescharacteristic
260C2×C3#6491711yescharacteristic
261C5#6767611yescharacteristic, complemented, Sylow
262C22—21no—
263C22—21no—
264C22—52no—
265C22—52no—
266C22—102no—
267C22—102no—
268C22—202no—
269C22—201no—
270C22—201no—
271C22#9122211yescharacteristic
272C22#9122311yescharacteristic
273C22#9123511yescharacteristic
274C4—21no—
275C4—102no—
276C4—1204no—
277C4#9122811yescharacteristic
278C4#9124011yescharacteristic
279C3#1059876411yescharacteristic, complemented, Sylow
280C2—21no—
281C2—21no—
282C2—102no—
283C2—102no—
284C2#1174676411yescharacteristic
285C2#1174676611yescharacteristic
286C2#1174678811yescharacteristic
subgroupG/H
centerC22#91235
commutatorC2×C3×C4×C5C42
FrattiniC22×C4C2×ΓL(1,𝔽16)
FittingC22×C3×C5×C8C4
socleC22×C3×C5#95
prime radicalC23×C3×D5D4
cosocleC22×C3×C4×D5C22
hypercenterC22×C8ΓL(1,𝔽16)
hypocenterC3×C5#1289
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
7. C22×C3×C5×C8C4semidirect
8. C22×C3×#168C4semidirect
9. #91235C4semidirect
35. C22×C5×C8Dic3semidirect
36. C22×#168Dic3semidirect
63. C22×C3×C8AGL(1,𝔽5)semidirect
142. C22×C8ΓL(1,𝔽16)semidirect
208. C3×C5#1289semidirect
261. C5#67676semidirect
279. C3#10598764semidirect

Maximal subgroups

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

subgroupindex
#117524242
C2×C3×#39522
#105987643
#676765