Finite Group Databasebeta

← #14276952

Subgroups — #14276952

Order 1920.

328 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
2C3×#65683C211yesmaximal, characteristic
3#11746871C211yesmaximal, characteristic
4#10598585—31nomaximal, Hall
5C2×C4⋊ΓL(1,𝔽16)—21no—
6C2×C4×ΓL(1,𝔽16)C412yes—
7C3×C42×D5C2211yescharacteristic
8#91235—21no—
9C3×#4003—21no—
10C3×#4002C412yescomplemented
11C3×C4⋊#170—21no—
12C4⋊(C3⋊#168)C411yescharacteristic, complemented
13C5×C4⋊(C3⋊C8)C411yescharacteristic, complemented
14#90955—21no—
15#90952—21no—
16#67749—51nomaximal, Hall
17C4×C4⋊AGL(1,𝔽5)—31no—
18#65683D311yescharacteristic
19#65124—31no—
20C22×ΓL(1,𝔽16)—22no—
21C2×C3×C4×D5C2×C411yescharacteristic
22C2×C3×C4×D5D411yescharacteristic
23C2×C3×C4×D5D411yescharacteristic
24C4⋊ΓL(1,𝔽16)—22no—
25C4⋊ΓL(1,𝔽16)—22no—
26C4×ΓL(1,𝔽16)—42no—
27C2×C3×#170—22no—
28C3×C42×C5C2×C411yescharacteristic
29C2×C3⋊#168—21no—
30C2×C5×C3⋊C8—21no—
31C3×C4×Dic5C2×C411yescharacteristic
32C3×C4×Dic5D411yescharacteristic
33C3×C4×Dic5Q811yescharacteristic
34C3×#432—51no—
35C4×C4⋊Dic3—51no—
36#4926—51no—
37C2×C4⋊AGL(1,𝔽5)—61no—
38C2×C4×AGL(1,𝔽5)—32no—
39C42×D5C2×D311yescharacteristic
40(C2×AGL(1,𝔽5))⋊C4—61no—
41#4003—21no—
42#4002Dic312yescomplemented
43C4⋊#170—21no—
44C5×C4⋊C8—31no—
45#3952—61no—
46#3947—61no—
47C4⋊#168—31no—
48#1285—151noSylow
49C22×C3×D5C22⋊C411yescharacteristic
50C2×ΓL(1,𝔽16)—24no—
51C2×ΓL(1,𝔽16)—42no—
52C2×ΓL(1,𝔽16)—42no—
53C2×C3×C4×C5C4211yescharacteristic
54C2×C3×C4×C5C22⋊C411yescharacteristic
55C2×C3×C4×C5C4⋊C411yescharacteristic
56C2×C3×Dic5C22⋊C411yescharacteristic
57C2×C3×Dic5C4⋊C411yescharacteristic
58C2×C3×Dic5C4⋊C411yescharacteristic
59C3×C4×D5—21no—
60C3×C4×D5—21no—
61C3×C4×D5—21no—
62C3×C4×D5—21no—
63C3×C4×D5—41no—
64C3×#170—42no—
65C3⋊#168—41no—
66C5×C3⋊C8—41no—
67C2×C3×C42—51no—
68C2×C4⋊Dic3—101no—
69C2×C4×Dic3—52no—
70C3×C4⋊C8—101no—
71C3×C22⋊C8—101no—
72C3×#97—52no—
73C3⋊#95—101no—
74#830—101no—
75C4⋊(C3⋊C8)—51no—
76C4⋊(C3⋊C8)AGL(1,𝔽5)11yescharacteristic, complemented
77#803—101no—
78C22×AGL(1,𝔽5)—62no—
79C2×C4×D5C4×D311yescharacteristic
80C2×C4×D5D1211yescharacteristic
81C2×C4×D5#6711yescharacteristic
82C2×#170—22no—
83C4⋊AGL(1,𝔽5)—62no—
84C4⋊AGL(1,𝔽5)—62no—
85C4×AGL(1,𝔽5)—122no—
86C2×C5×C8—61no—
87C42×C5C4×D311yescharacteristic
88C4×Dic5Dic611yescharacteristic
89C4×Dic5C2×Dic311yescharacteristic
90C4×Dic5#6711yescharacteristic
91C2×#168—61no—
92#432—51no—
93#432—151no—
94C4×C4⋊C4—151no—
95C22×C3×C5#9511yescharacteristic
96C2×C3×D5—21no—
97C2×C3×D5—21no—
98C2×C3×D5#10011yescharacteristic
99C2×C3×D5#10111yescharacteristic
100ΓL(1,𝔽16)—44no—
101C3×C4×C5—21no—
102C3×C4×C5—21no—
103C3×C4×C5#10411yescharacteristic
104C3×C4×C5#10411yescharacteristic
105C3×Dic5—21no—
106C3×Dic5—21no—
107C3×Dic5—21no—
108C22×C3×C4—51no—
109C22×C3×C4—51no—
110C22×C3×C4—51no—
111C22×Dic3—102no—
112C2×C3×C8—102no—
113C3×C42—51no—
114C3×C42—51no—
115C3×C42—51no—
116C3×C42C2×AGL(1,𝔽5)11yescharacteristic
117C4⋊Dic3—102no—
118C4⋊Dic3—102no—
119C4×Dic3—202no—
120C2×C3⋊C8—21no—
121C2×C3⋊C8—101no—
122C22×D5(C2×D3)⋊C411yescharacteristic
123C2×AGL(1,𝔽5)—64no—
124C2×AGL(1,𝔽5)—122no—
125C2×AGL(1,𝔽5)—122no—
126C2×C4×C5C4×Dic311yescharacteristic
127C2×C4×C5Dic3⋊C411yescharacteristic
128C2×C4×C5(C2×D3)⋊C411yescharacteristic
129C2×Dic5Dic3⋊C411yescharacteristic
130C2×Dic5C4⋊Dic311yescharacteristic
131C2×Dic5#21711yescharacteristic
132C4×D5—21no—
133C4×D5—21no—
134C4×D5—21no—
135C4×D5—21no—
136C4×D5—41no—
137#170—42no—
138C5×C8—121no—
139#168—121no—
140C2×C4⋊C4—301no—
141C2×C42—51no—
142C2×C42—152no—
143C4⋊C8—31no—
144C4⋊C8—101no—
145C4⋊C8—151no—
146C22⋊C8—101no—
147C22⋊C8—301no—
148#97—52no—
149#97—301no—
150#95—301no—
151C2×C3×C5H(3,ℤ/4)11yescharacteristic
152C2×C3×C5#33911yescharacteristic
153C2×C3×C5#34311yescharacteristic
154C3×D5—21no—
155C3×D5—21no—
156C23×C3—51no—
157C2×C3×C4—51no—
158C2×C3×C4—51no—
159C2×C3×C4—51no—
160C2×C3×C4—101no—
161C2×C3×C4—101no—
162C2×C3×C4—101no—
163C2×C3×C4—101no—
164C2×C3×C4—201no—
165C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
166C2×C3×C4C4⋊AGL(1,𝔽5)11yescharacteristic
167C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
168C2×Dic3—104no—
169C2×Dic3—202no—
170C2×Dic3—202no—
171C3×C8—202no—
172C3⋊C8—41no—
173C3⋊C8—201no—
174C22×C5C3⋊#9511yescharacteristic
175C2×D5—21no—
176C2×D5—21no—
177C2×D5#82311yescharacteristic
178C2×D5#82411yescharacteristic
179AGL(1,𝔽5)—124no—
180C4×C5—21no—
181C4×C5—21no—
182C4×C5#82511yescharacteristic
183C4×C5#82511yescharacteristic
184Dic5—21no—
185Dic5—21no—
186Dic5—21no—
187C22×C4—51no—
188C22×C4—51no—
189C22×C4—51no—
190C22×C4—302no—
191C2×C8—61no—
192C2×C8—102no—
193C2×C8—301no—
194C4⋊C4—302no—
195C4⋊C4—302no—
196C42—51no—
197C42—51no—
198C42—51no—
199C42—602no—
200C42D3×AGL(1,𝔽5)11yescharacteristic
201C3×C5#128511yescharacteristic, complemented, Hall
202C22×C3—51no—
203C22×C3—51no—
204C22×C3—101no—
205C22×C3—101no—
206C22×C3(C2×AGL(1,𝔽5))⋊C411yescharacteristic
207C3×C4—21no—
208C3×C4—21no—
209C3×C4—101no—
210C3×C4—101no—
211C3×C4—101no—
212C3×C4#400811yescharacteristic
213C3×C4#401011yescharacteristic
214Dic3—204no—
215C2×C5#453111yescharacteristic
216C2×C5#455411yescharacteristic
217C2×C5#455511yescharacteristic
218D5—21no—
219D5—21no—
220C23—51no—
221C2×C4—51no—
222C2×C4—51no—
223C2×C4—51no—
224C2×C4—101no—
225C2×C4—101no—
226C2×C4—101no—
227C2×C4—101no—
228C2×C4—201no—
229C2×C4—304no—
230C2×C4—602no—
231C2×C4—602no—
232C2×C4Dic3×AGL(1,𝔽5)11yescharacteristic
233C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
234C2×C4Dic3⋊AGL(1,𝔽5)11yescharacteristic
235C8—121no—
236C8—202no—
237C8—601no—
238C2×C3—101no—
239C2×C3—101no—
240C2×C3#6487511yescharacteristic
241C2×C3#6489511yescharacteristic
242C2×C3#6489911yescharacteristic
243C5#6774911yescharacteristic, complemented, Sylow
244C22—51no—
245C22—51no—
246C22—101no—
247C22—101no—
248C22(C2×AGL(1,𝔽5))⋊Dic311yescharacteristic
249C4—21no—
250C4—21no—
251C4—101no—
252C4—101no—
253C4—101no—
254C4—604no—
255C4#9115511yescharacteristic
256C4#9115711yescharacteristic
257C3#1059858511yescharacteristic, complemented, Sylow
258C2—101no—
259C2—101no—
260C2#1174651911yescharacteristic
261C2#1174652411yescharacteristic
262C2#1174653511yescharacteristic
subgroupG/H
centerC22(C2×AGL(1,𝔽5))⋊Dic3
commutatorC2×C3×C4×C5C42
Frattini, hypercenterC42D3×AGL(1,𝔽5)
FittingC3×C42×C5C2×C4
socleC22×C3×C5#95
prime radicalC22×C3×D5C22⋊C4
cosocleC3×C42×D5C22
hypocenterC3×C5#1285
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
10. C3×#4002C4semidirect
12. C4⋊(C3⋊#168)C4semidirect
13. C5×C4⋊(C3⋊C8)C4semidirect
42. #4002Dic3semidirect
76. C4⋊(C3⋊C8)AGL(1,𝔽5)semidirect
201. C3×C5#1285semidirect
243. C5#67749semidirect
257. C3#10598585semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊ΓL(1,𝔽16)2
C3×#656832
#117468712
#105985853
#677495