Finite Group Databasebeta

← #14307415

Subgroups — #14307415

Order 1920.

568 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 133 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×#66022C211yesmaximal, characteristic
2C42×ΓL(1,𝔽16)C211yesmaximal, characteristic
3#11752341C211yesmaximal, characteristic
4#11746746C214yesmaximal
5#10604587—31nomaximal, Hall
6C2×C4×ΓL(1,𝔽16)—22no—
7C2×C4×ΓL(1,𝔽16)C2212yes—
8C2×#6758C2212yes—
9C3×C5×#125C411yescharacteristic, complemented
10C3×#4042—22no—
11C3×#4038C2212yes—
12C3×#4034C412yescomplemented
13C3×#4033—22no—
14C3×C42×D5C2211yescharacteristic
15C3×#4016C411yescharacteristic, complemented
16#91233—22no—
17#71839—51nomaximal, Hall
18#66022D311yescharacteristic
19C42×AGL(1,𝔽5)—31no—
20#65682—31no—
21#64875—34no—
22C22×ΓL(1,𝔽16)—22no—
23C2×C3×C4×D5D412yes—
24C2×C3×C4×D5C2311yescharacteristic
25C2×C3×Dic10—22no—
26C2×C3⋊#170—22no—
27C4×ΓL(1,𝔽16)—28no—
28C4×ΓL(1,𝔽16)—22no—
29C4×ΓL(1,𝔽16)Q814yescomplemented
30#6758—24no—
31C3×C5×C4⋊C4—22no—
32C3×C5×C4⋊C4C2×C412yes—
33C3×C42×C5C2×C411yescharacteristic
34C3×#685—22no—
35C3×C4⋊Dic5C2×C412yes—
36C3×#683—22no—
37C3×#683—22no—
38C3×C4×Dic5—22no—
39C3×C4×Dic5C2×C411yescharacteristic
40C3×C4×Dic5D412yes—
41C3×#533—51no—
42C42×Dic3—51no—
43#4924—51no—
44#4532—54no—
45C2×C4×AGL(1,𝔽5)—32no—
46C2×C4×AGL(1,𝔽5)—62no—
47C2×#700—32no—
48C5×#125Dic311yescharacteristic, complemented
49#4042—22no—
50#4038C2×D312yes—
51#4034Dic312yescomplemented
52#4033—22no—
53C42×D5C2×D311yescharacteristic
54#4016Dic311yescharacteristic, complemented
55#4004—62no—
56#1994—151noSylow
57C22×C3×D5C2×D411yescharacteristic
58C2×ΓL(1,𝔽16)—28no—
59C2×ΓL(1,𝔽16)—24no—
60C2×C3×C4×C5—22no—
61C2×C3×C4×C5C22⋊C412yes—
62C2×C3×C4×C5C22×C411yescharacteristic
63C2×C3×Dic5—22no—
64C2×C3×Dic5C22⋊C412yes—
65C2×C3×Dic5C2×D411yescharacteristic
66C3×C4×D5—24no—
67C3×C4×D5C2×Q812yes—
68C3×C4×D5#4112yes—
69C3×Dic10—44no—
70C3⋊#170—42no—
71C3×C4⋊Q8—52no—
72C3×#125—51no—
73C3×#125AGL(1,𝔽5)11yescharacteristic, complemented
74C3×C22⋊Q8—102no—
75C3×C4×Q8—102no—
76C3×#117—52no—
77C2×C3×C42—51no—
78C2×C4×Dic3—52no—
79C2×C4×Dic3—102no—
80C2×#208—52no—
81C4⋊(C3⋊C8)—102no—
82C22×AGL(1,𝔽5)—62no—
83C2×C4×D5#6712yes—
84C2×C4×D5C22×D311yescharacteristic
85C2×Dic10—22no—
86C2×#170—62no—
87C4×AGL(1,𝔽5)—34no—
88C4×AGL(1,𝔽5)—68no—
89C4×AGL(1,𝔽5)—62no—
90#700—64no—
91C5×C4⋊C4—22no—
92C5×C4⋊C4C2×Dic312yes—
93C42×C5C2×Dic311yescharacteristic
94#685—22no—
95C4⋊Dic5C2×Dic312yes—
96#683—22no—
97#683—22no—
98C4×Dic5—22no—
99C4×Dic5C2×Dic311yescharacteristic
100C4×Dic5#6712yes—
101#533—51no—
102#423—151no—
103C43—151no—
104#339—154no—
105C22×C3×C5C2×C22⋊C411yescharacteristic
106C2×C3×D5—22no—
107C2×C3×D5C22⋊Q812yes—
108ΓL(1,𝔽16)—28no—
109C3×C4×C5—22no—
110C3×C4×C5—42no—
111C3×C4×C5C4×Q812yes—
112C3×Dic5—42no—
113C3×Dic5#10414yes—
114C3×Dic5#12411yescharacteristic
115C3×Dic5C4⋊Q811yescharacteristic
116C2×C3×Q8—102no—
117C22×C3×C4—51no—
118C22×C3×C4—52no—
119C22×Dic3—102no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—52no—
122C3×C4⋊C4—102no—
123C3×C4⋊C4—102no—
124C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
125C3×C22⋊C4—102no—
126C3×C42—52no—
127C3×C42—51no—
128C3×C42—102no—
129C3×C42C2×AGL(1,𝔽5)11yescharacteristic
130C4×Dic3—54no—
131C4×Dic3—102no—
132C4×Dic3—108no—
133#208—104no—
134C2×C3⋊C8—102no—
135C22×D5C2×#6711yescharacteristic
136C2×AGL(1,𝔽5)—68no—
137C2×AGL(1,𝔽5)—64no—
138C2×C4×C5—22no—
139C2×C4×C5#21712yes—
140C2×C4×C5C22×Dic311yescharacteristic
141C2×Dic5—22no—
142C2×Dic5#21712yes—
143C2×Dic5C2×#6711yescharacteristic
144C4×D5—24no—
145C4×D5D3×Q812yes—
146C4×D5#23912yes—
147Dic10—44no—
148#170—122no—
149C2×M16—152no—
150C4⋊Q8—52no—
151#125—51no—
152#125ΓL(1,𝔽16)11yescharacteristic, complemented
153C22⋊Q8—102no—
154C4×Q8—102no—
155#117—52no—
156C2×C42—51no—
157C2×C42—152no—
158C2×C42—302no—
159C4⋊C8—302no—
160C2×C3×C5#39111yescharacteristic
161C2×C3×C5C2×#10412yes—
162C3×D5—22no—
163C23×C3—51no—
164C3×Q8—204no—
165C2×C3×C4—22no—
166C2×C3×C4—52no—
167C2×C3×C4—51no—
168C2×C3×C4—52no—
169C2×C3×C4—52no—
170C2×C3×C4—104no—
171C2×C3×C4—102no—
172C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
173C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
174C2×Dic3—104no—
175C2×Dic3—108no—
176C3⋊C8—202no—
177C22×C5C2×#21711yescharacteristic
178C2×D5—22no—
179C2×D5#94612yes—
180AGL(1,𝔽5)—68no—
181C4×C5—22no—
182C4×C5—42no—
183C4×C5Q8×Dic312yes—
184Dic5—42no—
185Dic5#83714yes—
186Dic5#94411yescharacteristic
187Dic5#94711yescharacteristic
188C2×Q8—102no—
189C22×C4—51no—
190C22×C4—52no—
191C22×C4—302no—
192M16—304no—
193C2×C8—302no—
194C4⋊C4—22no—
195C4⋊C4—52no—
196C4⋊C4—102no—
197C4⋊C4—102no—
198C4⋊C4C2×ΓL(1,𝔽16)12yes—
199C22⋊C4—102no—
200C42—52no—
201C42—51no—
202C42—102no—
203C42—154no—
204C42—302no—
205C42—308no—
206C42C2×ΓL(1,𝔽16)11yescharacteristic
207C3×C5#199411yescharacteristic, complemented, Hall
208C22×C3—52no—
209C22×C3—102no—
210C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
211C3×C4—22no—
212C3×C4—42no—
213C3×C4—54no—
214C3×C4—51no—
215C3×C4—51no—
216C3×C4—202no—
217C3×C4Q8×AGL(1,𝔽5)12yes—
218Dic3—108no—
219C2×C5#522911yescharacteristic
220C2×C5C2×#83712yes—
221D5—22no—
222C23—51no—
223Q8—204no—
224C2×C4—22no—
225C2×C4—51no—
226C2×C4—52no—
227C2×C4—52no—
228C2×C4—52no—
229C2×C4—104no—
230C2×C4—102no—
231C2×C4—304no—
232C2×C4—308no—
233C2×C4#676312yes—
234C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
235C8—602no—
236C2×C3—102no—
237C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
238C2×C3D5⋊#10412yes—
239C5#7183911yescharacteristic, complemented, Sylow
240C22—52no—
241C22—102no—
242C22C2×#676311yescharacteristic
243C4—22no—
244C4—42no—
245C4—51no—
246C4—51no—
247C4—54no—
248C4—202no—
249C4—308no—
250C4Q8×ΓL(1,𝔽16)12yes—
251C3#1060458711yescharacteristic, complemented, Sylow
252C2—102no—
253C2#1175244911yescharacteristic
254C2D5⋊#83712yes—
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#125C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenter#125ΓL(1,𝔽16)
hypocenterC3×C5#1994
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
9. C3×C5×#125C4semidirect
12. C3×#4034C4semidirect
15. C3×#4016C4semidirect
29. C4×ΓL(1,𝔽16)Q8semidirect
48. C5×#125Dic3semidirect
51. #4034Dic3semidirect
54. #4016Dic3semidirect
73. C3×#125AGL(1,𝔽5)semidirect
152. #125ΓL(1,𝔽16)semidirect
207. C3×C5#1994semidirect
239. C5#71839semidirect
251. C3#10604587semidirect

Maximal subgroups

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

subgroupindex
C3×#660222
C42×ΓL(1,𝔽16)2
#117523412
#117467462
#106045873
#718395