Finite Group Databasebeta

← #14298854

Subgroups — #14298854

Order 1920.

432 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 126 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#11752348C211yesmaximal, characteristic
2C3×#65689C211yesmaximal, characteristic
3C4×D5×C3⋊C8C211yesmaximal, characteristic
4#11746529C214yesmaximal
5#10603846—31nomaximal, Hall
6C2×C4⋊ΓL(1,𝔽16)C2212yes—
7C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
8C3×C42×D5C2211yescharacteristic
9C2×D5×C3⋊C8C2212yes—
10#91235—22no—
11C3×(C2×AGL(1,𝔽5))⋊C4—22no—
12C4×C3⋊#168C411yescharacteristic, complemented
13C4×C5×C3⋊C8C411yescharacteristic, complemented
14Dic5×C3⋊C8C412yescomplemented
15#69547—51nomaximal, Hall
16#65689—31no—
17#65689D311yescharacteristic
18C4×C8×D5—31no—
19#64893—34no—
20C22×ΓL(1,𝔽16)—22no—
21C22×C3×AGL(1,𝔽5)—22no—
22C2×C3×C4×D5D412yes—
23C2×C3×C4×D5C2311yescharacteristic
24C4⋊ΓL(1,𝔽16)—24no—
25C3×C4⋊AGL(1,𝔽5)—24no—
26C3×C42×C5C2×C411yescharacteristic
27C2×C3⋊#168C2×C412yes—
28C2×C5×C3⋊C8C2×C412yes—
29C3×C4×Dic5C2×C411yescharacteristic
30C3×C4×Dic5Q812yes—
31D5×C3⋊C8—22no—
32D5×C3⋊C8—22no—
33C3×#382—51no—
34#4936—51no—
35C2×C4×C3⋊C8—51no—
36#4529—54no—
37C2×C4⋊AGL(1,𝔽5)—32no—
38C2×C4⋊AGL(1,𝔽5)C2×D312yes—
39C2×C8×D5—32no—
40C42×D5C2×D311yescharacteristic
41(C2×AGL(1,𝔽5))⋊C4—22no—
42(C2×AGL(1,𝔽5))⋊C4—62no—
43C4×C5×C8—31no—
44C8×Dic5—32no—
45C4×#168—31no—
46#1835—151noSylow
47C22×C3×D5C2×D411yescharacteristic
48C2×ΓL(1,𝔽16)—42no—
49C2×ΓL(1,𝔽16)—44no—
50C2×C3×AGL(1,𝔽5)—42no—
51C2×C3×AGL(1,𝔽5)—44no—
52C2×C3×C4×C5C4⋊C412yes—
53C2×C3×C4×C5C22×C411yescharacteristic
54C2×C3×Dic5C4⋊C412yes—
55C2×C3×Dic5C2×Q811yescharacteristic
56C3×C4×D5—24no—
57C3×C4×D5#4112yes—
58C3×C4×D5#4112yes—
59C3⋊#168—22no—
60C5×C3⋊C8—22no—
61C2×C3×C4⋊C4—52no—
62C2×C3×C42—51no—
63C2×C4⋊Dic3—52no—
64C22×C3⋊C8—52no—
65C3×#95—102no—
66C3⋊#95—102no—
67C4×C3⋊C8—51no—
68C4×C3⋊C8—52no—
69C4×C3⋊C8AGL(1,𝔽5)11yescharacteristic, complemented
70C22×AGL(1,𝔽5)—22no—
71C22×AGL(1,𝔽5)—62no—
72C2×C4×D5#6712yes—
73C2×C4×D5C22×D311yescharacteristic
74C4⋊AGL(1,𝔽5)—24no—
75C4⋊AGL(1,𝔽5)—64no—
76C2×C5×C8—32no—
77C42×C5C4×D311yescharacteristic
78C4×Dic5Dic612yes—
79C4×Dic5C4×D311yescharacteristic
80C2×#168—32no—
81C8×D5—62no—
82C8×D5—62no—
83C2×C4×C8—151no—
84#382—51no—
85#382—151no—
86#340—154no—
87C22×C3×C5C2×C4⋊C411yescharacteristic
88C2×C3×D5#12411yescharacteristic
89C2×C3×D5#12411yescharacteristic
90C2×C3×D5#13514yes—
91ΓL(1,𝔽16)—82no—
92C3×AGL(1,𝔽5)—82no—
93C3×C4×C5—22no—
94C3×C4×C5#11712yes—
95C3×Dic5—22no—
96C3×Dic5#12512yes—
97C22×C3×C4—51no—
98C22×C3×C4—52no—
99C22×C3×C4—102no—
100C22×Dic3—102no—
101C3×C4⋊C4—104no—
102C3×C42—52no—
103C3×C42—51no—
104C3×C42C2×AGL(1,𝔽5)11yescharacteristic
105C4⋊Dic3—104no—
106C2×C3⋊C8—52no—
107C2×C3⋊C8—102no—
108C2×C3⋊C8—102no—
109C2×C3⋊C8C2×AGL(1,𝔽5)12yes—
110C22×D5C2×#6711yescharacteristic
111C2×AGL(1,𝔽5)—42no—
112C2×AGL(1,𝔽5)—44no—
113C2×AGL(1,𝔽5)—122no—
114C2×AGL(1,𝔽5)—124no—
115C2×C4×C5Dic3⋊C412yes—
116C2×C4×C5C2×C4×D311yescharacteristic
117C2×Dic5Dic3⋊C412yes—
118C2×Dic5C2×Dic611yescharacteristic
119C4×D5—24no—
120C4×D5#23712yes—
121C4×D5#23912yes—
122C5×C8—62no—
123#168—62no—
124C22×C8—152no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4—152no—
127C2×C42—51no—
128C4×C8—31no—
129C4×C8—152no—
130C4×C8—151no—
131#95—102no—
132#95—302no—
133C2×C3×C5#38211yescharacteristic
134C2×C3×C5#42712yes—
135C3×D5#48812yes—
136C3×D5#48812yes—
137C23×C3—51no—
138C2×C3×C4—51no—
139C2×C3×C4—52no—
140C2×C3×C4—52no—
141C2×C3×C4—52no—
142C2×C3×C4—104no—
143C2×C3×C4—202no—
144C2×C3×C4—204no—
145C2×C3×C4C4⋊AGL(1,𝔽5)12yes—
146C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
147C2×Dic3—202no—
148C2×Dic3—204no—
149C3⋊C8—22no—
150C3⋊C8—102no—
151C22×C5C2×Dic3⋊C411yescharacteristic
152C2×D5#93611yescharacteristic
153C2×D5#94711yescharacteristic
154C2×D5#95014yes—
155AGL(1,𝔽5)—82no—
156AGL(1,𝔽5)—242no—
157C4×C5—22no—
158C4×C5#89212yes—
159Dic5—22no—
160Dic5#89012yes—
161C22×C4—51no—
162C22×C4—52no—
163C22×C4—102no—
164C22×C4—302no—
165C2×C8—32no—
166C2×C8—152no—
167C2×C8—302no—
168C2×C8—302no—
169C4⋊C4—104no—
170C4⋊C4—304no—
171C42—52no—
172C42—51no—
173C42D3×AGL(1,𝔽5)11yescharacteristic
174C3×C5#183511yescharacteristic, complemented, Hall
175C22×C3—51no—
176C22×C3—51no—
177C22×C3—54no—
178C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
179C3×C4—22no—
180C3×C4—52no—
181C3×C4—102no—
182C3×C4—402no—
183C3×C4#413012yes—
184Dic3—402no—
185C2×C5#497711yescharacteristic
186C2×C5#499812yes—
187D5#505612yes—
188D5#506812yes—
189C23—51no—
190C2×C4—51no—
191C2×C4—52no—
192C2×C4—52no—
193C2×C4—52no—
194C2×C4—104no—
195C2×C4—202no—
196C2×C4—204no—
197C2×C4—602no—
198C2×C4—604no—
199C2×C4Dic3⋊AGL(1,𝔽5)12yes—
200C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
201C8—62no—
202C8—302no—
203C2×C3—52no—
204C2×C3—52no—
205C2×C3#6568911yescharacteristic
206C2×C3#6572112yes—
207C5#6954711yescharacteristic, complemented, Sylow
208C22—54no—
209C22—51no—
210C22—51no—
211C22C2×Dic3⋊AGL(1,𝔽5)11yescharacteristic
212C4—22no—
213C4—52no—
214C4—102no—
215C4—402no—
216C4—1202no—
217C4#9190712yes—
218C3#1060384611yescharacteristic, complemented, Sylow
219C2—52no—
220C2—52no—
221C2#1175172612yes—
222C2#1175175111yescharacteristic
subgroupG/H
centerC22C2×Dic3⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C42×C5C2×C4
socleC22×C3×C5C2×C4⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1835
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
12. C4×C3⋊#168C4semidirect
13. C4×C5×C3⋊C8C4semidirect
14. Dic5×C3⋊C8C4semidirect
69. C4×C3⋊C8AGL(1,𝔽5)semidirect
174. C3×C5#1835semidirect
207. C5#69547semidirect
218. C3#10603846semidirect

Maximal subgroups

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

subgroupindex
#117523482
C3×#656892
C4×D5×C3⋊C82
#117465292
#106038463
#695475