Finite Group Databasebeta

← #14298868

Subgroups — #14298868

Order 1920.

448 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 137 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
2#11752347C211yesmaximal, characteristic
3C3×D5×#97C211yesmaximal, characteristic
4#11746764C214yesmaximal
5#10603882—31nomaximal, Hall
6C2×C4⋊ΓL(1,𝔽16)—22no—
7C2×C4⋊ΓL(1,𝔽16)C2212yes—
8C2×C4⋊ΓL(1,𝔽16)C2212yes—
9C2×C3×C8×D5C2212yes—
10C3×C42×D5C2211yescharacteristic
11#91235—22no—
12C3×C5×#97C411yescharacteristic, complemented
13C3×#3947C412yescomplemented
14C3×#3935C411yescharacteristic, complemented
15#69815—51nomaximal, Hall
16#65689—31no—
17#65688—31no—
18D5×#97D311yescharacteristic
19#64893—34no—
20C22×ΓL(1,𝔽16)—22no—
21C22×ΓL(1,𝔽16)—22no—
22C2×C3×C4×D5D412yes—
23C2×C3×C4×D5C2311yescharacteristic
24C4⋊ΓL(1,𝔽16)—24no—
25C4⋊ΓL(1,𝔽16)—24no—
26C4⋊ΓL(1,𝔽16)—44no—
27C2×C3×C5×C8C2×C412yes—
28C3×C42×C5C2×C411yescharacteristic
29C3×C4×Dic5C2×C411yescharacteristic
30C3×C4×Dic5Q812yes—
31C2×C3×#168C2×C412yes—
32C3×C8×D5—22no—
33C3×C8×D5—22no—
34C2×C3×#97—51no—
35#4936—51no—
36#4935—51no—
37#4551—54no—
38C2×C4⋊AGL(1,𝔽5)—32no—
39C2×C4⋊AGL(1,𝔽5)—32no—
40C2×C4⋊AGL(1,𝔽5)—62no—
41C2×C8×D5C2×D312yes—
42C42×D5C2×D311yescharacteristic
43(C2×AGL(1,𝔽5))⋊C4—62no—
44C5×#97Dic311yescharacteristic, complemented
45#3947Dic312yescomplemented
46#3935Dic311yescharacteristic, complemented
47#1836—151noSylow
48C22×C3×D5C2×D411yescharacteristic
49C2×ΓL(1,𝔽16)—42no—
50C2×ΓL(1,𝔽16)—42no—
51C2×ΓL(1,𝔽16)—44no—
52C2×ΓL(1,𝔽16)—42no—
53C2×ΓL(1,𝔽16)—42no—
54C2×C3×C4×C5C4⋊C412yes—
55C2×C3×C4×C5C22×C411yescharacteristic
56C2×C3×Dic5C4⋊C412yes—
57C2×C3×Dic5C2×Q811yescharacteristic
58C3×C4×D5—22no—
59C3×C4×D5—22no—
60C3×C4×D5#4112yes—
61C3×C4×D5#4112yes—
62C3×C5×C8—22no—
63C3×#168—22no—
64C22×C3×C8—52no—
65C2×C3×C42—51no—
66C2×C4⋊Dic3—52no—
67C2×C4⋊Dic3—52no—
68C2×C4⋊Dic3—102no—
69C3×#97—52no—
70C3×#97—51no—
71C3×#97AGL(1,𝔽5)11yescharacteristic, complemented
72C3⋊#95—102no—
73C22×AGL(1,𝔽5)—62no—
74C22×AGL(1,𝔽5)—62no—
75C2×C4×D5D1212yes—
76C2×C4×D5C22×D311yescharacteristic
77C4⋊AGL(1,𝔽5)—64no—
78C4⋊AGL(1,𝔽5)—64no—
79C4⋊AGL(1,𝔽5)—124no—
80C2×C5×C8C2×Dic312yes—
81C42×C5C2×Dic311yescharacteristic
82C4×Dic5Dic612yes—
83C4×Dic5C2×Dic311yescharacteristic
84C2×#168C2×Dic312yes—
85C8×D5—22no—
86C8×D5—22no—
87C2×#97—51no—
88#384—151no—
89#382—151no—
90#340—154no—
91C22×C3×C5C2×C4⋊C411yescharacteristic
92C2×C3×D5#12411yescharacteristic
93C2×C3×D5#12411yescharacteristic
94C2×C3×D5AGL(1,ℤ/8)12yes—
95C2×C3×D5#13712yes—
96ΓL(1,𝔽16)—82no—
97ΓL(1,𝔽16)—82no—
98C3×C4×C5—22no—
99C3×C4×C5#11712yes—
100C3×Dic5—22no—
101C3×Dic5#12512yes—
102C22×C3×C4—51no—
103C22×C3×C4—52no—
104C22×Dic3—102no—
105C22×Dic3—102no—
106C2×C3×C8—52no—
107C2×C3×C8—102no—
108C2×C3×C8—102no—
109C2×C3×C8C2×AGL(1,𝔽5)12yes—
110C3×C42—52no—
111C3×C42—51no—
112C3×C42C2×AGL(1,𝔽5)11yescharacteristic
113C4⋊Dic3—104no—
114C4⋊Dic3—104no—
115C4⋊Dic3—204no—
116C22×D5C2×D1211yescharacteristic
117C2×AGL(1,𝔽5)—122no—
118C2×AGL(1,𝔽5)—122no—
119C2×AGL(1,𝔽5)—124no—
120C2×AGL(1,𝔽5)—122no—
121C2×AGL(1,𝔽5)—122no—
122C2×C4×C5C4⋊Dic312yes—
123C2×C4×C5C22×Dic311yescharacteristic
124C2×Dic5C4⋊Dic312yes—
125C2×Dic5C2×Dic611yescharacteristic
126C4×D5—22no—
127C4×D5—22no—
128C4×D5#23512yes—
129C4×D5#23512yes—
130C5×C8—22no—
131#168—22no—
132C22×C8—52no—
133C2×C4⋊C4—152no—
134C2×C4⋊C4—152no—
135C2×C4⋊C4—302no—
136C2×C42—51no—
137#97—52no—
138#97—51no—
139#97ΓL(1,𝔽16)11yescharacteristic, complemented
140#95—302no—
141C2×C3×C5#38211yescharacteristic
142C2×C3×C5#42812yes—
143C3×D5#48912yes—
144C3×D5#49011yescharacteristic
145C3×D5#49111yescharacteristic
146C23×C3—51no—
147C2×C3×C4—51no—
148C2×C3×C4—52no—
149C2×C3×C4—52no—
150C2×C3×C4—52no—
151C2×C3×C4—102no—
152C2×C3×C4—102no—
153C2×C3×C4C4⋊AGL(1,𝔽5)12yes—
154C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
155C2×Dic3—202no—
156C2×Dic3—202no—
157C2×Dic3—204no—
158C2×Dic3—202no—
159C2×Dic3—202no—
160C3×C8—22no—
161C3×C8—102no—
162C22×C5C2×C4⋊Dic311yescharacteristic
163C2×D5#87511yescharacteristic
164C2×D5#87511yescharacteristic
165C2×D5#90812yes—
166C2×D5#90912yes—
167AGL(1,𝔽5)—242no—
168AGL(1,𝔽5)—242no—
169C4×C5—22no—
170C4×C5#92612yes—
171Dic5—22no—
172Dic5#87012yes—
173C22×C4—51no—
174C22×C4—52no—
175C22×C4—302no—
176C22×C4—302no—
177C2×C8—52no—
178C2×C8—102no—
179C2×C8—102no—
180C2×C8C2×ΓL(1,𝔽16)12yes—
181C4⋊C4—304no—
182C4⋊C4—304no—
183C4⋊C4—604no—
184C42—52no—
185C42—51no—
186C42C2×ΓL(1,𝔽16)11yescharacteristic
187C3×C5#183611yescharacteristic, complemented, Hall
188C22×C3—51no—
189C22×C3—51no—
190C22×C3—52no—
191C22×C3—52no—
192C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
193C3×C4—22no—
194C3×C4—52no—
195C3×C4—102no—
196C3×C4#413012yes—
197Dic3—402no—
198Dic3—402no—
199C2×C5#493611yescharacteristic
200C2×C5#512112yes—
201D5#470311yescharacteristic
202D5#471311yescharacteristic
203D5#471412yes—
204C23—51no—
205C2×C4—51no—
206C2×C4—52no—
207C2×C4—52no—
208C2×C4—52no—
209C2×C4—102no—
210C2×C4—102no—
211C2×C4—602no—
212C2×C4—602no—
213C2×C4—604no—
214C2×C4—602no—
215C2×C4—602no—
216C2×C4C4⋊ΓL(1,𝔽16)12yes—
217C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
218C8—22no—
219C8—102no—
220C2×C3—51no—
221C2×C3—52no—
222C2×C3—51no—
223C2×C3#6568911yescharacteristic
224C2×C3#6572712yes—
225C5#6981511yescharacteristic, complemented, Sylow
226C22—51no—
227C22—51no—
228C22—52no—
229C22—52no—
230C22C2×C4⋊ΓL(1,𝔽16)11yescharacteristic
231C4—22no—
232C4—52no—
233C4—102no—
234C4—1202no—
235C4—1202no—
236C4#9198912yes—
237C3#1060388211yescharacteristic, complemented, Sylow
238C2—51no—
239C2—52no—
240C2—51no—
241C2#1175234811yescharacteristic
242C2#1175238612yes—
subgroupG/H
centerC22C2×C4⋊ΓL(1,𝔽16)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#97C4
socleC22×C3×C5C2×C4⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenter#97ΓL(1,𝔽16)
hypocenterC3×C5#1836
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
12. C3×C5×#97C4semidirect
13. C3×#3947C4semidirect
14. C3×#3935C4semidirect
44. C5×#97Dic3semidirect
45. #3947Dic3semidirect
46. #3935Dic3semidirect
71. C3×#97AGL(1,𝔽5)semidirect
139. #97ΓL(1,𝔽16)semidirect
187. C3×C5#1836semidirect
225. C5#69815semidirect
237. C3#10603882semidirect

Maximal subgroups

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

subgroupindex
#117523482
#117523472
C3×D5×#972
#117467642
#106038823
#698155