Finite Group Databasebeta

← #14285231

Subgroups — #14285231

Order 1920.

664 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 109 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×D5×#127C211yesmaximal, characteristic
2#11752343C212yesmaximal, complemented
3#11746796C214yesmaximal, complemented
4#10604610—31nomaximal, Hall
5C2×C3×D4×D5—22no—
6C2×C3×D4×D5C412yes—
7C3×C5×#127C411yescharacteristic
8C3×#4086—22no—
9C3×#4086C2212yes—
10C3×#4020C411yescharacteristic
11C3×C42×D5C2211yescharacteristic
12#91233C2214yescomplemented
13#91232—22no—
14#91231—22no—
15#71760—51nomaximal, Hall
16D5×#127D311yescharacteristic
17#65684—32no—
18#64925—34no—
19C23×C3×D5—22no—
20C2×C3×C5×D4—22no—
21C2×C3×C5×D4C2×C412yes—
22C2×C3×C22⋊D5—22no—
23C2×C3×C22⋊D5—24no—
24C3×D4×D5—48no—
25C3×D4×D5—44no—
26C2×C3×D20—22no—
27C2×C3×D20C2×C412yes—
28C2×C3×C4×D5C2×C411yescharacteristic
29C2×C3×C4×D5D412yes—
30C2×C3⋊#170—24no—
31C3×C42×C5C2×C411yescharacteristic
32C3×C4×Dic5D412yes—
33C3×C4×Dic5C2311yescharacteristic
34C2×C3×#127—51no—
35#4926—52no—
36#4544—54no—
37C2×D4×D5—22no—
38C2×D4×D5Dic312yes—
39C5×#127Dic311yescharacteristic
40#4086—22no—
41#4086C2×D312yes—
42#4020Dic311yescharacteristic
43C42×D5C2×D311yescharacteristic
44#4004—34no—
45#4003—62no—
46#4002—62no—
47#1555—151noSylow
48C23×C3×C5—22no—
49C22×C3×D5—22no—
50C22×C3×D5—48no—
51C22×C3×D5—44no—
52C22×C3×D5C22⋊C411yescharacteristic
53C3×C5×D4—44no—
54C3×C5×D4—42no—
55C2×C3×C4×C5C22⋊C412yes—
56C2×C3×C4×C5C22×C411yescharacteristic
57C3×C22⋊D5—44no—
58C3×C22⋊D5—48no—
59C2×C3×Dic5C22⋊C411yescharacteristic
60C2×C3×Dic5C2×D412yes—
61C3×D20—44no—
62C3×D20—42no—
63C3×C4×D5—22no—
64C3×C4×D5—24no—
65C3⋊#170—44no—
66C22×C3×D4—52no—
67C22×C3×D4—102no—
68C3×#127—52no—
69C3×#127—51no—
70C3×#127—102no—
71C3×#127AGL(1,𝔽5)11yescharacteristic
72C2×C3×C42—51no—
73#830—102no—
74C4⋊(C3⋊C8)—54no—
75#803—102no—
76C23×D5—22no—
77C2×C5×D4—22no—
78C2×C5×D4C2×Dic312yes—
79C2×C22⋊D5—22no—
80C2×C22⋊D5—24no—
81D4×D5—44no—
82D4×D5—48no—
83C2×D20—22no—
84C2×D20C2×Dic312yes—
85C2×C4×D5C2×Dic311yescharacteristic
86C2×C4×D5#6712yes—
87C2×#170—64no—
88C42×C5C2×Dic311yescharacteristic
89C4×Dic5#6712yes—
90C4×Dic5C22×D311yescharacteristic
91C2×#127—51no—
92#432—152no—
93#331—154no—
94C22×C3×C5—42no—
95C22×C3×C5—44no—
96C22×C3×C5C2×C22⋊C411yescharacteristic
97C2×C3×D5—22no—
98C2×C3×D5—42no—
99C2×C3×D5—44no—
100C2×C3×D5—88no—
101C2×C3×D5#10012yes—
102C3×C4×C5—21no—
103C3×C4×C5—22no—
104C3×Dic5—21no—
105C3×Dic5AGL(1,ℤ/8)14yes—
106C24×C3—102no—
107C2×C3×D4—22no—
108C2×C3×D4—52no—
109C2×C3×D4—104no—
110C2×C3×D4—102no—
111C2×C3×D4—102no—
112C2×C3×D4—204no—
113C2×C3×D4—208no—
114C2×C3×D4C2×AGL(1,𝔽5)12yes—
115C22×C3×C4—52no—
116C22×C3×C4—51no—
117C3×C42—52no—
118C3×C42—51no—
119C3×C42C2×AGL(1,𝔽5)11yescharacteristic
120C2×C3⋊C8—104no—
121C23×C5—22no—
122C22×D5—22no—
123C22×D5—48no—
124C22×D5—44no—
125C22×D5#21711yescharacteristic
126C5×D4—42no—
127C5×D4—44no—
128C2×C4×C5#21712yes—
129C2×C4×C5C22×Dic311yescharacteristic
130C22⋊D5—44no—
131C22⋊D5—48no—
132C2×Dic5#21711yescharacteristic
133C2×Dic5C2×#6712yes—
134D20—42no—
135D20—44no—
136C4×D5—22no—
137C4×D5—24no—
138#170—124no—
139C22×D4—52no—
140C22×D4—102no—
141#127—52no—
142#127—51no—
143#127—102no—
144#127ΓL(1,𝔽16)11yescharacteristic
145C2×C42—51no—
146C4⋊C8—154no—
147C22⋊C8—302no—
148#97—302no—
149C2×C3×C5—82no—
150C2×C3×C5C2×#10011yescharacteristic
151C2×C3×C5#41812yes—
152C3×D5—22no—
153C3×D5—82no—
154C23×C3—22no—
155C23×C3—51no—
156C23×C3—102no—
157C23×C3—204no—
158C23×C3—208no—
159C3×D4—42no—
160C3×D4—44no—
161C3×D4—202no—
162C3×D4—204no—
163C3×D4—208no—
164C3×D4—204no—
165C2×C3×C4—52no—
166C2×C3×C4—51no—
167C2×C3×C4—104no—
168C2×C3×C4—102no—
169C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
170C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
171C3⋊C8—204no—
172C22×C5—44no—
173C22×C5—42no—
174C22×C5C2×#21711yescharacteristic
175C2×D5—22no—
176C2×D5—44no—
177C2×D5—42no—
178C2×D5—88no—
179C2×D5#83312yes—
180C4×C5—22no—
181C4×C5—21no—
182Dic5—21no—
183Dic5#93214yes—
184C24—102no—
185C2×D4—22no—
186C2×D4—52no—
187C2×D4—104no—
188C2×D4—102no—
189C2×D4—102no—
190C2×D4—204no—
191C2×D4—208no—
192C2×D4C2×ΓL(1,𝔽16)12yes—
193C22×C4—52no—
194C22×C4—51no—
195C2×C8—304no—
196C42—52no—
197C42—51no—
198C42C2×ΓL(1,𝔽16)11yescharacteristic
199C3×C5#155511yescharacteristic, complemented, Hall
200C22×C3—44no—
201C22×C3—42no—
202C22×C3—52no—
203C22×C3—102no—
204C22×C3—204no—
205C22×C3—202no—
206C22×C3—408no—
207C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
208C3×C4—22no—
209C3×C4—21no—
210C3×C4—54no—
211C3×C4—101no—
212C2×C5—82no—
213C2×C5#521512yes—
214C2×C5C2×#83311yescharacteristic
215D5—22no—
216D5—82no—
217C23—22no—
218C23—51no—
219C23—102no—
220C23—204no—
221C23—208no—
222D4—42no—
223D4—44no—
224D4—202no—
225D4—204no—
226D4—208no—
227D4—204no—
228C2×C4—52no—
229C2×C4—51no—
230C2×C4—104no—
231C2×C4—102no—
232C2×C4#676312yes—
233C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
234C8—604no—
235C2×C3—82no—
236C2×C3—102no—
237C2×C3—402no—
238C2×C3#6576712yes—
239C2×C3D5⋊#10011yescharacteristic
240C5#7176011yescharacteristic, complemented, Sylow
241C22—44no—
242C22—42no—
243C22—52no—
244C22—102no—
245C22—204no—
246C22—202no—
247C22—408no—
248C22C2×#676311yescharacteristic
249C4—22no—
250C4—21no—
251C4—54no—
252C4—101no—
253C3#1060461011yescharacteristic, complemented, Sylow
254C2—82no—
255C2—102no—
256C2—402no—
257C2#1175242612yes—
258C2D5⋊#83311yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#127C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×D5×#127C2
cosocleC3×C4×Dic5C23
hypercenter#127ΓL(1,𝔽16)
hypocenterC3×C5#1555
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
2. #11752343C2semidirect
3. #11746796C2semidirect
12. #91233C22semidirect
199. C3×C5#1555semidirect
240. C5#71760semidirect
253. C3#10604610semidirect

Maximal subgroups

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

subgroupindex
C3×D5×#1272
#117523432
#117467962
#106046103
#717605