Finite Group Databasebeta

← #14315678

Subgroups — #14315678

Order 1920.

292 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 66 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#11751689C212yesmaximal, complemented
2#11746834C211yesmaximal, characteristic
3C16×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
4C3×#64851C211yesmaximal, characteristic, complemented
5#11746461C212yesmaximal, complemented
6#10604136—31nomaximal, Hall
7#91916—22no—
8#91271C2212yes—
9C8×ΓL(1,𝔽16)C2211yescharacteristic
10D5⋊(C3⋊C16)C2211yescharacteristic
11C3×#3996C2212yes—
12#91157—22no—
13D240C411yescharacteristic, complemented
14C5×Dic24C411yescharacteristic, complemented
15C3×D5×C16C2211yescharacteristic
16#90945C412yescomplemented
17#69357—51nomaximal, Hall
18#65740—32no—
19#65208—31no—
20#64909—32no—
21#64851D311yescharacteristic, complemented
22C16×AGL(1,𝔽5)—31no—
23#6773—22no—
24C4×ΓL(1,𝔽16)D411yescharacteristic
25D5⋊(C3⋊C8)D411yescharacteristic
26C3×#700—22no—
27#6740—42no—
28#6738—42no—
29D120C2×C412yes—
30C5×Dic12C2×C412yes—
31C3×C8×D5C2311yescharacteristic
32#6660—22no—
33C3⋊#674—21no—
34C3×C5×C16C2×C411yescharacteristic
35C3×#672C2×C411yescharacteristic
36#4904—51no—
37#4895—52no—
38C3×#368—51no—
39Dic3×C16—51no—
40#4497—52no—
41#4133—62no—
42#4065—32no—
43#4010—62no—
44#3996C2×D312yes—
45C8×AGL(1,𝔽5)—31no—
46D5⋊C16—31no—
47C5×Q32—31no—
48#3960—32no—
49D80—31no—
50D5×C16C2×D311yescharacteristic
51#2178—151noSylow
52C2×ΓL(1,𝔽16)—21no—
53D60—22no—
54C5×Dic6—22no—
55C3×C4×D5C2×D411yescharacteristic
56#1202—42no—
57#1200—42no—
58C3⋊#170D811yescharacteristic
59C3⋊#170D811yescharacteristic
60C3×#170—42no—
61C3×C5×C8C22×C411yescharacteristic
62C3×#168#4111yescharacteristic
63#907—102no—
64#904—52no—
65C2×C3×C16—51no—
66C3×#108—52no—
67#825—102no—
68C8×Dic3—51no—
69C2×C3⋊C16—51no—
70Dic24AGL(1,𝔽5)11yescharacteristic, complemented
71#800—52no—
72D48—51no—
73#713—62no—
74C4×AGL(1,𝔽5)—31no—
75#700—22no—
76#700—122no—
77D5⋊C8—31no—
78D5⋊C8—122no—
79C5×Q16—32no—
80#688—62no—
81D40—32no—
82C8×D5C22×D311yescharacteristic
83#674—61no—
84C5×C16C4×D311yescharacteristic
85#672C4×D311yescharacteristic
86#508—151no—
87#443—152no—
88#368—51no—
89#359—152no—
90C4×C16—151no—
91C2×D15—42no—
92C2×C3×D5#13511yescharacteristic
93ΓL(1,𝔽16)—22no—
94C3×C4×C5C4×D411yescharacteristic
95C3×Dic5C2×D811yescharacteristic
96C5×Dic3—42no—
97#235—102no—
98C3×M16—102no—
99C2×C3×C8—51no—
100C4×Dic3—51no—
101C2×C3⋊C8—51no—
102Dic12C2×AGL(1,𝔽5)12yes—
103D24—52no—
104#204—102no—
105ΓL(1,𝔽25)—202no—
106D3×C8—202no—
107C3×C16—51no—
108C3×C16C2×AGL(1,𝔽5)11yescharacteristic
109C3⋊C16—101no—
110C2×AGL(1,𝔽5)—61no—
111C5×Q8—62no—
112D20—62no—
113D20—122no—
114C4×D5—122no—
115C4×D5D3×D411yescharacteristic
116#170—31no—
117#170—31no—
118#170—42no—
119C5×C8C2×C4×D311yescharacteristic
120#168#23911yescharacteristic
121#135—152no—
122#131—302no—
123Q32—31no—
124SD32—152no—
125D16—151no—
126C2×C16—51no—
127C2×C16—151no—
128#108—52no—
129#104—302no—
130C4×C8—151no—
131C2×C3×C5C4×D811yescharacteristic
132D15—82no—
133C3×D5—21no—
134C2×C3×C4—51no—
135#67—202no—
136C2×Dic3—101no—
137D12—102no—
138C4×D3—202no—
139Dic6—22no—
140C3×C8—51no—
141C3×C8—202no—
142C3×C8C22×AGL(1,𝔽5)11yescharacteristic
143C3⋊C8—51no—
144C3⋊C8—51no—
145C2×D5—122no—
146C2×D5#91911yescharacteristic
147AGL(1,𝔽5)—62no—
148C4×C5—122no—
149C4×C5#89311yescharacteristic
150Dic5D3×D811yescharacteristic
151#41—302no—
152Q16—32no—
153SD16—302no—
154D8—152no—
155M16—102no—
156M16—602no—
157C2×C8—51no—
158C2×C8—151no—
159C2×C8—602no—
160C42—151no—
161C16—51no—
162C16—301no—
163C16D3×AGL(1,𝔽5)11yescharacteristic
164C3×C5#217811yescharacteristic, complemented, Hall
165C22×C3—51no—
166C2×D3—202no—
167C3×C4—51no—
168C3×C4D4×AGL(1,𝔽5)11yescharacteristic
169Dic3—42no—
170Dic3—102no—
171C2×C5#487211yescharacteristic
172D5—21no—
173D5—242no—
174Q8—62no—
175D4—302no—
176D4—602no—
177C2×C4—51no—
178C2×C4—301no—
179C2×C4—602no—
180C8—51no—
181C8—151no—
182C8—151no—
183C8—202no—
184C8C2×D3×AGL(1,𝔽5)11yescharacteristic
185C2×C3—101no—
186C2×C3D8×AGL(1,𝔽5)11yescharacteristic
187D3—402no—
188C5#6935711yescharacteristic, complemented, Sylow
189C22—51no—
190C22—602no—
191C4—51no—
192C4—122no—
193C4—302no—
194C4#9192111yescharacteristic
195C3#1060413611yescharacteristic, complemented, Sylow
196C2—101no—
197C2—1202no—
198C2#1175166411yescharacteristic
subgroupG/H
centerC2#11751664
commutatorC3×C5×C8C22×C4
FrattiniC8C2×D3×AGL(1,𝔽5)
FittingC3×C5×C16C2×C4
socleC2×C3×C5C4×D8
prime radical#11746834C2
cosocleC3×C8×D5C23
hypercenterC16D3×AGL(1,𝔽5)
hypocenterC3×C5#2178
LCSG ▷ C3×C5×C8 ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
1. #11751689C2semidirect
3. C16×ΓL(1,𝔽16)C2semidirect
4. C3×#64851C2semidirect
5. #11746461C2semidirect
13. D240C4semidirect
14. C5×Dic24C4semidirect
16. #90945C4semidirect
21. #64851D3semidirect
70. Dic24AGL(1,𝔽5)semidirect
164. C3×C5#2178semidirect
188. C5#69357semidirect
195. C3#10604136semidirect

Maximal subgroups

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

subgroupindex
#117516892
#117468342
C16×ΓL(1,𝔽16)2
C3×#648512
#117464612
#106041363
#693575