Finite Group Databasebeta

← #72246743

Subgroups — #72246743

Order 2016.

780 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C23×C3⋊D21C211yesmaximal, characteristic, complemented
2C2×C32×C7×D4C211yesmaximal, characteristic, complemented
3C2×#92726C212yesmaximal, complemented
4#11759318C211yesmaximal, characteristic, complemented
5#11759313C212yesmaximal, complemented
6#10621420—34nomaximal
7C23×C32×C7C2211yescharacteristic
8C23×C32×C7C2211yescharacteristic, complemented
9C22×C3⋊D21—22no—
10C22×C3⋊D21—24no—
11C22×C3⋊D21C2212yes—
12C32×C7×D4—22no—
13C2×C32×C4×C7C2211yescharacteristic, complemented
14#92726—24no—
15C2×C3⋊Dic21C2212yescomplemented
16C23×D21—34no—
17C2×C3×C7×D4D314yescomplemented
18C2×#4268—38no—
19#66636—34no—
20#66631—38no—
21#64146—71nomaximal, Hall
22C22×C32×C7—22no—
23C22×C32×C7—22no—
24C22×C32×C7—21no—
25C22×C32×C7D412yescomplemented
26C22×C32×C7C2311yescharacteristic
27C2×C3⋊D21—28no—
28C2×C3⋊D21—22no—
29C2×C3⋊D21—22no—
30C2×C3⋊D21D414yescomplemented
31C32×C4×C7—21no—
32C3⋊Dic21—22no—
33#6500—91noHall
34C23×C3×C7C2×D314yes—
35C23×C3×C7C2×D314yescomplemented
36C22×D21—38no—
37C22×D21—616no—
38C22×D21—68no—
39C3×C7×D4—28no—
40C2×C3×C4×C7C2×D314yescomplemented
41#4268—616no—
42C2×Dic21—38no—
43C23×C3⋊D3—71no—
44C2×C32×D4D711yescharacteristic, complemented
45C2×#638—72no—
46#3757—71no—
47#3752—72no—
48C2×C32×C7—22no—
49C2×C32×C7—41no—
50C2×C32×C7C2×D412yes—
51C2×C32×C7C2×D411yescharacteristic
52C3⋊D21—24no—
53C23×D7—91no—
54C2×C7×D4C3⋊D311yescharacteristic, complemented
55C2×C22⋊D7—92no—
56C22⋊Dic7—91no—
57(C2×D7)⋊C4—92no—
58#937—214no—
59C22×C3×C7—28no—
60C22×C3×C7—28no—
61C22×C3×C7—24no—
62C22×C3×C7#6718yescomplemented
63C22×C3×C7C22×D314yes—
64C2×D21—316no—
65C2×D21—632no—
66C2×D21—68no—
67C2×D21—68no—
68C3×C4×C7—24no—
69Dic21—68no—
70C23×C32C2×D711yescharacteristic
71C23×C32C2×D711yescharacteristic, complemented
72C22×C3⋊D3—72no—
73C22×C3⋊D3—144no—
74C22×C3⋊D3—142no—
75C32×D4—22no—
76C2×C32×C4C2×D711yescharacteristic, complemented
77#638—144no—
78C2×C3⋊Dic3—72no—
79C32×C7#12011yescharacteristic, complemented, Hall
80C23×C7C2×C3⋊D311yescharacteristic
81C23×C7C2×C3⋊D311yescharacteristic, complemented
82C22×D7—92no—
83C22×D7—184no—
84C22×D7—182no—
85C7×D4—22no—
86C2×C4×C7C2×C3⋊D311yescharacteristic, complemented
87C22⋊D7—184no—
88C2×Dic7—92no—
89C23×D3—214no—
90C2×C3×D4D2114yescomplemented
91C2×#67—218no—
92#217—214no—
93(C2×D3)⋊C4—218no—
94C2×C3×C7—28no—
95C2×C3×C7—44no—
96C2×C3×C7D3×D418yes—
97C2×C3×C7C2×#6714yes—
98D21—616no—
99C22×C32—22no—
100C22×C32—22no—
101C22×C32—21no—
102C22×C32C22⋊D712yescomplemented
103C22×C32C22×D711yescharacteristic
104C2×C3⋊D3—74no—
105C2×C3⋊D3—148no—
106C2×C3⋊D3—142no—
107C2×C3⋊D3—142no—
108C32×C4—21no—
109C3⋊Dic3—142no—
110#120—631noSylow
111C22×C7—22no—
112C22×C7—22no—
113C22×C7—21no—
114C22×C7#63812yescomplemented
115C22×C7C22×C3⋊D311yescharacteristic
116C2×D7—94no—
117C2×D7—188no—
118C2×D7—182no—
119C2×D7—182no—
120C4×C7—21no—
121Dic7—182no—
122C23×C3C2×D2114yes—
123C23×C3C2×D2114yescomplemented
124C22×D3—218no—
125C22×D3—4216no—
126C22×D3—428no—
127C3×D4—28no—
128C2×C3×C4C2×D2114yescomplemented
129#67—4216no—
130C2×Dic3—218no—
131C3×C7#93714yescomplemented
132C2×C32—22no—
133C2×C32—41no—
134C2×C32D4×D712yes—
135C2×C32C2×C22⋊D711yescharacteristic
136C3⋊D3—144no—
137C24—631no—
138C2×D4—632no—
139C2×D4C3⋊D2111yescharacteristic, complemented
140C22⋊C4—631no—
141C22⋊C4—632no—
142C2×C7—22no—
143C2×C7—41no—
144C2×C7D4×C3⋊D312yes—
145C2×C7C2×#63811yescharacteristic
146D7—184no—
147C22×C3—28no—
148C22×C3—28no—
149C22×C3—24no—
150C22×C3#426818yescomplemented
151C22×C3C22×D2114yes—
152C2×D3—2116no—
153C2×D3—4232no—
154C2×D3—428no—
155C2×D3—428no—
156C3×C4—24no—
157Dic3—428no—
158C32#650011yescharacteristic, complemented, Sylow
159C23—632no—
160C23—1264no—
161C23—1262no—
162C23C2×C3⋊D2111yescharacteristic
163C23C2×C3⋊D2111yescharacteristic, complemented
164D4—22no—
165D4—1264no—
166C2×C4—632no—
167C2×C4C2×C3⋊D2111yescharacteristic, complemented
168C7#6414611yescharacteristic, complemented, Sylow
169C2×C3—28no—
170C2×C3—44no—
171C2×C3D4×D2118yes—
172C2×C3C2×#426814yes—
173D3—4216no—
174C22—22no—
175C22—22no—
176C22—21no—
177C22—634no—
178C22—1268no—
179C22—1262no—
180C22—1262no—
181C22#9272612yescomplemented
182C22C22×C3⋊D2111yescharacteristic
183C4—21no—
184C4—1262no—
185C3#1062142014yescomplemented
186C2—22no—
187C2—41no—
188C2—1264no—
189C2D4×C3⋊D2112yes—
190C2C2×#9272611yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊D21
commutator, socle, cosocleC22×C32×C7C23
FittingC2×C32×C7×D4C2
hypercenterC2×D4C3⋊D21
hypocenterC32×C7#120
LCSG ▷ C22×C32×C7 ▷ C32×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊D21C2semidirect
2. C2×C32×C7×D4C2semidirect
3. C2×#92726C2semidirect
4. #11759318C2semidirect
5. #11759313C2semidirect
8. C23×C32×C7C22semidirect
13. C2×C32×C4×C7C22semidirect
15. C2×C3⋊Dic21C22semidirect
17. C2×C3×C7×D4D3semidirect
25. C22×C32×C7D4semidirect
30. C2×C3⋊D21D4semidirect
35. C23×C3×C7C2×D3semidirect
40. C2×C3×C4×C7C2×D3semidirect
44. C2×C32×D4D7semidirect
54. C2×C7×D4C3⋊D3semidirect
62. C22×C3×C7#67semidirect
71. C23×C32C2×D7semidirect
76. C2×C32×C4C2×D7semidirect
79. C32×C7#120semidirect
81. C23×C7C2×C3⋊D3semidirect
86. C2×C4×C7C2×C3⋊D3semidirect
90. C2×C3×D4D21semidirect
102. C22×C32C22⋊D7semidirect
114. C22×C7#638semidirect
123. C23×C3C2×D21semidirect
128. C2×C3×C4C2×D21semidirect
131. C3×C7#937semidirect
139. C2×D4C3⋊D21semidirect
150. C22×C3#4268semidirect
158. C32#6500semidirect
163. C23C2×C3⋊D21semidirect
167. C2×C4C2×C3⋊D21semidirect
168. C7#64146semidirect
181. C22#92726semidirect
185. C3#10621420semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊D212
C2×C32×C7×D42
C2×#927262
#117593182
#117593132
#106214203
#641467