Finite Group Databasebeta

← #72246746

Subgroups — #72246746

Order 2016.

780 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 132 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×C7×#638C211yesmaximal, characteristic, complemented
3C2×#92678C212yesmaximal, complemented
4C32×C22⋊Dic7C211yesmaximal, characteristic, complemented
5#11759168C212yesmaximal, complemented
6#10621135—34nomaximal
7C23×C32×C7C2211yescharacteristic
8C22×C3⋊D21—22no—
9C22×C3⋊D21—24no—
10C22×C3⋊D21C2212yes—
11C22×C7×C3⋊D3C2211yescharacteristic, complemented
12C7×#638—22no—
13C2×C7×C3⋊Dic3C2211yescharacteristic, complemented
14C2×C32×Dic7C2212yescomplemented
15#92678—24no—
16C23×D21—34no—
17C2×C7×#67—34no—
18C2×#4247—38no—
19C3×C22⋊Dic7D314yescomplemented
20#66575—38no—
21#64091—71nomaximal, Hall
22C22×C32×C7—22no—
23C22×C32×C7D412yescomplemented
24C22×C32×C7C2311yescharacteristic
25C2×C3⋊D21—28no—
26C2×C3⋊D21—22no—
27C2×C3⋊D21—22no—
28C2×C3⋊D21D414yescomplemented
29C2×C7×C3⋊D3—22no—
30C2×C7×C3⋊D3—21no—
31C7×C3⋊Dic3—21no—
32C32×Dic7—22no—
33#6500—91noHall
34C23×C3×C7C2×D314yes—
35C22×D21—38no—
36C22×D21—616no—
37C22×D21—68no—
38C22×D3×C7—34no—
39C7×#67—68no—
40C2×C7×Dic3—34no—
41C2×C3×Dic7C2×D318yescomplemented
42#4247—616no—
43C23×C3⋊D3—71no—
44C2×#638D711yescharacteristic, complemented
45C2×C4⋊(C3⋊D3)—72no—
46C32×C22⋊C4—71no—
47#3752—72no—
48C2×C32×C7—22no—
49C2×C32×C7C2×D412yes—
50C2×C32×C7C2×D411yescharacteristic
51C3⋊D21—24no—
52C7×C3⋊D3—41no—
53C23×D7—91no—
54C2×C7×D4—91no—
55C2×C22⋊D7—92no—
56C22⋊Dic7C3⋊D311yescharacteristic, complemented
57(C2×D7)⋊C4—92no—
58#882—214no—
59C22×C3×C7—28no—
60C22×C3×C7D1218yescomplemented
61C22×C3×C7C22×D314yes—
62C2×D21—316no—
63C2×D21—632no—
64C2×D21—68no—
65C2×D21—68no—
66C2×D3×C7—64no—
67C2×D3×C7—68no—
68C3×Dic7—28no—
69C7×Dic3—64no—
70C23×C32C2×D711yescharacteristic
71C22×C3⋊D3—72no—
72C22×C3⋊D3—144no—
73C22×C3⋊D3—142no—
74C22×C3⋊D3C2×D711yescharacteristic, complemented
75C2×C32×C4—72no—
76#638—22no—
77C2×C3⋊Dic3C2×D711yescharacteristic, complemented
78C4⋊(C3⋊D3)—144no—
79C32×C7#12011yescharacteristic, complemented, Hall
80C23×C7—91no—
81C23×C7C2×C3⋊D311yescharacteristic
82C22×D7—92no—
83C22×D7—184no—
84C22×D7—182no—
85C7×D4—182no—
86C2×C4×C7—91no—
87C22⋊D7—184no—
88C2×Dic7C2×C3⋊D312yescomplemented
89C23×D3—214no—
90C2×#67—34no—
91C2×D12—218no—
92C3×C22⋊C4—74no—
93(C2×D3)⋊C4—218no—
94C2×C3×C7—28no—
95C2×C3×C7C2×D1214yes—
96C2×C3×C7D3×D418yes—
97D21—616no—
98D3×C7—124no—
99C22×C32—22no—
100C22×C32C22⋊D712yescomplemented
101C22×C32C22×D711yescharacteristic
102C2×C3⋊D3—22no—
103C2×C3⋊D3—21no—
104C2×C3⋊D3—74no—
105C2×C3⋊D3—148no—
106C2×C3⋊D3—142no—
107C2×C3⋊D3—142no—
108C32×C4—142no—
109C3⋊Dic3—21no—
110#120—631noSylow
111C22×C7—22no—
112C22×C7—181no—
113C22×C7—182no—
114C22×C7C4⋊(C3⋊D3)12yescomplemented
115C22×C7C22×C3⋊D311yescharacteristic
116C2×D7—94no—
117C2×D7—188no—
118C2×D7—182no—
119C2×D7—182no—
120C4×C7—181no—
121Dic7—22no—
122C23×C3D3×D714yes—
123C22×D3—34no—
124C22×D3—218no—
125C22×D3—4216no—
126C22×D3—428no—
127C2×C3×C4—78no—
128#67—68no—
129C2×Dic3—34no—
130D12—4216no—
131C3×C7#88214yescomplemented
132C2×C32—22no—
133C2×C32D4×D712yes—
134C2×C32C2×C22⋊D711yescharacteristic
135C3⋊D3—41no—
136C3⋊D3—144no—
137C24—631no—
138C2×D4—91no—
139C2×D4—632no—
140C22⋊C4—71no—
141C22⋊C4—632no—
142C2×C7—22no—
143C2×C7—361no—
144C2×C7C2×C4⋊(C3⋊D3)11yescharacteristic
145C2×C7D4×C3⋊D312yes—
146D7—184no—
147C22×C3—28no—
148C22×C3#424718yescomplemented
149C22×C3C2×D3×D714yes—
150C2×D3—64no—
151C2×D3—68no—
152C2×D3—2116no—
153C2×D3—4232no—
154C2×D3—428no—
155C2×D3—428no—
156C3×C4—148no—
157Dic3—64no—
158C32#650011yescharacteristic, complemented, Sylow
159C23—91no—
160C23—632no—
161C23—1264no—
162C23—1262no—
163C23D7×C3⋊D311yescharacteristic
164D4—182no—
165D4—1264no—
166C2×C4—72no—
167C2×C4—91no—
168C7#6409111yescharacteristic, complemented, Sylow
169C2×C3—28no—
170C2×C3C2×#424714yes—
171C2×C3#6669418yes—
172D3—124no—
173D3—4216no—
174C22—22no—
175C22—181no—
176C22—182no—
177C22—634no—
178C22—1268no—
179C22—1262no—
180C22—1262no—
181C22#9267812yescomplemented
182C22C2×D7×C3⋊D311yescharacteristic
183C4—142no—
184C4—181no—
185C3#1062113514yescomplemented
186C2—22no—
187C2—361no—
188C2—1264no—
189C2C2×#9267811yescharacteristic
190C2#1175958112yes—
subgroupG/H
center, FrattiniC22C2×D7×C3⋊D3
commutator, socle, cosocleC22×C32×C7C23
FittingC23×C32×C7C22
hypercenterC23D7×C3⋊D3
hypocenterC32×C7#120
LCSG ▷ C22×C32×C7 ▷ C32×C7
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊D21C2semidirect
2. C2×C7×#638C2semidirect
3. C2×#92678C2semidirect
4. C32×C22⋊Dic7C2semidirect
5. #11759168C2semidirect
11. C22×C7×C3⋊D3C22semidirect
13. C2×C7×C3⋊Dic3C22semidirect
14. C2×C32×Dic7C22semidirect
19. C3×C22⋊Dic7D3semidirect
23. C22×C32×C7D4semidirect
28. C2×C3⋊D21D4semidirect
41. C2×C3×Dic7C2×D3semidirect
44. C2×#638D7semidirect
56. C22⋊Dic7C3⋊D3semidirect
60. C22×C3×C7D12semidirect
74. C22×C3⋊D3C2×D7semidirect
77. C2×C3⋊Dic3C2×D7semidirect
79. C32×C7#120semidirect
88. C2×Dic7C2×C3⋊D3semidirect
100. C22×C32C22⋊D7semidirect
114. C22×C7C4⋊(C3⋊D3)semidirect
131. C3×C7#882semidirect
148. C22×C3#4247semidirect
158. C32#6500semidirect
168. C7#64091semidirect
181. C22#92678semidirect
185. C3#10621135semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊D212
C2×C7×#6382
C2×#926782
C32×C22⋊Dic72
#117591682
#106211353
#640917