Finite Group Databasebeta

← C3×#4247⋊C3 #13094709

Subgroups — C3×#4247⋊C3 #13094709

Order 1512.

240 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C2×C3×C3⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2C2×C3×D3×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C32×ΓL(1,𝔽8)⋊C4C211yesmaximal, characteristic, complemented
4C3×#4247C311yesmaximal, characteristic, complemented
5C3×(C22⋊D7)⋊C3—31nomaximal
6#4247⋊C3C313yesmaximal, direct product factor
7C2×C32×ΓL(1,𝔽8)C2211yescharacteristic
8C3×C3⋊AGL(1,𝔽7)—21no—
9C3×D3×ΓL(1,𝔽8)—21no—
10C2×C3×D21C2×C311yescharacteristic, complemented
11C2×C3×D3×C7C2×C311yescharacteristic, complemented
12C22×C3×ΓL(1,𝔽8)—31no—
13C2×C3⋊AGL(1,𝔽7)C2×C313yescomplemented
14C2×D3×ΓL(1,𝔽8)C2×C313yescomplemented
15C2×C3×AGL(1,𝔽7)—31no—
16C32×Dic7C2×C311yescharacteristic, complemented
17C3×ΓL(1,𝔽8)⋊C4—21no—
18C3×ΓL(1,𝔽8)⋊C4—23no—
19C3×ΓL(1,𝔽8)⋊C4D311yescharacteristic, complemented
20C3×ΓL(1,𝔽8)⋊C4C2×C313yescomplemented
21C32×D12—71nomaximal, Hall
22C32×ΓL(1,𝔽8)D411yescharacteristic, complemented, Hall
23C3×C22⋊D7—31no—
24#4247C3211yescharacteristic, complemented
25(C22⋊D7)⋊C3—33no—
26C2×C32×C7C22×C311yescharacteristic
27C3×D21—21no—
28C3×D3×C7—21no—
29C2×C3×ΓL(1,𝔽8)—21no—
30C2×C3×ΓL(1,𝔽8)—23no—
31C2×C3×ΓL(1,𝔽8)—61no—
32C2×C3×ΓL(1,𝔽8)C2×D311yescharacteristic
33C2×C3×ΓL(1,𝔽8)C22×C313yes—
34C3⋊AGL(1,𝔽7)—23no—
35D3×ΓL(1,𝔽8)—23no—
36C3×AGL(1,𝔽7)—61no—
37C2×C32×D3—71no—
38C2×C32×D3—71no—
39C33×C4—71no—
40C22×C3×C7—31no—
41C2×D21C2×C3211yescharacteristic, complemented
42C2×D3×C7C2×C3211yescharacteristic, complemented
43C2×C3×D7—31no—
44C22×ΓL(1,𝔽8)—33no—
45C2×AGL(1,𝔽7)—33no—
46C3×Dic7—21no—
47C3×Dic7C3×D311yescharacteristic, complemented
48C3×Dic7C2×C3211yescharacteristic, complemented
49ΓL(1,𝔽8)⋊C4—23no—
50ΓL(1,𝔽8)⋊C4C3×D313yescomplemented
51C32×D4—211no—
52C3×D12—73no—
53C3×D12—71no—
54C32×C7C3×D411yescharacteristic, complemented
55C3×ΓL(1,𝔽8)—21no—
56C3×ΓL(1,𝔽8)—23no—
57C3×ΓL(1,𝔽8)D1211yescharacteristic, complemented
58C3×ΓL(1,𝔽8)C3×D413yescomplemented
59C22⋊D7—31noHall
60C2×C33—71no—
61C32×D3—141no—
62C32×D3—141no—
63C2×C3×C7—21no—
64C2×C3×C7—61no—
65C2×C3×C7C2×C3×D311yescharacteristic
66C2×C3×C7C22×C3211yescharacteristic
67D21—21no—
68C3×D7—61no—
69D3×C7—21no—
70C2×ΓL(1,𝔽8)—23no—
71C2×ΓL(1,𝔽8)—63no—
72C2×ΓL(1,𝔽8)C2×C3×D313yes—
73AGL(1,𝔽7)—63no—
74C22×C32—211no—
75C22×C32—211no—
76C2×C3×D3—73no—
77C2×C3×D3—71no—
78C2×C3×D3—73no—
79C2×C3×D3AGL(1,𝔽7)11yescharacteristic, complemented
80C32×C4—71no—
81C32×C4—73no—
82C32×C4—71no—
83C32×C4—143no—
84C32×C4—141no—
85C22×C7—31no—
86C2×D7—31no—
87Dic7C32×D311yescharacteristic, complemented
88C33—71noSylow
89C3×D4—211no—
90C3×D4—213no—
91D12—71no—
92C3×C7—21no—
93C3×C7C3×D1211yescharacteristic, complemented
94C3×C7C32×D411yescharacteristic, complemented
95ΓL(1,𝔽8)—23no—
96ΓL(1,𝔽8)C3×D1213yescomplemented
97C2×C32—71no—
98C2×C32—73no—
99C2×C32—143no—
100C2×C32—141no—
101C2×C32—421no—
102C2×C32—421no—
103C2×C32C2×AGL(1,𝔽7)11yescharacteristic
104C3×D3—21no—
105C3×D3—143no—
106C3×D3—143no—
107C3×D3—141no—
108C2×C7—61no—
109C2×C7C2×C32×D311yescharacteristic
110D7—61no—
111C22×C3—31no—
112C22×C3—213no—
113C22×C3—213no—
114C22×C3—211no—
115C2×D3—71no—
116C2×D3C3×AGL(1,𝔽7)11yescharacteristic, complemented
117C3×C4—73no—
118C3×C4—71no—
119C3×C4—71no—
120C3×C4—141no—
121C3×C4—143no—
122C32—71no—
123C32—73no—
124C32—143no—
125C32—141no—
126C32(C22⋊D7)⋊C311yescharacteristic, complemented
127D4—211noSylow
128C7C32×D1211yescharacteristic, complemented, Sylow
129C2×C3—21no—
130C2×C3—61no—
131C2×C3—73no—
132C2×C3—143no—
133C2×C3—423no—
134C2×C3—423no—
135C2×C3—421no—
136C2×C3D3×AGL(1,𝔽7)11yescharacteristic
137C2×C3C2×C3×AGL(1,𝔽7)11yescharacteristic
138D3—21no—
139D3—141no—
140C22—31no—
141C22—211no—
142C4—71no—
143C3—21no—
144C3—73no—
145C3—143no—
146C3#4247⋊C311yescharacteristic, direct product factor
147C3C3×(C22⋊D7)⋊C311yescharacteristic, complemented
148C2—61no—
149C2—421no—
150C2C3×D3×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3D3×AGL(1,𝔽7)
commutator, cosocleC2×C3×C7C22×C32
FrattiniC2C3×D3×AGL(1,𝔽7)
Fitting, socleC2×C32×C7C22×C3
hypocenterC3×C7C32×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C3⋊AGL(1,𝔽7)C2semidirect
2. C2×C3×D3×ΓL(1,𝔽8)C2semidirect
3. C32×ΓL(1,𝔽8)⋊C4C2semidirect
4. C3×#4247C3semidirect
6. #4247⋊C3C3direct (N × H)
10. C2×C3×D21C2×C3semidirect
11. C2×C3×D3×C7C2×C3semidirect
13. C2×C3⋊AGL(1,𝔽7)C2×C3semidirect
14. C2×D3×ΓL(1,𝔽8)C2×C3semidirect
16. C32×Dic7C2×C3semidirect
19. C3×ΓL(1,𝔽8)⋊C4D3semidirect
20. C3×ΓL(1,𝔽8)⋊C4C2×C3semidirect
22. C32×ΓL(1,𝔽8)D4semidirect
24. #4247C32semidirect
41. C2×D21C2×C32semidirect
42. C2×D3×C7C2×C32semidirect
47. C3×Dic7C3×D3semidirect
48. C3×Dic7C2×C32semidirect
50. ΓL(1,𝔽8)⋊C4C3×D3semidirect
54. C32×C7C3×D4semidirect
57. C3×ΓL(1,𝔽8)D12semidirect
58. C3×ΓL(1,𝔽8)C3×D4semidirect
79. C2×C3×D3AGL(1,𝔽7)semidirect
87. Dic7C32×D3semidirect
93. C3×C7C3×D12semidirect
94. C3×C7C32×D4semidirect
96. ΓL(1,𝔽8)C3×D12semidirect
116. C2×D3C3×AGL(1,𝔽7)semidirect
126. C32(C22⋊D7)⋊C3semidirect
128. C7C32×D12semidirect
146. C3#4247⋊C3direct (N × H)
147. C3C3×(C22⋊D7)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊AGL(1,𝔽7)2
C2×C3×D3×ΓL(1,𝔽8)2
C32×ΓL(1,𝔽8)⋊C42
C3×#42473
C3×(C22⋊D7)⋊C33
#4247⋊C33
C32×D127