Finite Group Databasebeta

← #72246553

Subgroups — #72246553

Order 2352.

264 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 51 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×D21⋊D7C212yesmaximal, complemented
2C7×D84C211yesmaximal, characteristic, complemented
3C4×C7×D21C211yesmaximal, characteristic, complemented
4C3×C4⋊(C7⋊D7)C211yesmaximal, characteristic, complemented
5#11924253C212yesmaximal, complemented
6D7×D28—31nomaximal, Hall
7C2×C7×D21C2211yescharacteristic
8C2×C7×D21C2212yescomplemented
9C2×C3×C7⋊D7C2212yescomplemented
10D21⋊D7—24no—
11C3×C4×C72C2211yescharacteristic, complemented
12C7×Dic21C2211yescharacteristic, complemented
13C2×D72—32no—
14C4⋊(C7⋊D7)D311yescharacteristic, complemented
15C7×D28—31no—
16C4×C7×D7—31no—
17#87641—32no—
18#66681—71nomaximal
19D3×D28—71nomaximal
20C2×C3×C72C2311yescharacteristic
21C7×D21—22no—
22C7×D21D412yescomplemented
23C3×C7⋊D7—22no—
24C2×C7⋊D7C2×D312yescomplemented
25C2×C7×D7—31no—
26C2×C7×D7—32no—
27D72—64no—
28C4×C72C2×D311yescharacteristic, complemented
29C7×Dic7—31no—
30C2×D3×D7—72no—
31C2×D3×D7—72no—
32D84D711yescharacteristic, complemented
33C4×D21D711yescharacteristic, complemented
34C7×D12—71no—
35C4×D3×C7—71no—
36C3×D28—71no—
37C3×D28—71no—
38C3×D28—143no—
39#4246—72no—
40#4245—72no—
41C3×C72C2×D411yescharacteristic, complemented, Hall
42D4×D7—211no—
43C2×D28—211no—
44C2×C72C22×D311yescharacteristic
45C7⋊D7—22no—
46C7×D7—32no—
47C7×D7—62no—
48C2×D21C2×D711yescharacteristic
49C2×D21C2×D712yescomplemented
50C2×D3×C7—71no—
51C2×D3×C7—72no—
52C2×C3×D7—72no—
53C2×C3×D7—72no—
54C2×C3×D7—146no—
55D3×D7—144no—
56D3×D7—144no—
57C3×C4×C7—23no—
58C3×C4×C7C2×D711yescharacteristic, complemented
59C3×C4×C7C2×D711yescharacteristic, complemented
60Dic21C2×D711yescharacteristic, complemented
61C7×Dic3—71no—
62C22×D7—212no—
63C22×D7—212no—
64C7×D4—211no—
65C2×C4×C7—211no—
66C22⋊D7—212no—
67D28—31no—
68D28—71no—
69D28—71no—
70D28—143no—
71D28—212no—
72C4×D7—31no—
73C72D3×D411yescharacteristic, complemented, Sylow
74D3×D4—491noHall
75C2×C3×C7—23no—
76C2×C3×C7C22×D711yescharacteristic
77C2×C3×C7C22×D711yescharacteristic
78D21—22no—
79D21D2812yescomplemented
80C3×D7—142no—
81C3×D7—142no—
82C3×D7—286no—
83D3×C7—72no—
84D3×C7—142no—
85C22×C7—211no—
86C22×C7—212no—
87C2×D7—31no—
88C2×D7—32no—
89C2×D7—72no—
90C2×D7—72no—
91C2×D7—146no—
92C2×D7—424no—
93C2×D7—424no—
94C4×C7—23no—
95C4×C7—211no—
96C4×C7D3×D711yescharacteristic, complemented
97C4×C7D3×D711yescharacteristic, complemented
98Dic7—31no—
99C22×D3—492no—
100C3×D4—491no—
101#67—492no—
102D12—71no—
103C4×D3—71no—
104C3×C7—23no—
105C3×C7C2×D2811yescharacteristic, complemented
106C3×C7D4×D711yescharacteristic, complemented
107C2×D4—1471noSylow
108C2×C7—23no—
109C2×C7—212no—
110C2×C7—422no—
111C2×C7C2×D3×D711yescharacteristic
112C2×C7C2×D3×D711yescharacteristic
113D7—32no—
114D7—62no—
115D7—142no—
116D7—142no—
117D7—286no—
118C22×C3—492no—
119C2×D3—71no—
120C2×D3—72no—
121C2×D3—984no—
122C3×C4D7211yescharacteristic, complemented
123Dic3—71no—
124C23—1472no—
125D4—211no—
126D4—491no—
127D4—1472no—
128C2×C4—211no—
129C7—23no—
130C7D3×D2811yescharacteristic, complemented
131C7#6668111yescharacteristic, complemented
132C2×C3—982no—
133C2×C3C2×D7211yescharacteristic
134D3—72no—
135D3—142no—
136C22—211no—
137C22—212no—
138C22—2944no—
139C22—492no—
140C4—211no—
141C4D21⋊D711yescharacteristic, complemented
142C3D7×D2811yescharacteristic, complemented, Sylow
143C2—212no—
144C2—422no—
145C2—982no—
146C2C2×D21⋊D711yescharacteristic
subgroupG/H
center, FrattiniC2C2×D21⋊D7
commutator, socle, cosocleC2×C3×C72C23
FittingC3×C4×C72C22
hypercenterC4D21⋊D7
hypocenterC3×C72C2×D4
LCSG ▷ C2×C3×C72 ▷ C3×C72
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×D21⋊D7C2semidirect
2. C7×D84C2semidirect
3. C4×C7×D21C2semidirect
4. C3×C4⋊(C7⋊D7)C2semidirect
5. #11924253C2semidirect
8. C2×C7×D21C22semidirect
9. C2×C3×C7⋊D7C22semidirect
11. C3×C4×C72C22semidirect
12. C7×Dic21C22semidirect
14. C4⋊(C7⋊D7)D3semidirect
22. C7×D21D4semidirect
24. C2×C7⋊D7C2×D3semidirect
28. C4×C72C2×D3semidirect
32. D84D7semidirect
33. C4×D21D7semidirect
41. C3×C72C2×D4semidirect
49. C2×D21C2×D7semidirect
58. C3×C4×C7C2×D7semidirect
59. C3×C4×C7C2×D7semidirect
60. Dic21C2×D7semidirect
73. C72D3×D4semidirect
79. D21D28semidirect
96. C4×C7D3×D7semidirect
97. C4×C7D3×D7semidirect
105. C3×C7C2×D28semidirect
106. C3×C7D4×D7semidirect
122. C3×C4D72semidirect
130. C7D3×D28semidirect
131. C7#66681semidirect
141. C4D21⋊D7semidirect
142. C3D7×D28semidirect

Maximal subgroups

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

subgroupindex
C2×D21⋊D72
C7×D842
C4×C7×D212
C3×C4⋊(C7⋊D7)2
#119242532
D7×D283
#666817
D3×D287