Finite Group Databasebeta

← #28409213

Subgroups — #28409213

Order 14528.

142 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 39 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#20300335C211yesmaximal, characteristic
2#20300319C211yesmaximal, characteristic, complemented
3#20300286C211yesmaximal, characteristic, complemented
4#20300215C211yesmaximal, characteristic, complemented
5#20300214C211yesmaximal, characteristic, complemented
6#20300206C211yesmaximal, characteristic, complemented
7#137×C227C211yesmaximal, characteristic, complemented
8#16813884—21no—
9#16813884C2211yescharacteristic
10#16813883—22no—
11#16813883—21no—
12C2×Dic454—21no—
13C2×Dic454C2211yescharacteristic
14Q8×D227—21no—
15#16813874—21no—
16#16813871—21no—
17#16813868C2211yescharacteristic
18Dic227⋊C4—22no—
19C4×Dic227C2211yescharacteristic
20#41×C227C2211yescharacteristic
21C2×Q8×C227C2211yescharacteristic
22Q16×C227—21no—
23SD16×C227—21no—
24M16×C227C2211yescharacteristic
25Dic454—22no—
26Dic454—21no—
27Dic454—21no—
28Dic454D411yescharacteristic
29D908D411yescharacteristic
30C22⋊D227—22no—
31C22⋊D227—21no—
32C2×Dic227—22no—
33C2×Dic227—21no—
34C2×Dic227D412yes—
35C4×D227—22no—
36C4×D227—21no—
37C227⋊C8—21no—
38Q8×C227—21no—
39Q8×C227—21no—
40Q8×C227D411yescharacteristic
41D4×C227—21no—
42D4×C227D411yescharacteristic
43C2×C4×C227—21no—
44C2×C4×C227—21no—
45C2×C4×C227C2311yescharacteristic
46C8×C227—21no—
47C22×C227—21no—
48C22×C227C2×D411yescharacteristic
49C2×D227—21no—
50C4×C227—21no—
51C4×C227—41no—
52C4×C227C2×D411yescharacteristic
53C4×C227C2×D411yescharacteristic
54Dic227—22no—
55Dic227—21no—
56Dic227—22no—
57C2×C227—21no—
58C2×C227—41no—
59C2×C227#12011yescharacteristic
60D227—41no—
61C227#45611yescharacteristic, complemented, Sylow
62#456—2271nomaximal, Sylow
632−1+4—2271no—
64#137—2271no—
65#137D22711yescharacteristic, complemented
66C4⋊Q8—2271no—
67#104—2271no—
68#104—2271no—
69#101—2271no—
70#41—4541no—
71#41—4541no—
72#41—4542no—
73#41—2271no—
74#41C2×D22711yescharacteristic
75C2×Q8—4541no—
76C2×Q8—4541no—
77C2×Q8—2271no—
78C2×Q8C2×D22711yescharacteristic
79Q16—21no—
80Q16—4541no—
81SD16—21no—
82SD16—4541no—
83M16—2271no—
84M16C2×D22711yescharacteristic
85C4⋊C4—4542no—
86C42—2271no—
87Q8—21no—
88Q8—21no—
89Q8—4542no—
90Q8—4541no—
91Q8—4541no—
92Q8—2271no—
93Q8C22⋊D22711yescharacteristic
94D4—21no—
95D4—4542no—
96D4—4541no—
97D4—2271no—
98D4C22⋊D22711yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—4542no—
102C2×C4—4542no—
103C2×C4—4541no—
104C2×C4—4541no—
105C2×C4—2272no—
106C2×C4C22×D22711yescharacteristic
107C8—21no—
108C8—4541no—
109C22—21no—
110C22—4541no—
111C22D4×D22711yescharacteristic
112C4—21no—
113C4—41no—
114C4—4542no—
115C4—4541no—
116C4—4542no—
117C4D4×D22711yescharacteristic
118C4C2×C22⋊D22711yescharacteristic
119C2—21no—
120C2—41no—
121C2—9081no—
122C2#2030025611yescharacteristic
subgroupG/H
centerC2#20300256
commutator, cosocleC2×C4×C227C23
FrattiniC2×C4C22×D227
Fitting#137×C227C2
socleC2×C227#120
prime radical#20300335C2
hypercenter#137D227
hypocenterC227#456
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #20300319C2semidirect
3. #20300286C2semidirect
4. #20300215C2semidirect
5. #20300214C2semidirect
6. #20300206C2semidirect
7. #137×C227C2semidirect
61. C227#456semidirect
65. #137D227semidirect

Maximal subgroups

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

subgroupindex
#203003352
#203003192
#203002862
#203002152
#203002142
#203002062
#137×C2272
#456227