Finite Group Databasebeta

← #28383769

Subgroups — #28383769

Order 14272.

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

normal subgroups only

This group has 160 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#20297349C211yesmaximal, characteristic
2#20297287C212yesmaximal, complemented
3#20297219C211yesmaximal, characteristic, complemented
4#20297218C212yesmaximal, complemented
5#123×C223C211yesmaximal, characteristic, complemented
6#16813170—21no—
7#16813169—22no—
8C2×C22⋊D223—21no—
9C2×C22⋊D223C2212yes—
10D4×D223—22no—
11Dic223⋊C4—22no—
12C22⋊Dic223C2212yescomplemented
13(C2×D223)⋊C4—22no—
14C2×D4×C223C2211yescharacteristic
15C22×C4×C223C2211yescharacteristic, complemented
16C4⋊C4×C223—21no—
17C22⋊C4×C223—21no—
18C22⋊C4×C223C2211yescharacteristic, complemented
19C22×D223—21no—
20C22×D223D412yes—
21Dic446—21no—
22D892—21no—
23C22⋊D223—22no—
24C22⋊D223—22no—
25C22⋊D223—22no—
26C2×Dic223—21no—
27C2×Dic223—22no—
28C2×Dic223D412yes—
29C4×D223—22no—
30C23×C223D411yescharacteristic
31C23×C223C2311yescharacteristic
32D4×C223—22no—
33C2×C4×C223—21no—
34C2×C4×C223—21no—
35C2×C4×C223—22no—
36C2×C4×C223D411yescharacteristic
37C22×C223—21no—
38C22×C223—41no—
39C22×C223—41no—
40C22×C223C2×D412yes—
41C22×C223C2×D411yescharacteristic
42C2×D223—22no—
43C2×D223—42no—
44C2×D223—81no—
45C4×C223—21no—
46C4×C223—41no—
47C4×C223—41no—
48Dic223—22no—
49Dic223—42no—
50C2×C223—22no—
51C2×C223—21no—
52C2×C223—41no—
53C2×C223#12011yescharacteristic
54D223—42no—
55C223#45811yescharacteristic, complemented, Sylow
56#458—2231nomaximal, Sylow
57H(5,𝔽2)—2231no—
58#123—2232no—
59#123D22311yescharacteristic, complemented
60#99—2231no—
61#99—2232no—
62#41—4462no—
63#41—4461no—
64C2×D4—4462no—
65C2×D4—4461no—
66C2×D4—2232no—
67C2×D4C2×D22311yescharacteristic
68C22×C4C2×D22311yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—4462no—
71C22⋊C4—21no—
72C22⋊C4—4462no—
73C22⋊C4—2232no—
74C22⋊C4C2×D22311yescharacteristic, complemented
75C23—4461no—
76C23—2232no—
77C23C22⋊D22311yescharacteristic
78C23C22×D22311yescharacteristic
79Q8—4461no—
80D4—22no—
81D4—4461no—
82D4—4462no—
83D4—4462no—
84D4—4462no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—4462no—
89C2×C4—4462no—
90C2×C4—4461no—
91C2×C4—2232no—
92C2×C4C22⋊D22311yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—8922no—
97C22—4462no—
98C22—17841no—
99C22D4×D22312yes—
100C22C2×C22⋊D22311yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—8922no—
105C4—4462no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—8922no—
110C2#2029727211yescharacteristic
subgroupG/H
centerC2#20297272
commutator, cosocleC23×C223C23
FrattiniC23C22×D223
Fitting#123×C223C2
socleC2×C223#120
prime radical#20297349C2
hypercenter#123D223
hypocenterC223#458
LCSG ▷ C23×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #20297287C2semidirect
3. #20297219C2semidirect
4. #20297218C2semidirect
5. #123×C223C2semidirect
12. C22⋊Dic223C22semidirect
15. C22×C4×C223C22semidirect
18. C22⋊C4×C223C22semidirect
55. C223#458semidirect
59. #123D223semidirect
68. C22×C4C2×D223semidirect
74. C22⋊C4C2×D223semidirect

Maximal subgroups

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

subgroupindex
#202973492
#202972872
#202972192
#202972182
#123×C2232
#458223