Finite Group Databasebeta

← #28383761

Subgroups — #28383761

Order 14272.

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#20297351C211yesmaximal, characteristic
2#20297335C211yesmaximal, characteristic, complemented
3#20297302C211yesmaximal, characteristic, complemented
4#20297231C211yesmaximal, characteristic, complemented
5#20297230C211yesmaximal, characteristic, complemented
6#20297222C211yesmaximal, characteristic, complemented
7#137×C223C211yesmaximal, characteristic, complemented
8#16813170—21no—
9#16813170C2211yescharacteristic
10#16813169—22no—
11#16813169—21no—
12C2×Dic446—21no—
13C2×Dic446C2211yescharacteristic
14Q8×D223—21no—
15#16813160—21no—
16#16813157—21no—
17#16813154C2211yescharacteristic
18Dic223⋊C4—22no—
19C4×Dic223C2211yescharacteristic
20#41×C223C2211yescharacteristic
21C2×Q8×C223C2211yescharacteristic
22Q16×C223—21no—
23SD16×C223—21no—
24M16×C223C2211yescharacteristic
25Dic446—22no—
26Dic446—21no—
27Dic446—21no—
28Dic446D411yescharacteristic
29D892D411yescharacteristic
30C22⋊D223—22no—
31C22⋊D223—21no—
32C2×Dic223—22no—
33C2×Dic223—21no—
34C2×Dic223D412yes—
35C4×D223—22no—
36C4×D223—21no—
37C223⋊C8—21no—
38Q8×C223—21no—
39Q8×C223—21no—
40Q8×C223D411yescharacteristic
41D4×C223—21no—
42D4×C223D411yescharacteristic
43C2×C4×C223—21no—
44C2×C4×C223—21no—
45C2×C4×C223C2311yescharacteristic
46C8×C223—21no—
47C22×C223—21no—
48C22×C223C2×D411yescharacteristic
49C2×D223—21no—
50C4×C223—21no—
51C4×C223—41no—
52C4×C223C2×D411yescharacteristic
53C4×C223C2×D411yescharacteristic
54Dic223—22no—
55Dic223—21no—
56Dic223—22no—
57C2×C223—21no—
58C2×C223—41no—
59C2×C223#12011yescharacteristic
60D223—41no—
61C223#45611yescharacteristic, complemented, Sylow
62#456—2231nomaximal, Sylow
632−1+4—2231no—
64#137—2231no—
65#137D22311yescharacteristic, complemented
66C4⋊Q8—2231no—
67#104—2231no—
68#104—2231no—
69#101—2231no—
70#41—4461no—
71#41—4461no—
72#41—4462no—
73#41—2231no—
74#41C2×D22311yescharacteristic
75C2×Q8—4461no—
76C2×Q8—4461no—
77C2×Q8—2231no—
78C2×Q8C2×D22311yescharacteristic
79Q16—21no—
80Q16—4461no—
81SD16—21no—
82SD16—4461no—
83M16—2231no—
84M16C2×D22311yescharacteristic
85C4⋊C4—4462no—
86C42—2231no—
87Q8—21no—
88Q8—21no—
89Q8—4462no—
90Q8—4461no—
91Q8—4461no—
92Q8—2231no—
93Q8C22⋊D22311yescharacteristic
94D4—21no—
95D4—4462no—
96D4—4461no—
97D4—2231no—
98D4C22⋊D22311yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—4462no—
102C2×C4—4462no—
103C2×C4—4461no—
104C2×C4—4461no—
105C2×C4—2232no—
106C2×C4C22×D22311yescharacteristic
107C8—21no—
108C8—4461no—
109C22—21no—
110C22—4461no—
111C22D4×D22311yescharacteristic
112C4—21no—
113C4—41no—
114C4—4462no—
115C4—4461no—
116C4—4462no—
117C4D4×D22311yescharacteristic
118C4C2×C22⋊D22311yescharacteristic
119C2—21no—
120C2—41no—
121C2—8921no—
122C2#2029727211yescharacteristic
subgroupG/H
centerC2#20297272
commutator, cosocleC2×C4×C223C23
FrattiniC2×C4C22×D223
Fitting#137×C223C2
socleC2×C223#120
prime radical#20297351C2
hypercenter#137D223
hypocenterC223#456
LCSG ▷ C2×C4×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #20297335C2semidirect
3. #20297302C2semidirect
4. #20297231C2semidirect
5. #20297230C2semidirect
6. #20297222C2semidirect
7. #137×C223C2semidirect
61. C223#456semidirect
65. #137D223semidirect

Maximal subgroups

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

subgroupindex
#202973512
#202973352
#202973022
#202972312
#202972302
#202972222
#137×C2232
#456223