Finite Group Databasebeta

← #28383756

Subgroups — #28383756

Order 14272.

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

normal subgroups only

This group has 152 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#20297350C211yesmaximal, characteristic
2#20297335C211yesmaximal, characteristic, complemented
3#20297293C211yesmaximal, characteristic, complemented
4#20297231C211yesmaximal, characteristic, complemented
5#20297230C211yesmaximal, characteristic, complemented
6#20297220C211yesmaximal, characteristic, complemented
7#137×C223C211yesmaximal, characteristic, complemented
8#16813171—21no—
9#16813170—21no—
10#16813170C2211yescharacteristic
11C2×D892—21no—
12C2×D892C2211yescharacteristic
13D4×D223—22no—
14D4×D223—21no—
15#16813160—21no—
16#16813157—21no—
17#16813154C2211yescharacteristic, complemented
18(C2×D223)⋊C4—22no—
19C4×Dic223C2211yescharacteristic, complemented
20#41×C223C2211yescharacteristic
21C2×Q8×C223C2211yescharacteristic, complemented
22Q16×C223—21no—
23SD16×C223—21no—
24M16×C223C2211yescharacteristic, complemented
25C22×D223—22no—
26C22×D223D412yes—
27Dic446D411yescharacteristic
28D892—22no—
29D892—21no—
30D892—21no—
31D892D411yescharacteristic
32C22⋊D223—22no—
33C22⋊D223—21no—
34C2×Dic223—21no—
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—22no—
50C2×D223—21no—
51C2×D223—42no—
52C2×D223—82no—
53C4×C223—21no—
54C4×C223—41no—
55C4×C223C2×D411yescharacteristic
56C4×C223C2×D411yescharacteristic
57Dic223—21no—
58Dic223—41no—
59C2×C223—21no—
60C2×C223—41no—
61C2×C223#12011yescharacteristic
62D223—42no—
63D223—41no—
64C223#45511yescharacteristic, complemented, Sylow
65#455—2231nomaximal, Sylow
66H(5,𝔽2)—2231no—
67#137—2231no—
68#137D22311yescharacteristic, complemented
69#124—2231no—
70#104—2231no—
71#104—2231no—
72#100—2231no—
73#41—4461no—
74#41—4461no—
75#41—2231no—
76#41C2×D22311yescharacteristic
77C2×Q8C2×D22311yescharacteristic, complemented
78C2×D4—4461no—
79C2×D4—4462no—
80C2×D4—4461no—
81C2×D4—2231no—
82Q16—21no—
83Q16—4461no—
84SD16—21no—
85SD16—4461no—
86M16—2231no—
87M16C2×D22311yescharacteristic, complemented
88C22⋊C4—4462no—
89C42—2231no—
90C23—4462no—
91C23—2232no—
92Q8—21no—
93Q8—21no—
94Q8—2231no—
95Q8C22⋊D22311yescharacteristic
96D4—21no—
97D4—4462no—
98D4—4462no—
99D4—4461no—
100D4—4461no—
101D4—4461no—
102D4—2231no—
103D4C22⋊D22311yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—4462no—
107C2×C4—4461no—
108C2×C4—4461no—
109C2×C4C22×D22311yescharacteristic
110C8—21no—
111C8—4461no—
112C22—21no—
113C22—8922no—
114C22—4462no—
115C22—4461no—
116C22—17842no—
117C22D4×D22311yescharacteristic
118C4—21no—
119C4—41no—
120C4—8921no—
121C4—4461no—
122C4D4×D22311yescharacteristic
123C4C2×C22⋊D22311yescharacteristic
124C2—21no—
125C2—41no—
126C2—8922no—
127C2—8921no—
128C2#2029727211yescharacteristic
subgroupG/H
centerC2#20297272
commutator, cosocleC2×C4×C223C23
FrattiniC2×C4C22×D223
Fitting#137×C223C2
socleC2×C223#120
prime radical#20297350C2
hypercenter#137D223
hypocenterC223#455
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. #20297293C2semidirect
4. #20297231C2semidirect
5. #20297230C2semidirect
6. #20297220C2semidirect
7. #137×C223C2semidirect
17. #16813154C22semidirect
19. C4×Dic223C22semidirect
21. C2×Q8×C223C22semidirect
24. M16×C223C22semidirect
64. C223#455semidirect
68. #137D223semidirect
77. C2×Q8C2×D223semidirect
87. M16C2×D223semidirect

Maximal subgroups

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

subgroupindex
#202973502
#202973352
#202972932
#202972312
#202972302
#202972202
#137×C2232
#455223