Finite Group Databasebeta

← #42121992

Subgroups — #42121992

Order 27712.

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#28357267C211yesmaximal, characteristic
2#28357251C211yesmaximal, characteristic, complemented
3#28357218C211yesmaximal, characteristic, complemented
4#28357147C211yesmaximal, characteristic, complemented
5#28357146C211yesmaximal, characteristic, complemented
6#28357138C211yesmaximal, characteristic, complemented
7#137×C433C211yesmaximal, characteristic, complemented
8#20294097—21no—
9#20294097C2211yescharacteristic
10#20294096—22no—
11#20294096—21no—
12C2×Dic866—21no—
13C2×Dic866C2211yescharacteristic
14Q8×D433—21no—
15#20294087—21no—
16#20294084—21no—
17#20294081C2211yescharacteristic
18#20294076—22no—
19C4×Dic433C2211yescharacteristic
20#41×C433C2211yescharacteristic
21C2×Q8×C433C2211yescharacteristic
22Q16×C433—21no—
23SD16×C433—21no—
24M16×C433C2211yescharacteristic
25Dic866—22no—
26Dic866—21no—
27Dic866—21no—
28Dic866D411yescharacteristic
29D1732D411yescharacteristic
30C22⋊D433—22no—
31C22⋊D433—21no—
32C2×Dic433—22no—
33C2×Dic433—21no—
34C2×Dic433D412yes—
35C4×D433—22no—
36C4×D433—21no—
37#16812358—21no—
38Q8×C433—21no—
39Q8×C433—21no—
40Q8×C433D411yescharacteristic
41D4×C433—21no—
42D4×C433D411yescharacteristic
43C2×C4×C433—21no—
44C2×C4×C433—21no—
45C2×C4×C433C2311yescharacteristic
46C8×C433—21no—
47C22×C433—21no—
48C22×C433C2×D411yescharacteristic
49C2×D433—21no—
50C4×C433—21no—
51C4×C433—41no—
52C4×C433C2×D411yescharacteristic
53C4×C433C2×D411yescharacteristic
54Dic433—22no—
55Dic433—21no—
56Dic433—22no—
57C2×C433—21no—
58C2×C433—41no—
59C2×C433#12011yescharacteristic
60D433—41no—
61C433#45611yescharacteristic, complemented, Sylow
62#456—4331nomaximal, Sylow
632−1+4—4331no—
64#137—4331no—
65#137D43311yescharacteristic, complemented
66C4⋊Q8—4331no—
67#104—4331no—
68#104—4331no—
69#101—4331no—
70#41—8661no—
71#41—8661no—
72#41—8662no—
73#41—4331no—
74#41C2×D43311yescharacteristic
75C2×Q8—8661no—
76C2×Q8—8661no—
77C2×Q8—4331no—
78C2×Q8C2×D43311yescharacteristic
79Q16—21no—
80Q16—8661no—
81SD16—21no—
82SD16—8661no—
83M16—4331no—
84M16C2×D43311yescharacteristic
85C4⋊C4—8662no—
86C42—4331no—
87Q8—21no—
88Q8—21no—
89Q8—8662no—
90Q8—8661no—
91Q8—8661no—
92Q8—4331no—
93Q8C22⋊D43311yescharacteristic
94D4—21no—
95D4—8662no—
96D4—8661no—
97D4—4331no—
98D4C22⋊D43311yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—8662no—
102C2×C4—8662no—
103C2×C4—8661no—
104C2×C4—8661no—
105C2×C4—4332no—
106C2×C4C22×D43311yescharacteristic
107C8—21no—
108C8—8661no—
109C22—21no—
110C22—8661no—
111C22D4×D43311yescharacteristic
112C4—21no—
113C4—41no—
114C4—8662no—
115C4—8661no—
116C4—8662no—
117C4D4×D43311yescharacteristic
118C4C2×C22⋊D43311yescharacteristic
119C2—21no—
120C2—41no—
121C2—17321no—
122C2#2835718811yescharacteristic
subgroupG/H
centerC2#28357188
commutator, cosocleC2×C4×C433C23
FrattiniC2×C4C22×D433
Fitting#137×C433C2
socleC2×C433#120
prime radical#28357267C2
hypercenter#137D433
hypocenterC433#456
LCSG ▷ C2×C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #28357251C2semidirect
3. #28357218C2semidirect
4. #28357147C2semidirect
5. #28357146C2semidirect
6. #28357138C2semidirect
7. #137×C433C2semidirect
61. C433#456semidirect
65. #137D433semidirect

Maximal subgroups

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

subgroupindex
#283572672
#283572512
#283572182
#283571472
#283571462
#283571382
#137×C4332
#456433