Finite Group Databasebeta

← #42121985

Subgroups — #42121985

Order 27712.

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

normal subgroups only

This group has 146 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, complemented
2#28357248C211yesmaximal, characteristic, complemented
3#28357245C211yesmaximal, characteristic, complemented
4#28357208C211yesmaximal, characteristic, complemented
5#28357148C211yesmaximal, characteristic, complemented
6#28357138C211yesmaximal, characteristic, complemented
7#104×C433C211yesmaximal, characteristic, complemented
8#20294097—21no—
9#20294097C2211yescharacteristic
10#20294096—22no—
11#20294096—21no—
12C2×Dic866—21no—
13C2×Dic866C2211yescharacteristic
14Q8×D433—21no—
15C2×D1732C2211yescharacteristic
16#20294086—21no—
17#20294084—21no—
18D3464—21no—
19#20294082—21no—
20#20294081C2211yescharacteristic
21#20294074—22no—
22#41×C433C2211yescharacteristic
23M16×C433C2211yescharacteristic
24C42×C433C2211yescharacteristic
25C22×D433—21no—
26Dic866—22no—
27Dic866—21no—
28Dic866—21no—
29Dic866D411yescharacteristic
30D1732—21no—
31D1732—21no—
32D1732D411yescharacteristic
33C22⋊D433—22no—
34C22⋊D433—21no—
35C2×Dic433—22no—
36C2×Dic433D412yes—
37C4×D433—22no—
38C4×D433—21no—
39#16812358—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—
50C2×D433—42no—
51C2×D433—41no—
52C4×C433—21no—
53C4×C433—41no—
54C4×C433C2×D411yescharacteristic
55C4×C433C2×D411yescharacteristic
56Dic433—21no—
57Dic433—22no—
58C2×C433—21no—
59C2×C433—41no—
60C2×C433#12011yescharacteristic
61D433—41no—
62D433—81no—
63C433#45411yescharacteristic, complemented, Sylow
64#454—4331nomaximal, Sylow
652−1+4—4331no—
66AGL(1,ℤ/8)—4331no—
67AGL(1,ℤ/8)—4331no—
68#124—4331no—
69#104—4331no—
70#104D43311yescharacteristic, complemented
71#101—4331no—
72#41—8661no—
73#41—8661no—
74#41—8662no—
75#41—4331no—
76#41C2×D43311yescharacteristic
77C2×Q8—8661no—
78C2×Q8—8661no—
79C2×Q8—4331no—
80C2×D4—4331no—
81SD16—8661no—
82SD16—8661no—
83D8—8661no—
84D8—8661no—
85M16—4331no—
86M16C2×D43311yescharacteristic
87C22⋊C4—8662no—
88C42C2×D43311yescharacteristic
89C23—8661no—
90Q8—8662no—
91Q8—8661no—
92Q8—8661no—
93Q8—4331no—
94Q8D173211yescharacteristic
95D4—21no—
96D4—8662no—
97D4—8661no—
98D4—8661no—
99D4—8661no—
100D4—4331no—
101D4D173211yescharacteristic
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—8662no—
105C2×C4—8662no—
106C2×C4—8661no—
107C2×C4—4332no—
108C2×C4C22×D43311yescharacteristic
109C8—21no—
110C8—8661no—
111C22—21no—
112C22—8661no—
113C22—17322no—
114C22—17321no—
115C22D4×D43311yescharacteristic
116C4—21no—
117C4—41no—
118C4—8661no—
119C4—8662no—
120C4D4×D43311yescharacteristic
121C4C2×D173211yescharacteristic
122C2—21no—
123C2—41no—
124C2—34641no—
125C2—17321no—
126C2#2835719011yescharacteristic
subgroupG/H
centerC2#28357190
commutator, cosocleC2×C4×C433C23
FrattiniC2×C4C22×D433
Fitting#104×C433C2
socleC2×C433#120
hypercenter#104D433
hypocenterC433#454
LCSG ▷ C2×C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
1. #28357267C2semidirect
2. #28357248C2semidirect
3. #28357245C2semidirect
4. #28357208C2semidirect
5. #28357148C2semidirect
6. #28357138C2semidirect
7. #104×C433C2semidirect
63. C433#454semidirect
70. #104D433semidirect

Maximal subgroups

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

subgroupindex
#283572672
#283572482
#283572452
#283572082
#283571482
#283571382
#104×C4332
#454433