Finite Group Databasebeta

← #42122609

Subgroups — #42122609

Order 27712.

262 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#20294097C211yesmaximal, characteristic, complemented
2#28357251C212yesmaximal, complemented
3#28357250C212yesmaximal, complemented
4#28357246C212yesmaximal, complemented
5#28357245C212yesmaximal, complemented
6#28357240C214yesmaximal, complemented
7C2×#20294081C211yesmaximal, characteristic, complemented
8C2×#41×C433C211yesmaximal, characteristic, complemented
9#20294097—21no—
10#20294097—21no—
11#20294097C2212yes—
12#20294097C2212yescomplemented
13C2×Dic866C2211yescharacteristic, complemented
14C2×D1732C2211yescharacteristic, complemented
15C2×C22⋊D433—21no—
16C2×C4×D433—21no—
17#20294087C2214yescomplemented
18#20294085C2214yescomplemented
19#20294084C2214yescomplemented
20#20294082C2214yescomplemented
21#20294081C2212yescomplemented
22#20294081C2212yescomplemented
23C2×#16812358C2212yescomplemented
24#41×C433—21no—
25#41×C433—21no—
26#41×C433C2212yes—
27#41×C433C2212yescomplemented
28C2×Q8×C433C2211yescharacteristic, complemented
29C2×D4×C433—21no—
30C2×D4×C433C2211yescharacteristic, complemented
31C22×C4×C433—21no—
32C22×C4×C433C2211yescharacteristic
33C22×D433—21no—
34Dic866—21no—
35Dic866C2312yes—
36D1732—21no—
37D1732C2312yes—
38C22⋊D433—22no—
39C22⋊D433—22no—
40C2×Dic433—21no—
41C4×D433—22no—
42C4×D433—22no—
43#16812358C2314yes—
44C23×C433—21no—
45C23×C433D411yescharacteristic
46Q8×C433—21no—
47Q8×C433C2312yes—
48D4×C433—22no—
49D4×C433—22no—
50D4×C433—21no—
51D4×C433C2312yes—
52C2×C4×C433—21no—
53C2×C4×C433—22no—
54C2×C4×C433—22no—
55C2×C4×C433D411yescharacteristic
56C2×C4×C433D411yescharacteristic
57C2×C4×C433D411yescharacteristic
58C2×C4×C433C2311yescharacteristic
59C2×C4×C433C2311yescharacteristic
60C2×C4×C433C2311yescharacteristic
61C22×C433—22no—
62C22×C433—41no—
63C22×C433—42no—
64C22×C433C2×D411yescharacteristic
65C22×C433C2×D411yescharacteristic
66C22×C433C2×D411yescharacteristic
67C2×D433—22no—
68C2×D433—42no—
69C4×C433—22no—
70C4×C433C2×D411yescharacteristic
71C4×C433C2×D411yescharacteristic
72C4×C433C2×D411yescharacteristic
73C4×C433C2411yescharacteristic
74Dic433—22no—
75C2×C433—21no—
76C2×C433—21no—
77C2×C433—21no—
78C2×C433—42no—
79C2×C433C22×D411yescharacteristic
80D433—42no—
81C433#57511yescharacteristic, complemented, Sylow
82#575—4331nomaximal, Sylow
83C2×#41—4331no—
84C2×#41D43311yescharacteristic, complemented
85#137—4332no—
86#137—4332no—
87AGL(1,ℤ/8)—4332no—
88AGL(1,ℤ/8)—4332no—
89#135—4334no—
90C2×M16—4331no—
91#41—21no—
92#41—21no—
93#41—8661no—
94#41—8661no—
95#41—4332no—
96#41—4332no—
97#41C2×D43312yes—
98#41C2×D43312yescomplemented
99C2×Q8—4331no—
100C2×Q8C2×D43311yescharacteristic, complemented
101C2×D4—21no—
102C2×D4—8661no—
103C2×D4—4331no—
104C2×D4C2×D43311yescharacteristic, complemented
105C22×C4—21no—
106C22×C4—8661no—
107C22×C4C2×D43311yescharacteristic
108Q16—4334no—
109SD16—4334no—
110SD16—4334no—
111D8—4334no—
112M16—4332no—
113M16—4332no—
114C2×C8—4332no—
115C23—21no—
116C23—8661no—
117C23C22⋊D43311yescharacteristic
118Q8—21no—
119Q8—8661no—
120Q8—4332no—
121Q8C22×D43312yes—
122D4—22no—
123D4—22no—
124D4—21no—
125D4—8662no—
126D4—8662no—
127D4—8661no—
128D4—4332no—
129D4C22×D43312yes—
130C2×C4—21no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—8661no—
134C2×C4—8662no—
135C2×C4—8662no—
136C2×C4C22⋊D43311yescharacteristic
137C2×C4C22⋊D43311yescharacteristic
138C2×C4C22⋊D43311yescharacteristic
139C2×C4C22×D43311yescharacteristic
140C2×C4C22×D43311yescharacteristic
141C2×C4C22×D43311yescharacteristic
142C8—4334no—
143C22—22no—
144C22—41no—
145C22—42no—
146C22—8662no—
147C22—17322no—
148C22C2×C22⋊D43311yescharacteristic
149C22C2×C22⋊D43311yescharacteristic
150C22C2×C22⋊D43311yescharacteristic
151C4—22no—
152C4—8662no—
153C4C2×C22⋊D43311yescharacteristic
154C4C2×C22⋊D43311yescharacteristic
155C4C2×C22⋊D43311yescharacteristic
156C4C23×D43311yescharacteristic
157C2—21no—
158C2—21no—
159C2—21no—
160C2—42no—
161C2—17322no—
162C2C22×C22⋊D43311yescharacteristic
subgroupG/H
centerC4C2×C22⋊D433
commutator, cosocleC4×C433C24
FrattiniC4C23×D433
FittingC2×#41×C433C2
socleC2×C433C22×D4
hypercenterC2×#41D433
hypocenterC433#575
LCSG ▷ C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#20294097C2semidirect
2. #28357251C2semidirect
3. #28357250C2semidirect
4. #28357246C2semidirect
5. #28357245C2semidirect
6. #28357240C2semidirect
7. C2×#20294081C2semidirect
8. C2×#41×C433C2semidirect
12. #20294097C22semidirect
13. C2×Dic866C22semidirect
14. C2×D1732C22semidirect
17. #20294087C22semidirect
18. #20294085C22semidirect
19. #20294084C22semidirect
20. #20294082C22semidirect
21. #20294081C22semidirect
22. #20294081C22semidirect
23. C2×#16812358C22semidirect
27. #41×C433C22semidirect
28. C2×Q8×C433C22semidirect
30. C2×D4×C433C22semidirect
81. C433#575semidirect
84. C2×#41D433semidirect
98. #41C2×D433semidirect
100. C2×Q8C2×D433semidirect
104. C2×D4C2×D433semidirect

Maximal subgroups

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

subgroupindex
C2×#202940972
#283572512
#283572502
#283572462
#283572452
#283572402
C2×#202940812
C2×#41×C4332
#575433