Finite Group Databasebeta

← #42122608

Subgroups — #42122608

Order 27712.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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#41×D433C212yesmaximal, complemented
2#28357252C212yesmaximal, complemented
3#28357249C212yesmaximal, complemented
4#28357247C212yesmaximal, complemented
5#28357244C212yesmaximal, complemented
6#28357242C211yesmaximal, characteristic, complemented
7#28357240C212yesmaximal, complemented
8C2×#20294080C211yesmaximal, characteristic, complemented
9#135×C433C211yesmaximal, characteristic, complemented
10#20294098C2212yes—
11#20294097—22no—
12#20294097C2212yescomplemented
13#20294096—22no—
14#20294096C2212yes—
15Q8×D433C2212yescomplemented
16D4×D433—22no—
17D4×D433C2212yescomplemented
18C2×C4×D433—22no—
19C2×C4×D433C2211yescharacteristic
20Dic1732C2211yescharacteristic, complemented
21#20294087C2212yescomplemented
22#20294086C2212yescomplemented
23#20294085C2212yescomplemented
24#20294084C2212yescomplemented
25D3464C2211yescharacteristic, complemented
26#20294082C2212yescomplemented
27#20294080C2212yescomplemented
28#20294080C2212yescomplemented
29C2×#16812358C2211yescharacteristic, complemented
30#41×C433C2212yescomplemented
31Q16×C433C2211yescharacteristic, complemented
32SD16×C433C2212yescomplemented
33D8×C433C2211yescharacteristic, complemented
34C2×C8×C433C2211yescharacteristic, complemented
35C22×D433—22no—
36C22×D433D411yescharacteristic
37Dic866—22no—
38Dic866C2312yes—
39D1732—22no—
40D1732C2312yes—
41C22⋊D433—22no—
42C22⋊D433—22no—
43C22⋊D433—22no—
44C2×Dic433—22no—
45C2×Dic433D411yescharacteristic
46C4×D433—22no—
47C4×D433—22no—
48C4×D433—22no—
49C4×D433D412yes—
50C4×D433C2312yes—
51#16812358C2312yes—
52Q8×C433C2312yes—
53D4×C433—22no—
54D4×C433C2312yes—
55C2×C4×C433—22no—
56C2×C4×C433C2311yescharacteristic
57C8×C433C2311yescharacteristic
58C8×C433C2311yescharacteristic
59C22×C433—22no—
60C22×C433C2×D411yescharacteristic
61C2×D433—22no—
62C2×D433—41no—
63C2×D433—44no—
64C2×D433C2×D412yes—
65C4×C433—22no—
66C4×C433C2×D411yescharacteristic
67C4×C433C2411yescharacteristic
68Dic433—22no—
69Dic433C2×D412yes—
70C2×C433—21no—
71C2×C433—42no—
72C2×C433C22×D411yescharacteristic
73D433—22no—
74D433—42no—
75C433#57511yescharacteristic, complemented, Sylow
76#575—4331nomaximal, Sylow
77C2×#41—4332no—
78#137—4332no—
79#137—4332no—
80AGL(1,ℤ/8)—4332no—
81AGL(1,ℤ/8)—4332no—
82#135—4332no—
83#135—4331no—
84#135D43311yescharacteristic, complemented
85C2×M16—4331no—
86#41—8662no—
87#41—8662no—
88#41—4332no—
89#41—4332no—
90#41—4332no—
91#41C2×D43312yescomplemented
92C2×Q8—4332no—
93C2×D4—8662no—
94C2×D4—4332no—
95C22×C4—8662no—
96C22×C4—4331no—
97Q16—4332no—
98Q16—4331no—
99Q16C2×D43311yescharacteristic, complemented
100SD16—4332no—
101SD16—4332no—
102SD16—4332no—
103SD16C2×D43312yescomplemented
104D8—4332no—
105D8—4331no—
106D8C2×D43311yescharacteristic, complemented
107M16—4332no—
108M16—4332no—
109C2×C8—4331no—
110C2×C8C2×D43311yescharacteristic, complemented
111C23—8662no—
112C23—4331no—
113Q8—8662no—
114Q8—4332no—
115Q8C22×D43312yes—
116D4—22no—
117D4—8662no—
118D4—8662no—
119D4—8662no—
120D4—8662no—
121D4—4332no—
122D4C22×D43312yes—
123C2×C4—22no—
124C2×C4—8662no—
125C2×C4—8662no—
126C2×C4—8662no—
127C2×C4—8662no—
128C2×C4—4331no—
129C2×C4—4332no—
130C2×C4—4332no—
131C2×C4C22×D43311yescharacteristic
132C8—4332no—
133C8C22×D43311yescharacteristic
134C8C22×D43311yescharacteristic
135C22—22no—
136C22—8662no—
137C22—4332no—
138C22—17321no—
139C22—17324no—
140C22D4×D43311yescharacteristic
141C4—22no—
142C4—8662no—
143C4—4332no—
144C4D4×D43311yescharacteristic
145C4C23×D43311yescharacteristic
146C2—21no—
147C2—42no—
148C2—8662no—
149C2—17322no—
150C2C2×D4×D43311yescharacteristic
subgroupG/H
centerC4D4×D433
commutator, cosocleC4×C433C24
FrattiniC4C23×D433
Fitting#135×C433C2
socleC2×C433C22×D4
hypercenter#135D433
hypocenterC433#575
LCSG ▷ C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D433C2semidirect
2. #28357252C2semidirect
3. #28357249C2semidirect
4. #28357247C2semidirect
5. #28357244C2semidirect
6. #28357242C2semidirect
7. #28357240C2semidirect
8. C2×#20294080C2semidirect
9. #135×C433C2semidirect
12. #20294097C22semidirect
15. Q8×D433C22semidirect
17. D4×D433C22semidirect
20. Dic1732C22semidirect
21. #20294087C22semidirect
22. #20294086C22semidirect
23. #20294085C22semidirect
24. #20294084C22semidirect
25. D3464C22semidirect
26. #20294082C22semidirect
27. #20294080C22semidirect
28. #20294080C22semidirect
29. C2×#16812358C22semidirect
30. #41×C433C22semidirect
31. Q16×C433C22semidirect
32. SD16×C433C22semidirect
33. D8×C433C22semidirect
34. C2×C8×C433C22semidirect
75. C433#575semidirect
84. #135D433semidirect
91. #41C2×D433semidirect
99. Q16C2×D433semidirect
103. SD16C2×D433semidirect
106. D8C2×D433semidirect
110. C2×C8C2×D433semidirect

Maximal subgroups

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

subgroupindex
#41×D4332
#283572522
#283572492
#283572472
#283572442
#283572422
#283572402
C2×#202940802
#135×C4332
#575433