Finite Group Databasebeta

← #38762698

Subgroups — #38762698

Order 25408.

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#27163142C211yesmaximal, characteristic
2#27163127C211yesmaximal, characteristic, complemented
3#27163085C211yesmaximal, characteristic, complemented
4#27163023C211yesmaximal, characteristic, complemented
5#27163022C211yesmaximal, characteristic, complemented
6#27163012C211yesmaximal, characteristic, complemented
7#137×C397C211yesmaximal, characteristic, complemented
8#20263732—21no—
9#20263731—21no—
10#20263731C2211yescharacteristic
11C2×D1588—21no—
12C2×D1588C2211yescharacteristic
13D4×D397—22no—
14D4×D397—21no—
15#20263721—21no—
16#20263718—21no—
17#20263715C2211yescharacteristic, complemented
18#20263708—22no—
19C4×Dic397C2211yescharacteristic, complemented
20#41×C397C2211yescharacteristic
21C2×Q8×C397C2211yescharacteristic, complemented
22Q16×C397—21no—
23SD16×C397—21no—
24M16×C397C2211yescharacteristic, complemented
25C22×D397—22no—
26C22×D397D412yes—
27Dic794D411yescharacteristic
28D1588—22no—
29D1588—21no—
30D1588—21no—
31D1588D411yescharacteristic
32C22⋊D397—22no—
33C22⋊D397—21no—
34C2×Dic397—21no—
35C4×D397—22no—
36C4×D397—21no—
37#15691787—21no—
38Q8×C397—21no—
39Q8×C397—21no—
40Q8×C397D411yescharacteristic
41D4×C397—21no—
42D4×C397D411yescharacteristic
43C2×C4×C397—21no—
44C2×C4×C397—21no—
45C2×C4×C397C2311yescharacteristic
46C8×C397—21no—
47C22×C397—21no—
48C22×C397C2×D411yescharacteristic
49C2×D397—22no—
50C2×D397—21no—
51C2×D397—42no—
52C2×D397—82no—
53C4×C397—21no—
54C4×C397—41no—
55C4×C397C2×D411yescharacteristic
56C4×C397C2×D411yescharacteristic
57Dic397—21no—
58Dic397—41no—
59C2×C397—21no—
60C2×C397—41no—
61C2×C397#12011yescharacteristic
62D397—42no—
63D397—41no—
64C397#45511yescharacteristic, complemented, Sylow
65#455—3971nomaximal, Sylow
66H(5,𝔽2)—3971no—
67#137—3971no—
68#137D39711yescharacteristic, complemented
69#124—3971no—
70#104—3971no—
71#104—3971no—
72#100—3971no—
73#41—7941no—
74#41—7941no—
75#41—3971no—
76#41C2×D39711yescharacteristic
77C2×Q8C2×D39711yescharacteristic, complemented
78C2×D4—7941no—
79C2×D4—7942no—
80C2×D4—7941no—
81C2×D4—3971no—
82Q16—21no—
83Q16—7941no—
84SD16—21no—
85SD16—7941no—
86M16—3971no—
87M16C2×D39711yescharacteristic, complemented
88C22⋊C4—7942no—
89C42—3971no—
90C23—7942no—
91C23—3972no—
92Q8—21no—
93Q8—21no—
94Q8—3971no—
95Q8C22⋊D39711yescharacteristic
96D4—21no—
97D4—7942no—
98D4—7942no—
99D4—7941no—
100D4—7941no—
101D4—7941no—
102D4—3971no—
103D4C22⋊D39711yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—7942no—
107C2×C4—7941no—
108C2×C4—7941no—
109C2×C4C22×D39711yescharacteristic
110C8—21no—
111C8—7941no—
112C22—21no—
113C22—7942no—
114C22—7941no—
115C22—15882no—
116C22—31762no—
117C22D4×D39711yescharacteristic
118C4—21no—
119C4—41no—
120C4—7941no—
121C4—15881no—
122C4D4×D39711yescharacteristic
123C4C2×C22⋊D39711yescharacteristic
124C2—21no—
125C2—41no—
126C2—15882no—
127C2—15881no—
128C2#2716306411yescharacteristic
subgroupG/H
centerC2#27163064
commutator, cosocleC2×C4×C397C23
FrattiniC2×C4C22×D397
Fitting#137×C397C2
socleC2×C397#120
prime radical#27163142C2
hypercenter#137D397
hypocenterC397#455
LCSG ▷ C2×C4×C397 ▷ C2×C397 ▷ C397
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #27163127C2semidirect
3. #27163085C2semidirect
4. #27163023C2semidirect
5. #27163022C2semidirect
6. #27163012C2semidirect
7. #137×C397C2semidirect
17. #20263715C22semidirect
19. C4×Dic397C22semidirect
21. C2×Q8×C397C22semidirect
24. M16×C397C22semidirect
64. C397#455semidirect
68. #137D397semidirect
77. C2×Q8C2×D397semidirect
87. M16C2×D397semidirect

Maximal subgroups

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

subgroupindex
#271631422
#271631272
#271630852
#271630232
#271630222
#271630122
#137×C3972
#455397