Finite Group Databasebeta

← #36377963

Subgroups — #36377963

Order 22592.

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#26015038C211yesmaximal, characteristic
2#26015023C211yesmaximal, characteristic, complemented
3#26014981C211yesmaximal, characteristic, complemented
4#26014919C211yesmaximal, characteristic, complemented
5#26014918C211yesmaximal, characteristic, complemented
6#26014908C211yesmaximal, characteristic, complemented
7#137×C353C211yesmaximal, characteristic, complemented
8#19150738—21no—
9#19150737—21no—
10#19150737C2211yescharacteristic
11C2×D1412—21no—
12C2×D1412C2211yescharacteristic
13D4×D353—22no—
14D4×D353—21no—
15#19150727—21no—
16#19150724—21no—
17#19150721C2211yescharacteristic, complemented
18#19150714—22no—
19C4×Dic353C2211yescharacteristic, complemented
20#41×C353C2211yescharacteristic
21C2×Q8×C353C2211yescharacteristic, complemented
22Q16×C353—21no—
23SD16×C353—21no—
24M16×C353C2211yescharacteristic, complemented
25C22×D353—22no—
26C22×D353D412yes—
27Dic706D411yescharacteristic
28D1412—22no—
29D1412—21no—
30D1412—21no—
31D1412D411yescharacteristic
32C22⋊D353—22no—
33C22⋊D353—21no—
34C2×Dic353—21no—
35C4×D353—22no—
36C4×D353—21no—
37#15667725—21no—
38Q8×C353—21no—
39Q8×C353—21no—
40Q8×C353D411yescharacteristic
41D4×C353—21no—
42D4×C353D411yescharacteristic
43C2×C4×C353—21no—
44C2×C4×C353—21no—
45C2×C4×C353C2311yescharacteristic
46C8×C353—21no—
47C22×C353—21no—
48C22×C353C2×D411yescharacteristic
49C2×D353—22no—
50C2×D353—21no—
51C2×D353—42no—
52C2×D353—82no—
53C4×C353—21no—
54C4×C353—41no—
55C4×C353C2×D411yescharacteristic
56C4×C353C2×D411yescharacteristic
57Dic353—21no—
58Dic353—41no—
59C2×C353—21no—
60C2×C353—41no—
61C2×C353#12011yescharacteristic
62D353—42no—
63D353—41no—
64C353#45511yescharacteristic, complemented, Sylow
65#455—3531nomaximal, Sylow
66H(5,𝔽2)—3531no—
67#137—3531no—
68#137D35311yescharacteristic, complemented
69#124—3531no—
70#104—3531no—
71#104—3531no—
72#100—3531no—
73#41—3531no—
74#41—7061no—
75#41—7061no—
76#41C2×D35311yescharacteristic
77C2×Q8C2×D35311yescharacteristic, complemented
78C2×D4—3531no—
79C2×D4—7061no—
80C2×D4—7062no—
81C2×D4—7061no—
82Q16—21no—
83Q16—7061no—
84SD16—21no—
85SD16—7061no—
86M16—3531no—
87M16C2×D35311yescharacteristic, complemented
88C22⋊C4—7062no—
89C42—3531no—
90C23—3532no—
91C23—7062no—
92Q8—21no—
93Q8—21no—
94Q8—3531no—
95Q8C22⋊D35311yescharacteristic
96D4—21no—
97D4—3531no—
98D4—7062no—
99D4—7062no—
100D4—7061no—
101D4—7061no—
102D4—7061no—
103D4C22⋊D35311yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—7062no—
107C2×C4—7061no—
108C2×C4—7061no—
109C2×C4C22×D35311yescharacteristic
110C8—21no—
111C8—7061no—
112C22—21no—
113C22—28242no—
114C22—14122no—
115C22—7061no—
116C22—7062no—
117C22D4×D35311yescharacteristic
118C4—21no—
119C4—41no—
120C4—14121no—
121C4—7061no—
122C4D4×D35311yescharacteristic
123C4C2×C22⋊D35311yescharacteristic
124C2—21no—
125C2—41no—
126C2—14121no—
127C2—14122no—
128C2#2601496011yescharacteristic
subgroupG/H
centerC2#26014960
commutator, cosocleC2×C4×C353C23
FrattiniC2×C4C22×D353
Fitting#137×C353C2
socleC2×C353#120
prime radical#26015038C2
hypercenter#137D353
hypocenterC353#455
LCSG ▷ C2×C4×C353 ▷ C2×C353 ▷ C353
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #26015023C2semidirect
3. #26014981C2semidirect
4. #26014919C2semidirect
5. #26014918C2semidirect
6. #26014908C2semidirect
7. #137×C353C2semidirect
17. #19150721C22semidirect
19. C4×Dic353C22semidirect
21. C2×Q8×C353C22semidirect
24. M16×C353C22semidirect
64. C353#455semidirect
68. #137D353semidirect
77. C2×Q8C2×D353semidirect
87. M16C2×D353semidirect

Maximal subgroups

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

subgroupindex
#260150382
#260150232
#260149812
#260149192
#260149182
#260149082
#137×C3532
#455353