Finite Group Databasebeta

← #35203399

Subgroups — #35203399

Order 20288.

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#23778826C211yesmaximal, characteristic
2#23778811C211yesmaximal, characteristic, complemented
3#23778769C211yesmaximal, characteristic, complemented
4#23778707C211yesmaximal, characteristic, complemented
5#23778706C211yesmaximal, characteristic, complemented
6#23778696C211yesmaximal, characteristic, complemented
7#137×C317C211yesmaximal, characteristic, complemented
8#19099575—21no—
9#19099574—21no—
10#19099574C2211yescharacteristic
11C2×D1268—21no—
12C2×D1268C2211yescharacteristic
13D4×D317—22no—
14D4×D317—21no—
15#19099564—21no—
16#19099561—21no—
17#19099558C2211yescharacteristic, complemented
18#19099551—22no—
19C4×Dic317C2211yescharacteristic, complemented
20#41×C317C2211yescharacteristic
21C2×Q8×C317C2211yescharacteristic, complemented
22Q16×C317—21no—
23SD16×C317—21no—
24M16×C317C2211yescharacteristic, complemented
25C22×D317—22no—
26C22×D317D412yes—
27Dic634D411yescharacteristic
28D1268—22no—
29D1268—21no—
30D1268—21no—
31D1268D411yescharacteristic
32C22⋊D317—22no—
33C22⋊D317—21no—
34C2×Dic317—21no—
35C4×D317—22no—
36C4×D317—21no—
37#14579261—21no—
38Q8×C317—21no—
39Q8×C317—21no—
40Q8×C317D411yescharacteristic
41D4×C317—21no—
42D4×C317D411yescharacteristic
43C2×C4×C317—21no—
44C2×C4×C317—21no—
45C2×C4×C317C2311yescharacteristic
46C8×C317—21no—
47C22×C317—21no—
48C22×C317C2×D411yescharacteristic
49C2×D317—22no—
50C2×D317—21no—
51C2×D317—42no—
52C2×D317—82no—
53C4×C317—21no—
54C4×C317—41no—
55C4×C317C2×D411yescharacteristic
56C4×C317C2×D411yescharacteristic
57Dic317—21no—
58Dic317—41no—
59C2×C317—21no—
60C2×C317—41no—
61C2×C317#12011yescharacteristic
62D317—42no—
63D317—41no—
64C317#45511yescharacteristic, complemented, Sylow
65#455—3171nomaximal, Sylow
66H(5,𝔽2)—3171no—
67#137—3171no—
68#137D31711yescharacteristic, complemented
69#124—3171no—
70#104—3171no—
71#104—3171no—
72#100—3171no—
73#41—3171no—
74#41—6341no—
75#41—6341no—
76#41C2×D31711yescharacteristic
77C2×Q8C2×D31711yescharacteristic, complemented
78C2×D4—3171no—
79C2×D4—6341no—
80C2×D4—6342no—
81C2×D4—6341no—
82Q16—21no—
83Q16—6341no—
84SD16—21no—
85SD16—6341no—
86M16—3171no—
87M16C2×D31711yescharacteristic, complemented
88C22⋊C4—6342no—
89C42—3171no—
90C23—3172no—
91C23—6342no—
92Q8—21no—
93Q8—21no—
94Q8—3171no—
95Q8C22⋊D31711yescharacteristic
96D4—21no—
97D4—3171no—
98D4—6342no—
99D4—6342no—
100D4—6341no—
101D4—6341no—
102D4—6341no—
103D4C22⋊D31711yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—6342no—
107C2×C4—6341no—
108C2×C4—6341no—
109C2×C4C22×D31711yescharacteristic
110C8—21no—
111C8—6341no—
112C22—21no—
113C22—6341no—
114C22—6342no—
115C22—25362no—
116C22—12682no—
117C22D4×D31711yescharacteristic
118C4—21no—
119C4—41no—
120C4—6341no—
121C4—12681no—
122C4D4×D31711yescharacteristic
123C4C2×C22⋊D31711yescharacteristic
124C2—21no—
125C2—41no—
126C2—12681no—
127C2—12682no—
128C2#2377874811yescharacteristic
subgroupG/H
centerC2#23778748
commutator, cosocleC2×C4×C317C23
FrattiniC2×C4C22×D317
Fitting#137×C317C2
socleC2×C317#120
prime radical#23778826C2
hypercenter#137D317
hypocenterC317#455
LCSG ▷ C2×C4×C317 ▷ C2×C317 ▷ C317
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #23778811C2semidirect
3. #23778769C2semidirect
4. #23778707C2semidirect
5. #23778706C2semidirect
6. #23778696C2semidirect
7. #137×C317C2semidirect
17. #19099558C22semidirect
19. C4×Dic317C22semidirect
21. C2×Q8×C317C22semidirect
24. M16×C317C22semidirect
64. C317#455semidirect
68. #137D317semidirect
77. C2×Q8C2×D317semidirect
87. M16C2×D317semidirect

Maximal subgroups

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

subgroupindex
#237788262
#237788112
#237787692
#237787072
#237787062
#237786962
#137×C3172
#455317