Finite Group Databasebeta

← #37526631

Subgroups — #37526631

Order 22976.

198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 198 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#26040060C211yesmaximal, characteristic, complemented
2#26039985C212yesmaximal, complemented
3#26039983C211yesmaximal, characteristic, complemented
4#26039929C212yesmaximal, complemented
5#99×C359C211yesmaximal, characteristic, complemented
6C23×D359C2211yescharacteristic, complemented
7#19153341—21no—
8#19153340—22no—
9C2×D1436—22no—
10C2×C22⋊D359—21no—
11C2×C22⋊D359—21no—
12C2×C22⋊D359C2212yes—
13D4×D359—22no—
14C22⋊Dic359C2211yescharacteristic, complemented
15(C2×D359)⋊C4—22no—
16(C2×D359)⋊C4—21no—
17C2×D4×C359C2211yescharacteristic
18C22⋊C4×C359C2212yescomplemented
19C22×D359—21no—
20C22×D359—22no—
21C22×D359—21no—
22C22×D359—42no—
23C22×D359D412yescomplemented
24Dic718—21no—
25D1436—21no—
26D1436—44no—
27C22⋊D359—22no—
28C22⋊D359—22no—
29C22⋊D359—22no—
30C22⋊D359—42no—
31C2×Dic359—21no—
32C2×Dic359—21no—
33C2×Dic359D412yescomplemented
34C4×D359—22no—
35C23×C359D411yescharacteristic, complemented
36C23×C359C2311yescharacteristic
37D4×C359—22no—
38C2×C4×C359—22no—
39C2×C4×C359D411yescharacteristic, complemented
40C22×C359—21no—
41C22×C359—41no—
42C22×C359—41no—
43C22×C359C2×D412yes—
44C22×C359C2×D411yescharacteristic
45C2×D359—22no—
46C2×D359—24no—
47C2×D359—22no—
48C2×D359—41no—
49C2×D359—42no—
50C2×D359—41no—
51C2×D359—42no—
52C2×D359—81no—
53C4×C359—21no—
54C4×C359—42no—
55Dic359—22no—
56Dic359—41no—
57C2×C359—21no—
58C2×C359—22no—
59C2×C359—41no—
60C2×C359#12011yescharacteristic
61D359—42no—
62D359—42no—
63C359B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—3591nomaximal, Sylow
65H(5,𝔽2)—3591no—
66#120—3591no—
67#120—3592no—
68#99—3592no—
69#99D35911yescharacteristic, complemented
70C24—3591no—
71#41—7182no—
72#41—7181no—
73C2×D4—3592no—
74C2×D4—7182no—
75C2×D4—7182no—
76C2×D4—7181no—
77C2×D4—7181no—
78C2×D4C2×D35911yescharacteristic
79C22⋊C4—3591no—
80C22⋊C4—7182no—
81C22⋊C4—7181no—
82C22⋊C4C2×D35912yescomplemented
83C23—3592no—
84C23—14362no—
85C23—7181no—
86C23—7182no—
87C23—7181no—
88C23D143611yescharacteristic, complemented
89C23C22×D35911yescharacteristic
90Q8—7181no—
91D4—22no—
92D4—14364no—
93D4—14362no—
94D4—7181no—
95D4—7182no—
96D4—7182no—
97D4—7182no—
98C2×C4—22no—
99C2×C4—3592no—
100C2×C4—7182no—
101C2×C4—7181no—
102C2×C4—7181no—
103C2×C4D143611yescharacteristic, complemented
104C22—21no—
105C22—41no—
106C22—41no—
107C22—28721no—
108C22—14362no—
109C22—14362no—
110C22—14361no—
111C22—14361no—
112C22—7184no—
113C22—7182no—
114C22—7182no—
115C22D4×D35912yes—
116C22C2×D143611yescharacteristic
117C4—21no—
118C4—42no—
119C4—14361no—
120C4—7182no—
121C2—22no—
122C2—21no—
123C2—41no—
124C2—14362no—
125C2—14362no—
126C2#2603998511yescharacteristic
subgroupG/H
centerC2#26039985
commutator, cosocleC23×C359C23
FrattiniC23C22×D359
Fitting#99×C359C2
socleC2×C359#120
hypercenter#99D359
hypocenterC359B(4,𝔽2)
LCSG ▷ C23×C359 ▷ C2×C359 ▷ C359
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #26040060C2semidirect
2. #26039985C2semidirect
3. #26039983C2semidirect
4. #26039929C2semidirect
5. #99×C359C2semidirect
6. C23×D359C22semidirect
14. C22⋊Dic359C22semidirect
18. C22⋊C4×C359C22semidirect
23. C22×D359D4semidirect
33. C2×Dic359D4semidirect
35. C23×C359D4semidirect
39. C2×C4×C359D4semidirect
63. C359B(4,𝔽2)semidirect
69. #99D359semidirect
82. C22⋊C4C2×D359semidirect
88. C23D1436semidirect
103. C2×C4D1436semidirect

Maximal subgroups

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

subgroupindex
#260400602
#260399852
#260399832
#260399292
#99×C3592
B(4,𝔽2)359