Finite Group Databasebeta

← #26011679

Subgroups — #26011679

Order 11072.

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#19149724C211yesmaximal, characteristic, complemented
2#19149649C212yesmaximal, complemented
3#19149647C211yesmaximal, characteristic, complemented
4#19149593C212yesmaximal, complemented
5#99×C173C211yesmaximal, characteristic, complemented
6C23×D173C2211yescharacteristic, complemented
7#14583826—21no—
8#14583825—22no—
9C2×D692—22no—
10C2×C22⋊D173—21no—
11C2×C22⋊D173—21no—
12C2×C22⋊D173C2212yes—
13D4×D173—22no—
14C22⋊Dic173C2211yescharacteristic, complemented
15#14583803—22no—
16#14583803—21no—
17C2×D4×C173C2211yescharacteristic
18C22⋊C4×C173C2212yescomplemented
19C22×D173—21no—
20C22×D173—22no—
21C22×D173—21no—
22C22×D173—42no—
23C22×D173D412yescomplemented
24Dic346—21no—
25D692—21no—
26D692—44no—
27C22⋊D173—22no—
28C22⋊D173—22no—
29C22⋊D173—22no—
30C22⋊D173—42no—
31C2×Dic173—21no—
32C2×Dic173—21no—
33C2×Dic173D412yescomplemented
34C4×D173—22no—
35C23×C173D411yescharacteristic, complemented
36C23×C173C2311yescharacteristic
37D4×C173—22no—
38C2×C4×C173—22no—
39C2×C4×C173D411yescharacteristic, complemented
40C22×C173—21no—
41C22×C173—41no—
42C22×C173—41no—
43C22×C173C2×D412yes—
44C22×C173C2×D411yescharacteristic
45C2×D173—22no—
46C2×D173—24no—
47C2×D173—22no—
48C2×D173—41no—
49C2×D173—42no—
50C2×D173—41no—
51C2×D173—42no—
52C2×D173—81no—
53C4×C173—21no—
54C4×C173—42no—
55Dic173—22no—
56Dic173—41no—
57C2×C173—21no—
58C2×C173—22no—
59C2×C173—41no—
60C2×C173#12011yescharacteristic
61D173—42no—
62D173—42no—
63C173B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—1731nomaximal, Sylow
65H(5,𝔽2)—1731no—
66#120—1731no—
67#120—1732no—
68#99—1732no—
69#99D17311yescharacteristic, complemented
70C24—1731no—
71#41—3462no—
72#41—3461no—
73C2×D4—3462no—
74C2×D4—3462no—
75C2×D4—3461no—
76C2×D4—3461no—
77C2×D4—1732no—
78C2×D4C2×D17311yescharacteristic
79C22⋊C4—3462no—
80C22⋊C4—3461no—
81C22⋊C4—1731no—
82C22⋊C4C2×D17312yescomplemented
83C23—3461no—
84C23—3462no—
85C23—3461no—
86C23—1732no—
87C23—6922no—
88C23D69211yescharacteristic, complemented
89C23C22×D17311yescharacteristic
90Q8—3461no—
91D4—22no—
92D4—3461no—
93D4—3462no—
94D4—3462no—
95D4—3462no—
96D4—6924no—
97D4—6922no—
98C2×C4—22no—
99C2×C4—3462no—
100C2×C4—3461no—
101C2×C4—3461no—
102C2×C4—1732no—
103C2×C4D69211yescharacteristic, complemented
104C22—21no—
105C22—41no—
106C22—41no—
107C22—3464no—
108C22—3462no—
109C22—3462no—
110C22—13841no—
111C22—6922no—
112C22—6922no—
113C22—6921no—
114C22—6921no—
115C22D4×D17312yes—
116C22C2×D69211yescharacteristic
117C4—21no—
118C4—42no—
119C4—3462no—
120C4—6921no—
121C2—22no—
122C2—21no—
123C2—41no—
124C2—6922no—
125C2—6922no—
126C2#1914964911yescharacteristic
subgroupG/H
centerC2#19149649
commutator, cosocleC23×C173C23
FrattiniC23C22×D173
Fitting#99×C173C2
socleC2×C173#120
hypercenter#99D173
hypocenterC173B(4,𝔽2)
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #19149724C2semidirect
2. #19149649C2semidirect
3. #19149647C2semidirect
4. #19149593C2semidirect
5. #99×C173C2semidirect
6. C23×D173C22semidirect
14. C22⋊Dic173C22semidirect
18. C22⋊C4×C173C22semidirect
23. C22×D173D4semidirect
33. C2×Dic173D4semidirect
35. C23×C173D4semidirect
39. C2×C4×C173D4semidirect
63. C173B(4,𝔽2)semidirect
69. #99D173semidirect
82. C22⋊C4C2×D173semidirect
88. C23D692semidirect
103. C2×C4D692semidirect

Maximal subgroups

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

subgroupindex
#191497242
#191496492
#191496472
#191495932
#99×C1732
B(4,𝔽2)173