Finite Group Databasebeta

← #17986187

Subgroups — #17986187

Order 4544.

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#14556173C211yesmaximal, characteristic, complemented
2#14556098C212yesmaximal, complemented
3#14556096C211yesmaximal, characteristic, complemented
4#14556042C212yesmaximal, complemented
5#99×C71C211yesmaximal, characteristic, complemented
6C23×D71C2211yescharacteristic, complemented
7#11765865—21no—
8#11765864—22no—
9C2×D284—22no—
10C2×C22⋊D71—21no—
11C2×C22⋊D71—21no—
12C2×C22⋊D71C2212yes—
13D4×D71—22no—
14C22⋊Dic71C2211yescharacteristic, complemented
15(C2×D71)⋊C4—21no—
16(C2×D71)⋊C4—22no—
17C2×D4×C71C2211yescharacteristic
18C22⋊C4×C71C2212yescomplemented
19C23×C71D411yescharacteristic, complemented
20C23×C71C2311yescharacteristic
21C22×D71—21no—
22C22×D71—22no—
23C22×D71—21no—
24C22×D71—42no—
25C22×D71D412yescomplemented
26D4×C71—22no—
27C2×C4×C71—22no—
28C2×C4×C71D411yescharacteristic, complemented
29C22⋊D71—22no—
30C22⋊D71—22no—
31C22⋊D71—22no—
32C22⋊D71—42no—
33C2×Dic71—21no—
34C2×Dic71—21no—
35C2×Dic71D412yescomplemented
36D284—21no—
37D284—44no—
38C4×D71—22no—
39Dic142—21no—
40C22×C71—21no—
41C22×C71—41no—
42C22×C71—41no—
43C22×C71C2×D412yes—
44C22×C71C2×D411yescharacteristic
45C2×D71—24no—
46C2×D71—22no—
47C2×D71—22no—
48C2×D71—41no—
49C2×D71—42no—
50C2×D71—41no—
51C2×D71—42no—
52C2×D71—81no—
53C4×C71—21no—
54C4×C71—42no—
55Dic71—22no—
56Dic71—41no—
57C2×C71—21no—
58C2×C71—22no—
59C2×C71—41no—
60C2×C71#12011yescharacteristic
61D71—42no—
62D71—42no—
63C71B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—711nomaximal, Sylow
65H(5,𝔽2)—711no—
66#120—711no—
67#120—712no—
68#99—712no—
69#99D7111yescharacteristic, complemented
70C24—711no—
71#41—1422no—
72#41—1421no—
73C2×D4—712no—
74C2×D4—1422no—
75C2×D4—1422no—
76C2×D4—1421no—
77C2×D4—1421no—
78C2×D4C2×D7111yescharacteristic
79C22⋊C4—711no—
80C22⋊C4—1422no—
81C22⋊C4—1421no—
82C22⋊C4C2×D7112yescomplemented
83C23—2842no—
84C23—712no—
85C23—1421no—
86C23—1422no—
87C23—1421no—
88C23D28411yescharacteristic, complemented
89C23C22×D7111yescharacteristic
90Q8—1421no—
91D4—22no—
92D4—2844no—
93D4—2842no—
94D4—1421no—
95D4—1422no—
96D4—1422no—
97D4—1422no—
98C2×C4—22no—
99C2×C4—712no—
100C2×C4—1422no—
101C2×C4—1421no—
102C2×C4—1421no—
103C2×C4D28411yescharacteristic, complemented
104C22—21no—
105C22—41no—
106C22—41no—
107C22—2842no—
108C22—2842no—
109C22—2841no—
110C22—2841no—
111C22—5681no—
112C22—1424no—
113C22—1422no—
114C22—1422no—
115C22D4×D7112yes—
116C22C2×D28411yescharacteristic
117C4—21no—
118C4—42no—
119C4—2841no—
120C4—1422no—
121C2—22no—
122C2—21no—
123C2—41no—
124C2—2842no—
125C2—2842no—
126C2#1455609811yescharacteristic
subgroupG/H
centerC2#14556098
commutator, cosocleC23×C71C23
FrattiniC23C22×D71
Fitting#99×C71C2
socleC2×C71#120
hypercenter#99D71
hypocenterC71B(4,𝔽2)
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #14556173C2semidirect
2. #14556098C2semidirect
3. #14556096C2semidirect
4. #14556042C2semidirect
5. #99×C71C2semidirect
6. C23×D71C22semidirect
14. C22⋊Dic71C22semidirect
18. C22⋊C4×C71C22semidirect
19. C23×C71D4semidirect
25. C22×D71D4semidirect
28. C2×C4×C71D4semidirect
35. C2×Dic71D4semidirect
63. C71B(4,𝔽2)semidirect
69. #99D71semidirect
82. C22⋊C4C2×D71semidirect
88. C23D284semidirect
103. C2×C4D284semidirect

Maximal subgroups

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

subgroupindex
#145561732
#145560982
#145560962
#145560422
#99×C712
B(4,𝔽2)71