Finite Group Databasebeta

← D4×C11×AGL(1,𝔽5) #13183338

Subgroups — D4×C11×AGL(1,𝔽5) #13183338

Order 1760.

188 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C22×C11×AGL(1,𝔽5)C212yesmaximal, complemented
2D4×D5×C11C211yesmaximal, characteristic
3C11×C22⋊AGL(1,𝔽5)C212yesmaximal, complemented
4C11×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
5C4×C11×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
6C22×D5×C11C2212yes—
7C2×C11×AGL(1,𝔽5)—24no—
8C2×C11×AGL(1,𝔽5)C2211yescharacteristic
9C2×C11×AGL(1,𝔽5)C2212yes—
10C2×C11×AGL(1,𝔽5)C2211yescharacteristic
11C5×D4×C11C411yescharacteristic, complemented
12C11×C22⋊D5C412yescomplemented
13C11×D20C411yescharacteristic, complemented
14C4×D5×C11C2211yescharacteristic
15C4×D4×C11—51nomaximal, Hall
16C22×C5×C11C2×C412yescomplemented
17C2×D5×C11—22no—
18C2×D5×C11—22no—
19C2×D5×C11C2×C412yescomplemented
20C2×D5×C11C2311yescharacteristic
21C11×AGL(1,𝔽5)—22no—
22C11×AGL(1,𝔽5)—21no—
23C11×AGL(1,𝔽5)D412yesdirect product factor
24C4×C5×C11C2×C411yescharacteristic, complemented
25C11×Dic5C2×C411yescharacteristic, complemented
26C2×D4×C11—51no—
27C22×C4×C11—52no—
28C11×C4⋊C4—51no—
29C11×C22⋊C4—52no—
30C42×C11—51no—
31D4×AGL(1,𝔽5)C1111yesmaximal, characteristic, direct product factor, Hall
32C2×C5×C11—22no—
33C2×C5×C11C22×C411yescharacteristic
34D5×C11—22no—
35D5×C11C2×D411yescharacteristic
36D5×C11#4111yescharacteristic
37C23×C11—52no—
38D4×C11—52no—
39D4×C11—51no—
40D4×C11AGL(1,𝔽5)11yescharacteristic, direct product factor
41C2×C4×C11—51no—
42C2×C4×C11—51no—
43C2×C4×C11—51no—
44C2×C4×C11—52no—
45C2×C4×C11—104no—
46C22×AGL(1,𝔽5)C2×C1112yescomplemented
47D4×D5C2×C1111yescharacteristic
48C22⋊AGL(1,𝔽5)C2×C1112yescomplemented
49C4⋊AGL(1,𝔽5)C2×C1111yescharacteristic, complemented
50C4×AGL(1,𝔽5)C2×C1111yescharacteristic, complemented
51C5×C11C4×D411yescharacteristic, complemented, Hall
52C22×C11—51no—
53C22×C11—52no—
54C22×C11—102no—
55C22×C11—102no—
56C22×C11C2×AGL(1,𝔽5)12yescomplemented
57C4×C11—52no—
58C4×C11—51no—
59C4×C11—102no—
60C4×C11—101no—
61C4×C11C2×AGL(1,𝔽5)11yescharacteristic, complemented
62C22×D5C22×C1112yes—
63C2×AGL(1,𝔽5)—24no—
64C2×AGL(1,𝔽5)C22×C1111yescharacteristic
65C2×AGL(1,𝔽5)C22×C1112yes—
66C2×AGL(1,𝔽5)C22×C1111yescharacteristic
67C5×D4C4×C1111yescharacteristic, complemented
68C22⋊D5C4×C1112yescomplemented
69D20C4×C1111yescharacteristic, complemented
70C4×D5C22×C1111yescharacteristic
71C4×D4—51noSylow
72C2×C11—22no—
73C2×C11—51no—
74C2×C11—51no—
75C2×C11—102no—
76C2×C11C22×AGL(1,𝔽5)11yescharacteristic
77C22×C5C2×C4×C1112yescomplemented
78C2×D5—22no—
79C2×D5—22no—
80C2×D5C2×C4×C1112yescomplemented
81C2×D5C23×C1111yescharacteristic
82AGL(1,𝔽5)—22no—
83AGL(1,𝔽5)—21no—
84AGL(1,𝔽5)D4×C1112yesdirect product factor
85C4×C5C2×C4×C1111yescharacteristic, complemented
86Dic5C2×C4×C1111yescharacteristic, complemented
87C2×D4—51no—
88C22×C4—52no—
89C4⋊C4—51no—
90C22⋊C4—52no—
91C42—51no—
92C11D4×AGL(1,𝔽5)11yescharacteristic, direct product factor, Sylow
93C2×C5—22no—
94C2×C5C22×C4×C1111yescharacteristic
95D5—22no—
96D5C2×D4×C1111yescharacteristic
97D5C11×#4111yescharacteristic
98C23—52no—
99D4—52no—
100D4—51no—
101D4C11×AGL(1,𝔽5)11yescharacteristic, direct product factor
102C2×C4—51no—
103C2×C4—51no—
104C2×C4—51no—
105C2×C4—52no—
106C2×C4—104no—
107C5C4×D4×C1111yescharacteristic, complemented, Sylow
108C22—51no—
109C22—52no—
110C22—102no—
111C22—102no—
112C22C2×C11×AGL(1,𝔽5)12yescomplemented
113C4—52no—
114C4—51no—
115C4—102no—
116C4—101no—
117C4C2×C11×AGL(1,𝔽5)11yescharacteristic, complemented
118C2—22no—
119C2—51no—
120C2—51no—
121C2—102no—
122C2C22×C11×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C11C22×AGL(1,𝔽5)
commutatorC2×C5C22×C4×C11
FrattiniC2C22×C11×AGL(1,𝔽5)
FittingC5×D4×C11C4
socleC2×C5×C11C22×C4
prime radicalD4×D5×C11C2
cosocleC2×D5C23×C11
hypercenterD4×C11AGL(1,𝔽5)
hypocenterC5C4×D4×C11
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C11 ◁ D4×C11

Complements

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

NH ≅ G/Ntype
1. C22×C11×AGL(1,𝔽5)C2semidirect
3. C11×C22⋊AGL(1,𝔽5)C2semidirect
4. C11×C4⋊AGL(1,𝔽5)C2semidirect
5. C4×C11×AGL(1,𝔽5)C2semidirect
11. C5×D4×C11C4semidirect
12. C11×C22⋊D5C4semidirect
13. C11×D20C4semidirect
16. C22×C5×C11C2×C4semidirect
19. C2×D5×C11C2×C4semidirect
23. C11×AGL(1,𝔽5)D4direct (N × H)
24. C4×C5×C11C2×C4semidirect
25. C11×Dic5C2×C4semidirect
31. D4×AGL(1,𝔽5)C11direct (N × H)
40. D4×C11AGL(1,𝔽5)direct (N × H)
46. C22×AGL(1,𝔽5)C2×C11semidirect
48. C22⋊AGL(1,𝔽5)C2×C11semidirect
49. C4⋊AGL(1,𝔽5)C2×C11semidirect
50. C4×AGL(1,𝔽5)C2×C11semidirect
51. C5×C11C4×D4semidirect
56. C22×C11C2×AGL(1,𝔽5)semidirect
61. C4×C11C2×AGL(1,𝔽5)semidirect
67. C5×D4C4×C11semidirect
68. C22⋊D5C4×C11semidirect
69. D20C4×C11semidirect
77. C22×C5C2×C4×C11semidirect
80. C2×D5C2×C4×C11semidirect
84. AGL(1,𝔽5)D4×C11direct (N × H)
85. C4×C5C2×C4×C11semidirect
86. Dic5C2×C4×C11semidirect
92. C11D4×AGL(1,𝔽5)direct (N × H)
101. D4C11×AGL(1,𝔽5)direct (N × H)
107. C5C4×D4×C11semidirect
112. C22C2×C11×AGL(1,𝔽5)semidirect
117. C4C2×C11×AGL(1,𝔽5)semidirect

Maximal subgroups

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

subgroupindex
C22×C11×AGL(1,𝔽5)2
D4×D5×C112
C11×C22⋊AGL(1,𝔽5)2
C11×C4⋊AGL(1,𝔽5)2
C4×C11×AGL(1,𝔽5)2
C4×D4×C115
D4×AGL(1,𝔽5)11