Finite Group Databasebeta

← #31823629

Subgroups — #31823629

Order 17984.

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

normal subgroups only

This group has 168 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#22582925C211yesmaximal, characteristic, complemented
2#22582905C211yesmaximal, characteristic, complemented
3#22582875C211yesmaximal, characteristic, complemented
4#22582806C211yesmaximal, characteristic, complemented
5#22582805C211yesmaximal, characteristic, complemented
6#22582795C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C281C211yesmaximal, characteristic, complemented
8#17985139—21no—
9#17985138—21no—
10#17985138C2211yescharacteristic
11C2×D1124—21no—
12C2×D1124C2211yescharacteristic
13C2×C22⋊D281—22no—
14D4×D281—22no—
15D4×D281—21no—
16#17985126—21no—
17#17985123—21no—
18#17985122C2211yescharacteristic, complemented
19C4×Dic281C2211yescharacteristic, complemented
20#41×C281C2211yescharacteristic
21C2×D4×C281C2211yescharacteristic, complemented
22SD16×C281—21no—
23D8×C281—21no—
24M16×C281C2211yescharacteristic, complemented
25C22×D281—22no—
26C22×D281D412yes—
27Dic562D411yescharacteristic, complemented
28D1124—22no—
29D1124—21no—
30D1124—21no—
31D1124D411yescharacteristic, complemented
32C22⋊D281—22no—
33C22⋊D281—21no—
34C22⋊D281—44no—
35C2×Dic281—21no—
36C4×D281—22no—
37C4×D281—21no—
38#14555886—21no—
39C23×C281—21no—
40Q8×C281D411yescharacteristic, complemented
41D4×C281—21no—
42D4×C281—21no—
43D4×C281—21no—
44D4×C281D411yescharacteristic, complemented
45C2×C4×C281—21no—
46C2×C4×C281C2311yescharacteristic
47C8×C281—21no—
48C22×C281—21no—
49C22×C281—42no—
50C22×C281—41no—
51C22×C281C2×D411yescharacteristic
52C2×D281—22no—
53C2×D281—21no—
54C2×D281—42no—
55C2×D281—82no—
56C4×C281—21no—
57C4×C281C2×D411yescharacteristic
58C4×C281C2×D411yescharacteristic
59Dic281—22no—
60Dic281—21no—
61C2×C281—21no—
62C2×C281—41no—
63C2×C281—81no—
64C2×C281#12011yescharacteristic
65D281—42no—
66D281—41no—
67C281#45311yescharacteristic, complemented, Sylow
68#453—2811nomaximal, Sylow
69H(5,𝔽2)—2811no—
70AGL(1,ℤ/8)—2811no—
71AGL(1,ℤ/8)D28111yescharacteristic, complemented
72#127—2811no—
73#104—2811no—
74#104—2811no—
75#100—2811no—
76#41—2811no—
77#41—5621no—
78#41—5621no—
79#41C2×D28111yescharacteristic
80C2×D4—2811no—
81C2×D4—5621no—
82C2×D4—5622no—
83C2×D4—5621no—
84C2×D4—5622no—
85C2×D4C2×D28111yescharacteristic, complemented
86SD16—21no—
87SD16—5621no—
88D8—21no—
89D8—5621no—
90M16—2811no—
91M16C2×D28111yescharacteristic, complemented
92C42—2811no—
93C23—21no—
94C23—2812no—
95C23—5622no—
96Q8—2811no—
97Q8C22⋊D28111yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—2811no—
102D4—5622no—
103D4—5622no—
104D4—5621no—
105D4—5621no—
106D4—5621no—
107D4—11244no—
108D4C22⋊D28111yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—5622no—
111C2×C4—5621no—
112C2×C4—5621no—
113C2×C4C22×D28111yescharacteristic
114C8—21no—
115C8—5621no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—5621no—
120C22—5622no—
121C22—11242no—
122C22—22482no—
123C22D4×D28111yescharacteristic
124C4—21no—
125C4—5622no—
126C4—5621no—
127C4D4×D28111yescharacteristic
128C4C2×C22⋊D28111yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—11242no—
133C2—11241no—
134C2#2258284711yescharacteristic
subgroupG/H
centerC2#22582847
commutator, cosocleC2×C4×C281C23
FrattiniC2×C4C22×D281
FittingAGL(1,ℤ/8)×C281C2
socleC2×C281#120
hypercenterAGL(1,ℤ/8)D281
hypocenterC281#453
LCSG ▷ C2×C4×C281 ▷ C2×C281 ▷ C281
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22582925C2semidirect
2. #22582905C2semidirect
3. #22582875C2semidirect
4. #22582806C2semidirect
5. #22582805C2semidirect
6. #22582795C2semidirect
7. AGL(1,ℤ/8)×C281C2semidirect
18. #17985122C22semidirect
19. C4×Dic281C22semidirect
21. C2×D4×C281C22semidirect
24. M16×C281C22semidirect
27. Dic562D4semidirect
31. D1124D4semidirect
40. Q8×C281D4semidirect
44. D4×C281D4semidirect
67. C281#453semidirect
71. AGL(1,ℤ/8)D281semidirect
85. C2×D4C2×D281semidirect
91. M16C2×D281semidirect
97. Q8C22⋊D281semidirect
108. D4C22⋊D281semidirect

Maximal subgroups

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

subgroupindex
#225829252
#225829052
#225828752
#225828062
#225828052
#225827952
AGL(1,ℤ/8)×C2812
#453281