Finite Group Databasebeta

← #36377953

Subgroups — #36377953

Order 22592.

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#26015038C211yesmaximal, characteristic, complemented
2#26015018C211yesmaximal, characteristic, complemented
3#26014988C211yesmaximal, characteristic, complemented
4#26014919C211yesmaximal, characteristic, complemented
5#26014918C211yesmaximal, characteristic, complemented
6#26014908C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C353C211yesmaximal, characteristic, complemented
8#19150738—21no—
9#19150737—21no—
10#19150737C2211yescharacteristic
11C2×D1412—21no—
12C2×D1412C2211yescharacteristic
13C2×C22⋊D353—22no—
14D4×D353—22no—
15D4×D353—21no—
16#19150725—21no—
17#19150722—21no—
18#19150721C2211yescharacteristic, complemented
19C4×Dic353C2211yescharacteristic, complemented
20#41×C353C2211yescharacteristic
21C2×D4×C353C2211yescharacteristic, complemented
22SD16×C353—21no—
23D8×C353—21no—
24M16×C353C2211yescharacteristic, complemented
25C22×D353—22no—
26C22×D353D412yes—
27Dic706D411yescharacteristic, complemented
28D1412—22no—
29D1412—21no—
30D1412—21no—
31D1412D411yescharacteristic, complemented
32C22⋊D353—22no—
33C22⋊D353—21no—
34C22⋊D353—44no—
35C2×Dic353—21no—
36C4×D353—22no—
37C4×D353—21no—
38#15667725—21no—
39C23×C353—21no—
40Q8×C353D411yescharacteristic, complemented
41D4×C353—21no—
42D4×C353—21no—
43D4×C353—21no—
44D4×C353D411yescharacteristic, complemented
45C2×C4×C353—21no—
46C2×C4×C353C2311yescharacteristic
47C8×C353—21no—
48C22×C353—21no—
49C22×C353—42no—
50C22×C353—41no—
51C22×C353C2×D411yescharacteristic
52C2×D353—22no—
53C2×D353—21no—
54C2×D353—42no—
55C2×D353—82no—
56C4×C353—21no—
57C4×C353C2×D411yescharacteristic
58C4×C353C2×D411yescharacteristic
59Dic353—22no—
60Dic353—21no—
61C2×C353—21no—
62C2×C353—41no—
63C2×C353—81no—
64C2×C353#12011yescharacteristic
65D353—42no—
66D353—41no—
67C353#45311yescharacteristic, complemented, Sylow
68#453—3531nomaximal, Sylow
69H(5,𝔽2)—3531no—
70AGL(1,ℤ/8)—3531no—
71AGL(1,ℤ/8)D35311yescharacteristic, complemented
72#127—3531no—
73#104—3531no—
74#104—3531no—
75#100—3531no—
76#41—3531no—
77#41—7061no—
78#41—7061no—
79#41C2×D35311yescharacteristic
80C2×D4—3531no—
81C2×D4—7061no—
82C2×D4—7062no—
83C2×D4—7062no—
84C2×D4—7061no—
85C2×D4C2×D35311yescharacteristic, complemented
86SD16—21no—
87SD16—7061no—
88D8—21no—
89D8—7061no—
90M16—3531no—
91M16C2×D35311yescharacteristic, complemented
92C42—3531no—
93C23—21no—
94C23—3532no—
95C23—7062no—
96Q8—3531no—
97Q8C22⋊D35311yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—3531no—
102D4—14124no—
103D4—7062no—
104D4—7062no—
105D4—7061no—
106D4—7061no—
107D4—7061no—
108D4C22⋊D35311yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—7062no—
111C2×C4—7061no—
112C2×C4—7061no—
113C2×C4C22×D35311yescharacteristic
114C8—21no—
115C8—7061no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—28242no—
120C22—14122no—
121C22—7061no—
122C22—7062no—
123C22D4×D35311yescharacteristic
124C4—21no—
125C4—7062no—
126C4—7061no—
127C4D4×D35311yescharacteristic
128C4C2×C22⋊D35311yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—14121no—
133C2—14122no—
134C2#2601496011yescharacteristic
subgroupG/H
centerC2#26014960
commutator, cosocleC2×C4×C353C23
FrattiniC2×C4C22×D353
FittingAGL(1,ℤ/8)×C353C2
socleC2×C353#120
hypercenterAGL(1,ℤ/8)D353
hypocenterC353#453
LCSG ▷ C2×C4×C353 ▷ C2×C353 ▷ C353
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26015038C2semidirect
2. #26015018C2semidirect
3. #26014988C2semidirect
4. #26014919C2semidirect
5. #26014918C2semidirect
6. #26014908C2semidirect
7. AGL(1,ℤ/8)×C353C2semidirect
18. #19150721C22semidirect
19. C4×Dic353C22semidirect
21. C2×D4×C353C22semidirect
24. M16×C353C22semidirect
27. Dic706D4semidirect
31. D1412D4semidirect
40. Q8×C353D4semidirect
44. D4×C353D4semidirect
67. C353#453semidirect
71. AGL(1,ℤ/8)D353semidirect
85. C2×D4C2×D353semidirect
91. M16C2×D353semidirect
97. Q8C22⋊D353semidirect
108. D4C22⋊D353semidirect

Maximal subgroups

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

subgroupindex
#260150382
#260150182
#260149882
#260149192
#260149182
#260149082
AGL(1,ℤ/8)×C3532
#453353