Finite Group Databasebeta

← #36352887

Subgroups — #36352887

Order 22144.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#26012239C211yesmaximal, characteristic
2#26011861C211yesmaximal, characteristic, complemented
3#26011851C212yesmaximal, complemented
4#26011230C212yesmaximal, complemented
5#330×C173C211yesmaximal, characteristic, complemented
6#19149679C2211yescharacteristic
7#19149667C2212yes—
8#19149664—22no—
9#19149657—22no—
10#19149654—22no—
11#19149604—22no—
12#19149602—22no—
13#19149601—22no—
14#19149590C2211yescharacteristic
15#125×C173C2211yescharacteristic
16#97×C173C2212yes—
17C2×Dic346—21no—
18C2×Dic346D412yes—
19C2×D692—22no—
20C2×D692D412yes—
21C2×C4×D173—22no—
22C2×#13063436—21no—
23C4⋊Dic173—22no—
24#14583805—22no—
25#14583803—22no—
26#14583803—22no—
27#14583803—22no—
28C2×C8×C173—22no—
29C4⋊C4×C173—22no—
30C4⋊C4×C173D412yes—
31C42×C173C2311yescharacteristic
32C22×D173—22no—
33Dic346—42no—
34Dic346—42no—
35D692—42no—
36D692—82no—
37C2×Dic173—22no—
38C2×Dic173—41no—
39C4×D173—82no—
40#13063436—41no—
41C2×C4×C173—22no—
42C2×C4×C173C2×D412yes—
43C2×C4×C173C2×D411yescharacteristic
44C8×C173—42no—
45C22×C173#12011yescharacteristic
46C2×D173—42no—
47C2×D173—42no—
48C2×D173—42no—
49C4×C173—21no—
50C4×C173—22no—
51C4×C173—42no—
52Dic173—42no—
53Dic173—81no—
54C2×C173#45411yescharacteristic
55C2×C173#45512yes—
56D173—82no—
57C173#166211yescharacteristic, complemented, Sylow
58#1662—1731nomaximal, Sylow
59#562—1731no—
60#491—1731no—
61#489—1732no—
62#330D17311yescharacteristic, complemented
63#329—1732no—
64C4⋊Q8—1731no—
65#125C2×D17311yescharacteristic
66#124—1732no—
67#123—3462no—
68C22⋊Q8—3462no—
69#121—3462no—
70#103—3462no—
71#103—3462no—
72#102—3462no—
73#97—1731no—
74#97C2×D17312yes—
75C2×Q8—3461no—
76C2×Q8—1732no—
77C2×D4—3462no—
78C2×D4—1732no—
79C22×C4—3462no—
80C2×C8—22no—
81C2×C8—3461no—
82C4⋊C4—22no—
83C4⋊C4—3462no—
84C4⋊C4—3462no—
85C4⋊C4D69212yes—
86C22⋊C4—3462no—
87C22⋊C4—3462no—
88C22⋊C4—3462no—
89C42C22×D17311yescharacteristic
90C23—3462no—
91Q8—6922no—
92Q8—6922no—
93D4—13842no—
94D4—6922no—
95C2×C4—22no—
96C2×C4—3462no—
97C2×C4—13842no—
98C2×C4—6921no—
99C2×C4D4×D17312yes—
100C2×C4C2×D69211yescharacteristic
101C8—42no—
102C8—6921no—
103C22—6922no—
104C22—6922no—
105C22—6922no—
106C22#1914964911yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—13841no—
111C4—6922no—
112C2—13842no—
113C2#2601166611yescharacteristic
114C2#2601167012yes—
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#330×C173C2
socleC22×C173#120
prime radical#26012239C2
hypercenter#330D173
hypocenterC173#1662
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #26011861C2semidirect
3. #26011851C2semidirect
4. #26011230C2semidirect
5. #330×C173C2semidirect
57. C173#1662semidirect
62. #330D173semidirect

Maximal subgroups

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

subgroupindex
#260122392
#260118612
#260118512
#260112302
#330×C1732
#1662173