Finite Group Databasebeta

← #36352884

Subgroups — #36352884

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#26011853C212yesmaximal, complemented
3#26011233C211yesmaximal, characteristic, complemented
4#26011230C212yesmaximal, complemented
5#491×C173C211yesmaximal, characteristic, complemented
6#19149667C2212yes—
7#19149664—22no—
8#19149657—22no—
9#19149654—22no—
10#19149602—22no—
11#19149599—22no—
12#19149590C2212yes—
13C4⋊Q8×C173C2211yescharacteristic
14#125×C173C2211yescharacteristic
15#103×C173—22no—
16#97×C173C2211yescharacteristic
17C2×Dic346D412yes—
18C2×D692—22no—
19C2×D692D412yes—
20C2×C4×D173—22no—
21C2×#13063436—22no—
22#14583805—22no—
23#14583803—22no—
24#14583803—22no—
25#14583803—22no—
26C2×Q8×C173—21no—
27C2×C8×C173—21no—
28C4⋊C4×C173—22no—
29C4⋊C4×C173—22no—
30C4⋊C4×C173D412yes—
31C42×C173C2311yescharacteristic
32C22×D173—22no—
33Dic346—42no—
34D692—42no—
35D692—82no—
36C2×Dic173—22no—
37C4×D173—82no—
38#13063436—42no—
39Q8×C173—42no—
40C2×C4×C173—22no—
41C2×C4×C173—41no—
42C2×C4×C173C2×D412yes—
43C2×C4×C173C2×D411yescharacteristic
44C8×C173—41no—
45C22×C173#12011yescharacteristic
46C2×D173—42no—
47C2×D173—42no—
48C2×D173—42no—
49C4×C173—21no—
50C4×C173—22no—
51C4×C173—42no—
52C4×C173—81no—
53Dic173—42no—
54C2×C173#45411yescharacteristic
55C2×C173#45512yes—
56D173—82no—
57C173#166211yescharacteristic, complemented, Sylow
58#1662—1731nomaximal, Sylow
59#562—1731no—
60#491D17311yescharacteristic, complemented
61#489—1732no—
62#330—1731no—
63#329—1732no—
64C4⋊Q8C2×D17311yescharacteristic
65#125C2×D17311yescharacteristic
66#124—1732no—
67#123—3462no—
68C22⋊Q8—3462no—
69#121—3462no—
70#103—22no—
71#103—3462no—
72#102—3462no—
73#97—1732no—
74#97C2×D17311yescharacteristic
75C2×Q8—21no—
76C2×Q8—1732no—
77C2×D4—3462no—
78C2×D4—1732no—
79C22×C4—3462no—
80C2×C8—21no—
81C2×C8—3462no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—3462no—
85C4⋊C4C22⋊D17312yes—
86C22⋊C4—3462no—
87C22⋊C4—3462no—
88C22⋊C4—3462no—
89C42C22×D17311yescharacteristic
90C23—3462no—
91Q8—42no—
92Q8—6922no—
93D4—13842no—
94D4—6922no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—3462no—
98C2×C4—13842no—
99C2×C4D4×D17312yes—
100C2×C4C2×C22⋊D17311yescharacteristic
101C8—41no—
102C8—6922no—
103C22—6922no—
104C22—6922no—
105C22—6922no—
106C22#1914964711yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—81no—
111C4—6922no—
112C2—13842no—
113C2#2601166311yescharacteristic
114C2#2601166712yes—
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#491×C173C2
socleC22×C173#120
prime radical#26012239C2
hypercenter#491D173
hypocenterC173#1662
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #26011853C2semidirect
3. #26011233C2semidirect
4. #26011230C2semidirect
5. #491×C173C2semidirect
57. C173#1662semidirect
60. #491D173semidirect

Maximal subgroups

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

subgroupindex
#260122392
#260118532
#260112332
#260112302
#491×C1732
#1662173