Finite Group Databasebeta

← #36352156

Subgroups — #36352156

Order 22144.

226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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#26012047C211yesmaximal, characteristic
2#26011443C212yesmaximal
3#26011233C211yesmaximal, characteristic, complemented
4#26011232C212yesmaximal, complemented
5#330×C173C211yesmaximal, characteristic, complemented
6C2×#14583809—22no—
7#19149678C412yes—
8#19149671C2211yescharacteristic
9#19149657—22no—
10#19149644—22no—
11#19149637C412yes—
12C42×D173C2211yescharacteristic
13#19149590C2212yes—
14C8×Dic173—22no—
15#125×C173C2211yescharacteristic
16#97×C173C2212yes—
17C2×Dic346—22no—
18C2×C4×D173C2×C411yescharacteristic
19C2×C4×D173D412yes—
20#14583809—28no—
21C2×#13063436—22no—
22C4⋊Dic173C2×C412yes—
23#14583805—22no—
24#14583805—22no—
25#14583803—22no—
26C4×Dic173—22no—
27C4×Dic173C2×C411yescharacteristic
28C4×Dic173D412yes—
29C2×C8×C173—22no—
30C4⋊C4×C173—22no—
31C4⋊C4×C173C2×C412yes—
32C42×C173C2311yescharacteristic
33C22×D173C22⋊C411yescharacteristic
34Dic346—44no—
35C2×Dic173—22no—
36C2×Dic173C22⋊C411yescharacteristic
37C2×Dic173C22⋊C412yes—
38C4×D173—22no—
39C4×D173—24no—
40#13063436—24no—
41C2×C4×C173—22no—
42C2×C4×C173C22×C411yescharacteristic
43C2×C4×C173C2×D412yes—
44C8×C173—24no—
45C22×C173C2×C22⋊C411yescharacteristic
46C2×D173—21no—
47C2×D173—22no—
48C4×C173—21no—
49C4×C173—22no—
50C4×C173—42no—
51Dic173—42no—
52Dic173#10112yes—
53Dic173#10414yes—
54C2×C173C2×#10111yescharacteristic
55C2×C173C2×#10412yes—
56D173—41no—
57C173#152611yescharacteristic, complemented, Sylow
58#1526—1731nomaximal, Sylow
59#533—1731no—
60C4×M16—1732no—
61#330—1731no—
62#330—1732no—
63#330D17311yescharacteristic, complemented
64C2×M16—3462no—
65C4⋊Q8—1732no—
66#125—1731no—
67#125C2×D17311yescharacteristic
68C22⋊Q8—3462no—
69C4×Q8—3462no—
70#117—1732no—
71C2×C42—1731no—
72#97—1732no—
73#97C2×D17312yes—
74C4×C8—3462no—
75C2×Q8—3462no—
76C22×C4—1731no—
77C22×C4—1732no—
78M16—3468no—
79C2×C8—22no—
80C2×C8—3462no—
81C4⋊C4—22no—
82C4⋊C4—3462no—
83C4⋊C4—3462no—
84C4⋊C4—1732no—
85C4⋊C4C4×D17312yes—
86C22⋊C4—3462no—
87C42—3462no—
88C42—1731no—
89C42—1732no—
90C42C22×D17311yescharacteristic
91C23—1731no—
92Q8—6924no—
93C2×C4—22no—
94C2×C4—3464no—
95C2×C4—3462no—
96C2×C4—3462no—
97C2×C4—1731no—
98C2×C4—1732no—
99C2×C4C2×C4×D17311yescharacteristic
100C2×C4D4×D17312yes—
101C8—24no—
102C8—3464no—
103C22—3461no—
104C22—3462no—
105C22C22⋊C4×D17311yescharacteristic
106C4—21no—
107C4—22no—
108C4—42no—
109C4—1732no—
110C4—1734no—
111C4—6922no—
112C2—6921no—
113C2#101×D17311yescharacteristic
114C2#104×D17312yes—
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC42C22×D173
Fitting#330×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC22×D173C22⋊C4
cosocleC42×C173C23
hypercenter#330D173
hypocenterC173#1526
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
3. #26011233C2semidirect
4. #26011232C2semidirect
5. #330×C173C2semidirect
57. C173#1526semidirect
63. #330D173semidirect

Maximal subgroups

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

subgroupindex
#260120472
#260114432
#260112332
#260112322
#330×C1732
#1526173