Finite Group Databasebeta

← #36368824

Subgroups — #36368824

Order 22144.

300 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 64 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#26012511C212yesmaximal, complemented
2#26012335C214yesmaximal, complemented
3C2×#19149641C211yesmaximal, characteristic
4#19149746—22no—
5C22⋊#13063443C2214yes—
6#19149734—22no—
7C22×D692C411yescharacteristic
8C22×C4×D173C412yes—
9#19149641—22no—
10#19149641—22no—
11C2×#14583803C2212yes—
12C2×C4×Dic173C2211yescharacteristic
13C2×C4⋊C4×C173C411yescharacteristic
14C2×#13063443—24no—
15C23×D173C2×C412yes—
16C2×D692—24no—
17C2×D692—22no—
18C22×Dic173C2×C411yescharacteristic
19C22×Dic173C2311yescharacteristic
20C2×C4×D173—22no—
21C2×C4×D173—22no—
22C2×C4×D173—44no—
23#14583803—24no—
24C4×Dic173D412yes—
25C4×Dic173D412yes—
26C22×C4×C173C2×C411yescharacteristic
27C22×C4×C173C2×C412yes—
28C4⋊C4×C173—22no—
29#13063443—44no—
30C22×D173—22no—
31C22×D173—22no—
32C22×D173—42no—
33C22×D173—42no—
34C22×D173M1614yes—
35D692—28no—
36C2×Dic173—22no—
37C2×Dic173—41no—
38C2×Dic173C22⋊C411yescharacteristic
39C2×Dic173C22⋊C411yescharacteristic
40C2×Dic173C2×D412yes—
41C4×D173—44no—
42C4×D173—44no—
43C23×C173C22×C411yescharacteristic
44C2×C4×C173—22no—
45C2×C4×C173—22no—
46C2×C4×C173—42no—
47C2×C4×C173C22⋊C412yes—
48C22×C173—21no—
49C22×C173—21no—
50C22×C173C2×C22⋊C411yescharacteristic
51C22×C173C2×M1612yes—
52C2×D173—24no—
53C2×D173—44no—
54C2×D173—42no—
55C2×D173—42no—
56C2×D173—44no—
57C4×C173—22no—
58C4×C173—42no—
59Dic173—22no—
60Dic173—21no—
61Dic173#10012yes—
62C2×C173—22no—
63C2×C173#40711yescharacteristic
64C2×C173#41011yescharacteristic
65C2×C173C2×#10011yescharacteristic
66D173—44no—
67C173#146611yescharacteristic, complemented, Sylow
68#1466—1731nomaximal, Sylow
69C2×C4×D4—1731no—
70#432—1732no—
71#323—1734no—
72C22×D4—1731no—
73C23×C4—1732no—
74C4×D4—3462no—
75C4×D4—3462no—
76C2×C4⋊C4#1062192411yescharacteristic
77C2×C22⋊C4—1732no—
78C2×C42—1731no—
79C4⋊C8—3462no—
80C22⋊C8—1734no—
81#97—3462no—
82C24—1732no—
83C2×D4—3464no—
84C2×D4—3462no—
85C22×C4—3462no—
86C22×C4—3462no—
87C22×C4—1731no—
88C22×C4—1731no—
89C22×C4—6924no—
90C22×C4C2×#1062192412yes—
91C22×C4C2×#1062192411yescharacteristic
92C2×C8—3464no—
93C4⋊C4—22no—
94C22⋊C4—3464no—
95C42—1732no—
96C42—1732no—
97C23—3462no—
98C23—3462no—
99C23—1734no—
100C23—6922no—
101C23—6922no—
102C23C22×#1062192411yescharacteristic
103D4—3468no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—42no—
107C2×C4—3462no—
108C2×C4—1731no—
109C2×C4—1732no—
110C2×C4—1731no—
111C2×C4—6924no—
112C2×C4—6924no—
113C2×C4—6921no—
114C2×C4C22⋊#1062192412yes—
115C8—6924no—
116C22—21no—
117C22—21no—
118C22—3464no—
119C22—6924no—
120C22—6924no—
121C22—6922no—
122C22—6922no—
123C22C2×C22⋊#1062192411yescharacteristic
124C22D173⋊M1612yes—
125C4—22no—
126C4—42no—
127C4—3462no—
128C4—3461no—
129C4—1732no—
130C2—22no—
131C2—6924no—
132C2#2601246311yescharacteristic
133C2#2601247211yescharacteristic
134C2D173⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×D692C4
cosocleC22×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1466
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #26012511C2semidirect
2. #26012335C2semidirect
67. C173#1466semidirect

Maximal subgroups

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

subgroupindex
#260125112
#260123352
C2×#191496412
#1466173