Finite Group Databasebeta

← #36368827

Subgroups — #36368827

Order 22144.

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

normal subgroups only

This group has 236 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
2#26012337C214yesmaximal
3C2×#19149644C211yesmaximal, characteristic
4#19149746—22no—
5C22⋊#13063443C2214yes—
6#19149734—22no—
7C22×Dic346C411yescharacteristic
8#19149644—22no—
9#19149644—22no—
10C2×#14583805C2212yes—
11C2×C4×Dic173C412yes—
12C2×C4×Dic173C2211yescharacteristic
13C2×C4⋊C4×C173C411yescharacteristic
14C2×#13063443—24no—
15C2×Dic346—24no—
16C2×Dic346—22no—
17C22×Dic173C2×C411yescharacteristic
18C22×Dic173C2×C412yes—
19C22×Dic173C2311yescharacteristic
20#14583805—24no—
21C4×Dic173—22no—
22C4×Dic173—22no—
23C4×Dic173D412yes—
24C4×Dic173D412yes—
25C22×C4×C173C2×C411yescharacteristic
26C22×C4×C173C2×C412yes—
27C4⋊C4×C173—22no—
28#13063443—44no—
29Dic346—28no—
30C2×Dic173—22no—
31C2×Dic173—22no—
32C2×Dic173—22no—
33C2×Dic173—41no—
34C2×Dic173C22⋊C411yescharacteristic
35C2×Dic173C22⋊C411yescharacteristic
36C2×Dic173M1614yes—
37C2×Dic173C2×D412yes—
38C23×C173C22×C411yescharacteristic
39C2×C4×C173—22no—
40C2×C4×C173—22no—
41C2×C4×C173—42no—
42C2×C4×C173C22⋊C412yes—
43C22×C173—21no—
44C22×C173—21no—
45C22×C173C2×C22⋊C411yescharacteristic
46C22×C173C2×M1612yes—
47C4×C173—22no—
48C4×C173—42no—
49Dic173—24no—
50Dic173—22no—
51Dic173—21no—
52Dic173#10112yes—
53C2×C173—22no—
54C2×C173#40711yescharacteristic
55C2×C173#41011yescharacteristic
56C2×C173C2×#10111yescharacteristic
57C173#146711yescharacteristic, complemented, Sylow
58#1467—1731nomaximal, Sylow
59C2×C4×Q8—1731no—
60#432—1732no—
61#324—1734no—
62C22×Q8—1731no—
63C4×Q8—3462no—
64C4×Q8—3462no—
65C2×C4⋊C4—1732no—
66C2×C4⋊C4#1062192411yescharacteristic
67C2×C42—1732no—
68C2×C42—1731no—
69C4⋊C8—3462no—
70C22⋊C8—1734no—
71#97—3462no—
72C2×Q8—3464no—
73C2×Q8—3462no—
74C22×C4—1731no—
75C22×C4—1732no—
76C22×C4—1731no—
77C22×C4C2×#1062192412yes—
78C22×C4C2×#1062192411yescharacteristic
79C2×C8—3464no—
80C4⋊C4—22no—
81C4⋊C4—3464no—
82C42—3462no—
83C42—3462no—
84C42—1732no—
85C42—1732no—
86C23C22×#1062192411yescharacteristic
87Q8—3468no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—42no—
91C2×C4—3462no—
92C2×C4—3462no—
93C2×C4—3462no—
94C2×C4—1731no—
95C2×C4—1732no—
96C2×C4—1734no—
97C2×C4—1731no—
98C2×C4—6921no—
99C2×C4C22⋊#1062192412yes—
100C8—6924no—
101C22—21no—
102C22—21no—
103C22C2×C22⋊#1062192411yescharacteristic
104C22D173⋊M1612yes—
105C4—22no—
106C4—42no—
107C4—3464no—
108C4—3462no—
109C4—3461no—
110C4—1732no—
111C2—22no—
112C2#2601246311yescharacteristic
113C2#2601247111yescharacteristic
114C2D173⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutator, prime radicalC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
cosocleC22×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1467
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
57. C173#1467semidirect

Maximal subgroups

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

subgroupindex
#260125112
#260123372
C2×#191496442
#1467173