Finite Group Databasebeta

← #36368508

Subgroups — #36368508

Order 22144.

158 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 34 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#26012334C211yesmaximal, characteristic
2#26012334C211yesmaximal, characteristic
3#26011364C211yesmaximal, characteristic
4C22⋊#13063443—21no—
5C22⋊#13063443—21no—
6C2×#14583805—21no—
7C2×C22⋊Dic173C411yescharacteristic
8C2×C22⋊Dic173C2211yescharacteristic
9#19149586—21no—
10C23×C4×C173C411yescharacteristic
11C2×#13063443—41no—
12C2×#13063443—41no—
13C22×Dic173—21no—
14C22×Dic173D411yescharacteristic
15C22×Dic173D411yescharacteristic
16#14583805—42no—
17C22⋊Dic173—21no—
18C22⋊Dic173—41no—
19C22⋊Dic173C812yescomplemented
20C24×C173C2×C411yescharacteristic
21C22×C4×C173—21no—
22C22×C4×C173—42no—
23C22×C4×C173—41no—
24#13063443—81no—
25#13063443—81no—
26C2×Dic173—22no—
27C2×Dic173—21no—
28C2×Dic173—21no—
29C2×Dic173—41no—
30C2×Dic173—41no—
31C2×Dic173—41no—
32C2×Dic173—41no—
33C23×C173—21no—
34C23×C173—21no—
35C23×C173—41no—
36C23×C173—41no—
37C23×C173C22⋊C411yescharacteristic
38C23×C173C2×C811yescharacteristic
39C23×C173M1611yescharacteristic
40C2×C4×C173—22no—
41C2×C4×C173—41no—
42C2×C4×C173—42no—
43C2×C4×C173—41no—
44C2×C4×C173—42no—
45C22×C173—21no—
46C22×C173—21no—
47C22×C173—21no—
48C22×C173—21no—
49C22×C173—42no—
50C22×C173—41no—
51C22×C173—41no—
52C22×C173—41no—
53C22×C173—41no—
54C22×C173C22⋊C811yescharacteristic
55C22×C173#9911yescharacteristic
56C22×C173#10111yescharacteristic
57C4×C173—42no—
58Dic173—42no—
59Dic173—41no—
60Dic173—41no—
61C2×C173—21no—
62C2×C173—21no—
63C2×C173—41no—
64C2×C173—41no—
65C2×C173#32411yescharacteristic
66C2×C173AΣL(1,𝔽16)11yescharacteristic
67C2×C173#35211yescharacteristic
68C173#132611yescharacteristic, complemented, Sylow
69#1326—1731nomaximal, Sylow
70#385—1731no—
71#323—1731no—
72#323—1731no—
73C23×C4#1062192411yescharacteristic
74C2×C4⋊C4—3461no—
75C2×C22⋊C4—1731no—
76C2×C22⋊C4—1731no—
77C22⋊C8—3461no—
78C22⋊C8—3461no—
79#95—3461no—
80C24C2×#1062192411yescharacteristic
81C22×C4—21no—
82C22×C4—42no—
83C22×C4—41no—
84C22×C4—3461no—
85C22×C4—1731no—
86C22×C4—1731no—
87C2×C8—6921no—
88C2×C8—6921no—
89C4⋊C4—6922no—
90C22⋊C4—3461no—
91C22⋊C4—1732no—
92C22⋊C4—6921no—
93C23—21no—
94C23—21no—
95C23—41no—
96C23—41no—
97C23C22⋊#1062192411yescharacteristic
98C23D173⋊C811yescharacteristic
99C23#1458383611yescharacteristic
100C2×C4—22no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—3462no—
106C2×C4—3461no—
107C2×C4—3461no—
108C2×C4—6921no—
109C2×C4—6921no—
110C2×C4—6921no—
111C2×C4—6921no—
112C8—13841no—
113C8—13841no—
114C22—21no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—41no—
119C22—41no—
120C22—42no—
121C22—41no—
122C22—41no—
123C22#1914973611yescharacteristic
124C22#1914973711yescharacteristic
125C22#1914974011yescharacteristic
126C4—42no—
127C4—6922no—
128C4—6921no—
129C4—6921no—
130C2—21no—
131C2—21no—
132C2—41no—
133C2—41no—
134C2#2601233711yescharacteristic
135C2#2601239111yescharacteristic
136C2#2601239311yescharacteristic
subgroupG/H
centerC22#19149737
commutatorC23×C173C2×C8
FrattiniC24C2×#10621924
FittingC23×C4×C173C4
socleC22×C173#99
prime radicalC24×C173C2×C4
cosocleC2×C22⋊Dic173C22
hypercenterC23×C4#10621924
hypocenterC173#1326
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
19. C22⋊Dic173C8semidirect
68. C173#1326semidirect

Maximal subgroups

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

subgroupindex
#260123342
#260113642
#1326173