Finite Group Databasebeta

← #36368465

Subgroups — #36368465

Order 22144.

200 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 44 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#26012463C211yesmaximal, characteristic
2#26012433C211yesmaximal, characteristic
3#26011453C211yesmaximal, characteristic
4C2×#14583835C412yescomplemented
5C2×C4⋊#10621924—21no—
6C2×C22⋊#10621924C412yes—
7#19149736—21no—
8#19149736—21no—
9(C2×#10621924)⋊C4—21no—
10C22×C4×D173C2211yescharacteristic
11C2×#14583809—21no—
12C22⋊#13063436C411yescharacteristic, complemented
13#19149591—21no—
14C22⋊C8×C173C411yescharacteristic, complemented
15C22×#10621924—22no—
16#14583835—42no—
17C2×#13063443—22no—
18C4⋊#10621924—42no—
19C22⋊#10621924—42no—
20C23×D173C2×C411yescharacteristic
21C22×Dic173C2×C411yescharacteristic
22C2×C4×D173—42no—
23C2×C4×D173D411yescharacteristic
24C2×C4×D173D411yescharacteristic
25C2×C4×D173D411yescharacteristic
26C2×C4×D173Q811yescharacteristic
27#14583809—42no—
28C2×#13063436—21no—
29C22×C4×C173C2×C411yescharacteristic
30C2×C8×C173—21no—
31C2×#10621924—42no—
32C2×#10621924—42no—
33C2×#10621924—42no—
34#13063443—42no—
35C22×D173—21no—
36C22×D173—21no—
37C22×D173—41no—
38C22×D173—41no—
39C22×D173C22⋊C411yescharacteristic
40C22×D173C4⋊C411yescharacteristic
41C2×Dic173—22no—
42C2×Dic173C22⋊C411yescharacteristic
43C2×Dic173C4⋊C411yescharacteristic
44C4×D173—22no—
45C4×D173—22no—
46C4×D173—41no—
47C4×D173—41no—
48#13063436—41no—
49C23×C173C4211yescharacteristic
50C2×C4×C173—41no—
51C2×C4×C173C22⋊C411yescharacteristic
52C2×C4×C173C4⋊C411yescharacteristic
53C8×C173—41no—
54C22×C173—21no—
55C22×C173—21no—
56C22×C173—21no—
57C22×C173#9511yescharacteristic
58C2×D173—21no—
59C2×D173—21no—
60C2×D173—21no—
61C2×D173—21no—
62C2×D173—21no—
63C2×D173—42no—
64C2×D173—42no—
65C2×D173#10011yescharacteristic
66C2×D173#10111yescharacteristic
67#10621924—82no—
68C4×C173—21no—
69C4×C173—21no—
70Dic173—21no—
71Dic173#9912yes—
72C2×C173—41no—
73C2×C173#34211yescharacteristic
74C2×C173#34311yescharacteristic
75C2×C173#34411yescharacteristic
76D173—21no—
77D173—21no—
78D173—41no—
79C173#130511yescharacteristic, complemented, Sylow
80#1305—1731nomaximal, Sylow
81#407—1731no—
82#407—1731no—
83#380—1731no—
84C23×C4—1731no—
85C2×M16—3461no—
86C2×M16—1732no—
87C2×C4⋊C4—3461no—
88C2×C22⋊C4—1732no—
89C22⋊C8—3461no—
90C22⋊C8—3461no—
91C22⋊C8—3461no—
92C22⋊C8—1731no—
93C22⋊C8#1062192411yescharacteristic, complemented
94#95—3461no—
95C24—1731no—
96C22×C4—3462no—
97C22×C4—1731no—
98C22×C4—1731no—
99C22×C4—1731no—
100C22×C4—1731no—
101C22×C4—1731no—
102C22×C4—6922no—
103C22×C4C2×#1062192411yescharacteristic
104M16—6922no—
105M16—6922no—
106C2×C8—21no—
107C2×C8—3462no—
108C2×C8—3461no—
109C4⋊C4—6922no—
110C22⋊C4—6922no—
111C23—3461no—
112C23—3461no—
113C23—1731no—
114C23—1731no—
115C23—6921no—
116C23—6921no—
117C23C4×#1062192411yescharacteristic
118C2×C4—41no—
119C2×C4—3462no—
120C2×C4—3462no—
121C2×C4—3462no—
122C2×C4—1731no—
123C2×C4—1731no—
124C2×C4—6921no—
125C2×C4—6921no—
126C2×C4—6922no—
127C2×C4—6922no—
128C2×C4—6922no—
129C2×C4C22⋊#1062192411yescharacteristic
130C2×C4C4⋊#1062192411yescharacteristic
131C8—41no—
132C8—6921no—
133C8—6922no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—3461no—
138C22—3461no—
139C22—3461no—
140C22—3461no—
141C22—3461no—
142C22—1731no—
143C22—1731no—
144C22—6922no—
145C22—6922no—
146C22(C2×#10621924)⋊C411yescharacteristic
147C4—21no—
148C4—21no—
149C4—3461no—
150C4—13842no—
151C4—1732no—
152C2—41no—
153C2—3461no—
154C2—3461no—
155C2—6921no—
156C2#2601236811yescharacteristic
157C2#2601237011yescharacteristic
158C2#2601237311yescharacteristic
subgroupG/H
centerC22(C2×#10621924)⋊C4
commutatorC23×C173C42
FrattiniC22×C4C2×#10621924
FittingC22⋊C8×C173C4
socleC22×C173#95
prime radicalC23×D173C2×C4
cosocleC22×C4×D173C22
hypercenterC22⋊C8#10621924
hypocenterC173#1305
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×#14583835C4semidirect
12. C22⋊#13063436C4semidirect
14. C22⋊C8×C173C4semidirect
79. C173#1305semidirect
93. C22⋊C8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124632
#260124332
#260114532
#1305173