Finite Group Databasebeta

← #36368904

Subgroups — #36368904

Order 22144.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012512C211yesmaximal, characteristic, complemented
2C4×#14583835C211yesmaximal, characteristic, complemented
3#26012339C214yesmaximal, complemented
4C2×#19149641C211yesmaximal, characteristic
5C2×#14583835—21no—
6C4⋊#13063443C2212yescomplemented
7C22⋊#13063443—21no—
8#19149734—21no—
9#19149734—21no—
10C4×#13063443C2212yescomplemented
11C22×D692C411yescharacteristic
12C22×C4×D173—21no—
13#19149641—21no—
14#19149641—21no—
15#19149641C412yes—
16#19149641C2212yes—
17C2×#14583803—21no—
18C2×C4×Dic173C2211yescharacteristic
19C2×C4⋊C4×C173C411yescharacteristic
20#14583835—24no—
21C2×#13063443—22no—
22C2×#13063443—22no—
23C23×D173—21no—
24C2×D692—24no—
25C2×D692—21no—
26C2×D692C2×C412yes—
27C22×Dic173D411yescharacteristic
28C22×Dic173D411yescharacteristic
29C2×C4×D173—22no—
30C2×C4×D173—22no—
31C2×C4×D173—44no—
32#14583803—22no—
33#14583803—22no—
34C4×Dic173C2×C411yescharacteristic
35C4×Dic173D411yescharacteristic
36C4×Dic173D411yescharacteristic
37C4×Dic173C2311yescharacteristic
38C22×C4×C173—21no—
39C22×C4×C173C2×C411yescharacteristic
40C4⋊C4×C173—21no—
41C4⋊C4×C173C2×C412yes—
42#13063443—24no—
43#13063443—42no—
44C22×D173—24no—
45C22×D173—22no—
46C22×D173—42no—
47C22×D173—42no—
48D692—24no—
49D692—22no—
50D692M1614yescomplemented
51C2×Dic173—22no—
52C2×Dic173—22no—
53C2×Dic173C22⋊C411yescharacteristic
54C2×Dic173C22⋊C411yescharacteristic
55C2×Dic173C2×D411yescharacteristic
56C2×Dic173C2×D411yescharacteristic
57C4×D173—44no—
58C4×D173—44no—
59C23×C173C22⋊C411yescharacteristic
60C2×C4×C173—22no—
61C2×C4×C173—22no—
62C2×C4×C173—42no—
63C2×C4×C173C22⋊C411yescharacteristic
64C2×C4×C173C22×C411yescharacteristic
65C22×C173—21no—
66C22×C173—21no—
67C22×C173#10412yes—
68C22×C173C2×C22⋊C411yescharacteristic
69C2×D173—24no—
70C2×D173—48no—
71C2×D173—42no—
72C2×D173—42no—
73C4×C173—21no—
74C4×C173—42no—
75C4×C173C2×M1612yes—
76Dic173—21no—
77Dic173—21no—
78Dic173—21no—
79Dic173AGL(1,ℤ/8)12yes—
80C2×C173—22no—
81C2×C173#40711yescharacteristic
82C2×C173#41811yescharacteristic
83C2×C173C2×#10411yescharacteristic
84D173—44no—
85C173#149011yescharacteristic, complemented, Sylow
86#1490—1731nomaximal, Sylow
87C2×C4×D4—1731no—
88#432—1731no—
89C4×M16—1731no—
90D4⋊C8—1734no—
91C22×D4—1731no—
92C23×C4—3461no—
93C2×M16—3461no—
94C4×D4—3461no—
95C4×D4—3461no—
96C4×D4—1732no—
97C4×D4—1732no—
98C2×C4⋊C4#1062192411yescharacteristic
99C2×C22⋊C4—3461no—
100C2×C42—1731no—
101C4⋊C8—1732no—
102C22⋊C8—3461no—
103#97—3461no—
104#97—3461no—
105C4×C8—1732no—
106C24—3461no—
107C2×D4—3464no—
108C2×D4—3461no—
109C2×D4—1732no—
110C22×C4—21no—
111C22×C4—3462no—
112C22×C4—3462no—
113C22×C4—1731no—
114C22×C4—1731no—
115C22×C4—6924no—
116C22×C4C2×#1062192411yescharacteristic
117M16—3464no—
118C2×C8—3462no—
119C2×C8—3462no—
120C4⋊C4—21no—
121C4⋊C4C2×#1062192412yes—
122C22⋊C4—3462no—
123C22⋊C4—3462no—
124C42—1731no—
125C42—1731no—
126C42—1731no—
127C42—1731no—
128C23—3464no—
129C23—3462no—
130C23—6922no—
131C23—6922no—
132C23C22⋊#1062192411yescharacteristic
133D4—3464no—
134D4—3462no—
135D4—1734no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—3462no—
140C2×C4—3462no—
141C2×C4—1731no—
142C2×C4—1731no—
143C2×C4—1731no—
144C2×C4—1731no—
145C2×C4—6924no—
146C2×C4—6924no—
147C2×C4C22⋊#1062192411yescharacteristic
148C2×C4C22×#1062192411yescharacteristic
149C8—3464no—
150C8—6922no—
151C22—21no—
152C22—21no—
153C22—3464no—
154C22—6928no—
155C22—6922no—
156C22—6922no—
157C22#1914974412yes—
158C22C2×C22⋊#1062192411yescharacteristic
159C4—21no—
160C4—42no—
161C4—3461no—
162C4—3461no—
163C4—3461no—
164C4—1732no—
165C4D173⋊M1612yes—
166C2—22no—
167C2—6924no—
168C2#2601246311yescharacteristic
169C2#2601248311yescharacteristic
170C2C2×#1914974411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×D692C4
cosocleC4×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1490
LCSG ▷ C2×C4×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. #26012512C2semidirect
2. C4×#14583835C2semidirect
3. #26012339C2semidirect
6. C4⋊#13063443C22semidirect
10. C4×#13063443C22semidirect
50. D692M16semidirect
85. C173#1490semidirect

Maximal subgroups

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

subgroupindex
#260125122
C4×#145838352
#260123392
C2×#191496412
#1490173