Finite Group Databasebeta

← #36368864

Subgroups — #36368864

Order 22144.

312 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 84 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#26012507C211yesmaximal, characteristic, complemented
2#26012507C211yesmaximal, characteristic, complemented
3#26012339C214yesmaximal, complemented
4C2×#19149641C211yesmaximal, characteristic
5C4⋊#13063443—21no—
6C4⋊#13063443C2212yescomplemented
7C22⋊#13063443—21no—
8C22⋊#13063443—21no—
9C4×#13063443—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
20C2×#13063443—22no—
21C2×#13063443—22no—
22C23×D173—21no—
23C2×D692—21no—
24C2×D692—22no—
25C2×D692C814yescomplemented
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—42no—
43#13063443—42no—
44C22×D173—24no—
45C22×D173—22no—
46C22×D173—42no—
47C22×D173—42no—
48D692—24no—
49D692—22no—
50D692C2×C814yes—
51C2×Dic173—22no—
52C2×Dic173—41no—
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—42no—
62C2×C4×C173C22⋊C411yescharacteristic
63C2×C4×C173C2×C812yes—
64C2×C4×C173M1612yes—
65C2×C4×C173C22×C411yescharacteristic
66C22×C173—21no—
67C22×C173—21no—
68C22×C173C22⋊C812yes—
69C22×C173C2×C22⋊C411yescharacteristic
70C2×D173—24no—
71C2×D173—48no—
72C2×D173—42no—
73C2×D173—42no—
74C4×C173—42no—
75C4×C173C22⋊C812yes—
76C4×C173C22×C811yescharacteristic
77C4×C173C2×M1611yescharacteristic
78Dic173—21no—
79Dic173—21no—
80Dic173—21no—
81Dic173AGL(1,ℤ/8)12yes—
82C2×C173—22no—
83C2×C173C2×C22⋊C811yescharacteristic
84C2×C173#41811yescharacteristic
85C2×C173#42111yescharacteristic
86D173—44no—
87C173#147811yescharacteristic, complemented, Sylow
88#1478—1731nomaximal, Sylow
89C2×C4×D4—1731no—
90#431—1731no—
91#431—1731no—
92D4⋊C8—1734no—
93C22×D4—1731no—
94C23×C4—3461no—
95C4×D4—3461no—
96C4×D4—3461no—
97C4×D4—1732no—
98C4×D4—1732no—
99C2×C4⋊C4#1062192411yescharacteristic
100C2×C22⋊C4—3461no—
101C2×C42—1731no—
102C4⋊C8—3461no—
103C4⋊C8—1732no—
104C22⋊C8—3461no—
105C22⋊C8—3461no—
106C4×C8—3461no—
107C4×C8—1732no—
108C24—3461no—
109C2×D4—3461no—
110C2×D4—3462no—
111C2×D4—1734no—
112C2×D4—1732no—
113C22×C4—21no—
114C22×C4—3462no—
115C22×C4—3462no—
116C22×C4—1731no—
117C22×C4—1731no—
118C22×C4—6924no—
119C22×C4C2×#1062192411yescharacteristic
120C2×C8—3462no—
121C2×C8—3462no—
122C4⋊C4—21no—
123C4⋊C4C2×#1062192412yes—
124C22⋊C4—3462no—
125C22⋊C4—3462no—
126C42—1731no—
127C42—1731no—
128C42—1731no—
129C42—1731no—
130C23—3464no—
131C23—3462no—
132C23—6922no—
133C23—6922no—
134C23C22⋊#1062192411yescharacteristic
135D4—3464no—
136D4—3462no—
137D4—1734no—
138C2×C4—22no—
139C2×C4—42no—
140C2×C4—3462no—
141C2×C4—1731no—
142C2×C4—1731no—
143C2×C4—1731no—
144C2×C4—1731no—
145C2×C4—6924no—
146C2×C4—6924no—
147C2×C4—6921no—
148C2×C4C22⋊#1062192411yescharacteristic
149C2×C4D173⋊C812yes—
150C2×C4#1458383612yes—
151C2×C4C22×#1062192411yescharacteristic
152C8—6922no—
153C8—6922no—
154C22—21no—
155C22—21no—
156C22—3464no—
157C22—6928no—
158C22—6922no—
159C22—6922no—
160C22#1914973612yes—
161C22C2×C22⋊#1062192411yescharacteristic
162C4—42no—
163C4—3461no—
164C4—3461no—
165C4—3461no—
166C4—1732no—
167C4#1914973612yes—
168C4C2×D173⋊C811yescharacteristic
169C4C2×#1458383611yescharacteristic
170C2—22no—
171C2—6924no—
172C2C2×#1914973611yescharacteristic
173C2#2601248311yescharacteristic
174C2#2601249011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC4×C173C22×C8
FrattiniC2×C4C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×D692C4
cosocleC4×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1478
LCSG ▷ 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. #26012507C2semidirect
2. #26012507C2semidirect
3. #26012339C2semidirect
6. C4⋊#13063443C22semidirect
10. C4×#13063443C22semidirect
25. C2×D692C8semidirect
87. C173#1478semidirect

Maximal subgroups

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

subgroupindex
#260125072
#260123392
C2×#191496412
#1478173