Finite Group Databasebeta

← #36352088

Subgroups — #36352088

Order 22144.

242 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 81 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#26011548C211yesmaximal, characteristic, complemented
2C4×C4⋊Dic173C211yesmaximal, characteristic, complemented
3#26011228C212yesmaximal, complemented
4#26011225C212yesmaximal
5#529×C173C211yesmaximal, characteristic
6C2×C4⋊Dic173—21no—
7C2×C4×Dic173—21no—
8C4⋊#13063436—21no—
9C22⋊#13063436C2212yes—
10#19149590—21no—
11#19149586C2212yes—
12#126×C173—21no—
13#125×C173C411yescharacteristic, complemented
14#124×C173C411yescharacteristic, complemented
15#123×C173—21no—
16C22⋊Q8×C173C2211yescharacteristic
17#121×C173C2211yescharacteristic
18C4×Q8×C173C411yescharacteristic, complemented
19C4×D4×C173—21no—
20C4×D4×C173C411yescharacteristic, complemented
21#117×C173—21no—
22C2×C42×C173C2211yescharacteristic
23C22×Dic173—22no—
24C2×#13063436—22no—
25C4⋊Dic173—24no—
26C4×Dic173—42no—
27C2×Q8×C173C2×C411yescharacteristic
28C2×D4×C173—21no—
29C2×D4×C173C2×C411yescharacteristic
30C22×C4×C173—21no—
31C22×C4×C173D411yescharacteristic
32C22×C4×C173D411yescharacteristic
33C22×C4×C173C2311yescharacteristic
34C4⋊C4×C173—21no—
35C4⋊C4×C173—21no—
36C4⋊C4×C173—21no—
37C4⋊C4×C173—21no—
38C4⋊C4×C173C2×C411yescharacteristic
39C4⋊C4×C173C2×C411yescharacteristic
40C22⋊C4×C173—21no—
41C22⋊C4×C173—21no—
42C22⋊C4×C173—21no—
43C22⋊C4×C173—21no—
44C22⋊C4×C173—21no—
45C42×C173—21no—
46C42×C173C2×C411yescharacteristic
47C42×C173C2×C411yescharacteristic
48C42×C173D411yescharacteristic
49C42×C173D411yescharacteristic
50C2×Dic173—24no—
51C2×Dic173—42no—
52C2×Dic173—42no—
53#13063436—42no—
54C23×C173—21no—
55C23×C173C2×D411yescharacteristic
56Q8×C173—41no—
57D4×C173—41no—
58D4×C173—41no—
59D4×C173—41no—
60C2×C4×C173—21no—
61C2×C4×C173—21no—
62C2×C4×C173—22no—
63C2×C4×C173—21no—
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—41no—
67C2×C4×C173—42no—
68C2×C4×C173C22⋊C411yescharacteristic
69C2×C4×C173C22⋊C411yescharacteristic
70C2×C4×C173C22⋊C411yescharacteristic
71C2×C4×C173C22⋊C411yescharacteristic
72C2×C4×C173C22×C411yescharacteristic
73C2×C4×C173C2×D411yescharacteristic
74C22×C173—21no—
75C22×C173—21no—
76C22×C173—41no—
77C22×C173—41no—
78C22×C173—41no—
79C22×C173C2×C22⋊C411yescharacteristic
80C22×C173AGL(1,ℤ/8)11yescharacteristic
81C22×C173#13711yescharacteristic
82C4×C173—21no—
83C4×C173—21no—
84C4×C173—21no—
85C4×C173—21no—
86C4×C173—21no—
87C4×C173—41no—
88C4×C173—41no—
89C4×C173—41no—
90C4×C173#10412yes—
91Dic173—44no—
92C2×C173—21no—
93C2×C173—21no—
94C2×C173—81no—
95C2×C173#41011yescharacteristic
96C2×C173#41711yescharacteristic
97C2×C173C2×#10411yescharacteristic
98C173#150911yescharacteristic, complemented, Sylow
99#1509—1731nomaximal, Sylow
100#529D17311yescharacteristic
101#432—1731no—
102C4×C4⋊C4—1731no—
103#328—1732no—
104#327—1732no—
105#126—21no—
106#125Dic17311yescharacteristic, complemented
107#124Dic17311yescharacteristic, complemented
108#123—21no—
109C22⋊Q8C2×D17311yescharacteristic
110#121C2×D17311yescharacteristic
111C4×Q8Dic17311yescharacteristic, complemented
112C4×D4—21no—
113C4×D4Dic17311yescharacteristic, complemented
114#117—21no—
115C2×C4⋊C4—3461no—
116C2×C42—3461no—
117C2×C42C2×D17311yescharacteristic
118C4⋊C8—3461no—
119C22⋊C8—1732no—
120#97—3461no—
121#95—1732no—
122C2×Q8C2×Dic17311yescharacteristic
123C2×D4—21no—
124C2×D4C2×Dic17311yescharacteristic
125C22×C4—21no—
126C22×C4—3462no—
127C22×C4C22⋊D17311yescharacteristic
128C22×C4C22⋊D17311yescharacteristic
129C22×C4C22×D17311yescharacteristic
130C2×C8—3462no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—3464no—
136C4⋊C4C2×Dic17311yescharacteristic
137C4⋊C4C2×Dic17311yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C22⋊C4—21no—
143C42—21no—
144C42—6922no—
145C42C2×Dic17311yescharacteristic
146C42C2×Dic17311yescharacteristic
147C42C22⋊D17311yescharacteristic
148C42C22⋊D17311yescharacteristic
149C23—21no—
150C23C2×C22⋊D17311yescharacteristic
151Q8—41no—
152D4—41no—
153D4—41no—
154D4—41no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—3464no—
164C2×C4—6922no—
165C2×C4—6922no—
166C2×C4C22⋊Dic17311yescharacteristic
167C2×C4C22⋊Dic17311yescharacteristic
168C2×C4C22⋊Dic17311yescharacteristic
169C2×C4C22⋊Dic17311yescharacteristic
170C2×C4C22×Dic17311yescharacteristic
171C2×C4C2×C22⋊D17311yescharacteristic
172C8—6922no—
173C22—21no—
174C22—21no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22C2×C22⋊Dic17311yescharacteristic
179C22#1914970411yescharacteristic
180C22#1914970911yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—41no—
188C4—41no—
189C4—6924no—
190C4#1914960612yes—
191C2—21no—
192C2—21no—
193C2—81no—
194C2#2601146511yescharacteristic
195C2#2601149211yescharacteristic
196C2C2×#1914960611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#529×C173C2
socleC22×C173C2×C22⋊C4
prime radical#121×C173C22
cosocleC22×C4×C173C23
hypercenter#529D173
hypocenterC173#1509
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #26011548C2semidirect
2. C4×C4⋊Dic173C2semidirect
3. #26011228C2semidirect
13. #125×C173C4semidirect
14. #124×C173C4semidirect
18. C4×Q8×C173C4semidirect
20. C4×D4×C173C4semidirect
98. C173#1509semidirect
106. #125Dic173semidirect
107. #124Dic173semidirect
111. C4×Q8Dic173semidirect
113. C4×D4Dic173semidirect

Maximal subgroups

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

subgroupindex
#260115482
C4×C4⋊Dic1732
#260112282
#260112252
#529×C1732
#1509173