Finite Group Databasebeta

← C22×#19149705 #36368296

Subgroups — C22×#19149705 #36368296

Order 22144.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 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
1C22×#14583826C211yesmaximal, characteristic, complemented
2C2×#19149705C2124yesmaximal, direct product factor
3C22×#14583814C212yesmaximal, complemented
4C22×#14583811C212yesmaximal, complemented
5C22×#14583810C211yesmaximal, characteristic, complemented
6C23×D4×C173C211yesmaximal, characteristic, complemented
7C2×#14583826—26no—
8C2×#14583826C22112yescomplemented
9C22×Dic346C2211yescharacteristic, complemented
10C22×D692C2211yescharacteristic, complemented
11C22×C22⋊D173—21no—
12C22×C4×D173—21no—
13#19149705C22164yesdirect product factor
14C2×#14583814C22124yescomplemented
15C2×#14583811C22124yescomplemented
16C2×#14583810C22112yescomplemented
17C22×#13063436C2212yescomplemented
18C25×C173—21no—
19C22×D4×C173—21no—
20C22×D4×C173—26no—
21C22×D4×C173C2212yes—
22C22×D4×C173C22112yescomplemented
23C23×C4×C173C2211yescharacteristic
24C23×D173—21no—
25#14583826—224no—
26#14583826C23116yescomplemented
27C2×Dic346—23no—
28C2×Dic346C2316yescomplemented
29C2×D692—23no—
30C2×D692C2316yescomplemented
31C2×C22⋊D173—212no—
32C22×Dic173—21no—
33C2×C4×D173—212no—
34#14583814C23132yescomplemented
35#14583811C23132yescomplemented
36#14583810C23116yescomplemented
37C2×#13063436C23112yescomplemented
38C24×C173—22no—
39C24×C173—212no—
40C24×C173—48no—
41C24×C173D411yescharacteristic
42C2×D4×C173—26no—
43C2×D4×C173—212no—
44C2×D4×C173—224no—
45C2×D4×C173C23112yes—
46C2×D4×C173C23116yescomplemented
47C22×C4×C173D411yescharacteristic
48C22×C4×C173D416yes—
49C22×C4×C173C2311yescharacteristic
50C22×C4×C173C2316yes—
51C22×D173—26no—
52C22×D173—44no—
53Dic346—26no—
54Dic346C2414yescomplemented
55D692—26no—
56D692C2414yescomplemented
57C22⋊D173—216no—
58C2×Dic173—26no—
59C4×D173—216no—
60#13063436C2418yescomplemented
61C23×C173—24no—
62C23×C173—212no—
63C23×C173—216no—
64C23×C173—448no—
65C23×C173—48no—
66C23×C173C2×D416yes—
67C23×C173C2×D411yescharacteristic
68D4×C173—212no—
69D4×C173—216no—
70D4×C173C2418yes—
71C2×C4×C173C2×D416yes—
72C2×C4×C173C2×D413yes—
73C2×C4×C173C2×D4112yes—
74C2×C4×C173C2413yes—
75C2×C4×C173C2414yes—
76C22×C173—212no—
77C22×C173—28no—
78C22×C173—432no—
79C22×C173—424no—
80C22×C173AGL(1,ℤ/8)14yescomplemented
81C22×C173C22×D414yes—
82C22×C173C22×D413yes—
83C2×D173—24no—
84C2×D173—412no—
85C4×C173C22×D414yes—
86C4×C173C22×D413yes—
87C4×C173C2511yescharacteristic
88Dic173—24no—
89C2×C173—24no—
90C2×C173—48no—
91C2×C173C2×AGL(1,ℤ/8)16yescomplemented
92C2×C173C23×D411yescharacteristic
93D173—44no—
94C173C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1731nomaximal, Sylow
96C22×#41—1731no—
97C23×D4D17311yescharacteristic, complemented
98C2×AGL(1,ℤ/8)—17324no—
99C22×SD16—1732no—
100C22×D8—1732no—
101C22×M16—1731no—
102C25—21no—
103C2×#41—3466no—
104C2×#41—17312no—
105C22×Q8—1731no—
106C22×D4—26no—
107C22×D4—21no—
108C22×D4—3461no—
109C22×D4—1731no—
110C22×D4C2×D17312yes—
111C22×D4C2×D173112yescomplemented
112C23×C4—3461no—
113C23×C4C2×D17311yescharacteristic
114AGL(1,ℤ/8)—17364no—
115C2×SD16—17324no—
116C2×D8—17324no—
117C2×M16—17312no—
118C22×C8—1732no—
119C24—212no—
120C24—22no—
121C24—48no—
122C24—3461no—
123C24C22⋊D17311yescharacteristic
124#41—34624no—
125#41—17316no—
126C2×Q8—3463no—
127C2×Q8—1736no—
128C2×D4—26no—
129C2×D4—212no—
130C2×D4—224no—
131C2×D4—34612no—
132C2×D4—3463no—
133C2×D4—1736no—
134C2×D4C22×D173112yes—
135C2×D4C22×D173116yescomplemented
136C22×C4—3461no—
137C22×C4—34612no—
138C22×C4C22⋊D17316yes—
139C22×C4C22⋊D17311yescharacteristic
140C22×C4C22×D17316yes—
141C22×C4C22×D17311yescharacteristic
142SD16—17332no—
143D8—17332no—
144M16—17316no—
145C2×C8—17312no—
146C23—24no—
147C23—212no—
148C23—216no—
149C23—448no—
150C23—48no—
151C23—3466no—
152C23—6924no—
153C23C2×C22⋊D17316yes—
154C23C2×C22⋊D17311yescharacteristic
155Q8—3466no—
156Q8—1734no—
157D4—212no—
158D4—216no—
159D4—34616no—
160D4—3466no—
161D4—1734no—
162D4C23×D17318yes—
163C2×C4—3466no—
164C2×C4—34616no—
165C2×C4C2×C22⋊D17316yes—
166C2×C4C2×C22⋊D17313yes—
167C2×C4C2×C22⋊D173112yes—
168C2×C4C23×D17313yes—
169C2×C4C23×D17314yes—
170C8—1738no—
171C22—212no—
172C22—28no—
173C22—432no—
174C22—424no—
175C22—3464no—
176C22—69212no—
177C22#1914970514yesdirect product factor
178C22C22×C22⋊D17314yes—
179C22C22×C22⋊D17313yes—
180C4—3464no—
181C4C22×C22⋊D17314yes—
182C4C22×C22⋊D17313yes—
183C4C24×D17311yescharacteristic
184C2—24no—
185C2—48no—
186C2—6924no—
187C2C2×#1914970516yesdirect product factor
188C2C23×C22⋊D17311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
FittingC23×D4×C173C2
socleC23×C173C2×D4
hypercenterC23×D4D173
hypocenterC173C22×AGL(1,ℤ/8)
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14583826C2semidirect
2. C2×#19149705C2direct (N × H)
3. C22×#14583814C2semidirect
4. C22×#14583811C2semidirect
5. C22×#14583810C2semidirect
6. C23×D4×C173C2semidirect
8. C2×#14583826C22semidirect
9. C22×Dic346C22semidirect
10. C22×D692C22semidirect
13. #19149705C22direct (N × H)
14. C2×#14583814C22semidirect
15. C2×#14583811C22semidirect
16. C2×#14583810C22semidirect
17. C22×#13063436C22semidirect
22. C22×D4×C173C22semidirect
26. #14583826C23semidirect
28. C2×Dic346C23semidirect
30. C2×D692C23semidirect
34. #14583814C23semidirect
35. #14583811C23semidirect
36. #14583810C23semidirect
37. C2×#13063436C23semidirect
46. C2×D4×C173C23semidirect
54. Dic346C24semidirect
56. D692C24semidirect
60. #13063436C24semidirect
80. C22×C173AGL(1,ℤ/8)semidirect
91. C2×C173C2×AGL(1,ℤ/8)semidirect
94. C173C22×AGL(1,ℤ/8)semidirect
97. C23×D4D173semidirect
111. C22×D4C2×D173semidirect
135. C2×D4C22×D173semidirect
177. C22#19149705direct (N × H)
187. C2C2×#19149705direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838262
C2×#191497052
C22×#145838142
C22×#145838112
C22×#145838102
C23×D4×C1732
C22×AGL(1,ℤ/8)173