Finite Group Databasebeta

← C22×#19149707 #36368298

Subgroups — C22×#19149707 #36368298

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
2C23×D692C211yesmaximal, characteristic, complemented
3C2×#19149707C2124yesmaximal, direct product factor
4C22×#14583815C212yesmaximal, complemented
5C22×D1384C212yesmaximal, complemented
6C22×M16×C173C211yesmaximal, characteristic, complemented
7C24×D173—21no—
8C2×#14583826—26no—
9C2×#14583826C22112yescomplemented
10C22×Dic346C2211yescharacteristic, complemented
11C22×D692—26no—
12C22×D692—21no—
13C22×D692C2212yes—
14C22×D692C22112yescomplemented
15C22×D692C2211yescharacteristic, complemented
16C22×C22⋊D173—21no—
17C22×C4×D173—21no—
18#19149707C22164yesdirect product factor
19C2×#14583815C22124yescomplemented
20C2×D1384C22124yescomplemented
21C23×C4×C173C2211yescharacteristic
22C2×M16×C173C22112yescomplemented
23C22×C8×C173C2212yescomplemented
24C23×D173—21no—
25C23×D173—22no—
26C23×D173—212no—
27C23×D173—48no—
28#14583826—224no—
29#14583826C23116yescomplemented
30C2×Dic346—23no—
31C2×Dic346C2316yescomplemented
32C2×D692—26no—
33C2×D692—212no—
34C2×D692—224no—
35C2×D692—23no—
36C2×D692C23112yes—
37C2×D692C23116yescomplemented
38C2×D692C2316yescomplemented
39C2×C22⋊D173—212no—
40C22×Dic173—21no—
41C2×C4×D173—212no—
42#14583815C23132yescomplemented
43D1384C23132yescomplemented
44C24×C173D411yescharacteristic
45C22×C4×C173D411yescharacteristic
46C22×C4×C173D416yes—
47C22×C4×C173C2311yescharacteristic
48C22×C4×C173C2316yes—
49M16×C173C23116yescomplemented
50C2×C8×C173C23112yescomplemented
51C22×D173—26no—
52C22×D173—212no—
53C22×D173—216no—
54C22×D173—448no—
55C22×D173—44no—
56C22×D173—48no—
57Dic346—26no—
58Dic346C2414yescomplemented
59D692—212no—
60D692—216no—
61D692—26no—
62D692C2418yes—
63D692C2414yescomplemented
64C22⋊D173—216no—
65C2×Dic173—26no—
66C4×D173—216no—
67C23×C173—24no—
68C23×C173C2×D416yes—
69C23×C173C2×D411yescharacteristic
70C2×C4×C173C2×D416yes—
71C2×C4×C173C2×D413yes—
72C2×C4×C173C2×D4112yes—
73C2×C4×C173C2413yes—
74C2×C4×C173C2414yes—
75C8×C173C2418yescomplemented
76C22×C173—212no—
77C22×C173AGL(1,ℤ/8)14yescomplemented
78C22×C173C22×D414yes—
79C22×C173C22×D413yes—
80C2×D173—24no—
81C2×D173—28no—
82C2×D173—432no—
83C2×D173—412no—
84C2×D173—424no—
85C4×C173C22×D414yes—
86C4×C173C22×D413yes—
87C4×C173C2511yescharacteristic
88Dic173—24no—
89C2×C173—24no—
90C2×C173C2×AGL(1,ℤ/8)16yescomplemented
91C2×C173C23×D411yescharacteristic
92D173—44no—
93D173—48no—
94C173C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1731nomaximal, Sylow
96C22×#41—1731no—
97C23×D4—1731no—
98C2×AGL(1,ℤ/8)—17324no—
99C22×SD16—1732no—
100C22×D8—1732no—
101C22×M16D17311yescharacteristic, complemented
102C25—3461no—
103C2×#41—3466no—
104C2×#41—17312no—
105C22×Q8—1731no—
106C22×D4—3461no—
107C22×D4—3466no—
108C22×D4—3461no—
109C22×D4—1731no—
110C22×D4—1732no—
111C22×D4—17312no—
112C23×C4—3461no—
113C23×C4C2×D17311yescharacteristic
114AGL(1,ℤ/8)—17364no—
115C2×SD16—17324no—
116C2×D8—17324no—
117C2×M16C2×D173112yescomplemented
118C22×C8C2×D17312yescomplemented
119C24—34612no—
120C24—3461no—
121C24—3462no—
122C24—6928no—
123C24D69211yescharacteristic
124#41—34624no—
125#41—17316no—
126C2×Q8—3463no—
127C2×Q8—1736no—
128C2×D4—34612no—
129C2×D4—34612no—
130C2×D4—3466no—
131C2×D4—34624no—
132C2×D4—3463no—
133C2×D4—17312no—
134C2×D4—1736no—
135C2×D4—17316no—
136C22×C4—3461no—
137C22×C4—34612no—
138C22×C4D69216yes—
139C22×C4D69211yescharacteristic
140C22×C4C22×D17316yes—
141C22×C4C22×D17311yescharacteristic
142SD16—17332no—
143D8—17332no—
144M16C22×D173116yescomplemented
145C2×C8C22×D173112yescomplemented
146C23—24no—
147C23—3466no—
148C23—34616no—
149C23—34612no—
150C23—69248no—
151C23—6924no—
152C23—6928no—
153C23C2×D69216yes—
154C23C2×D69211yescharacteristic
155Q8—3466no—
156Q8—1734no—
157D4—34616no—
158D4—34616no—
159D4—34612no—
160D4—3466no—
161D4—1738no—
162D4—1734no—
163C2×C4—3466no—
164C2×C4—34616no—
165C2×C4C2×D69216yes—
166C2×C4C2×D69213yes—
167C2×C4C2×D692112yes—
168C2×C4C23×D17313yes—
169C2×C4C23×D17314yes—
170C8C23×D17318yescomplemented
171C22—212no—
172C22—3464no—
173C22—3468no—
174C22—69232no—
175C22—69224no—
176C22—69212no—
177C22#1914970714yesdirect product factor
178C22C22×D69214yes—
179C22C22×D69213yes—
180C4—3464no—
181C4C22×D69214yes—
182C4C22×D69213yes—
183C4C24×D17311yescharacteristic
184C2—24no—
185C2—6924no—
186C2—6928no—
187C2C2×#1914970716yesdirect product factor
188C2C23×D69211yescharacteristic
subgroupG/H
centerC23C2×D692
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
FittingC22×M16×C173C2
socleC23×C173C2×D4
hypercenterC22×M16D173
hypocenterC173C22×AGL(1,ℤ/8)
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×#14583826C2semidirect
2. C23×D692C2semidirect
3. C2×#19149707C2direct (N × H)
4. C22×#14583815C2semidirect
5. C22×D1384C2semidirect
6. C22×M16×C173C2semidirect
9. C2×#14583826C22semidirect
10. C22×Dic346C22semidirect
14. C22×D692C22semidirect
15. C22×D692C22semidirect
18. #19149707C22direct (N × H)
19. C2×#14583815C22semidirect
20. C2×D1384C22semidirect
22. C2×M16×C173C22semidirect
23. C22×C8×C173C22semidirect
29. #14583826C23semidirect
31. C2×Dic346C23semidirect
37. C2×D692C23semidirect
38. C2×D692C23semidirect
42. #14583815C23semidirect
43. D1384C23semidirect
49. M16×C173C23semidirect
50. C2×C8×C173C23semidirect
58. Dic346C24semidirect
63. D692C24semidirect
75. C8×C173C24semidirect
77. C22×C173AGL(1,ℤ/8)semidirect
90. C2×C173C2×AGL(1,ℤ/8)semidirect
94. C173C22×AGL(1,ℤ/8)semidirect
101. C22×M16D173semidirect
117. C2×M16C2×D173semidirect
118. C22×C8C2×D173semidirect
144. M16C22×D173semidirect
145. C2×C8C22×D173semidirect
170. C8C23×D173semidirect
177. C22#19149707direct (N × H)
187. C2C2×#19149707direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838262
C23×D6922
C2×#191497072
C22×#145838152
C22×D13842
C22×M16×C1732
C22×AGL(1,ℤ/8)173