Finite Group Databasebeta

← C22×#19149706 #36368297

Subgroups — C22×#19149706 #36368297

Order 22144.

1720 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 525 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×#14583827C211yesmaximal, characteristic, complemented
2C22×D4×D173C211yesmaximal, characteristic, complemented
3C2×#19149706C2124yesmaximal, direct product factor
4C22×#14583813C211yesmaximal, characteristic, complemented
5C22×D1384C211yesmaximal, characteristic, complemented
6C22×#14583811C211yesmaximal, characteristic, complemented
7C22×#14583809C211yesmaximal, characteristic, complemented
8C22×SD16×C173C211yesmaximal, characteristic, complemented
9C24×D173—21no—
10C2×#14583827—26no—
11C2×#14583827C22112yescomplemented
12C22×D692—21no—
13C22×D692C2211yescharacteristic
14C22×D692C2211yescharacteristic, complemented
15C22×C22⋊D173—21no—
16C2×D4×D173—26no—
17C2×D4×D173C22112yescomplemented
18C22×C4×D173—21no—
19C22×C4×D173C2211yescharacteristic
20#19149706C22164yesdirect product factor
21C2×#14583813C22112yescomplemented
22C2×D1384C22112yescomplemented
23C2×#14583811C22112yescomplemented
24C2×#14583809C22112yescomplemented
25C22×#13063436C2211yescharacteristic, complemented
26C22×Q8×C173C2211yescharacteristic, complemented
27C22×D4×C173C2211yescharacteristic
28C2×SD16×C173C22112yescomplemented
29C22×C8×C173C2211yescharacteristic, complemented
30C23×D173—212no—
31C23×D173—21no—
32C23×D173—21no—
33C23×D173—48no—
34C23×D173D411yescharacteristic
35#14583827—224no—
36#14583827C23116yescomplemented
37C2×D692—212no—
38C2×D692—23no—
39C2×D692—23no—
40C2×D692C2316yes—
41C2×D692C2316yescomplemented
42C2×C22⋊D173—212no—
43D4×D173—224no—
44D4×D173C23116yescomplemented
45C22×Dic173D411yescharacteristic
46C2×C4×D173—212no—
47C2×C4×D173D416yes—
48C2×C4×D173C2316yes—
49#14583813C23116yescomplemented
50D1384C23116yescomplemented
51#14583811C23116yescomplemented
52#14583809C23116yescomplemented
53C2×#13063436C2316yescomplemented
54C24×C173—21no—
55C2×Q8×C173—23no—
56C2×Q8×C173C2316yescomplemented
57C2×D4×C173—23no—
58C2×D4×C173C2316yes—
59C22×C4×C173—21no—
60C22×C4×C173C2311yescharacteristic
61SD16×C173C23116yescomplemented
62C2×C8×C173C2316yescomplemented
63C22×D173—24no—
64C22×D173—26no—
65C22×D173—26no—
66C22×D173—216no—
67C22×D173—448no—
68C22×D173—44no—
69C22×D173—44no—
70C22×D173C2×D416yes—
71D692—216no—
72D692—26no—
73D692—26no—
74D692C2414yes—
75D692C2414yescomplemented
76C22⋊D173—216no—
77C2×Dic173C2×D416yes—
78C4×D173—216no—
79C4×D173C2×D4112yes—
80C4×D173C2414yes—
81#13063436C2414yescomplemented
82C23×C173—26no—
83C23×C173—44no—
84C23×C173C2×D411yescharacteristic
85Q8×C173—26no—
86Q8×C173C2414yescomplemented
87D4×C173—26no—
88D4×C173C2414yes—
89C2×C4×C173—26no—
90C2×C4×C173C2×D413yes—
91C2×C4×C173C2413yes—
92C8×C173C2414yescomplemented
93C22×C173—24no—
94C22×C173—412no—
95C22×C173AGL(1,ℤ/8)14yescomplemented
96C22×C173C22×D413yes—
97C2×D173—212no—
98C2×D173—24no—
99C2×D173—24no—
100C2×D173—432no—
101C2×D173—412no—
102C2×D173—412no—
103C2×D173C22×D414yes—
104C4×C173—24no—
105C4×C173C22×D413yes—
106C4×C173C2511yescharacteristic
107Dic173C22×D414yes—
108C2×C173—44no—
109C2×C173C2×AGL(1,ℤ/8)16yescomplemented
110C2×C173C23×D411yescharacteristic
111D173—24no—
112D173—44no—
113D173—44no—
114C173C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
115C22×AGL(1,ℤ/8)—1731nomaximal, Sylow
116C22×#41—1731no—
117C23×D4—1731no—
118C2×AGL(1,ℤ/8)—17324no—
119C22×SD16—1731no—
120C22×SD16D17311yescharacteristic, complemented
121C22×D8—1731no—
122C22×D8—1731no—
123C22×M16—1731no—
124C25—3461no—
125C2×#41—3466no—
126C2×#41—17312no—
127C22×Q8C2×D17311yescharacteristic, complemented
128C22×D4—3461no—
129C22×D4—3466no—
130C22×D4—3461no—
131C22×D4—17312no—
132C22×D4—1731no—
133C22×D4—1731no—
134C22×D4C2×D17311yescharacteristic
135C23×C4—3461no—
136C23×C4—1731no—
137AGL(1,ℤ/8)—17364no—
138C2×SD16—17312no—
139C2×SD16C2×D173112yescomplemented
140C2×D8—17312no—
141C2×D8—17312no—
142C2×M16—17312no—
143C22×C8—1731no—
144C22×C8C2×D17311yescharacteristic, complemented
145C24—21no—
146C24—34612no—
147C24—3461no—
148C24—3461no—
149C24—1731no—
150C24—6928no—
151#41—34624no—
152#41—17316no—
153C2×Q8—23no—
154C2×Q8C22×D17316yescomplemented
155C2×D4—23no—
156C2×D4—34612no—
157C2×D4—34624no—
158C2×D4—34612no—
159C2×D4—3463no—
160C2×D4—3463no—
161C2×D4—17316no—
162C2×D4—1736no—
163C2×D4—1736no—
164C2×D4C22×D17316yes—
165C22×C4—21no—
166C22×C4—34612no—
167C22×C4—1731no—
168C22×C4—1736no—
169C22×C4—1736no—
170C22×C4C22×D17311yescharacteristic
171SD16—17316no—
172SD16C22×D173116yescomplemented
173D8—17316no—
174D8—17316no—
175M16—17316no—
176C2×C8—1736no—
177C2×C8C22×D17316yescomplemented
178C23—26no—
179C23—44no—
180C23—34616no—
181C23—3464no—
182C23—3466no—
183C23—3466no—
184C23—1736no—
185C23—69248no—
186C23—6924no—
187C23—6924no—
188C23D4×D17311yescharacteristic
189Q8—26no—
190Q8C23×D17314yescomplemented
191D4—26no—
192D4—34616no—
193D4—34616no—
194D4—3466no—
195D4—3466no—
196D4—1734no—
197D4—1734no—
198D4C23×D17314yes—
199C2×C4—26no—
200C2×C4—34616no—
201C2×C4—1736no—
202C2×C4—17312no—
203C2×C4—1734no—
204C2×C4D4×D17313yes—
205C2×C4C23×D17313yes—
206C8—1734no—
207C8C23×D17314yescomplemented
208C22—24no—
209C22—412no—
210C22—34612no—
211C22—3464no—
212C22—3464no—
213C22—1734no—
214C22—69232no—
215C22—69212no—
216C22—69212no—
217C22#1914970614yesdirect product factor
218C22C2×D4×D17313yes—
219C4—24no—
220C4—1734no—
221C4C2×D4×D17313yes—
222C4C24×D17311yescharacteristic
223C2—44no—
224C2—3464no—
225C2—6924no—
226C2—6924no—
227C2C2×#1914970616yesdirect product factor
228C2C22×D4×D17311yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
FittingC22×SD16×C173C2
socleC23×C173C2×D4
hypercenterC22×SD16D173
hypocenterC173C22×AGL(1,ℤ/8)
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14583827C2semidirect
2. C22×D4×D173C2semidirect
3. C2×#19149706C2direct (N × H)
4. C22×#14583813C2semidirect
5. C22×D1384C2semidirect
6. C22×#14583811C2semidirect
7. C22×#14583809C2semidirect
8. C22×SD16×C173C2semidirect
11. C2×#14583827C22semidirect
14. C22×D692C22semidirect
17. C2×D4×D173C22semidirect
20. #19149706C22direct (N × H)
21. C2×#14583813C22semidirect
22. C2×D1384C22semidirect
23. C2×#14583811C22semidirect
24. C2×#14583809C22semidirect
25. C22×#13063436C22semidirect
26. C22×Q8×C173C22semidirect
28. C2×SD16×C173C22semidirect
29. C22×C8×C173C22semidirect
36. #14583827C23semidirect
41. C2×D692C23semidirect
44. D4×D173C23semidirect
49. #14583813C23semidirect
50. D1384C23semidirect
51. #14583811C23semidirect
52. #14583809C23semidirect
53. C2×#13063436C23semidirect
56. C2×Q8×C173C23semidirect
61. SD16×C173C23semidirect
62. C2×C8×C173C23semidirect
75. D692C24semidirect
81. #13063436C24semidirect
86. Q8×C173C24semidirect
92. C8×C173C24semidirect
95. C22×C173AGL(1,ℤ/8)semidirect
109. C2×C173C2×AGL(1,ℤ/8)semidirect
114. C173C22×AGL(1,ℤ/8)semidirect
120. C22×SD16D173semidirect
127. C22×Q8C2×D173semidirect
139. C2×SD16C2×D173semidirect
144. C22×C8C2×D173semidirect
154. C2×Q8C22×D173semidirect
172. SD16C22×D173semidirect
177. C2×C8C22×D173semidirect
190. Q8C23×D173semidirect
207. C8C23×D173semidirect
217. C22#19149706direct (N × H)
227. C2C2×#19149706direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838272
C22×D4×D1732
C2×#191497062
C22×#145838132
C22×D13842
C22×#145838112
C22×#145838092
C22×SD16×C1732
C22×AGL(1,ℤ/8)173