Finite Group Databasebeta

← C22×#22548743 #44565100

Subgroups — C22×#22548743 #44565100

Order 32128.

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×#16860810C211yesmaximal, characteristic, complemented
2C23×D1004C211yesmaximal, characteristic, complemented
3C2×#22548743C2124yesmaximal, direct product factor
4C22×#16860799C212yesmaximal, complemented
5C22×D2008C212yesmaximal, complemented
6C22×M16×C251C211yesmaximal, characteristic, complemented
7C24×D251—21no—
8C2×#16860810—26no—
9C2×#16860810C22112yescomplemented
10C22×Dic502C2211yescharacteristic, complemented
11C22×D1004—21no—
12C22×D1004—26no—
13C22×D1004C2212yes—
14C22×D1004C22112yescomplemented
15C22×D1004C2211yescharacteristic, complemented
16C22×C22⋊D251—21no—
17C22×C4×D251—21no—
18#22548743C22164yesdirect product factor
19C2×#16860799C22124yescomplemented
20C2×D2008C22124yescomplemented
21C23×C4×C251C2211yescharacteristic
22C2×M16×C251C22112yescomplemented
23C22×C8×C251C2212yescomplemented
24C23×D251—21no—
25C23×D251—22no—
26C23×D251—212no—
27C23×D251—48no—
28#16860810—224no—
29#16860810C23116yescomplemented
30C2×Dic502—23no—
31C2×Dic502C2316yescomplemented
32C2×D1004—26no—
33C2×D1004—212no—
34C2×D1004—224no—
35C2×D1004—23no—
36C2×D1004C23112yes—
37C2×D1004C23116yescomplemented
38C2×D1004C2316yescomplemented
39C2×C22⋊D251—212no—
40C22×Dic251—21no—
41C2×C4×D251—212no—
42#16860799C23132yescomplemented
43D2008C23132yescomplemented
44C24×C251D411yescharacteristic
45C22×C4×C251D411yescharacteristic
46C22×C4×C251D416yes—
47C22×C4×C251C2311yescharacteristic
48C22×C4×C251C2316yes—
49M16×C251C23116yescomplemented
50C2×C8×C251C23112yescomplemented
51C22×D251—26no—
52C22×D251—212no—
53C22×D251—216no—
54C22×D251—448no—
55C22×D251—44no—
56C22×D251—48no—
57Dic502—26no—
58Dic502C2414yescomplemented
59D1004—212no—
60D1004—216no—
61D1004—26no—
62D1004C2418yes—
63D1004C2414yescomplemented
64C22⋊D251—216no—
65C2×Dic251—26no—
66C4×D251—216no—
67C23×C251—24no—
68C23×C251C2×D416yes—
69C23×C251C2×D411yescharacteristic
70C2×C4×C251C2×D416yes—
71C2×C4×C251C2×D413yes—
72C2×C4×C251C2×D4112yes—
73C2×C4×C251C2413yes—
74C2×C4×C251C2414yes—
75C8×C251C2418yescomplemented
76C22×C251—212no—
77C22×C251AGL(1,ℤ/8)14yescomplemented
78C22×C251C22×D414yes—
79C22×C251C22×D413yes—
80C2×D251—24no—
81C2×D251—28no—
82C2×D251—432no—
83C2×D251—412no—
84C2×D251—424no—
85C4×C251C22×D414yes—
86C4×C251C22×D413yes—
87C4×C251C2511yescharacteristic
88Dic251—24no—
89C2×C251—24no—
90C2×C251C2×AGL(1,ℤ/8)16yescomplemented
91C2×C251C23×D411yescharacteristic
92D251—44no—
93D251—48no—
94C251C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—2511nomaximal, Sylow
96C22×#41—2511no—
97C23×D4—2511no—
98C2×AGL(1,ℤ/8)—25124no—
99C22×SD16—2512no—
100C22×D8—2512no—
101C22×M16D25111yescharacteristic, complemented
102C25—5021no—
103C2×#41—5026no—
104C2×#41—25112no—
105C22×Q8—2511no—
106C22×D4—5021no—
107C22×D4—5026no—
108C22×D4—5021no—
109C22×D4—2512no—
110C22×D4—2511no—
111C22×D4—25112no—
112C23×C4—5021no—
113C23×C4C2×D25111yescharacteristic
114AGL(1,ℤ/8)—25164no—
115C2×SD16—25124no—
116C2×D8—25124no—
117C2×M16C2×D251112yescomplemented
118C22×C8C2×D25112yescomplemented
119C24—10048no—
120C24—50212no—
121C24—5021no—
122C24—5022no—
123C24D100411yescharacteristic
124#41—50224no—
125#41—25116no—
126C2×Q8—5023no—
127C2×Q8—2516no—
128C2×D4—50212no—
129C2×D4—50212no—
130C2×D4—5026no—
131C2×D4—50224no—
132C2×D4—5023no—
133C2×D4—2516no—
134C2×D4—25112no—
135C2×D4—25116no—
136C22×C4—5021no—
137C22×C4—50212no—
138C22×C4D100416yes—
139C22×C4D100411yescharacteristic
140C22×C4C22×D25116yes—
141C22×C4C22×D25111yescharacteristic
142SD16—25132no—
143D8—25132no—
144M16C22×D251116yescomplemented
145C2×C8C22×D251112yescomplemented
146C23—24no—
147C23—100448no—
148C23—10048no—
149C23—10044no—
150C23—5026no—
151C23—50216no—
152C23—50212no—
153C23C2×D100416yes—
154C23C2×D100411yescharacteristic
155Q8—5026no—
156Q8—2514no—
157D4—50216no—
158D4—50216no—
159D4—50212no—
160D4—5026no—
161D4—2514no—
162D4—2518no—
163C2×C4—5026no—
164C2×C4—50216no—
165C2×C4C2×D100416yes—
166C2×C4C2×D100413yes—
167C2×C4C2×D1004112yes—
168C2×C4C23×D25113yes—
169C2×C4C23×D25114yes—
170C8C23×D25118yescomplemented
171C22—212no—
172C22—100432no—
173C22—100424no—
174C22—100412no—
175C22—5024no—
176C22—5028no—
177C22#2254874314yesdirect product factor
178C22C22×D100414yes—
179C22C22×D100413yes—
180C4—5024no—
181C4C22×D100414yes—
182C4C22×D100413yes—
183C4C24×D25111yescharacteristic
184C2—24no—
185C2—10048no—
186C2—10044no—
187C2C2×#2254874316yesdirect product factor
188C2C23×D100411yescharacteristic
subgroupG/H
centerC23C2×D1004
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
FittingC22×M16×C251C2
socleC23×C251C2×D4
hypercenterC22×M16D251
hypocenterC251C22×AGL(1,ℤ/8)
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×#16860810C2semidirect
2. C23×D1004C2semidirect
3. C2×#22548743C2direct (N × H)
4. C22×#16860799C2semidirect
5. C22×D2008C2semidirect
6. C22×M16×C251C2semidirect
9. C2×#16860810C22semidirect
10. C22×Dic502C22semidirect
14. C22×D1004C22semidirect
15. C22×D1004C22semidirect
18. #22548743C22direct (N × H)
19. C2×#16860799C22semidirect
20. C2×D2008C22semidirect
22. C2×M16×C251C22semidirect
23. C22×C8×C251C22semidirect
29. #16860810C23semidirect
31. C2×Dic502C23semidirect
37. C2×D1004C23semidirect
38. C2×D1004C23semidirect
42. #16860799C23semidirect
43. D2008C23semidirect
49. M16×C251C23semidirect
50. C2×C8×C251C23semidirect
58. Dic502C24semidirect
63. D1004C24semidirect
75. C8×C251C24semidirect
77. C22×C251AGL(1,ℤ/8)semidirect
90. C2×C251C2×AGL(1,ℤ/8)semidirect
94. C251C22×AGL(1,ℤ/8)semidirect
101. C22×M16D251semidirect
117. C2×M16C2×D251semidirect
118. C22×C8C2×D251semidirect
144. M16C22×D251semidirect
145. C2×C8C22×D251semidirect
170. C8C23×D251semidirect
177. C22#22548743direct (N × H)
187. C2C2×#22548743direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168608102
C23×D10042
C2×#225487432
C22×#168607992
C22×D20082
C22×M16×C2512
C22×AGL(1,ℤ/8)251