Finite Group Databasebeta

← C22×#22548741 #44565098

Subgroups — C22×#22548741 #44565098

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
2C2×#22548741C2124yesmaximal, direct product factor
3C22×#16860798C212yesmaximal, complemented
4C22×#16860795C212yesmaximal, complemented
5C22×#16860794C211yesmaximal, characteristic, complemented
6C23×D4×C251C211yesmaximal, characteristic, complemented
7C2×#16860810—26no—
8C2×#16860810C22112yescomplemented
9C22×Dic502C2211yescharacteristic, complemented
10C22×D1004C2211yescharacteristic, complemented
11C22×C22⋊D251—21no—
12C22×C4×D251—21no—
13#22548741C22164yesdirect product factor
14C2×#16860798C22124yescomplemented
15C2×#16860795C22124yescomplemented
16C2×#16860794C22112yescomplemented
17C22×C251⋊C8C2212yescomplemented
18C25×C251—21no—
19C22×D4×C251—21no—
20C22×D4×C251—26no—
21C22×D4×C251C2212yes—
22C22×D4×C251C22112yescomplemented
23C23×C4×C251C2211yescharacteristic
24C23×D251—21no—
25#16860810—224no—
26#16860810C23116yescomplemented
27C2×Dic502—23no—
28C2×Dic502C2316yescomplemented
29C2×D1004—23no—
30C2×D1004C2316yescomplemented
31C2×C22⋊D251—212no—
32C22×Dic251—21no—
33C2×C4×D251—212no—
34#16860798C23132yescomplemented
35#16860795C23132yescomplemented
36#16860794C23116yescomplemented
37C2×C251⋊C8C23112yescomplemented
38C24×C251—22no—
39C24×C251—212no—
40C24×C251—48no—
41C24×C251D411yescharacteristic
42C2×D4×C251—26no—
43C2×D4×C251—212no—
44C2×D4×C251—224no—
45C2×D4×C251C23112yes—
46C2×D4×C251C23116yescomplemented
47C22×C4×C251D411yescharacteristic
48C22×C4×C251D416yes—
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2316yes—
51C22×D251—26no—
52C22×D251—44no—
53Dic502—26no—
54Dic502C2414yescomplemented
55D1004—26no—
56D1004C2414yescomplemented
57C22⋊D251—216no—
58C2×Dic251—26no—
59C4×D251—216no—
60C251⋊C8C2418yescomplemented
61C23×C251—24no—
62C23×C251—212no—
63C23×C251—216no—
64C23×C251—448no—
65C23×C251—48no—
66C23×C251C2×D416yes—
67C23×C251C2×D411yescharacteristic
68D4×C251—212no—
69D4×C251—216no—
70D4×C251C2418yes—
71C2×C4×C251C2×D416yes—
72C2×C4×C251C2×D413yes—
73C2×C4×C251C2×D4112yes—
74C2×C4×C251C2413yes—
75C2×C4×C251C2414yes—
76C22×C251—212no—
77C22×C251—28no—
78C22×C251—432no—
79C22×C251—424no—
80C22×C251AGL(1,ℤ/8)14yescomplemented
81C22×C251C22×D414yes—
82C22×C251C22×D413yes—
83C2×D251—24no—
84C2×D251—412no—
85C4×C251C22×D414yes—
86C4×C251C22×D413yes—
87C4×C251C2511yescharacteristic
88Dic251—24no—
89C2×C251—24no—
90C2×C251—48no—
91C2×C251C2×AGL(1,ℤ/8)16yescomplemented
92C2×C251C23×D411yescharacteristic
93D251—44no—
94C251C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—2511nomaximal, Sylow
96C22×#41—2511no—
97C23×D4D25111yescharacteristic, complemented
98C2×AGL(1,ℤ/8)—25124no—
99C22×SD16—2512no—
100C22×D8—2512no—
101C22×M16—2511no—
102C25—21no—
103C2×#41—5026no—
104C2×#41—25112no—
105C22×Q8—2511no—
106C22×D4—26no—
107C22×D4—21no—
108C22×D4—5021no—
109C22×D4—2511no—
110C22×D4C2×D25112yes—
111C22×D4C2×D251112yescomplemented
112C23×C4—5021no—
113C23×C4C2×D25111yescharacteristic
114AGL(1,ℤ/8)—25164no—
115C2×SD16—25124no—
116C2×D8—25124no—
117C2×M16—25112no—
118C22×C8—2512no—
119C24—212no—
120C24—22no—
121C24—48no—
122C24—5021no—
123C24C22⋊D25111yescharacteristic
124#41—50224no—
125#41—25116no—
126C2×Q8—5023no—
127C2×Q8—2516no—
128C2×D4—26no—
129C2×D4—212no—
130C2×D4—224no—
131C2×D4—50212no—
132C2×D4—5023no—
133C2×D4—2516no—
134C2×D4C22×D251112yes—
135C2×D4C22×D251116yescomplemented
136C22×C4—5021no—
137C22×C4—50212no—
138C22×C4C22⋊D25116yes—
139C22×C4C22⋊D25111yescharacteristic
140C22×C4C22×D25116yes—
141C22×C4C22×D25111yescharacteristic
142SD16—25132no—
143D8—25132no—
144M16—25116no—
145C2×C8—25112no—
146C23—24no—
147C23—212no—
148C23—216no—
149C23—448no—
150C23—48no—
151C23—10044no—
152C23—5026no—
153C23C2×C22⋊D25116yes—
154C23C2×C22⋊D25111yescharacteristic
155Q8—5026no—
156Q8—2514no—
157D4—212no—
158D4—216no—
159D4—50216no—
160D4—5026no—
161D4—2514no—
162D4C23×D25118yes—
163C2×C4—5026no—
164C2×C4—50216no—
165C2×C4C2×C22⋊D25116yes—
166C2×C4C2×C22⋊D25113yes—
167C2×C4C2×C22⋊D251112yes—
168C2×C4C23×D25113yes—
169C2×C4C23×D25114yes—
170C8—2518no—
171C22—212no—
172C22—28no—
173C22—432no—
174C22—424no—
175C22—100412no—
176C22—5024no—
177C22#2254874114yesdirect product factor
178C22C22×C22⋊D25114yes—
179C22C22×C22⋊D25113yes—
180C4—5024no—
181C4C22×C22⋊D25114yes—
182C4C22×C22⋊D25113yes—
183C4C24×D25111yescharacteristic
184C2—24no—
185C2—48no—
186C2—10044no—
187C2C2×#2254874116yesdirect product factor
188C2C23×C22⋊D25111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D251
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
FittingC23×D4×C251C2
socleC23×C251C2×D4
hypercenterC23×D4D251
hypocenterC251C22×AGL(1,ℤ/8)
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#16860810C2semidirect
2. C2×#22548741C2direct (N × H)
3. C22×#16860798C2semidirect
4. C22×#16860795C2semidirect
5. C22×#16860794C2semidirect
6. C23×D4×C251C2semidirect
8. C2×#16860810C22semidirect
9. C22×Dic502C22semidirect
10. C22×D1004C22semidirect
13. #22548741C22direct (N × H)
14. C2×#16860798C22semidirect
15. C2×#16860795C22semidirect
16. C2×#16860794C22semidirect
17. C22×C251⋊C8C22semidirect
22. C22×D4×C251C22semidirect
26. #16860810C23semidirect
28. C2×Dic502C23semidirect
30. C2×D1004C23semidirect
34. #16860798C23semidirect
35. #16860795C23semidirect
36. #16860794C23semidirect
37. C2×C251⋊C8C23semidirect
46. C2×D4×C251C23semidirect
54. Dic502C24semidirect
56. D1004C24semidirect
60. C251⋊C8C24semidirect
80. C22×C251AGL(1,ℤ/8)semidirect
91. C2×C251C2×AGL(1,ℤ/8)semidirect
94. C251C22×AGL(1,ℤ/8)semidirect
97. C23×D4D251semidirect
111. C22×D4C2×D251semidirect
135. C2×D4C22×D251semidirect
177. C22#22548741direct (N × H)
187. C2C2×#22548741direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168608102
C2×#225487412
C22×#168607982
C22×#168607952
C22×#168607942
C23×D4×C2512
C22×AGL(1,ℤ/8)251