Finite Group Databasebeta

← C22×#22548740 #44565097

Subgroups — C22×#22548740 #44565097

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
1C23×D1004C211yesmaximal, characteristic, complemented
2C2×#22548740C2124yesmaximal, direct product factor
3C22×#16860797C212yesmaximal, complemented
4C22×#16860795C212yesmaximal, complemented
5C22×#16860794C211yesmaximal, characteristic, complemented
6C22×#41×C251C211yesmaximal, characteristic, complemented
7C24×D251—21no—
8C22×D1004—21no—
9C22×D1004—26no—
10C22×D1004C2212yes—
11C22×D1004C22112yescomplemented
12#22548740C22164yesdirect product factor
13C2×#16860797C22124yescomplemented
14C2×#16860795C22124yescomplemented
15C2×#16860794C22112yescomplemented
16C22×C251⋊C8C2212yescomplemented
17C2×#41×C251—26no—
18C2×#41×C251C22112yescomplemented
19C22×Q8×C251C2211yescharacteristic, complemented
20C22×D4×C251—21no—
21C22×D4×C251C2211yescharacteristic, complemented
22C23×C4×C251—21no—
23C23×C4×C251C2211yescharacteristic
24C23×D251—22no—
25C23×D251—212no—
26C23×D251—48no—
27C2×D1004—26no—
28C2×D1004—212no—
29C2×D1004—224no—
30C2×D1004C23112yes—
31C2×D1004C23116yescomplemented
32#16860797C23132yescomplemented
33#16860795C23132yescomplemented
34#16860794C23116yescomplemented
35C2×C251⋊C8C23112yescomplemented
36C24×C251—21no—
37C24×C251D411yescharacteristic
38#41×C251—224no—
39#41×C251C23116yescomplemented
40C2×Q8×C251—23no—
41C2×Q8×C251C2316yescomplemented
42C2×D4×C251—212no—
43C2×D4×C251—23no—
44C2×D4×C251C2316yescomplemented
45C22×C4×C251—21no—
46C22×C4×C251—212no—
47C22×C4×C251D411yescharacteristic
48C22×C4×C251D416yes—
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2316yes—
51C22×D251—212no—
52C22×D251—216no—
53C22×D251—448no—
54C22×D251—48no—
55D1004—212no—
56D1004—216no—
57D1004C2418yes—
58C251⋊C8C2418yescomplemented
59C23×C251—24no—
60C23×C251—26no—
61C23×C251—44no—
62C23×C251C2×D416yes—
63C23×C251C2×D411yescharacteristic
64Q8×C251—26no—
65Q8×C251C2414yescomplemented
66D4×C251—216no—
67D4×C251—26no—
68D4×C251C2414yescomplemented
69C2×C4×C251—26no—
70C2×C4×C251—216no—
71C2×C4×C251C2×D416yes—
72C2×C4×C251C2×D413yes—
73C2×C4×C251C2×D4112yes—
74C2×C4×C251C2413yes—
75C2×C4×C251C2414yes—
76C22×C251—212no—
77C22×C251—24no—
78C22×C251—412no—
79C22×C251AGL(1,ℤ/8)14yescomplemented
80C22×C251C22×D414yes—
81C22×C251C22×D413yes—
82C2×D251—28no—
83C2×D251—432no—
84C2×D251—424no—
85C4×C251—24no—
86C4×C251C22×D414yes—
87C4×C251C22×D413yes—
88C4×C251C2511yescharacteristic
89C2×C251—24no—
90C2×C251—44no—
91C2×C251C2×AGL(1,ℤ/8)16yescomplemented
92C2×C251C23×D411yescharacteristic
93D251—48no—
94C251C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—2511nomaximal, Sylow
96C22×#41D25111yescharacteristic, complemented
97C23×D4—2511no—
98C2×AGL(1,ℤ/8)—25124no—
99C22×SD16—2512no—
100C22×D8—2512no—
101C22×M16—2511no—
102C25—5021no—
103C2×#41—26no—
104C2×#41C2×D251112yescomplemented
105C22×Q8C2×D25111yescharacteristic, complemented
106C22×D4—21no—
107C22×D4—5026no—
108C22×D4—5021no—
109C22×D4—2512no—
110C22×D4—25112no—
111C22×D4C2×D25111yescharacteristic, complemented
112C23×C4—21no—
113C23×C4C2×D25111yescharacteristic
114AGL(1,ℤ/8)—25164no—
115C2×SD16—25124no—
116C2×D8—25124no—
117C2×M16—25112no—
118C22×C8—2512no—
119C24—21no—
120C24—10048no—
121C24—50212no—
122C24—5022no—
123C24C22⋊D25111yescharacteristic
124#41—224no—
125#41C22×D251116yescomplemented
126C2×Q8—23no—
127C2×Q8C22×D25116yescomplemented
128C2×D4—212no—
129C2×D4—23no—
130C2×D4—50212no—
131C2×D4—5026no—
132C2×D4—50224no—
133C2×D4—25112no—
134C2×D4—25116no—
135C2×D4C22×D25116yescomplemented
136C22×C4—21no—
137C22×C4—212no—
138C22×C4C22⋊D25116yes—
139C22×C4C22⋊D25111yescharacteristic
140C22×C4C22×D25116yes—
141C22×C4C22×D25111yescharacteristic
142SD16—25132no—
143D8—25132no—
144M16—25116no—
145C2×C8—25112no—
146C23—24no—
147C23—26no—
148C23—44no—
149C23—100448no—
150C23—10048no—
151C23—50216no—
152C23—50212no—
153C23C2×C22⋊D25116yes—
154C23C2×C22⋊D25111yescharacteristic
155Q8—26no—
156Q8C23×D25114yescomplemented
157D4—216no—
158D4—26no—
159D4—50216no—
160D4—50212no—
161D4—2518no—
162D4C23×D25114yescomplemented
163C2×C4—26no—
164C2×C4—216no—
165C2×C4C2×C22⋊D25116yes—
166C2×C4C2×C22⋊D25113yes—
167C2×C4C2×C22⋊D251112yes—
168C2×C4C23×D25113yes—
169C2×C4C23×D25114yes—
170C8—2518no—
171C22—212no—
172C22—24no—
173C22—412no—
174C22—100432no—
175C22—100424no—
176C22—5028no—
177C22#2254874014yesdirect product factor
178C22C22×C22⋊D25114yes—
179C22C22×C22⋊D25113yes—
180C4—24no—
181C4C22×C22⋊D25114yes—
182C4C22×C22⋊D25113yes—
183C4C24×D25111yescharacteristic
184C2—24no—
185C2—44no—
186C2—10048no—
187C2C2×#2254874016yesdirect product factor
188C2C23×C22⋊D25111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D251
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
FittingC22×#41×C251C2
socleC23×C251C2×D4
hypercenterC22×#41D251
hypocenterC251C22×AGL(1,ℤ/8)
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C23×D1004C2semidirect
2. C2×#22548740C2direct (N × H)
3. C22×#16860797C2semidirect
4. C22×#16860795C2semidirect
5. C22×#16860794C2semidirect
6. C22×#41×C251C2semidirect
11. C22×D1004C22semidirect
12. #22548740C22direct (N × H)
13. C2×#16860797C22semidirect
14. C2×#16860795C22semidirect
15. C2×#16860794C22semidirect
16. C22×C251⋊C8C22semidirect
18. C2×#41×C251C22semidirect
19. C22×Q8×C251C22semidirect
21. C22×D4×C251C22semidirect
31. C2×D1004C23semidirect
32. #16860797C23semidirect
33. #16860795C23semidirect
34. #16860794C23semidirect
35. C2×C251⋊C8C23semidirect
39. #41×C251C23semidirect
41. C2×Q8×C251C23semidirect
44. C2×D4×C251C23semidirect
58. C251⋊C8C24semidirect
65. Q8×C251C24semidirect
68. D4×C251C24semidirect
79. C22×C251AGL(1,ℤ/8)semidirect
91. C2×C251C2×AGL(1,ℤ/8)semidirect
94. C251C22×AGL(1,ℤ/8)semidirect
96. C22×#41D251semidirect
104. C2×#41C2×D251semidirect
105. C22×Q8C2×D251semidirect
111. C22×D4C2×D251semidirect
125. #41C22×D251semidirect
127. C2×Q8C22×D251semidirect
135. C2×D4C22×D251semidirect
156. Q8C23×D251semidirect
162. D4C23×D251semidirect
177. C22#22548740direct (N × H)
187. C2C2×#22548740direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D10042
C2×#225487402
C22×#168607972
C22×#168607952
C22×#168607942
C22×#41×C2512
C22×AGL(1,ℤ/8)251