Finite Group Databasebeta

← #44558825

Subgroups — #44558825

Order 32128.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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
1C2×#22548760C211yesmaximal, characteristic
2#30632162C211yesmaximal, characteristic, complemented
3#30631660C214yesmaximal, complemented
4C2×#22548629C218yesmaximal, complemented
5#518×C251C211yesmaximal, characteristic, complemented
6#22548760—24no—
7#22548760—24no—
8C2×#16860810C2212yes—
9C2×#16860809C414yescomplemented
10C22×C22⋊D251C2214yes—
11C2×D4×D251C414yescomplemented
12D4×Dic251—24no—
13#22548674C2212yescomplemented
14C2×C22⋊Dic251C2214yescomplemented
15#22548629C22116yescomplemented
16C22×D4×C251C2211yescharacteristic
17C4×D4×C251—24no—
18#117×C251C2212yescomplemented
19C2×C22⋊C4×C251C2214yescomplemented
20C23×D251C2×C414yescomplemented
21#16860810—24no—
22#16860810—24no—
23#16860809—216no—
24C2×Dic502C2×C412yescomplemented
25C2×D1004C2×C412yescomplemented
26C2×C22⋊D251—28no—
27C2×C22⋊D251—28no—
28C2×C22⋊D251C2×C418yescomplemented
29C2×C22⋊D251C2314yes—
30D4×D251—216no—
31C22×Dic251—24no—
32C22×Dic251C2×C414yescomplemented
33C2×C4×D251C2×C414yescomplemented
34C4⋊Dic251—22no—
35C22⋊Dic251—24no—
36C22⋊Dic251C2314yes—
37C4×Dic251—22no—
38C24×C251C2312yes—
39C2×D4×C251C2×C414yes—
40C2×D4×C251D414yes—
41C2×D4×C251D414yes—
42C22×C4×C251—24no—
43C22×C4×C251C2311yescharacteristic
44C4⋊C4×C251—22no—
45C22⋊C4×C251—24no—
46C22⋊C4×C251C2314yes—
47C42×C251—22no—
48C22×D251—28no—
49C22×D251—84no—
50C22×D251C22×C414yes—
51Dic502—24no—
52D1004—24no—
53C22⋊D251—216no—
54C22⋊D251—24no—
55C22⋊D251—24no—
56C2×Dic251—28no—
57C2×Dic251—24no—
58C2×Dic251—84no—
59C2×Dic251C22×C414yes—
60C4×D251—28no—
61C23×C251—82no—
62C23×C251C22×C414yes—
63C23×C251C2×D414yes—
64C23×C251C2×D414yes—
65C23×C251C2411yescharacteristic
66D4×C251C22⋊C4116yes—
67C2×C4×C251—24no—
68C2×C4×C251—84no—
69C2×C4×C251C22×C412yes—
70C2×C4×C251C2×D412yes—
71C2×C4×C251C2×D412yes—
72C22×C251—44no—
73C22×C251—41no—
74C22×C251—44no—
75C22×C251—44no—
76C22×C251C2×C22⋊C418yes—
77C22×C251C23×C411yescharacteristic
78C22×C251C22×D411yescharacteristic
79C22×C251C22×D411yescharacteristic
80C2×D251—24no—
81C2×D251—44no—
82C2×D251—88no—
83C4×C251—44no—
84C4×C251C2×C22⋊C414yes—
85Dic251—24no—
86Dic251—44no—
87C2×C251—21no—
88C2×C251—21no—
89C2×C251—21no—
90C2×C251—28no—
91C2×C251C22×C22⋊C411yescharacteristic
92D251—44no—
93C251#288311yescharacteristic, complemented, Sylow
94#2883—2511nomaximal, Sylow
95C2×H(5,𝔽2)—2511no—
96#518—2511no—
97#518D25111yescharacteristic, complemented
98#410—2514no—
99C2×#99—2518no—
100H(5,𝔽2)—5024no—
101H(5,𝔽2)—5024no—
102C2×#41—2514no—
103C2×#41—2512no—
104C22×D4—2514no—
105C22×D4—2514no—
106C22×D4C2×D25111yescharacteristic
107C4×D4—24no—
108C4×D4—5024no—
109#117—2512no—
110#117C2×D25112yescomplemented
111C2×C22⋊C4—2514no—
112C2×C22⋊C4C2×D25114yescomplemented
113#99—25116no—
114C24—2514no—
115C24C22×D25112yes—
116#41—50216no—
117#41—5024no—
118#41—5024no—
119C2×Q8—2512no—
120C2×D4—5028no—
121C2×D4—5028no—
122C2×D4—50216no—
123C2×D4—2518no—
124C2×D4—2514no—
125C2×D4—2512no—
126C2×D4C4×D25114yes—
127C2×D4C22⋊D25114yes—
128C2×D4D100414yes—
129C22×C4—24no—
130C22×C4—5024no—
131C22×C4—2514no—
132C22×C4—2514no—
133C22×C4C22×D25111yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—5022no—
136C22⋊C4—24no—
137C22⋊C4—5024no—
138C22⋊C4—2514no—
139C22⋊C4C22×D25114yes—
140C42—22no—
141C42—5022no—
142C23—82no—
143C23—20084no—
144C23—5028no—
145C23—2514no—
146C23C2×C4×D25114yes—
147C23C2×C22⋊D25114yes—
148C23C2×D100414yes—
149C23C23×D25111yescharacteristic
150Q8—5024no—
151D4—5024no—
152D4—5024no—
153D4—50216no—
154D4—5024no—
155D4(C2×D251)⋊C4116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—20084no—
159C2×C4—5024no—
160C2×C4—5028no—
161C2×C4—5028no—
162C2×C4—2514no—
163C2×C4C2×C4×D25112yes—
164C2×C4C2×C22⋊D25112yes—
165C2×C4C2×D100412yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—20088no—
171C22—10044no—
172C22—5024no—
173C22C2×(C2×D251)⋊C418yes—
174C22C22×C4×D25111yescharacteristic
175C22C22×C22⋊D25111yescharacteristic
176C22C22×D100411yescharacteristic
177C4—44no—
178C4—10044no—
179C4—5024no—
180C4C2×(C2×D251)⋊C414yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—10044no—
186C2C22×(C2×D251)⋊C411yescharacteristic
subgroupG/H
centerC2C22×(C2×D251)⋊C4
commutatorC22×C251C23×C4
FrattiniC23C23×D251
Fitting#518×C251C2
socleC2×C251C22×C22⋊C4
prime radicalC2×#22548760C2
cosocleC23×C251C24
hypercenter#518D251
hypocenterC251#2883
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #30632162C2semidirect
3. #30631660C2semidirect
4. C2×#22548629C2semidirect
5. #518×C251C2semidirect
9. C2×#16860809C4semidirect
11. C2×D4×D251C4semidirect
13. #22548674C22semidirect
14. C2×C22⋊Dic251C22semidirect
15. #22548629C22semidirect
18. #117×C251C22semidirect
19. C2×C22⋊C4×C251C22semidirect
20. C23×D251C2×C4semidirect
24. C2×Dic502C2×C4semidirect
25. C2×D1004C2×C4semidirect
28. C2×C22⋊D251C2×C4semidirect
32. C22×Dic251C2×C4semidirect
33. C2×C4×D251C2×C4semidirect
93. C251#2883semidirect
97. #518D251semidirect
110. #117C2×D251semidirect
112. C2×C22⋊C4C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487602
#306321622
#306316602
C2×#225486292
#518×C2512
#2883251