Finite Group Databasebeta

← #44408194

Subgroups — #44408194

Order 29056.

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×#20300333C211yesmaximal, characteristic
2#28409507C211yesmaximal, characteristic, complemented
3#28409005C214yesmaximal, complemented
4C2×#20300202C218yesmaximal, complemented
5#518×C227C211yesmaximal, characteristic, complemented
6#20300333—24no—
7#20300333—24no—
8C2×#16813884C2212yes—
9C2×#16813883C414yescomplemented
10C22×C22⋊D227C2214yes—
11C2×D4×D227C414yescomplemented
12D4×Dic227—24no—
13#20300247C2212yescomplemented
14C2×C22⋊Dic227C2214yescomplemented
15#20300202C22116yescomplemented
16C22×D4×C227C2211yescharacteristic
17C4×D4×C227—24no—
18#117×C227C2212yescomplemented
19C2×C22⋊C4×C227C2214yescomplemented
20C23×D227C2×C414yescomplemented
21#16813884—24no—
22#16813884—24no—
23#16813883—216no—
24C2×Dic454C2×C412yescomplemented
25C2×D908C2×C412yescomplemented
26C2×C22⋊D227—28no—
27C2×C22⋊D227—28no—
28C2×C22⋊D227C2×C418yescomplemented
29C2×C22⋊D227C2314yes—
30D4×D227—216no—
31C22×Dic227—24no—
32C22×Dic227C2×C414yescomplemented
33C2×C4×D227C2×C414yescomplemented
34C4⋊Dic227—22no—
35C22⋊Dic227—24no—
36C22⋊Dic227C2314yes—
37C4×Dic227—22no—
38C24×C227C2312yes—
39C2×D4×C227C2×C414yes—
40C2×D4×C227D414yes—
41C2×D4×C227D414yes—
42C22×C4×C227—24no—
43C22×C4×C227C2311yescharacteristic
44C4⋊C4×C227—22no—
45C22⋊C4×C227—24no—
46C22⋊C4×C227C2314yes—
47C42×C227—22no—
48C22×D227—28no—
49C22×D227—84no—
50C22×D227C22×C414yes—
51Dic454—24no—
52D908—24no—
53C22⋊D227—216no—
54C22⋊D227—24no—
55C22⋊D227—24no—
56C2×Dic227—28no—
57C2×Dic227—24no—
58C2×Dic227—84no—
59C2×Dic227C22×C414yes—
60C4×D227—28no—
61C23×C227—82no—
62C23×C227C22×C414yes—
63C23×C227C2×D414yes—
64C23×C227C2×D414yes—
65C23×C227C2411yescharacteristic
66D4×C227C22⋊C4116yes—
67C2×C4×C227—24no—
68C2×C4×C227—84no—
69C2×C4×C227C22×C412yes—
70C2×C4×C227C2×D412yes—
71C2×C4×C227C2×D412yes—
72C22×C227—44no—
73C22×C227—41no—
74C22×C227—44no—
75C22×C227—44no—
76C22×C227C2×C22⋊C418yes—
77C22×C227C23×C411yescharacteristic
78C22×C227C22×D411yescharacteristic
79C22×C227C22×D411yescharacteristic
80C2×D227—24no—
81C2×D227—44no—
82C2×D227—88no—
83C4×C227—44no—
84C4×C227C2×C22⋊C414yes—
85Dic227—24no—
86Dic227—44no—
87C2×C227—21no—
88C2×C227—21no—
89C2×C227—21no—
90C2×C227—28no—
91C2×C227C22×C22⋊C411yescharacteristic
92D227—44no—
93C227#288311yescharacteristic, complemented, Sylow
94#2883—2271nomaximal, Sylow
95C2×H(5,𝔽2)—2271no—
96#518—2271no—
97#518D22711yescharacteristic, complemented
98#410—2274no—
99C2×#99—2278no—
100H(5,𝔽2)—4544no—
101H(5,𝔽2)—4544no—
102C2×#41—2274no—
103C2×#41—2272no—
104C22×D4—2274no—
105C22×D4—2274no—
106C22×D4C2×D22711yescharacteristic
107C4×D4—24no—
108C4×D4—4544no—
109#117—2272no—
110#117C2×D22712yescomplemented
111C2×C22⋊C4—2274no—
112C2×C22⋊C4C2×D22714yescomplemented
113#99—22716no—
114C24—2274no—
115C24C22×D22712yes—
116#41—45416no—
117#41—4544no—
118#41—4544no—
119C2×Q8—2272no—
120C2×D4—4548no—
121C2×D4—4548no—
122C2×D4—45416no—
123C2×D4—2278no—
124C2×D4—2274no—
125C2×D4—2272no—
126C2×D4C4×D22714yes—
127C2×D4C22⋊D22714yes—
128C2×D4D90814yes—
129C22×C4—24no—
130C22×C4—4544no—
131C22×C4—2274no—
132C22×C4—2274no—
133C22×C4C22×D22711yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—4542no—
136C22⋊C4—24no—
137C22⋊C4—4544no—
138C22⋊C4—2274no—
139C22⋊C4C22×D22714yes—
140C42—22no—
141C42—4542no—
142C23—82no—
143C23—18164no—
144C23—4548no—
145C23—2274no—
146C23C2×C4×D22714yes—
147C23C2×C22⋊D22714yes—
148C23C2×D90814yes—
149C23C23×D22711yescharacteristic
150Q8—4544no—
151D4—4544no—
152D4—4544no—
153D4—45416no—
154D4—4544no—
155D4(C2×D227)⋊C4116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—18164no—
159C2×C4—4544no—
160C2×C4—4548no—
161C2×C4—4548no—
162C2×C4—2274no—
163C2×C4C2×C4×D22712yes—
164C2×C4C2×C22⋊D22712yes—
165C2×C4C2×D90812yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—18168no—
171C22—9084no—
172C22—4544no—
173C22C2×(C2×D227)⋊C418yes—
174C22C22×C4×D22711yescharacteristic
175C22C22×C22⋊D22711yescharacteristic
176C22C22×D90811yescharacteristic
177C4—44no—
178C4—9084no—
179C4—4544no—
180C4C2×(C2×D227)⋊C414yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—9084no—
186C2C22×(C2×D227)⋊C411yescharacteristic
subgroupG/H
centerC2C22×(C2×D227)⋊C4
commutatorC22×C227C23×C4
FrattiniC23C23×D227
Fitting#518×C227C2
socleC2×C227C22×C22⋊C4
prime radicalC2×#20300333C2
cosocleC23×C227C24
hypercenter#518D227
hypocenterC227#2883
LCSG ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #28409507C2semidirect
3. #28409005C2semidirect
4. C2×#20300202C2semidirect
5. #518×C227C2semidirect
9. C2×#16813883C4semidirect
11. C2×D4×D227C4semidirect
13. #20300247C22semidirect
14. C2×C22⋊Dic227C22semidirect
15. #20300202C22semidirect
18. #117×C227C22semidirect
19. C2×C22⋊C4×C227C22semidirect
20. C23×D227C2×C4semidirect
24. C2×Dic454C2×C4semidirect
25. C2×D908C2×C4semidirect
28. C2×C22⋊D227C2×C4semidirect
32. C22×Dic227C2×C4semidirect
33. C2×C4×D227C2×C4semidirect
93. C227#2883semidirect
97. #518D227semidirect
110. #117C2×D227semidirect
112. C2×C22⋊C4C2×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003332
#284095072
#284090052
C2×#203002022
#518×C2272
#2883227