Finite Group Databasebeta

← #44408193

Subgroups — #44408193

Order 29056.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#20300334C211yesmaximal, characteristic
2#28409506C212yesmaximal, complemented
3#28409006C211yesmaximal, characteristic, complemented
4#28409005C212yesmaximal, complemented
5#99×D227C218yesmaximal, complemented
6#410×C227C211yesmaximal, characteristic, complemented
7#20300334—24no—
8#20300334—24no—
9C2×#16813885C412yescomplemented
10C2×#16813884C412yescomplemented
11C2×#16813884C2211yescharacteristic
12C22×D908C412yescomplemented
13C22×D908C2211yescharacteristic
14C2×D4×D227C412yescomplemented
15C2×D4×D227C2214yes—
16C4×D908—24no—
17#20300250—24no—
18#20300247C2212yescomplemented
19C22⋊C4×D227C2218yescomplemented
20#20300203C2214yescomplemented
21#20300202C2218yescomplemented
22C2×#41×C227C2211yescharacteristic
23#117×C227C2212yescomplemented
24#99×C227C2214yescomplemented
25C23×D227C2×C414yescomplemented
26C23×D227C2312yes—
27#16813885—28no—
28#16813884—24no—
29#16813884—24no—
30#16813884—22no—
31#16813884—22no—
32C2×Dic454C2×C411yescharacteristic, complemented
33C2×D908—24no—
34C2×D908—24no—
35C2×D908C2×C412yes—
36C2×D908C2×C412yes—
37C2×D908C2×C414yescomplemented
38C2×D908C2×C411yescharacteristic, complemented
39C2×D908D414yes—
40C2×D908D414yes—
41C2×C22⋊D227C2×C414yescomplemented
42C2×C22⋊D227C2312yes—
43D4×D227—216no—
44D4×D227—28no—
45C2×C4×D227—28no—
46C2×C4×D227C2×C412yescomplemented
47C2×C4×D227C2×C414yescomplemented
48C4⋊Dic227—22no—
49C22⋊Dic227C2314yes—
50(C2×D227)⋊C4—28no—
51C4×Dic227—22no—
52#41×C227—22no—
53#41×C227—22no—
54C2×Q8×C227C2×C411yescharacteristic, complemented
55C2×D4×C227C2×C411yescharacteristic, complemented
56C2×D4×C227C2312yes—
57C22×C4×C227C2×C412yescomplemented
58C22×C4×C227C2311yescharacteristic
59C4⋊C4×C227—22no—
60C22⋊C4×C227C2314yes—
61C42×C227—22no—
62C22×D227—28no—
63C22×D227—82no—
64C22×D227—84no—
65C22×D227C22×C412yes—
66C22×D227C22×C414yes—
67C22×D227C2×D418yes—
68Dic454—21no—
69Dic454—21no—
70D908—28no—
71D908—21no—
72D908—21no—
73D908C22⋊C418yes—
74D908C22⋊C414yes—
75D908C22⋊C414yes—
76C22⋊D227—24no—
77C22⋊D227—28no—
78C2×Dic227—24no—
79C2×Dic227C22×C412yes—
80C4×D227—28no—
81C4×D227—24no—
82C4×D227—88no—
83C23×C227C22×C412yes—
84C23×C227C2411yescharacteristic
85Q8×C227—22no—
86D4×C227—24no—
87D4×C227—22no—
88C2×C4×C227—22no—
89C2×C4×C227—22no—
90C2×C4×C227—24no—
91C2×C4×C227C22×C412yes—
92C2×C4×C227C22×C411yescharacteristic
93C2×C4×C227C22×C411yescharacteristic
94C2×C4×C227C2×D412yes—
95C2×C4×C227C2×D412yes—
96C22×C227—22no—
97C22×C227—41no—
98C22×C227—42no—
99C22×C227C23×C411yescharacteristic
100C22×C227C22×D412yes—
101C2×D227—22no—
102C2×D227—48no—
103C2×D227—44no—
104C2×D227—42no—
105C2×D227—88no—
106C2×D227C2×C22⋊C418yes—
107C4×C227—22no—
108C4×C227—44no—
109C4×C227C2×C22⋊C411yescharacteristic
110C4×C227C2×C22⋊C411yescharacteristic
111C4×C227C2×C22⋊C412yes—
112Dic227—22no—
113Dic227—44no—
114C2×C227—22no—
115C2×C227—21no—
116C2×C227—42no—
117C2×C227C22×C22⋊C411yescharacteristic
118D227—28no—
119D227—42no—
120C227#288311yescharacteristic, complemented, Sylow
121#2883—2271nomaximal, Sylow
122C2×H(5,𝔽2)—2271no—
123#518—2272no—
124#410—2271no—
125#410—2272no—
126#410D22711yescharacteristic, complemented
127C2×#99—2278no—
128H(5,𝔽2)—4544no—
129H(5,𝔽2)—4544no—
130C2×#41—2272no—
131C2×#41—2272no—
132C2×#41—2271no—
133C2×#41C2×D22711yescharacteristic
134C22×D4—2272no—
135C22×D4—2274no—
136C22×D4—2272no—
137C22×D4—2271no—
138C4×D4—4544no—
139C4×D4—4544no—
140#117—2272no—
141#117C2×D22712yescomplemented
142C2×C22⋊C4—2278no—
143#99—2274no—
144#99—2278no—
145#99C2×D22714yescomplemented
146C24—2274no—
147C24—2272no—
148#41—22no—
149#41—22no—
150#41—4544no—
151#41—4544no—
152#41—4548no—
153#41—4542no—
154#41—4542no—
155C2×Q8—2271no—
156C2×Q8C4×D22711yescharacteristic, complemented
157C2×D4—4544no—
158C2×D4—4544no—
159C2×D4—45416no—
160C2×D4—4548no—
161C2×D4—2274no—
162C2×D4—2272no—
163C2×D4—2274no—
164C2×D4—2271no—
165C2×D4—2272no—
166C2×D4—2272no—
167C2×D4—2274no—
168C2×D4—2274no—
169C2×D4C4×D22711yescharacteristic, complemented
170C2×D4C22×D22712yes—
171C22×C4—4548no—
172C22×C4—2274no—
173C22×C4—2272no—
174C22×C4C4×D22712yescomplemented
175C22×C4C22×D22711yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—4542no—
178C22⋊C4—4548no—
179C22⋊C4—2274no—
180C22⋊C4C22×D22714yes—
181C42—22no—
182C42—4542no—
183C23—18164no—
184C23—18162no—
185C23—4548no—
186C23—2272no—
187C23—2274no—
188C23—2278no—
189C23C2×C4×D22712yes—
190C23C23×D22711yescharacteristic
191Q8—22no—
192Q8—4541no—
193Q8—4541no—
194D4—24no—
195D4—22no—
196D4—4548no—
197D4—4548no—
198D4—4544no—
199D4—4541no—
200D4—4541no—
201D4—2274no—
202D4—2274no—
203D4—2278no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—18168no—
208C2×C4—4548no—
209C2×C4—4544no—
210C2×C4—4544no—
211C2×C4—2272no—
212C2×C4C2×C4×D22712yes—
213C2×C4C2×C4×D22711yescharacteristic
214C2×C4C2×C4×D22711yescharacteristic
215C2×C4D4×D22712yes—
216C2×C4D4×D22712yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—18168no—
221C22—9082no—
222C22—9088no—
223C22—9084no—
224C22—4542no—
225C22—2278no—
226C22C22×C4×D22711yescharacteristic
227C22C2×D4×D22712yes—
228C4—22no—
229C4—44no—
230C4—9084no—
231C4—4542no—
232C4C22⋊C4×D22711yescharacteristic
233C4C22⋊C4×D22711yescharacteristic
234C4C22⋊C4×D22712yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—9082no—
239C2—4548no—
240C2C2×C22⋊C4×D22711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D227
commutatorC22×C227C23×C4
FrattiniC23C23×D227
Fitting#410×C227C2
socleC2×C227C22×C22⋊C4
prime radicalC2×#20300334C2
cosocleC23×C227C24
hypercenter#410D227
hypocenterC227#2883
LCSG ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #28409506C2semidirect
3. #28409006C2semidirect
4. #28409005C2semidirect
5. #99×D227C2semidirect
6. #410×C227C2semidirect
9. C2×#16813885C4semidirect
10. C2×#16813884C4semidirect
12. C22×D908C4semidirect
14. C2×D4×D227C4semidirect
18. #20300247C22semidirect
19. C22⋊C4×D227C22semidirect
20. #20300203C22semidirect
21. #20300202C22semidirect
23. #117×C227C22semidirect
24. #99×C227C22semidirect
25. C23×D227C2×C4semidirect
32. C2×Dic454C2×C4semidirect
37. C2×D908C2×C4semidirect
38. C2×D908C2×C4semidirect
41. C2×C22⋊D227C2×C4semidirect
46. C2×C4×D227C2×C4semidirect
47. C2×C4×D227C2×C4semidirect
54. C2×Q8×C227C2×C4semidirect
55. C2×D4×C227C2×C4semidirect
57. C22×C4×C227C2×C4semidirect
120. C227#2883semidirect
126. #410D227semidirect
141. #117C2×D227semidirect
145. #99C2×D227semidirect
156. C2×Q8C4×D227semidirect
169. C2×D4C4×D227semidirect
174. C22×C4C4×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003342
#284095062
#284090062
#284090052
#99×D2272
#410×C2272
#2883227