Finite Group Databasebeta

← #44409090

Subgroups — #44409090

Order 29056.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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, complemented
2#28409830C212yesmaximal, complemented
3#28409209C214yesmaximal, complemented
4#28409199C214yesmaximal, complemented
5#28409050C212yesmaximal, complemented
6C2×#20300205C211yesmaximal, characteristic, complemented
7#535×C227C211yesmaximal, characteristic, complemented
8#20300333—24no—
9#20300333—22no—
10#20300333C2214yes—
11C2×#16813884C2212yes—
12C2×#16813883—22no—
13C22×C22⋊D227—22no—
14C2×D4×D227—22no—
15#20300319C2214yescomplemented
16#20300318—22no—
17#20300314C2214yescomplemented
18#20300313—22no—
19#20300308—24no—
20C2×#16813868C2212yescomplemented
21#20300216C2218yescomplemented
22#20300205C2214yescomplemented
23C2×#41×C227C2211yescharacteristic, complemented
24C22×D4×C227C2211yescharacteristic
25#127×C227C2212yescomplemented
26#124×C227C2212yescomplemented
27#121×C227—22no—
28#120×C227—22no—
29#117×C227C2211yescharacteristic, complemented
30C23×D227—22no—
31#16813884—22no—
32#16813884—22no—
33#16813884D414yescomplemented
34#16813884C2314yes—
35#16813883—28no—
36#16813883—28no—
37C2×Dic454D412yescomplemented
38C2×D908D412yescomplemented
39C2×C22⋊D227—24no—
40C2×C22⋊D227—28no—
41C2×C22⋊D227—22no—
42C2×C22⋊D227—28no—
43D4×D227—28no—
44D4×D227—28no—
45C22×Dic227—22no—
46C2×C4×D227—22no—
47#16813874—24no—
48#16813872—24no—
49#16813871—24no—
50#16813869—24no—
51#16813868—22no—
52#16813868C2314yescomplemented
53C2×C227⋊C8—22no—
54C24×C227D412yes—
55#41×C227—22no—
56#41×C227—22no—
57C2×Q8×C227C2311yescharacteristic, complemented
58C2×D4×C227—21no—
59C2×D4×C227—24no—
60C2×D4×C227—22no—
61C2×D4×C227—22no—
62C2×D4×C227D412yes—
63C2×D4×C227C2312yes—
64C2×D4×C227C2311yescharacteristic, complemented
65C22×C4×C227—21no—
66C22×C4×C227C2311yescharacteristic
67C4⋊C4×C227—21no—
68C22⋊C4×C227—21no—
69C22⋊C4×C227—24no—
70C42×C227C2312yescomplemented
71C22×D227—22no—
72C22×D227—28no—
73C22×D227—84no—
74Dic454—22no—
75Dic454C2×D414yes—
76D908—22no—
77D908C2×D414yes—
78C22⋊D227—216no—
79C22⋊D227—24no—
80C22⋊D227—24no—
81C2×Dic227—28no—
82C2×Dic227—22no—
83C4×D227—24no—
84C4×D227—24no—
85C227⋊C8—24no—
86C23×C227—22no—
87C23×C227—22no—
88C23×C227—21no—
89C23×C227—82no—
90C23×C227C2×D414yes—
91C23×C227C2×D411yescharacteristic
92Q8×C227—22no—
93D4×C227—24no—
94D4×C227—24no—
95D4×C227—22no—
96D4×C227—41no—
97D4×C227—41no—
98D4×C227—84no—
99C2×C4×C227—22no—
100C2×C4×C227—21no—
101C2×C4×C227—42no—
102C2×C4×C227C2×D411yescharacteristic
103C2×C4×C227C2×D412yes—
104C2×C4×C227C2×D412yes—
105C2×C4×C227C2411yescharacteristic
106C22×C227—24no—
107C22×C227—41no—
108C22×C227—44no—
109C22×C227—44no—
110C22×C227—42no—
111C22×C227—41no—
112C22×C227—82no—
113C22×C227#12011yescharacteristic
114C22×C227#12011yescharacteristic
115C22×C227C22×D411yescharacteristic
116C2×D227—24no—
117C2×D227—44no—
118C2×D227—88no—
119C4×C227—42no—
120C4×C227—41no—
121C4×C227#12012yes—
122C4×C227C22×D412yes—
123Dic227—24no—
124C2×C227—21no—
125C2×C227—21no—
126C2×C227—21no—
127C2×C227—44no—
128C2×C227—81no—
129C2×C227C2×#12011yescharacteristic
130D227—44no—
131C227#301911yescharacteristic, complemented, Sylow
132#3019—2271nomaximal, Sylow
133C2×H(5,𝔽2)—2271no—
134#575—2272no—
135#535D22711yescharacteristic, complemented
136#455—2274no—
137#453—2274no—
138#421—2272no—
139C2×#100—2271no—
140H(5,𝔽2)—4544no—
141H(5,𝔽2)—4542no—
142H(5,𝔽2)—2274no—
143C2×#41—4542no—
144C2×#41—2272no—
145C2×#41C2×D22711yescharacteristic, complemented
146C22×D4—4542no—
147C22×D4—4542no—
148C22×D4C2×D22711yescharacteristic
149#137—4542no—
150#137—2274no—
151AGL(1,ℤ/8)—4542no—
152AGL(1,ℤ/8)—2274no—
153#135—4544no—
154C2×M16—2272no—
155#127C2×D22712yescomplemented
156#124C2×D22712yescomplemented
157#121—22no—
158#120—22no—
159#117C2×D22711yescharacteristic, complemented
160#104—2278no—
161#100—2274no—
162C24—4542no—
163C24C22⋊D22712yes—
164#41—22no—
165#41—22no—
166#41—4548no—
167#41—4548no—
168#41—4542no—
169#41—4542no—
170#41—2274no—
171#41—2274no—
172C2×Q8—2272no—
173C2×Q8C22×D22711yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—4548no—
179C2×D4—4548no—
180C2×D4—4544no—
181C2×D4—4548no—
182C2×D4—4542no—
183C2×D4—4548no—
184C2×D4—2272no—
185C2×D4C22⋊D22712yes—
186C2×D4C22×D22712yes—
187C2×D4C22×D22711yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—4542no—
190C22×C4—4542no—
191C22×C4C22×D22711yescharacteristic
192Q16—4544no—
193SD16—4544no—
194SD16—4544no—
195D8—4544no—
196M16—4542no—
197M16—2274no—
198C2×C8—4542no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D22712yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—18164no—
208C23—4548no—
209C23—4542no—
210C23C2×C22⋊D22714yes—
211C23C2×C22⋊D22711yescharacteristic
212Q8—22no—
213Q8—4542no—
214Q8—2274no—
215D4—24no—
216D4—22no—
217D4—24no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—45416no—
222D4—4544no—
223D4—4544no—
224D4—4542no—
225D4—2274no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—4548no—
230C2×C4—4542no—
231C2×C4—4544no—
232C2×C4—4544no—
233C2×C4D4×D22712yes—
234C2×C4D4×D22712yes—
235C2×C4C2×C22⋊D22711yescharacteristic
236C2×C4C23×D22711yescharacteristic
237C8—4544no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—18168no—
246C22—9084no—
247C22—4544no—
248C22#2030025611yescharacteristic
249C22#2030025611yescharacteristic
250C22C22×C22⋊D22711yescharacteristic
251C4—42no—
252C4—41no—
253C4—4544no—
254C4#2030025612yes—
255C4C2×D4×D22712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—9084no—
262C2C2×#2030025611yescharacteristic
subgroupG/H
centerC2C2×#20300256
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#535×C227C2
socleC2×C227C2×#120
hypercenter#535D227
hypocenterC227#3019
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#20300333C2semidirect
2. #28409830C2semidirect
3. #28409209C2semidirect
4. #28409199C2semidirect
5. #28409050C2semidirect
6. C2×#20300205C2semidirect
7. #535×C227C2semidirect
15. #20300319C22semidirect
17. #20300314C22semidirect
20. C2×#16813868C22semidirect
21. #20300216C22semidirect
22. #20300205C22semidirect
23. C2×#41×C227C22semidirect
25. #127×C227C22semidirect
26. #124×C227C22semidirect
29. #117×C227C22semidirect
33. #16813884D4semidirect
37. C2×Dic454D4semidirect
38. C2×D908D4semidirect
52. #16813868C23semidirect
57. C2×Q8×C227C23semidirect
64. C2×D4×C227C23semidirect
70. C42×C227C23semidirect
131. C227#3019semidirect
135. #535D227semidirect
145. C2×#41C2×D227semidirect
155. #127C2×D227semidirect
156. #124C2×D227semidirect
159. #117C2×D227semidirect
173. C2×Q8C22×D227semidirect
187. C2×D4C22×D227semidirect
202. C42C22×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003332
#284098302
#284092092
#284091992
#284090502
C2×#203002052
#535×C2272
#3019227