Finite Group Databasebeta

← #44409135

Subgroups — #44409135

Order 29056.

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

normal subgroups only

This group has 772 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
1#28409868C211yesmaximal, characteristic, complemented
2#28409599C212yesmaximal, complemented
3#28409218C218yesmaximal, complemented
4#28409006C211yesmaximal, characteristic, complemented
5#28409005C212yesmaximal, complemented
6#535×C227C211yesmaximal, characteristic, complemented
7#20300336—21no—
8#20300335—22no—
9#20300334—22no—
10#20300333—21no—
11#20300333C2214yes—
12C2×#16813884—22no—
13C2×#16813884C2212yes—
14#41×D227—24no—
15#20300284—22no—
16#20300275—22no—
17#20300259—24no—
18#20300257C2218yescomplemented
19#20300247C2212yescomplemented
20#20300203C2214yescomplemented
21#20300202C2218yescomplemented
22C2×#41×C227C2211yescharacteristic
23C22×D4×C227C2211yescharacteristic, complemented
24#127×C227—21no—
25#124×C227—21no—
26#121×C227—22no—
27#120×C227C2214yescomplemented
28#117×C227C2211yescharacteristic, complemented
29#16813885—24no—
30#16813884—24no—
31#16813884—24no—
32#16813884—22no—
33#16813884—22no—
34#16813884—24no—
35#16813884—22no—
36#16813884—22no—
37#16813883—24no—
38#16813883—28no—
39C2×Dic454—22no—
40C2×Dic454D412yescomplemented
41Q8×D227—24no—
42C2×D908—22no—
43C2×D908D412yescomplemented
44C2×C22⋊D227—28no—
45C2×C22⋊D227—24no—
46C2×C22⋊D227C2314yes—
47D4×D227—24no—
48D4×D227—28no—
49C2×C4×D227—24no—
50C2×C4×D227D414yescomplemented
51C4⋊Dic227—22no—
52C22⋊Dic227—28no—
53C22⋊Dic227C2314yes—
54C4×Dic227—22no—
55C24×C227C2312yes—
56#41×C227—22no—
57#41×C227—22no—
58C2×Q8×C227D411yescharacteristic, complemented
59C2×D4×C227—24no—
60C2×D4×C227—21no—
61C2×D4×C227—21no—
62C2×D4×C227—22no—
63C2×D4×C227—22no—
64C2×D4×C227D411yescharacteristic, complemented
65C2×D4×C227C2312yes—
66C22×C4×C227D412yescomplemented
67C22×C4×C227C2311yescharacteristic
68C4⋊C4×C227—21no—
69C22⋊C4×C227—24no—
70C22⋊C4×C227C2312yes—
71C42×C227—21no—
72C22×D227—24no—
73C22×D227C2×D414yes—
74Dic454—24no—
75Dic454—22no—
76Dic454—22no—
77D908—24no—
78D908—22no—
79D908—22no—
80C22⋊D227—28no—
81C22⋊D227—24no—
82C22⋊D227—24no—
83C22⋊D227—88no—
84C2×Dic227—24no—
85C2×Dic227—24no—
86C2×Dic227C2×D414yes—
87C4×D227—28no—
88C4×D227—24no—
89C4×D227—24no—
90C23×C227—22no—
91C23×C227—24no—
92C23×C227—82no—
93C23×C227C2×D412yes—
94C23×C227C2411yescharacteristic
95Q8×C227—22no—
96D4×C227—24no—
97D4×C227—22no—
98D4×C227—41no—
99D4×C227—41no—
100D4×C227—42no—
101D4×C227—84no—
102C2×C4×C227—22no—
103C2×C4×C227—22no—
104C2×C4×C227—22no—
105C2×C4×C227C2×D412yes—
106C2×C4×C227C2×D412yes—
107C2×C4×C227C2×D412yes—
108C2×C4×C227C2×D411yescharacteristic
109C2×C4×C227C2×D411yescharacteristic
110C22×C227—22no—
111C22×C227—41no—
112C22×C227—42no—
113C22×C227—44no—
114C22×C227—48no—
115C22×C227—42no—
116C22×C227C22×D412yes—
117C22×C227C22×D411yescharacteristic
118C2×D227—24no—
119C2×D227—44no—
120C2×D227—84no—
121C4×C227—22no—
122C4×C227—42no—
123C4×C227#12011yescharacteristic
124C4×C227#12011yescharacteristic
125C4×C227#12012yes—
126Dic227—24no—
127Dic227—44no—
128C2×C227—22no—
129C2×C227—21no—
130C2×C227—42no—
131C2×C227—82no—
132C2×C227C2×#12011yescharacteristic
133D227—44no—
134C227#302711yescharacteristic, complemented, Sylow
135#3027—2271nomaximal, Sylow
136#585—2271no—
137#535—2272no—
138#535D22711yescharacteristic, complemented
139B(4,𝔽2)—2278no—
140#410—2271no—
141#410—2272no—
1422−1+4—4542no—
1432−1+4—4541no—
144H(5,𝔽2)—4542no—
145H(5,𝔽2)—4541no—
146H(5,𝔽2)—2274no—
147C2×#41—4544no—
148C2×#41—4542no—
149C2×#41—2272no—
150C2×#41C2×D22711yescharacteristic
151C22×D4C2×D22711yescharacteristic, complemented
152#127—21no—
153#127—4542no—
154#124—21no—
155#124—4542no—
156#121—22no—
157#121—4544no—
158#120—2278no—
159#120C2×D22714yescomplemented
160#117—2272no—
161#117C2×D22711yescharacteristic, complemented
162#99—2274no—
163#99—2278no—
164C24C22×D22712yes—
165#41—22no—
166#41—22no—
167#41—4544no—
168#41—4544no—
169#41—4544no—
170#41—4544no—
171#41—4548no—
172#41—4542no—
173#41—4542no—
174#41—4542no—
175#41—4544no—
176#41—4542no—
177C2×Q8—4542no—
178C2×Q8—4544no—
179C2×Q8—2272no—
180C2×Q8C22⋊D22711yescharacteristic, complemented
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—4542no—
187C2×D4—4544no—
188C2×D4—4548no—
189C2×D4—4548no—
190C2×D4—4544no—
191C2×D4—2274no—
192C2×D4—2272no—
193C2×D4C22⋊D22711yescharacteristic, complemented
194C2×D4C22×D22712yes—
195C22×C4—4544no—
196C22×C4—2274no—
197C22×C4C22⋊D22712yescomplemented
198C22×C4C22×D22711yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—4542no—
201C22⋊C4—24no—
202C22⋊C4—4548no—
203C22⋊C4—2274no—
204C22⋊C4C22×D22712yes—
205C42—21no—
206C42—4542no—
207C23—24no—
208C23—22no—
209C23—82no—
210C23—4544no—
211C23—2274no—
212C23C2×C22⋊D22712yes—
213C23C23×D22711yescharacteristic
214Q8—22no—
215Q8—4544no—
216Q8—4542no—
217Q8—4542no—
218D4—24no—
219D4—22no—
220D4—41no—
221D4—41no—
222D4—42no—
223D4—84no—
224D4—18168no—
225D4—4544no—
226D4—4548no—
227D4—4544no—
228D4—4544no—
229D4—4542no—
230D4—4542no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—4548no—
235C2×C4—4544no—
236C2×C4—4544no—
237C2×C4—4544no—
238C2×C4—4544no—
239C2×C4—2274no—
240C2×C4D4×D22712yes—
241C2×C4D4×D22712yes—
242C2×C4C2×C22⋊D22712yes—
243C2×C4C2×C22⋊D22711yescharacteristic
244C2×C4C2×C22⋊D22711yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—44no—
249C22—48no—
250C22—42no—
251C22—18164no—
252C22—9084no—
253C22—4544no—
254C22C2×D4×D22712yes—
255C22C22×C22⋊D22711yescharacteristic
256C4—22no—
257C4—42no—
258C4—9084no—
259C4—4544no—
260C4#2030025611yescharacteristic
261C4#2030025611yescharacteristic
262C4#2030025612yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—9084no—
268C2C2×#2030025611yescharacteristic
subgroupG/H
centerC2C2×#20300256
commutator, cosocleC23×C227C24
FrattiniC23C23×D227
Fitting#535×C227C2
socleC2×C227C2×#120
hypercenter#535D227
hypocenterC227#3027
LCSG ▷ C23×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #28409868C2semidirect
2. #28409599C2semidirect
3. #28409218C2semidirect
4. #28409006C2semidirect
5. #28409005C2semidirect
6. #535×C227C2semidirect
18. #20300257C22semidirect
19. #20300247C22semidirect
20. #20300203C22semidirect
21. #20300202C22semidirect
23. C22×D4×C227C22semidirect
27. #120×C227C22semidirect
28. #117×C227C22semidirect
40. C2×Dic454D4semidirect
43. C2×D908D4semidirect
50. C2×C4×D227D4semidirect
58. C2×Q8×C227D4semidirect
64. C2×D4×C227D4semidirect
66. C22×C4×C227D4semidirect
134. C227#3027semidirect
138. #535D227semidirect
151. C22×D4C2×D227semidirect
159. #120C2×D227semidirect
161. #117C2×D227semidirect
180. C2×Q8C22⋊D227semidirect
193. C2×D4C22⋊D227semidirect
197. C22×C4C22⋊D227semidirect

Maximal subgroups

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

subgroupindex
#284098682
#284095992
#284092182
#284090062
#284090052
#535×C2272
#3027227