Finite Group Databasebeta

← #44401733

Subgroups — #44401733

Order 29056.

270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 41 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#28409775C211yesmaximal, characteristic, complemented
2#28409219C211yesmaximal, characteristic, complemented
3#28409210C211yesmaximal, characteristic, complemented
4#28408826C211yesmaximal, characteristic, complemented
5#28408819C211yesmaximal, characteristic, complemented
6#28408818C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C227C211yesmaximal, characteristic, complemented
8#20300318—21no—
9#20300275—21no—
10#20300275C2211yescharacteristic
11#20300274—21no—
12#20300274—21no—
13#20300274—21no—
14#20300257—21no—
15#20300256—21no—
16#20300256—21no—
17#20300256C2211yescharacteristic
18#20300215—21no—
19#20300205C2211yescharacteristic
20#20300203—21no—
21#20300203C2211yescharacteristic
22H(5,𝔽2)×C227C2211yescharacteristic
23#120×C227—21no—
24#120×C227C2211yescharacteristic
25#99×C227—21no—
26#99×C227C2211yescharacteristic
27C23×D227D411yescharacteristic, complemented
28C2×Dic454—21no—
29C2×Dic454D411yescharacteristic, complemented
30C2×C22⋊D227—21no—
31C2×C22⋊D227—21no—
32C2×C22⋊D227—21no—
33C2×C22⋊D227—41no—
34#16813874—41no—
35#16813872—41no—
36#16813868—21no—
37C22⋊Dic227—21no—
38C22⋊Dic227—21no—
39C22⋊Dic227—21no—
40C22⋊Dic227—21no—
41C22⋊Dic227—21no—
42C22⋊Dic227D411yescharacteristic, complemented
43(C2×D227)⋊C4—21no—
44(C2×D227)⋊C4—21no—
45(C2×D227)⋊C4—21no—
46(C2×D227)⋊C4—41no—
47C4×Dic227—21no—
48C4×Dic227D411yescharacteristic, complemented
49C24×C227D411yescharacteristic, complemented
50#41×C227—21no—
51#41×C227—41no—
52C2×D4×C227—21no—
53C2×D4×C227—21no—
54C2×D4×C227—21no—
55C2×D4×C227—41no—
56C2×D4×C227—41no—
57C2×D4×C227C2311yescharacteristic
58C22⋊C4×C227—21no—
59C22⋊C4×C227—21no—
60C22⋊C4×C227—41no—
61C22⋊C4×C227D411yescharacteristic, complemented
62C22×D227—21no—
63C22×D227—41no—
64C22×D227—81no—
65Dic454—41no—
66Dic454—81no—
67C22⋊D227—81no—
68C22⋊D227—81no—
69C22⋊D227—81no—
70C22⋊D227—82no—
71C2×Dic227—21no—
72C2×Dic227—21no—
73C2×Dic227—21no—
74C2×Dic227—41no—
75C227⋊C8—41no—
76C23×C227—21no—
77C23×C227—21no—
78C23×C227—21no—
79C23×C227—41no—
80C23×C227—81no—
81C23×C227C2×D411yescharacteristic
82C23×C227C2×D411yescharacteristic
83Q8×C227—21no—
84D4×C227—21no—
85D4×C227—41no—
86D4×C227—41no—
87D4×C227—41no—
88D4×C227—41no—
89D4×C227—81no—
90D4×C227—82no—
91C2×C4×C227—21no—
92C2×C4×C227—21no—
93C2×C4×C227—21no—
94C2×C4×C227—41no—
95C2×C4×C227—41no—
96C2×C4×C227C2×D411yescharacteristic
97C22×C227—21no—
98C22×C227—21no—
99C22×C227—41no—
100C22×C227—41no—
101C22×C227—41no—
102C22×C227—41no—
103C22×C227—42no—
104C22×C227—41no—
105C22×C227—41no—
106C22×C227—81no—
107C22×C227—81no—
108C22×C227—81no—
109C22×C227#12011yescharacteristic
110C2×D227—41no—
111C2×D227—41no—
112C2×D227—42no—
113C2×D227—41no—
114C2×D227—81no—
115C4×C227—21no—
116C4×C227—41no—
117C4×C227—41no—
118C4×C227—81no—
119Dic227—41no—
120Dic227—41no—
121Dic227—41no—
122Dic227—81no—
123C2×C227—21no—
124C2×C227—41no—
125C2×C227—41no—
126C2×C227—81no—
127C2×C227—81no—
128C2×C227B(4,𝔽2)11yescharacteristic
129D227—81no—
130C227#219911yescharacteristic, complemented, Sylow
131#2199—2271nomaximal, Sylow
132H(3,𝔽4)—2271no—
133B(4,𝔽2)—2271no—
134B(4,𝔽2)D22711yescharacteristic, complemented
135#455—2271no—
136#354—2271no—
137AΣL(1,𝔽16)—2271no—
138AΣL(1,𝔽16)—2271no—
139H(5,𝔽2)C2×D22711yescharacteristic
140#137—4541no—
141#124—4541no—
142#124—4541no—
143#124—4541no—
144#124—4541no—
145#124—2271no—
146#120—21no—
147#120—4541no—
148#120—4541no—
149#120—4541no—
150#120—2271no—
151#120C2×D22711yescharacteristic
152#104—4541no—
153#100—2271no—
154#99—21no—
155#99—4541no—
156#99—2271no—
157#99C2×D22711yescharacteristic
158C24—2271no—
159C24C22⋊D22711yescharacteristic, complemented
160#41—21no—
161#41—41no—
162C2×Q8—4541no—
163C2×Q8—2271no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—9081no—
170C2×D4—4541no—
171C2×D4—4541no—
172C2×D4—4541no—
173C2×D4C22×D22711yescharacteristic
174Q16—9081no—
175SD16—9081no—
176M16—4541no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—41no—
180C22⋊C4—9081no—
181C22⋊C4—4541no—
182C22⋊C4—4541no—
183C22⋊C4—4541no—
184C22⋊C4—4541no—
185C22⋊C4—4541no—
186C22⋊C4—4541no—
187C22⋊C4—4541no—
188C22⋊C4—4541no—
189C22⋊C4—2271no—
190C22⋊C4C22⋊D22711yescharacteristic, complemented
191C42—4541no—
192C42—2271no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—41no—
197C23—81no—
198C23—18161no—
199C23—9081no—
200C23—4541no—
201C23D4×D22711yescharacteristic
202C23C2×C22⋊D22711yescharacteristic
203Q8—21no—
204Q8—18161no—
205Q8—9081no—
206D4—21no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—81no—
212D4—82no—
213D4—18162no—
214D4—18161no—
215D4—18161no—
216D4—18161no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—9081no—
223C2×C4—4541no—
224C2×C4—4541no—
225C2×C4—4541no—
226C2×C4D4×D22711yescharacteristic
227C8—9081no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—18161no—
241C22—9081no—
242C22—9081no—
243C22—9082no—
244C22—9081no—
245C22#2030025611yescharacteristic
246C4—21no—
247C4—41no—
248C4—41no—
249C4—81no—
250C4—18161no—
251C4—9081no—
252C4—9081no—
253C4—9081no—
254C2—21no—
255C2—41no—
256C2—41no—
257C2—81no—
258C2—81no—
259C2—18161no—
260C2#2840921811yescharacteristic
subgroupG/H
centerC2#28409218
commutator, cosocleC2×D4×C227C23
FrattiniC2×D4C22×D227
FittingB(4,𝔽2)×C227C2
socleC2×C227B(4,𝔽2)
hypercenterB(4,𝔽2)D227
hypocenterC227#2199
LCSG ▷ C2×D4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #28409775C2semidirect
2. #28409219C2semidirect
3. #28409210C2semidirect
4. #28408826C2semidirect
5. #28408819C2semidirect
6. #28408818C2semidirect
7. B(4,𝔽2)×C227C2semidirect
27. C23×D227D4semidirect
29. C2×Dic454D4semidirect
42. C22⋊Dic227D4semidirect
48. C4×Dic227D4semidirect
49. C24×C227D4semidirect
61. C22⋊C4×C227D4semidirect
130. C227#2199semidirect
134. B(4,𝔽2)D227semidirect
159. C24C22⋊D227semidirect
190. C22⋊C4C22⋊D227semidirect

Maximal subgroups

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

subgroupindex
#284097752
#284092192
#284092102
#284088262
#284088192
#284088182
B(4,𝔽2)×C2272
#2199227