Finite Group Databasebeta

← #44401743

Subgroups — #44401743

Order 29056.

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

normal subgroups only

This group has 260 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#28409771C211yesmaximal, characteristic, complemented
2#28409223C211yesmaximal, characteristic, complemented
3#28409218C211yesmaximal, characteristic, complemented
4#28409202C211yesmaximal, characteristic, complemented
5#28408824C211yesmaximal, characteristic, complemented
6#28408823C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C227C211yesmaximal, characteristic, complemented
8#20300333C2211yescharacteristic
9#20300316—21no—
10#20300284C2211yescharacteristic
11#20300281—21no—
12#20300272—21no—
13#20300270—21no—
14#20300269C2211yescharacteristic
15#20300261—21no—
16#20300260—21no—
17#20300259—21no—
18#20300257—21no—
19#20300257—21no—
20#20300214—21no—
21#20300203C2211yescharacteristic
22#20300202—21no—
23#20300202—21no—
24#120×C227C2211yescharacteristic
25#100×C227C2211yescharacteristic
26#99×C227C2211yescharacteristic
27#16813884—21no—
28#16813883—41no—
29C2×D908D411yescharacteristic, complemented
30C2×C22⋊D227—21no—
31C2×C22⋊D227—21no—
32C2×C22⋊D227—21no—
33C2×C22⋊D227—21no—
34C2×C22⋊D227—21no—
35C2×C22⋊D227—21no—
36C2×C22⋊D227—41no—
37D4×D227—41no—
38C22×Dic227D411yescharacteristic, complemented
39C2×C4×D227—21no—
40#16813873—41no—
41D1816—41no—
42C4⋊Dic227—21no—
43C4⋊Dic227—41no—
44Dic227⋊C4—21no—
45Dic227⋊C4—21no—
46C22⋊Dic227—21no—
47C22⋊Dic227—21no—
48C22⋊Dic227—21no—
49C22⋊Dic227—21no—
50C22⋊Dic227—41no—
51C22⋊Dic227D411yescharacteristic, complemented
52(C2×D227)⋊C4—21no—
53(C2×D227)⋊C4—41no—
54C4×Dic227D411yescharacteristic, complemented
55C24×C227D411yescharacteristic
56C2×D4×C227—21no—
57C2×D4×C227C2311yescharacteristic
58M16×C227—21no—
59C22⋊C4×C227—21no—
60C22⋊C4×C227—21no—
61C22⋊C4×C227D411yescharacteristic
62C22×D227—21no—
63C22×D227—21no—
64C22×D227—21no—
65Dic454—21no—
66D908—21no—
67D908—41no—
68C22⋊D227—41no—
69C22⋊D227—41no—
70C22⋊D227—41no—
71C22⋊D227—81no—
72C22⋊D227—81no—
73C22⋊D227—81no—
74C22⋊D227—81no—
75C22⋊D227—82no—
76C2×Dic227—21no—
77C2×Dic227—21no—
78C2×Dic227—21no—
79C2×Dic227—21no—
80C2×Dic227—21no—
81C2×Dic227—41no—
82C2×Dic227—41no—
83C4×D227—41no—
84C4×D227—81no—
85C23×C227—21no—
86C23×C227—41no—
87C23×C227—81no—
88C23×C227C2×D411yescharacteristic
89C23×C227C2×D411yescharacteristic
90D4×C227—41no—
91D4×C227—81no—
92C2×C4×C227—21no—
93C2×C4×C227—41no—
94C2×C4×C227C2×D411yescharacteristic
95C8×C227—41no—
96C22×C227—21no—
97C22×C227—21no—
98C22×C227—41no—
99C22×C227—41no—
100C22×C227—41no—
101C22×C227—41no—
102C22×C227—42no—
103C22×C227—41no—
104C22×C227—81no—
105C22×C227#12011yescharacteristic
106C2×D227—41no—
107C2×D227—41no—
108C2×D227—41no—
109C2×D227—41no—
110C2×D227—81no—
111C2×D227—81no—
112C4×C227—21no—
113C4×C227—41no—
114C4×C227—81no—
115Dic227—41no—
116Dic227—41no—
117Dic227—41no—
118Dic227—41no—
119Dic227—81no—
120C2×C227—21no—
121C2×C227—41no—
122C2×C227—41no—
123C2×C227—81no—
124C2×C227B(4,𝔽2)11yescharacteristic
125D227—81no—
126D227—81no—
127C227#220011yescharacteristic, complemented, Sylow
128#2200—2271nomaximal, Sylow
129#560—2271no—
130#458—2271no—
131B(4,𝔽2)—2271no—
132#453—2271no—
133#353—2271no—
134#352—2271no—
135AΣL(1,𝔽16)D22711yescharacteristic, complemented
136H(5,𝔽2)—2271no—
137AGL(1,ℤ/8)—4541no—
138#127—2271no—
139#126—4541no—
140#123—4541no—
141#123—4541no—
142#123—2271no—
143#121—4541no—
144#121—4541no—
145#121—4541no—
146#120—4541no—
147#120—4541no—
148#120C2×D22711yescharacteristic
149#104—4541no—
150#100C2×D22711yescharacteristic
151#99—4541no—
152#99—4541no—
153#99—2271no—
154#99C2×D22711yescharacteristic
155C24D90811yescharacteristic
156#41—9081no—
157#41—4541no—
158C2×D4—21no—
159C2×D4—9081no—
160C2×D4—9081no—
161C2×D4—4541no—
162C2×D4—4541no—
163C2×D4—4541no—
164C2×D4—4541no—
165C2×D4—4541no—
166C2×D4—4541no—
167C2×D4—2271no—
168C2×D4C22×D22711yescharacteristic
169C22×C4—4541no—
170C22×C4—2271no—
171SD16—9081no—
172D8—9081no—
173M16—21no—
174C4⋊C4—9081no—
175C4⋊C4—4541no—
176C4⋊C4—4541no—
177C4⋊C4—4541no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—9081no—
181C22⋊C4—9081no—
182C22⋊C4—4541no—
183C22⋊C4—4541no—
184C22⋊C4—4541no—
185C22⋊C4—4541no—
186C22⋊C4—4541no—
187C22⋊C4—2271no—
188C22⋊C4D90811yescharacteristic
189C42—2271no—
190C23—21no—
191C23—41no—
192C23—81no—
193C23—4541no—
194C23—4541no—
195C23—4541no—
196C23D4×D22711yescharacteristic
197C23C2×D90811yescharacteristic
198Q8—4541no—
199D4—41no—
200D4—81no—
201D4—18162no—
202D4—18161no—
203D4—18161no—
204D4—18161no—
205D4—18161no—
206D4—9081no—
207D4—9081no—
208D4—9081no—
209D4—9081no—
210D4—4541no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—18161no—
214C2×C4—9081no—
215C2×C4—9081no—
216C2×C4—9081no—
217C2×C4—4541no—
218C2×C4—4541no—
219C2×C4—4541no—
220C2×C4—4541no—
221C2×C4—4541no—
222C2×C4D4×D22711yescharacteristic
223C8—41no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—41no—
232C22—81no—
233C22—18161no—
234C22—18161no—
235C22—9081no—
236C22—9081no—
237C22—9081no—
238C22—9081no—
239C22#2030025811yescharacteristic
240C4—21no—
241C4—41no—
242C4—81no—
243C4—18161no—
244C4—9081no—
245C4—9081no—
246C4—9081no—
247C4—9081no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—18161no—
253C2—18161no—
254C2#2840922011yescharacteristic
subgroupG/H
centerC2#28409220
commutator, cosocleC2×D4×C227C23
FrattiniC2×D4C22×D227
FittingAΣL(1,𝔽16)×C227C2
socleC2×C227B(4,𝔽2)
hypercenterAΣL(1,𝔽16)D227
hypocenterC227#2200
LCSG ▷ C2×D4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
1. #28409771C2semidirect
2. #28409223C2semidirect
3. #28409218C2semidirect
4. #28409202C2semidirect
5. #28408824C2semidirect
6. #28408823C2semidirect
7. AΣL(1,𝔽16)×C227C2semidirect
29. C2×D908D4semidirect
38. C22×Dic227D4semidirect
51. C22⋊Dic227D4semidirect
54. C4×Dic227D4semidirect
127. C227#2200semidirect
135. AΣL(1,𝔽16)D227semidirect

Maximal subgroups

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

subgroupindex
#284097712
#284092232
#284092182
#284092022
#284088242
#284088232
AΣL(1,𝔽16)×C2272
#2200227