Finite Group Databasebeta

← #44401372

Subgroups — #44401372

Order 29056.

264 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 55 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#28409538C211yesmaximal, characteristic
2#28409013C211yesmaximal, characteristic, complemented
3#28409004C211yesmaximal, characteristic, complemented
4#28408823C211yesmaximal, characteristic, complemented
5#28408821C211yesmaximal, characteristic, complemented
6#28408819C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C227C211yesmaximal, characteristic, complemented
8C2×#16813883C2211yescharacteristic
9C23×Dic227C411yescharacteristic, complemented
10M16×D227—21no—
11#20300272C411yescharacteristic, complemented
12#20300269C2211yescharacteristic
13#20300265—21no—
14#20300260—21no—
15#20300257C411yescharacteristic, complemented
16D4×Dic227—21no—
17(C22⋊D227)⋊C4—21no—
18#20300244—21no—
19#20300244C411yescharacteristic, complemented
20#20300206—21no—
21#20300205C2211yescharacteristic
22#20300203C2211yescharacteristic
23#20300202—21no—
24#120×C227C2211yescharacteristic
25#100×C227C2211yescharacteristic
26#99×C227C2211yescharacteristic
27#16813883—41no—
28#16813883—41no—
29C2×Dic454D411yescharacteristic
30C2×C22⋊D227—21no—
31C2×C22⋊D227C2×C411yescharacteristic
32C2×C22⋊D227D411yescharacteristic
33C22×Dic227—21no—
34C22×Dic227—21no—
35C22×Dic227—41no—
36C22×Dic227—41no—
37C22×Dic227C2×C411yescharacteristic
38C22×Dic227C2×C411yescharacteristic
39C22×Dic227D411yescharacteristic
40C2×C4×D227D411yescharacteristic
41#16813868—21no—
42#16813867—41no—
43C8×D227—41no—
44C4⋊Dic227—21no—
45Dic227⋊C4—21no—
46Dic227⋊C4—21no—
47Dic227⋊C4—41no—
48C22⋊Dic227—21no—
49C22⋊Dic227—21no—
50C22⋊Dic227—21no—
51C22⋊Dic227C2×C411yescharacteristic
52(C2×D227)⋊C4—21no—
53C4×Dic227—21no—
54C4×Dic227—41no—
55C24×C227C2×C411yescharacteristic
56C2×D4×C227—21no—
57C2×D4×C227C2311yescharacteristic
58M16×C227—21no—
59C22⋊C4×C227—21no—
60C22⋊C4×C227—21no—
61C22⋊C4×C227C2×C411yescharacteristic
62C22×D227C22⋊C411yescharacteristic
63Dic454—41no—
64C22⋊D227—41no—
65C22⋊D227—41no—
66C22⋊D227—42no—
67C2×Dic227—21no—
68C2×Dic227—21no—
69C2×Dic227—21no—
70C2×Dic227—21no—
71C2×Dic227—21no—
72C2×Dic227—21no—
73C2×Dic227—42no—
74C2×Dic227—41no—
75C2×Dic227—41no—
76C2×Dic227—41no—
77C2×Dic227—41no—
78C2×Dic227C22⋊C411yescharacteristic
79C2×Dic227C22⋊C411yescharacteristic
80C2×Dic227C22⋊C411yescharacteristic
81C4×D227—21no—
82C4×D227—21no—
83C227⋊C8—41no—
84C23×C227—21no—
85C23×C227—42no—
86C23×C227—41no—
87C23×C227C22×C411yescharacteristic
88C23×C227C2×D411yescharacteristic
89D4×C227—41no—
90D4×C227—42no—
91C2×C4×C227—21no—
92C2×C4×C227—41no—
93C2×C4×C227C2×D411yescharacteristic
94C8×C227—41no—
95C22×C227—22no—
96C22×C227—21no—
97C22×C227—21no—
98C22×C227—41no—
99C22×C227—41no—
100C22×C227—41no—
101C22×C227—42no—
102C22×C227—42no—
103C22×C227—41no—
104C22×C227C2×C22⋊C411yescharacteristic
105C2×D227—21no—
106C2×D227—41no—
107C4×C227—21no—
108C4×C227—41no—
109C4×C227—81no—
110Dic227—21no—
111Dic227—21no—
112Dic227—41no—
113Dic227—41no—
114Dic227—81no—
115Dic227#9911yescharacteristic
116Dic227#9911yescharacteristic
117C2×C227—21no—
118C2×C227—41no—
119C2×C227—42no—
120C2×C227—41no—
121C2×C227C2×#9911yescharacteristic
122D227—41no—
123C227#212011yescharacteristic, complemented, Sylow
124#2120—2271nomaximal, Sylow
125#525—2271no—
126#413—2271no—
127#410—2271no—
128#352—2271no—
129#352—2271no—
130AΣL(1,𝔽16)—2271no—
131AΣL(1,𝔽16)D22711yescharacteristic, complemented
132C2×#41—2271no—
133C23×C4—2271no—
134C2×M16—4541no—
135#123—2271no—
136#123—2271no—
137C22⋊Q8—4541no—
138#121—4541no—
139#120—2271no—
140#120C2×D22711yescharacteristic
141C4×D4—4541no—
142C4×D4—4541no—
143#117—4541no—
144#117—2271no—
145#101—4541no—
146#100—2271no—
147#100C2×D22711yescharacteristic
148#99—4541no—
149#99—2271no—
150#99C2×D22711yescharacteristic
151C24C4×D22711yescharacteristic
152#41—9081no—
153#41—9081no—
154C2×Q8—2271no—
155C2×D4—21no—
156C2×D4—4541no—
157C2×D4—2271no—
158C2×D4—2271no—
159C2×D4C22×D22711yescharacteristic
160C22×C4—9081no—
161C22×C4—9081no—
162C22×C4—4541no—
163C22×C4—4541no—
164C22×C4—2271no—
165C22×C4—2271no—
166C22×C4—2271no—
167C22×C4—2271no—
168M16—21no—
169M16—9081no—
170M16—4541no—
171C2×C8—9081no—
172C4⋊C4—9081no—
173C4⋊C4—4541no—
174C4⋊C4—4541no—
175C4⋊C4—4541no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—4541no—
179C22⋊C4—4541no—
180C22⋊C4—4541no—
181C22⋊C4—4541no—
182C22⋊C4—2271no—
183C22⋊C4C4×D22711yescharacteristic
184C42—9081no—
185C42—4541no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23—2271no—
190C23C2×C4×D22711yescharacteristic
191C23D4×D22711yescharacteristic
192Q8—9081no—
193D4—41no—
194D4—42no—
195D4—9082no—
196D4—9081no—
197D4—9081no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—9081no—
201C2×C4—9082no—
202C2×C4—9081no—
203C2×C4—9081no—
204C2×C4—9081no—
205C2×C4—4541no—
206C2×C4—4541no—
207C2×C4—4541no—
208C2×C4—4541no—
209C2×C4—4541no—
210C2×C4—4541no—
211C2×C4—4541no—
212C2×C4—4541no—
213C2×C4—2271no—
214C2×C4—2271no—
215C2×C4—2271no—
216C2×C4D4×D22711yescharacteristic
217C8—41no—
218C8—9081no—
219C22—22no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—9081no—
229C22—4541no—
230C22C22⋊C4×D22711yescharacteristic
231C4—21no—
232C4—41no—
233C4—81no—
234C4—18161no—
235C4—9081no—
236C4—9081no—
237C4—4541no—
238C4—4541no—
239C4—2271no—
240C4—2271no—
241C2—21no—
242C2—41no—
243C2—42no—
244C2—41no—
245C2—9081no—
246C2#99×D22711yescharacteristic
subgroupG/H
centerC2#99×D227
commutatorC23×C227C22×C4
FrattiniC2×D4C22×D227
FittingAΣL(1,𝔽16)×C227C2
socleC2×C227C2×#99
prime radical#28409538C2
cosocleC2×D4×C227C23
hypercenterAΣL(1,𝔽16)D227
hypocenterC227#2120
LCSG ▷ C23×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
2. #28409013C2semidirect
3. #28409004C2semidirect
4. #28408823C2semidirect
5. #28408821C2semidirect
6. #28408819C2semidirect
7. AΣL(1,𝔽16)×C227C2semidirect
9. C23×Dic227C4semidirect
11. #20300272C4semidirect
15. #20300257C4semidirect
19. #20300244C4semidirect
123. C227#2120semidirect
131. AΣL(1,𝔽16)D227semidirect

Maximal subgroups

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

subgroupindex
#284095382
#284090132
#284090042
#284088232
#284088212
#284088192
AΣL(1,𝔽16)×C2272
#2120227