Finite Group Databasebeta

← #44552003

Subgroups — #44552003

Order 32128.

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#30632193C211yesmaximal, characteristic
2#30631668C211yesmaximal, characteristic, complemented
3#30631659C211yesmaximal, characteristic, complemented
4#30631478C211yesmaximal, characteristic, complemented
5#30631476C211yesmaximal, characteristic, complemented
6#30631474C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C251C211yesmaximal, characteristic, complemented
8C2×#16860809C2211yescharacteristic
9C23×Dic251C411yescharacteristic, complemented
10M16×D251—21no—
11#22548699C411yescharacteristic, complemented
12#22548696C2211yescharacteristic
13#22548692—21no—
14#22548687—21no—
15#22548684C411yescharacteristic, complemented
16D4×Dic251—21no—
17(C22⋊D251)⋊C4—21no—
18#22548671—21no—
19#22548671C411yescharacteristic, complemented
20#22548633—21no—
21#22548632C2211yescharacteristic
22#22548630C2211yescharacteristic
23#22548629—21no—
24#120×C251C2211yescharacteristic
25#100×C251C2211yescharacteristic
26#99×C251C2211yescharacteristic
27#16860809—41no—
28#16860809—41no—
29C2×Dic502D411yescharacteristic
30C2×C22⋊D251—21no—
31C2×C22⋊D251C2×C411yescharacteristic
32C2×C22⋊D251D411yescharacteristic
33C22×Dic251—21no—
34C22×Dic251—21no—
35C22×Dic251—41no—
36C22×Dic251—41no—
37C22×Dic251C2×C411yescharacteristic
38C22×Dic251C2×C411yescharacteristic
39C22×Dic251D411yescharacteristic
40C2×C4×D251D411yescharacteristic
41#16860794—21no—
42#16860793—41no—
43C8×D251—41no—
44C4⋊Dic251—21no—
45Dic251⋊C4—21no—
46Dic251⋊C4—21no—
47Dic251⋊C4—41no—
48C22⋊Dic251—21no—
49C22⋊Dic251—21no—
50C22⋊Dic251—21no—
51C22⋊Dic251C2×C411yescharacteristic
52(C2×D251)⋊C4—21no—
53C4×Dic251—21no—
54C4×Dic251—41no—
55C24×C251C2×C411yescharacteristic
56C2×D4×C251—21no—
57C2×D4×C251C2311yescharacteristic
58M16×C251—21no—
59C22⋊C4×C251—21no—
60C22⋊C4×C251—21no—
61C22⋊C4×C251C2×C411yescharacteristic
62C22×D251C22⋊C411yescharacteristic
63Dic502—41no—
64C22⋊D251—41no—
65C22⋊D251—41no—
66C22⋊D251—42no—
67C2×Dic251—21no—
68C2×Dic251—21no—
69C2×Dic251—21no—
70C2×Dic251—21no—
71C2×Dic251—21no—
72C2×Dic251—21no—
73C2×Dic251—42no—
74C2×Dic251—41no—
75C2×Dic251—41no—
76C2×Dic251—41no—
77C2×Dic251—41no—
78C2×Dic251C22⋊C411yescharacteristic
79C2×Dic251C22⋊C411yescharacteristic
80C2×Dic251C22⋊C411yescharacteristic
81C4×D251—21no—
82C4×D251—21no—
83C251⋊C8—41no—
84C23×C251—21no—
85C23×C251—42no—
86C23×C251—41no—
87C23×C251C22×C411yescharacteristic
88C23×C251C2×D411yescharacteristic
89D4×C251—41no—
90D4×C251—42no—
91C2×C4×C251—21no—
92C2×C4×C251—41no—
93C2×C4×C251C2×D411yescharacteristic
94C8×C251—41no—
95C22×C251—22no—
96C22×C251—21no—
97C22×C251—21no—
98C22×C251—41no—
99C22×C251—41no—
100C22×C251—41no—
101C22×C251—42no—
102C22×C251—42no—
103C22×C251—41no—
104C22×C251C2×C22⋊C411yescharacteristic
105C2×D251—21no—
106C2×D251—41no—
107C4×C251—21no—
108C4×C251—41no—
109C4×C251—81no—
110Dic251—21no—
111Dic251—21no—
112Dic251—41no—
113Dic251—41no—
114Dic251—81no—
115Dic251#9911yescharacteristic
116Dic251#9911yescharacteristic
117C2×C251—21no—
118C2×C251—41no—
119C2×C251—42no—
120C2×C251—41no—
121C2×C251C2×#9911yescharacteristic
122D251—41no—
123C251#212011yescharacteristic, complemented, Sylow
124#2120—2511nomaximal, Sylow
125#525—2511no—
126#413—2511no—
127#410—2511no—
128#352—2511no—
129#352—2511no—
130AΣL(1,𝔽16)—2511no—
131AΣL(1,𝔽16)D25111yescharacteristic, complemented
132C2×#41—2511no—
133C23×C4—2511no—
134C2×M16—5021no—
135#123—2511no—
136#123—2511no—
137C22⋊Q8—5021no—
138#121—5021no—
139#120—2511no—
140#120C2×D25111yescharacteristic
141C4×D4—5021no—
142C4×D4—5021no—
143#117—5021no—
144#117—2511no—
145#101—5021no—
146#100—2511no—
147#100C2×D25111yescharacteristic
148#99—5021no—
149#99—2511no—
150#99C2×D25111yescharacteristic
151C24C4×D25111yescharacteristic
152#41—10041no—
153#41—10041no—
154C2×Q8—2511no—
155C2×D4—21no—
156C2×D4—5021no—
157C2×D4—2511no—
158C2×D4—2511no—
159C2×D4C22×D25111yescharacteristic
160C22×C4—10041no—
161C22×C4—10041no—
162C22×C4—5021no—
163C22×C4—5021no—
164C22×C4—2511no—
165C22×C4—2511no—
166C22×C4—2511no—
167C22×C4—2511no—
168M16—21no—
169M16—10041no—
170M16—5021no—
171C2×C8—10041no—
172C4⋊C4—10041no—
173C4⋊C4—5021no—
174C4⋊C4—5021no—
175C4⋊C4—5021no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—5021no—
179C22⋊C4—5021no—
180C22⋊C4—5021no—
181C22⋊C4—5021no—
182C22⋊C4—2511no—
183C22⋊C4C4×D25111yescharacteristic
184C42—10041no—
185C42—5021no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23—2511no—
190C23C2×C4×D25111yescharacteristic
191C23D4×D25111yescharacteristic
192Q8—10041no—
193D4—41no—
194D4—42no—
195D4—10042no—
196D4—10041no—
197D4—10041no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—10041no—
201C2×C4—10042no—
202C2×C4—10041no—
203C2×C4—10041no—
204C2×C4—10041no—
205C2×C4—5021no—
206C2×C4—5021no—
207C2×C4—5021no—
208C2×C4—5021no—
209C2×C4—5021no—
210C2×C4—5021no—
211C2×C4—5021no—
212C2×C4—5021no—
213C2×C4—2511no—
214C2×C4—2511no—
215C2×C4—2511no—
216C2×C4D4×D25111yescharacteristic
217C8—41no—
218C8—10041no—
219C22—22no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—10041no—
229C22—5021no—
230C22C22⋊C4×D25111yescharacteristic
231C4—21no—
232C4—41no—
233C4—81no—
234C4—20081no—
235C4—10041no—
236C4—10041no—
237C4—5021no—
238C4—5021no—
239C4—2511no—
240C4—2511no—
241C2—21no—
242C2—41no—
243C2—42no—
244C2—41no—
245C2—10041no—
246C2#99×D25111yescharacteristic
subgroupG/H
centerC2#99×D251
commutatorC23×C251C22×C4
FrattiniC2×D4C22×D251
FittingAΣL(1,𝔽16)×C251C2
socleC2×C251C2×#99
prime radical#30632193C2
cosocleC2×D4×C251C23
hypercenterAΣL(1,𝔽16)D251
hypocenterC251#2120
LCSG ▷ C23×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #30631668C2semidirect
3. #30631659C2semidirect
4. #30631478C2semidirect
5. #30631476C2semidirect
6. #30631474C2semidirect
7. AΣL(1,𝔽16)×C251C2semidirect
9. C23×Dic251C4semidirect
11. #22548699C4semidirect
15. #22548684C4semidirect
19. #22548671C4semidirect
123. C251#2120semidirect
131. AΣL(1,𝔽16)D251semidirect

Maximal subgroups

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

subgroupindex
#306321932
#306316682
#306316592
#306314782
#306314762
#306314742
AΣL(1,𝔽16)×C2512
#2120251