Finite Group Databasebeta

← #44552008

Subgroups — #44552008

Order 32128.

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

normal subgroups only

This group has 280 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#30632247C211yesmaximal, characteristic
2#100×D251C211yesmaximal, characteristic, complemented
3#99×D251C211yesmaximal, characteristic, complemented
4#30631478C211yesmaximal, characteristic, complemented
5#30631476C211yesmaximal, characteristic, complemented
6#30631474C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C251C211yesmaximal, characteristic, complemented
8C22×C22⋊D251C411yescharacteristic, complemented
9C2×D4×D251C2211yescharacteristic
10M16×D251—21no—
11#22548699C411yescharacteristic, complemented
12#22548697—21no—
13#22548696C2211yescharacteristic
14#22548688—21no—
15#22548686—21no—
16#22548684C411yescharacteristic, complemented
17#22548683—21no—
18C22⋊C4×D251—21no—
19C22⋊C4×D251C411yescharacteristic, complemented
20#22548632C2211yescharacteristic
21#22548631—21no—
22#22548630C2211yescharacteristic
23#22548629—21no—
24#120×C251C2211yescharacteristic
25#100×C251C2211yescharacteristic
26#99×C251C2211yescharacteristic
27C23×D251C2×C411yescharacteristic
28C23×D251D411yescharacteristic
29C2×D1004D411yescharacteristic
30C2×C22⋊D251—21no—
31C2×C22⋊D251—21no—
32C2×C22⋊D251—21no—
33C2×C22⋊D251—41no—
34C2×C22⋊D251—41no—
35C2×C22⋊D251C2×C411yescharacteristic
36C2×C22⋊D251D411yescharacteristic
37D4×D251—41no—
38D4×D251—41no—
39C22×Dic251C2×C411yescharacteristic, complemented
40C2×C4×D251—21no—
41C2×C4×D251—41no—
42C2×C4×D251D411yescharacteristic
43#16860794—21no—
44#16860793—41no—
45C8×D251—41no—
46C4⋊Dic251—21no—
47Dic251⋊C4—21no—
48C22⋊Dic251—21no—
49C22⋊Dic251—21no—
50C22⋊Dic251—21no—
51C22⋊Dic251C2×C411yescharacteristic, complemented
52(C2×D251)⋊C4—21no—
53(C2×D251)⋊C4—21no—
54(C2×D251)⋊C4—41no—
55C24×C251C2×C411yescharacteristic, complemented
56C2×D4×C251—21no—
57C2×D4×C251C2311yescharacteristic
58M16×C251—21no—
59C22⋊C4×C251—21no—
60C22⋊C4×C251—21no—
61C22⋊C4×C251C2×C411yescharacteristic, complemented
62C22×D251—21no—
63C22×D251—21no—
64C22×D251—21no—
65C22×D251—21no—
66C22×D251—81no—
67C22×D251—81no—
68C22×D251C22⋊C411yescharacteristic
69C22×D251C22⋊C411yescharacteristic
70C22×D251C22⋊C411yescharacteristic
71D1004—41no—
72C22⋊D251—41no—
73C22⋊D251—41no—
74C22⋊D251—42no—
75C22⋊D251—41no—
76C22⋊D251—41no—
77C22⋊D251—81no—
78C2×Dic251—21no—
79C2×Dic251—21no—
80C2×Dic251—41no—
81C2×Dic251—41no—
82C2×Dic251C22⋊C411yescharacteristic
83C4×D251—21no—
84C4×D251—21no—
85C4×D251—81no—
86C4×D251—82no—
87C251⋊C8—41no—
88C23×C251—21no—
89C23×C251—41no—
90C23×C251—81no—
91C23×C251C22×C411yescharacteristic
92C23×C251C2×D411yescharacteristic
93D4×C251—41no—
94D4×C251—81no—
95C2×C4×C251—21no—
96C2×C4×C251—41no—
97C2×C4×C251C2×D411yescharacteristic
98C8×C251—41no—
99C22×C251—21no—
100C22×C251—21no—
101C22×C251—41no—
102C22×C251—41no—
103C22×C251—41no—
104C22×C251—41no—
105C22×C251—41no—
106C22×C251—82no—
107C22×C251C2×C22⋊C411yescharacteristic
108C2×D251—21no—
109C2×D251—21no—
110C2×D251—41no—
111C2×D251—42no—
112C2×D251—41no—
113C2×D251—41no—
114C2×D251—81no—
115C2×D251—81no—
116C2×D251—81no—
117C2×D251#9911yescharacteristic
118C2×D251#9911yescharacteristic
119C4×C251—21no—
120C4×C251—41no—
121C4×C251—81no—
122Dic251—21no—
123Dic251—41no—
124Dic251—41no—
125Dic251—81no—
126C2×C251—21no—
127C2×C251—41no—
128C2×C251—41no—
129C2×C251—81no—
130C2×C251C2×#9911yescharacteristic
131D251—21no—
132D251—21no—
133D251—41no—
134D251—41no—
135C251#212111yescharacteristic, complemented, Sylow
136#2121—2511nomaximal, Sylow
137#534—2511no—
138C2×#100—2511no—
139C2×#99—2511no—
140#352—2511no—
141#352—2511no—
142AΣL(1,𝔽16)—2511no—
143AΣL(1,𝔽16)D25111yescharacteristic, complemented
144C22×D4—2511no—
145C22×D4—2511no—
146C2×M16—5021no—
147#123—5021no—
148#123—2511no—
149#123—2511no—
150#121—5021no—
151#121—5021no—
152#120—5021no—
153#120—2511no—
154#120C2×D25111yescharacteristic
155C2×C22⋊C4—5021no—
156C2×C22⋊C4—2511no—
157#100—5021no—
158#100—2511no—
159#100C2×D25111yescharacteristic
160#99—5021no—
161#99—2511no—
162#99C2×D25111yescharacteristic
163C24—2511no—
164C24—2511no—
165C24C4×D25111yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—10041no—
168C2×D4—10041no—
169C2×D4—10041no—
170C2×D4—10041no—
171C2×D4—5021no—
172C2×D4—5021no—
173C2×D4—5021no—
174C2×D4—2511no—
175C2×D4—2511no—
176C2×D4—2511no—
177C2×D4C22×D25111yescharacteristic
178C22×C4—10041no—
179C22×C4—5021no—
180C22×C4—2511no—
181C22×C4—2511no—
182M16—21no—
183M16—10041no—
184M16—5021no—
185C2×C8—10041no—
186C4⋊C4—5021no—
187C4⋊C4—5021no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—10041no—
191C22⋊C4—5021no—
192C22⋊C4—5021no—
193C22⋊C4—5021no—
194C22⋊C4—5021no—
195C22⋊C4—5021no—
196C22⋊C4—2511no—
197C22⋊C4C4×D25111yescharacteristic, complemented
198C23—21no—
199C23—41no—
200C23—81no—
201C23—20081no—
202C23—20081no—
203C23—5021no—
204C23—5021no—
205C23—5021no—
206C23—5021no—
207C23—2511no—
208C23—2511no—
209C23—2511no—
210C23C2×C4×D25111yescharacteristic
211C23D4×D25111yescharacteristic
212D4—41no—
213D4—81no—
214D4—20081no—
215D4—10041no—
216D4—10042no—
217D4—10041no—
218D4—10041no—
219D4—10041no—
220D4—10041no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—20081no—
224C2×C4—20082no—
225C2×C4—10041no—
226C2×C4—10041no—
227C2×C4—5021no—
228C2×C4—5021no—
229C2×C4—5021no—
230C2×C4—5021no—
231C2×C4—2511no—
232C2×C4D4×D25111yescharacteristic
233C8—41no—
234C8—10041no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—82no—
243C22—20081no—
244C22—20081no—
245C22—20081no—
246C22—10041no—
247C22—10041no—
248C22—10041no—
249C22—10042no—
250C22—5021no—
251C22—5021no—
252C22—2511no—
253C22—2511no—
254C22C22⋊C4×D25111yescharacteristic
255C4—21no—
256C4—41no—
257C4—81no—
258C4—20081no—
259C4—10041no—
260C4—10041no—
261C4—5021no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—81no—
266C2—10041no—
267C2—10041no—
268C2—5021no—
269C2—5021no—
270C2#99×D25111yescharacteristic
subgroupG/H
centerC2#99×D251
commutatorC23×C251C22×C4
FrattiniC2×D4C22×D251
FittingAΣL(1,𝔽16)×C251C2
socleC2×C251C2×#99
prime radical#30632247C2
cosocleC2×D4×C251C23
hypercenterAΣL(1,𝔽16)D251
hypocenterC251#2121
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. #100×D251C2semidirect
3. #99×D251C2semidirect
4. #30631478C2semidirect
5. #30631476C2semidirect
6. #30631474C2semidirect
7. AΣL(1,𝔽16)×C251C2semidirect
8. C22×C22⋊D251C4semidirect
11. #22548699C4semidirect
16. #22548684C4semidirect
19. C22⋊C4×D251C4semidirect
39. C22×Dic251C2×C4semidirect
51. C22⋊Dic251C2×C4semidirect
55. C24×C251C2×C4semidirect
61. C22⋊C4×C251C2×C4semidirect
135. C251#2121semidirect
143. AΣL(1,𝔽16)D251semidirect
165. C24C4×D251semidirect
197. C22⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
#306322472
#100×D2512
#99×D2512
#306314782
#306314762
#306314742
AΣL(1,𝔽16)×C2512
#2121251