Finite Group Databasebeta

← #30640378

Subgroups — #30640378

Order 16256.

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

normal subgroups only

This group has 266 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#22550441C211yesmaximal, characteristic
2#22549857C211yesmaximal, characteristic, complemented
3#22549848C211yesmaximal, characteristic, complemented
4#22549664C211yesmaximal, characteristic, complemented
5#22549663C211yesmaximal, characteristic, complemented
6#22549662C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C127C211yesmaximal, characteristic, complemented
8C2×#14523247C2211yescharacteristic
9C22×C22⋊D127C411yescharacteristic, complemented
10M16×D127—21no—
11#16861205—21no—
12#16861195—21no—
13#16861182—21no—
14#16861181—21no—
15#16861179C411yescharacteristic, complemented
16#16861178C411yescharacteristic, complemented
17#16861177C2211yescharacteristic
18#16861165—21no—
19#16861165C411yescharacteristic, complemented
20#16861127—21no—
21#16861126C2211yescharacteristic
22#16861124C2211yescharacteristic
23#16861123—21no—
24#120×C127C2211yescharacteristic
25#100×C127C2211yescharacteristic
26#99×C127C2211yescharacteristic
27C23×D127C2×C411yescharacteristic
28#14523247—41no—
29#14523247—41no—
30C2×Dic254D411yescharacteristic
31C2×D508—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—41no—
37C2×C22⋊D127—41no—
38C2×C22⋊D127C2×C411yescharacteristic
39C2×C22⋊D127D411yescharacteristic
40C22×Dic127C2×C411yescharacteristic
41C22×Dic127D411yescharacteristic
42C2×C4×D127D411yescharacteristic
43#14523232—21no—
44#14523231—41no—
45C8×D127—41no—
46Dic127⋊C4—21no—
47Dic127⋊C4—41no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50C22⋊Dic127C2×C411yescharacteristic
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—21no—
53C4×Dic127—21no—
54C4×Dic127—41no—
55C24×C127C2×C411yescharacteristic
56C2×D4×C127—21no—
57C2×D4×C127C2311yescharacteristic
58M16×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127C2×C411yescharacteristic
62C22×D127—21no—
63C22×D127—41no—
64C22×D127—81no—
65C22×D127C22⋊C411yescharacteristic
66Dic254—41no—
67D508—81no—
68C22⋊D127—41no—
69C22⋊D127—41no—
70C22⋊D127—42no—
71C22⋊D127—41no—
72C22⋊D127—41no—
73C22⋊D127—81no—
74C22⋊D127—81no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—21no—
80C2×Dic127—41no—
81C2×Dic127C22⋊C411yescharacteristic
82C2×Dic127C22⋊C411yescharacteristic
83C2×Dic127C22⋊C411yescharacteristic
84C4×D127—21no—
85C4×D127—21no—
86C127⋊C8—41no—
87C23×C127—21no—
88C23×C127—41no—
89C23×C127—81no—
90C23×C127C22×C411yescharacteristic
91C23×C127C2×D411yescharacteristic
92D4×C127—41no—
93D4×C127—81no—
94C2×C4×C127—21no—
95C2×C4×C127—41no—
96C2×C4×C127C2×D411yescharacteristic
97C8×C127—41no—
98C22×C127—21no—
99C22×C127—21no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—82no—
106C22×C127C2×C22⋊C411yescharacteristic
107C2×D127—21no—
108C2×D127—41no—
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—41no—
112C2×D127—82no—
113C4×C127—21no—
114C4×C127—41no—
115C4×C127—81no—
116Dic127—21no—
117Dic127—21no—
118Dic127—41no—
119Dic127—81no—
120Dic127#9911yescharacteristic
121Dic127#9911yescharacteristic
122C2×C127—21no—
123C2×C127—41no—
124C2×C127—41no—
125C2×C127—81no—
126C2×C127C2×#9911yescharacteristic
127D127—41no—
128D127—81no—
129#2122—1271nomaximal, Sylow
130C127#212211yescharacteristic, complemented, Sylow
131#535—1271no—
132#413—1271no—
133#410—1271no—
134AΣL(1,𝔽16)—1271no—
135AΣL(1,𝔽16)—1271no—
136AΣL(1,𝔽16)—1271no—
137AΣL(1,𝔽16)D12711yescharacteristic, complemented
138C2×#41—1271no—
139C22×D4—1271no—
140C2×M16—2541no—
141#127—2541no—
142#124—2541no—
143#121—2541no—
144#121—2541no—
145#120—1271no—
146#120—1271no—
147#120—1271no—
148#120C2×D12711yescharacteristic
149#117—1271no—
150#117—2541no—
151#101—2541no—
152#100—1271no—
153#100C2×D12711yescharacteristic
154#99—1271no—
155#99—2541no—
156#99C2×D12711yescharacteristic
157C24—1271no—
158C24C4×D12711yescharacteristic
159#41—5081no—
160#41—5081no—
161C2×Q8—1271no—
162C2×D4—21no—
163C2×D4—1271no—
164C2×D4—1271no—
165C2×D4—5081no—
166C2×D4—5081no—
167C2×D4—2541no—
168C2×D4—2541no—
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4C22×D12711yescharacteristic
173C22×C4—1271no—
174C22×C4—1271no—
175C22×C4—1271no—
176M16—21no—
177M16—5081no—
178M16—2541no—
179C2×C8—5081no—
180C4⋊C4—5081no—
181C4⋊C4—2541no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—1271no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C22⋊C4—2541no—
188C22⋊C4—2541no—
189C22⋊C4C4×D12711yescharacteristic
190C42—5081no—
191C42—2541no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—1271no—
196C23—10161no—
197C23—5081no—
198C23—2541no—
199C23C2×C4×D12711yescharacteristic
200C23D4×D12711yescharacteristic
201Q8—5081no—
202D4—41no—
203D4—81no—
204D4—10161no—
205D4—10161no—
206D4—10161no—
207D4—5081no—
208D4—5081no—
209D4—5081no—
210D4—5081no—
211D4—5082no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—1271no—
215C2×C4—1271no—
216C2×C4—1271no—
217C2×C4—5081no—
218C2×C4—2541no—
219C2×C4—2541no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4—2541no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4D4×D12711yescharacteristic
226C8—41no—
227C8—5081no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—82no—
236C22—10162no—
237C22—5081no—
238C22—5081no—
239C22—5081no—
240C22—5081no—
241C22—2541no—
242C22C22⋊C4×D12711yescharacteristic
243C4—21no—
244C4—41no—
245C4—81no—
246C4—1271no—
247C4—1271no—
248C4—10161no—
249C4—5081no—
250C4—2541no—
251C4—2541no—
252C2—21no—
253C2—41no—
254C2—41no—
255C2—81no—
256C2—10161no—
257C2—5081no—
258C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
FittingAΣL(1,𝔽16)×C127C2
socleC2×C127C2×#99
prime radical#22550441C2
cosocleC2×D4×C127C23
hypercenterAΣL(1,𝔽16)D127
hypocenterC127#2122
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #22549857C2semidirect
3. #22549848C2semidirect
4. #22549664C2semidirect
5. #22549663C2semidirect
6. #22549662C2semidirect
7. AΣL(1,𝔽16)×C127C2semidirect
9. C22×C22⋊D127C4semidirect
15. #16861179C4semidirect
16. #16861178C4semidirect
19. #16861165C4semidirect
130. C127#2122semidirect
137. AΣL(1,𝔽16)D127semidirect

Maximal subgroups

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

subgroupindex
#225504412
#225498572
#225498482
#225496642
#225496632
#225496622
AΣL(1,𝔽16)×C1272
#2122127