Finite Group Databasebeta

← #30640732

Subgroups — #30640732

Order 16256.

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

normal subgroups only

This group has 270 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#22550619C211yesmaximal, characteristic, complemented
2#22550064C211yesmaximal, characteristic, complemented
3#22550062C211yesmaximal, characteristic, complemented
4#22550056C211yesmaximal, characteristic, complemented
5#22549670C211yesmaximal, characteristic, complemented
6#22549664C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C127C211yesmaximal, characteristic, complemented
8#16861254C2211yescharacteristic
9#16861242—21no—
10#16861196—21no—
11#16861196C2211yescharacteristic
12#16861195—21no—
13#16861195—21no—
14#16861195—21no—
15#16861179—21no—
16#16861178—21no—
17#16861178—21no—
18#16861177—21no—
19#16861177C2211yescharacteristic
20#16861135—21no—
21#16861124C2211yescharacteristic
22#16861123—21no—
23#16861123—21no—
24#120×C127C2211yescharacteristic
25#100×C127C2211yescharacteristic
26#99×C127C2211yescharacteristic
27C23×D127D411yescharacteristic, complemented
28#14523248—21no—
29#14523247—41no—
30C2×Dic254—21no—
31C2×Dic254D411yescharacteristic, complemented
32C2×D508—41no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—21no—
37C2×C22⋊D127—21no—
38C2×C22⋊D127—41no—
39D4×D127—41no—
40Dic508—41no—
41#14523237—41no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—21no—
45C22⋊Dic127—21no—
46C22⋊Dic127—21no—
47C22⋊Dic127—41no—
48C22⋊Dic127D411yescharacteristic, complemented
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—41no—
53C4×Dic127—21no—
54C4×Dic127D411yescharacteristic, complemented
55C24×C127D411yescharacteristic, complemented
56C2×D4×C127—21no—
57C2×D4×C127C2311yescharacteristic
58M16×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127D411yescharacteristic, complemented
62C22×D127—21no—
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—41no—
66C22×D127—81no—
67Dic254—21no—
68Dic254—41no—
69Dic254—81no—
70D508—21no—
71D508—82no—
72C22⋊D127—41no—
73C22⋊D127—41no—
74C22⋊D127—41no—
75C22⋊D127—81no—
76C22⋊D127—81no—
77C22⋊D127—81no—
78C22⋊D127—82no—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C2×Dic127—41no—
85C4×D127—41no—
86C23×C127—21no—
87C23×C127—41no—
88C23×C127—81no—
89C23×C127C2×D411yescharacteristic
90C23×C127C2×D411yescharacteristic
91D4×C127—41no—
92D4×C127—81no—
93C2×C4×C127—21no—
94C2×C4×C127—41no—
95C2×C4×C127C2×D411yescharacteristic
96C8×C127—41no—
97C22×C127—21no—
98C22×C127—21no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—42no—
104C22×C127—41no—
105C22×C127—81no—
106C22×C127#12011yescharacteristic
107C2×D127—41no—
108C2×D127—41no—
109C2×D127—42no—
110C2×D127—41no—
111C2×D127—41no—
112C2×D127—81no—
113C2×D127—81no—
114C2×D127—81no—
115C4×C127—21no—
116C4×C127—41no—
117C4×C127—81no—
118Dic127—41no—
119Dic127—41no—
120Dic127—41no—
121Dic127—41no—
122Dic127—81no—
123C2×C127—21no—
124C2×C127—41no—
125C2×C127—41no—
126C2×C127—81no—
127C2×C127B(4,𝔽2)11yescharacteristic
128D127—81no—
129D127—81no—
130#2199—1271nomaximal, Sylow
131C127#219911yescharacteristic, complemented, Sylow
132H(3,𝔽4)—1271no—
133B(4,𝔽2)—1271no—
134B(4,𝔽2)—1271no—
135#455—1271no—
136#354—1271no—
137AΣL(1,𝔽16)—1271no—
138AΣL(1,𝔽16)D12711yescharacteristic, complemented
139H(5,𝔽2)—1271no—
140#137—2541no—
141#124—1271no—
142#124—2541no—
143#124—2541no—
144#124—2541no—
145#124—2541no—
146#120—1271no—
147#120—2541no—
148#120—2541no—
149#120—2541no—
150#120—2541no—
151#120C2×D12711yescharacteristic
152#104—2541no—
153#100C2×D12711yescharacteristic
154#99—1271no—
155#99—2541no—
156#99—2541no—
157#99C2×D12711yescharacteristic
158C24—1271no—
159C24D50811yescharacteristic, complemented
160#41—5081no—
161#41—2541no—
162C2×Q8—1271no—
163C2×Q8—2541no—
164C2×D4—21no—
165C2×D4—5081no—
166C2×D4—5081no—
167C2×D4—5081no—
168C2×D4—2541no—
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4C22×D12711yescharacteristic
174Q16—5081no—
175SD16—5081no—
176M16—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—1271no—
180C22⋊C4—5081no—
181C22⋊C4—5081no—
182C22⋊C4—2541no—
183C22⋊C4—2541no—
184C22⋊C4—2541no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C22⋊C4—2541no—
188C22⋊C4—2541no—
189C22⋊C4—2541no—
190C22⋊C4D50811yescharacteristic, complemented
191C42—1271no—
192C42—2541no—
193C23—21no—
194C23—41no—
195C23—81no—
196C23—10161no—
197C23—5081no—
198C23—2541no—
199C23—2541no—
200C23—2541no—
201C23D4×D12711yescharacteristic
202C23C2×D50811yescharacteristic
203Q8—10161no—
204Q8—5081no—
205Q8—2541no—
206D4—41no—
207D4—81no—
208D4—10162no—
209D4—10162no—
210D4—10161no—
211D4—10161no—
212D4—10161no—
213D4—5081no—
214D4—5081no—
215D4—5081no—
216D4—2541no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—5081no—
220C2×C4—5081no—
221C2×C4—2541no—
222C2×C4—2541no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4—2541no—
226C2×C4D4×D12711yescharacteristic
227C8—41no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22—81no—
237C22—10161no—
238C22—10161no—
239C22—10161no—
240C22—5081no—
241C22—5081no—
242C22—5082no—
243C22—5081no—
244C22—5081no—
245C22#1686117911yescharacteristic
246C4—21no—
247C4—41no—
248C4—81no—
249C4—10161no—
250C4—5081no—
251C4—5081no—
252C4—5081no—
253C4—5081no—
254C2—21no—
255C2—41no—
256C2—41no—
257C2—81no—
258C2—10161no—
259C2—10161no—
260C2#2255006411yescharacteristic
subgroupG/H
centerC2#22550064
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
FittingAΣL(1,𝔽16)×C127C2
socleC2×C127B(4,𝔽2)
hypercenterAΣL(1,𝔽16)D127
hypocenterC127#2199
LCSG ▷ C2×D4×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
1. #22550619C2semidirect
2. #22550064C2semidirect
3. #22550062C2semidirect
4. #22550056C2semidirect
5. #22549670C2semidirect
6. #22549664C2semidirect
7. AΣL(1,𝔽16)×C127C2semidirect
27. C23×D127D4semidirect
31. C2×Dic254D4semidirect
48. C22⋊Dic127D4semidirect
54. C4×Dic127D4semidirect
55. C24×C127D4semidirect
61. C22⋊C4×C127D4semidirect
131. C127#2199semidirect
138. AΣL(1,𝔽16)D127semidirect
159. C24D508semidirect
190. C22⋊C4D508semidirect

Maximal subgroups

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

subgroupindex
#225506192
#225500642
#225500622
#225500562
#225496702
#225496642
AΣL(1,𝔽16)×C1272
#2199127