Finite Group Databasebeta

← #30640739

Subgroups — #30640739

Order 16256.

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

normal subgroups only

This group has 260 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#22550615C211yesmaximal, characteristic, complemented
2#22550067C211yesmaximal, characteristic, complemented
3#22550062C211yesmaximal, characteristic, complemented
4#22550046C211yesmaximal, characteristic, complemented
5#22549668C211yesmaximal, characteristic, complemented
6#22549667C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C127C211yesmaximal, characteristic, complemented
8#16861254C2211yescharacteristic
9#16861237—21no—
10#16861205C2211yescharacteristic
11#16861202—21no—
12#16861193—21no—
13#16861191—21no—
14#16861190C2211yescharacteristic
15#16861182—21no—
16#16861181—21no—
17#16861180—21no—
18#16861178—21no—
19#16861178—21no—
20#16861135—21no—
21#16861124C2211yescharacteristic
22#16861123—21no—
23#16861123—21no—
24#120×C127C2211yescharacteristic
25#100×C127C2211yescharacteristic
26#99×C127C2211yescharacteristic
27#14523248—21no—
28#14523247—41no—
29C2×D508D411yescharacteristic, complemented
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—41no—
37D4×D127—41no—
38C22×Dic127D411yescharacteristic, complemented
39C2×C4×D127—21no—
40#14523237—41no—
41D1016—41no—
42C4⋊Dic127—21no—
43C4⋊Dic127—41no—
44Dic127⋊C4—21no—
45Dic127⋊C4—21no—
46C22⋊Dic127—21no—
47C22⋊Dic127—21no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50C22⋊Dic127—41no—
51C22⋊Dic127D411yescharacteristic, complemented
52(C2×D127)⋊C4—21no—
53(C2×D127)⋊C4—41no—
54C4×Dic127D411yescharacteristic, complemented
55C24×C127D411yescharacteristic
56C2×D4×C127—21no—
57C2×D4×C127C2311yescharacteristic
58M16×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127D411yescharacteristic
62C22×D127—21no—
63C22×D127—21no—
64C22×D127—21no—
65Dic254—21no—
66D508—21no—
67D508—41no—
68C22⋊D127—41no—
69C22⋊D127—41no—
70C22⋊D127—41no—
71C22⋊D127—81no—
72C22⋊D127—81no—
73C22⋊D127—81no—
74C22⋊D127—81no—
75C22⋊D127—82no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—41no—
82C2×Dic127—41no—
83C4×D127—41no—
84C4×D127—81no—
85C23×C127—21no—
86C23×C127—41no—
87C23×C127—81no—
88C23×C127C2×D411yescharacteristic
89C23×C127C2×D411yescharacteristic
90D4×C127—41no—
91D4×C127—81no—
92C2×C4×C127—21no—
93C2×C4×C127—41no—
94C2×C4×C127C2×D411yescharacteristic
95C8×C127—41no—
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—42no—
103C22×C127—41no—
104C22×C127—81no—
105C22×C127#12011yescharacteristic
106C2×D127—41no—
107C2×D127—41no—
108C2×D127—41no—
109C2×D127—41no—
110C2×D127—81no—
111C2×D127—81no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—81no—
115Dic127—41no—
116Dic127—41no—
117Dic127—41no—
118Dic127—41no—
119Dic127—81no—
120C2×C127—21no—
121C2×C127—41no—
122C2×C127—41no—
123C2×C127—81no—
124C2×C127B(4,𝔽2)11yescharacteristic
125D127—81no—
126D127—81no—
127#2200—1271nomaximal, Sylow
128C127#220011yescharacteristic, complemented, Sylow
129#560—1271no—
130#458—1271no—
131B(4,𝔽2)—1271no—
132#453—1271no—
133#353—1271no—
134#352—1271no—
135AΣL(1,𝔽16)D12711yescharacteristic, complemented
136H(5,𝔽2)—1271no—
137AGL(1,ℤ/8)—2541no—
138#127—1271no—
139#126—2541no—
140#123—1271no—
141#123—2541no—
142#123—2541no—
143#121—2541no—
144#121—2541no—
145#121—2541no—
146#120—2541no—
147#120—2541no—
148#120C2×D12711yescharacteristic
149#104—2541no—
150#100C2×D12711yescharacteristic
151#99—1271no—
152#99—2541no—
153#99—2541no—
154#99C2×D12711yescharacteristic
155C24D50811yescharacteristic
156#41—5081no—
157#41—2541no—
158C2×D4—21no—
159C2×D4—1271no—
160C2×D4—5081no—
161C2×D4—5081no—
162C2×D4—2541no—
163C2×D4—2541no—
164C2×D4—2541no—
165C2×D4—2541no—
166C2×D4—2541no—
167C2×D4—2541no—
168C2×D4C22×D12711yescharacteristic
169C22×C4—1271no—
170C22×C4—2541no—
171SD16—5081no—
172D8—5081no—
173M16—21no—
174C4⋊C4—5081no—
175C4⋊C4—2541no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—1271no—
181C22⋊C4—5081no—
182C22⋊C4—5081no—
183C22⋊C4—2541no—
184C22⋊C4—2541no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C22⋊C4—2541no—
188C22⋊C4D50811yescharacteristic
189C42—1271no—
190C23—21no—
191C23—41no—
192C23—81no—
193C23—2541no—
194C23—2541no—
195C23—2541no—
196C23D4×D12711yescharacteristic
197C23C2×D50811yescharacteristic
198Q8—2541no—
199D4—41no—
200D4—81no—
201D4—10162no—
202D4—10161no—
203D4—10161no—
204D4—10161no—
205D4—10161no—
206D4—5081no—
207D4—5081no—
208D4—5081no—
209D4—5081no—
210D4—2541no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—10161no—
214C2×C4—5081no—
215C2×C4—5081no—
216C2×C4—5081no—
217C2×C4—2541no—
218C2×C4—2541no—
219C2×C4—2541no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4D4×D12711yescharacteristic
223C8—41no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—41no—
232C22—81no—
233C22—10161no—
234C22—10161no—
235C22—5081no—
236C22—5081no—
237C22—5081no—
238C22—5081no—
239C22#1686117911yescharacteristic
240C4—21no—
241C4—41no—
242C4—81no—
243C4—10161no—
244C4—5081no—
245C4—5081no—
246C4—5081no—
247C4—5081no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—10161no—
253C2—10161no—
254C2#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#2200
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. #22550615C2semidirect
2. #22550067C2semidirect
3. #22550062C2semidirect
4. #22550046C2semidirect
5. #22549668C2semidirect
6. #22549667C2semidirect
7. AΣL(1,𝔽16)×C127C2semidirect
29. C2×D508D4semidirect
38. C22×Dic127D4semidirect
51. C22⋊Dic127D4semidirect
54. C4×Dic127D4semidirect
128. C127#2200semidirect
135. AΣL(1,𝔽16)D127semidirect

Maximal subgroups

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

subgroupindex
#225506152
#225500672
#225500622
#225500462
#225496682
#225496672
AΣL(1,𝔽16)×C1272
#2200127