Finite Group Databasebeta

← #30640722

Subgroups — #30640722

Order 16256.

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

normal subgroups only

This group has 258 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#22550557C211yesmaximal, characteristic, complemented
2#22550067C211yesmaximal, characteristic, complemented
3#22550062C211yesmaximal, characteristic
4#22550056C211yesmaximal, characteristic, complemented
5#22549670C211yesmaximal, characteristic, complemented
6#22549667C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C127C211yesmaximal, characteristic, complemented
8#16861254C2211yescharacteristic
9#16861242—21no—
10#16861202—21no—
11#16861196C2211yescharacteristic
12#16861193—21no—
13#16861190—21no—
14#16861190C2211yescharacteristic
15#16861186—21no—
16#16861178—21no—
17D4×Dic127—21no—
18#16861165—21no—
19C2×C22⋊Dic127—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×Dic254D411yescharacteristic, complemented
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C2×C22⋊D127—41no—
33D4×D127—41no—
34C22×Dic127—21no—
35C22×Dic127—21no—
36C22×Dic127D411yescharacteristic, complemented
37Dic508—41no—
38#14523237—41no—
39C4⋊Dic127—21no—
40C4⋊Dic127—41no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44C22⋊Dic127—21no—
45C22⋊Dic127—21no—
46C22⋊Dic127—21no—
47C22⋊Dic127—21no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50C22⋊Dic127—41no—
51C22⋊Dic127D411yescharacteristic, complemented
52(C2×D127)⋊C4—41no—
53C4×Dic127—21no—
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—
64Dic254—21no—
65Dic254—41no—
66D508—21no—
67C22⋊D127—41no—
68C22⋊D127—41no—
69C22⋊D127—41no—
70C22⋊D127—82no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—42no—
78C2×Dic127—41no—
79C2×Dic127—41no—
80C2×Dic127—81no—
81C4×D127—41no—
82C23×C127—21no—
83C23×C127—42no—
84C23×C127—41no—
85C23×C127C2×D411yescharacteristic
86C23×C127C2×D411yescharacteristic
87D4×C127—41no—
88D4×C127—42no—
89C2×C4×C127—21no—
90C2×C4×C127—41no—
91C2×C4×C127C2×D411yescharacteristic
92C8×C127—41no—
93C22×C127—22no—
94C22×C127—21no—
95C22×C127—21no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127—42no—
100C22×C127—41no—
101C22×C127—81no—
102C22×C127#12011yescharacteristic
103C2×D127—41no—
104C2×D127—81no—
105C2×D127—81no—
106C4×C127—21no—
107C4×C127—41no—
108C4×C127—81no—
109Dic127—41no—
110Dic127—41no—
111Dic127—41no—
112Dic127—41no—
113Dic127—41no—
114Dic127—81no—
115C2×C127—21no—
116C2×C127—41no—
117C2×C127—41no—
118C2×C127—42no—
119C2×C127B(4,𝔽2)11yescharacteristic
120D127—81no—
121#2197—1271nomaximal, Sylow
122C127#219711yescharacteristic, complemented, Sylow
123#551—1271no—
124#458—1271no—
125B(4,𝔽2)—1271no—
126#455—1271no—
127#354—1271no—
128#352—1271no—
129AΣL(1,𝔽16)D12711yescharacteristic, complemented
130H(5,𝔽2)—1271no—
131#137—2541no—
132#126—2541no—
133#124—1271no—
134#123—1271no—
135#123—2541no—
136#123—2541no—
137C22⋊Q8—2541no—
138#120—2541no—
139#120C2×D12711yescharacteristic
140C4×D4—2541no—
141#117—2541no—
142C2×C22⋊C4—2541no—
143#104—2541no—
144#100C2×D12711yescharacteristic
145#99—1271no—
146#99—2541no—
147#99—2541no—
148#99C2×D12711yescharacteristic
149C24D50811yescharacteristic
150#41—5081no—
151#41—2541no—
152C2×Q8—1271no—
153C2×D4—21no—
154C2×D4—5081no—
155C2×D4—5081no—
156C2×D4—2541no—
157C2×D4—2541no—
158C2×D4C22×D12711yescharacteristic
159C22×C4—1271no—
160C22×C4—2541no—
161C22×C4—2541no—
162Q16—5081no—
163SD16—5081no—
164M16—21no—
165C4⋊C4—5081no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C4⋊C4—2541no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—1271no—
173C22⋊C4—5081no—
174C22⋊C4—5081no—
175C22⋊C4—2541no—
176C22⋊C4—2541no—
177C22⋊C4—2541no—
178C22⋊C4—2541no—
179C22⋊C4—2541no—
180C22⋊C4—2541no—
181C22⋊C4D50811yescharacteristic
182C42—1271no—
183C42—2541no—
184C23—21no—
185C23—41no—
186C23—42no—
187C23—2541no—
188C23—2541no—
189C23D4×D12711yescharacteristic
190C23C2×D50811yescharacteristic
191Q8—5081no—
192Q8—2541no—
193D4—41no—
194D4—42no—
195D4—10162no—
196D4—5081no—
197D4—5081no—
198D4—5081no—
199D4—2541no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—10161no—
203C2×C4—5081no—
204C2×C4—5081no—
205C2×C4—5082no—
206C2×C4—5081no—
207C2×C4—2541no—
208C2×C4—2541no—
209C2×C4—2541no—
210C2×C4—2541no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4D4×D12711yescharacteristic
214C8—41no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22—10161no—
225C22—10161no—
226C22—5081no—
227C22#1686117911yescharacteristic
228C4—21no—
229C4—41no—
230C4—81no—
231C4—10161no—
232C4—5081no—
233C4—5081no—
234C4—5081no—
235C4—5081no—
236C4—5081no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—10161no—
242C2#2255006411yescharacteristic
subgroupG/H
centerC2#22550064
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
FittingAΣL(1,𝔽16)×C127C2
socleC2×C127B(4,𝔽2)
prime radical#22550062C2
hypercenterAΣL(1,𝔽16)D127
hypocenterC127#2197
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. #22550557C2semidirect
2. #22550067C2semidirect
4. #22550056C2semidirect
5. #22549670C2semidirect
6. #22549667C2semidirect
7. AΣL(1,𝔽16)×C127C2semidirect
29. C2×Dic254D4semidirect
36. C22×Dic127D4semidirect
51. C22⋊Dic127D4semidirect
54. C4×Dic127D4semidirect
122. C127#2197semidirect
129. AΣL(1,𝔽16)D127semidirect

Maximal subgroups

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

subgroupindex
#225505572
#225500672
#225500622
#225500562
#225496702
#225496672
AΣL(1,𝔽16)×C1272
#2197127