Finite Group Databasebeta

← #30640715

Subgroups — #30640715

Order 16256.

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

normal subgroups only

This group has 354 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#22550508C211yesmaximal, characteristic, complemented
2#22550064C211yesmaximal, characteristic, complemented
3#22550062C211yesmaximal, characteristic, complemented
4#22550046C211yesmaximal, characteristic, complemented
5#22549668C211yesmaximal, characteristic, complemented
6#22549664C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C127C211yesmaximal, characteristic, complemented
8#16861254C2211yescharacteristic, complemented
9C22×C22⋊D127—21no—
10C2×D4×D127—21no—
11#16861237—21no—
12#16861205C2211yescharacteristic
13#16861182—21no—
14#16861181—21no—
15#16861179—21no—
16#16861179—21no—
17#16861178—21no—
18#16861177C2211yescharacteristic
19(C22⋊D127)⋊C4—21no—
20#16861135—21no—
21#16861124C2211yescharacteristic, complemented
22#16861123—21no—
23#16861123—21no—
24#120×C127C2211yescharacteristic
25#100×C127C2211yescharacteristic, complemented
26#99×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C23×D127D411yescharacteristic, complemented
29#14523248—21no—
30#14523247—41no—
31C2×D508—21no—
32C2×D508—41no—
33C2×D508D411yescharacteristic, complemented
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—21no—
37C2×C22⋊D127—21no—
38C2×C22⋊D127—21no—
39C2×C22⋊D127—21no—
40C2×C22⋊D127—21no—
41C2×C22⋊D127—44no—
42C2×C22⋊D127—41no—
43D4×D127—44no—
44D4×D127—41no—
45C22×Dic127—21no—
46C2×C4×D127—21no—
47#14523237—41no—
48D1016—41no—
49Dic127⋊C4—21no—
50C22⋊Dic127—21no—
51C22⋊Dic127—41no—
52C22⋊Dic127D411yescharacteristic, complemented
53(C2×D127)⋊C4—21no—
54(C2×D127)⋊C4—21no—
55(C2×D127)⋊C4—41no—
56C4×Dic127D411yescharacteristic, complemented
57C24×C127D411yescharacteristic, complemented
58C2×D4×C127—21no—
59C2×D4×C127C2311yescharacteristic
60M16×C127—21no—
61C22⋊C4×C127—21no—
62C22⋊C4×C127—21no—
63C22⋊C4×C127D411yescharacteristic, complemented
64C22×D127—21no—
65C22×D127—21no—
66C22×D127—21no—
67C22×D127—21no—
68C22×D127—42no—
69C22×D127—42no—
70C22×D127—41no—
71C22×D127—42no—
72C22×D127—81no—
73Dic254—21no—
74D508—21no—
75D508—41no—
76D508—42no—
77D508—82no—
78C22⋊D127—24no—
79C22⋊D127—42no—
80C22⋊D127—42no—
81C22⋊D127—42no—
82C22⋊D127—41no—
83C22⋊D127—41no—
84C22⋊D127—42no—
85C22⋊D127—41no—
86C22⋊D127—82no—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127—21no—
90C2×Dic127—21no—
91C2×Dic127—42no—
92C2×Dic127—41no—
93C4×D127—42no—
94C4×D127—41no—
95C23×C127—21no—
96C23×C127—41no—
97C23×C127—42no—
98C23×C127C2×D411yescharacteristic
99C23×C127C2×D411yescharacteristic
100D4×C127—41no—
101D4×C127—42no—
102C2×C4×C127—21no—
103C2×C4×C127—41no—
104C2×C4×C127C2×D411yescharacteristic
105C8×C127—41no—
106C22×C127—22no—
107C22×C127—21no—
108C22×C127—21no—
109C22×C127—41no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127—42no—
113C22×C127—41no—
114C22×C127—81no—
115C22×C127#12011yescharacteristic
116C2×D127—22no—
117C2×D127—41no—
118C2×D127—41no—
119C2×D127—42no—
120C2×D127—41no—
121C2×D127—41no—
122C2×D127—41no—
123C2×D127—41no—
124C2×D127—81no—
125C2×D127—84no—
126C2×D127—81no—
127C2×D127—81no—
128C4×C127—21no—
129C4×C127—41no—
130C4×C127—81no—
131Dic127—22no—
132Dic127—41no—
133Dic127—41no—
134Dic127—81no—
135C2×C127—21no—
136C2×C127—41no—
137C2×C127—41no—
138C2×C127—42no—
139C2×C127B(4,𝔽2)11yescharacteristic
140D127—42no—
141D127—81no—
142D127—81no—
143#2196—1271nomaximal, Sylow
144C127#219611yescharacteristic, complemented, Sylow
145D42—1271no—
146B(4,𝔽2)—1271no—
147B(4,𝔽2)—1271no—
148#453—1271no—
149#353—1271no—
150AΣL(1,𝔽16)—1271no—
151AΣL(1,𝔽16)D12711yescharacteristic, complemented
152H(5,𝔽2)—1271no—
153C22×D4—2541no—
154C22×D4—2541no—
155AGL(1,ℤ/8)—2541no—
156#127—1271no—
157#121—2541no—
158#121—2541no—
159#120—1271no—
160#120—2541no—
161#120—2541no—
162#120—2541no—
163#120C2×D12711yescharacteristic
164C4×D4—2541no—
165#104—2541no—
166#100C2×D12711yescharacteristic, complemented
167#99—1271no—
168#99—2541no—
169#99—2541no—
170#99C2×D12711yescharacteristic, complemented
171C24—1271no—
172C24—2541no—
173C24D50811yescharacteristic, complemented
174#41—5081no—
175#41—2541no—
176C2×D4—21no—
177C2×D4—1271no—
178C2×D4—5081no—
179C2×D4—5084no—
180C2×D4—5081no—
181C2×D4—5081no—
182C2×D4—5084no—
183C2×D4—2541no—
184C2×D4—2541no—
185C2×D4—2541no—
186C2×D4—2541no—
187C2×D4—2541no—
188C2×D4—2541no—
189C2×D4—2541no—
190C2×D4—2541no—
191C2×D4C22×D12711yescharacteristic
192C22×C4—2541no—
193C22×C4—2541no—
194SD16—5081no—
195D8—5081no—
196M16—21no—
197C4⋊C4—2541no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—1271no—
201C22⋊C4—5081no—
202C22⋊C4—5081no—
203C22⋊C4—2541no—
204C22⋊C4—2541no—
205C22⋊C4—2541no—
206C22⋊C4D50811yescharacteristic, complemented
207C42—1271no—
208C23—21no—
209C23—41no—
210C23—42no—
211C23—10161no—
212C23—5082no—
213C23—5081no—
214C23—5082no—
215C23—5082no—
216C23—2541no—
217C23—2541no—
218C23—2541no—
219C23—2541no—
220C23D4×D12711yescharacteristic
221C23C2×D50811yescharacteristic
222Q8—2541no—
223D4—41no—
224D4—42no—
225D4—10162no—
226D4—10162no—
227D4—5081no—
228D4—5082no—
229D4—5082no—
230D4—5081no—
231D4—5081no—
232D4—5082no—
233D4—5081no—
234D4—5082no—
235D4—5082no—
236D4—2541no—
237D4—2544no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—5081no—
241C2×C4—5082no—
242C2×C4—5081no—
243C2×C4—5082no—
244C2×C4—2541no—
245C2×C4—2541no—
246C2×C4—2541no—
247C2×C4—2541no—
248C2×C4D4×D12711yescharacteristic
249C8—41no—
250C22—22no—
251C22—21no—
252C22—21no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—41no—
258C22—81no—
259C22—10161no—
260C22—10161no—
261C22—10164no—
262C22—10161no—
263C22—5081no—
264C22—5081no—
265C22—5081no—
266C22—5082no—
267C22—5081no—
268C22—5081no—
269C22—5081no—
270C22—2542no—
271C22#1686117911yescharacteristic
272C4—21no—
273C4—41no—
274C4—81no—
275C4—10161no—
276C4—5081no—
277C4—5081no—
278C4—2542no—
279C2—21no—
280C2—41no—
281C2—41no—
282C2—42no—
283C2—10161no—
284C2—10161no—
285C2—5082no—
286C2#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#2196
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. #22550508C2semidirect
2. #22550064C2semidirect
3. #22550062C2semidirect
4. #22550046C2semidirect
5. #22549668C2semidirect
6. #22549664C2semidirect
7. AΣL(1,𝔽16)×C127C2semidirect
8. #16861254C22semidirect
21. #16861124C22semidirect
25. #100×C127C22semidirect
26. #99×C127C22semidirect
28. C23×D127D4semidirect
33. C2×D508D4semidirect
52. C22⋊Dic127D4semidirect
56. C4×Dic127D4semidirect
57. C24×C127D4semidirect
63. C22⋊C4×C127D4semidirect
144. C127#2196semidirect
151. AΣL(1,𝔽16)D127semidirect
166. #100C2×D127semidirect
170. #99C2×D127semidirect
173. C24D508semidirect
206. C22⋊C4D508semidirect

Maximal subgroups

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

subgroupindex
#225505082
#225500642
#225500622
#225500462
#225496682
#225496642
AΣL(1,𝔽16)×C1272
#2196127