Finite Group Databasebeta

← #30637676

Subgroups — #30637676

Order 16256.

320 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550026C211yesmaximal, characteristic, complemented
2#22550009C211yesmaximal, characteristic, complemented
3#22549785C211yesmaximal, characteristic, complemented
4#22549613C211yesmaximal, characteristic, complemented
5#22549610C211yesmaximal, characteristic, complemented
6#22549600C211yesmaximal, characteristic, complemented
7#538×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C22⋊D127—21no—
10C2×#14523235—21no—
11C2×#14523233—21no—
12C2×(C2×D127)⋊C4—21no—
13#16861129—21no—
14#16861129—21no—
15C22⋊(C127⋊C8)C2211yescharacteristic
16C22⋊(C127⋊C8)C2211yescharacteristic
17C127⋊#95C2211yescharacteristic
18#126×C127—21no—
19#124×C127—21no—
20#123×C127—21no—
21C22⋊Q8×C127C2211yescharacteristic
22#121×C127—21no—
23#121×C127C2211yescharacteristic
24#120×C127—21no—
25C4×D4×C127—21no—
26C2×C22⋊C4×C127C2211yescharacteristic
27C23×D127—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—42no—
31C2×C22⋊D127—42no—
32C2×C22⋊D127—42no—
33C2×C22⋊D127—42no—
34C22×Dic127—21no—
35#14523235—42no—
36#14523233—42no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39(C2×D127)⋊C4—42no—
40C24×C127D411yescharacteristic
41C2×Q8×C127D411yescharacteristic, complemented
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127D411yescharacteristic, complemented
46C22×C4×C127—21no—
47C22×C4×C127D411yescharacteristic
48C22×C4×C127C2311yescharacteristic
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127D411yescharacteristic, complemented
52C4⋊C4×C127D411yescharacteristic, complemented
53C22⋊C4×C127—21no—
54C22⋊C4×C127—21no—
55C22⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C42×C127—21no—
61C22×D127—22no—
62C22×D127—41no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66C22×D127—41no—
67C22×D127—41no—
68D508—22no—
69D508—22no—
70D508—41no—
71D508—41no—
72C22⋊D127—48no—
73C2×Dic127—22no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C127⋊C8—41no—
77C127⋊C8—41no—
78C23×C127—21no—
79C23×C127—21no—
80C23×C127—21no—
81C23×C127—21no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—41no—
86D4×C127—41no—
87D4×C127—41no—
88D4×C127—41no—
89D4×C127—81no—
90D4×C127—81no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127—81no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—42no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127—81no—
110C22×C127#12011yescharacteristic
111C22×C127#13511yescharacteristic
112C22×C127AGL(1,ℤ/8)11yescharacteristic
113C2×D127—42no—
114C2×D127—42no—
115C2×D127—42no—
116C2×D127—81no—
117C2×D127—81no—
118C2×D127—81no—
119C2×D127—81no—
120C4×C127—21no—
121C4×C127—21no—
122C4×C127—41no—
123C4×C127—41no—
124C4×C127—41no—
125C4×C127—41no—
126Dic127—42no—
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—81no—
130C2×C127—81no—
131C2×C127#44911yescharacteristic
132C2×C127#45311yescharacteristic
133C2×C127B(4,𝔽2)11yescharacteristic
134D127—82no—
135#1600—1271nomaximal, Sylow
136C127#160011yescharacteristic, complemented, Sylow
137#538D12711yescharacteristic, complemented
138#450—1271no—
139#447—1271no—
140#392—1271no—
141#328—1271no—
142#327—1271no—
143#323—1271no—
144C22×D4—1271no—
145C22×D4—2541no—
146C2×SD16—2541no—
147C2×D8—2541no—
148#126—21no—
149#124—21no—
150#123—21no—
151C22⋊Q8C2×D12711yescharacteristic
152#121—21no—
153#121C2×D12711yescharacteristic
154#120—21no—
155C4×D4—21no—
156C2×C22⋊C4—2541no—
157C2×C22⋊C4C2×D12711yescharacteristic
158#102—2541no—
159#102—2541no—
160C22⋊C8—1271no—
161C22⋊C8—1271no—
162#95—1271no—
163C24—2541no—
164C24C22⋊D12711yescharacteristic
165C2×Q8C22⋊D12711yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—5082no—
170C2×D4—5082no—
171C2×D4—5082no—
172C2×D4—5082no—
173C2×D4—2541no—
174C2×D4—2541no—
175C2×D4C22⋊D12711yescharacteristic, complemented
176C22×C4—21no—
177C22×C4—2541no—
178C22×C4C22⋊D12711yescharacteristic
179C22×C4C22×D12711yescharacteristic
180SD16—5082no—
181D8—5082no—
182C2×C8—2541no—
183C2×C8—2541no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4C22⋊D12711yescharacteristic, complemented
187C4⋊C4C22⋊D12711yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—5082no—
196C42—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23—5081no—
204C23—5081no—
205C23—5081no—
206C23—5081no—
207C23—5081no—
208C23—5081no—
209C23—2542no—
210C23C2×C22⋊D12711yescharacteristic
211Q8—41no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—81no—
216D4—81no—
217D4—5081no—
218D4—5081no—
219D4—5088no—
220D4—2542no—
221D4—2542no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4—5081no—
230C2×C4—5081no—
231C2×C4—2542no—
232C2×C4C2×C22⋊D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C8—5081no—
235C8—5081no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—81no—
246C22—10161no—
247C22—10161no—
248C22—10161no—
249C22—10161no—
250C22—5082no—
251C22—5082no—
252C22—5082no—
253C22#1686117811yescharacteristic
254C22#1686122911yescharacteristic
255C22#1686123411yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—5082no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—81no—
267C2—10162no—
268C2#2255002111yescharacteristic
269C2#2255004411yescharacteristic
270C2#2255006311yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#538×C127C2
socleC22×C127#120
hypercenter#538D127
hypocenterC127#1600
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #22550026C2semidirect
2. #22550009C2semidirect
3. #22549785C2semidirect
4. #22549613C2semidirect
5. #22549610C2semidirect
6. #22549600C2semidirect
7. #538×C127C2semidirect
41. C2×Q8×C127D4semidirect
45. C2×D4×C127D4semidirect
51. C4⋊C4×C127D4semidirect
52. C4⋊C4×C127D4semidirect
136. C127#1600semidirect
137. #538D127semidirect
165. C2×Q8C22⋊D127semidirect
175. C2×D4C22⋊D127semidirect
186. C4⋊C4C22⋊D127semidirect
187. C4⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225500262
#225500092
#225497852
#225496132
#225496102
#225496002
#538×C1272
#1600127