Finite Group Databasebeta

← #30639760

Subgroups — #30639760

Order 16256.

394 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#16861240C211yesmaximal, characteristic, complemented
2#22550445C211yesmaximal, characteristic
3C2×#16861198C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#419×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×D508C2211yescharacteristic
10#16861240—22no—
11#16861240—22no—
12C2×#14523238—21no—
13C2×#14523235—21no—
14C2×#14523232C2211yescharacteristic
15#16861206—21no—
16#16861198—22no—
17#16861198—22no—
18#16861197—21no—
19#16861184—22no—
20#16861179—22no—
21C2×(C2×D127)⋊C4—22no—
22C2×C4×Dic127C2211yescharacteristic
23#16861135—22no—
24#16861125—22no—
25C22×Q8×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27#117×C127C2211yescharacteristic
28#103×C127—22no—
29C23×D127D412yes—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34C2×Dic254D411yescharacteristic
35C2×D508—24no—
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508—21no—
42C2×D508D411yescharacteristic
43C2×C22⋊D127—21no—
44C22×Dic127—21no—
45C2×C4×D127—21no—
46#14523238—24no—
47#14523235—24no—
48#14523232—21no—
49#14523232D412yescomplemented
50C2×C127⋊C8—21no—
51(C2×D127)⋊C4—24no—
52(C2×D127)⋊C4—28no—
53C4×Dic127—21no—
54C4×Dic127D412yescomplemented
55C2×Q8×C127—21no—
56C2×Q8×C127—21no—
57C2×Q8×C127—21no—
58C2×Q8×C127—21no—
59C2×Q8×C127—21no—
60C2×Q8×C127D412yescomplemented
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63M16×C127—21no—
64M16×C127—21no—
65C2×C8×C127—21no—
66C4⋊C4×C127D412yes—
67C22⋊C4×C127—21no—
68C42×C127—21no—
69C22×D127—22no—
70C22×D127—22no—
71C22×D127—21no—
72C22×D127—22no—
73C22×D127—44no—
74Dic254—21no—
75Dic254—21no—
76D508—21no—
77D508—21no—
78D508—41no—
79D508—41no—
80D508—41no—
81D508—41no—
82D508—84no—
83C22⋊D127—41no—
84C22⋊D127—41no—
85C2×Dic127—22no—
86C2×Dic127—21no—
87C2×Dic127—42no—
88C4×D127—41no—
89C4×D127—41no—
90C127⋊C8—22no—
91C23×C127C2×D411yescharacteristic
92Q8×C127—22no—
93Q8×C127—22no—
94Q8×C127—41no—
95Q8×C127—41no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—41no—
99C2×C4×C127—41no—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127#4111yescharacteristic
106C8×C127—41no—
107C22×C127—22no—
108C22×C127#12011yescharacteristic
109C22×C127#12011yescharacteristic
110C22×C127#12111yescharacteristic
111C2×D127—44no—
112C2×D127—41no—
113C2×D127—42no—
114C2×D127—42no—
115C2×D127—44no—
116C2×D127—82no—
117C4×C127—42no—
118C4×C127—42no—
119C4×C127#12011yescharacteristic
120C4×C127#12111yescharacteristic
121C4×C127#12111yescharacteristic
122C4×C127#12711yescharacteristic
123Dic127—42no—
124Dic127—41no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127#39211yescharacteristic
128C2×C127#45512yes—
129D127—81no—
130D127—82no—
131#2004—1271nomaximal, Sylow
132C127#200411yescharacteristic, complemented, Sylow
133C2×#137—1271no—
134#535—1271no—
135C2×#124—1271no—
136C2×#104—1271no—
137#419D12711yescharacteristic, complemented
138C2×#100—1271no—
139#339—1271no—
140C2×#41—1271no—
141C22×Q8C2×D12711yescharacteristic
142C22×D4—1271no—
143#137—2542no—
144#137—2542no—
145C2×Q16—2541no—
146C2×SD16—2541no—
147C2×M16—1271no—
148C2×M16C2×D12711yescharacteristic
149#127—2541no—
150#124—2541no—
151#124—2542no—
152#124—2542no—
153#121—2542no—
154#120—2542no—
155#117C2×D12711yescharacteristic
156C2×C22⋊C4—2542no—
157C2×C42—1271no—
158#104—2542no—
159#103—22no—
160#100—2542no—
161C24—1272no—
162#41—2541no—
163#41—2541no—
164#41—2541no—
165#41—2541no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—1271no—
172C2×Q8D50812yescomplemented
173C2×D4—1271no—
174C2×D4—2544no—
175C2×D4—2541no—
176C2×D4—2541no—
177C2×D4—2541no—
178C2×D4—2541no—
179C2×D4—2541no—
180C2×D4—2541no—
181C2×D4—2541no—
182C22×C4—21no—
183C22×C4—2541no—
184C22×C4—2541no—
185C22×C4C22×D12711yescharacteristic
186Q16—2544no—
187SD16—2544no—
188M16—21no—
189M16—21no—
190M16—1272no—
191M16—2541no—
192C2×C8—21no—
193C2×C8—2541no—
194C4⋊C4C22⋊D12712yes—
195C22⋊C4—21no—
196C22⋊C4—2548no—
197C22⋊C4—2544no—
198C42—21no—
199C42—1272no—
200C42—2541no—
201C23—5084no—
202C23—2542no—
203C23—2542no—
204C23—2541no—
205C23—2542no—
206C23D4×D12711yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211Q8—2541no—
212Q8—2541no—
213D4—10164no—
214D4—5081no—
215D4—5081no—
216D4—5081no—
217D4—5081no—
218D4—5081no—
219D4—5081no—
220D4—2541no—
221D4—2541no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—5082no—
227C2×C4—5081no—
228C2×C4—5081no—
229C2×C4—2541no—
230C2×C4—2542no—
231C2×C4D4×D12711yescharacteristic
232C2×C4D4×D12711yescharacteristic
233C2×C4D4×D12711yescharacteristic
234C2×C4C2×C22⋊D12711yescharacteristic
235C2×C4C2×D50811yescharacteristic
236C2×C4#1452324811yescharacteristic
237C8—41no—
238C8—2542no—
239C22—22no—
240C22—10162no—
241C22—5084no—
242C22—5081no—
243C22—5082no—
244C22—5082no—
245C22—5084no—
246C22#1686117711yescharacteristic
247C22#1686117911yescharacteristic
248C22#1686118211yescharacteristic
249C4—42no—
250C4—42no—
251C4—5082no—
252C4—5081no—
253C4#1686117911yescharacteristic
254C4#1686118211yescharacteristic
255C4#1686118311yescharacteristic
256C4#1686120511yescharacteristic
257C2—21no—
258C2—21no—
259C2—10162no—
260C2—10161no—
261C2#2254978511yescharacteristic
262C2#2255005212yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC22×C127#120
prime radical#22550445C2
hypercenter#419D127
hypocenterC127#2004
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. C2×#16861198C2semidirect
4. C2×#16861135C2semidirect
5. C2×#16861125C2semidirect
6. #22549635C2semidirect
7. #419×C127C2semidirect
49. #14523232D4semidirect
54. C4×Dic127D4semidirect
60. C2×Q8×C127D4semidirect
132. C127#2004semidirect
137. #419D127semidirect
172. C2×Q8D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225504452
C2×#168611982
C2×#168611352
C2×#168611252
#225496352
#419×C1272
#2004127