Finite Group Databasebeta

← #30649084

Subgroups — #30649084

Order 16256.

404 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550504C211yesmaximal, characteristic, complemented
2#22550240C212yesmaximal, complemented
3#22550235C212yesmaximal, complemented
4#22550230C212yesmaximal, complemented
5#22550222C212yesmaximal, complemented
6#22549881C211yesmaximal, characteristic, complemented
7#22549877C211yesmaximal, characteristic, complemented
8#22549872C211yesmaximal, characteristic, complemented
9C2×#16861130C211yesmaximal, characteristic, complemented
10#22549832C211yesmaximal, characteristic, complemented
11#535×C127C211yesmaximal, characteristic
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861209C2212yescomplemented
15#16861208—21no—
16#16861201C2212yescomplemented
17#16861200—21no—
18#16861186—22no—
19#16861168C2211yescharacteristic
20C2×C4⋊Dic127C2211yescharacteristic
21#16861133C2211yescharacteristic, complemented
22#16861133C2211yescharacteristic, complemented
23#16861133C2212yescomplemented
24#16861130C2212yes—
25#16861130C2212yes—
26#16861130C2212yes—
27#16861130C2212yes—
28#16861130C2211yescharacteristic, complemented
29#16861130C2211yescharacteristic, complemented
30#16861130C2212yescomplemented
31#16861120C2212yescomplemented
32C4×C127⋊C8C2212yescomplemented
33C2×#41×C127C2211yescharacteristic
34C22×D4×C127C2211yescharacteristic
35#127×C127C2212yes—
36#124×C127C2212yes—
37#121×C127—21no—
38#121×C127—21no—
39#120×C127—22no—
40#117×C127C2211yescharacteristic
41C2×Dic254—22no—
42C22×Dic127—21no—
43#14523232—22no—
44C2×C127⋊C8—22no—
45C2×C127⋊C8C2311yescharacteristic
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2312yes—
48C4⋊Dic127—22no—
49C4⋊Dic127—21no—
50C4⋊Dic127C2311yescharacteristic
51C4⋊Dic127C2312yes—
52C4⋊Dic127C2311yescharacteristic
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55C22⋊Dic127—21no—
56C4×Dic127—21no—
57C24×C127—21no—
58#41×C127—22no—
59#41×C127—22no—
60C2×Q8×C127C2311yescharacteristic
61C2×D4×C127—21no—
62C2×D4×C127—21no—
63C2×D4×C127—22no—
64C2×D4×C127—22no—
65C2×D4×C127—22no—
66C2×D4×C127—22no—
67C2×D4×C127C2312yes—
68C2×D4×C127C2311yescharacteristic
69C22×C4×C127—21no—
70C22×C4×C127C2311yescharacteristic
71C4⋊C4×C127D411yescharacteristic
72C4⋊C4×C127D411yescharacteristic
73C22⋊C4×C127—22no—
74C22⋊C4×C127—22no—
75C22⋊C4×C127D412yes—
76C42×C127C2312yes—
77Dic254—82no—
78C2×Dic127—21no—
79C2×Dic127—21no—
80C2×Dic127—22no—
81C2×Dic127—81no—
82C127⋊C8—21no—
83C127⋊C8—21no—
84C127⋊C8—22no—
85C23×C127—22no—
86C23×C127—21no—
87C23×C127—42no—
88C23×C127—81no—
89C23×C127—81no—
90C23×C127C2×D411yescharacteristic
91Q8×C127—21no—
92Q8×C127—21no—
93D4×C127—22no—
94D4×C127—22no—
95D4×C127—21no—
96D4×C127—21no—
97D4×C127—42no—
98D4×C127—42no—
99D4×C127—82no—
100D4×C127—82no—
101C2×C4×C127—21no—
102C2×C4×C127—42no—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D412yes—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127#4112yes—
107C2×C4×C127#4112yes—
108C2×C4×C127C2411yescharacteristic
109C22×C127—21no—
110C22×C127—21no—
111C22×C127—41no—
112C22×C127—42no—
113C22×C127—41no—
114C22×C127—42no—
115C22×C127—41no—
116C22×C127—42no—
117C22×C127—84no—
118C22×C127#12412yes—
119C22×C127C22×D411yescharacteristic
120C4×C127—22no—
121C4×C127—22no—
122C4×C127—41no—
123C4×C127#12412yes—
124C4×C127C2×#4111yescharacteristic
125C4×C127C2×#4111yescharacteristic
126Dic127—42no—
127Dic127—41no—
128Dic127—41no—
129C2×C127—22no—
130C2×C127—81no—
131C2×C127—82no—
132C2×C127C2×#12411yescharacteristic
133C2×C127#57611yescharacteristic
134C2×C127#57711yescharacteristic
135#3136—1271nomaximal, Sylow
136C127#313611yescharacteristic, complemented, Sylow
137#544—1271no—
138#535D12711yescharacteristic
139#490—1272no—
140#489—1272no—
141#488—1272no—
142#486—1272no—
143#418—1271no—
144#417—1271no—
145#416—1271no—
146C2×#102—1271no—
147#405—1271no—
148C2×#41C2×D12711yescharacteristic
149C22×D4C2×D12711yescharacteristic
150C2×M16—1271no—
151C22×C8—1271no—
152C4⋊Q8—1272no—
153C4⋊Q8—2541no—
154#127C2×D12712yes—
155#125—1272no—
156#125—2541no—
157#124C2×D12712yes—
158C22⋊Q8—2542no—
159#121—21no—
160#121—21no—
161#120—22no—
162#117—1271no—
163#117C2×D12711yescharacteristic
164C2×C4⋊C4—1271no—
165#103—1271no—
166#103—1271no—
167#103—1272no—
168#102—1272no—
169#102—1272no—
170#102—1271no—
171#102—1271no—
172#102—1272no—
173#102—1272no—
174#102—1272no—
175#97—1272no—
176C4×C8—1272no—
177C24—21no—
178#41—22no—
179#41—22no—
180C2×Q8—2542no—
181C2×Q8C22×D12711yescharacteristic
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4C22×D12712yes—
189C2×D4C22×D12711yescharacteristic
190C22×C4—21no—
191C22×C4—2541no—
192C22×C4C22×D12711yescharacteristic
193M16—2542no—
194C2×C8—1271no—
195C2×C8—1272no—
196C2×C8—1271no—
197C2×C8—2542no—
198C4⋊C4—1272no—
199C4⋊C4—1271no—
200C4⋊C4—1271no—
201C4⋊C4—2542no—
202C4⋊C4—2542no—
203C4⋊C4—2541no—
204C4⋊C4—2542no—
205C4⋊C4C22⋊D12711yescharacteristic
206C4⋊C4C22⋊D12711yescharacteristic
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—2541no—
210C22⋊C4C22⋊D12712yes—
211C42—2541no—
212C42C22×D12712yes—
213C23—22no—
214C23—21no—
215C23—42no—
216C23—81no—
217C23—81no—
218C23C2×C22⋊D12711yescharacteristic
219Q8—21no—
220Q8—21no—
221Q8—10162no—
222D4—22no—
223D4—22no—
224D4—21no—
225D4—21no—
226D4—42no—
227D4—42no—
228D4—82no—
229D4—82no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—10161no—
233C2×C4—2541no—
234C2×C4—2541no—
235C2×C4—2542no—
236C2×C4C2×C22⋊D12712yes—
237C2×C4C2×C22⋊D12712yes—
238C2×C4C2×C22⋊D12711yescharacteristic
239C2×C4#1452324712yes—
240C2×C4#1452324712yes—
241C2×C4C23×D12711yescharacteristic
242C8—2541no—
243C8—2541no—
244C8—2542no—
245C22—21no—
246C22—21no—
247C22—41no—
248C22—42no—
249C22—41no—
250C22—42no—
251C22—41no—
252C22—42no—
253C22—84no—
254C22#1686119612yes—
255C22C22×C22⋊D12711yescharacteristic
256C4—22no—
257C4—22no—
258C4—41no—
259C4—5082no—
260C4—5081no—
261C4—5081no—
262C4#1686119612yes—
263C4C2×#1452324711yescharacteristic
264C4C2×#1452324711yescharacteristic
265C2—22no—
266C2—82no—
267C2—81no—
268C2C2×#1686119611yescharacteristic
269C2#2255067911yescharacteristic
270C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#535×C127C2
socleC22×C127C22×D4
hypercenter#535D127
hypocenterC127#3136
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22550504C2semidirect
2. #22550240C2semidirect
3. #22550235C2semidirect
4. #22550230C2semidirect
5. #22550222C2semidirect
6. #22549881C2semidirect
7. #22549877C2semidirect
8. #22549872C2semidirect
9. C2×#16861130C2semidirect
10. #22549832C2semidirect
14. #16861209C22semidirect
16. #16861201C22semidirect
21. #16861133C22semidirect
22. #16861133C22semidirect
23. #16861133C22semidirect
28. #16861130C22semidirect
29. #16861130C22semidirect
30. #16861130C22semidirect
31. #16861120C22semidirect
32. C4×C127⋊C8C22semidirect
136. C127#3136semidirect

Maximal subgroups

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

subgroupindex
#225505042
#225502402
#225502352
#225502302
#225502222
#225498812
#225498772
#225498722
C2×#168611302
#225498322
#535×C1272
#3136127