Finite Group Databasebeta

← #30648251

Subgroups — #30648251

Order 16256.

490 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C22×#14523235C211yesmaximal, characteristic, complemented
3#22550445C211yesmaximal, characteristic
4#22550098C214yesmaximal, complemented
5#22550076C214yesmaximal, complemented
6#22549903C211yesmaximal, characteristic, complemented
7#22549880C211yesmaximal, characteristic, complemented
8#22549870C211yesmaximal, characteristic, complemented
9#519×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×D508C2211yescharacteristic
12#16861240—24no—
13C2×#14523238C2212yescomplemented
14C2×#14523235—24no—
15C2×#14523235C2214yes—
16C2×#14523235C2212yescomplemented
17C2×#14523232C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19#16861206C2212yes—
20#16861197C2212yes—
21#16861184—22no—
22#16861179—22no—
23C4⋊(C127⋊C8)C2214yescomplemented
24#16861132C2212yescomplemented
25#16861129C2214yes—
26#16861129C2212yescomplemented
27C22×Q8×C127C2211yescharacteristic
28C4×Q8×C127—22no—
29C4×Q8×C127C2214yescomplemented
30#117×C127—22no—
31#117×C127C2211yescharacteristic
32#117×C127C2211yescharacteristic
33C23×D127—21no—
34#14523248—22no—
35#14523248—22no—
36C2×Dic254C2311yescharacteristic
37C2×D508—22no—
38C2×D508—21no—
39C2×D508—24no—
40C2×D508—22no—
41C2×D508C2312yes—
42C2×D508C2311yescharacteristic
43C2×C22⋊D127—21no—
44C2×C4×D127—21no—
45#14523238—24no—
46#14523235—28no—
47#14523235—24no—
48#14523232—22no—
49C2×C127⋊C8—22no—
50C2×C127⋊C8C2312yes—
51C2×C127⋊C8C2312yes—
52(C2×D127)⋊C4—24no—
53C2×Q8×C127—21no—
54C2×Q8×C127—22no—
55C2×Q8×C127D414yescomplemented
56C2×Q8×C127C2312yes—
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59C4⋊C4×C127—24no—
60C4⋊C4×C127D412yes—
61C4⋊C4×C127C2312yes—
62C22⋊C4×C127—21no—
63C22⋊C4×C127D412yes—
64C42×C127—24no—
65C42×C127—21no—
66C42×C127C2312yes—
67C22×D127—22no—
68C22×D127—21no—
69C22×D127—42no—
70C22×D127—82no—
71Dic254—21no—
72Dic254—21no—
73D508—22no—
74D508—21no—
75D508—22no—
76D508—21no—
77D508—42no—
78D508—84no—
79C22⋊D127—42no—
80C2×Dic127—21no—
81C4×D127—42no—
82C127⋊C8—22no—
83C127⋊C8—22no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—24no—
86Q8×C127—22no—
87Q8×C127C2×D414yes—
88C2×C4×C127—24no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127C2×D414yes—
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D412yes—
94C2×C4×C127#4112yes—
95C2×C4×C127C2411yescharacteristic
96C22×C127—22no—
97C22×C127#12112yes—
98C22×C127C22×D411yescharacteristic
99C2×D127—41no—
100C2×D127—42no—
101C2×D127—42no—
102C2×D127—44no—
103C2×D127—84no—
104C4×C127—24no—
105C4×C127—24no—
106C4×C127—42no—
107C4×C127#12112yes—
108C4×C127C22×D411yescharacteristic
109C4×C127C2×#4111yescharacteristic
110Dic127—41no—
111C2×C127—22no—
112C2×C127C2×#12111yescharacteristic
113C2×C127#57712yes—
114D127—81no—
115D127—82no—
116#3043—1271nomaximal, Sylow
117C127#304311yescharacteristic, complemented, Sylow
118C2×#137—1271no—
119C22×SD16—1271no—
120#535—1271no—
121#519D12711yescharacteristic, complemented
122#464—1274no—
123#460—1274no—
124#424—1271no—
125#418—1271no—
126#416—1271no—
127C2×#41—1271no—
128C22×Q8C2×D12711yescharacteristic
129C22×D4—1271no—
130#137—2544no—
131C2×Q16—1272no—
132C2×SD16—1274no—
133C2×SD16—1272no—
134C2×SD16—2544no—
135C2×M16—1271no—
136C22×C8—1271no—
137#127—1272no—
138#124—1272no—
139#121—2542no—
140#120—2542no—
141C4×Q8—22no—
142C4×Q8C2×D12714yescomplemented
143#117—22no—
144#117C2×D12711yescharacteristic
145#117C2×D12711yescharacteristic
146C4⋊C8—1274no—
147#103—1272no—
148#102—1272no—
149#102—1274no—
150C24—2541no—
151#41—2542no—
152#41—2542no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—1271no—
156C2×Q8D50814yescomplemented
157C2×Q8C22×D12712yes—
158C2×D4—1272no—
159C2×D4—1271no—
160C2×D4—2541no—
161C2×D4—2544no—
162C2×D4—2541no—
163C2×D4—2542no—
164C2×D4—2542no—
165C22×C4—21no—
166C22×C4—2541no—
167C22×C4C22×D12711yescharacteristic
168Q16—2544no—
169SD16—2548no—
170SD16—2544no—
171M16—2542no—
172C2×C8—1272no—
173C2×C8—1272no—
174C2×C8—2542no—
175C4⋊C4—24no—
176C4⋊C4C22⋊D12712yes—
177C4⋊C4C22×D12712yes—
178C22⋊C4—21no—
179C22⋊C4—2544no—
180C22⋊C4C22⋊D12712yes—
181C42—24no—
182C42—21no—
183C42C22×D12712yes—
184C23—10162no—
185C23—5082no—
186C23—2542no—
187C23—2541no—
188C23C2×C22⋊D12711yescharacteristic
189Q8—22no—
190Q8—24no—
191Q8—2541no—
192Q8—2541no—
193Q8C2×D50814yes—
194D4—10164no—
195D4—5082no—
196D4—5082no—
197D4—2542no—
198D4—2541no—
199D4—2542no—
200D4—2541no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—5082no—
205C2×C4—2541no—
206C2×C4C2×C22⋊D12714yes—
207C2×C4C2×C22⋊D12711yescharacteristic
208C2×C4C2×D50812yes—
209C2×C4#1452324812yes—
210C2×C4C23×D12711yescharacteristic
211C8—2542no—
212C8—2542no—
213C22—22no—
214C22—10164no—
215C22—5081no—
216C22—5082no—
217C22—5082no—
218C22—5084no—
219C22#1686118312yes—
220C22C22×C22⋊D12711yescharacteristic
221C4—24no—
222C4—24no—
223C4—42no—
224C4—5081no—
225C4#1686118312yes—
226C4C22×D50811yescharacteristic
227C4C2×#1452324811yescharacteristic
228C2—22no—
229C2—10162no—
230C2—10161no—
231C2C2×#1686118311yescharacteristic
232C2#2255068412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#519×C127C2
socleC22×C127C22×D4
prime radical#22550445C2
hypercenter#519D127
hypocenterC127#3043
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
2. C22×#14523235C2semidirect
4. #22550098C2semidirect
5. #22550076C2semidirect
6. #22549903C2semidirect
7. #22549880C2semidirect
8. #22549870C2semidirect
9. #519×C127C2semidirect
13. C2×#14523238C22semidirect
16. C2×#14523235C22semidirect
23. C4⋊(C127⋊C8)C22semidirect
24. #16861132C22semidirect
26. #16861129C22semidirect
29. C4×Q8×C127C22semidirect
55. C2×Q8×C127D4semidirect
117. C127#3043semidirect
121. #519D127semidirect
142. C4×Q8C2×D127semidirect
156. C2×Q8D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C22×#145232352
#225504452
#225500982
#225500762
#225499032
#225498802
#225498702
#519×C1272
#3043127