Finite Group Databasebeta

← #30648375

Subgroups — #30648375

Order 16256.

524 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14523233C211yesmaximal, characteristic, complemented
2C2×#16861216C211yesmaximal, characteristic, complemented
3#22550440C211yesmaximal, characteristic, complemented
4#22550127C214yesmaximal, complemented
5#22550110C214yesmaximal, complemented
6#22549912C211yesmaximal, characteristic, complemented
7#22549881C211yesmaximal, characteristic, complemented
8#22549880C211yesmaximal, characteristic, complemented
9#535×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C2×#14523233—24no—
12C2×#14523233C2212yes—
13C2×#14523233C2212yes—
14#16861216—24no—
15C2×#14523232C2211yescharacteristic
16C2×#14523232C2212yescomplemented
17C22×C127⋊C8C2211yescharacteristic
18C22×C127⋊C8C2212yescomplemented
19#16861205—21no—
20#16861198—21no—
21#16861183C2214yescomplemented
22#16861177—22no—
23#16861168C2211yescharacteristic
24#16861142C2212yescomplemented
25#16861140C2212yescomplemented
26#16861130C2214yescomplemented
27#16861129C2214yescomplemented
28C2×#41×C127C2211yescharacteristic
29C22×D4×C127C2211yescharacteristic
30#127×C127—21no—
31#124×C127—21no—
32#121×C127C2214yescomplemented
33#120×C127—22no—
34#117×C127C2211yescharacteristic
35C23×D127—21no—
36C2×D508—22no—
37C2×D508—21no—
38C2×D508—22no—
39C2×D508C2312yes—
40C2×C22⋊D127—24no—
41#14523233—24no—
42#14523233—24no—
43#14523232—24no—
44#14523232—22no—
45C2×C127⋊C8—24no—
46C2×C127⋊C8D414yescomplemented
47C2×C127⋊C8C2312yes—
48C2×C127⋊C8C2311yescharacteristic
49C2×C127⋊C8C2311yescharacteristic
50C4⋊Dic127C2312yes—
51C22⋊Dic127—21no—
52(C2×D127)⋊C4—24no—
53C4×Dic127—21no—
54C24×C127—21no—
55#41×C127—22no—
56#41×C127—22no—
57C2×Q8×C127D411yescharacteristic
58C2×D4×C127—24no—
59C2×D4×C127—22no—
60C2×D4×C127—21no—
61C2×D4×C127—22no—
62C2×D4×C127D411yescharacteristic
63C2×D4×C127D412yes—
64C2×D4×C127C2312yes—
65C22×C4×C127C2312yes—
66C22×C4×C127C2311yescharacteristic
67C4⋊C4×C127C2312yes—
68C22⋊C4×C127—21no—
69C22⋊C4×C127—24no—
70C42×C127—21no—
71C22×D127—22no—
72C22×D127—42no—
73C22×D127—82no—
74D508—22no—
75D508—22no—
76D508—42no—
77C22⋊D127—84no—
78C2×Dic127—22no—
79C127⋊C8—22no—
80C127⋊C8C2×D412yes—
81C127⋊C8C2×D412yes—
82C23×C127—22no—
83C23×C127—42no—
84C23×C127—82no—
85C23×C127C2×D412yes—
86C23×C127C2×D411yescharacteristic
87Q8×C127—22no—
88D4×C127—24no—
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—42no—
93D4×C127—84no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127C2×D412yes—
98C2×C4×C127C2×D412yes—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4112yes—
101C2×C4×C127C2411yescharacteristic
102C22×C127—22no—
103C22×C127—22no—
104C22×C127—24no—
105C22×C127—42no—
106C22×C127—44no—
107C22×C127—84no—
108C22×C127#12112yes—
109C22×C127C22×D411yescharacteristic
110C2×D127—42no—
111C2×D127—44no—
112C2×D127—84no—
113C4×C127—22no—
114C4×C127—42no—
115C4×C127#12112yes—
116C4×C127C22×D411yescharacteristic
117C4×C127C2×#4111yescharacteristic
118Dic127—42no—
119C2×C127—22no—
120C2×C127—42no—
121C2×C127—82no—
122C2×C127C2×#12111yescharacteristic
123C2×C127#57612yes—
124D127—82no—
125#3058—1271nomaximal, Sylow
126C127#305811yescharacteristic, complemented, Sylow
127C22×D8—1271no—
128C2×#131—1271no—
129#535—1271no—
130#535D12711yescharacteristic, complemented
131#469—1274no—
132#466—1274no—
133#427—1271no—
134#418—1271no—
135#418—1271no—
136C2×#41C2×D12711yescharacteristic
137C22×D4—1271no—
138C22×D4C2×D12711yescharacteristic
139C2×D8—1272no—
140C2×D8—1272no—
141C2×D8—2544no—
142#131—2544no—
143C2×M16—1271no—
144C2×M16—1272no—
145C22×C8—1271no—
146C22×C8—1272no—
147#127—21no—
148#127—2541no—
149#124—21no—
150#124—2541no—
151#121—1274no—
152#121C2×D12714yescomplemented
153#120—22no—
154#120—2542no—
155#117—1271no—
156#117C2×D12711yescharacteristic
157#107—1272no—
158#106—1272no—
159#102—1274no—
160#102—1274no—
161C24—21no—
162C24—2541no—
163#41—22no—
164#41—22no—
165C2×Q8C22⋊D12711yescharacteristic
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—1272no—
171C2×D4—2544no—
172C2×D4—2542no—
173C2×D4—2541no—
174C2×D4—2542no—
175C2×D4C22⋊D12712yes—
176C2×D4C22⋊D12711yescharacteristic
177C2×D4C22×D12712yes—
178C22×C4C22×D12712yes—
179C22×C4C22×D12711yescharacteristic
180D8—2544no—
181D8—2544no—
182M16—2544no—
183M16—2542no—
184C2×C8—1274no—
185C2×C8—1271no—
186C2×C8—1271no—
187C2×C8—1272no—
188C2×C8—2544no—
189C4⋊C4—1272no—
190C4⋊C4C22×D12712yes—
191C22⋊C4—21no—
192C22⋊C4—24no—
193C22⋊C4—2541no—
194C22⋊C4—2544no—
195C42—21no—
196C42—2541no—
197C23—22no—
198C23—42no—
199C23—82no—
200C23—10162no—
201C23—5082no—
202C23—2542no—
203C23C2×C22⋊D12712yes—
204C23C2×C22⋊D12711yescharacteristic
205Q8—22no—
206D4—24no—
207D4—22no—
208D4—22no—
209D4—22no—
210D4—42no—
211D4—84no—
212D4—10164no—
213D4—5082no—
214D4—2542no—
215D4—2542no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—2542no—
220C2×C4D4×D12712yes—
221C2×C4C2×C22⋊D12712yes—
222C2×C4C2×C22⋊D12711yescharacteristic
223C2×C4#1452324712yes—
224C2×C4C23×D12711yescharacteristic
225C8—1272no—
226C8—1272no—
227C8—2542no—
228C22—22no—
229C22—22no—
230C22—24no—
231C22—42no—
232C22—44no—
233C22—84no—
234C22—10164no—
235C22—5082no—
236C22—5084no—
237C22#1686118112yes—
238C22C22×C22⋊D12711yescharacteristic
239C4—22no—
240C4—42no—
241C4—5082no—
242C4#1686118112yes—
243C4C2×D4×D12711yescharacteristic
244C4C2×#1452324711yescharacteristic
245C2—22no—
246C2—42no—
247C2—82no—
248C2—10162no—
249C2C2×#1686118111yescharacteristic
250C2#2255067912yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#535×C127C2
socleC22×C127C22×D4
hypercenter#535D127
hypocenterC127#3058
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C22×#14523233C2semidirect
2. C2×#16861216C2semidirect
3. #22550440C2semidirect
4. #22550127C2semidirect
5. #22550110C2semidirect
6. #22549912C2semidirect
7. #22549881C2semidirect
8. #22549880C2semidirect
9. #535×C127C2semidirect
16. C2×#14523232C22semidirect
18. C22×C127⋊C8C22semidirect
21. #16861183C22semidirect
24. #16861142C22semidirect
25. #16861140C22semidirect
26. #16861130C22semidirect
27. #16861129C22semidirect
32. #121×C127C22semidirect
46. C2×C127⋊C8D4semidirect
126. C127#3058semidirect
130. #535D127semidirect
152. #121C2×D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×#168612162
#225504402
#225501272
#225501102
#225499122
#225498812
#225498802
#535×C1272
#3058127