Finite Group Databasebeta

← #30648109

Subgroups — #30648109

Order 16256.

704 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 161 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×#16861239C212yesmaximal, complemented
2#22550447C211yesmaximal, characteristic, complemented
3#22550059C214yesmaximal, complemented
4#22550054C214yesmaximal, complemented
5#22549893C212yesmaximal, complemented
6#22549860C211yesmaximal, characteristic, complemented
7#585×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic, complemented
9#16861239—24no—
10#16861239C2218yescomplemented
11C2×#14523238—22no—
12C2×#14523236—22no—
13C2×#14523232C2212yescomplemented
14#16861207C2212yescomplemented
15#16861196C2212yescomplemented
16#16861190—22no—
17#16861188—22no—
18#16861168C2211yescharacteristic, complemented
19#16861136C2218yescomplemented
20#16861128C2212yescomplemented
21#16861126C2212yescomplemented
222−1+4×C127—22no—
232−1+4×C127C2212yes—
24H(5,𝔽2)×C127—22no—
25H(5,𝔽2)×C127—22no—
26H(5,𝔽2)×C127C2212yes—
27C2×#41×C127—24no—
28C2×#41×C127—22no—
29C2×#41×C127C2212yes—
30C2×#41×C127C2211yescharacteristic
31C2×Dic254—21no—
32C2×Dic254—22no—
33C2×Dic254—22no—
34C2×Dic254C2312yes—
35C22×Dic127—21no—
36#14523238—28no—
37#14523236—28no—
38#14523232—22no—
39#14523232C2314yes—
40C2×C127⋊C8—22no—
41C4⋊Dic127—21no—
42Dic127⋊C4—24no—
43C22⋊Dic127—24no—
44C22⋊Dic127—21no—
45C4×Dic127C2312yes—
46#41×C127—24no—
47#41×C127—28no—
48#41×C127—24no—
49#41×C127—24no—
50#41×C127—24no—
51#41×C127—22no—
52#41×C127—22no—
53#41×C127—24no—
54#41×C127—22no—
55#41×C127—22no—
56#41×C127D414yes—
57#41×C127C2314yes—
58C2×Q8×C127—22no—
59C2×Q8×C127—24no—
60C2×Q8×C127D412yes—
61C2×Q8×C127C2311yescharacteristic
62C2×D4×C127—28no—
63C2×D4×C127—22no—
64C2×D4×C127—24no—
65C2×D4×C127—22no—
66C2×D4×C127—24no—
67C2×D4×C127D412yes—
68C2×D4×C127D412yes—
69C2×D4×C127C2311yescharacteristic
70C22×C4×C127—24no—
71C22×C4×C127—22no—
72C22×C4×C127D412yes—
73C22×C4×C127C2311yescharacteristic
74Dic254—24no—
75Dic254—41no—
76Dic254—41no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—42no—
80C127⋊C8—24no—
81C23×C127—24no—
82C23×C127—22no—
83C23×C127C2×D412yes—
84C23×C127C2×D411yescharacteristic
85Q8×C127—24no—
86Q8×C127—22no—
87Q8×C127—22no—
88Q8×C127C2×D414yes—
89D4×C127—28no—
90D4×C127—24no—
91D4×C127—24no—
92D4×C127—24no—
93D4×C127—24no—
94D4×C127—22no—
95D4×C127—22no—
96D4×C127C2×D414yes—
97C2×C4×C127—24no—
98C2×C4×C127—28no—
99C2×C4×C127—22no—
100C2×C4×C127—24no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—24no—
104C2×C4×C127C2×D412yes—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D412yes—
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2411yescharacteristic
109C22×C127—22no—
110C22×C127—24no—
111C22×C127—41no—
112C22×C127—42no—
113C22×C127—82no—
114C22×C127—84no—
115C22×C127#12012yes—
116C22×C127C22×D411yescharacteristic
117C4×C127—24no—
118C4×C127—22no—
119C4×C127#12011yescharacteristic
120C4×C127#12011yescharacteristic
121C4×C127C22×D412yes—
122Dic127—42no—
123Dic127—42no—
124C2×C127—21no—
125C2×C127—22no—
126C2×C127—42no—
127C2×C127—44no—
128C2×C127C2×#12011yescharacteristic
129#3022—1271nomaximal, Sylow
130C127#302211yescharacteristic, complemented, Sylow
131#585D12711yescharacteristic
132C2×#137—1272no—
133#536—1271no—
134#456—1274no—
135#455—1274no—
136#421—1272no—
137#413—1271no—
1382−1+4—22no—
1392−1+4C2×D12712yes—
140H(5,𝔽2)—22no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D12712yes—
143C2×#41—24no—
144C2×#41—22no—
145C2×#41C2×D12712yes—
146C2×#41C2×D12711yescharacteristic
147C22×Q8—1271no—
148#137—1278no—
149#137—2544no—
150C2×Q16—2542no—
151C2×SD16—2542no—
152C2×M16—1272no—
153C4⋊Q8—1272no—
154#124—1272no—
155#123—2542no—
156C22⋊Q8—2542no—
157#117—1271no—
158#104—1278no—
159#101—1272no—
160#100—1272no—
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41C22⋊D12714yes—
172#41C22×D12714yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—1272no—
176C2×Q8—2542no—
177C2×Q8—2542no—
178C2×Q8—2541no—
179C2×Q8C22⋊D12712yes—
180C2×Q8C22×D12711yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4C22⋊D12712yes—
187C2×D4C22⋊D12712yes—
188C2×D4C22×D12711yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—2541no—
192C22×C4C22⋊D12712yes—
193C22×C4C22×D12711yescharacteristic
194Q16—2548no—
195SD16—2548no—
196M16—1274no—
197M16—2542no—
198C2×C8—2542no—
199C4⋊C4—2544no—
200C4⋊C4—2541no—
201C22⋊C4—2541no—
202C22⋊C4—2544no—
203C42—1272no—
204C23—24no—
205C23—22no—
206C23C2×C22⋊D12712yes—
207C23C2×C22⋊D12711yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—5081no—
212Q8—5081no—
213Q8—2544no—
214Q8C2×C22⋊D12714yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×C22⋊D12714yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—5082no—
231C2×C4—2542no—
232C2×C4—2542no—
233C2×C4C2×C22⋊D12712yes—
234C2×C4C2×C22⋊D12712yes—
235C2×C4C2×C22⋊D12712yes—
236C2×C4C2×C22⋊D12711yescharacteristic
237C2×C4C23×D12711yescharacteristic
238C8—2544no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22#1686117812yes—
246C22C22×C22⋊D12711yescharacteristic
247C4—24no—
248C4—22no—
249C4—5082no—
250C4—5082no—
251C4#1686117811yescharacteristic
252C4#1686117811yescharacteristic
253C4C22×C22⋊D12712yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#1686117811yescharacteristic
subgroupG/H
centerC2C2×#16861178
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#585×C127C2
socleC2×C127C2×#120
hypercenter#585D127
hypocenterC127#3022
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. #22550447C2semidirect
3. #22550059C2semidirect
4. #22550054C2semidirect
5. #22549893C2semidirect
6. #22549860C2semidirect
8. C22×Dic254C22semidirect
10. #16861239C22semidirect
13. C2×#14523232C22semidirect
14. #16861207C22semidirect
15. #16861196C22semidirect
18. #16861168C22semidirect
19. #16861136C22semidirect
20. #16861128C22semidirect
21. #16861126C22semidirect
130. C127#3022semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
#225504472
#225500592
#225500542
#225498932
#225498602
#585×C1272
#3022127