Finite Group Databasebeta

← #30641902

Subgroups — #30641902

Order 16256.

520 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16861198C211yesmaximal, characteristic
2#22549783C211yesmaximal, characteristic, complemented
3#22549767C212yesmaximal, complemented
4#22549765C212yesmaximal, complemented
5#22549754C212yesmaximal
6#22549750C212yesmaximal, complemented
7#22549747C212yesmaximal, complemented
8#22549740C212yesmaximal, complemented
9#391×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×C4×D127C2212yes—
12#16861198C418yescomplemented
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4C2212yes—
15C2×Dic127⋊C4C2212yes—
16C2×(C2×D127)⋊C4C2212yes—
17C2×(C2×D127)⋊C4C2212yes—
18C2×(C2×D127)⋊C4C2212yes—
19C2×(C2×D127)⋊C4C2212yes—
20C2×(C2×D127)⋊C4C2212yes—
21C2×C4×Dic127C2211yescharacteristic
22C2×C4×Dic127C2211yescharacteristic
23C127⋊#95C2212yes—
24C127⋊#95C2212yes—
25C127⋊#95C2212yes—
26C127⋊#95C2212yes—
27C22×Q8×C127C2211yescharacteristic
28C2×C4⋊C4×C127C2211yescharacteristic
29C2×C42×C127C2211yescharacteristic
30#95×C127C2212yes—
31#95×C127C2212yes—
32C23×D127C2312yes—
33C2×D508—41no—
34C2×D508—41no—
35C2×D508C2×C414yes—
36C22×Dic127C2312yes—
37C22×Dic127C2312yes—
38C22×Dic127C2312yes—
39C2×C4×D127—42no—
40C2×C4×D127—42no—
41C2×C4×D127—42no—
42C4⋊Dic127—41no—
43Dic127⋊C4—42no—
44Dic127⋊C4—42no—
45(C2×D127)⋊C4—42no—
46(C2×D127)⋊C4—42no—
47(C2×D127)⋊C4—42no—
48(C2×D127)⋊C4C2×C418yescomplemented
49(C2×D127)⋊C4C2×C418yescomplemented
50C4×Dic127—41no—
51C4×Dic127C2×C414yescomplemented
52C2×Q8×C127—41no—
53C2×Q8×C127—41no—
54C2×Q8×C127C2×C414yescomplemented
55C22×C4×C127C2312yes—
56C22×C4×C127C2312yes—
57C22×C4×C127C2311yescharacteristic
58C22×C4×C127C2311yescharacteristic
59C22×C4×C127C2311yescharacteristic
60C4⋊C4×C127—41no—
61C42×C127—41no—
62C22×D127—22no—
63C22×D127—22no—
64C22×D127—22no—
65C22×D127—22no—
66C22×D127—22no—
67C22×D127—22no—
68C22×D127C22×C414yes—
69D508—44no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127C22×C414yes—
79C4×D127—48no—
80C23×C127C2411yescharacteristic
81Q8×C127—44no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127C22×C412yes—
95C2×C4×C127C22×C412yes—
96C2×C4×C127C22×C412yes—
97C22×C127C23×C411yescharacteristic
98C22×C127H(5,𝔽2)11yescharacteristic
99C22×C127H(5,𝔽2)11yescharacteristic
100C22×C127H(5,𝔽2)12yes—
101C22×C1272−1+412yes—
102C2×D127—24no—
103C2×D127—24no—
104C2×D127—24no—
105C2×D127—42no—
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—42no—
109C4×C127—22no—
110C4×C127—22no—
111C4×C127—22no—
112C4×C127—42no—
113C4×C127—42no—
114Dic127—24no—
115Dic127—42no—
116Dic127—42no—
117C2×C127#51811yescharacteristic
118C2×C127#52012yes—
119C2×C127H(3,𝔽4)11yescharacteristic
120C2×C127#56212yes—
121C2×C127#56311yescharacteristic
122D127—44no—
123#2379—1271nomaximal, Sylow
124C127#237911yescharacteristic, complemented, Sylow
125C2×#124—1271no—
126#391—1271no—
127#391D12711yescharacteristic, complemented
128#388—1272no—
129#388—1272no—
130#386—1272no—
131#385—1272no—
132#385—1272no—
133#382—1272no—
134C22×Q8C2×D12711yescharacteristic
135C22×D4—1271no—
136C23×C4—1272no—
137#124—1278no—
138C2×C4⋊C4—1272no—
139C2×C4⋊C4—1272no—
140C2×C4⋊C4—1271no—
141C2×C4⋊C4C2×D12711yescharacteristic
142C2×C22⋊C4—1272no—
143C2×C22⋊C4—1272no—
144C2×C22⋊C4—1272no—
145C2×C22⋊C4—1272no—
146C2×C22⋊C4—1272no—
147C2×C42—1271no—
148C2×C42—1271no—
149C2×C42C2×D12711yescharacteristic
150#95—1272no—
151#95—1272no—
152#95—1272no—
153#95—1272no—
154#95C2×D12712yes—
155#95C2×D12712yes—
156C24—1272no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C4×D12714yescomplemented
160C2×D4—1274no—
161C2×D4—5081no—
162C2×D4—5081no—
163C22×C4—1272no—
164C22×C4—1272no—
165C22×C4—1272no—
166C22×C4—5082no—
167C22×C4—5082no—
168C22×C4—5082no—
169C22×C4C22×D12712yes—
170C22×C4C22×D12712yes—
171C22×C4C22×D12711yescharacteristic
172C22×C4C22×D12711yescharacteristic
173C22×C4C22×D12711yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—5082no—
176C4⋊C4—5082no—
177C4⋊C4—5081no—
178C22⋊C4—1278no—
179C22⋊C4—1278no—
180C22⋊C4—5082no—
181C22⋊C4—5082no—
182C22⋊C4—5082no—
183C42—41no—
184C42—1274no—
185C42—5081no—
186C23—1274no—
187C23—2542no—
188C23—2542no—
189C23—2542no—
190C23—2542no—
191C23—2542no—
192C23—2542no—
193C23C23×D12711yescharacteristic
194Q8—44no—
195D4—5084no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—1274no—
209C2×C4—5088no—
210C2×C4—2542no—
211C2×C4—2542no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4—2542no—
216C2×C4—2542no—
217C2×C4—2542no—
218C2×C4C2×C4×D12712yes—
219C2×C4C2×C4×D12712yes—
220C2×C4C2×C4×D12712yes—
221C22—5082no—
222C22—5082no—
223C22—5082no—
224C22—5082no—
225C22—2544no—
226C22—2544no—
227C22—2544no—
228C22C22×C4×D12711yescharacteristic
229C22#1686125411yescharacteristic
230C22#1686125411yescharacteristic
231C22#1686125512yes—
232C22#1686125712yes—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—5082no—
239C4—5082no—
240C4—2544no—
241C2—5084no—
242C2#2255034911yescharacteristic
243C2#2255035812yes—
244C2#2255061811yescharacteristic
245C2#2255062212yes—
246C2#2255062511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#391×C127C2
socle, cosocleC23×C127C24
prime radicalC22×D508C22
hypercenter#391D127
hypocenterC127#2379
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #22549783C2semidirect
3. #22549767C2semidirect
4. #22549765C2semidirect
6. #22549750C2semidirect
7. #22549747C2semidirect
8. #22549740C2semidirect
9. #391×C127C2semidirect
12. #16861198C4semidirect
48. (C2×D127)⋊C4C2×C4semidirect
49. (C2×D127)⋊C4C2×C4semidirect
51. C4×Dic127C2×C4semidirect
54. C2×Q8×C127C2×C4semidirect
124. C127#2379semidirect
127. #391D127semidirect
159. C2×Q8C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611982
#225497832
#225497672
#225497652
#225497542
#225497502
#225497472
#225497402
#391×C1272
#2379127