Finite Group Databasebeta

← #30637235

Subgroups — #30637235

Order 16256.

334 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 77 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#22550387C211yesmaximal, characteristic
2C2×C127⋊#95C211yesmaximal, characteristic, complemented
3#22549614C212yesmaximal, complemented
4#22549611C212yesmaximal, complemented
5#407×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C411yescharacteristic, complemented
7C23×Dic127—21no—
8#16861192—21no—
9#16861188C411yescharacteristic, complemented
10#16861188C2211yescharacteristic
11#16861183C411yescharacteristic, complemented
12#16861183C2211yescharacteristic
13#16861178—21no—
14C4×C22⋊D127—21no—
15C4×C22⋊D127—21no—
16#16861168C411yescharacteristic, complemented
17C127⋊#95—21no—
18C127⋊#95C2212yes—
19C23×C4×C127C2211yescharacteristic
20C2×M16×C127—21no—
21C22⋊C8×C127—21no—
22C22⋊C8×C127C2212yes—
23#14523248—42no—
24C2×Dic254C2×C411yescharacteristic, complemented
25C2×D508C2×C411yescharacteristic, complemented
26C2×C22⋊D127—21no—
27C2×C22⋊D127—21no—
28C2×C22⋊D127—21no—
29C22×Dic127—24no—
30C22×Dic127—22no—
31C22×Dic127—42no—
32C22×Dic127—42no—
33C2×C4×D127—21no—
34C4⋊Dic127C2×C411yescharacteristic, complemented
35C4⋊Dic127C2×C411yescharacteristic, complemented
36Dic127⋊C4—21no—
37Dic127⋊C4—21no—
38C22⋊Dic127—21no—
39C22⋊Dic127—21no—
40C22⋊Dic127—21no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43C4×Dic127—21no—
44C24×C127D411yescharacteristic
45C22×C4×C127—42no—
46C22×C4×C127C2×C411yescharacteristic
47C22×C4×C127C2×C411yescharacteristic
48C22×C4×C127D411yescharacteristic
49C22×C4×C127D411yescharacteristic
50C22×C4×C127D411yescharacteristic
51C22×C4×C127C2311yescharacteristic
52M16×C127—42no—
53C2×C8×C127—22no—
54C22×D127—21no—
55Dic254—41no—
56D508—41no—
57C22⋊D127—42no—
58C22⋊D127—42no—
59C22⋊D127—41no—
60C22⋊D127—41no—
61C2×Dic127—24no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—48no—
66C2×Dic127—42no—
67C2×Dic127—42no—
68C4×D127—42no—
69C23×C127—22no—
70C23×C127—22no—
71C23×C127—21no—
72C23×C127—21no—
73C23×C127C22⋊C411yescharacteristic
74C23×C127C22⋊C411yescharacteristic
75C23×C127C2×D411yescharacteristic
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127C22⋊C411yescharacteristic
83C2×C4×C127C22⋊C411yescharacteristic
84C2×C4×C127C22×C411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C8×C127—42no—
87C22×C127—24no—
88C22×C127—24no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127#9912yes—
96C22×C127#10412yes—
97C22×C127C2×C22⋊C411yescharacteristic
98C22×C127AGL(1,ℤ/8)11yescharacteristic
99C22×C127#13711yescharacteristic
100C2×D127—41no—
101C2×D127—41no—
102C2×D127—41no—
103C4×C127—21no—
104C4×C127—21no—
105C4×C127—21no—
106C4×C127—21no—
107Dic127—44no—
108Dic127—41no—
109Dic127—41no—
110Dic127—41no—
111C2×C127—21no—
112C2×C127—22no—
113C2×C127—22no—
114C2×C127—21no—
115C2×C127C2×#9911yescharacteristic
116C2×C127#41711yescharacteristic
117C2×C127C2×#10411yescharacteristic
118D127—81no—
119#1504—1271nomaximal, Sylow
120C127#150411yescharacteristic, complemented, Sylow
121#525—1271no—
122#407D12711yescharacteristic, complemented
123C2×#95—1271no—
124#328—1272no—
125#327—1272no—
126C2×#41—1271no—
127C23×C4—2541no—
128C23×C4C2×D12711yescharacteristic
129C2×M16—21no—
130#123—2541no—
131C22⋊Q8—1271no—
132C22⋊Q8—1271no—
133#121—1271no—
134#121—1271no—
135#120—2541no—
136C4×D4—2541no—
137C4×D4—2541no—
138#117—1271no—
139C22⋊C8—21no—
140C22⋊C8C2×D12712yes—
141#95—1272no—
142#95—2541no—
143C24D50811yescharacteristic
144#41—5082no—
145C2×Q8—1271no—
146C2×D4—1271no—
147C2×D4—2541no—
148C2×D4—2541no—
149C2×D4—2541no—
150C22×C4—42no—
151C22×C4—5082no—
152C22×C4—5082no—
153C22×C4—2544no—
154C22×C4—2542no—
155C22×C4—2541no—
156C22×C4C4×D12711yescharacteristic
157C22×C4C4×D12711yescharacteristic
158C22×C4C22⋊D12711yescharacteristic
159C22×C4C22⋊D12711yescharacteristic
160C22×C4D50811yescharacteristic
161C22×C4C22×D12711yescharacteristic
162M16—42no—
163C2×C8—22no—
164C4⋊C4—1271no—
165C4⋊C4—1271no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C22⋊C4—2541no—
169C22⋊C4—2541no—
170C22⋊C4—2541no—
171C22⋊C4—2541no—
172C22⋊C4—2541no—
173C42—2541no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23—2541no—
179C23(C2×D127)⋊C411yescharacteristic
180C23(C2×D127)⋊C411yescharacteristic
181C23C2×D50811yescharacteristic
182Q8—5081no—
183D4—5081no—
184D4—5081no—
185D4—5081no—
186D4—5082no—
187D4—5082no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—5082no—
195C2×C4—5088no—
196C2×C4—5082no—
197C2×C4—5082no—
198C2×C4—2544no—
199C2×C4—2541no—
200C2×C4—2541no—
201C2×C4—2541no—
202C2×C4(C2×D127)⋊C411yescharacteristic
203C2×C4(C2×D127)⋊C411yescharacteristic
204C2×C4C2×C4×D12711yescharacteristic
205C2×C4C2×C22⋊D12711yescharacteristic
206C8—42no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—5081no—
216C22—5081no—
217C22—5081no—
218C22#1686112312yes—
219C22#1686113512yes—
220C22C2×(C2×D127)⋊C411yescharacteristic
221C22#1686123711yescharacteristic
222C22#1686124211yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—5084no—
228C4—5081no—
229C4—5081no—
230C4—5081no—
231C2—21no—
232C2—21no—
233C2—22no—
234C2—22no—
235C2—10161no—
236C2C2×#1686112311yescharacteristic
237C2#2254987811yescharacteristic
238C2C2×#1686113511yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550387C2
cosocleC22×C4×C127C23
hypercenter#407D127
hypocenterC127#1504
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×C127⋊#95C2semidirect
3. #22549614C2semidirect
4. #22549611C2semidirect
5. #407×C127C2semidirect
6. C2×#14523248C4semidirect
9. #16861188C4semidirect
11. #16861183C4semidirect
16. #16861168C4semidirect
24. C2×Dic254C2×C4semidirect
25. C2×D508C2×C4semidirect
34. C4⋊Dic127C2×C4semidirect
35. C4⋊Dic127C2×C4semidirect
120. C127#1504semidirect
122. #407D127semidirect

Maximal subgroups

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

subgroupindex
#225503872
C2×C127⋊#952
#225496142
#225496112
#407×C1272
#1504127