Finite Group Databasebeta

← #30637236

Subgroups — #30637236

Order 16256.

272 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 63 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#22550421C211yesmaximal, characteristic
2#22549930C211yesmaximal, characteristic, complemented
3#22549721C211yesmaximal, characteristic, complemented
4#22549611C211yesmaximal, characteristic, complemented
5#22549610C211yesmaximal, characteristic, complemented
6#22549609C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9#16861208C411yescharacteristic, complemented
10#16861205C411yescharacteristic, complemented
11#16861195—21no—
12#16861183C2211yescharacteristic
13#16861181—21no—
14D4×Dic127C411yescharacteristic, complemented
15#16861171C411yescharacteristic, complemented
16(C22⋊D127)⋊C4—21no—
17C2×C4×Dic127—21no—
18C2×C4×Dic127C2211yescharacteristic
19Dic127⋊C8—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21#16861119—21no—
22C127⋊#95—21no—
23C127⋊#95C2211yescharacteristic
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27#14523247—42no—
28#14523247—42no—
29C2×Dic254—21no—
30C2×D508C2×C411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C22×Dic127—21no—
35C22×Dic127—21no—
36C22×Dic127D411yescharacteristic
37C22×Dic127D411yescharacteristic
38C2×C4×D127—21no—
39C2×C127⋊C8—21no—
40C4⋊Dic127C2×C411yescharacteristic
41Dic127⋊C4—21no—
42C22⋊Dic127—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45C4×Dic127—24no—
46C4×Dic127C2×C411yescharacteristic
47C4×Dic127C2×C411yescharacteristic
48C4×Dic127D411yescharacteristic
49C4×Dic127D411yescharacteristic
50C2×D4×C127—21no—
51C2×D4×C127C2×C411yescharacteristic
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C4⋊C4×C127C2×C411yescharacteristic
56C22⋊C4×C127—21no—
57C22×D127—21no—
58Dic254—42no—
59D508—41no—
60C22⋊D127—42no—
61C22⋊D127—41no—
62C22⋊D127—41no—
63C22⋊D127—42no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—21no—
69C2×Dic127—42no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127C22⋊C411yescharacteristic
73C2×Dic127C22⋊C411yescharacteristic
74C2×Dic127C22⋊C411yescharacteristic
75C2×Dic127C22⋊C411yescharacteristic
76C4×D127—42no—
77C127⋊C8—41no—
78C23×C127—21no—
79C23×C127C2×D411yescharacteristic
80D4×C127—41no—
81D4×C127—41no—
82D4×C127—41no—
83C2×C4×C127—22no—
84C2×C4×C127—21no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127C22×C411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C8×C127—41no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—41no—
94C22×C127—41no—
95C22×C127—41no—
96C22×C127C2×C22⋊C411yescharacteristic
97C22×C127AGL(1,ℤ/8)11yescharacteristic
98C22×C127AGL(1,ℤ/8)11yescharacteristic
99C2×D127—41no—
100C2×D127—41no—
101C2×D127—41no—
102C4×C127—21no—
103C4×C127—21no—
104C4×C127—42no—
105C4×C127—41no—
106Dic127—21no—
107Dic127—21no—
108Dic127—21no—
109Dic127—42no—
110Dic127—41no—
111Dic127#10412yes—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—81no—
115C2×C127#41011yescharacteristic
116C2×C127#41811yescharacteristic
117C2×C127C2×#10411yescharacteristic
118D127—81no—
119#1505—1271nomaximal, Sylow
120C127#150511yescharacteristic, complemented, Sylow
121#532—1271no—
122#432—1271no—
123#376—1271no—
124#327—1271no—
125#327—1271no—
126#327—1271no—
127#327D12711yescharacteristic, complemented
128C2×#41—2541no—
129C4⋊Q8—1271no—
130#127—1271no—
131#124—2541no—
132#121—1271no—
133#121—2541no—
134#121C2×D12711yescharacteristic
135C4×D4—1271no—
136C4×D4—1271no—
137C4×D4—2541no—
138C2×C42—1271no—
139C2×C42—2541no—
140C4⋊C8—2541no—
141C22⋊C8—1271no—
142C22⋊C8C2×D12711yescharacteristic
143#97—2541no—
144#95—1271no—
145#95—2541no—
146#95C2×D12711yescharacteristic
147#41—5082no—
148#41—5082no—
149C2×Q8—2541no—
150C2×D4—21no—
151C2×D4—1271no—
152C2×D4—2541no—
153C2×D4—2541no—
154C2×D4—2541no—
155C2×D4C4×D12711yescharacteristic
156C22×C4—21no—
157C22×C4—1271no—
158C22×C4—1271no—
159C22×C4—2541no—
160C22×C4—2541no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163C2×C8—21no—
164C2×C8—2541no—
165C4⋊C4—1271no—
166C4⋊C4—2541no—
167C4⋊C4C4×D12711yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—2541no—
170C22⋊C4—2541no—
171C22⋊C4—2541no—
172C42—1271no—
173C42—1271no—
174C42—1271no—
175C42—1271no—
176C42—2544no—
177C23—21no—
178C23—2541no—
179C23D4×D12711yescharacteristic
180Q8—5082no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—5081no—
185D4—5082no—
186D4—5081no—
187D4—5081no—
188D4—5082no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—1271no—
195C2×C4—1271no—
196C2×C4—1271no—
197C2×C4—1271no—
198C2×C4—5082no—
199C2×C4—5081no—
200C2×C4—5081no—
201C2×C4—5082no—
202C2×C4—2542no—
203C2×C4—2542no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4—2541no—
207C2×C4C2×C4×D12711yescharacteristic
208C2×C4D4×D12711yescharacteristic
209C8—41no—
210C8—5081no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—5081no—
217C22—5081no—
218C22—5081no—
219C22C22⋊C4×D12711yescharacteristic
220C22#1686123311yescharacteristic
221C22#1686123611yescharacteristic
222C4—21no—
223C4—21no—
224C4—42no—
225C4—41no—
226C4—1272no—
227C4—5082no—
228C4—5081no—
229C4—2541no—
230C4—2541no—
231C4—2541no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—10161no—
236C2#2254984811yescharacteristic
237C2#2254987911yescharacteristic
238C2#104×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550421C2
cosocleC22×C4×C127C23
hypercenter#327D127
hypocenterC127#1505
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22549930C2semidirect
3. #22549721C2semidirect
4. #22549611C2semidirect
5. #22549610C2semidirect
6. #22549609C2semidirect
7. #327×C127C2semidirect
9. #16861208C4semidirect
10. #16861205C4semidirect
14. D4×Dic127C4semidirect
15. #16861171C4semidirect
120. C127#1505semidirect
127. #327D127semidirect

Maximal subgroups

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

subgroupindex
#225504212
#225499302
#225497212
#225496112
#225496102
#225496092
#327×C1272
#1505127