Finite Group Databasebeta

← #30637717

Subgroups — #30637717

Order 16256.

280 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550434C211yesmaximal, characteristic
2#22550190C211yesmaximal, characteristic, complemented
3#22550187C211yesmaximal, characteristic, complemented
4#22549803C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C2×D4×D127—21no—
9#16861194—21no—
10#16861191—21no—
11#16861183C2211yescharacteristic
12#16861181—21no—
13#16861179—21no—
14#16861177—21no—
15C2×C4⋊Dic127C2211yescharacteristic, complemented
16C2×(C2×D127)⋊C4—21no—
17C2×(C2×D127)⋊C4C2211yescharacteristic
18#16861143—21no—
19#16861142—21no—
20#16861131—21no—
21#16861130—21no—
22C22⋊(C127⋊C8)C2211yescharacteristic, complemented
23C127⋊#95—21no—
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic, complemented
26#95×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C23×D127D411yescharacteristic
29C2×D508—21no—
30C2×D508D411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34D4×D127—42no—
35D4×D127—42no—
36C22×Dic127—21no—
37C22×Dic127D411yescharacteristic
38C2×C4×D127—21no—
39C2×C127⋊C8—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42C4⋊Dic127D411yescharacteristic
43Dic127⋊C4—21no—
44C22⋊Dic127—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—42no—
50C2×D4×C127—21no—
51C2×D4×C127D411yescharacteristic
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C4⋊C4×C127D411yescharacteristic
56C22⋊C4×C127—21no—
57C22×D127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127—42no—
61C22×D127—41no—
62C22×D127—41no—
63C22×D127—81no—
64C22×D127—81no—
65C22×D127D812yes—
66D508—41no—
67D508—42no—
68C22⋊D127—42no—
69C22⋊D127—41no—
70C22⋊D127—41no—
71C22⋊D127—81no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C4×D127—42no—
79C127⋊C8—41no—
80C23×C127—21no—
81C23×C127C2×D411yescharacteristic
82D4×C127—41no—
83D4×C127—41no—
84D4×C127—41no—
85C2×C4×C127—21no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C8×C127—41no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127#12011yescharacteristic
99C22×C127C2×D811yescharacteristic
100C22×C127AGL(1,ℤ/8)11yescharacteristic
101C2×D127—22no—
102C2×D127—41no—
103C2×D127—41no—
104C2×D127—42no—
105C2×D127—41no—
106C2×D127—41no—
107C2×D127—41no—
108C2×D127—81no—
109C2×D127—84no—
110C4×C127—21no—
111C4×C127—21no—
112C4×C127—41no—
113C4×C127—81no—
114Dic127—41no—
115Dic127—41no—
116Dic127—81no—
117C2×C127—21no—
118C2×C127—21no—
119C2×C127—81no—
120C2×C127#44711yescharacteristic
121C2×C127#45511yescharacteristic
122C2×C127#45811yescharacteristic
123D127—42no—
124D127—81no—
125#1607—1271nomaximal, Sylow
126C127#160711yescharacteristic, complemented, Sylow
127#534—1271no—
128#480—1271no—
129#480—1271no—
130#397—1271no—
131#327—1271no—
132#327D12711yescharacteristic, complemented
133#323—1271no—
134C22×D4—2541no—
135#123—2541no—
136#123—2541no—
137#121—1271no—
138#121—2541no—
139#121C2×D12711yescharacteristic
140#120—2541no—
141#120—2541no—
142C2×C4⋊C4—1271no—
143C2×C22⋊C4—1271no—
144C2×C22⋊C4—2541no—
145#107—2541no—
146#107—2541no—
147#102—2541no—
148#102—2541no—
149C22⋊C8—1271no—
150C22⋊C8C2×D12711yescharacteristic, complemented
151#95—2541no—
152#95C2×D12711yescharacteristic, complemented
153C24—1271no—
154C24—2541no—
155C2×D4—21no—
156C2×D4—1271no—
157C2×D4—5082no—
158C2×D4—5082no—
159C2×D4—2541no—
160C2×D4—2541no—
161C2×D4—2541no—
162C2×D4—2541no—
163C2×D4D50811yescharacteristic
164C22×C4—21no—
165C22×C4—1271no—
166C22×C4—2541no—
167C22×C4—2541no—
168C22×C4C22×D12711yescharacteristic
169C2×C8—21no—
170C2×C8—2541no—
171C4⋊C4—1271no—
172C4⋊C4—2541no—
173C4⋊C4—2541no—
174C4⋊C4—2541no—
175C4⋊C4D50811yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—5082no—
178C22⋊C4—2541no—
179C22⋊C4—2541no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C22⋊C4—2541no—
183C23—21no—
184C23—1272no—
185C23—10161no—
186C23—10161no—
187C23—5081no—
188C23—5081no—
189C23—5082no—
190C23—2541no—
191C23—2541no—
192C23—2541no—
193C23D4×D12711yescharacteristic
194D4—41no—
195D4—41no—
196D4—41no—
197D4—10161no—
198D4—5081no—
199D4—5082no—
200D4—5082no—
201D4—5081no—
202D4—5081no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—5082no—
209C2×C4—5081no—
210C2×C4—5081no—
211C2×C4—5081no—
212C2×C4—5081no—
213C2×C4—2541no—
214C2×C4—2541no—
215C2×C4D4×D12711yescharacteristic
216C2×C4C2×D50811yescharacteristic
217C8—41no—
218C8—5081no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—10161no—
225C22—10164no—
226C22—5082no—
227C22—5081no—
228C22—5081no—
229C22—5081no—
230C22—5081no—
231C22—5081no—
232C22—2542no—
233C22#1686117911yescharacteristic
234C22D8×D12711yescharacteristic
235C22#1686123611yescharacteristic
236C4—21no—
237C4—21no—
238C4—41no—
239C4—81no—
240C4—10161no—
241C4—5081no—
242C4—5081no—
243C2—21no—
244C2—21no—
245C2—81no—
246C2—10161no—
247C2—5082no—
248C2#2255001111yescharacteristic
249C2#2255005511yescharacteristic
250C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
prime radical#22550434C2
hypercenter#327D127
hypocenterC127#1607
LCSG ▷ C22×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. #22550190C2semidirect
3. #22550187C2semidirect
4. #22549803C2semidirect
5. #22549609C2semidirect
6. #22549599C2semidirect
7. #327×C127C2semidirect
15. C2×C4⋊Dic127C22semidirect
22. C22⋊(C127⋊C8)C22semidirect
25. C22⋊C8×C127C22semidirect
26. #95×C127C22semidirect
126. C127#1607semidirect
132. #327D127semidirect
150. C22⋊C8C2×D127semidirect
152. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504342
#225501902
#225501872
#225498032
#225496092
#225495992
#327×C1272
#1607127