Finite Group Databasebeta

← #30637232

Subgroups — #30637232

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
2#22549840C211yesmaximal, characteristic, complemented
3#22549612C212yesmaximal, complemented
4#22549609C212yesmaximal, complemented
5C2×#95×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C411yescharacteristic, complemented
7C2×#14523232—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
17C22⋊(C127⋊C8)—21no—
18C22⋊(C127⋊C8)C2212yes—
19C23×C4×C127—21no—
20C23×C4×C127C2211yescharacteristic
21#95×C127—21no—
22#95×C127C2212yes—
23#14523248—42no—
24C2×Dic254C2×C411yescharacteristic, complemented
25C2×D508C2×C411yescharacteristic, complemented
26C2×C22⋊D127—21no—
27C2×C22⋊D127—21no—
28C2×C22⋊D127—21no—
29C2×C4×D127—21no—
30#14523232—42no—
31C2×C127⋊C8—22no—
32C4⋊Dic127C2×C411yescharacteristic, complemented
33C4⋊Dic127C2×C411yescharacteristic, complemented
34Dic127⋊C4—21no—
35Dic127⋊C4—21no—
36C22⋊Dic127—21no—
37C22⋊Dic127—21no—
38C22⋊Dic127—21no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41C4×Dic127—21no—
42C24×C127D411yescharacteristic
43C22×C4×C127—24no—
44C22×C4×C127—22no—
45C22×C4×C127—42no—
46C22×C4×C127—42no—
47C22×C4×C127—42no—
48C22×C4×C127C2×C411yescharacteristic
49C22×C4×C127C2×C411yescharacteristic
50C22×C4×C127D411yescharacteristic
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C22×D127—21no—
55Dic254—41no—
56D508—41no—
57C22⋊D127—42no—
58C22⋊D127—42no—
59C22⋊D127—41no—
60C22⋊D127—41no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C4×D127—42no—
65C127⋊C8—42no—
66C23×C127—22no—
67C23×C127—22no—
68C23×C127—21no—
69C23×C127—21no—
70C23×C127C22⋊C411yescharacteristic
71C23×C127C22⋊C411yescharacteristic
72C23×C127C2×D411yescharacteristic
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—24no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—48no—
81C2×C4×C127—42no—
82C2×C4×C127—42no—
83C2×C4×C127C22⋊C411yescharacteristic
84C2×C4×C127C22⋊C411yescharacteristic
85C2×C4×C127C22×C411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
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—
107C4×C127—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#407—1271no—
123C2×#95D12711yescharacteristic, complemented
124#328—1272no—
125#327—1272no—
126C2×#41—1271no—
127C23×C4—21no—
128C23×C4C2×D12711yescharacteristic
129C2×M16—2541no—
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—1272no—
140C22⋊C8—2541no—
141#95—21no—
142#95C2×D12712yes—
143C24C22⋊D12711yescharacteristic
144#41—5082no—
145C2×Q8—1271no—
146C2×D4—1271no—
147C2×D4—2541no—
148C2×D4—2541no—
149C2×D4—2541no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—2541no—
156C22×C4C4×D12711yescharacteristic
157C22×C4C4×D12711yescharacteristic
158C22×C4C22⋊D12711yescharacteristic
159C22×C4D50811yescharacteristic
160C22×C4D50811yescharacteristic
161C22×C4C22×D12711yescharacteristic
162M16—5082no—
163C2×C8—2542no—
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×C22⋊D12711yescharacteristic
182Q8—5081no—
183D4—5081no—
184D4—5081no—
185D4—5081no—
186D4—5082no—
187D4—5082no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—48no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—5082no—
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×D50811yescharacteristic
206C8—5082no—
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#1686113712yes—
220C22C2×(C2×D127)⋊C411yescharacteristic
221C22#1686123511yescharacteristic
222C22#1686124011yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—44no—
228C4—5081no—
229C4—5081no—
230C4—5081no—
231C2—21no—
232C2—21no—
233C2—22no—
234C2—22no—
235C2—10161no—
236C2C2×#1686112311yescharacteristic
237C2#2254987511yescharacteristic
238C2C2×#1686113711yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#95×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550387C2
cosocleC22×C4×C127C23
hypercenterC2×#95D127
hypocenterC127#1504
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. #22549840C2semidirect
3. #22549612C2semidirect
4. #22549609C2semidirect
5. C2×#95×C127C2semidirect
6. C2×#14523248C4semidirect
9. #16861188C4semidirect
11. #16861183C4semidirect
16. #16861168C4semidirect
24. C2×Dic254C2×C4semidirect
25. C2×D508C2×C4semidirect
32. C4⋊Dic127C2×C4semidirect
33. C4⋊Dic127C2×C4semidirect
120. C127#1504semidirect
123. C2×#95D127semidirect

Maximal subgroups

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

subgroupindex
#225503872
#225498402
#225496122
#225496092
C2×#95×C1272
#1504127