Finite Group Databasebeta

← #30640397

Subgroups — #30640397

Order 16256.

286 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550443C211yesmaximal, characteristic
2C2×#16861124C211yesmaximal, characteristic, complemented
3#22549670C212yesmaximal, complemented
4#22549668C212yesmaximal, complemented
5C2×#99×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C411yescharacteristic, complemented
7#16861205C411yescharacteristic, complemented
8#16861205C2211yescharacteristic
9#16861196C411yescharacteristic, complemented
10#16861196C2211yescharacteristic
11#16861180—21no—
12#16861180—21no—
13#16861178—21no—
14#16861178—21no—
15#16861168C411yescharacteristic, complemented
16C2×C22⋊Dic127—21no—
17#16861124—21no—
18#16861124C2212yes—
19C22×D4×C127C2211yescharacteristic
20C2×C22⋊C4×C127—21no—
21#99×C127—21no—
22#99×C127C2212yes—
23#14523248—41no—
24#14523248—41no—
25C2×Dic254C2×C411yescharacteristic, complemented
26C2×D508C2×C411yescharacteristic, complemented
27C2×C22⋊D127—21no—
28C2×C22⋊D127—21no—
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C22×Dic127—41no—
33C2×C4×D127—21no—
34C4⋊Dic127—21no—
35C22⋊Dic127—21no—
36C22⋊Dic127—21no—
37C22⋊Dic127—21no—
38C22⋊Dic127—21no—
39C22⋊Dic127—21no—
40C22⋊Dic127—22no—
41C22⋊Dic127—41no—
42C4×Dic127C2×C411yescharacteristic, complemented
43C4×Dic127C2×C411yescharacteristic, complemented
44C24×C127D411yescharacteristic
45C24×C127D411yescharacteristic
46C2×D4×C127—41no—
47C2×D4×C127—41no—
48C2×D4×C127C2×C411yescharacteristic
49C2×D4×C127D411yescharacteristic
50C2×D4×C127D411yescharacteristic
51C2×D4×C127C2311yescharacteristic
52C22×C4×C127—41no—
53C22×C4×C127C2×C411yescharacteristic
54C22⋊C4×C127—22no—
55C22⋊C4×C127—41no—
56C22×D127—21no—
57Dic254—41no—
58D508—41no—
59C22⋊D127—41no—
60C22⋊D127—41no—
61C22⋊D127—81no—
62C22⋊D127—81no—
63C22⋊D127—81no—
64C22⋊D127—81no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—42no—
69C2×Dic127—82no—
70C4×D127—42no—
71C23×C127—21no—
72C23×C127—21no—
73C23×C127—21no—
74C23×C127—21no—
75C23×C127—81no—
76C23×C127—81no—
77C23×C127C22⋊C411yescharacteristic
78C23×C127C22⋊C411yescharacteristic
79C23×C127C22⋊C411yescharacteristic
80C23×C127C2×D411yescharacteristic
81C23×C127C2×D411yescharacteristic
82D4×C127—41no—
83D4×C127—41no—
84D4×C127—41no—
85D4×C127—41no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—42no—
89C2×C4×C127—82no—
90C2×C4×C127C22⋊C411yescharacteristic
91C2×C4×C127C22×C411yescharacteristic
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—21no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—42no—
102C22×C127—42no—
103C22×C127—81no—
104C22×C127—81no—
105C22×C127#9911yescharacteristic
106C22×C127#9911yescharacteristic
107C22×C127C2×C22⋊C411yescharacteristic
108C2×D127—41no—
109C2×D127—41no—
110C2×D127—41no—
111C4×C127—21no—
112C4×C127—21no—
113C4×C127—82no—
114Dic127—41no—
115Dic127—41no—
116Dic127—41no—
117Dic127—82no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127—21no—
121C2×C127—41no—
122C2×C127—41no—
123C2×C127—41no—
124C2×C127—41no—
125C2×C127C2×#9911yescharacteristic
126D127—81no—
127#2127—1271nomaximal, Sylow
128C127#212711yescharacteristic, complemented, Sylow
129#535—1271no—
130C2×#99—1271no—
131C2×#99D12711yescharacteristic, complemented
132#354—1272no—
133#353—1272no—
134C2×#41—1271no—
135C22×D4C2×D12711yescharacteristic
136#127—1271no—
137#127—1271no—
138#124—1271no—
139#124—1271no—
140#121—2541no—
141#121—2541no—
142#120—2541no—
143#120—2541no—
144#117—1271no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—2541no—
147#99—21no—
148#99—1272no—
149#99—2541no—
150#99C2×D12712yes—
151C24C22⋊D12711yescharacteristic
152C24D50811yescharacteristic
153#41—5081no—
154#41—5081no—
155C2×Q8—1271no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—1271no—
159C2×D4—2541no—
160C2×D4—2541no—
161C2×D4—2541no—
162C2×D4—2541no—
163C2×D4—2541no—
164C2×D4C4×D12711yescharacteristic
165C2×D4C22⋊D12711yescharacteristic
166C2×D4D50811yescharacteristic
167C2×D4C22×D12711yescharacteristic
168C22×C4—41no—
169C22×C4—5081no—
170C22×C4—2541no—
171C22×C4C4×D12711yescharacteristic
172C4⋊C4—2541no—
173C22⋊C4—22no—
174C22⋊C4—41no—
175C22⋊C4—5081no—
176C22⋊C4—2542no—
177C22⋊C4—2541no—
178C22⋊C4—2541no—
179C22⋊C4—2541no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C42—1271no—
183C42—1271no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—81no—
189C23—81no—
190C23—2541no—
191C23(C2×D127)⋊C411yescharacteristic
192C23(C2×D127)⋊C411yescharacteristic
193C23(C2×D127)⋊C411yescharacteristic
194C23C2×C22⋊D12711yescharacteristic
195C23C2×D50811yescharacteristic
196Q8—5081no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—10161no—
202D4—10161no—
203D4—10161no—
204D4—10161no—
205D4—5081no—
206D4—5081no—
207D4—5081no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—10162no—
213C2×C4—5082no—
214C2×C4—5082no—
215C2×C4—2541no—
216C2×C4—2541no—
217C2×C4—2541no—
218C2×C4(C2×D127)⋊C411yescharacteristic
219C2×C4C2×C4×D12711yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—5081no—
234C22—5081no—
235C22—5081no—
236C22#1686112311yescharacteristic
237C22#1686112311yescharacteristic
238C22C2×(C2×D127)⋊C411yescharacteristic
239C4—21no—
240C4—21no—
241C4—82no—
242C4—10162no—
243C4—5081no—
244C4—5081no—
245C4—5081no—
246C2—21no—
247C2—21no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—41no—
252C2—41no—
253C2—10161no—
254C2C2×#1686112311yescharacteristic
subgroupG/H
centerC2C2×#16861123
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
FittingC2×#99×C127C2
socleC2×C127C2×#99
prime radical#22550443C2
cosocleC2×D4×C127C23
hypercenterC2×#99D127
hypocenterC127#2127
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#16861124C2semidirect
3. #22549670C2semidirect
4. #22549668C2semidirect
5. C2×#99×C127C2semidirect
6. C2×#14523248C4semidirect
7. #16861205C4semidirect
9. #16861196C4semidirect
15. #16861168C4semidirect
25. C2×Dic254C2×C4semidirect
26. C2×D508C2×C4semidirect
42. C4×Dic127C2×C4semidirect
43. C4×Dic127C2×C4semidirect
128. C127#2127semidirect
131. C2×#99D127semidirect

Maximal subgroups

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

subgroupindex
#225504432
C2×#168611242
#225496702
#225496682
C2×#99×C1272
#2127127