Finite Group Databasebeta

← #327×D127 #30637205

Subgroups — #327×D127 #30637205

Order 16256.

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

normal subgroups only

This group has 388 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#121×D127C211yesmaximal, characteristic
2C22⋊C8×D127C211yesmaximal, characteristic, complemented
3#95×D127C211yesmaximal, characteristic, complemented
4#22549611C211yesmaximal, characteristic, complemented
5#22549610C211yesmaximal, characteristic, complemented
6#22549609C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C2×D4×D127—21no—
9C2×D4×D127C411yescharacteristic, complemented
10C22×C4×D127—21no—
11C22×C4×D127C2211yescharacteristic
12C2×C8×D127—21no—
13#16861184C411yescharacteristic, complemented
14#16861183C2211yescharacteristic
15#16861182—21no—
16#16861181—21no—
17#16861180C411yescharacteristic, complemented
18C4⋊C4×D127C411yescharacteristic, complemented
19C22⋊C4×D127—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic, complemented
21(C2×D127)⋊C8—21no—
22C127⋊#95—21no—
23C127⋊#95C2211yescharacteristic, complemented
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic, complemented
26#95×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C23×D127D411yescharacteristic
29C2×D508—21no—
30C2×D508C2×C411yescharacteristic, complemented
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34D4×D127—42no—
35D4×D127—42no—
36D4×D127—42no—
37C22×Dic127—21no—
38C22×Dic127D411yescharacteristic
39C2×C4×D127—24no—
40C2×C4×D127—21no—
41C2×C4×D127—42no—
42C2×C4×D127—42no—
43C2×C4×D127—42no—
44C2×C4×D127C2×C411yescharacteristic
45C2×C4×D127C2×C411yescharacteristic
46C2×C4×D127D411yescharacteristic
47C2×C4×D127D411yescharacteristic
48C2×C127⋊C8—21no—
49C8×D127—42no—
50C4⋊Dic127C2×C411yescharacteristic, complemented
51Dic127⋊C4—21no—
52C22⋊Dic127—21no—
53(C2×D127)⋊C4—21no—
54(C2×D127)⋊C4—21no—
55C2×D4×C127—21no—
56C2×D4×C127C2×C411yescharacteristic, complemented
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59C2×C8×C127—21no—
60C4⋊C4×C127C2×C411yescharacteristic, complemented
61C22⋊C4×C127—21no—
62C22×D127—22no—
63C22×D127—22no—
64C22×D127—21no—
65C22×D127—42no—
66C22×D127—42no—
67C22×D127—42no—
68C22×D127C22⋊C411yescharacteristic
69C22×D127C22⋊C411yescharacteristic
70C22×D127D812yes—
71C22×D127SD1612yes—
72D508—41no—
73D508—42no—
74C22⋊D127—42no—
75C22⋊D127—41no—
76C22⋊D127—41no—
77C22⋊D127—42no—
78C2×Dic127—22no—
79C2×Dic127—21no—
80C2×Dic127—41no—
81C2×Dic127—41no—
82C2×Dic127—41no—
83C2×Dic127C22⋊C411yescharacteristic
84C2×Dic127C22⋊C411yescharacteristic
85C4×D127—22no—
86C4×D127—22no—
87C4×D127—41no—
88C4×D127—41no—
89C4×D127—48no—
90C4×D127—42no—
91C127⋊C8—41no—
92C23×C127—21no—
93C23×C127C2×D411yescharacteristic
94D4×C127—41no—
95D4×C127—41no—
96D4×C127—41no—
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127C22×C411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C8×C127—41no—
105C22×C127—21no—
106C22×C127—21no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127—41no—
110C22×C127C2×C22⋊C411yescharacteristic
111C22×C127C2×D811yescharacteristic
112C22×C127C2×SD1611yescharacteristic
113C2×D127—21no—
114C2×D127—21no—
115C2×D127—24no—
116C2×D127—24no—
117C2×D127—41no—
118C2×D127—41no—
119C2×D127—41no—
120C2×D127—84no—
121C2×D127#9912yes—
122C2×D127#10212yes—
123C2×D127#10212yes—
124C2×D127#10412yes—
125C4×C127—21no—
126C4×C127—21no—
127C4×C127—42no—
128C4×C127—41no—
129Dic127—21no—
130Dic127—21no—
131Dic127—42no—
132Dic127—41no—
133C2×C127—21no—
134C2×C127—21no—
135C2×C127—81no—
136C2×C127C2×#9911yescharacteristic
137C2×C127C2×#10211yescharacteristic
138C2×C127C2×#10411yescharacteristic
139D127—22no—
140D127—81no—
141D127#32714yesdirect product factor
142C2×#327—1271nomaximal, Sylow
143C127C2×#32711yescharacteristic, complemented, Sylow
144C2×#121—1271no—
145C2×C22⋊C8—1271no—
146C2×#95—1271no—
147#327—1271no—
148#327—1271no—
149#327—1271no—
150#327D12711yescharacteristic, direct product factor
151C22×D4—1271no—
152C22×D4—2541no—
153C23×C4—1271no—
154C23×C4—2541no—
155C22×C8—2541no—
156#121—1271no—
157#121—1271no—
158#121—1271no—
159#121—2541no—
160#121—2541no—
161#121C2×D12711yescharacteristic
162C2×C4⋊C4—1271no—
163C2×C22⋊C4—2541no—
164C22⋊C8—1271no—
165C22⋊C8—2541no—
166C22⋊C8C2×D12711yescharacteristic, complemented
167#95—1271no—
168#95—2541no—
169#95C2×D12711yescharacteristic, complemented
170C24—1271no—
171C24—2541no—
172C2×D4—21no—
173C2×D4—1271no—
174C2×D4—5082no—
175C2×D4—5082no—
176C2×D4—5082no—
177C2×D4—2541no—
178C2×D4—2541no—
179C2×D4—2541no—
180C2×D4—2541no—
181C2×D4C4×D12711yescharacteristic, complemented
182C22×C4—21no—
183C22×C4—1271no—
184C22×C4—1271no—
185C22×C4—1271no—
186C22×C4—1271no—
187C22×C4—1271no—
188C22×C4—5082no—
189C22×C4—5082no—
190C22×C4—5082no—
191C22×C4—2541no—
192C22×C4—2544no—
193C22×C4—2541no—
194C22×C4C22×D12711yescharacteristic
195C2×C8—21no—
196C2×C8—5082no—
197C2×C8—2541no—
198C4⋊C4—1271no—
199C4⋊C4—2541no—
200C4⋊C4C4×D12711yescharacteristic, complemented
201C22⋊C4—21no—
202C22⋊C4—2541no—
203C22⋊C4—2541no—
204C22⋊C4—2541no—
205C23—21no—
206C23—1271no—
207C23—1272no—
208C23—1272no—
209C23—1271no—
210C23—5082no—
211C23—5082no—
212C23—5082no—
213C23—2542no—
214C23—2542no—
215C23—2541no—
216C23D4×D12711yescharacteristic
217D4—41no—
218D4—41no—
219D4—41no—
220D4—5081no—
221D4—5082no—
222D4—5082no—
223D4—5081no—
224D4—5081no—
225D4—5082no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—1271no—
232C2×C4—1271no—
233C2×C4—5081no—
234C2×C4—5081no—
235C2×C4—5082no—
236C2×C4—5088no—
237C2×C4—5081no—
238C2×C4—5081no—
239C2×C4—5081no—
240C2×C4—2542no—
241C2×C4—2542no—
242C2×C4—2542no—
243C2×C4—2541no—
244C2×C4C2×C4×D12711yescharacteristic
245C2×C4D4×D12711yescharacteristic
246C8—41no—
247C8—5081no—
248C22—21no—
249C22—21no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—1272no—
254C22—1272no—
255C22—1272no—
256C22—1272no—
257C22—10164no—
258C22—5081no—
259C22—5081no—
260C22—5081no—
261C22—2544no—
262C22—2544no—
263C22—2541no—
264C22—2541no—
265C22C22⋊C4×D12711yescharacteristic
266C22D8×D12711yescharacteristic
267C22SD16×D12711yescharacteristic
268C4—21no—
269C4—21no—
270C4—42no—
271C4—41no—
272C4—5082no—
273C4—5081no—
274C4—2541no—
275C4—2541no—
276C2—21no—
277C2—21no—
278C2—81no—
279C2—1274no—
280C2—10161no—
281C2—2542no—
282C2#99×D12711yescharacteristic
283C2#102×D12711yescharacteristic
284C2#104×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127C2×C22⋊C4
prime radical#121×D127C2
cosocleC22×C4×C127C23
hypercenter#327D127
hypocenterC127C2×#327
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. C22⋊C8×D127C2semidirect
3. #95×D127C2semidirect
4. #22549611C2semidirect
5. #22549610C2semidirect
6. #22549609C2semidirect
7. #327×C127C2semidirect
9. C2×D4×D127C4semidirect
13. #16861184C4semidirect
17. #16861180C4semidirect
18. C4⋊C4×D127C4semidirect
20. C22⋊(C127⋊C8)C22semidirect
23. C127⋊#95C22semidirect
25. C22⋊C8×C127C22semidirect
26. #95×C127C22semidirect
30. C2×D508C2×C4semidirect
50. C4⋊Dic127C2×C4semidirect
56. C2×D4×C127C2×C4semidirect
60. C4⋊C4×C127C2×C4semidirect
141. D127#327direct (N × H)
143. C127C2×#327semidirect
150. #327D127direct (N × H)
166. C22⋊C8C2×D127semidirect
169. #95C2×D127semidirect
181. C2×D4C4×D127semidirect
200. C4⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
#121×D1272
C22⋊C8×D1272
#95×D1272
#225496112
#225496102
#225496092
#327×C1272
C2×#327127