Finite Group Databasebeta

← #30639049

Subgroups — #30639049

Order 16256.

412 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#16861180C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic, complemented
4C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
5#22549639C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127—21no—
9C22×C4×D127C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11#16861180—22no—
12#16861180C414yescomplemented
13#16861167—22no—
14#16861167—22no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C4⋊Dic127C2211yescharacteristic
17C2×C22⋊Dic127—21no—
18C2×(C2×D127)⋊C4—21no—
19C2×C4×Dic127—21no—
20#16861130—22no—
21(C2×D127)⋊C8—22no—
22C22×D4×C127C2211yescharacteristic
23C22×C8×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25#102×C127—22no—
26C23×D127D411yescharacteristic
27C2×C22⋊D127—24no—
28C2×C22⋊D127—42no—
29C2×C22⋊D127—42no—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127D411yescharacteristic
33C2×C4×D127—21no—
34C2×C4×D127—21no—
35C2×C4×D127—21no—
36C2×C4×D127—21no—
37C2×C4×D127C2×C412yes—
38C2×C4×D127D412yes—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C4⋊Dic127—21no—
43C4⋊Dic127—21no—
44C4⋊Dic127C2×C412yes—
45C4⋊Dic127D412yescomplemented
46C22⋊Dic127—24no—
47(C2×D127)⋊C4—24no—
48C4×Dic127—24no—
49C24×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127C2×C412yes—
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58C2×C8×C127—21no—
59C2×C8×C127—21no—
60C2×C8×C127—21no—
61C4⋊C4×C127—21no—
62C4⋊C4×C127D412yescomplemented
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—21no—
66C22×D127—21no—
67C22×D127—21no—
68C22×D127—21no—
69C22×D127C22⋊C412yes—
70C22⋊D127—48no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C2×Dic127C22⋊C412yes—
80C4×D127—22no—
81C4×D127—22no—
82C4×D127—22no—
83C4×D127—22no—
84C127⋊C8—41no—
85C23×C127—22no—
86C23×C127—41no—
87C23×C127—41no—
88C23×C127—41no—
89C23×C127—41no—
90C23×C127—41no—
91C23×C127—41no—
92C23×C127C2×D411yescharacteristic
93D4×C127—22no—
94D4×C127—22no—
95D4×C127—41no—
96D4×C127—41no—
97C2×C4×C127—22no—
98C2×C4×C127—41no—
99C2×C4×C127—41no—
100C2×C4×C127C22×C411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127#4111yescharacteristic
105C2×C4×C127#4111yescharacteristic
106C8×C127—41no—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127—42no—
110C22×C127—81no—
111C22×C127—81no—
112C22×C127—81no—
113C22×C127—81no—
114C22×C127C2×C22⋊C411yescharacteristic
115C22×C127#12011yescharacteristic
116C22×C127#12311yescharacteristic
117C22×C127#13511yescharacteristic
118C22×C127#13511yescharacteristic
119C22×C127AGL(1,ℤ/8)11yescharacteristic
120C22×C127AGL(1,ℤ/8)11yescharacteristic
121C2×D127—22no—
122C2×D127—22no—
123C2×D127—22no—
124C2×D127—41no—
125C2×D127—41no—
126C2×D127—41no—
127C2×D127—41no—
128C4×C127—42no—
129C4×C127C4×D411yescharacteristic
130C4×C127C4×D411yescharacteristic
131C4×C127#12111yescharacteristic
132C4×C127#12111yescharacteristic
133Dic127—22no—
134Dic127—42no—
135Dic127—42no—
136C2×C127—82no—
137C2×C127#38611yescharacteristic
138C2×C127#41611yescharacteristic
139C2×C127#41811yescharacteristic
140C2×C127#44911yescharacteristic
141C2×C127#44911yescharacteristic
142C2×C127#48211yescharacteristic
143C2×C127#48211yescharacteristic
144D127—42no—
145#1875—1271nomaximal, Sylow
146C127#187511yescharacteristic, complemented, Sylow
147C2×#121—1271no—
148C2×#117—1271no—
149C2×#102—1271no—
150C2×#102D12711yescharacteristic, complemented
151C2×C22⋊C8—1271no—
152#340—1271no—
153#340—1271no—
154C22×D4—2541no—
155C22×D4C2×D12711yescharacteristic
156C23×C4—1271no—
157C22×C8—1271no—
158C22×C8C2×D12711yescharacteristic
159#121—1274no—
160#121—2542no—
161#117—2542no—
162#117—2542no—
163C2×C4⋊C4—1271no—
164C2×C4⋊C4—1271no—
165C2×C4⋊C4C2×D12711yescharacteristic
166C2×C22⋊C4—2541no—
167C2×C22⋊C4—2541no—
168C2×C42—2541no—
169#102—22no—
170#102—2542no—
171C22⋊C8—2542no—
172C24—21no—
173C24—1271no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—5082no—
180C2×D4—5082no—
181C2×D4—2544no—
182C2×D4C4×D12712yes—
183C22×C4—21no—
184C22×C4—1271no—
185C22×C4—1272no—
186C22×C4—1272no—
187C22×C4—2541no—
188C22×C4—2541no—
189C22×C4—2541no—
190C22×C4—2541no—
191C22×C4—2541no—
192C22×C4—2541no—
193C22×C4C22×D12711yescharacteristic
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—2541no—
198C2×C8—2541no—
199C2×C8—2541no—
200C4⋊C4—21no—
201C4⋊C4—1272no—
202C4⋊C4—1272no—
203C4⋊C4—2541no—
204C4⋊C4—2541no—
205C4⋊C4C22⋊D12712yescomplemented
206C22⋊C4—2544no—
207C22⋊C4—2544no—
208C42—2544no—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—1272no—
217C23—2541no—
218C23—2541no—
219C23—2541no—
220C23—2541no—
221C23—2541no—
222C23—2541no—
223C23D4×D12711yescharacteristic
224D4—22no—
225D4—22no—
226D4—41no—
227D4—41no—
228D4—5088no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—1272no—
233C2×C4—5081no—
234C2×C4—5081no—
235C2×C4—5081no—
236C2×C4—5081no—
237C2×C4—2542no—
238C2×C4—2542no—
239C2×C4—2541no—
240C2×C4—2541no—
241C2×C4—2542no—
242C2×C4—2542no—
243C2×C4—2542no—
244C2×C4—2542no—
245C2×C4C2×C4×D12711yescharacteristic
246C2×C4D4×D12711yescharacteristic
247C2×C4D4×D12711yescharacteristic
248C2×C4C2×C22⋊D12711yescharacteristic
249C2×C4#1452324711yescharacteristic
250C2×C4#1452324811yescharacteristic
251C8—41no—
252C8—5081no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22—5081no—
261C22—5081no—
262C22—5081no—
263C22—5081no—
264C22—2542no—
265C22—2542no—
266C22—2542no—
267C22C22⋊C4×D12711yescharacteristic
268C22#1686117711yescharacteristic
269C22#1686119111yescharacteristic
270C22#1686122811yescharacteristic
271C22#1686123011yescharacteristic
272C22#1686123311yescharacteristic
273C22#1686123611yescharacteristic
274C4—42no—
275C4—5082no—
276C4—5082no—
277C4—2542no—
278C4(C22⋊D127)⋊C411yescharacteristic
279C4C4×C22⋊D12711yescharacteristic
280C4#1686118011yescharacteristic
281C4#1686118211yescharacteristic
282C2—82no—
283C2—5082no—
284C2#2254975211yescharacteristic
285C2#2254986911yescharacteristic
286C2#2254987911yescharacteristic
287C2#2255001811yescharacteristic
288C2#2255002011yescharacteristic
289C2#2255019811yescharacteristic
290C2#2255019911yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861180C2
cosocleC22×C4×C127C23
hypercenterC2×#102D127
hypocenterC127#1875
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16861167C2semidirect
3. C2×#16861130C2semidirect
4. C2×(C2×D127)⋊C8C2semidirect
5. #22549639C2semidirect
6. #22549638C2semidirect
7. C2×#102×C127C2semidirect
12. #16861180C4semidirect
45. C4⋊Dic127D4semidirect
62. C4⋊C4×C127D4semidirect
146. C127#1875semidirect
150. C2×#102D127semidirect
205. C4⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611802
C2×#168611672
C2×#168611302
C2×(C2×D127)⋊C82
#225496392
#225496382
C2×#102×C1272
#1875127