Finite Group Databasebeta

← #30639109

Subgroups — #30639109

Order 16256.

370 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861235C211yesmaximal, characteristic
2#22550357C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic, complemented
5#22549645C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7#390×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—22no—
10#16861235C414yescomplemented
11C2×#14523236—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861174—21no—
16#16861171—21no—
17C4×C22⋊D127—22no—
18#16861168C2211yescharacteristic
19#16861167—21no—
20C4⋊C4×D127—21no—
21#16861137—22no—
22#16861130—21no—
23#16861130—21no—
24C22×D4×C127C2211yescharacteristic
25C2×C4⋊C4×C127C2211yescharacteristic
26C2×C22⋊C4×C127—22no—
27C2×C42×C127C2211yescharacteristic
28#14523248—21no—
29#14523248—21no—
30#14523248C2×C412yes—
31#14523248D412yes—
32C2×Dic254D411yescharacteristic
33C2×D508D411yescharacteristic
34C2×C22⋊D127—21no—
35C2×C4×D127—22no—
36C2×C4×D127—21no—
37#14523236—24no—
38#14523233—24no—
39#14523232—21no—
40#14523232—21no—
41#14523232—21no—
42#14523232C2×C412yescomplemented
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C4⋊Dic127D411yescharacteristic
46C4⋊Dic127D411yescharacteristic
47Dic127⋊C4—22no—
48C22⋊Dic127—21no—
49(C2×D127)⋊C4—22no—
50C4×Dic127—22no—
51C4×Dic127—21no—
52C24×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127C2×C412yescomplemented
59C22×C4×C127—21no—
60C22×C4×C127D412yes—
61C22×C4×C127C2311yescharacteristic
62C4⋊C4×C127—21no—
63C4⋊C4×C127—21no—
64C22⋊C4×C127—44no—
65C42×C127—21no—
66C42×C127—21no—
67C22×D127—21no—
68Dic254—21no—
69Dic254C22⋊C412yescomplemented
70D508—21no—
71D508C22⋊C412yescomplemented
72C22⋊D127—22no—
73C22⋊D127—22no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C4×D127—22no—
78C4×D127—22no—
79C4×D127—82no—
80C127⋊C8—22no—
81C127⋊C8—41no—
82C23×C127—22no—
83C23×C127—42no—
84C23×C127—42no—
85C23×C127—41no—
86C23×C127—41no—
87C23×C127C2×D411yescharacteristic
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—41no—
91D4×C127—41no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127—42no—
97C2×C4×C127C22×C411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C22×C127—21no—
104C22×C127—21no—
105C22×C127—44no—
106C22×C127—42no—
107C22×C127—82no—
108C22×C127—81no—
109C22×C127—81no—
110C22×C127C4×D411yescharacteristic
111C22×C127#12011yescharacteristic
112C22×C127#12111yescharacteristic
113C2×D127—22no—
114C2×D127—41no—
115C2×D127—41no—
116C4×C127—42no—
117C4×C127—42no—
118C4×C127C2×C22⋊C411yescharacteristic
119C4×C127C4×D411yescharacteristic
120C4×C127#12111yescharacteristic
121C4×C127#12311yescharacteristic
122Dic127—22no—
123Dic127—41no—
124Dic127—41no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—82no—
128C2×C127#38611yescharacteristic
129C2×C127#45311yescharacteristic
130C2×C127#45411yescharacteristic
131D127—42no—
132#1883—1271nomaximal, Sylow
133C127#188311yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1271no—
135#520—1271no—
136C2×#104—1271no—
137#418—1271no—
138#390D12711yescharacteristic, complemented
139#343—1271no—
140#339—1271no—
141C2×#41—1271no—
142C22×D4C2×D12711yescharacteristic
143AGL(1,ℤ/8)—1274no—
144AGL(1,ℤ/8)—2542no—
145C2×SD16—2541no—
146C2×D8—2541no—
147C2×M16—1271no—
148C2×M16—1271no—
149C4×Q8—2541no—
150C4×D4—2541no—
151C4×D4—2542no—
152#117—1271no—
153#117—2541no—
154C2×C4⋊C4—2541no—
155C2×C4⋊C4C2×D12711yescharacteristic
156C2×C22⋊C4—22no—
157C2×C42C2×D12711yescharacteristic
158#104—2542no—
159#102—2541no—
160#102—2541no—
161C24—21no—
162#41—1272no—
163#41—1272no—
164#41—2541no—
165#41—2541no—
166C2×Q8—1271no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—1271no—
173C2×D4—2541no—
174C2×D4C4×D12712yescomplemented
175C22×C4—21no—
176C22×C4—2542no—
177C22×C4—2541no—
178C22×C4C22⋊D12712yes—
179C22×C4C22×D12711yescharacteristic
180SD16—2544no—
181D8—2544no—
182M16—1272no—
183M16—2541no—
184M16—2541no—
185M16—2541no—
186C2×C8—2541no—
187C2×C8—2541no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—1271no—
191C4⋊C4—1271no—
192C4⋊C4—2542no—
193C22⋊C4—44no—
194C22⋊C4—2542no—
195C22⋊C4—2541no—
196C42—21no—
197C42—21no—
198C42—2542no—
199C42—2541no—
200C23—22no—
201C23—42no—
202C23—42no—
203C23—41no—
204C23—41no—
205C23—2541no—
206C23C2×C22⋊D12711yescharacteristic
207Q8—1272no—
208Q8—2541no—
209D4—22no—
210D4—22no—
211D4—41no—
212D4—41no—
213D4—1272no—
214D4—2542no—
215D4—2542no—
216D4—2541no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—10162no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4—2541no—
226C2×C4—2542no—
227C2×C4—2542no—
228C2×C4C2×C4×D12711yescharacteristic
229C2×C4D4×D12711yescharacteristic
230C2×C4D4×D12711yescharacteristic
231C2×C4D4×D12711yescharacteristic
232C2×C4#1452324711yescharacteristic
233C2×C4#1452324811yescharacteristic
234C8—5081no—
235C8—2542no—
236C22—21no—
237C22—21no—
238C22—44no—
239C22—42no—
240C22—82no—
241C22—81no—
242C22—81no—
243C22—5081no—
244C22—5081no—
245C22—2542no—
246C22C4×C22⋊D12711yescharacteristic
247C22#1686117711yescharacteristic
248C22#1686118011yescharacteristic
249C4—42no—
250C4—42no—
251C4—5081no—
252C4—5081no—
253C4—2542no—
254C4C22⋊C4×D12711yescharacteristic
255C4(C22⋊D127)⋊C411yescharacteristic
256C4#1686118211yescharacteristic
257C4#1686119111yescharacteristic
258C2—21no—
259C2—21no—
260C2—82no—
261C2—5082no—
262C2#2254975211yescharacteristic
263C2#2255004311yescharacteristic
264C2#2255004811yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC22×C127#120
prime radicalC2×#16861235C2
cosocleC22×C4×C127C23
hypercenter#390D127
hypocenterC127#1883
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #22550357C2semidirect
3. C2×#16861137C2semidirect
4. #22549881C2semidirect
5. #22549645C2semidirect
6. #22549636C2semidirect
7. #390×C127C2semidirect
10. #16861235C4semidirect
42. #14523232C2×C4semidirect
58. C2×D4×C127C2×C4semidirect
69. Dic254C22⋊C4semidirect
71. D508C22⋊C4semidirect
133. C127#1883semidirect
138. #390D127semidirect
174. C2×D4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225503572
C2×#168611372
#225498812
#225496452
#225496362
#390×C1272
#1883127