Finite Group Databasebeta

← #30641863

Subgroups — #30641863

Order 16256.

552 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 185 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×#16861190C211yesmaximal, characteristic
2#22549779C211yesmaximal, characteristic
3#22549764C212yesmaximal, complemented
4#22549764C212yesmaximal, complemented
5#22549763C212yesmaximal, complemented
6#22549727C212yesmaximal, complemented
7C22⋊C4×Dic127C212yesmaximal, complemented
8C4×C22⋊Dic127C212yesmaximal, complemented
9#386×C127C211yesmaximal, characteristic
10C23×Dic127C2211yescharacteristic
11#16861190C418yescomplemented
12C2×C4⋊Dic127C2211yescharacteristic
13C2×Dic127⋊C4C2212yes—
14C2×Dic127⋊C4C2212yes—
15C2×C22⋊Dic127C2212yes—
16C2×C22⋊Dic127C2212yes—
17C2×C22⋊Dic127C2212yes—
18C2×C22⋊Dic127C2211yescharacteristic
19C2×C4×Dic127C2212yes—
20C2×C4×Dic127C2212yes—
21C2×C4×Dic127C2212yes—
22C2×C4×Dic127C2211yescharacteristic
23C127⋊#95C2212yes—
24C127⋊#95C2212yes—
25C127⋊#95C2212yes—
26C127⋊#95C2212yes—
27C22×D4×C127C2211yescharacteristic
28C23×C4×C127C2211yescharacteristic
29C2×C22⋊C4×C127C2212yes—
30C2×C22⋊C4×C127C2211yescharacteristic
31#95×C127C2212yes—
32C22×Dic127—41no—
33C22×Dic127—41no—
34C22×Dic127C2×C414yescomplemented
35C22×Dic127C2312yes—
36C22×Dic127C2312yes—
37C22×Dic127C2312yes—
38C22×Dic127C2312yes—
39C4⋊Dic127—41no—
40Dic127⋊C4—24no—
41Dic127⋊C4C2×C418yescomplemented
42C22⋊Dic127—24no—
43C22⋊Dic127—24no—
44C22⋊Dic127C2×C418yes—
45C22⋊Dic127C2×C414yescomplemented
46C4×Dic127—24no—
47C4×Dic127—42no—
48C4×Dic127—41no—
49C4×Dic127—42no—
50C24×C127C2311yescharacteristic
51C24×C127C2311yescharacteristic
52C2×D4×C127—41no—
53C2×D4×C127—41no—
54C2×D4×C127C2×C414yes—
55C22×C4×C127—24no—
56C22×C4×C127—41no—
57C22×C4×C127C2312yes—
58C22×C4×C127C2311yescharacteristic
59C22×C4×C127C2312yes—
60C22⋊C4×C127—24no—
61C22⋊C4×C127—41no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—44no—
72C2×Dic127C22×C414yes—
73C2×Dic127C22×C414yes—
74C2×Dic127#4114yes—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127—22no—
78C23×C127—22no—
79C23×C127—21no—
80C23×C127—21no—
81C23×C127—21no—
82C23×C127—21no—
83C23×C127C22×C412yes—
84C23×C127C22×C412yes—
85C23×C127C2411yescharacteristic
86D4×C127—44no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—44no—
95C2×C4×C127C22×C412yes—
96C2×C4×C127#4114yes—
97C22×C127—24no—
98C22×C127—24no—
99C22×C127—22no—
100C22×C127—22no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—42no—
104C22×C127—42no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127C23×C411yescharacteristic
108C22×C127C2×#4112yes—
109C22×C127C2×#4112yes—
110C22×C127H(5,𝔽2)11yescharacteristic
111C22×C127H(5,𝔽2)11yescharacteristic
112C4×C127—22no—
113C4×C127—24no—
114C4×C127—42no—
115Dic127—24no—
116Dic127—24no—
117Dic127—24no—
118Dic127—42no—
119C2×C127—42no—
120C2×C127—42no—
121C2×C127C4×#4112yes—
122C2×C127#51811yescharacteristic
123C2×C127#55111yescharacteristic
124C2×C127#55312yes—
125C2×C127#55911yescharacteristic
126#2374—1271nomaximal, Sylow
127C127#237411yescharacteristic, complemented, Sylow
128C2×#123—1271no—
129#390—1271no—
130#388—1272no—
131#388—1272no—
132#387—1272no—
133#386D12711yescharacteristic
134C4×C4⋊C4—1272no—
135C4×C22⋊C4—1272no—
136C4×C22⋊C4—1272no—
137C22×D4C2×D12711yescharacteristic
138C23×C4—1271no—
139C23×C4C2×D12711yescharacteristic
140#123—1278no—
141C2×C4⋊C4—1272no—
142C2×C4⋊C4—1271no—
143C2×C4⋊C4—1272no—
144C2×C22⋊C4—1272no—
145C2×C22⋊C4—1271no—
146C2×C22⋊C4—1272no—
147C2×C22⋊C4—1272no—
148C2×C22⋊C4C2×D12712yes—
149C2×C22⋊C4C2×D12711yescharacteristic
150C2×C42—1272no—
151C2×C42—1272no—
152C2×C42—1271no—
153C2×C42—1272no—
154#95—1272no—
155#95—1272no—
156#95—1272no—
157#95—1272no—
158#95C2×D12712yes—
159C24C22×D12711yescharacteristic
160C24C22×D12711yescharacteristic
161C2×D4—41no—
162C2×D4—41no—
163C2×D4C4×D12714yes—
164C22×C4—24no—
165C22×C4—41no—
166C22×C4—1274no—
167C22×C4—1272no—
168C22×C4—1272no—
169C22×C4—1272no—
170C22×C4—1272no—
171C22×C4—5081no—
172C22×C4—5081no—
173C22×C4C22×D12712yes—
174C22×C4C22×D12711yescharacteristic
175C22×C4C22×D12712yes—
176C4⋊C4—1278no—
177C4⋊C4—5081no—
178C4⋊C4—2544no—
179C22⋊C4—24no—
180C22⋊C4—41no—
181C22⋊C4—1278no—
182C22⋊C4—1274no—
183C22⋊C4—2544no—
184C22⋊C4—2544no—
185C42—5082no—
186C42—5081no—
187C42—5082no—
188C42—2544no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23C2×C4×D12712yes—
198C23C2×C4×D12712yes—
199C23C23×D12711yescharacteristic
200D4—44no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—44no—
209C2×C4—1274no—
210C2×C4—1274no—
211C2×C4—1274no—
212C2×C4—5084no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4—2542no—
216C2×C4—2542no—
217C2×C4—2542no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4—2542no—
222C2×C4C2×C4×D12712yes—
223C2×C4#1452324714yes—
224C22—24no—
225C22—24no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—41no—
233C22—41no—
234C22C22×C4×D12711yescharacteristic
235C22#41×D12712yes—
236C22C2×#1452324712yes—
237C22#1686125411yescharacteristic
238C22#1686125411yescharacteristic
239C4—24no—
240C4—22no—
241C4—42no—
242C4—5082no—
243C4—2544no—
244C4—2544no—
245C4—2544no—
246C2—42no—
247C2—42no—
248C2C4×#1452324712yes—
249C2#2255034911yescharacteristic
250C2#2255055711yescharacteristic
251C2#2255057312yes—
252C2#2255061011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#386×C127C2
socle, cosocleC23×C127C24
prime radicalC22×D4×C127C22
hypercenter#386D127
hypocenterC127#2374
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #22549764C2semidirect
4. #22549764C2semidirect
5. #22549763C2semidirect
6. #22549727C2semidirect
7. C22⋊C4×Dic127C2semidirect
8. C4×C22⋊Dic127C2semidirect
11. #16861190C4semidirect
34. C22×Dic127C2×C4semidirect
41. Dic127⋊C4C2×C4semidirect
45. C22⋊Dic127C2×C4semidirect
127. C127#2374semidirect

Maximal subgroups

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

subgroupindex
C2×#168611902
#225497792
#225497642
#225497632
#225497272
C22⋊C4×Dic1272
C4×C22⋊Dic1272
#386×C1272
#2374127