Finite Group Databasebeta

← #30641889

Subgroups — #30641889

Order 16256.

576 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16861183C211yesmaximal, characteristic
2#22549776C212yesmaximal, complemented
3#22549768C212yesmaximal, complemented
4#22549759C212yesmaximal, complemented
5#22549754C212yesmaximal, complemented
6#22549752C212yesmaximal, complemented
7#22549750C212yesmaximal, complemented
8#22549736C211yesmaximal, characteristic, complemented
9#381×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×C22⋊D127C2212yes—
12C22×C4×D127C2212yes—
13#16861183C418yescomplemented
14C2×C4⋊Dic127C2212yes—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4C2212yes—
17C2×C22⋊Dic127C2211yescharacteristic
18C2×C22⋊Dic127C2211yescharacteristic
19C2×(C2×D127)⋊C4C2212yes—
20C2×(C2×D127)⋊C4C2212yes—
21C2×(C2×D127)⋊C4C2212yes—
22C2×(C2×D127)⋊C4C2212yes—
23C2×C4×Dic127C2212yes—
24C127⋊#95C2212yes—
25C127⋊#95C2212yes—
26C127⋊#95C2212yes—
27C23×C4×C127C2211yescharacteristic
28C2×C22⋊C4×C127C2212yes—
29#95×C127C2211yescharacteristic
30#95×C127C2211yescharacteristic
31#95×C127C2212yes—
32C23×D127C2312yes—
33C2×D508—42no—
34C2×D508C2×C414yes—
35C2×C22⋊D127—42no—
36C2×C22⋊D127—42no—
37C2×C22⋊D127C2×C418yescomplemented
38C22×Dic127C2312yes—
39C22×Dic127C2312yes—
40C22×Dic127C2312yes—
41C2×C4×D127—42no—
42C2×C4×D127—42no—
43C2×C4×D127—42no—
44C4⋊Dic127—42no—
45C4⋊Dic127C2×C414yescomplemented
46Dic127⋊C4—42no—
47C22⋊Dic127—41no—
48C22⋊Dic127—41no—
49(C2×D127)⋊C4—42no—
50(C2×D127)⋊C4—42no—
51(C2×D127)⋊C4—42no—
52(C2×D127)⋊C4C2×C418yescomplemented
53C4×Dic127—42no—
54C24×C127C2311yescharacteristic
55C22×C4×C127—42no—
56C22×C4×C127C2×C414yescomplemented
57C22×C4×C127C2312yes—
58C22×C4×C127C2312yes—
59C22×C4×C127C2311yescharacteristic
60C22×C4×C127C2311yescharacteristic
61C22⋊C4×C127—42no—
62C22×D127—22no—
63C22×D127—22no—
64C22×D127—22no—
65C22×D127—22no—
66C22×D127—22no—
67C22×D127—22no—
68C22×D127C22×C414yes—
69D508—44no—
70C22⋊D127—48no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127C22×C414yes—
80C4×D127—48no—
81C23×C127—22no—
82C23×C127—21no—
83C23×C127—22no—
84C23×C127—21no—
85C23×C127C22×C412yes—
86C23×C127C2411yescharacteristic
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—44no—
96C2×C4×C127C22×C412yes—
97C2×C4×C127C22×C412yes—
98C22×C127—22no—
99C22×C127—22no—
100C22×C127—22no—
101C22×C127—42no—
102C22×C127—42no—
103C22×C127C23×C411yescharacteristic
104C22×C127H(5,𝔽2)12yes—
105C22×C127H(5,𝔽2)12yes—
106C22×C127H(5,𝔽2)11yescharacteristic
107C22×C1272−1+411yescharacteristic
108C2×D127—24no—
109C2×D127—24no—
110C2×D127—24no—
111C2×D127—42no—
112C2×D127—42no—
113C2×D127—42no—
114C2×D127—42no—
115C4×C127—22no—
116C4×C127—22no—
117C4×C127—42no—
118C4×C127—42no—
119Dic127—24no—
120Dic127—42no—
121Dic127—42no—
122C2×C127—42no—
123C2×C127#51811yescharacteristic
124C2×C127#51811yescharacteristic
125C2×C127#52011yescharacteristic
126C2×C127#56012yes—
127C2×C127#56212yes—
128D127—44no—
129#2377—1271nomaximal, Sylow
130C127#237711yescharacteristic, complemented, Sylow
131C2×#121—1271no—
132#390—1272no—
133#388—1272no—
134#387—1272no—
135#386—1272no—
136#386—1272no—
137#385—1272no—
138#381D12711yescharacteristic, complemented
139#380—1271no—
140C22×D4—1272no—
141C22×D4—1271no—
142C23×C4—1272no—
143C23×C4C2×D12711yescharacteristic
144#121—1278no—
145C2×C4⋊C4—1271no—
146C2×C4⋊C4—1272no—
147C2×C4⋊C4—1272no—
148C2×C22⋊C4—1272no—
149C2×C22⋊C4—1271no—
150C2×C22⋊C4—1271no—
151C2×C22⋊C4—1272no—
152C2×C22⋊C4—1272no—
153C2×C22⋊C4—1272no—
154C2×C22⋊C4C2×D12712yes—
155C2×C42—1272no—
156#95—1272no—
157#95—1272no—
158#95—1272no—
159#95C2×D12711yescharacteristic
160#95C2×D12712yes—
161#95C2×D12711yescharacteristic
162C24—1272no—
163C24C22×D12711yescharacteristic
164C2×D4—1278no—
165C2×D4—1274no—
166C2×D4—5082no—
167C2×D4—5082no—
168C2×D4—5082no—
169C22×C4—42no—
170C22×C4—1272no—
171C22×C4—1272no—
172C22×C4—1272no—
173C22×C4—5082no—
174C22×C4—5082no—
175C22×C4—5082no—
176C22×C4C4×D12714yescomplemented
177C22×C4C22×D12712yes—
178C22×C4C22×D12712yes—
179C22×C4C22×D12711yescharacteristic
180C22×C4C22×D12711yescharacteristic
181C4⋊C4—1274no—
182C4⋊C4—5082no—
183C4⋊C4—5082no—
184C22⋊C4—42no—
185C22⋊C4—1278no—
186C22⋊C4—5082no—
187C22⋊C4—5082no—
188C22⋊C4—5082no—
189C22⋊C4—5081no—
190C22⋊C4—5081no—
191C42—5082no—
192C23—22no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—1274no—
197C23—2542no—
198C23—2542no—
199C23—2542no—
200C23—2542no—
201C23—2542no—
202C23—2542no—
203C23C2×C4×D12712yes—
204C23C23×D12711yescharacteristic
205D4—5084no—
206D4—5088no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—1274no—
217C2×C4—5088no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4—2542no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4—2542no—
225C2×C4—2542no—
226C2×C4C2×C4×D12712yes—
227C2×C4C2×C4×D12712yes—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—42no—
232C22—42no—
233C22—5082no—
234C22—5082no—
235C22—5082no—
236C22—5082no—
237C22—2544no—
238C22—2544no—
239C22—2544no—
240C22C22×C4×D12711yescharacteristic
241C22#1686125412yes—
242C22#1686125512yes—
243C22#1686125511yescharacteristic
244C22#1686125611yescharacteristic
245C4—22no—
246C4—22no—
247C4—42no—
248C4—42no—
249C4—5082no—
250C4—5082no—
251C4—2544no—
252C2—42no—
253C2—5084no—
254C2#2255034911yescharacteristic
255C2#2255035011yescharacteristic
256C2#2255035611yescharacteristic
257C2#2255061612yes—
258C2#2255062312yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#381×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861183C2
hypercenter#381D127
hypocenterC127#2377
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #22549776C2semidirect
3. #22549768C2semidirect
4. #22549759C2semidirect
5. #22549754C2semidirect
6. #22549752C2semidirect
7. #22549750C2semidirect
8. #22549736C2semidirect
9. #381×C127C2semidirect
13. #16861183C4semidirect
37. C2×C22⋊D127C2×C4semidirect
45. C4⋊Dic127C2×C4semidirect
52. (C2×D127)⋊C4C2×C4semidirect
56. C22×C4×C127C2×C4semidirect
130. C127#2377semidirect
138. #381D127semidirect
176. C22×C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
#225497762
#225497682
#225497592
#225497542
#225497522
#225497502
#225497362
#381×C1272
#2377127