Finite Group Databasebeta

← #30641492

Subgroups — #30641492

Order 16256.

596 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 177 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#22549765C212yesmaximal, complemented
4#22549761C212yesmaximal, complemented
5#22549754C212yesmaximal, complemented
6#22549752C212yesmaximal, complemented
7C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
8#22549723C212yesmaximal, complemented
9#381×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×C22⋊D127C2212yes—
12C22×C4×D127C2212yes—
13#16861183C418yescomplemented
14C2×C4⋊Dic127C2211yescharacteristic
15C2×Dic127⋊C4C2212yes—
16C2×C22⋊Dic127C2212yes—
17C2×(C2×D127)⋊C4C2212yes—
18C2×(C2×D127)⋊C4C2212yes—
19C2×(C2×D127)⋊C4C2212yes—
20C2×(C2×D127)⋊C4C2212yes—
21C2×C4×Dic127C2212yes—
22C2×C4×Dic127C2211yescharacteristic
23C2×C4×Dic127C2211yescharacteristic
24C127⋊#95C2212yes—
25C127⋊#95C2212yes—
26C127⋊#95C2212yes—
27C23×C4×C127C2211yescharacteristic
28C2×C22⋊C4×C127C2211yescharacteristic
29C2×C22⋊C4×C127C2211yescharacteristic
30#95×C127C2212yes—
31#95×C127C2212yes—
32C23×D127C2312yes—
33C2×D508—41no—
34C2×D508—41no—
35C2×D508C2×C414yes—
36C2×C22⋊D127—42no—
37C2×C22⋊D127—42no—
38C2×C22⋊D127C2×C418yescomplemented
39C22×Dic127C2312yes—
40C22×Dic127C2312yes—
41C22×Dic127C2312yes—
42C2×C4×D127—24no—
43C2×C4×D127—42no—
44C2×C4×D127—42no—
45C4⋊Dic127C2×C414yescomplemented
46Dic127⋊C4—42no—
47C22⋊Dic127—24no—
48(C2×D127)⋊C4—24no—
49(C2×D127)⋊C4—42no—
50(C2×D127)⋊C4—42no—
51(C2×D127)⋊C4C2×C418yescomplemented
52C4×Dic127—41no—
53C4×Dic127—41no—
54C4×Dic127—42no—
55C24×C127C2311yescharacteristic
56C22×C4×C127—41no—
57C22×C4×C127—41no—
58C22×C4×C127C2×C414yescomplemented
59C22×C4×C127C2312yes—
60C22×C4×C127C2312yes—
61C22×C4×C127C2311yescharacteristic
62C22×C4×C127C2311yescharacteristic
63C22⋊C4×C127—41no—
64C22⋊C4×C127—41no—
65C22×D127—22no—
66C22×D127—22no—
67C22×D127—22no—
68C22×D127—22no—
69C22×D127—22no—
70C22×D127—22no—
71C22×D127C22×C414yes—
72D508—44no—
73C22⋊D127—48no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127C22×C414yes—
82C2×Dic127#4114yes—
83C4×D127—48no—
84C23×C127—22no—
85C23×C127—21no—
86C23×C127—21no—
87C23×C127—22no—
88C23×C127C22×C412yes—
89C23×C127C2411yescharacteristic
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—44no—
101C2×C4×C127C22×C412yes—
102C2×C4×C127C22×C412yes—
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—22no—
106C22×C127—42no—
107C22×C127—42no—
108C22×C127C23×C411yescharacteristic
109C22×C127C2×#4112yes—
110C22×C127H(5,𝔽2)11yescharacteristic
111C22×C127H(5,𝔽2)11yescharacteristic
112C22×C127H(5,𝔽2)12yes—
113C2×D127—24no—
114C2×D127—24no—
115C2×D127—24no—
116C2×D127—42no—
117C2×D127—42no—
118C2×D127—42no—
119C2×D127—42no—
120C4×C127—22no—
121C4×C127—22no—
122C4×C127—42no—
123C4×C127—42no—
124Dic127—24no—
125Dic127—24no—
126Dic127—42no—
127C2×C127—42no—
128C2×C127C4×#4111yescharacteristic
129C2×C127#51811yescharacteristic
130C2×C127#51811yescharacteristic
131C2×C127#53812yes—
132C2×C127#54012yes—
133D127—44no—
134#2333—1271nomaximal, Sylow
135C127#233311yescharacteristic, complemented, Sylow
136C2×#121—1271no—
137#390—1272no—
138#388—1272no—
139#387—1272no—
140#386—1272no—
141#386—1272no—
142#381D12711yescharacteristic, complemented
143C4×C22⋊C4—1271no—
144C4×C22⋊C4—1272no—
145C22×D4—1272no—
146C22×D4—1271no—
147C23×C4—1272no—
148C23×C4C2×D12711yescharacteristic
149#121—1278no—
150C2×C4⋊C4—1271no—
151C2×C4⋊C4—1272no—
152C2×C22⋊C4—1272no—
153C2×C22⋊C4—1272no—
154C2×C22⋊C4—1272no—
155C2×C22⋊C4—1272no—
156C2×C22⋊C4—1272no—
157C2×C22⋊C4C2×D12711yescharacteristic
158C2×C22⋊C4C2×D12711yescharacteristic
159C2×C42—1271no—
160C2×C42—1272no—
161C2×C42—1271no—
162#95—1272no—
163#95—1272no—
164#95—1272no—
165#95C2×D12712yes—
166#95C2×D12712yes—
167C24—1272no—
168C24C22×D12711yescharacteristic
169C2×D4—1278no—
170C2×D4—1274no—
171C2×D4—5081no—
172C2×D4—5081no—
173C2×D4—5082no—
174C2×D4—5082no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—1272no—
178C22×C4—1272no—
179C22×C4—1272no—
180C22×C4—5082no—
181C22×C4—5082no—
182C22×C4—2544no—
183C22×C4C4×D12714yescomplemented
184C22×C4C22×D12712yes—
185C22×C4C22×D12712yes—
186C22×C4C22×D12711yescharacteristic
187C22×C4C22×D12711yescharacteristic
188C4⋊C4—1274no—
189C4⋊C4—5082no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—1278no—
193C22⋊C4—5082no—
194C22⋊C4—5082no—
195C22⋊C4—2544no—
196C22⋊C4—2544no—
197C42—5081no—
198C42—5081no—
199C42—5082no—
200C23—22no—
201C23—21no—
202C23—21no—
203C23—22no—
204C23—1274no—
205C23—2542no—
206C23—2542no—
207C23—2542no—
208C23—2542no—
209C23—2542no—
210C23—2542no—
211C23C2×C4×D12712yes—
212C23C23×D12711yescharacteristic
213D4—5084no—
214D4—5088no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—44no—
226C2×C4—1274no—
227C2×C4—1274no—
228C2×C4—5088no—
229C2×C4—2542no—
230C2×C4—2542no—
231C2×C4—2542no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4—2542no—
235C2×C4—2542no—
236C2×C4C2×C4×D12712yes—
237C2×C4C2×C4×D12712yes—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—42no—
242C22—42no—
243C22—5082no—
244C22—5082no—
245C22—5082no—
246C22—5082no—
247C22—2544no—
248C22—2544no—
249C22—2544no—
250C22C22×C4×D12711yescharacteristic
251C22#41×D12712yes—
252C22#1686125411yescharacteristic
253C22#1686125411yescharacteristic
254C22#1686125512yes—
255C4—22no—
256C4—22no—
257C4—42no—
258C4—42no—
259C4—5082no—
260C4—2544no—
261C4—2544no—
262C2—42no—
263C2—5084no—
264C2#2255034411yescharacteristic
265C2#2255034911yescharacteristic
266C2#2255035011yescharacteristic
267C2#2255046212yes—
268C2#2255047812yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#381×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861183C2
hypercenter#381D127
hypocenterC127#2333
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #22549776C2semidirect
3. #22549765C2semidirect
4. #22549761C2semidirect
5. #22549754C2semidirect
6. #22549752C2semidirect
7. C4×C22⋊Dic127C2semidirect
8. #22549723C2semidirect
9. #381×C127C2semidirect
13. #16861183C4semidirect
38. C2×C22⋊D127C2×C4semidirect
45. C4⋊Dic127C2×C4semidirect
51. (C2×D127)⋊C4C2×C4semidirect
58. C22×C4×C127C2×C4semidirect
135. C127#2333semidirect
142. #381D127semidirect
183. 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
#225497652
#225497612
#225497542
#225497522
C4×C22⋊Dic1272
#225497232
#381×C1272
#2333127