Finite Group Databasebeta

← #30647055

Subgroups — #30647055

Order 16256.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16861193C211yesmaximal, characteristic
2C22×C4×Dic127C211yesmaximal, characteristic, complemented
3#22549766C212yesmaximal, complemented
4#22549765C212yesmaximal, complemented
5#22549763C212yesmaximal, complemented
6#22549762C212yesmaximal, complemented
7#22549755C211yesmaximal, characteristic
8#22549753C211yesmaximal, characteristic
9#22549748C211yesmaximal, characteristic, complemented
10#22549737C211yesmaximal, characteristic, complemented
11#385×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127C2211yescharacteristic
13C23×Dic127C2211yescharacteristic
14C23×Dic127C2211yescharacteristic
15#16861193—24no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×C4⋊Dic127C2211yescharacteristic
18C2×Dic127⋊C4C2212yes—
19C2×C22⋊Dic127C2211yescharacteristic
20C2×(C2×D127)⋊C4C2211yescharacteristic
21C2×(C2×D127)⋊C4C2212yes—
22C2×(C2×D127)⋊C4C2212yes—
23C2×(C2×D127)⋊C4C2211yescharacteristic
24C2×C4×Dic127—24no—
25C2×C4×Dic127C2212yes—
26C2×C4×Dic127C2212yes—
27C127⋊#95C2212yes—
28C127⋊#95C2212yes—
29C127⋊#95C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C23×C4×C127C2211yescharacteristic
33C2×C4⋊C4×C127C2212yes—
34C2×C22⋊C4×C127C2211yescharacteristic
35C2×C22⋊C4×C127C2211yescharacteristic
36#95×C127C2212yes—
37C23×D127C2311yescharacteristic
38C2×C22⋊D127—22no—
39C2×C22⋊D127—22no—
40C2×C22⋊D127—22no—
41C22×Dic127—24no—
42C22×Dic127—22no—
43C22×Dic127—22no—
44C22×Dic127—22no—
45C22×Dic127D414yes—
46C22×Dic127C2311yescharacteristic
47C22×Dic127C2312yes—
48C22×Dic127C2311yescharacteristic
49C22×Dic127C2311yescharacteristic
50C22×Dic127C2312yes—
51C4⋊Dic127—22no—
52C4⋊Dic127—22no—
53Dic127⋊C4—24no—
54C22⋊Dic127—22no—
55(C2×D127)⋊C4—22no—
56(C2×D127)⋊C4—22no—
57(C2×D127)⋊C4—24no—
58(C2×D127)⋊C4—24no—
59C4×Dic127—24no—
60C4×Dic127—24no—
61C24×C127C2311yescharacteristic
62C22×C4×C127—22no—
63C22×C4×C127—24no—
64C22×C4×C127C2311yescharacteristic
65C22×C4×C127C2312yes—
66C22×C4×C127C2311yescharacteristic
67C22×C4×C127C2312yes—
68C4⋊C4×C127—24no—
69C22⋊C4×C127—22no—
70C22⋊C4×C127—22no—
71C22×D127—21no—
72C22×D127—22no—
73C22×D127—22no—
74C22×D127—21no—
75C22×D127—21no—
76C22⋊D127—44no—
77C2×Dic127—28no—
78C2×Dic127—28no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C2×Dic127—21no—
85C2×Dic127—21no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127—21no—
90C2×Dic127—21no—
91C2×Dic127C2×D412yes—
92C2×Dic127C2×D412yes—
93C2×Dic127#4114yes—
94C23×C127—22no—
95C23×C127—22no—
96C23×C127C2×D412yes—
97C23×C127#4112yes—
98C23×C127#4112yes—
99C23×C127C2411yescharacteristic
100C2×C4×C127—28no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127#4114yes—
111C22×C127—22no—
112C22×C127—22no—
113C22×C127—24no—
114C22×C127—24no—
115C22×C127#12314yes—
116C22×C127C22×D411yescharacteristic
117C22×C127C2×#4112yes—
118C22×C127C2×#4112yes—
119C22×C127C2×#4111yescharacteristic
120C22×C127C2×#4111yescharacteristic
121C2×D127—41no—
122C2×D127—41no—
123C2×D127—41no—
124C2×D127—42no—
125C2×D127—42no—
126C4×C127—24no—
127C4×C127—41no—
128C4×C127—41no—
129C4×C127—42no—
130Dic127—22no—
131Dic127—22no—
132Dic127—24no—
133Dic127—41no—
134Dic127—42no—
135C2×C127—24no—
136C2×C127C2×#12311yescharacteristic
137C2×C127#52511yescharacteristic
138C2×C127#52511yescharacteristic
139C2×C127#52912yes—
140C2×C127#52912yes—
141D127—81no—
142#2853—1271nomaximal, Sylow
143C127#285311yescharacteristic, complemented, Sylow
144C2×#123—1271no—
145C22×C42—1271no—
146#388—1272no—
147#388—1272no—
148#387—1272no—
149#387—1272no—
150#386—1271no—
151#386—1271no—
152#385—1271no—
153#385D12711yescharacteristic, complemented
154#381—1271no—
155C22×D4—1271no—
156C23×C4—1271no—
157C23×C4—1271no—
158C23×C4C2×D12711yescharacteristic
159#123—2544no—
160C2×C4⋊C4—1271no—
161C2×C4⋊C4—1271no—
162C2×C4⋊C4—1272no—
163C2×C4⋊C4C2×D12712yes—
164C2×C22⋊C4—1271no—
165C2×C22⋊C4—1271no—
166C2×C22⋊C4—1271no—
167C2×C22⋊C4—1272no—
168C2×C22⋊C4—1272no—
169C2×C22⋊C4C2×D12711yescharacteristic
170C2×C22⋊C4C2×D12711yescharacteristic
171C2×C42—1272no—
172C2×C42—1272no—
173C2×C42—2544no—
174#95—1272no—
175#95—1272no—
176#95—1272no—
177#95—1272no—
178#95—1272no—
179#95C2×D12712yes—
180C24—1271no—
181C24C22×D12711yescharacteristic
182C2×D4—2542no—
183C2×D4—2542no—
184C2×D4—2542no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—1274no—
188C22×C4—1271no—
189C22×C4—1271no—
190C22×C4—1272no—
191C22×C4—1272no—
192C22×C4—1271no—
193C22×C4—2544no—
194C22×C4—2542no—
195C22×C4—2542no—
196C22×C4—2542no—
197C22×C4C22×D12712yes—
198C22×C4C22×D12711yescharacteristic
199C22×C4C22×D12711yescharacteristic
200C22×C4C22×D12712yes—
201C4⋊C4—24no—
202C4⋊C4—2544no—
203C4⋊C4—2542no—
204C4⋊C4—2542no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—2544no—
208C22⋊C4—2542no—
209C22⋊C4—2542no—
210C22⋊C4—2542no—
211C22⋊C4—2544no—
212C42—2544no—
213C42—2544no—
214C23—22no—
215C23—22no—
216C23—2541no—
217C23—2542no—
218C23—2542no—
219C23—2541no—
220C23—2541no—
221C23D4×D12712yes—
222C23#1452324812yes—
223C23#1452324912yes—
224C23C23×D12711yescharacteristic
225D4—5084no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—1272no—
237C2×C4—1272no—
238C2×C4—1274no—
239C2×C4—2541no—
240C2×C4—2542no—
241C2×C4—2542no—
242C2×C4—2541no—
243C2×C4—2542no—
244C2×C4—2541no—
245C2×C4—2548no—
246C2×C4—2548no—
247C2×C4—2542no—
248C2×C4—2542no—
249C2×C4—2541no—
250C2×C4—2541no—
251C2×C4—2541no—
252C2×C4—2541no—
253C2×C4#1452324714yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—5081no—
259C22—5081no—
260C22—5081no—
261C22—5082no—
262C22—5082no—
263C22#1686119414yes—
264C22C2×D4×D12711yescharacteristic
265C22#41×D12712yes—
266C22C2×#1452324712yes—
267C22C2×#1452324811yescharacteristic
268C22C2×#1452324911yescharacteristic
269C4—24no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—5081no—
274C4—5082no—
275C4—2542no—
276C4—2542no—
277C4—2544no—
278C2—24no—
279C2—10161no—
280C2C2×#1686119411yescharacteristic
281C2#2255038211yescharacteristic
282C2#2255038411yescharacteristic
283C2#2255040612yes—
284C2#2255041212yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#385×C127C2
prime radicalC22×C22⋊D127C22
hypercenter#385D127
hypocenterC127#2853
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic127C2semidirect
3. #22549766C2semidirect
4. #22549765C2semidirect
5. #22549763C2semidirect
6. #22549762C2semidirect
9. #22549748C2semidirect
10. #22549737C2semidirect
11. #385×C127C2semidirect
143. C127#2853semidirect
153. #385D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611932
C22×C4×Dic1272
#225497662
#225497652
#225497632
#225497622
#225497552
#225497532
#225497482
#225497372
#385×C1272
#2853127