Finite Group Databasebeta

← #30641971

Subgroups — #30641971

Order 16256.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16861171C211yesmaximal, characteristic, complemented
2C2×#16861171C212yesmaximal, complemented
3C2×#16861167C211yesmaximal, characteristic, complemented
4#22549802C211yesmaximal, characteristic, complemented
5#22549798C212yesmaximal, complemented
6#22549794C212yesmaximal, complemented
7#22549793C211yesmaximal, characteristic, complemented
8#22549786C211yesmaximal, characteristic
9#22549727C211yesmaximal, characteristic, complemented
10#22549723C212yesmaximal, complemented
11#398×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×D508C2212yes—
14C22×C4×D127C2212yes—
15C22×C4×D127C2211yescharacteristic
16#16861171—28no—
17#16861171—24no—
18#16861167—24no—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4C2212yes—
21C2×(C2×D127)⋊C4C2212yes—
22C2×(C2×D127)⋊C4C2211yescharacteristic
23C2×(C2×D127)⋊C4C2212yes—
24C2×(C2×D127)⋊C4C2212yes—
25C2×(C2×D127)⋊C4C2211yescharacteristic
26C2×(C2×D127)⋊C4C2212yes—
27C2×(C2×D127)⋊C4C2212yes—
28C2×C4×Dic127C2211yescharacteristic
29C2×C4×Dic127C2212yes—
30C2×C4×Dic127C2211yescharacteristic
31C127⋊#95C2211yescharacteristic
32C127⋊#95C2212yes—
33C2×C4⋊C4×C127C2211yescharacteristic
34C2×C4⋊C4×C127C2212yes—
35C2×C4⋊C4×C127C2211yescharacteristic
36#95×C127C2211yescharacteristic
37#95×C127C2212yes—
38C23×D127C2311yescharacteristic
39C23×D127C2312yes—
40C2×D508—24no—
41C2×D508—24no—
42C2×D508—24no—
43C2×D508—22no—
44C2×D508—24no—
45C22×Dic127C2312yes—
46C22×Dic127C2311yescharacteristic
47C22×Dic127C2311yescharacteristic
48C22×Dic127C2311yescharacteristic
49C2×C4×D127—22no—
50C2×C4×D127—22no—
51C2×C4×D127—24no—
52C2×C4×D127—24no—
53C2×C4×D127—24no—
54C2×C4×D127D414yescomplemented
55C4⋊Dic127—22no—
56Dic127⋊C4—24no—
57(C2×D127)⋊C4—24no—
58(C2×D127)⋊C4—24no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—24no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—24no—
63(C2×D127)⋊C4—24no—
64C4×Dic127—22no—
65C4×Dic127—24no—
66C4×Dic127D414yescomplemented
67C22×C4×C127C2312yes—
68C22×C4×C127C2311yescharacteristic
69C22×C4×C127C2311yescharacteristic
70C22×C4×C127C2311yescharacteristic
71C22×C4×C127C2312yes—
72C4⋊C4×C127—22no—
73C4⋊C4×C127—24no—
74C4⋊C4×C127—22no—
75C22×D127—21no—
76C22×D127—22no—
77C22×D127—22no—
78C22×D127—22no—
79C22×D127—21no—
80C22×D127—22no—
81C22×D127—22no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127—22no—
85C22×D127C2×D412yes—
86C22×D127#4114yes—
87D508—28no—
88D508—216no—
89C2×Dic127—22no—
90C2×Dic127—22no—
91C2×Dic127—21no—
92C2×Dic127—22no—
93C2×Dic127—22no—
94C2×Dic127—22no—
95C2×Dic127C2×D412yes—
96C2×Dic127C2×D412yes—
97C2×Dic127C2×D412yes—
98C2×Dic127#4112yes—
99C4×D127—28no—
100C4×D127—48no—
101C23×C127C2411yescharacteristic
102C2×C4×C127—22no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127C2×D412yes—
113C2×C4×C127C2×D412yes—
114C2×C4×C127#4114yes—
115C22×C127C22×D411yescharacteristic
116C22×C127C22×D411yescharacteristic
117C22×C127C2×#4111yescharacteristic
118C22×C127C2×#4112yes—
119C22×C127C2×#4112yes—
120C2×D127—22no—
121C2×D127—24no—
122C2×D127—24no—
123C2×D127—24no—
124C2×D127—24no—
125C2×D127—42no—
126C2×D127—41no—
127C2×D127—42no—
128C2×D127—42no—
129C2×D127—42no—
130C2×D127—41no—
131C2×D127—42no—
132C4×C127—22no—
133C4×C127—22no—
134C4×C127—24no—
135C4×C127—42no—
136C4×C127—41no—
137Dic127—22no—
138Dic127—42no—
139Dic127—41no—
140Dic127#12114yes—
141C2×C127C2×#12111yescharacteristic
142C2×C127#52511yescharacteristic
143C2×C127#52512yes—
144C2×C127#52912yes—
145C2×C127#53211yescharacteristic
146D127—42no—
147D127—44no—
148#2392—1271nomaximal, Sylow
149C127#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—1271no—
151C2×C4×D4—1272no—
152C2×#117—1271no—
153#398D12711yescharacteristic, complemented
154#397—1271no—
155#396—1272no—
156#394—1271no—
157#394—1272no—
158#392—1271no—
159C4×C4⋊C4—1271no—
160C4×C22⋊C4—1272no—
161C22×D4—1271no—
162C22×D4—1272no—
163C23×C4—1271no—
164C23×C4—1272no—
165C4×D4—2544no—
166C4×D4—2548no—
167#117—2544no—
168C2×C4⋊C4—1271no—
169C2×C4⋊C4—1272no—
170C2×C4⋊C4C2×D12711yescharacteristic
171C2×C4⋊C4C2×D12712yes—
172C2×C4⋊C4C2×D12711yescharacteristic
173C2×C22⋊C4—1271no—
174C2×C22⋊C4—1271no—
175C2×C22⋊C4—1272no—
176C2×C22⋊C4—1272no—
177C2×C22⋊C4—1272no—
178C2×C22⋊C4—1272no—
179C2×C22⋊C4—1272no—
180C2×C42—1271no—
181C2×C42—1271no—
182C2×C42—1272no—
183#95—1272no—
184#95—1271no—
185#95C2×D12712yes—
186#95C2×D12711yescharacteristic
187C24—1271no—
188C24—1272no—
189C2×D4—2544no—
190C2×D4—2544no—
191C2×D4—2544no—
192C2×D4—2542no—
193C2×D4—2544no—
194C22×C4—1271no—
195C22×C4—1271no—
196C22×C4—1274no—
197C22×C4—1272no—
198C22×C4—1271no—
199C22×C4—2544no—
200C22×C4—2544no—
201C22×C4—2544no—
202C22×C4—2542no—
203C22×C4—2542no—
204C22×C4C22×D12712yes—
205C22×C4C22×D12711yescharacteristic
206C22×C4C22×D12711yescharacteristic
207C22×C4C22×D12711yescharacteristic
208C22×C4C22×D12712yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—2542no—
213C4⋊C4—2544no—
214C22⋊C4—2542no—
215C22⋊C4—2544no—
216C22⋊C4—2544no—
217C22⋊C4—2544no—
218C22⋊C4—2542no—
219C22⋊C4—2544no—
220C22⋊C4—2544no—
221C42—1274no—
222C42—2542no—
223C42—2544no—
224C23—1272no—
225C23—1274no—
226C23—2541no—
227C23—2542no—
228C23—2542no—
229C23—2542no—
230C23—2541no—
231C23—2542no—
232C23—2542no—
233C23—2542no—
234C23—2542no—
235C23—2542no—
236C23C23×D12711yescharacteristic
237D4—2548no—
238D4—25416no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—1272no—
250C2×C4—1272no—
251C2×C4—1272no—
252C2×C4—1272no—
253C2×C4—5088no—
254C2×C4—2548no—
255C2×C4—2542no—
256C2×C4—2541no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4—2542no—
260C2×C4—2542no—
261C2×C4D4×D12712yes—
262C2×C4D4×D12712yes—
263C2×C4#1452324914yes—
264C22—5082no—
265C22—5081no—
266C22—5082no—
267C22—5082no—
268C22—5082no—
269C22—5081no—
270C22—5082no—
271C22—2542no—
272C22—2544no—
273C22—2544no—
274C22—2544no—
275C22—2544no—
276C22C2×D4×D12711yescharacteristic
277C22C2×D4×D12711yescharacteristic
278C22#41×D12711yescharacteristic
279C22#41×D12712yes—
280C22C2×#1452324912yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—1274no—
287C4—5082no—
288C4—5081no—
289C4—2542no—
290C2—5084no—
291C2—5082no—
292C2#121×D12711yescharacteristic
293C2#2255038411yescharacteristic
294C2#2255038912yes—
295C2#2255041112yes—
296C2#2255042711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#398×C127C2
prime radical#22549786C2
hypercenter#398D127
hypocenterC127#2392
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#16861171C2semidirect
2. C2×#16861171C2semidirect
3. C2×#16861167C2semidirect
4. #22549802C2semidirect
5. #22549798C2semidirect
6. #22549794C2semidirect
7. #22549793C2semidirect
9. #22549727C2semidirect
10. #22549723C2semidirect
11. #398×C127C2semidirect
54. C2×C4×D127D4semidirect
66. C4×Dic127D4semidirect
149. C127#2392semidirect
153. #398D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611712
C2×#168611672
#225498022
#225497982
#225497942
#225497932
#225497862
#225497272
#225497232
#398×C1272
#2392127