Finite Group Databasebeta

← #30641969

Subgroups — #30641969

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×D4×Dic127C211yesmaximal, characteristic, complemented
2C2×(C22⋊D127)⋊C4C212yesmaximal, complemented
3C2×#16861167C211yesmaximal, characteristic, complemented
4#22549806C211yesmaximal, characteristic, complemented
5#22549803C211yesmaximal, characteristic, complemented
6#22549799C212yesmaximal, complemented
7#22549791C212yesmaximal, complemented
8#22549784C211yesmaximal, characteristic
9#22549727C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic127C212yesmaximal, complemented
11#394×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127C2212yes—
13C23×Dic127C2212yes—
14C22×C4×D127C2211yescharacteristic
15D4×Dic127—24no—
16(C22⋊D127)⋊C4—28no—
17#16861167—24no—
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4C2212yes—
21C2×Dic127⋊C4C2212yes—
22C2×C22⋊Dic127C2212yes—
23C2×C22⋊Dic127C2212yes—
24C2×(C2×D127)⋊C4C2211yescharacteristic
25C2×(C2×D127)⋊C4C2211yescharacteristic
26C2×(C2×D127)⋊C4C2212yes—
27C2×C4×Dic127C2211yescharacteristic
28C2×C4×Dic127C2212yes—
29C2×C4×Dic127C2211yescharacteristic
30C127⋊#95C2211yescharacteristic
31C127⋊#95C2212yes—
32C127⋊#95C2212yes—
33C22×D4×C127C2211yescharacteristic
34C2×C4⋊C4×C127C2211yescharacteristic
35C2×C22⋊C4×C127C2212yes—
36C2×C22⋊C4×C127C2212yes—
37#95×C127C2211yescharacteristic
38C23×D127C2311yescharacteristic
39C2×C22⋊D127—24no—
40C2×C22⋊D127—24no—
41C2×C22⋊D127—24no—
42C22×Dic127—24no—
43C22×Dic127—24no—
44C22×Dic127—24no—
45C22×Dic127C2312yes—
46C22×Dic127C2311yescharacteristic
47C22×Dic127C2311yescharacteristic
48C22×Dic127C2312yes—
49C22×Dic127C2311yescharacteristic
50C2×C4×D127—22no—
51C2×C4×D127—22no—
52C2×C4×D127D414yescomplemented
53C4⋊Dic127—22no—
54C4⋊Dic127—22no—
55Dic127⋊C4—24no—
56Dic127⋊C4—24no—
57C22⋊Dic127—24no—
58C22⋊Dic127—24no—
59(C2×D127)⋊C4—24no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—22no—
62C4×Dic127—22no—
63C4×Dic127—24no—
64C4×Dic127D414yescomplemented
65C24×C127C2312yes—
66C2×D4×C127—24no—
67C2×D4×C127—22no—
68C22×C4×C127C2312yes—
69C22×C4×C127C2311yescharacteristic
70C22×C4×C127C2311yescharacteristic
71C22×C4×C127C2311yescharacteristic
72C4⋊C4×C127—22no—
73C22⋊C4×C127—24no—
74C22⋊C4×C127—24no—
75C22×D127—21no—
76C22×D127—21no—
77C22×D127—22no—
78C22×D127—22no—
79C22×D127C2×D412yes—
80C22⋊D127—216no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—21no—
85C2×Dic127—22no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127—48no—
90C2×Dic127C2×D412yes—
91C2×Dic127C2×D412yes—
92C2×Dic127C2×D412yes—
93C2×Dic127#4114yes—
94C2×Dic127#4112yes—
95C4×D127—28no—
96C23×C127—22no—
97C23×C127—22no—
98C23×C127—22no—
99C23×C127—22no—
100C23×C127—22no—
101C23×C127—22no—
102C23×C127#4114yes—
103C23×C127C2411yescharacteristic
104D4×C127—28no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127C2×D412yes—
114C2×C4×C127C2×D412yes—
115C22×C127—24no—
116C22×C127—24no—
117C22×C127—24no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127—42no—
122C22×C127C22×D411yescharacteristic
123C22×C127C22×D411yescharacteristic
124C22×C127C2×#4111yescharacteristic
125C22×C127C2×#4112yes—
126C22×C127C2×#4112yes—
127C2×D127—22no—
128C2×D127—24no—
129C2×D127—41no—
130C2×D127—42no—
131C2×D127—41no—
132C4×C127—22no—
133C4×C127—22no—
134C4×C127—41no—
135C4×C127—42no—
136Dic127—24no—
137Dic127—22no—
138Dic127—42no—
139Dic127—41no—
140Dic127#12114yes—
141C2×C127—44no—
142C2×C127C2×#12111yescharacteristic
143C2×C127#52511yescharacteristic
144C2×C127#52512yes—
145C2×C127#52912yes—
146C2×C127#53211yescharacteristic
147D127—42no—
148#2392—1271nomaximal, Sylow
149C127#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—1271no—
151C2×C4×D4—1272no—
152C2×#117—1271no—
153#398—1271no—
154#397—1271no—
155#396—1272no—
156#394—1272no—
157#394D12711yescharacteristic, complemented
158#392—1271no—
159C4×C4⋊C4—1271no—
160C4×C22⋊C4—1272no—
161C22×D4—1272no—
162C22×D4C2×D12711yescharacteristic
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⋊C4—1271no—
171C2×C4⋊C4—1272no—
172C2×C4⋊C4C2×D12711yescharacteristic
173C2×C22⋊C4—1272no—
174C2×C22⋊C4—1272no—
175C2×C22⋊C4—1271no—
176C2×C22⋊C4—1271no—
177C2×C22⋊C4—1272no—
178C2×C22⋊C4C2×D12712yes—
179C2×C22⋊C4C2×D12712yes—
180C2×C42—1271no—
181C2×C42—1271no—
182C2×C42—1272no—
183#95—1272no—
184#95—1272no—
185#95—1271no—
186#95C2×D12711yescharacteristic
187C24—1271no—
188C24C22×D12712yes—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—2544no—
192C2×D4—2544no—
193C2×D4—2544no—
194C22×C4—1271no—
195C22×C4—1272no—
196C22×C4—1271no—
197C22×C4—1274no—
198C22×C4—1271no—
199C22×C4—1272no—
200C22×C4—2542no—
201C22×C4—2542no—
202C22×C4—2544no—
203C22×C4—2544no—
204C22×C4—2544no—
205C22×C4C22×D12711yescharacteristic
206C22×C4C22×D12712yes—
207C22×C4C22×D12711yescharacteristic
208C22×C4C22×D12711yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—2544no—
211C4⋊C4—2542no—
212C4⋊C4—2544no—
213C4⋊C4—2542no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—2542no—
217C22⋊C4—2544no—
218C22⋊C4—2544no—
219C22⋊C4—2544no—
220C22⋊C4—2542no—
221C42—1274no—
222C42—2542no—
223C42—2544no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—1272no—
231C23—2541no—
232C23—2541no—
233C23—2542no—
234C23—2542no—
235C23#1452324714yes—
236C23C23×D12711yescharacteristic
237D4—28no—
238D4—25416no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—1272no—
248C2×C4—1272no—
249C2×C4—1272no—
250C2×C4—1272no—
251C2×C4—1274no—
252C2×C4—5088no—
253C2×C4—2548no—
254C2×C4—2541no—
255C2×C4—2542no—
256C2×C4—2542no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4—2542no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4D4×D12712yes—
263C2×C4D4×D12712yes—
264C22—24no—
265C22—24no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—5081no—
272C22—5082no—
273C22—5081no—
274C22—2542no—
275C22—2544no—
276C22C2×D4×D12711yescharacteristic
277C22C2×D4×D12711yescharacteristic
278C22#41×D12711yescharacteristic
279C22#41×D12712yes—
280C22C2×#1452324712yes—
281C4—22no—
282C4—22no—
283C4—41no—
284C4—42no—
285C4—1274no—
286C4—5082no—
287C4—5081no—
288C4—2544no—
289C4—2542no—
290C2—44no—
291C2—5082no—
292C2#121×D12711yescharacteristic
293C2#2255038211yescharacteristic
294C2#2255038612yes—
295C2#2255040312yes—
296C2#2255042511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#394×C127C2
prime radical#22549784C2
hypercenter#394D127
hypocenterC127#2392
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic127C2semidirect
2. C2×(C22⋊D127)⋊C4C2semidirect
3. C2×#16861167C2semidirect
4. #22549806C2semidirect
5. #22549803C2semidirect
6. #22549799C2semidirect
7. #22549791C2semidirect
9. #22549727C2semidirect
10. C22⋊C4×Dic127C2semidirect
11. #394×C127C2semidirect
52. C2×C4×D127D4semidirect
64. C4×Dic127D4semidirect
149. C127#2392semidirect
157. #394D127semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1272
C2×(C22⋊D127)⋊C42
C2×#168611672
#225498062
#225498032
#225497992
#225497912
#225497842
#225497272
C22⋊C4×Dic1272
#394×C1272
#2392127