Finite Group Databasebeta

← #30647051

Subgroups — #30647051

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×#16861190C211yesmaximal, characteristic
2C22×C4×Dic127C211yesmaximal, characteristic, complemented
3#22549770C212yesmaximal, complemented
4#22549769C212yesmaximal, complemented
5#22549759C212yesmaximal, complemented
6#22549758C212yesmaximal, complemented
7#22549756C211yesmaximal, characteristic
8#22549751C211yesmaximal, characteristic, complemented
9#22549749C211yesmaximal, characteristic, complemented
10#22549737C211yesmaximal, characteristic, complemented
11#386×C127C211yesmaximal, characteristic
12C23×Dic127C2211yescharacteristic
13C23×Dic127C2211yescharacteristic
14#16861190—24no—
15C2×C4⋊Dic127C2212yes—
16C2×Dic127⋊C4C2212yes—
17C2×Dic127⋊C4C2211yescharacteristic
18C2×Dic127⋊C4C2211yescharacteristic
19C2×C22⋊Dic127C2211yescharacteristic
20C2×C22⋊Dic127C2212yes—
21C2×C22⋊Dic127C2211yescharacteristic
22C2×C22⋊Dic127C2211yescharacteristic
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C4×Dic127—24no—
25C2×C4×Dic127C2212yes—
26C2×C4×Dic127C2212yes—
27C127⋊#95C2212yes—
28C127⋊#95C2212yes—
29C127⋊#95C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C22×D4×C127C2211yescharacteristic
33C23×C4×C127C2211yescharacteristic
34C2×C22⋊C4×C127C2212yes—
35C2×C22⋊C4×C127C2211yescharacteristic
36#95×C127C2212yes—
37C22×Dic127—24no—
38C22×Dic127—22no—
39C22×Dic127—22no—
40C22×Dic127—22no—
41C22×Dic127D414yes—
42C22×Dic127C2312yes—
43C22×Dic127C2311yescharacteristic
44C22×Dic127C2311yescharacteristic
45C22×Dic127C2311yescharacteristic
46C22×Dic127C2311yescharacteristic
47C22×Dic127C2312yes—
48C4⋊Dic127—24no—
49Dic127⋊C4—22no—
50Dic127⋊C4—22no—
51Dic127⋊C4—24no—
52C22⋊Dic127—22no—
53C22⋊Dic127—22no—
54C22⋊Dic127—24no—
55C22⋊Dic127—22no—
56C22⋊Dic127—22no—
57C4×Dic127—24no—
58C4×Dic127—24no—
59C24×C127C2311yescharacteristic
60C24×C127C2311yescharacteristic
61C2×D4×C127—22no—
62C2×D4×C127—22no—
63C2×D4×C127—22no—
64C22×C4×C127—22no—
65C22×C4×C127—24no—
66C22×C4×C127C2311yescharacteristic
67C22×C4×C127C2312yes—
68C22×C4×C127C2312yes—
69C22⋊C4×C127—22no—
70C22⋊C4×C127—24no—
71C2×Dic127—28no—
72C2×Dic127—28no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127C2×D412yes—
87C2×Dic127C2×D412yes—
88C2×Dic127#4114yes—
89C23×C127—22no—
90C23×C127—22no—
91C23×C127—21no—
92C23×C127—22no—
93C23×C127—21no—
94C23×C127—21no—
95C23×C127—22no—
96C23×C127C2×D412yes—
97C23×C127#4112yes—
98C23×C127#4112yes—
99C23×C127C2411yescharacteristic
100D4×C127—44no—
101C2×C4×C127—28no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—21no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127#4114yes—
111C22×C127—22no—
112C22×C127—22no—
113C22×C127—24no—
114C22×C127—24no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127#12314yes—
121C22×C127C22×D411yescharacteristic
122C22×C127C2×#4112yes—
123C22×C127C2×#4111yescharacteristic
124C22×C127C2×#4112yes—
125C22×C127C2×#4111yescharacteristic
126C4×C127—24no—
127C4×C127—41no—
128C4×C127—42no—
129Dic127—22no—
130Dic127—22no—
131Dic127—24no—
132Dic127—41no—
133Dic127—41no—
134Dic127—42no—
135C2×C127—24no—
136C2×C127—81no—
137C2×C127C2×#12311yescharacteristic
138C2×C127#52511yescharacteristic
139C2×C127#52511yescharacteristic
140C2×C127#52912yes—
141C2×C127#52912yes—
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#386D12711yescharacteristic
152#385—1271no—
153#385—1271no—
154#381—1271no—
155C22×D4C2×D12711yescharacteristic
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⋊C4—1272no—
164C2×C22⋊C4—1271no—
165C2×C22⋊C4—1271no—
166C2×C22⋊C4—1271no—
167C2×C22⋊C4—1271no—
168C2×C22⋊C4—1272no—
169C2×C22⋊C4C2×D12711yescharacteristic
170C2×C22⋊C4C2×D12712yes—
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—
180C24C22×D12711yescharacteristic
181C24C22×D12711yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
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—1271no—
194C22×C4—2544no—
195C22×C4—2542no—
196C22×C4—2542no—
197C22×C4—2542no—
198C22×C4C22×D12712yes—
199C22×C4C22×D12711yescharacteristic
200C22×C4C22×D12712yes—
201C4⋊C4—2542no—
202C4⋊C4—2542no—
203C4⋊C4—2544no—
204C4⋊C4—2544no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—2544no—
208C22⋊C4—2542no—
209C22⋊C4—2542no—
210C22⋊C4—2542no—
211C22⋊C4—2542no—
212C42—2544no—
213C42—2544no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23D4×D12712yes—
222C23#1452324712yes—
223C23#1452324812yes—
224C23C23×D12711yescharacteristic
225D4—44no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—1272no—
236C2×C4—1272no—
237C2×C4—1274no—
238C2×C4—2541no—
239C2×C4—2541no—
240C2×C4—2542no—
241C2×C4—2542no—
242C2×C4—2542no—
243C2×C4—2542no—
244C2×C4—2542no—
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—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22#1686119114yes—
264C22C2×D4×D12711yescharacteristic
265C22#41×D12712yes—
266C22C2×#1452324711yescharacteristic
267C22C2×#1452324712yes—
268C22C2×#1452324811yescharacteristic
269C4—24no—
270C4—41no—
271C4—42no—
272C4—5081no—
273C4—5081no—
274C4—5082no—
275C4—2542no—
276C4—2542no—
277C4—2544no—
278C2—24no—
279C2—81no—
280C2C2×#1686119111yescharacteristic
281C2#2255038211yescharacteristic
282C2#2255038411yescharacteristic
283C2#2255040612yes—
284C2#2255040712yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#386×C127C2
prime radicalC22×D4×C127C22
hypercenter#386D127
hypocenterC127#2853
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic127C2semidirect
3. #22549770C2semidirect
4. #22549769C2semidirect
5. #22549759C2semidirect
6. #22549758C2semidirect
8. #22549751C2semidirect
9. #22549749C2semidirect
10. #22549737C2semidirect
143. C127#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#168611902
C22×C4×Dic1272
#225497702
#225497692
#225497592
#225497582
#225497562
#225497512
#225497492
#225497372
#386×C1272
#2853127