Finite Group Databasebeta

← #30648897

Subgroups — #30648897

Order 16256.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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×C4×Dic254C211yesmaximal, characteristic, complemented
2#22550209C211yesmaximal, characteristic, complemented
3#22550192C211yesmaximal, characteristic
4#22550180C212yesmaximal, complemented
5#22550095C212yesmaximal, complemented
6#22550031C211yesmaximal, characteristic, complemented
7#22550014C211yesmaximal, characteristic
8#22549981C212yesmaximal, complemented
9#22549976C212yesmaximal, complemented
10#22549933C211yesmaximal, characteristic, complemented
11#529×C127C211yesmaximal, characteristic
12C22×Dic254C2211yescharacteristic
13C2×#14523238C2212yes—
14C2×#14523236C2212yes—
15C4×Dic254—21no—
16C4×Dic254—21no—
17C4×Dic254C2212yes—
18C4×Dic254C2212yes—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4—21no—
21C2×C4×Dic127—21no—
22#16861142C2212yes—
23#16861140C2212yes—
24C4⋊(C127⋊C8)C2212yes—
25#16861133C2212yes—
26#16861132C2212yes—
27#16861132C2212yes—
28#16861130C2212yes—
29C22⋊(C127⋊C8)C2211yescharacteristic
30C22⋊(C127⋊C8)C2211yescharacteristic
31#16861120C2212yes—
32#126×C127—21no—
33#125×C127C2211yescharacteristic
34#124×C127C2211yescharacteristic
35#123×C127—21no—
36C22⋊Q8×C127C2211yescharacteristic
37#121×C127C2211yescharacteristic
38C4×Q8×C127C2211yescharacteristic
39C4×D4×C127—21no—
40C4×D4×C127C2211yescharacteristic
41#117×C127—21no—
42C2×C42×C127C2211yescharacteristic
43C2×Dic254—24no—
44C2×Dic254—21no—
45C2×Dic254C2312yes—
46C22×Dic127—21no—
47C22×Dic127—21no—
48#14523238—24no—
49#14523236—24no—
50C2×C127⋊C8C2312yes—
51C2×C127⋊C8C2312yes—
52C4⋊Dic127—21no—
53C4⋊Dic127C2312yes—
54Dic127⋊C4—22no—
55Dic127⋊C4—22no—
56C4×Dic127—22no—
57C4×Dic127—22no—
58C2×Q8×C127C2311yescharacteristic
59C2×D4×C127—21no—
60C2×D4×C127C2311yescharacteristic
61C22×C4×C127—21no—
62C22×C4×C127D411yescharacteristic
63C22×C4×C127D411yescharacteristic
64C22×C4×C127C2311yescharacteristic
65C4⋊C4×C127—21no—
66C4⋊C4×C127—21no—
67C4⋊C4×C127—21no—
68C4⋊C4×C127—21no—
69C4⋊C4×C127C2311yescharacteristic
70C4⋊C4×C127C2311yescharacteristic
71C22⋊C4×C127—21no—
72C22⋊C4×C127—21no—
73C22⋊C4×C127—21no—
74C22⋊C4×C127—21no—
75C22⋊C4×C127—21no—
76C42×C127—21no—
77C42×C127D411yescharacteristic
78C42×C127D411yescharacteristic
79C42×C127C2311yescharacteristic
80C42×C127C2311yescharacteristic
81Dic254—24no—
82Dic254—22no—
83Dic254#4114yes—
84C2×Dic127—24no—
85C2×Dic127—22no—
86C2×Dic127—22no—
87C2×Dic127—42no—
88C127⋊C8—22no—
89C127⋊C8—22no—
90C23×C127—21no—
91C23×C127C2×D411yescharacteristic
92Q8×C127—22no—
93D4×C127—22no—
94D4×C127—42no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—41no—
101C2×C4×C127—42no—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2411yescharacteristic
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127#13712yes—
114C22×C127C22×D411yescharacteristic
115C4×C127—21no—
116C4×C127—21no—
117C4×C127—21no—
118C4×C127—21no—
119C4×C127—21no—
120C4×C127—41no—
121C4×C127—41no—
122C4×C127—41no—
123C4×C127C2×#4112yes—
124Dic127—24no—
125Dic127—42no—
126C2×C127—22no—
127C2×C127—81no—
128C2×C127#52511yescharacteristic
129C2×C127C2×#13711yescharacteristic
130C2×C127#57511yescharacteristic
131#3117—1271nomaximal, Sylow
132C127#311711yescharacteristic, complemented, Sylow
133#529D12711yescharacteristic
134C2×C4×Q8—1271no—
135#484—1271no—
136#481—1271no—
137#479—1272no—
138#464—1272no—
139#451—1271no—
140#448—1271no—
141#441—1272no—
142#440—1272no—
143#432—1271no—
144C22×Q8—1271no—
145C2×Q16—1272no—
146C2×SD16—1272no—
147#126—21no—
148#125C2×D12711yescharacteristic
149#124C2×D12711yescharacteristic
150#123—21no—
151C22⋊Q8C2×D12711yescharacteristic
152#121C2×D12711yescharacteristic
153C4×Q8—1272no—
154C4×Q8—1272no—
155C4×Q8—2541no—
156C4×Q8—2541no—
157C4×Q8C2×D12711yescharacteristic
158C4×D4—21no—
159C4×D4C2×D12711yescharacteristic
160#117—21no—
161C2×C4⋊C4—1271no—
162C2×C4⋊C4—2541no—
163C2×C42—2541no—
164C2×C42C2×D12711yescharacteristic
165#107—1272no—
166#106—1272no—
167C4⋊C8—1272no—
168#103—1272no—
169#103—1272no—
170#103—1272no—
171#102—1272no—
172C22⋊C8—1271no—
173C22⋊C8—1271no—
174#97—1272no—
175C2×Q8—1272no—
176C2×Q8—2544no—
177C2×Q8—2541no—
178C2×Q8C22×D12711yescharacteristic
179C2×D4—21no—
180C2×D4C22×D12711yescharacteristic
181C22×C4—21no—
182C22×C4—2541no—
183C22×C4—2541no—
184C22×C4C22⋊D12711yescharacteristic
185C22×C4C22⋊D12711yescharacteristic
186C22×C4C22×D12711yescharacteristic
187Q16—2544no—
188SD16—2544no—
189C2×C8—1272no—
190C2×C8—1272no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—1272no—
196C4⋊C4—2542no—
197C4⋊C4—2542no—
198C4⋊C4—2541no—
199C4⋊C4C22×D12711yescharacteristic
200C4⋊C4C22×D12711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C42—21no—
207C42—2542no—
208C42—2542no—
209C42C22⋊D12711yescharacteristic
210C42C22⋊D12711yescharacteristic
211C42C22×D12711yescharacteristic
212C42C22×D12711yescharacteristic
213C23—21no—
214C23C2×C22⋊D12711yescharacteristic
215Q8—22no—
216Q8—1274no—
217Q8—2544no—
218Q8—2542no—
219D4—22no—
220D4—42no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—5082no—
229C2×C4—2542no—
230C2×C4—2544no—
231C2×C4—2542no—
232C2×C4C2×C22⋊D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C2×C4C2×C22⋊D12711yescharacteristic
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4C2×C22⋊D12711yescharacteristic
237C2×C4C23×D12711yescharacteristic
238C8—2542no—
239C8—2542no—
240C22—21no—
241C22—21no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22#1686123912yes—
246C22C22×C22⋊D12711yescharacteristic
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—41no—
253C4—41no—
254C4—41no—
255C4—5082no—
256C4—2544no—
257C4#41×D12712yes—
258C2—22no—
259C2—81no—
260C2#2255038311yescharacteristic
261C2C2×#1686123911yescharacteristic
262C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#529×C127C2
socleC22×C127C22×D4
prime radical#121×C127C22
hypercenter#529D127
hypocenterC127#3117
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic254C2semidirect
2. #22550209C2semidirect
4. #22550180C2semidirect
5. #22550095C2semidirect
6. #22550031C2semidirect
8. #22549981C2semidirect
9. #22549976C2semidirect
10. #22549933C2semidirect
132. C127#3117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2542
#225502092
#225501922
#225501802
#225500952
#225500312
#225500142
#225499812
#225499762
#225499332
#529×C1272
#3117127