Finite Group Databasebeta

← #30641987

Subgroups — #30641987

Order 16256.

560 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 157 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×C22⋊D127C211yesmaximal, characteristic
2#22549814C211yesmaximal, characteristic, complemented
3#22549811C211yesmaximal, characteristic, complemented
4#22549807C211yesmaximal, characteristic, complemented
5#22549804C212yesmaximal, complemented
6#22549802C211yesmaximal, characteristic, complemented
7#22549791C212yesmaximal
8C4×Dic127⋊C4C212yesmaximal, complemented
9C4×(C2×D127)⋊C4C212yesmaximal, complemented
10C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
11C2×#117×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127C2211yescharacteristic
13C22×C4×D127C2211yescharacteristic
14C4×C22⋊D127—24no—
15C2×Dic127⋊C4C2212yes—
16C2×Dic127⋊C4C2212yes—
17C2×Dic127⋊C4C2211yescharacteristic
18C2×C22⋊Dic127C2212yes—
19C2×C22⋊Dic127C2211yescharacteristic
20C2×(C2×D127)⋊C4C2212yes—
21C2×(C2×D127)⋊C4C2212yes—
22C2×(C2×D127)⋊C4C2211yescharacteristic
23C2×C4×Dic127C2211yescharacteristic
24C2×C4×Dic127C2212yes—
25C127⋊#95C2212yes—
26C127⋊#95C2212yes—
27C127⋊#95C2211yescharacteristic
28C127⋊#95C2212yes—
29C127⋊#95C2212yes—
30C127⋊#95C2211yescharacteristic
31C23×C4×C127C2211yescharacteristic
32#117×C127—24no—
33C2×C4⋊C4×C127C2211yescharacteristic
34C2×C4⋊C4×C127C2211yescharacteristic
35C2×C22⋊C4×C127C2212yes—
36C2×C42×C127C2212yes—
37C23×D127C2311yescharacteristic
38C2×C22⋊D127—22no—
39C2×C22⋊D127—22no—
40C2×C22⋊D127—22no—
41C22×Dic127C2311yescharacteristic
42C22×Dic127C2312yes—
43C22×Dic127C2311yescharacteristic
44C22×Dic127C2311yescharacteristic
45C22×Dic127C2312yes—
46C2×C4×D127—22no—
47C2×C4×D127—24no—
48Dic127⋊C4—24no—
49Dic127⋊C4—24no—
50Dic127⋊C4—22no—
51C22⋊Dic127—24no—
52C22⋊Dic127—22no—
53(C2×D127)⋊C4—24no—
54(C2×D127)⋊C4—24no—
55(C2×D127)⋊C4—22no—
56C4×Dic127—22no—
57C4×Dic127—24no—
58C24×C127C2311yescharacteristic
59C22×C4×C127—24no—
60C22×C4×C127D414yes—
61C22×C4×C127C2312yes—
62C22×C4×C127C2312yes—
63C22×C4×C127C2311yescharacteristic
64C22×C4×C127C2311yescharacteristic
65C4⋊C4×C127—22no—
66C4⋊C4×C127—22no—
67C22⋊C4×C127—24no—
68C42×C127—24no—
69C22×D127—21no—
70C22×D127—22no—
71C22×D127—22no—
72C22×D127—21no—
73C22×D127#4112yes—
74C22⋊D127—28no—
75C2×Dic127—21no—
76C2×Dic127—22no—
77C2×Dic127—22no—
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×Dic127#4114yes—
87C2×Dic127#4112yes—
88C4×D127—44no—
89C23×C127—22no—
90C23×C127—22no—
91C23×C127—21no—
92C23×C127—21no—
93C23×C127C2×D412yes—
94C23×C127C2411yescharacteristic
95C2×C4×C127—22no—
96C2×C4×C127—28no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D412yes—
105C2×C4×C127#4114yes—
106C22×C127—22no—
107C22×C127—22no—
108C22×C127—22no—
109C22×C127—42no—
110C22×C127—42no—
111C22×C127C22×D411yescharacteristic
112C22×C127C2×#4111yescharacteristic
113C22×C127C2×#4111yescharacteristic
114C22×C127C2×#4112yes—
115C22×C127C2×#4112yes—
116C2×D127—22no—
117C2×D127—24no—
118C2×D127—41no—
119C2×D127—42no—
120C2×D127—41no—
121C4×C127—22no—
122C4×C127—42no—
123C4×C127—42no—
124C4×C127#12314yes—
125Dic127—22no—
126Dic127—24no—
127Dic127—41no—
128Dic127—42no—
129Dic127—41no—
130C2×C127—42no—
131C2×C127C2×#12311yescharacteristic
132C2×C127#52511yescharacteristic
133C2×C127#52912yes—
134C2×C127#52912yes—
135C2×C127#53211yescharacteristic
136D127—42no—
137#2395—1271nomaximal, Sylow
138C127#239511yescharacteristic, complemented, Sylow
139C2×C4×D4—1271no—
140C2×#117D12711yescharacteristic, complemented
141#400—1271no—
142#399—1271no—
143#398—1271no—
144#397—1271no—
145#397—1272no—
146#394—1272no—
147C4×C4⋊C4—1272no—
148C4×C22⋊C4—1271no—
149C4×C22⋊C4—1272no—
150C22×D4—1271no—
151C23×C4—1271no—
152C23×C4C2×D12711yescharacteristic
153C4×D4—2544no—
154#117—24no—
155C2×C4⋊C4—1271no—
156C2×C4⋊C4—1272no—
157C2×C4⋊C4—1272no—
158C2×C4⋊C4C2×D12711yescharacteristic
159C2×C4⋊C4C2×D12711yescharacteristic
160C2×C22⋊C4—1271no—
161C2×C22⋊C4—1272no—
162C2×C22⋊C4—1271no—
163C2×C22⋊C4—1272no—
164C2×C22⋊C4—1272no—
165C2×C22⋊C4C2×D12712yes—
166C2×C42—1271no—
167C2×C42—1272no—
168C2×C42C2×D12712yes—
169#95—1272no—
170#95—1271no—
171#95—1272no—
172#95—1271no—
173#95—1272no—
174#95—1272no—
175C24—1271no—
176C24C22×D12711yescharacteristic
177C2×D4—2542no—
178C2×D4—2542no—
179C2×D4—2542no—
180C22×C4—24no—
181C22×C4—1272no—
182C22×C4—1271no—
183C22×C4—1272no—
184C22×C4—1271no—
185C22×C4—1271no—
186C22×C4—2542no—
187C22×C4—2544no—
188C22×C4C22⋊D12714yes—
189C22×C4C22×D12712yes—
190C22×C4C22×D12712yes—
191C22×C4C22×D12711yescharacteristic
192C22×C4C22×D12711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—2544no—
196C4⋊C4—2544no—
197C4⋊C4—2542no—
198C22⋊C4—24no—
199C22⋊C4—2544no—
200C22⋊C4—2544no—
201C22⋊C4—2544no—
202C22⋊C4—2542no—
203C22⋊C4—2542no—
204C42—24no—
205C42—2542no—
206C42—2544no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—1272no—
212C23—2541no—
213C23—2542no—
214C23—2541no—
215C23—2542no—
216C23C2×C22⋊D12712yes—
217C23C23×D12711yescharacteristic
218D4—2548no—
219C2×C4—22no—
220C2×C4—28no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—1274no—
228C2×C4—1272no—
229C2×C4—5084no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4—2542no—
235C2×C4—2542no—
236C2×C4—2542no—
237C2×C4—2541no—
238C2×C4—2542no—
239C2×C4—2541no—
240C2×C4—2542no—
241C2×C4C2×C22⋊D12712yes—
242C2×C4C2×C22⋊D12712yes—
243C2×C4#1452324814yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—5081no—
250C22—5082no—
251C22—5081no—
252C22—2542no—
253C22—2544no—
254C22C22×C22⋊D12711yescharacteristic
255C22#41×D12711yescharacteristic
256C22#41×D12711yescharacteristic
257C22#41×D12712yes—
258C22C2×#1452324812yes—
259C4—22no—
260C4—42no—
261C4—42no—
262C4—5081no—
263C4—5082no—
264C4—5081no—
265C4—2542no—
266C4—2544no—
267C4#1686119214yes—
268C2—42no—
269C2—5082no—
270C2C2×#1686119211yescharacteristic
271C2#2255038311yescharacteristic
272C2#2255040412yes—
273C2#2255040412yes—
274C2#2255042211yescharacteristic
subgroupG/H
centerC22×C4C22×D127
commutator, socle, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC2×#117×C127C2
prime radicalC22×C22⋊D127C22
hypercenterC2×#117D127
hypocenterC127#2395
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22549814C2semidirect
3. #22549811C2semidirect
4. #22549807C2semidirect
5. #22549804C2semidirect
6. #22549802C2semidirect
8. C4×Dic127⋊C4C2semidirect
9. C4×(C2×D127)⋊C4C2semidirect
10. C4×C22⋊Dic127C2semidirect
11. C2×#117×C127C2semidirect
138. C127#2395semidirect
140. C2×#117D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1272
#225498142
#225498112
#225498072
#225498042
#225498022
#225497912
C4×Dic127⋊C42
C4×(C2×D127)⋊C42
C4×C22⋊Dic1272
C2×#117×C1272
#2395127