Finite Group Databasebeta

← C2×#22550075 #30648161

Subgroups — C2×#22550075 #30648161

Order 16256.

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

normal subgroups only

This group has 584 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
1C22×#14523237C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3C2×#16861205C211yesmaximal, characteristic
4C2×#16861174C211yesmaximal, characteristic, complemented
5#22550075C218yesmaximal, direct product factor
6C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
7C2×#16861129C211yesmaximal, characteristic, complemented
8C2×#102×C127C211yesmaximal, characteristic, complemented
9C22×Dic254C2211yescharacteristic
10C22×D508C2211yescharacteristic
11C22×C22⋊D127—21no—
12C22×C22⋊D127—21no—
13C2×#14523237—24no—
14C2×#14523237C2214yescomplemented
15C2×#14523236—24no—
16C2×#14523236C2214yescomplemented
17C22×C127⋊C8C2211yescharacteristic
18#16861205—22no—
19#16861205C2214yes—
20#16861174—22no—
21#16861174C2214yescomplemented
22C2×Dic127⋊C4—21no—
23C2×C4×Dic127—21no—
24C2×C4×Dic127C2211yescharacteristic
25Dic127⋊C8C2214yescomplemented
26#16861129C2214yescomplemented
27C22×D4×C127C2211yescharacteristic
28C22×C8×C127C2211yescharacteristic
29C2×C4⋊C4×C127C2211yescharacteristic
30#102×C127C2214yescomplemented
31C23×D127—21no—
32C2×Dic254—21no—
33C2×Dic254—22no—
34C2×Dic254D414yescomplemented
35C2×Dic254C2312yes—
36C2×D508—22no—
37C2×D508—21no—
38C2×D508—22no—
39C2×D508C2312yes—
40C2×C22⋊D127—24no—
41C2×C22⋊D127—24no—
42C2×C22⋊D127—44no—
43C2×C22⋊D127—44no—
44C22×Dic127—21no—
45C22×Dic127D411yescharacteristic
46C22×Dic127D411yescharacteristic
47#14523237—28no—
48#14523236—28no—
49C2×C127⋊C8—22no—
50C2×C127⋊C8C2312yes—
51Dic127⋊C4—24no—
52C4×Dic127—24no—
53C4×Dic127D412yes—
54C4×Dic127C2312yes—
55C24×C127—21no—
56C2×D4×C127—22no—
57C2×D4×C127—21no—
58C2×D4×C127—22no—
59C2×D4×C127C2312yes—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62C2×C8×C127—22no—
63C2×C8×C127C2312yes—
64C4⋊C4×C127—21no—
65C4⋊C4×C127C2312yes—
66C22×D127—22no—
67C22×D127—42no—
68C22×D127—42no—
69C22×D127—42no—
70Dic254—24no—
71Dic254—22no—
72Dic254C2×D414yescomplemented
73D508—22no—
74D508—22no—
75D508—42no—
76C22⋊D127—48no—
77C22⋊D127—48no—
78C2×Dic127—24no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127SD1614yescomplemented
82C2×Dic127C2×D412yes—
83C2×Dic127C2×D412yes—
84C127⋊C8—22no—
85C23×C127—22no—
86C23×C127—42no—
87C23×C127—42no—
88C23×C127—42no—
89C23×C127C2×D411yescharacteristic
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—42no—
93C2×C4×C127—22no—
94C2×C4×C127—42no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D412yes—
97C2×C4×C127#4112yes—
98C2×C4×C127C2411yescharacteristic
99C8×C127—22no—
100C22×C127—42no—
101C22×C127—42no—
102C22×C127—42no—
103C22×C127—84no—
104C22×C127#12112yes—
105C22×C127C2×SD1612yes—
106C22×C127AGL(1,ℤ/8)12yes—
107C22×C127C22×D411yescharacteristic
108C2×D127—42no—
109C2×D127—42no—
110C2×D127—42no—
111C2×D127—84no—
112C4×C127—42no—
113C4×C127#12112yes—
114C4×C127C22×D411yescharacteristic
115C4×C127C2×#4111yescharacteristic
116Dic127—24no—
117Dic127—22no—
118Dic127C2×SD1614yescomplemented
119C2×C127—82no—
120C2×C127#46014yescomplemented
121C2×C127C2×#12111yescharacteristic
122C2×C127C22×SD1611yescharacteristic
123C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
124D127—82no—
125C2×#460—1271nomaximal, Sylow
126C127C2×#46011yescharacteristic, complemented, Sylow
127C22×SD16—1271no—
128C22×SD16—1271no—
129C2×#127—1271no—
130C2×C4×Q8—1271no—
131#460—1278no—
132C2×C4⋊C8—1271no—
133C2×#102—1271no—
134C2×#102D12711yescharacteristic, complemented
135C22×Q8—1271no—
136C22×D4—1271no—
137C22×D4—2541no—
138C22×D4—2541no—
139C22×D4C2×D12711yescharacteristic
140C2×SD16—1274no—
141C2×SD16—1274no—
142C2×SD16—2544no—
143C2×SD16—2544no—
144C22×C8—1271no—
145C22×C8C2×D12711yescharacteristic
146#127—1274no—
147#127—2542no—
148C4×Q8—1274no—
149C4×Q8—2542no—
150C2×C4⋊C4—2541no—
151C2×C4⋊C4C2×D12711yescharacteristic
152C2×C42—1271no—
153C2×C42—2541no—
154C4⋊C8—1274no—
155#102—1274no—
156#102C2×D12714yescomplemented
157C24—21no—
158C24—2541no—
159C2×Q8—1274no—
160C2×Q8—1272no—
161C2×Q8—2541no—
162C2×Q8—2542no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—1272no—
167C2×D4—5084no—
168C2×D4—5084no—
169C2×D4—2544no—
170C2×D4—2544no—
171C2×D4—2541no—
172C2×D4—2542no—
173C2×D4—2542no—
174C2×D4C22×D12712yes—
175C22×C4—21no—
176C22×C4—1271no—
177C22×C4—1271no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180SD16—2548no—
181SD16—2548no—
182C2×C8—22no—
183C2×C8—1272no—
184C2×C8—2542no—
185C2×C8C22×D12712yes—
186C4⋊C4—21no—
187C4⋊C4—2544no—
188C4⋊C4C22×D12712yes—
189C42—1272no—
190C42—1272no—
191C42—2544no—
192C23—22no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—5082no—
197C23—5082no—
198C23—5082no—
199C23—2542no—
200C23D4×D12711yescharacteristic
201Q8—1274no—
202Q8—2542no—
203Q8—2544no—
204D4—22no—
205D4—22no—
206D4—42no—
207D4—5088no—
208D4—5088no—
209D4—5082no—
210D4—2542no—
211D4—2542no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—1274no—
215C2×C4—1272no—
216C2×C4—1272no—
217C2×C4—2542no—
218C2×C4—2542no—
219C2×C4—2544no—
220C2×C4D4×D12712yes—
221C2×C4D4×D12711yescharacteristic
222C2×C4#1452324812yes—
223C2×C4C23×D12711yescharacteristic
224C8—22no—
225C8—2542no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—84no—
230C22—10164no—
231C22—5082no—
232C22—5082no—
233C22—5082no—
234C22#1686118212yes—
235C22SD16×D12712yes—
236C22#1686123312yes—
237C22C2×D4×D12711yescharacteristic
238C4—42no—
239C4—1274no—
240C4—2544no—
241C4—2542no—
242C4#1686118212yes—
243C4C2×D4×D12711yescharacteristic
244C4C2×#1452324811yescharacteristic
245C2—82no—
246C2—10162no—
247C2#2255007514yesdirect product factor
248C2C2×#1686118211yescharacteristic
249C2C2×SD16×D12711yescharacteristic
250C2C2×#1686123311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861205C2
hypercenterC2×#102D127
hypocenterC127C2×#460
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#14523237C2semidirect
2. C22×#14523236C2semidirect
4. C2×#16861174C2semidirect
5. #22550075C2direct (N × H)
6. C2×Dic127⋊C8C2semidirect
7. C2×#16861129C2semidirect
8. C2×#102×C127C2semidirect
14. C2×#14523237C22semidirect
16. C2×#14523236C22semidirect
21. #16861174C22semidirect
25. Dic127⋊C8C22semidirect
26. #16861129C22semidirect
30. #102×C127C22semidirect
34. C2×Dic254D4semidirect
72. Dic254C2×D4semidirect
81. C2×Dic127SD16semidirect
118. Dic127C2×SD16semidirect
120. C2×C127#460semidirect
126. C127C2×#460semidirect
134. C2×#102D127semidirect
156. #102C2×D127semidirect
247. C2#22550075direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232372
C22×#145232362
C2×#168612052
C2×#168611742
#225500752
C2×Dic127⋊C82
C2×#168611292
C2×#102×C1272
C2×#460127