Finite Group Databasebeta

← C2×#22550074 #30648160

Subgroups — C2×#22550074 #30648160

Order 16256.

584 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14523235C211yesmaximal, characteristic, complemented
2C2×#16861205C211yesmaximal, characteristic
3C2×Q8×Dic127C211yesmaximal, characteristic, complemented
4#22550074C218yesmaximal, direct product factor
5C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
6C2×#16861131C211yesmaximal, characteristic, complemented
7C2×#16861130C211yesmaximal, characteristic, complemented
8C22×SD16×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11C22×C22⋊D127—21no—
12C2×#14523235—24no—
13C2×#14523235C2214yescomplemented
14C22×C127⋊C8C2211yescharacteristic
15#16861205—22no—
16#16861205C2214yes—
17Q8×Dic127—22no—
18Q8×Dic127C2214yescomplemented
19C2×C4⋊Dic127—21no—
20C2×C4⋊Dic127C2211yescharacteristic
21C2×C4×Dic127—21no—
22C2×C4×Dic127C2211yescharacteristic
23Dic127⋊C8C2214yescomplemented
24#16861131C2214yescomplemented
25#16861130C2214yescomplemented
26C22×Q8×C127C2211yescharacteristic
27C22×D4×C127C2211yescharacteristic
28C2×SD16×C127—24no—
29C2×SD16×C127C2214yescomplemented
30C22×C8×C127C2211yescharacteristic
31C23×D127—21no—
32C2×D508—22no—
33C2×D508—21no—
34C2×D508—22no—
35C2×D508C2312yes—
36C2×C22⋊D127—24no—
37C2×C22⋊D127—24no—
38C2×C22⋊D127—44no—
39C2×C22⋊D127—44no—
40C22×Dic127—21no—
41C22×Dic127D411yescharacteristic
42C22×Dic127D411yescharacteristic
43#14523235—28no—
44C2×C127⋊C8—22no—
45C2×C127⋊C8C2312yes—
46C4⋊Dic127—24no—
47C4⋊Dic127—21no—
48C4⋊Dic127C2312yes—
49C4×Dic127—24no—
50C4×Dic127D412yes—
51C4×Dic127C2312yes—
52C24×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—22no—
55C2×Q8×C127D414yescomplemented
56C2×Q8×C127C2312yes—
57C2×D4×C127—22no—
58C2×D4×C127—21no—
59C2×D4×C127—22no—
60C2×D4×C127C2312yes—
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63SD16×C127—28no—
64C2×C8×C127—22no—
65C2×C8×C127C2312yes—
66C22×D127—22no—
67C22×D127—42no—
68C22×D127—42no—
69C22×D127—42no—
70D508—22no—
71D508—22no—
72D508—42no—
73C22⋊D127—48no—
74C22⋊D127—48no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—42no—
78C2×Dic127SD1614yescomplemented
79C2×Dic127C2×D412yes—
80C2×Dic127C2×D412yes—
81C127⋊C8—22no—
82C23×C127—22no—
83C23×C127—42no—
84C23×C127—42no—
85C23×C127—42no—
86C23×C127C2×D411yescharacteristic
87Q8×C127—24no—
88Q8×C127—22no—
89Q8×C127C2×D414yescomplemented
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—42no—
93C2×C4×C127—24no—
94C2×C4×C127—22no—
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—24no—
113C4×C127#12112yes—
114C4×C127C22×D411yescharacteristic
115C4×C127C2×#4111yescharacteristic
116Dic127—22no—
117Dic127—42no—
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×SD16D12711yescharacteristic, complemented
129C2×#127—1271no—
130C2×C4×Q8—1271no—
131#460—1278no—
132C2×C4⋊C8—1271no—
133C2×#102—1271no—
134C2×#102—1271no—
135C22×Q8C2×D12711yescharacteristic
136C22×D4—1271no—
137C22×D4—2541no—
138C22×D4—2541no—
139C22×D4C2×D12711yescharacteristic
140C2×SD16—24no—
141C2×SD16—1274no—
142C2×SD16—2544no—
143C2×SD16C2×D12714yescomplemented
144C22×C8—1271no—
145C22×C8C2×D12711yescharacteristic
146#127—1274no—
147#127—2542no—
148C4×Q8—1274no—
149C4×Q8—2542no—
150C2×C4⋊C4—1271no—
151C2×C4⋊C4—2541no—
152C2×C42—1271no—
153C2×C42—2541no—
154C4⋊C8—1274no—
155#102—1274no—
156#102—1274no—
157C24—21no—
158C24—2541no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8C22⋊D12714yescomplemented
162C2×Q8C22×D12712yes—
163C2×D4—21no—
164C2×D4—22no—
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—28no—
181SD16—2548no—
182C2×C8—22no—
183C2×C8—1272no—
184C2×C8—2542no—
185C2×C8C22×D12712yes—
186C4⋊C4—1272no—
187C4⋊C4—2544no—
188C4⋊C4—2541no—
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—22no—
202Q8—24no—
203Q8C2×C22⋊D12714yescomplemented
204D4—22no—
205D4—22no—
206D4—42no—
207D4—5088no—
208D4—5088no—
209D4—5082no—
210D4—2542no—
211D4—2542no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—1274no—
215C2×C4—1272no—
216C2×C4—1272no—
217C2×C4—5082no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4D4×D12711yescharacteristic
221C2×C4C2×C22⋊D12712yes—
222C2×C4#1452324712yes—
223C2×C4C23×D12711yescharacteristic
224C8—22no—
225C8—2542no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—84no—
230C22—10164no—
231C22—5082no—
232C22—5082no—
233C22—5082no—
234C22#1686118112yes—
235C22SD16×D12712yes—
236C22#1686123612yes—
237C22C2×D4×D12711yescharacteristic
238C4—24no—
239C4—1274no—
240C4—5082no—
241C4—2542no—
242C4#1686118112yes—
243C4C22×C22⋊D12711yescharacteristic
244C4C2×#1452324711yescharacteristic
245C2—82no—
246C2—10162no—
247C2#2255007414yesdirect product factor
248C2C2×#1686118111yescharacteristic
249C2C2×SD16×D12711yescharacteristic
250C2C2×#1686123611yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×SD16×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861205C2
hypercenterC22×SD16D127
hypocenterC127C2×#460
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14523235C2semidirect
3. C2×Q8×Dic127C2semidirect
4. #22550074C2direct (N × H)
5. C2×Dic127⋊C8C2semidirect
6. C2×#16861131C2semidirect
7. C2×#16861130C2semidirect
8. C22×SD16×C127C2semidirect
13. C2×#14523235C22semidirect
18. Q8×Dic127C22semidirect
23. Dic127⋊C8C22semidirect
24. #16861131C22semidirect
25. #16861130C22semidirect
29. C2×SD16×C127C22semidirect
55. C2×Q8×C127D4semidirect
78. C2×Dic127SD16semidirect
89. Q8×C127C2×D4semidirect
118. Dic127C2×SD16semidirect
120. C2×C127#460semidirect
126. C127C2×#460semidirect
128. C22×SD16D127semidirect
143. C2×SD16C2×D127semidirect
161. C2×Q8C22⋊D127semidirect
203. Q8C2×C22⋊D127semidirect
247. C2#22550074direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232352
C2×#168612052
C2×Q8×Dic1272
#225500742
C2×Dic127⋊C82
C2×#168611312
C2×#168611302
C22×SD16×C1272
C2×#460127