Finite Group Databasebeta

← C2×#22550051 #30648068

Subgroups — C2×#22550051 #30648068

Order 16256.

728 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861257C211yesmaximal, characteristic, complemented
2C2×#16861237C212yesmaximal, complemented
3C2×#16861198C211yesmaximal, characteristic, complemented
4#22550051C218yesmaximal, direct product factor
5C2×#16861135C212yesmaximal, complemented
6C2×#101×C127C211yesmaximal, characteristic, complemented
7#16861257—24no—
8#16861257—22no—
9#16861257C2214yescomplemented
10C2×#14523249—22no—
11C2×#14523248—22no—
12C2×#14523248C2212yes—
13C2×Q8×D127—22no—
14C22×D508C2211yescharacteristic
15#16861237—24no—
16#16861237C2218yescomplemented
17C2×#14523237—22no—
18C2×D1016—22no—
19#16861198—22no—
20#16861198C2214yescomplemented
21C2×(C2×D127)⋊C4—22no—
22C2×C4×Dic127C2211yescharacteristic
23#16861135C2218yescomplemented
24C22×Q8×C127C2211yescharacteristic
25C2×M16×C127C2212yes—
26#101×C127C2214yescomplemented
27C23×D127—21no—
28#14523249—28no—
29#14523249—28no—
30#14523248—28no—
31#14523248—28no—
32#14523248—22no—
33#14523248—22no—
34#14523248D414yes—
35#14523248C2314yes—
36C2×Dic254—22no—
37C2×Dic254D412yes—
38Q8×D127—28no—
39Q8×D127—28no—
40C2×D508—22no—
41C2×D508—22no—
42C2×D508—22no—
43C2×D508—21no—
44C2×D508D412yes—
45C2×D508C2312yes—
46C2×C22⋊D127—22no—
47C22×Dic127—21no—
48C2×C4×D127—24no—
49C2×C4×D127—22no—
50#14523237—28no—
51D1016—28no—
52(C2×D127)⋊C4—28no—
53C4×Dic127—21no—
54C4×Dic127C2312yes—
55C2×Q8×C127—22no—
56C2×Q8×C127—22no—
57C2×Q8×C127D412yes—
58C2×Q8×C127C2312yes—
59C22×C4×C127D412yes—
60C22×C4×C127C2311yescharacteristic
61M16×C127—22no—
62M16×C127C2314yes—
63C2×C8×C127—22no—
64C22×D127—22no—
65C22×D127—22no—
66C22×D127—44no—
67C22×D127—41no—
68C22×D127—41no—
69Dic254—28no—
70Dic254—22no—
71Dic254C2×D414yes—
72D508—28no—
73D508—22no—
74D508—24no—
75D508—42no—
76D508C2×D414yes—
77C22⋊D127—24no—
78C22⋊D127—24no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—42no—
82C4×D127—216no—
83C4×D127—24no—
84C4×D127—24no—
85C23×C127C2×D411yescharacteristic
86Q8×C127—24no—
87Q8×C127—24no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127C2×D412yes—
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D414yes—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2411yescharacteristic
95C8×C127—24no—
96C22×C127—22no—
97C22×C127#12011yescharacteristic
98C22×C127#12011yescharacteristic
99C22×C127C22×D411yescharacteristic
100C2×D127—24no—
101C2×D127—44no—
102C2×D127—44no—
103C2×D127—42no—
104C2×D127—82no—
105C2×D127—82no—
106C4×C127—24no—
107C4×C127#12012yes—
108C4×C127C22×D412yes—
109Dic127—24no—
110Dic127—42no—
111C2×C127—21no—
112C2×C127—21no—
113C2×C127#45412yescomplemented
114C2×C127C2×#12011yescharacteristic
115D127—44no—
116D127—82no—
117C2×#454—1271nomaximal, Sylow
118C127C2×#45411yescharacteristic, complemented, Sylow
119C2×2−1+4—1271no—
120C2×AGL(1,ℤ/8)—1272no—
121C2×#124—1271no—
122#454—1278no—
123C2×#104—1272no—
124C2×#101D12711yescharacteristic, complemented
1252−1+4—1274no—
1262−1+4—2544no—
1272−1+4—2542no—
128C2×#41—1272no—
129C2×#41—2542no—
130C2×#41—2542no—
131C22×Q8—2542no—
132C22×Q8C2×D12711yescharacteristic
133C22×D4—1271no—
134AGL(1,ℤ/8)—1278no—
135AGL(1,ℤ/8)—2544no—
136C2×SD16—2542no—
137C2×D8—2542no—
138C2×M16C2×D12712yes—
139#124—1274no—
140#124—2542no—
141C2×C22⋊C4—2542no—
142C2×C42—1271no—
143#104—1278no—
144#101C2×D12714yescomplemented
145C24—2541no—
146#41—1274no—
147#41—1274no—
148#41—2548no—
149#41—2548no—
150#41—2548no—
151#41—2542no—
152#41—2542no—
153#41—2548no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1272no—
157C2×Q8—2542no—
158C2×Q8—2548no—
159C2×Q8—2548no—
160C2×Q8D50812yes—
161C2×Q8C22×D12712yes—
162C2×D4—1272no—
163C2×D4—1272no—
164C2×D4—2542no—
165C2×D4—2542no—
166C2×D4—2542no—
167C2×D4—2542no—
168C2×D4—2541no—
169C22×C4—2544no—
170C22×C4—2541no—
171C22×C4—2542no—
172C22×C4D50812yes—
173C22×C4C22×D12711yescharacteristic
174SD16—2548no—
175D8—2548no—
176M16—22no—
177M16C22×D12714yes—
178C2×C8—22no—
179C22⋊C4—2548no—
180C42—1272no—
181C42—2541no—
182C23—5084no—
183C23—5081no—
184C23—5081no—
185C23—2542no—
186C23—2542no—
187C23C2×D50811yescharacteristic
188Q8—24no—
189Q8—24no—
190Q8—1274no—
191Q8—2548no—
192Q8—2542no—
193D4—1274no—
194D4—5082no—
195D4—2548no—
196D4—2544no—
197D4—2544no—
198D4—2544no—
199D4—2542no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—5082no—
203C2×C4—25416no—
204C2×C4—2542no—
205C2×C4—2542no—
206C2×C4—2544no—
207C2×C4—2544no—
208C2×C4D4×D12712yes—
209C2×C4D4×D12712yes—
210C2×C4C2×D50814yes—
211C2×C4C2×D50811yescharacteristic
212C2×C4C23×D12711yescharacteristic
213C8—24no—
214C22—22no—
215C22—10162no—
216C22—10162no—
217C22—5084no—
218C22—5082no—
219C22—5084no—
220C22—2544no—
221C22#1686117911yescharacteristic
222C22#1686117911yescharacteristic
223C22C22×D50811yescharacteristic
224C4—24no—
225C4—5082no—
226C4—2544no—
227C4#1686117912yes—
228C4C2×D4×D12712yes—
229C2—21no—
230C2—21no—
231C2—10162no—
232C2—5084no—
233C2#2255005112yesdirect product factor
234C2C2×#1686117911yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#101×C127C2
socleC22×C127#120
hypercenterC2×#101D127
hypocenterC127C2×#454
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
1. C2×#16861257C2semidirect
2. C2×#16861237C2semidirect
3. C2×#16861198C2semidirect
4. #22550051C2direct (N × H)
5. C2×#16861135C2semidirect
6. C2×#101×C127C2semidirect
9. #16861257C22semidirect
16. #16861237C22semidirect
20. #16861198C22semidirect
23. #16861135C22semidirect
26. #101×C127C22semidirect
113. C2×C127#454semidirect
118. C127C2×#454semidirect
124. C2×#101D127semidirect
144. #101C2×D127semidirect
233. C2#22550051direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612572
C2×#168612372
C2×#168611982
#225500512
C2×#168611352
C2×#101×C1272
C2×#454127