Finite Group Databasebeta

← C2×#22550049 #30648066

Subgroups — C2×#22550049 #30648066

Order 16256.

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

normal subgroups only

This group has 728 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
1C2×#16861234C212yesmaximal, complemented
2C2×#16861198C211yesmaximal, characteristic, complemented
3#22550049C218yesmaximal, direct product factor
4C2×#16861136C212yesmaximal, complemented
5C2×#16861128C211yesmaximal, characteristic, complemented
6C2×2−1+4×C127C211yesmaximal, characteristic, complemented
7C22×D508C2211yescharacteristic
8#16861234—24no—
9#16861234C2218yescomplemented
10C2×#14523235—22no—
11C2×#14523233—22no—
12C2×#14523232C2212yes—
13#16861198—22no—
14#16861198C2214yescomplemented
15C2×(C2×D127)⋊C4—22no—
16C2×C4×Dic127C2211yescharacteristic
17#16861136C2218yescomplemented
18#16861128C2214yescomplemented
192−1+4×C127—24no—
202−1+4×C127—22no—
212−1+4×C127C2214yescomplemented
22C2×#41×C127—22no—
23C2×#41×C127—22no—
24C2×#41×C127C2212yes—
25C22×Q8×C127—22no—
26C22×Q8×C127C2211yescharacteristic
27C23×D127—21no—
28C2×D508—22no—
29C2×D508—22no—
30C2×D508—21no—
31C2×D508C2312yes—
32C22×Dic127—21no—
33#14523235—28no—
34#14523233—28no—
35#14523232—22no—
36#14523232C2314yes—
37C2×C127⋊C8—22no—
38(C2×D127)⋊C4—28no—
39C4×Dic127—21no—
40C4×Dic127C2312yes—
41#41×C127—28no—
42#41×C127—28no—
43#41×C127—28no—
44#41×C127—28no—
45#41×C127—22no—
46#41×C127—22no—
47#41×C127D414yes—
48#41×C127C2314yes—
49C2×Q8×C127—22no—
50C2×Q8×C127—28no—
51C2×Q8×C127—28no—
52C2×Q8×C127—22no—
53C2×Q8×C127—22no—
54C2×Q8×C127D412yes—
55C2×Q8×C127D412yes—
56C2×Q8×C127C2312yes—
57C2×D4×C127—22no—
58C2×D4×C127—22no—
59C2×D4×C127D412yes—
60C22×C4×C127—24no—
61C22×C4×C127—22no—
62C22×C4×C127D412yes—
63C22×C4×C127C2311yescharacteristic
64C22×D127—22no—
65C22×D127—44no—
66C22×D127—41no—
67C22×D127—41no—
68D508—24no—
69D508—42no—
70C2×Dic127—22no—
71C2×Dic127—42no—
72C127⋊C8—24no—
73C23×C127—22no—
74C23×C127C2×D411yescharacteristic
75Q8×C127—28no—
76Q8×C127—22no—
77Q8×C127—24no—
78Q8×C127—24no—
79Q8×C127C2×D414yes—
80D4×C127—28no—
81D4×C127—24no—
82D4×C127—24no—
83D4×C127—22no—
84D4×C127C2×D414yes—
85C2×C4×C127—216no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—24no—
90C2×C4×C127—24no—
91C2×C4×C127C2×D414yes—
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D412yes—
95C2×C4×C127C2411yescharacteristic
96C22×C127—22no—
97C22×C127—24no—
98C22×C127—44no—
99C22×C127#12011yescharacteristic
100C22×C127#12011yescharacteristic
101C22×C127C22×D411yescharacteristic
102C2×D127—44no—
103C2×D127—42no—
104C2×D127—82no—
105C2×D127—82no—
106C4×C127—24no—
107C4×C127—24no—
108C4×C127#12012yes—
109C4×C127C22×D412yes—
110Dic127—42no—
111C2×C127—21no—
112C2×C127—21no—
113C2×C127—44no—
114C2×C127#45412yescomplemented
115C2×C127C2×#12011yescharacteristic
116D127—82no—
117C2×#454—1271nomaximal, Sylow
118C127C2×#45411yescharacteristic, complemented, Sylow
119C2×2−1+4D12711yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—1272no—
121C2×#124—1271no—
122#454—1278no—
123C2×#104—1272no—
124C2×#101—1271no—
1252−1+4—24no—
1262−1+4—22no—
1272−1+4C2×D12714yescomplemented
128C2×#41—22no—
129C2×#41—22no—
130C2×#41C2×D12712yes—
131C22×Q8—22no—
132C22×Q8C2×D12711yescharacteristic
133C22×D4—1271no—
134AGL(1,ℤ/8)—1278no—
135AGL(1,ℤ/8)—2544no—
136C2×SD16—2542no—
137C2×D8—2542no—
138C2×M16—1272no—
139#124—1274no—
140#124—2542no—
141C2×C22⋊C4—2542no—
142C2×C42—1271no—
143#104—1278no—
144#101—1274no—
145C24—2541no—
146#41—28no—
147#41—28no—
148#41—28no—
149#41—28no—
150#41—22no—
151#41—22no—
152#41C22⋊D12714yes—
153#41C22×D12714yes—
154C2×Q8—22no—
155C2×Q8—28no—
156C2×Q8—28no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8C22⋊D12712yes—
160C2×Q8C22⋊D12712yes—
161C2×Q8C22×D12712yes—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—1272no—
165C2×D4—2541no—
166C2×D4—2542no—
167C2×D4—2542no—
168C2×D4C22⋊D12712yes—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—2541no—
172C22×C4C22⋊D12712yes—
173C22×C4C22×D12711yescharacteristic
174SD16—2548no—
175D8—2548no—
176M16—1274no—
177M16—2542no—
178C2×C8—2542no—
179C22⋊C4—2548no—
180C42—1272no—
181C42—2541no—
182C23—22no—
183C23—5084no—
184C23—5081no—
185C23—5081no—
186C23—2542no—
187C23C2×C22⋊D12711yescharacteristic
188Q8—28no—
189Q8—24no—
190Q8—22no—
191Q8—24no—
192Q8C2×C22⋊D12714yes—
193D4—28no—
194D4—24no—
195D4—24no—
196D4—22no—
197D4—5082no—
198D4—2544no—
199D4C2×C22⋊D12714yes—
200C2×C4—216no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—5082no—
207C2×C4—2542no—
208C2×C4C2×C22⋊D12714yes—
209C2×C4C2×C22⋊D12712yes—
210C2×C4C2×C22⋊D12712yes—
211C2×C4C2×C22⋊D12711yescharacteristic
212C2×C4C23×D12711yescharacteristic
213C8—2544no—
214C22—22no—
215C22—24no—
216C22—44no—
217C22—10162no—
218C22—10162no—
219C22—5084no—
220C22—5082no—
221C22#1686117811yescharacteristic
222C22#1686117811yescharacteristic
223C22C22×C22⋊D12711yescharacteristic
224C4—24no—
225C4—24no—
226C4—5082no—
227C4#1686117812yes—
228C4C22×C22⋊D12712yes—
229C2—21no—
230C2—21no—
231C2—44no—
232C2—10162no—
233C2#2255004912yesdirect product factor
234C2C2×#1686117811yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×2−1+4×C127C2
socleC22×C127#120
hypercenterC2×2−1+4D127
hypocenterC127C2×#454
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×#16861198C2semidirect
3. #22550049C2direct (N × H)
4. C2×#16861136C2semidirect
5. C2×#16861128C2semidirect
6. C2×2−1+4×C127C2semidirect
9. #16861234C22semidirect
14. #16861198C22semidirect
17. #16861136C22semidirect
18. #16861128C22semidirect
21. 2−1+4×C127C22semidirect
114. C2×C127#454semidirect
118. C127C2×#454semidirect
119. C2×2−1+4D127semidirect
127. 2−1+4C2×D127semidirect
233. C2#22550049direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×#168611982
#225500492
C2×#168611362
C2×#168611282
C2×2−1+4×C1272
C2×#454127