Finite Group Databasebeta

← C2×#22550200 #30648614

Subgroups — C2×#22550200 #30648614

Order 16256.

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

normal subgroups only

This group has 468 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×#16861183C211yesmaximal, characteristic
2#22550200C218yesmaximal, direct product factor
3C2×#16861142C211yesmaximal, characteristic, complemented
4C2×#16861140C211yesmaximal, characteristic, complemented
5C2×#16861129C211yesmaximal, characteristic, complemented
6C2×#16861129C211yesmaximal, characteristic, complemented
7C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
8C2×#117×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11C22×C127⋊C8C2211yescharacteristic
12C22×C127⋊C8C2211yescharacteristic
13#16861183—22no—
14#16861183C2214yes—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×(C2×D127)⋊C4—21no—
17#16861142C2214yescomplemented
18#16861140C2214yescomplemented
19#16861129C2214yescomplemented
20#16861129C2214yescomplemented
21C22⋊(C127⋊C8)C2214yescomplemented
22C23×C4×C127C2211yescharacteristic
23#117×C127—22no—
24#117×C127C2214yescomplemented
25C2×C4⋊C4×C127C2211yescharacteristic
26C2×C4⋊C4×C127C2211yescharacteristic
27C2×C22⋊C4×C127—21no—
28C2×C42×C127—21no—
29C23×D127—21no—
30C2×D508—21no—
31C2×D508—22no—
32C2×D508—22no—
33C2×D508C2312yes—
34C2×C22⋊D127—24no—
35C2×C22⋊D127—44no—
36C22×Dic127—21no—
37C2×C127⋊C8—22no—
38C2×C127⋊C8—22no—
39C2×C127⋊C8C2312yes—
40C2×C127⋊C8C2312yes—
41C4⋊Dic127—21no—
42C4⋊Dic127C2312yes—
43(C2×D127)⋊C4—24no—
44C24×C127D411yescharacteristic
45C22×C4×C127—22no—
46C22×C4×C127—22no—
47C22×C4×C127—21no—
48C22×C4×C127—21no—
49C22×C4×C127D411yescharacteristic
50C22×C4×C127D412yes—
51C22×C4×C127C2312yes—
52C22×C4×C127C2311yescharacteristic
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127C2312yes—
56C4⋊C4×C127C2312yes—
57C22⋊C4×C127—24no—
58C42×C127—24no—
59C22×D127—22no—
60C22×D127—42no—
61C22×D127—42no—
62C22×D127—42no—
63D508—22no—
64D508—22no—
65D508—42no—
66C22⋊D127—48no—
67C2×Dic127—22no—
68C2×Dic127—42no—
69C127⋊C8—22no—
70C127⋊C8—22no—
71C23×C127—22no—
72C23×C127—22no—
73C23×C127—22no—
74C23×C127C2×D412yes—
75C23×C127C2×D411yescharacteristic
76C2×C4×C127—22no—
77C2×C4×C127—24no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—42no—
83C2×C4×C127—42no—
84C2×C4×C127C2×D412yes—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4112yes—
87C2×C4×C127#4112yes—
88C2×C4×C127C2411yescharacteristic
89C22×C127—22no—
90C22×C127—22no—
91C22×C127—22no—
92C22×C127—44no—
93C22×C127#12312yes—
94C22×C127#13512yes—
95C22×C127AGL(1,ℤ/8)12yes—
96C22×C127C22×D411yescharacteristic
97C2×D127—42no—
98C2×D127—42no—
99C2×D127—42no—
100C2×D127—84no—
101C4×C127—22no—
102C4×C127—42no—
103C4×C127—42no—
104C4×C127#12312yes—
105C4×C127C2×#4111yescharacteristic
106C4×C127C2×#4111yescharacteristic
107Dic127—42no—
108C2×C127—42no—
109C2×C127#48214yescomplemented
110C2×C127C2×#12311yescharacteristic
111C2×C127C2×#13511yescharacteristic
112C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
113D127—82no—
114C2×#482—1271nomaximal, Sylow
115C127C2×#48211yescharacteristic, complemented, Sylow
116C2×#121—1271no—
117C2×#117D12711yescharacteristic, complemented
118#482—1278no—
119C2×#107—1271no—
120C2×#106—1271no—
121C2×#102—1271no—
122C2×#102—1271no—
123C2×C22⋊C8—1271no—
124C22×D4—1271no—
125C22×D4—2541no—
126C23×C4C2×D12711yescharacteristic
127C22×C8—1271no—
128C22×C8—1271no—
129#121—1274no—
130#121—2542no—
131#117—22no—
132#117C2×D12714yescomplemented
133C2×C4⋊C4—1271no—
134C2×C4⋊C4C2×D12711yescharacteristic
135C2×C4⋊C4C2×D12711yescharacteristic
136C2×C22⋊C4—21no—
137C2×C22⋊C4—2541no—
138C2×C42—21no—
139#107—1274no—
140#106—1274no—
141#102—1274no—
142#102—1274no—
143C22⋊C8—1274no—
144C24—2541no—
145C24C22⋊D12711yescharacteristic
146C2×D4—1272no—
147C2×D4—5084no—
148C2×D4—2544no—
149C2×D4—2541no—
150C2×D4—2542no—
151C2×D4—2542no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—2541no—
157C22×C4C22⋊D12712yes—
158C22×C4C22⋊D12711yescharacteristic
159C22×C4C22×D12712yes—
160C22×C4C22×D12711yescharacteristic
161C2×C8—1272no—
162C2×C8—1272no—
163C2×C8—2542no—
164C2×C8—2542no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—1272no—
168C4⋊C4—2541no—
169C4⋊C4C22×D12712yes—
170C4⋊C4C22×D12712yes—
171C22⋊C4—24no—
172C22⋊C4—2544no—
173C42—24no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—5082no—
178C23—5082no—
179C23—5082no—
180C23—2542no—
181C23C2×C22⋊D12712yes—
182C23C2×C22⋊D12711yescharacteristic
183D4—5088no—
184D4—5082no—
185D4—2542no—
186D4—2542no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—5082no—
196C2×C4—2542no—
197C2×C4C2×C22⋊D12712yes—
198C2×C4C2×C22⋊D12711yescharacteristic
199C2×C4#1452324812yes—
200C2×C4#1452324812yes—
201C2×C4C23×D12711yescharacteristic
202C8—2542no—
203C8—2542no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—44no—
208C22—10164no—
209C22—5082no—
210C22—5082no—
211C22—5082no—
212C22#1686119212yes—
213C22#1686122912yes—
214C22#1686123412yes—
215C22C22×C22⋊D12711yescharacteristic
216C4—22no—
217C4—42no—
218C4—42no—
219C4—5082no—
220C4#1686119212yes—
221C4C2×#1452324811yescharacteristic
222C4C2×#1452324811yescharacteristic
223C2—42no—
224C2—10162no—
225C2#2255020014yesdirect product factor
226C2C2×#1686119211yescharacteristic
227C2C2×#1686122911yescharacteristic
228C2C2×#1686123411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#117×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861183C2
hypercenterC2×#117D127
hypocenterC127C2×#482
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22550200C2direct (N × H)
3. C2×#16861142C2semidirect
4. C2×#16861140C2semidirect
5. C2×#16861129C2semidirect
6. C2×#16861129C2semidirect
7. C2×C22⋊(C127⋊C8)C2semidirect
8. C2×#117×C127C2semidirect
17. #16861142C22semidirect
18. #16861140C22semidirect
19. #16861129C22semidirect
20. #16861129C22semidirect
21. C22⋊(C127⋊C8)C22semidirect
24. #117×C127C22semidirect
109. C2×C127#482semidirect
115. C127C2×#482semidirect
117. C2×#117D127semidirect
132. #117C2×D127semidirect
225. C2#22550200direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168611832
#225502002
C2×#168611422
C2×#168611402
C2×#168611292
C2×C22⋊(C127⋊C8)2
C2×#117×C1272
C2×#482127