Finite Group Databasebeta

← C2×#22550199 #30648613

Subgroups — C2×#22550199 #30648613

Order 16256.

468 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861180C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic, complemented
3#22550199C218yesmaximal, direct product factor
4C2×#16861142C211yesmaximal, characteristic, complemented
5C2×#16861141C211yesmaximal, characteristic, complemented
6C2×#16861130C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
8C2×#102×C127C211yesmaximal, characteristic, complemented
9C22×C22⋊D127—21no—
10C22×C4×D127C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861180—22no—
13#16861180C2214yes—
14#16861167—22no—
15#16861167C2214yescomplemented
16C2×C4⋊Dic127C2211yescharacteristic
17C2×C4⋊Dic127C2211yescharacteristic
18C2×C22⋊Dic127—21no—
19C2×(C2×D127)⋊C4—21no—
20C2×C4×Dic127—21no—
21#16861142C2214yescomplemented
22#16861141C2214yescomplemented
23#16861130C2214yescomplemented
24(C2×D127)⋊C8C2214yescomplemented
25C22×D4×C127C2211yescharacteristic
26C22×C8×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28#102×C127C2214yescomplemented
29C23×D127D411yescharacteristic
30C2×C22⋊D127—24no—
31C2×C22⋊D127—44no—
32C22×Dic127—21no—
33C22×Dic127—21no—
34C22×Dic127D411yescharacteristic
35C2×C4×D127—22no—
36C2×C4×D127—22no—
37C2×C4×D127D412yes—
38C2×C4×D127C2312yes—
39C2×C127⋊C8—22no—
40C2×C127⋊C8C2312yes—
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43C4⋊Dic127C2312yes—
44C4⋊Dic127C2312yes—
45C22⋊Dic127—24no—
46(C2×D127)⋊C4—24no—
47C4×Dic127—24no—
48C24×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—22no—
51C2×D4×C127—22no—
52C2×D4×C127C2312yes—
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55C2×C8×C127—22no—
56C2×C8×C127C2312yes—
57C4⋊C4×C127—21no—
58C4⋊C4×C127C2312yes—
59C22×D127—22no—
60C22×D127—22no—
61C22×D127—22no—
62C22×D127C2×D412yes—
63C22⋊D127—48no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—42no—
68C2×Dic127—42no—
69C2×Dic127C2×D412yes—
70C4×D127—24no—
71C4×D127—22no—
72C4×D127—22no—
73C127⋊C8—22no—
74C23×C127—22no—
75C23×C127—42no—
76C23×C127—42no—
77C23×C127—42no—
78C23×C127C2×D411yescharacteristic
79D4×C127—22no—
80D4×C127—22no—
81D4×C127—42no—
82C2×C4×C127—22no—
83C2×C4×C127—42no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127#4112yes—
86C2×C4×C127#4112yes—
87C2×C4×C127C2411yescharacteristic
88C8×C127—22no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127—42no—
92C22×C127—84no—
93C22×C127#12312yes—
94C22×C127#13512yes—
95C22×C127AGL(1,ℤ/8)12yes—
96C22×C127C22×D411yescharacteristic
97C2×D127—22no—
98C2×D127—22no—
99C2×D127—22no—
100C2×D127—44no—
101C4×C127—42no—
102C4×C127#12312yes—
103C4×C127C2×#4111yescharacteristic
104C4×C127C2×#4111yescharacteristic
105Dic127—22no—
106Dic127—42no—
107Dic127—42no—
108C2×C127—82no—
109C2×C127#48214yescomplemented
110C2×C127C2×#12311yescharacteristic
111C2×C127C2×#13511yescharacteristic
112C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
113D127—42no—
114C2×#482—1271nomaximal, Sylow
115C127C2×#48211yescharacteristic, complemented, Sylow
116C2×#121—1271no—
117C2×#117—1271no—
118#482—1278no—
119C2×#107—1271no—
120C2×#106—1271no—
121C2×#102—1271no—
122C2×#102D12711yescharacteristic, complemented
123C2×C22⋊C8—1271no—
124C22×D4—2541no—
125C22×D4C2×D12711yescharacteristic
126C23×C4—1271no—
127C22×C8—1271no—
128C22×C8C2×D12711yescharacteristic
129#121—1274no—
130#121—2542no—
131#117—1274no—
132#117—2542no—
133C2×C4⋊C4—1271no—
134C2×C4⋊C4—1271no—
135C2×C4⋊C4C2×D12711yescharacteristic
136C2×C22⋊C4—2541no—
137C2×C22⋊C4—2541no—
138C2×C42—2541no—
139#107—1274no—
140#106—1274no—
141#102—1274no—
142#102C2×D12714yescomplemented
143C22⋊C8—1274no—
144C24—21no—
145C24—1271no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—5084no—
150C2×D4—2544no—
151C2×D4C22×D12712yes—
152C22×C4—21no—
153C22×C4—1271no—
154C22×C4—1272no—
155C22×C4—1272no—
156C22×C4—2541no—
157C22×C4—2541no—
158C22×C4—2542no—
159C22×C4—2542no—
160C22×C4C22×D12711yescharacteristic
161C2×C8—22no—
162C2×C8—1272no—
163C2×C8—2542no—
164C2×C8C22×D12712yes—
165C4⋊C4—21no—
166C4⋊C4—1272no—
167C4⋊C4—1272no—
168C4⋊C4—2541no—
169C4⋊C4—2541no—
170C4⋊C4C22×D12712yes—
171C22⋊C4—2544no—
172C22⋊C4—2544no—
173C42—2544no—
174C23—22no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—1272no—
179C23—2542no—
180C23—2542no—
181C23—2542no—
182C23D4×D12711yescharacteristic
183D4—22no—
184D4—22no—
185D4—42no—
186D4—5088no—
187C2×C4—22no—
188C2×C4—42no—
189C2×C4—1272no—
190C2×C4—5082no—
191C2×C4—5082no—
192C2×C4—2542no—
193C2×C4—2542no—
194C2×C4—2542no—
195C2×C4—2544no—
196C2×C4—2542no—
197C2×C4—2542no—
198C2×C4D4×D12711yescharacteristic
199C2×C4#1452324712yes—
200C2×C4#1452324812yes—
201C2×C4C23×D12711yescharacteristic
202C8—22no—
203C8—2542no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—84no—
208C22—5084no—
209C22—2542no—
210C22—2542no—
211C22—2542no—
212C22#1686119112yes—
213C22#1686123012yes—
214C22#1686123312yes—
215C22C2×D4×D12711yescharacteristic
216C4—42no—
217C4—5082no—
218C4—5082no—
219C4—2542no—
220C4#1686119112yes—
221C4C2×#1452324711yescharacteristic
222C4C2×#1452324811yescharacteristic
223C2—82no—
224C2—5082no—
225C2#2255019914yesdirect product factor
226C2C2×#1686119111yescharacteristic
227C2C2×#1686123011yescharacteristic
228C2C2×#1686123311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861180C2
hypercenterC2×#102D127
hypocenterC127C2×#482
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
2. C2×#16861167C2semidirect
3. #22550199C2direct (N × H)
4. C2×#16861142C2semidirect
5. C2×#16861141C2semidirect
6. C2×#16861130C2semidirect
7. C2×(C2×D127)⋊C8C2semidirect
8. C2×#102×C127C2semidirect
15. #16861167C22semidirect
21. #16861142C22semidirect
22. #16861141C22semidirect
23. #16861130C22semidirect
24. (C2×D127)⋊C8C22semidirect
28. #102×C127C22semidirect
109. C2×C127#482semidirect
115. C127C2×#482semidirect
122. C2×#102D127semidirect
142. #102C2×D127semidirect
225. C2#22550199direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168611802
C2×#168611672
#225501992
C2×#168611422
C2×#168611412
C2×#168611302
C2×(C2×D127)⋊C82
C2×#102×C1272
C2×#482127