Finite Group Databasebeta

← C2×#22550188 #30648600

Subgroups — C2×#22550188 #30648600

Order 16256.

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

normal subgroups only

This group has 512 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×#16861180C211yesmaximal, characteristic
2C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
3#22550188C218yesmaximal, direct product factor
4C2×#16861143C211yesmaximal, characteristic, complemented
5C2×#16861142C211yesmaximal, characteristic, complemented
6C2×#16861130C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
8C2×#102×C127C211yesmaximal, characteristic, complemented
9C22×C22⋊D127—21no—
10C22×C4×D127—21no—
11C22×C4×D127C2211yescharacteristic
12C22×C127⋊C8C2211yescharacteristic
13#16861180—22no—
14#16861180C2214yes—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C4⋊Dic127C2211yescharacteristic
17C2×Dic127⋊C4—21no—
18C4⋊C4×D127—22no—
19C4⋊C4×D127C2214yescomplemented
20C2×C22⋊Dic127—21no—
21#16861143C2214yescomplemented
22#16861142C2214yescomplemented
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—24no—
36C2×C4×D127—22no—
37C2×C4×D127—22no—
38C2×C4×D127—44no—
39C2×C4×D127D412yes—
40C2×C4×D127C2312yes—
41C2×C127⋊C8—22no—
42C2×C127⋊C8C2312yes—
43C4⋊Dic127—21no—
44C4⋊Dic127—21no—
45C4⋊Dic127C2312yes—
46C4⋊Dic127C2312yes—
47Dic127⋊C4—24no—
48C22⋊Dic127—24no—
49C24×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—22no—
52C2×D4×C127—22no—
53C2×D4×C127C2312yes—
54C22×C4×C127—21no—
55C22×C4×C127C2311yescharacteristic
56C2×C8×C127—22no—
57C2×C8×C127C2312yes—
58C4⋊C4×C127—21no—
59C4⋊C4×C127C2312yes—
60C22×D127—22no—
61C22×D127—22no—
62C22×D127D814yes—
63C22×D127C2×D412yes—
64C22⋊D127—48no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—42no—
69C2×Dic127—42no—
70C2×Dic127C2×D412yes—
71C4×D127—24no—
72C4×D127—22no—
73C4×D127—22no—
74C4×D127—48no—
75C127⋊C8—22no—
76C23×C127—22no—
77C23×C127—42no—
78C23×C127—42no—
79C23×C127—42no—
80C23×C127C2×D411yescharacteristic
81D4×C127—22no—
82D4×C127—22no—
83D4×C127—42no—
84C2×C4×C127—22no—
85C2×C4×C127—42no—
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127#4112yes—
88C2×C4×C127#4112yes—
89C2×C4×C127C2411yescharacteristic
90C8×C127—22no—
91C22×C127—42no—
92C22×C127—42no—
93C22×C127—42no—
94C22×C127—84no—
95C22×C127#12312yes—
96C22×C127C2×D812yes—
97C22×C127#13712yes—
98C22×C127C22×D411yescharacteristic
99C2×D127—22no—
100C2×D127—22no—
101C2×D127—28no—
102C2×D127C2×D814yes—
103C4×C127—42no—
104C4×C127#12312yes—
105C4×C127C2×#4111yescharacteristic
106C4×C127C2×#4111yescharacteristic
107Dic127—22no—
108Dic127—42no—
109Dic127—42no—
110C2×C127—82no—
111C2×C127#48014yescomplemented
112C2×C127C2×#12311yescharacteristic
113C2×C127C22×D811yescharacteristic
114C2×C127C2×#13711yescharacteristic
115D127—24no—
116C2×#480—1271nomaximal, Sylow
117C127C2×#48011yescharacteristic, complemented, Sylow
118C2×#121—1271no—
119C22×C4⋊C4—1271no—
120#480—1278no—
121C2×#107—1271no—
122C2×#107—1271no—
123C2×#102—1271no—
124C2×#102D12711yescharacteristic, complemented
125C2×C22⋊C8—1271no—
126C22×D4—2541no—
127C22×D4C2×D12711yescharacteristic
128C23×C4—1271no—
129C23×C4—2541no—
130C22×C8—1271no—
131C22×C8C2×D12711yescharacteristic
132#121—1274no—
133#121—2542no—
134C2×C4⋊C4—1271no—
135C2×C4⋊C4—1271no—
136C2×C4⋊C4—1274no—
137C2×C4⋊C4—2542no—
138C2×C4⋊C4—2541no—
139C2×C4⋊C4C2×D12711yescharacteristic
140C2×C22⋊C4—2541no—
141#107—1274no—
142#107—1274no—
143#102—1274no—
144#102C2×D12714yescomplemented
145C22⋊C8—1274no—
146C24—21no—
147C24—1271no—
148C2×D4—21no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—5084no—
152C2×D4—2544no—
153C2×D4C22×D12712yes—
154C22×C4—21no—
155C22×C4—1271no—
156C22×C4—1272no—
157C22×C4—1272no—
158C22×C4—5084no—
159C22×C4—2544no—
160C22×C4—2541no—
161C22×C4—2541no—
162C22×C4—2542no—
163C22×C4—2542no—
164C22×C4C22×D12711yescharacteristic
165C2×C8—22no—
166C2×C8—1272no—
167C2×C8—2542no—
168C2×C8C22×D12712yes—
169C4⋊C4—21no—
170C4⋊C4—1272no—
171C4⋊C4—1272no—
172C4⋊C4—2544no—
173C4⋊C4—2541no—
174C4⋊C4—2541no—
175C4⋊C4C22×D12712yes—
176C22⋊C4—2544no—
177C23—22no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—1274no—
182C23—1272no—
183C23—2542no—
184C23—2542no—
185C23D4×D12711yescharacteristic
186D4—22no—
187D4—22no—
188D4—42no—
189D4—5088no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—1272no—
193C2×C4—5088no—
194C2×C4—5082no—
195C2×C4—5082no—
196C2×C4—2542no—
197C2×C4—2542no—
198C2×C4—2542no—
199C2×C4—2544no—
200C2×C4—2542no—
201C2×C4—2542no—
202C2×C4D4×D12711yescharacteristic
203C2×C4#1452324712yes—
204C2×C4#1452324812yes—
205C2×C4C23×D12711yescharacteristic
206C8—22no—
207C8—2542no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—84no—
212C22—1274no—
213C22—2542no—
214C22—2542no—
215C22—2548no—
216C22#1686119112yes—
217C22D8×D12712yes—
218C22#1686123812yes—
219C22C2×D4×D12711yescharacteristic
220C4—42no—
221C4—5082no—
222C4—5082no—
223C4—2542no—
224C4#1686119112yes—
225C4C2×#1452324711yescharacteristic
226C4C2×#1452324811yescharacteristic
227C2—82no—
228C2—2544no—
229C2#2255018814yesdirect product factor
230C2C2×#1686119111yescharacteristic
231C2C2×D8×D12711yescharacteristic
232C2C2×#1686123811yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861180C2
hypercenterC2×#102D127
hypocenterC127C2×#480
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×C4⋊C4×D127C2semidirect
3. #22550188C2direct (N × H)
4. C2×#16861143C2semidirect
5. C2×#16861142C2semidirect
6. C2×#16861130C2semidirect
7. C2×(C2×D127)⋊C8C2semidirect
8. C2×#102×C127C2semidirect
19. C4⋊C4×D127C22semidirect
21. #16861143C22semidirect
22. #16861142C22semidirect
23. #16861130C22semidirect
24. (C2×D127)⋊C8C22semidirect
28. #102×C127C22semidirect
111. C2×C127#480semidirect
117. C127C2×#480semidirect
124. C2×#102D127semidirect
144. #102C2×D127semidirect
229. C2#22550188direct (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×C4⋊C4×D1272
#225501882
C2×#168611432
C2×#168611422
C2×#168611302
C2×(C2×D127)⋊C82
C2×#102×C1272
C2×#480127