Finite Group Databasebeta

← C2×#22550013 #30647916

Subgroups — C2×#22550013 #30647916

Order 16256.

792 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×Q8×D127C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3C2×#16861180C211yesmaximal, characteristic
4#22550013C218yesmaximal, direct product factor
5C2×#16861134C211yesmaximal, characteristic, complemented
6C2×#16861133C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
8C22×SD16×C127C211yesmaximal, characteristic, complemented
9C22×Dic254—21no—
10C22×Dic254C2211yescharacteristic
11C2×Q8×D127—24no—
12C2×Q8×D127—22no—
13C2×Q8×D127—24no—
14C2×Q8×D127C2214yescomplemented
15C22×C22⋊D127—21no—
16C22×C4×D127—21no—
17C22×C4×D127C2211yescharacteristic
18C2×#14523236—24no—
19C2×#14523236C2214yescomplemented
20C22×C127⋊C8C2211yescharacteristic
21#16861180—22no—
22#16861180C2214yes—
23C2×C4⋊Dic127C2211yescharacteristic
24C2×C22⋊Dic127—21no—
25#16861134C2214yescomplemented
26#16861133C2214yescomplemented
27(C2×D127)⋊C8C2214yescomplemented
28C22×Q8×C127C2211yescharacteristic
29C22×D4×C127C2211yescharacteristic
30C2×SD16×C127—24no—
31C2×SD16×C127C2214yescomplemented
32C22×C8×C127C2211yescharacteristic
33C23×D127D411yescharacteristic
34C2×Dic254—28no—
35C2×Dic254—24no—
36C2×Dic254—21no—
37C2×Dic254—22no—
38C2×Dic254D414yescomplemented
39C2×Dic254C2312yes—
40Q8×D127—216no—
41Q8×D127—28no—
42Q8×D127—28no—
43C2×C22⋊D127—24no—
44C2×C22⋊D127—44no—
45C22×Dic127—21no—
46C22×Dic127—21no—
47C22×Dic127D411yescharacteristic
48C2×C4×D127—28no—
49C2×C4×D127—24no—
50C2×C4×D127—22no—
51C2×C4×D127—22no—
52C2×C4×D127D412yes—
53C2×C4×D127C2312yes—
54#14523236—28no—
55C2×C127⋊C8—22no—
56C2×C127⋊C8C2312yes—
57C4⋊Dic127—21no—
58C4⋊Dic127C2312yes—
59C22⋊Dic127—24no—
60C24×C127—21no—
61C2×Q8×C127—22no—
62C2×Q8×C127—21no—
63C2×Q8×C127D414yescomplemented
64C2×Q8×C127C2312yes—
65C2×D4×C127—22no—
66C2×D4×C127—22no—
67C2×D4×C127—21no—
68C2×D4×C127C2312yes—
69C22×C4×C127—21no—
70C22×C4×C127C2311yescharacteristic
71SD16×C127—28no—
72C2×C8×C127—22no—
73C2×C8×C127C2312yes—
74C22×D127—22no—
75C22×D127—22no—
76C22×D127SD1614yescomplemented
77C22×D127C2×D412yes—
78Dic254—216no—
79Dic254—24no—
80Dic254—22no—
81Dic254C2×D414yescomplemented
82C22⋊D127—48no—
83C2×Dic127—24no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127—22no—
87C2×Dic127—42no—
88C2×Dic127C2×D412yes—
89C4×D127—216no—
90C4×D127—24no—
91C4×D127—22no—
92C4×D127—22no—
93C127⋊C8—22no—
94C23×C127—22no—
95C23×C127—42no—
96C23×C127—42no—
97C23×C127—42no—
98C23×C127C2×D411yescharacteristic
99Q8×C127—24no—
100Q8×C127—22no—
101Q8×C127C2×D414yescomplemented
102D4×C127—22no—
103D4×C127—22no—
104D4×C127—42no—
105C2×C4×C127—24no—
106C2×C4×C127—22no—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D412yes—
109C2×C4×C127C2×D412yes—
110C2×C4×C127C2411yescharacteristic
111C8×C127—22no—
112C22×C127—42no—
113C22×C127—42no—
114C22×C127—42no—
115C22×C127—84no—
116C22×C127#12012yes—
117C22×C127C2×SD1612yes—
118C22×C127#13712yes—
119C22×C127C22×D411yescharacteristic
120C2×D127—22no—
121C2×D127—22no—
122C2×D127—28no—
123C2×D127C2×SD1614yescomplemented
124C4×C127—24no—
125C4×C127#12012yes—
126C4×C127C22×D411yescharacteristic
127C4×C127C22×D411yescharacteristic
128Dic127—24no—
129Dic127—22no—
130Dic127—42no—
131C2×C127—82no—
132C2×C127#44814yescomplemented
133C2×C127C2×#12011yescharacteristic
134C2×C127C22×SD1611yescharacteristic
135C2×C127C2×#13711yescharacteristic
136D127—24no—
137C2×#448—1271nomaximal, Sylow
138C127C2×#44811yescharacteristic, complemented, Sylow
139C23×Q8—1271no—
140C22×SD16—1271no—
141C22×SD16D12711yescharacteristic, complemented
142C2×#121—1271no—
143#448—1278no—
144C2×#103—1271no—
145C2×#103—1271no—
146C2×C22⋊C8—1271no—
147C22×Q8—1274no—
148C22×Q8—1271no—
149C22×Q8—2541no—
150C22×Q8—2542no—
151C22×Q8—2544no—
152C22×Q8—2544no—
153C22×Q8C2×D12711yescharacteristic
154C22×D4—2541no—
155C22×D4C2×D12711yescharacteristic
156C23×C4—1271no—
157C23×C4—2541no—
158C2×SD16—24no—
159C2×SD16—1274no—
160C2×SD16—2544no—
161C2×SD16C2×D12714yescomplemented
162C22×C8—1271no—
163C22×C8C2×D12711yescharacteristic
164#121—1274no—
165#121—2542no—
166C2×C4⋊C4—1271no—
167C2×C22⋊C4—2541no—
168#103—1274no—
169#103—1274no—
170C22⋊C8—1274no—
171C24—21no—
172C24—1271no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—1274no—
176C2×Q8—1272no—
177C2×Q8—2544no—
178C2×Q8—2548no—
179C2×Q8—2548no—
180C2×Q8—25416no—
181C2×Q8—2548no—
182C2×Q8—2541no—
183C2×Q8—2542no—
184C2×Q8C22⋊D12714yescomplemented
185C2×Q8C22×D12712yes—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—5084no—
190C2×D4—2544no—
191C2×D4C22×D12712yes—
192C22×C4—21no—
193C22×C4—1271no—
194C22×C4—1272no—
195C22×C4—1272no—
196C22×C4—2544no—
197C22×C4—2548no—
198C22×C4—2541no—
199C22×C4—2541no—
200C22×C4—2542no—
201C22×C4—2542no—
202C22×C4C22×D12711yescharacteristic
203SD16—28no—
204SD16—2548no—
205C2×C8—22no—
206C2×C8—1272no—
207C2×C8—2542no—
208C2×C8C22×D12712yes—
209C4⋊C4—1272no—
210C4⋊C4—2541no—
211C22⋊C4—2544no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—1274no—
217C23—1272no—
218C23—2542no—
219C23—2542no—
220C23D4×D12711yescharacteristic
221Q8—22no—
222Q8—24no—
223Q8—1274no—
224Q8—25416no—
225Q8—2542no—
226Q8—2544no—
227Q8C2×C22⋊D12714yescomplemented
228D4—22no—
229D4—22no—
230D4—42no—
231D4—5088no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—1272no—
235C2×C4—5082no—
236C2×C4—25416no—
237C2×C4—2542no—
238C2×C4—2542no—
239C2×C4—2542no—
240C2×C4—2544no—
241C2×C4—2542no—
242C2×C4—2542no—
243C2×C4—2544no—
244C2×C4D4×D12712yes—
245C2×C4D4×D12711yescharacteristic
246C2×C4C2×C22⋊D12712yes—
247C2×C4C23×D12711yescharacteristic
248C8—22no—
249C8—2542no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—84no—
254C22—1274no—
255C22—2542no—
256C22—2542no—
257C22—2548no—
258C22#1686117712yes—
259C22SD16×D12712yes—
260C22#1686123812yes—
261C22C2×D4×D12711yescharacteristic
262C4—24no—
263C4—5082no—
264C4—2542no—
265C4—2544no—
266C4#1686117712yes—
267C4C2×D4×D12711yescharacteristic
268C4C22×C22⋊D12711yescharacteristic
269C2—82no—
270C2—2544no—
271C2#2255001314yesdirect product factor
272C2C2×#1686117711yescharacteristic
273C2C2×SD16×D12711yescharacteristic
274C2C2×#1686123811yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×SD16×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861180C2
hypercenterC22×SD16D127
hypocenterC127C2×#448
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D127C2semidirect
2. C22×#14523236C2semidirect
4. #22550013C2direct (N × H)
5. C2×#16861134C2semidirect
6. C2×#16861133C2semidirect
7. C2×(C2×D127)⋊C8C2semidirect
8. C22×SD16×C127C2semidirect
14. C2×Q8×D127C22semidirect
19. C2×#14523236C22semidirect
25. #16861134C22semidirect
26. #16861133C22semidirect
27. (C2×D127)⋊C8C22semidirect
31. C2×SD16×C127C22semidirect
38. C2×Dic254D4semidirect
63. C2×Q8×C127D4semidirect
76. C22×D127SD16semidirect
81. Dic254C2×D4semidirect
101. Q8×C127C2×D4semidirect
123. C2×D127C2×SD16semidirect
132. C2×C127#448semidirect
138. C127C2×#448semidirect
141. C22×SD16D127semidirect
161. C2×SD16C2×D127semidirect
184. C2×Q8C22⋊D127semidirect
227. Q8C2×C22⋊D127semidirect
271. C2#22550013direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D1272
C22×#145232362
C2×#168611802
#225500132
C2×#168611342
C2×#168611332
C2×(C2×D127)⋊C82
C22×SD16×C1272
C2×#448127