Finite Group Databasebeta

← C2×#22550079 #30648146

Subgroups — C2×#22550079 #30648146

Order 16256.

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

normal subgroups only

This group has 544 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
1C22×#14523236C211yesmaximal, characteristic
2C2×#16861207C211yesmaximal, characteristic, complemented
3C2×D4×Dic127C211yesmaximal, characteristic
4#22550079C218yesmaximal, direct product factor
5C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
6C2×#16861134C211yesmaximal, characteristic, complemented
7C2×#16861133C211yesmaximal, characteristic, complemented
8C22×SD16×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic
10C23×Dic127—21no—
11C2×#14523236—24no—
12C2×#14523236C2214yes—
13C22×C127⋊C8C2211yescharacteristic
14#16861207—22no—
15#16861207C2214yescomplemented
16D4×Dic127—22no—
17D4×Dic127C2214yes—
18C2×C4⋊Dic127C2211yescharacteristic
19C2×Dic127⋊C4—21no—
20C2×Dic127⋊C4—21no—
21C2×C22⋊Dic127—21no—
22C2×C4×Dic127C2211yescharacteristic
23Dic127⋊C8C2214yescomplemented
24#16861134C2214yescomplemented
25#16861133C2214yescomplemented
26C22×Q8×C127C2211yescharacteristic
27C22×D4×C127C2211yescharacteristic
28C2×SD16×C127—24no—
29C2×SD16×C127C2214yes—
30C22×C8×C127C2211yescharacteristic
31C2×Dic254—21no—
32C2×Dic254—22no—
33C2×Dic254—22no—
34C2×Dic254C2312yes—
35C22×Dic127—24no—
36C22×Dic127—21no—
37C22×Dic127—21no—
38C22×Dic127—44no—
39C22×Dic127D411yescharacteristic
40C22×Dic127D411yescharacteristic
41#14523236—28no—
42C2×C127⋊C8—22no—
43C2×C127⋊C8C2312yes—
44C4⋊Dic127—21no—
45C4⋊Dic127C2312yes—
46Dic127⋊C4—24no—
47Dic127⋊C4—24no—
48C22⋊Dic127—24no—
49C4×Dic127D412yes—
50C4×Dic127C2312yes—
51C24×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—22no—
54C2×Q8×C127—22no—
55C2×Q8×C127C2312yes—
56C2×D4×C127—21no—
57C2×D4×C127—22no—
58C2×D4×C127D414yes—
59C2×D4×C127C2312yes—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62SD16×C127—28no—
63C2×C8×C127—22no—
64C2×C8×C127C2312yes—
65Dic254—22no—
66Dic254—22no—
67Dic254—42no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—42no—
72C2×Dic127—42no—
73C2×Dic127—48no—
74C2×Dic127SD1614yescomplemented
75C2×Dic127C2×D412yes—
76C2×Dic127C2×D412yes—
77C127⋊C8—22no—
78C23×C127—24no—
79C23×C127—22no—
80C23×C127—42no—
81C23×C127—42no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—22no—
84Q8×C127—22no—
85Q8×C127—42no—
86D4×C127—24no—
87D4×C127—22no—
88D4×C127C2×D414yes—
89C2×C4×C127—22no—
90C2×C4×C127—42no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D412yes—
93C2×C4×C127#4112yes—
94C2×C4×C127C2411yescharacteristic
95C8×C127—22no—
96C22×C127—24no—
97C22×C127—42no—
98C22×C127—42no—
99C22×C127—48no—
100C22×C127#12112yes—
101C22×C127C2×SD1612yes—
102C22×C127#13712yes—
103C22×C127C22×D411yescharacteristic
104C4×C127—42no—
105C4×C127#12112yes—
106C4×C127C22×D411yescharacteristic
107C4×C127C2×#4111yescharacteristic
108Dic127—22no—
109Dic127—42no—
110Dic127—42no—
111Dic127C2×SD1614yescomplemented
112C2×C127—44no—
113C2×C127#46114yescomplemented
114C2×C127C2×#12111yescharacteristic
115C2×C127C22×SD1611yescharacteristic
116C2×C127C2×#13711yescharacteristic
117C2×#461—1271nomaximal, Sylow
118C127C2×#46111yescharacteristic, complemented, Sylow
119C22×SD16—1271no—
120C22×SD16D12711yescharacteristic
121C2×C4⋊Q8—1271no—
122C2×C4×D4—1271no—
123#461—1278no—
124C2×C4⋊C8—1271no—
125C2×#103—1271no—
126C2×#103—1271no—
127C22×Q8—1271no—
128C22×Q8C2×D12711yescharacteristic
129C22×D4C2×D12711yescharacteristic
130C23×C4—2541no—
131C2×SD16—24no—
132C2×SD16—1274no—
133C2×SD16—2544no—
134C2×SD16C2×D12714yes—
135C22×C8—1271no—
136C22×C8C2×D12711yescharacteristic
137C4⋊Q8—1274no—
138C4⋊Q8—2542no—
139C4×D4—1274no—
140C4×D4—2542no—
141C2×C4⋊C4—1271no—
142C2×C4⋊C4—2541no—
143C2×C4⋊C4—2541no—
144C2×C22⋊C4—2541no—
145C2×C42—1271no—
146C4⋊C8—1274no—
147#103—1274no—
148#103—1274no—
149C24—21no—
150C2×Q8—21no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—1272no—
154C2×Q8—2542no—
155C2×Q8—2542no—
156C2×Q8—2541no—
157C2×Q8C22×D12712yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4C22⋊D12714yes—
161C2×D4C22×D12712yes—
162C22×C4—21no—
163C22×C4—1271no—
164C22×C4—1271no—
165C22×C4—5084no—
166C22×C4—2544no—
167C22×C4—2541no—
168C22×C4—2541no—
169C22×C4C22×D12711yescharacteristic
170SD16—28no—
171SD16—2548no—
172C2×C8—22no—
173C2×C8—1272no—
174C2×C8—2542no—
175C2×C8C22×D12712yes—
176C4⋊C4—1272no—
177C4⋊C4—2544no—
178C4⋊C4—2544no—
179C4⋊C4—2541no—
180C22⋊C4—2544no—
181C42—1272no—
182C42—1272no—
183C23—22no—
184C23—24no—
185C23—42no—
186C23—42no—
187C23D4×D12711yescharacteristic
188Q8—22no—
189Q8—22no—
190Q8—42no—
191Q8—5082no—
192Q8—2542no—
193Q8—2542no—
194D4—22no—
195D4—24no—
196D4C2×C22⋊D12714yes—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—1274no—
200C2×C4—1272no—
201C2×C4—1272no—
202C2×C4—5088no—
203C2×C4—5082no—
204C2×C4—5082no—
205C2×C4—2542no—
206C2×C4—2542no—
207C2×C4—2542no—
208C2×C4D4×D12711yescharacteristic
209C2×C4C2×C22⋊D12712yes—
210C2×C4#1452324712yes—
211C2×C4C23×D12711yescharacteristic
212C8—22no—
213C8—2542no—
214C22—24no—
215C22—48no—
216C22—42no—
217C22—42no—
218C22#1686118112yes—
219C22SD16×D12712yes—
220C22#1686123812yes—
221C22C2×D4×D12711yescharacteristic
222C4—42no—
223C4—1274no—
224C4—5082no—
225C4—5082no—
226C4—2542no—
227C4#1686118112yes—
228C4C22×C22⋊D12711yescharacteristic
229C4C2×#1452324711yescharacteristic
230C2—44no—
231C2#2255007914yesdirect product factor
232C2C2×#1686118111yescharacteristic
233C2C2×SD16×D12711yescharacteristic
234C2C2×#1686123811yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×SD16×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenterC22×SD16D127
hypocenterC127C2×#461
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
2. C2×#16861207C2semidirect
4. #22550079C2direct (N × H)
5. C2×Dic127⋊C8C2semidirect
6. C2×#16861134C2semidirect
7. C2×#16861133C2semidirect
15. #16861207C22semidirect
23. Dic127⋊C8C22semidirect
24. #16861134C22semidirect
25. #16861133C22semidirect
74. C2×Dic127SD16semidirect
111. Dic127C2×SD16semidirect
113. C2×C127#461semidirect
118. C127C2×#461semidirect
231. C2#22550079direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168612072
C2×D4×Dic1272
#225500792
C2×Dic127⋊C82
C2×#168611342
C2×#168611332
C22×SD16×C1272
C2×#461127