Finite Group Databasebeta

← C2×#30631935 #44559955

Subgroups — C2×#30631935 #44559955

Order 32128.

544 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16860797C211yesmaximal, characteristic, complemented
2C22×#16860794C211yesmaximal, characteristic, complemented
3C2×#22548689C211yesmaximal, characteristic
4#30631935C218yesmaximal, direct product factor
5C2×#22548648C211yesmaximal, characteristic, complemented
6C2×#22548639C211yesmaximal, characteristic, complemented
7C2×#22548635C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C251C211yesmaximal, characteristic, complemented
9C22×D1004C2211yescharacteristic
10C22×C22⋊D251—21no—
11C2×#16860797—24no—
12C2×#16860797C2214yescomplemented
13C2×#16860794—24no—
14C2×#16860794C2214yescomplemented
15C22×C251⋊C8C2211yescharacteristic
16C22×C251⋊C8C2211yescharacteristic
17#22548689—22no—
18#22548689C2214yes—
19C2×C4⋊Dic251C2211yescharacteristic
20C2×(C2×D251)⋊C4—21no—
21#22548648C2214yescomplemented
22#22548639C2214yescomplemented
23#22548635C2214yescomplemented
24C22×Q8×C251C2211yescharacteristic
25C23×C4×C251C2211yescharacteristic
26C22⋊Q8×C251—22no—
27C22⋊Q8×C251C2214yescomplemented
28C2×C4⋊C4×C251—21no—
29C2×C4⋊C4×C251C2211yescharacteristic
30C2×C22⋊C4×C251—21no—
31C23×D251—21no—
32C2×D1004—22no—
33C2×D1004—21no—
34C2×D1004—22no—
35C2×D1004C2312yes—
36C2×C22⋊D251—24no—
37C2×C22⋊D251—44no—
38C22×Dic251—21no—
39#16860797—28no—
40#16860794—28no—
41C2×C251⋊C8—22no—
42C2×C251⋊C8D414yescomplemented
43C2×C251⋊C8C2312yes—
44C2×C251⋊C8C2312yes—
45C4⋊Dic251—21no—
46C4⋊Dic251C2312yes—
47(C2×D251)⋊C4—24no—
48C24×C251D411yescharacteristic
49C2×Q8×C251—21no—
50C2×Q8×C251—22no—
51C2×Q8×C251—22no—
52C2×Q8×C251C2312yes—
53C22×C4×C251—22no—
54C22×C4×C251—22no—
55C22×C4×C251—21no—
56C22×C4×C251—21no—
57C22×C4×C251D411yescharacteristic
58C22×C4×C251D412yes—
59C22×C4×C251C2312yes—
60C22×C4×C251C2311yescharacteristic
61C4⋊C4×C251—24no—
62C4⋊C4×C251—21no—
63C4⋊C4×C251C2312yes—
64C22⋊C4×C251—24no—
65C22×D251—22no—
66C22×D251—42no—
67C22×D251—42no—
68C22×D251—42no—
69D1004—22no—
70D1004—22no—
71D1004—42no—
72C22⋊D251—48no—
73C2×Dic251—22no—
74C2×Dic251—42no—
75C251⋊C8—22no—
76C251⋊C8C2×D414yescomplemented
77C23×C251—22no—
78C23×C251—22no—
79C23×C251—22no—
80C23×C251C2×D412yes—
81C23×C251C2×D411yescharacteristic
82Q8×C251—22no—
83Q8×C251—22no—
84Q8×C251—42no—
85C2×C4×C251—22no—
86C2×C4×C251—24no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—22no—
91C2×C4×C251—42no—
92C2×C4×C251—42no—
93C2×C4×C251C2×D412yes—
94C2×C4×C251C2×D412yes—
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251#4112yes—
97C2×C4×C251C2411yescharacteristic
98C22×C251—22no—
99C22×C251—22no—
100C22×C251—22no—
101C22×C251—44no—
102C22×C251#12112yes—
103C22×C251AGL(1,ℤ/8)12yes—
104C22×C251#13712yes—
105C22×C251C22×D411yescharacteristic
106C2×D251—42no—
107C2×D251—42no—
108C2×D251—42no—
109C2×D251—84no—
110C4×C251—22no—
111C4×C251—42no—
112C4×C251—42no—
113C4×C251#12112yes—
114C4×C251C22×D411yescharacteristic
115C4×C251C2×#4111yescharacteristic
116Dic251—42no—
117C2×C251—42no—
118C2×C251#46814yescomplemented
119C2×C251C2×#12111yescharacteristic
120C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C251C2×#13711yescharacteristic
122D251—82no—
123C251C2×#46811yescharacteristic, complemented, Sylow
124C2×#468—2511nomaximal, Sylow
125C22×SD16—2511no—
126C22×M16—2511no—
127C2×C22⋊Q8D25111yescharacteristic, complemented
128C2×#121—2511no—
129#468—2518no—
130C2×#107—2511no—
131C2×#103—2511no—
132C2×#102—2511no—
133C22×Q8C2×D25111yescharacteristic
134C22×D4—5021no—
135C22×D4—2511no—
136C23×C4C2×D25111yescharacteristic
137C2×SD16—5024no—
138C2×SD16—2514no—
139C2×M16—5024no—
140C2×M16—2514no—
141C22×C8—2511no—
142C22×C8—2511no—
143C22⋊Q8—22no—
144C22⋊Q8C2×D25114yescomplemented
145#121—5022no—
146#121—2514no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—2511no—
149C2×C4⋊C4C2×D25111yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—5021no—
152#107—2514no—
153#103—2514no—
154#102—2514no—
155C24—5021no—
156C24C22⋊D25111yescharacteristic
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D25112yes—
161C2×D4—10044no—
162C2×D4—5024no—
163C2×D4—5022no—
164C2×D4—5021no—
165C2×D4—5022no—
166C2×D4—2512no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—5021no—
172C22×C4C22⋊D25112yes—
173C22×C4C22⋊D25111yescharacteristic
174C22×C4C22×D25112yes—
175C22×C4C22×D25111yescharacteristic
176SD16—5028no—
177M16—5028no—
178C2×C8—5022no—
179C2×C8—2512no—
180C2×C8—2514no—
181C2×C8—2512no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—5021no—
185C4⋊C4—2512no—
186C4⋊C4C22×D25112yes—
187C22⋊C4—24no—
188C22⋊C4—5024no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—10042no—
193C23—10042no—
194C23—10042no—
195C23—5022no—
196C23C2×C22⋊D25112yes—
197C23C2×C22⋊D25111yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—10048no—
202D4—10042no—
203D4—5022no—
204D4—5022no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—10042no—
214C2×C4—5022no—
215C2×C4D4×D25112yes—
216C2×C4C2×C22⋊D25112yes—
217C2×C4C2×C22⋊D25111yescharacteristic
218C2×C4#1686080912yes—
219C2×C4C23×D25111yescharacteristic
220C8—5022no—
221C8—2514no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—44no—
226C22—20084no—
227C22—10042no—
228C22—10042no—
229C22—10042no—
230C22#2254868712yes—
231C22#2254874012yes—
232C22#2254874612yes—
233C22C22×C22⋊D25111yescharacteristic
234C4—22no—
235C4—42no—
236C4—42no—
237C4—10042no—
238C4#2254868712yes—
239C4C2×D4×D25111yescharacteristic
240C4C2×#1686080911yescharacteristic
241C2—42no—
242C2—20082no—
243C2#3063193514yesdirect product factor
244C2C2×#2254868711yescharacteristic
245C2C2×#2254874011yescharacteristic
246C2C2×#2254874611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×C22⋊Q8×C251C2
socleC23×C251C2×D4
prime radicalC2×#22548689C2
hypercenterC2×C22⋊Q8D251
hypocenterC251C2×#468
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#16860797C2semidirect
2. C22×#16860794C2semidirect
4. #30631935C2direct (N × H)
5. C2×#22548648C2semidirect
6. C2×#22548639C2semidirect
7. C2×#22548635C2semidirect
8. C2×C22⋊Q8×C251C2semidirect
12. C2×#16860797C22semidirect
14. C2×#16860794C22semidirect
21. #22548648C22semidirect
22. #22548639C22semidirect
23. #22548635C22semidirect
27. C22⋊Q8×C251C22semidirect
42. C2×C251⋊C8D4semidirect
76. C251⋊C8C2×D4semidirect
118. C2×C251#468semidirect
123. C251C2×#468semidirect
127. C2×C22⋊Q8D251semidirect
144. C22⋊Q8C2×D251semidirect
243. C2#30631935direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168607972
C22×#168607942
C2×#225486892
#306319352
C2×#225486482
C2×#225486392
C2×#225486352
C2×C22⋊Q8×C2512
C2×#468251