Finite Group Databasebeta

← C2×#30631938 #44559959

Subgroups — C2×#30631938 #44559959

Order 32128.

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

normal subgroups only

This group has 596 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×#16860795C211yesmaximal, characteristic, complemented
2C22×#16860794C211yesmaximal, characteristic, complemented
3C2×#22548689C211yesmaximal, characteristic, complemented
4#30631938C218yesmaximal, direct product factor
5C2×#22548646C211yesmaximal, characteristic, complemented
6C2×#22548636C211yesmaximal, characteristic, complemented
7C2×#22548635C211yesmaximal, characteristic, complemented
8C2×#121×C251C211yesmaximal, characteristic, complemented
9C22×D1004C2211yescharacteristic
10C22×C22⋊D251—21no—
11C2×#16860795—24no—
12C2×#16860795C2214yescomplemented
13C2×#16860794—24no—
14C2×#16860794C2214yescomplemented
15C22×C251⋊C8C2211yescharacteristic
16C22×C251⋊C8C2211yescharacteristic
17#22548689—22no—
18#22548689C2214yescomplemented
19C2×C4⋊Dic251C2211yescharacteristic
20C2×(C2×D251)⋊C4—21no—
21#22548646C2214yescomplemented
22#22548636C2214yescomplemented
23#22548635C2214yescomplemented
24C22×D4×C251—21no—
25C22×D4×C251C2211yescharacteristic
26C23×C4×C251C2211yescharacteristic
27#121×C251—22no—
28#121×C251C2214yescomplemented
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#16860795—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×C251—21no—
49C24×C251D411yescharacteristic
50C2×D4×C251—24no—
51C2×D4×C251—22no—
52C2×D4×C251—21no—
53C2×D4×C251—22no—
54C2×D4×C251—44no—
55C2×D4×C251C2312yes—
56C22×C4×C251—22no—
57C22×C4×C251—22no—
58C22×C4×C251—21no—
59C22×C4×C251D411yescharacteristic
60C22×C4×C251D412yes—
61C22×C4×C251C2312yes—
62C22×C4×C251C2311yescharacteristic
63C4⋊C4×C251—21no—
64C4⋊C4×C251C2312yes—
65C22⋊C4×C251—24no—
66C22×D251—22no—
67C22×D251—42no—
68C22×D251—42no—
69C22×D251—42no—
70D1004—22no—
71D1004—22no—
72D1004—42no—
73C22⋊D251—48no—
74C2×Dic251—22no—
75C2×Dic251—42no—
76C251⋊C8—22no—
77C251⋊C8C2×D414yescomplemented
78C23×C251—22no—
79C23×C251—22no—
80C23×C251—22no—
81C23×C251—22no—
82C23×C251—42no—
83C23×C251—42no—
84C23×C251—42no—
85C23×C251C2×D412yes—
86C23×C251C2×D411yescharacteristic
87D4×C251—22no—
88D4×C251—22no—
89D4×C251—48no—
90D4×C251—42no—
91C2×C4×C251—22no—
92C2×C4×C251—24no—
93C2×C4×C251—22no—
94C2×C4×C251—22no—
95C2×C4×C251—22no—
96C2×C4×C251—42no—
97C2×C4×C251C2×D412yes—
98C2×C4×C251C2×D412yes—
99C2×C4×C251C2×D411yescharacteristic
100C2×C4×C251#4112yes—
101C2×C4×C251C2411yescharacteristic
102C22×C251—22no—
103C22×C251—22no—
104C22×C251—22no—
105C22×C251—44no—
106C22×C251—42no—
107C22×C251—42no—
108C22×C251—42no—
109C22×C251—84no—
110C22×C251#12112yes—
111C22×C251AGL(1,ℤ/8)12yes—
112C22×C251AGL(1,ℤ/8)12yes—
113C22×C251C22×D411yescharacteristic
114C2×D251—42no—
115C2×D251—42no—
116C2×D251—42no—
117C2×D251—84no—
118C4×C251—22no—
119C4×C251—42no—
120C4×C251#12112yes—
121C4×C251C22×D411yescharacteristic
122C4×C251C2×#4111yescharacteristic
123Dic251—42no—
124C2×C251—42no—
125C2×C251—82no—
126C2×C251#46914yescomplemented
127C2×C251C2×#12111yescharacteristic
128C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
130D251—82no—
131C251C2×#46911yescharacteristic, complemented, Sylow
132C2×#469—2511nomaximal, Sylow
133C22×D8—2511no—
134C22×M16—2511no—
135C2×#121—2511no—
136C2×#121D25111yescharacteristic, complemented
137#469—2518no—
138C2×#106—2511no—
139C2×#102—2511no—
140C2×#102—2511no—
141C22×D4—21no—
142C22×D4—5021no—
143C22×D4—2511no—
144C22×D4C2×D25111yescharacteristic
145C23×C4C2×D25111yescharacteristic
146C2×D8—5024no—
147C2×D8—2514no—
148C2×M16—5024no—
149C2×M16—2514no—
150C22×C8—2511no—
151C22×C8—2511no—
152#121—22no—
153#121—5022no—
154#121—2514no—
155#121C2×D25114yescomplemented
156C2×C4⋊C4—2511no—
157C2×C4⋊C4C2×D25111yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—5021no—
160#106—2514no—
161#102—2514no—
162#102—2514no—
163C24—21no—
164C24—5021no—
165C24C22⋊D25111yescharacteristic
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—44no—
171C2×D4—10044no—
172C2×D4—5024no—
173C2×D4—5022no—
174C2×D4—5021no—
175C2×D4—5022no—
176C2×D4—2512no—
177C2×D4C22×D25112yes—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—5021no—
182C22×C4C22⋊D25112yes—
183C22×C4C22⋊D25111yescharacteristic
184C22×C4C22×D25112yes—
185C22×C4C22×D25111yescharacteristic
186D8—5028no—
187M16—5028no—
188C2×C8—5022no—
189C2×C8—2512no—
190C2×C8—2514no—
191C2×C8—2512no—
192C4⋊C4—21no—
193C4⋊C4—5021no—
194C4⋊C4—2512no—
195C4⋊C4C22×D25112yes—
196C22⋊C4—24no—
197C22⋊C4—5024no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—10042no—
206C23—10042no—
207C23—10042no—
208C23—5022no—
209C23C2×C22⋊D25112yes—
210C23C2×C22⋊D25111yescharacteristic
211D4—22no—
212D4—22no—
213D4—48no—
214D4—42no—
215D4—10048no—
216D4—10042no—
217D4—5022no—
218D4—5022no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—10042no—
226C2×C4—5022no—
227C2×C4D4×D25112yes—
228C2×C4C2×C22⋊D25112yes—
229C2×C4C2×C22⋊D25111yescharacteristic
230C2×C4#1686080912yes—
231C2×C4C23×D25111yescharacteristic
232C8—5022no—
233C8—2514no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—44no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—84no—
242C22—20084no—
243C22—10042no—
244C22—10042no—
245C22—10042no—
246C22#2254868712yes—
247C22#2254874012yes—
248C22#2254874112yes—
249C22C22×C22⋊D25111yescharacteristic
250C4—22no—
251C4—42no—
252C4—10042no—
253C4#2254868712yes—
254C4C2×D4×D25111yescharacteristic
255C4C2×#1686080911yescharacteristic
256C2—42no—
257C2—82no—
258C2—20082no—
259C2#3063193814yesdirect product factor
260C2C2×#2254868711yescharacteristic
261C2C2×#2254874011yescharacteristic
262C2C2×#2254874111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#121×C251C2
socleC23×C251C2×D4
hypercenterC2×#121D251
hypocenterC251C2×#469
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#16860795C2semidirect
2. C22×#16860794C2semidirect
3. C2×#22548689C2semidirect
4. #30631938C2direct (N × H)
5. C2×#22548646C2semidirect
6. C2×#22548636C2semidirect
7. C2×#22548635C2semidirect
8. C2×#121×C251C2semidirect
12. C2×#16860795C22semidirect
14. C2×#16860794C22semidirect
18. #22548689C22semidirect
21. #22548646C22semidirect
22. #22548636C22semidirect
23. #22548635C22semidirect
28. #121×C251C22semidirect
42. C2×C251⋊C8D4semidirect
77. C251⋊C8C2×D4semidirect
126. C2×C251#469semidirect
131. C251C2×#469semidirect
136. C2×#121D251semidirect
155. #121C2×D251semidirect
259. C2#30631938direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168607952
C22×#168607942
C2×#225486892
#306319382
C2×#225486462
C2×#225486362
C2×#225486352
C2×#121×C2512
C2×#469251