Finite Group Databasebeta

← C2×#28408819 #44401366

Subgroups — C2×#28408819 #44401366

Order 29056.

438 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 77 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×#20300205C211yesmaximal, characteristic, complemented
2C2×#20300203C211yesmaximal, characteristic, complemented
3#28408819C214yesmaximal, direct product factor
4C2×#120×C227C211yesmaximal, characteristic
5C2×#16813868—21no—
6C2×C22⋊Dic227—21no—
7#20300205—21no—
8#20300205C2212yescomplemented
9#20300203—21no—
10#20300203C2212yescomplemented
11C25×C227C411yescharacteristic, complemented
12C22×D4×C227—21no—
13C22×D4×C227C2211yescharacteristic
14#120×C227—21no—
15#120×C227—21no—
16#120×C227C412yescomplemented
17#120×C227C2212yes—
18C2×C22⋊C4×C227—21no—
19C2×C22⋊C4×C227C411yescharacteristic, complemented
20C22×Dic227—41no—
21#16813868—22no—
22#16813868—41no—
23C2×C227⋊C8—41no—
24C22⋊Dic227—22no—
25C22⋊Dic227—41no—
26C24×C227—21no—
27C24×C227—21no—
28C24×C227—44no—
29C24×C227—41no—
30C24×C227—41no—
31C24×C227C2×C411yescharacteristic
32C24×C227C2×C412yescomplemented
33C24×C227D411yescharacteristic
34C2×D4×C227—22no—
35C2×D4×C227—22no—
36C2×D4×C227—41no—
37C2×D4×C227—41no—
38C2×D4×C227—44no—
39C2×D4×C227C2×C411yescharacteristic
40C2×D4×C227D411yescharacteristic
41C2×D4×C227D411yescharacteristic
42C2×D4×C227C2311yescharacteristic
43C22×C4×C227—21no—
44C22×C4×C227D411yescharacteristic
45C22⋊C4×C227—21no—
46C22⋊C4×C227—22no—
47C22⋊C4×C227—22no—
48C22⋊C4×C227C2×C412yescomplemented
49C2×Dic227—42no—
50C2×Dic227—82no—
51C227⋊C8—42no—
52C23×C227—24no—
53C23×C227—21no—
54C23×C227—21no—
55C23×C227—21no—
56C23×C227—22no—
57C23×C227—21no—
58C23×C227—42no—
59C23×C227—44no—
60C23×C227—42no—
61C23×C227—42no—
62C23×C227—44no—
63C23×C227—42no—
64C23×C227—42no—
65C23×C227—48no—
66C23×C227—44no—
67C23×C227—41no—
68C23×C227—41no—
69C23×C227—44no—
70C23×C227C22⋊C411yescharacteristic
71C23×C227C22⋊C411yescharacteristic
72C23×C227C22⋊C411yescharacteristic
73C23×C227C22×C411yescharacteristic
74C23×C227C2×D411yescharacteristic
75D4×C227—41no—
76D4×C227—41no—
77D4×C227—41no—
78D4×C227—41no—
79D4×C227—44no—
80D4×C227—44no—
81C2×C4×C227—21no—
82C2×C4×C227—21no—
83C2×C4×C227—22no—
84C2×C4×C227—42no—
85C2×C4×C227C22⋊C411yescharacteristic
86C2×C4×C227C2×D411yescharacteristic
87C22×C227—22no—
88C22×C227—24no—
89C22×C227—21no—
90C22×C227—21no—
91C22×C227—21no—
92C22×C227—21no—
93C22×C227—41no—
94C22×C227—41no—
95C22×C227—41no—
96C22×C227—41no—
97C22×C227—42no—
98C22×C227—44no—
99C22×C227—42no—
100C22×C227—44no—
101C22×C227—48no—
102C22×C227—44no—
103C22×C227—44no—
104C22×C227—42no—
105C22×C227—42no—
106C22×C227—42no—
107C22×C227—81no—
108C22×C227#9911yescharacteristic
109C22×C227#9911yescharacteristic
110C22×C227C2×C22⋊C411yescharacteristic
111C4×C227—21no—
112C4×C227—21no—
113C4×C227—42no—
114Dic227—82no—
115C2×C227—21no—
116C2×C227—21no—
117C2×C227—41no—
118C2×C227—44no—
119C2×C227—41no—
120C2×C227—41no—
121C2×C227—41no—
122C2×C227AΣL(1,𝔽16)12yescomplemented
123C2×C227C2×#9911yescharacteristic
124C227C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
125C2×AΣL(1,𝔽16)—2271nomaximal, Sylow
126C2×#120D22711yescharacteristic
127C2×#100—2271no—
128C2×#99—2271no—
129AΣL(1,𝔽16)—2274no—
130C25Dic22711yescharacteristic, complemented
131C22×D4—21no—
132C22×D4C2×D22711yescharacteristic
133C2×M16—4541no—
134#120—21no—
135#120—21no—
136#120Dic22712yescomplemented
137#120C2×D22712yes—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—4541no—
140C2×C22⋊C4Dic22711yescharacteristic, complemented
141#100—4541no—
142#100—2272no—
143#99—4541no—
144#99—2272no—
145C24—21no—
146C24—21no—
147C24—44no—
148C24—41no—
149C24—41no—
150C24C2×Dic22711yescharacteristic
151C24C2×Dic22712yescomplemented
152C24C22⋊D22711yescharacteristic
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—44no—
158C2×D4C2×Dic22711yescharacteristic
159C2×D4C22⋊D22711yescharacteristic
160C2×D4C22⋊D22711yescharacteristic
161C2×D4C22×D22711yescharacteristic
162C22×C4—21no—
163C22×C4—9081no—
164C22×C4C22⋊D22711yescharacteristic
165M16—9081no—
166M16—4542no—
167C2×C8—9081no—
168C22⋊C4—21no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—9081no—
172C22⋊C4—4542no—
173C22⋊C4C2×Dic22712yescomplemented
174C23—24no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23—21no—
180C23—42no—
181C23—44no—
182C23—42no—
183C23—42no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23—44no—
188C23—48no—
189C23—44no—
190C23—41no—
191C23—41no—
192C23C22⋊Dic22711yescharacteristic
193C23C22⋊Dic22711yescharacteristic
194C23C22⋊Dic22711yescharacteristic
195C23C22×Dic22711yescharacteristic
196C23C2×C22⋊D22711yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—44no—
202D4—44no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—18162no—
208C2×C4—9082no—
209C2×C4C22⋊Dic22711yescharacteristic
210C2×C4C2×C22⋊D22711yescharacteristic
211C8—9082no—
212C22—22no—
213C22—24no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—42no—
223C22—44no—
224C22—42no—
225C22—44no—
226C22—44no—
227C22—48no—
228C22—42no—
229C22—42no—
230C22—44no—
231C22—42no—
232C22—81no—
233C22#2030020311yescharacteristic
234C22#2030020311yescharacteristic
235C22C2×C22⋊Dic22711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—18162no—
240C2—21no—
241C2—21no—
242C2—41no—
243C2—44no—
244C2—41no—
245C2—41no—
246C2—41no—
247C2#2840881912yesdirect product factor
248C2C2×#2030020311yescharacteristic
subgroupG/H
centerC22#20300203
commutatorC23×C227C22×C4
FrattiniC2×D4C22×D227
Fitting, prime radicalC2×#120×C227C2
socleC22×C227#99
cosocleC2×D4×C227C23
hypercenterC2×#120D227
hypocenterC227C2×AΣL(1,𝔽16)
LCSG ▷ C23×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#20300205C2semidirect
2. C2×#20300203C2semidirect
3. #28408819C2direct (N × H)
8. #20300205C22semidirect
10. #20300203C22semidirect
11. C25×C227C4semidirect
16. #120×C227C4semidirect
19. C2×C22⋊C4×C227C4semidirect
32. C24×C227C2×C4semidirect
48. C22⋊C4×C227C2×C4semidirect
122. C2×C227AΣL(1,𝔽16)semidirect
124. C227C2×AΣL(1,𝔽16)semidirect
130. C25Dic227semidirect
136. #120Dic227semidirect
140. C2×C22⋊C4Dic227semidirect
151. C24C2×Dic227semidirect
173. C22⋊C4C2×Dic227semidirect
247. C2#28408819direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#203002052
C2×#203002032
#284088192
C2×#120×C2272
C2×AΣL(1,𝔽16)227