Finite Group Databasebeta

← D4×#6105 #13126528

Subgroups — D4×#6105 #13126528

Order 1664.

546 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 149 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
1D42×C13C211yesmaximal, characteristic
2C22×#6105C212yesmaximal, complemented
3D4×Dic26C211yesmaximal, characteristic, complemented
4#11718879C212yesmaximal, complemented
5#11718737C211yesmaximal, characteristic, complemented
6#11718736C211yesmaximal, characteristic, complemented
7#11718702C212yesmaximal, complemented
8#11718701C212yesmaximal, complemented
9#11718680C211yesmaximal, characteristic, complemented
10C4×#6105C211yesmaximal, characteristic, complemented
11D4×#1055C211yesmaximal, characteristic, complemented
12C22×D4×C13—21no—
13C22×D4×C13C2212yes—
14C22×Dic26C2212yescomplemented
15C13×#127C2211yescharacteristic
16C13×#121—21no—
17C13×#121C2212yes—
18C13×#120—22no—
19C4×D4×C13C2211yescharacteristic
20C4×D4×C13C2211yescharacteristic
21D4×Dic13—21no—
22C2×#6105—28no—
23C2×#6105C2212yes—
24C2×#6105C2211yescharacteristic
25C2×#6105C2212yescomplemented
26C2×#6105C2212yescomplemented
27C2×#6105C2212yescomplemented
28#88178C2212yescomplemented
29C22×#1055C2212yescomplemented
30#88142—21no—
31#88132—22no—
32#88123C2211yescharacteristic, complemented
33C4×Dic26C2211yescharacteristic, complemented
34#88072C2211yescharacteristic
35#88072C2212yescomplemented
36C22⋊#1055C2212yescomplemented
37#88050C2211yescharacteristic, complemented
38#88050C2212yescomplemented
39#88048C2211yescharacteristic, complemented
40C4⋊#1055C2211yescharacteristic, complemented
41C4×#1055C2211yescharacteristic, complemented
42C24×C13—22no—
43C2×D4×C13—24no—
44C2×D4×C13—22no—
45C2×D4×C13—22no—
46C2×D4×C13—21no—
47C2×D4×C13—21no—
48C2×D4×C13—42no—
49C2×D4×C13—44no—
50C2×D4×C13D411yescharacteristic
51C2×D4×C13C2312yes—
52C2×D4×C13C2311yescharacteristic
53C22×C4×C13—21no—
54C22×C4×C13C2312yes—
55C22×Dic13—22no—
56C2×Dic26—24no—
57C2×Dic26—22no—
58C2×Dic26—42no—
59C2×Dic26C2312yes—
60C2×Dic26C2311yescharacteristic
61C13×C4⋊C4D411yescharacteristic
62C13×C4⋊C4C2311yescharacteristic
63C13×C22⋊C4—21no—
64C13×C22⋊C4—22no—
65C13×C22⋊C4D412yes—
66C42×C13C2311yescharacteristic
67C22⋊Dic13—22no—
68#6105—24no—
69#6105—24no—
70#6105—24no—
71#6105—42no—
72#6105D414yesdirect product factor
73C4⋊Dic13—21no—
74C4⋊Dic13—21no—
75C4⋊Dic13C2311yescharacteristic
76#6101—21no—
77#6101—22no—
78C4×Dic13—21no—
79C2×#1055—24no—
80C2×#1055C2311yescharacteristic
81C2×#1055C2312yes—
82C2×#1055C2311yescharacteristic
83D4×SD16—131nomaximal, Sylow
84C23×C13—21no—
85C23×C13—22no—
86C23×C13—44no—
87C23×C13—44no—
88C23×C13—82no—
89C23×C13C2×D412yes—
90D4×C13—22no—
91D4×C13—21no—
92D4×C13—22no—
93D4×C13—44no—
94D4×C13—44no—
95D4×C13—42no—
96D4×C13—81no—
97D4×C13SD1614yescomplemented
98D4×C13C2×D412yes—
99C2×C4×C13—22no—
100C2×C4×C13—22no—
101C2×C4×C13—21no—
102C2×C4×C13—42no—
103C2×C4×C13C2×D411yescharacteristic
104C2×C4×C13C2×D412yes—
105C2×C4×C13C2×D411yescharacteristic
106C2×C4×C13C2411yescharacteristic
107C2×Dic13—21no—
108C2×Dic13—21no—
109C2×Dic13—22no—
110C2×Dic13—44no—
111Dic26—21no—
112Dic26—22no—
113Dic26—22no—
114Dic26—44no—
115Dic26C2×D412yescomplemented
116#1055—22no—
117#1055—21no—
118#1055C2×D412yescomplemented
119C22×SD16—132no—
120D4×Q8—131no—
121D42D1311yescharacteristic
122#492—131no—
123#465—132no—
124#461—131no—
125#460—131no—
126#450—132no—
127#448—132no—
128C4×SD16—131no—
129C8×D4—131no—
130C22×C13—22no—
131C22×C13—22no—
132C22×C13—22no—
133C22×C13—41no—
134C22×C13—41no—
135C22×C13—42no—
136C22×C13—42no—
137C22×C13—42no—
138C22×C13—88no—
139C22×C13C2×SD1614yescomplemented
140C22×C13C22×D411yescharacteristic
141C4×C13—22no—
142C4×C13—21no—
143C4×C13—41no—
144C4×C13C2×SD1612yescomplemented
145C4×C13C22×D411yescharacteristic
146C4×C13H(5,𝔽2)11yescharacteristic
147Dic13—22no—
148Dic13—42no—
149Dic13—41no—
150C22×Q8—132no—
151C22×D4—21no—
152C22×D4C2×D1312yes—
153C2×SD16—131no—
154C2×SD16—132no—
155C2×SD16—132no—
156C2×SD16—132no—
157C2×SD16—132no—
158C2×SD16—268no—
159C22×C8—132no—
160C4⋊Q8—131no—
161C4⋊Q8—261no—
162#127C2×D1311yescharacteristic
163C22⋊Q8—132no—
164C22⋊Q8—262no—
165#121—21no—
166#121C2×D1312yes—
167#120—22no—
168C4×Q8—131no—
169C4×D4—261no—
170C4×D4C2×D1311yescharacteristic
171C4×D4C2×D1311yescharacteristic
172#106—131no—
173C4⋊C8—131no—
174#103—131no—
175#103—132no—
176#102—131no—
177#102—132no—
178C22⋊C8—132no—
179C4×C8—131no—
180C2×C13—24no—
181C2×C13—42no—
182C2×C13—82no—
183C2×C13D4211yescharacteristic
184C2×C13C22×SD1611yescharacteristic
185C2×C13#57711yescharacteristic
186C24—22no—
187C2×Q8—131no—
188C2×Q8—132no—
189C2×Q8—264no—
190C2×Q8—262no—
191C2×Q8—522no—
192C2×D4—24no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—42no—
198C2×D4—44no—
199C2×D4C22⋊D1311yescharacteristic
200C2×D4C22×D1312yes—
201C2×D4C22×D1311yescharacteristic
202C22×C4—21no—
203C22×C4—262no—
204C22×C4C22×D1312yes—
205SD16—134no—
206SD16—264no—
207SD16—264no—
208SD16—264no—
209SD16—522no—
210C2×C8—131no—
211C2×C8—131no—
212C2×C8—132no—
213C2×C8—264no—
214C4⋊C4—131no—
215C4⋊C4—261no—
216C4⋊C4—261no—
217C4⋊C4—262no—
218C4⋊C4—261no—
219C4⋊C4C22⋊D1311yescharacteristic
220C4⋊C4C22×D1311yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—22no—
223C22⋊C4—262no—
224C22⋊C4C22⋊D1312yes—
225C42—261no—
226C42C22×D1311yescharacteristic
227C13D4×SD1611yescharacteristic, complemented, Sylow
228C23—21no—
229C23—22no—
230C23—44no—
231C23—44no—
232C23—82no—
233C23C2×C22⋊D1312yes—
234Q8—132no—
235Q8—261no—
236Q8—262no—
237Q8—262no—
238Q8—524no—
239D4—22no—
240D4—21no—
241D4—22no—
242D4—44no—
243D4—44no—
244D4—42no—
245D4—81no—
246D4#610514yesdirect product factor
247D4D4×D1312yes—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—261no—
253C2×C4—261no—
254C2×C4—262no—
255C2×C4—524no—
256C2×C4C2×C22⋊D1311yescharacteristic
257C2×C4C2×C22⋊D1312yes—
258C2×C4C2×C22⋊D1311yescharacteristic
259C2×C4C23×D1311yescharacteristic
260C8—132no—
261C8—262no—
262C8—261no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—88no—
272C22C2×#610514yescomplemented
273C22C22×C22⋊D1311yescharacteristic
274C4—22no—
275C4—21no—
276C4—41no—
277C4—262no—
278C4—522no—
279C4—521no—
280C4C2×#610512yescomplemented
281C4C2×D4×D1311yescharacteristic
282C4#8825111yescharacteristic
283C2—24no—
284C2—42no—
285C2—82no—
286C2C22×#610511yescharacteristic
287C2D4×C22⋊D1311yescharacteristic
288C2#1171954111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
Fitting, prime radicalD42×C13C2
socleC22×C13C22×D4
hypercenterD42D13
hypocenterC13D4×SD16
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
2. C22×#6105C2semidirect
3. D4×Dic26C2semidirect
4. #11718879C2semidirect
5. #11718737C2semidirect
6. #11718736C2semidirect
7. #11718702C2semidirect
8. #11718701C2semidirect
9. #11718680C2semidirect
10. C4×#6105C2semidirect
11. D4×#1055C2semidirect
14. C22×Dic26C22semidirect
25. C2×#6105C22semidirect
26. C2×#6105C22semidirect
27. C2×#6105C22semidirect
28. #88178C22semidirect
29. C22×#1055C22semidirect
32. #88123C22semidirect
33. C4×Dic26C22semidirect
35. #88072C22semidirect
36. C22⋊#1055C22semidirect
37. #88050C22semidirect
38. #88050C22semidirect
39. #88048C22semidirect
40. C4⋊#1055C22semidirect
41. C4×#1055C22semidirect
72. #6105D4direct (N × H)
97. D4×C13SD16semidirect
115. Dic26C2×D4semidirect
118. #1055C2×D4semidirect
139. C22×C13C2×SD16semidirect
144. C4×C13C2×SD16semidirect
227. C13D4×SD16semidirect
246. D4#6105direct (N × H)
272. C22C2×#6105semidirect
280. C4C2×#6105semidirect

Maximal subgroups

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

subgroupindex
D42×C132
C22×#61052
D4×Dic262
#117188792
#117187372
#117187362
#117187022
#117187012
#117186802
C4×#61052
D4×#10552
D4×SD1613