Finite Group Databasebeta

← D4×#13133807 #28348159

Subgroups — D4×#13133807 #28348159

Order 13696.

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
1C22×#13133807C212yesmaximal, complemented
2D4×Dic214C211yesmaximal, characteristic, complemented
3D4×D428C211yesmaximal, characteristic
4#20293000C211yesmaximal, characteristic, complemented
5#20292856C212yesmaximal, complemented
6#20292833C211yesmaximal, characteristic, complemented
7#20292828C211yesmaximal, characteristic, complemented
8#20292780C212yesmaximal, complemented
9#20292768C212yesmaximal, complemented
10C4×#13133807C211yesmaximal, characteristic, complemented
11C8×D4×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2212yescomplemented
13C22×D428C2212yes—
14C2×D4×D107—21no—
15C2×#13133807—28no—
16C2×#13133807C2212yes—
17C2×#13133807C2211yescharacteristic
18C2×#13133807C2212yescomplemented
19C2×#13133807C2212yescomplemented
20C2×#13133807C2212yescomplemented
21#16812109C2211yescharacteristic, complemented
22#16812108—21no—
23#16812106C2211yescharacteristic
24#16812088C2212yescomplemented
25#16812086—22no—
26#16812084—21no—
27#16812083C2212yes—
28#16812079—22no—
29C4×Dic214C2211yescharacteristic, complemented
30C4×D428C2211yescharacteristic
31D4×Dic107—21no—
32#16812041C2211yescharacteristic, complemented
33#16812034C2211yescharacteristic, complemented
34#16812034C2212yescomplemented
35#16812031C2211yescharacteristic
36#16812031C2212yescomplemented
37C22×C8×C107C2212yescomplemented
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic, complemented
40C22⋊C8×C107C2212yescomplemented
41C4×C8×C107C2211yescharacteristic, complemented
42C23×D107—22no—
43C2×Dic214—24no—
44C2×Dic214—22no—
45C2×Dic214—42no—
46C2×Dic214C2312yes—
47C2×Dic214C2311yescharacteristic
48C2×D428—24no—
49C2×D428—22no—
50C2×D428—21no—
51C2×D428—21no—
52C2×D428—42no—
53C2×D428C2312yes—
54C2×D428C2311yescharacteristic
55C2×C22⋊D107—22no—
56D4×D107—44no—
57C22×Dic107—22no—
58C2×C4×D107—21no—
59#13133807—24no—
60#13133807—24no—
61#13133807—24no—
62#13133807—42no—
63#13133807D414yesdirect product factor
64C4⋊Dic107—21no—
65C4⋊Dic107—21no—
66C4⋊Dic107C2311yescharacteristic
67C4⋊Dic107C2311yescharacteristic
68Dic107⋊C4—21no—
69Dic107⋊C4—22no—
70C22⋊Dic107—22no—
71(C2×D107)⋊C4—21no—
72(C2×D107)⋊C4—22no—
73C4×Dic107—21no—
74C2×D4×C107D411yescharacteristic
75C22×C4×C107C2312yes—
76C2×C8×C107—24no—
77C2×C8×C107C2312yes—
78C2×C8×C107C2311yescharacteristic
79C2×C8×C107C2311yescharacteristic
80C4⋊C4×C107D411yescharacteristic
81C22⋊C4×C107D412yes—
82C42×C107C2311yescharacteristic
83C23×C107C2×D412yes—
84C22×D107—21no—
85C22×D107—22no—
86C22×D107—44no—
87C22×D107—44no—
88C22×D107—82no—
89D4×C107SD1614yescomplemented
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D412yes—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—44no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—21no—
100C2×Dic107—22no—
101C2×Dic107—44no—
102D428—22no—
103D428—21no—
104D428—22no—
105D428—44no—
106D428—42no—
107D428—81no—
108D428C2×D412yes—
109C4×D107—42no—
110Dic214—21no—
111Dic214—22no—
112Dic214—22no—
113Dic214—44no—
114Dic214C2×D412yescomplemented
115C8×C107—22no—
116C8×C107—21no—
117C8×C107C2×D412yescomplemented
118C22×C107—22no—
119C22×C107—22no—
120C22×C107C2×SD1614yescomplemented
121C22×C107C22×D411yescharacteristic
122C2×D107—22no—
123C2×D107—41no—
124C2×D107—41no—
125C2×D107—42no—
126C2×D107—42no—
127C2×D107—42no—
128C2×D107—88no—
129C4×C107—22no—
130C4×C107—21no—
131C4×C107C2×SD1612yescomplemented
132C4×C107C22×D411yescharacteristic
133C4×C107H(5,𝔽2)11yescharacteristic
134Dic107—22no—
135Dic107—42no—
136Dic107—41no—
137Dic107—41no—
138C2×C107—24no—
139C2×C107D4211yescharacteristic
140C2×C107C22×SD1611yescharacteristic
141C2×C107#57711yescharacteristic
142D107—42no—
143D107—82no—
144D4×SD16—1071nomaximal, Sylow
145C107D4×SD1611yescharacteristic, complemented, Sylow
146C22×SD16—1072no—
147D4×Q8—1071no—
148D42—1071no—
149#492—1071no—
150#465—1072no—
151#461—1071no—
152#460—1071no—
153#450—1072no—
154#448—1072no—
155C4×SD16—1071no—
156C8×D4D10711yescharacteristic, complemented
157C22×Q8—1072no—
158C22×D4—1072no—
159C22×D4—2141no—
160C2×SD16—1071no—
161C2×SD16—1072no—
162C2×SD16—1072no—
163C2×SD16—1072no—
164C2×SD16—1072no—
165C2×SD16—2148no—
166C22×C8C2×D10712yescomplemented
167C4⋊Q8—1071no—
168C4⋊Q8—2141no—
169#127—1071no—
170C22⋊Q8—1072no—
171C22⋊Q8—2142no—
172#121—1072no—
173#121—2141no—
174#120—2142no—
175C4×Q8—1071no—
176C4×D4—1071no—
177C4×D4—2141no—
178C4×D4C2×D10711yescharacteristic
179#106—1071no—
180C4⋊C8C2×D10711yescharacteristic, complemented
181#103—1071no—
182#103—1072no—
183#102—1071no—
184#102—1072no—
185C22⋊C8C2×D10712yescomplemented
186C4×C8C2×D10711yescharacteristic, complemented
187C24—2142no—
188C2×Q8—1071no—
189C2×Q8—1072no—
190C2×Q8—4282no—
191C2×Q8—2144no—
192C2×Q8—2142no—
193C2×D4—1071no—
194C2×D4—1072no—
195C2×D4—4282no—
196C2×D4—4284no—
197C2×D4—2144no—
198C2×D4—2142no—
199C2×D4—2141no—
200C2×D4—2142no—
201C2×D4—2141no—
202C2×D4D42811yescharacteristic
203C22×C4—2142no—
204C22×C4—2141no—
205C22×C4C22×D10712yes—
206SD16—1074no—
207SD16—4282no—
208SD16—2144no—
209SD16—2144no—
210SD16—2144no—
211C2×C8—24no—
212C2×C8C22×D10711yescharacteristic
213C2×C8C22×D10712yes—
214C2×C8C22×D10711yescharacteristic
215C4⋊C4—1071no—
216C4⋊C4—1071no—
217C4⋊C4—2141no—
218C4⋊C4—2141no—
219C4⋊C4—2142no—
220C4⋊C4—2141no—
221C4⋊C4D42811yescharacteristic
222C22⋊C4—2142no—
223C22⋊C4—2141no—
224C22⋊C4—2142no—
225C22⋊C4D42812yes—
226C42—2141no—
227C42C22×D10711yescharacteristic
228C23—8562no—
229C23—4284no—
230C23—4284no—
231C23—2142no—
232C23—2141no—
233C23C2×D42812yes—
234Q8—1072no—
235Q8—4284no—
236Q8—2142no—
237Q8—2142no—
238Q8—2141no—
239D4—8561no—
240D4—1072no—
241D4—4284no—
242D4—4284no—
243D4—4282no—
244D4—2142no—
245D4—2142no—
246D4—2141no—
247D4#1313380714yesdirect product factor
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—4284no—
251C2×C4—4282no—
252C2×C4—2141no—
253C2×C4—2141no—
254C2×C4—2142no—
255C2×C4—2141no—
256C2×C4C2×D42811yescharacteristic
257C2×C4C2×D42812yes—
258C2×C4C2×D42811yescharacteristic
259C2×C4C23×D10711yescharacteristic
260C8—22no—
261C8—21no—
262C8D4×D10712yescomplemented
263C22—22no—
264C22—22no—
265C22—8568no—
266C22—4282no—
267C22—4282no—
268C22—4282no—
269C22—4281no—
270C22—4281no—
271C22—2142no—
272C22C2×#1313380714yescomplemented
273C22C22×D42811yescharacteristic
274C4—21no—
275C4—22no—
276C4—4281no—
277C4—4281no—
278C4—4282no—
279C4—2142no—
280C4C2×#1313380712yescomplemented
281C4C2×D4×D10711yescharacteristic
282C4#1681215511yescharacteristic
283C2—24no—
284C2—8562no—
285C2—4282no—
286C2D4×D42811yescharacteristic
287C2C22×#1313380711yescharacteristic
288C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
prime radicalD4×D428C2
hypercenterC8×D4D107
hypocenterC107D4×SD16
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×#13133807C2semidirect
2. D4×Dic214C2semidirect
4. #20293000C2semidirect
5. #20292856C2semidirect
6. #20292833C2semidirect
7. #20292828C2semidirect
8. #20292780C2semidirect
9. #20292768C2semidirect
10. C4×#13133807C2semidirect
11. C8×D4×C107C2semidirect
12. C22×Dic214C22semidirect
18. C2×#13133807C22semidirect
19. C2×#13133807C22semidirect
20. C2×#13133807C22semidirect
21. #16812109C22semidirect
24. #16812088C22semidirect
29. C4×Dic214C22semidirect
32. #16812041C22semidirect
33. #16812034C22semidirect
34. #16812034C22semidirect
36. #16812031C22semidirect
37. C22×C8×C107C22semidirect
39. C4⋊C8×C107C22semidirect
40. C22⋊C8×C107C22semidirect
41. C4×C8×C107C22semidirect
63. #13133807D4direct (N × H)
89. D4×C107SD16semidirect
114. Dic214C2×D4semidirect
117. C8×C107C2×D4semidirect
120. C22×C107C2×SD16semidirect
131. C4×C107C2×SD16semidirect
145. C107D4×SD16semidirect
156. C8×D4D107semidirect
166. C22×C8C2×D107semidirect
180. C4⋊C8C2×D107semidirect
185. C22⋊C8C2×D107semidirect
186. C4×C8C2×D107semidirect
247. D4#13133807direct (N × H)
262. C8D4×D107semidirect
272. C22C2×#13133807semidirect
280. C4C2×#13133807semidirect

Maximal subgroups

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

subgroupindex
C22×#131338072
D4×Dic2142
D4×D4282
#202930002
#202928562
#202928332
#202928282
#202927802
#202927682
C4×#131338072
C8×D4×C1072
D4×SD16107