Finite Group Databasebeta

← D4×#13133805 #28348157

Subgroups — D4×#13133805 #28348157

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×#13133805C212yesmaximal, complemented
2D4×D428C211yesmaximal, characteristic
3#20293001C211yesmaximal, characteristic, complemented
4#20292858C212yesmaximal, complemented
5#20292829C211yesmaximal, characteristic, complemented
6#20292827C211yesmaximal, characteristic, complemented
7#20292777C212yesmaximal, complemented
8#20292766C212yesmaximal, complemented
9C4×#13133805C211yesmaximal, characteristic, complemented
10D4×C107⋊C8C211yesmaximal, characteristic, complemented
11D4×Q8×C107C211yesmaximal, characteristic, complemented
12C22×D428C2212yes—
13C2×D4×D107—21no—
14C2×#13133805—28no—
15C2×#13133805C2212yes—
16C2×#13133805C2211yescharacteristic
17C2×#13133805C2212yescomplemented
18C2×#13133805C2212yescomplemented
19C2×#13133805C2212yescomplemented
20C22×C107⋊C8C2212yescomplemented
21#16812106C2211yescharacteristic
22#16812084—21no—
23#16812083C2212yes—
24#16812079—22no—
25C4×D428C2211yescharacteristic
26#16812040C2211yescharacteristic, complemented
27C4⋊(C107⋊C8)C2211yescharacteristic, complemented
28#16812033C2211yescharacteristic, complemented
29#16812033C2212yescomplemented
30#16812029C2211yescharacteristic
31#16812029C2212yescomplemented
32C22⋊(C107⋊C8)C2212yescomplemented
33C4×C107⋊C8C2211yescharacteristic, complemented
34C22×Q8×C107C2212yescomplemented
35C4⋊Q8×C107—21no—
36C4⋊Q8×C107C2211yescharacteristic, complemented
37C22⋊Q8×C107—22no—
38C22⋊Q8×C107C2212yescomplemented
39C4×Q8×C107C2211yescharacteristic, complemented
40C4×D4×C107—21no—
41C4×D4×C107C2211yescharacteristic
42C23×D107—22no—
43C2×D428—24no—
44C2×D428—22no—
45C2×D428—21no—
46C2×D428—21no—
47C2×D428—42no—
48C2×D428C2312yes—
49C2×D428C2311yescharacteristic
50C2×C22⋊D107—22no—
51D4×D107—44no—
52C2×C4×D107—21no—
53#13133805—24no—
54#13133805—24no—
55#13133805—24no—
56#13133805—42no—
57#13133805D414yesdirect product factor
58C2×C107⋊C8—24no—
59C2×C107⋊C8C2312yes—
60C2×C107⋊C8C2311yescharacteristic
61C2×C107⋊C8C2311yescharacteristic
62C4⋊Dic107C2311yescharacteristic
63(C2×D107)⋊C4—21no—
64(C2×D107)⋊C4—22no—
65C2×Q8×C107—22no—
66C2×Q8×C107—24no—
67C2×Q8×C107—42no—
68C2×Q8×C107C2312yes—
69C2×Q8×C107C2311yescharacteristic
70C2×D4×C107D411yescharacteristic
71C22×C4×C107—22no—
72C22×C4×C107C2312yes—
73C4⋊C4×C107—21no—
74C4⋊C4×C107—21no—
75C4⋊C4×C107—21no—
76C4⋊C4×C107—22no—
77C4⋊C4×C107D411yescharacteristic
78C4⋊C4×C107C2311yescharacteristic
79C22⋊C4×C107—22no—
80C22⋊C4×C107D412yes—
81C42×C107—21no—
82C42×C107C2311yescharacteristic
83C23×C107C2×D412yes—
84C22×D107—21no—
85C22×D107—22no—
86C22×D107—44no—
87C22×D107—44no—
88C22×D107—82no—
89Q8×C107—21no—
90Q8×C107—22no—
91Q8×C107—22no—
92Q8×C107—44no—
93Q8×C107C2×D412yescomplemented
94D4×C107SD1614yescomplemented
95C2×C4×C107—21no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—21no—
99C2×C4×C107—22no—
100C2×C4×C107—44no—
101C2×C4×C107C2×D411yescharacteristic
102C2×C4×C107C2×D412yes—
103C2×C4×C107C2×D411yescharacteristic
104C2×C4×C107C2411yescharacteristic
105C22⋊D107—44no—
106C2×Dic107—21no—
107D428—22no—
108D428—21no—
109D428—22no—
110D428—44no—
111D428—42no—
112D428—81no—
113D428C2×D412yes—
114C4×D107—42no—
115C107⋊C8—22no—
116C107⋊C8—21no—
117C107⋊C8C2×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—22no—
131C4×C107—21no—
132C4×C107—42no—
133C4×C107—41no—
134C4×C107C2×SD1612yescomplemented
135C4×C107C22×D411yescharacteristic
136C4×C107H(5,𝔽2)11yescharacteristic
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×Q8D10711yescharacteristic, complemented
148D42—1071no—
149#492—1071no—
150#465—1072no—
151#461—1071no—
152#460—1071no—
153#450—1072no—
154#448—1072no—
155C4×SD16—1071no—
156C8×D4—1071no—
157C22×Q8C2×D10712yescomplemented
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×C8—1072no—
167C4⋊Q8—21no—
168C4⋊Q8C2×D10711yescharacteristic, complemented
169#127—1071no—
170C22⋊Q8—22no—
171C22⋊Q8C2×D10712yescomplemented
172#121—1072no—
173#121—2141no—
174#120—2142no—
175C4×Q8C2×D10711yescharacteristic, complemented
176C4×D4—21no—
177C4×D4—1071no—
178C4×D4C2×D10711yescharacteristic
179#106—1071no—
180C4⋊C8—1071no—
181#103—1071no—
182#103—1072no—
183#102—1071no—
184#102—1072no—
185C22⋊C8—1072no—
186C4×C8—1071no—
187C24—2142no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8C22×D10712yes—
192C2×Q8C22×D10711yescharacteristic
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×D4C22⋊D10711yescharacteristic
203C22×C4—22no—
204C22×C4—2141no—
205C22×C4C22×D10712yes—
206SD16—1074no—
207SD16—4282no—
208SD16—2144no—
209SD16—2144no—
210SD16—2144no—
211C2×C8—1071no—
212C2×C8—1071no—
213C2×C8—1072no—
214C2×C8—2144no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—1071no—
220C4⋊C4C22⋊D10711yescharacteristic
221C4⋊C4C22×D10711yescharacteristic
222C22⋊C4—22no—
223C22⋊C4—2141no—
224C22⋊C4—2142no—
225C22⋊C4C22⋊D10712yes—
226C42—21no—
227C42C22×D10711yescharacteristic
228C23—8562no—
229C23—4284no—
230C23—4284no—
231C23—2142no—
232C23—2141no—
233C23C2×C22⋊D10712yes—
234Q8—21no—
235Q8—22no—
236Q8—22no—
237Q8—44no—
238Q8D4×D10712yescomplemented
239D4—8561no—
240D4—1072no—
241D4—4284no—
242D4—4284no—
243D4—4282no—
244D4—2142no—
245D4—2142no—
246D4—2141no—
247D4#1313380514yesdirect product factor
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—4282no—
255C2×C4—2141no—
256C2×C4C2×C22⋊D10711yescharacteristic
257C2×C4C2×C22⋊D10712yes—
258C2×C4C2×C22⋊D10711yescharacteristic
259C2×C4C23×D10711yescharacteristic
260C8—1072no—
261C8—2142no—
262C8—2141no—
263C22—22no—
264C22—22no—
265C22—8568no—
266C22—4282no—
267C22—4282no—
268C22—4282no—
269C22—4281no—
270C22—4281no—
271C22—2142no—
272C22C2×#1313380514yescomplemented
273C22C22×C22⋊D10711yescharacteristic
274C4—21no—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—42no—
279C4—4281no—
280C4C2×#1313380512yescomplemented
281C4C2×D4×D10711yescharacteristic
282C4#1681215411yescharacteristic
283C2—24no—
284C2—8562no—
285C2—4282no—
286C2D4×C22⋊D10711yescharacteristic
287C2C22×#1313380511yescharacteristic
288C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingD4×Q8×C107C2
socleC22×C107C22×D4
prime radicalD4×D428C2
hypercenterD4×Q8D107
hypocenterC107D4×SD16
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13133805C2semidirect
3. #20293001C2semidirect
4. #20292858C2semidirect
5. #20292829C2semidirect
6. #20292827C2semidirect
7. #20292777C2semidirect
8. #20292766C2semidirect
9. C4×#13133805C2semidirect
10. D4×C107⋊C8C2semidirect
11. D4×Q8×C107C2semidirect
17. C2×#13133805C22semidirect
18. C2×#13133805C22semidirect
19. C2×#13133805C22semidirect
20. C22×C107⋊C8C22semidirect
26. #16812040C22semidirect
27. C4⋊(C107⋊C8)C22semidirect
28. #16812033C22semidirect
29. #16812033C22semidirect
31. #16812029C22semidirect
32. C22⋊(C107⋊C8)C22semidirect
33. C4×C107⋊C8C22semidirect
34. C22×Q8×C107C22semidirect
36. C4⋊Q8×C107C22semidirect
38. C22⋊Q8×C107C22semidirect
39. C4×Q8×C107C22semidirect
57. #13133805D4direct (N × H)
93. Q8×C107C2×D4semidirect
94. D4×C107SD16semidirect
117. C107⋊C8C2×D4semidirect
120. C22×C107C2×SD16semidirect
134. C4×C107C2×SD16semidirect
145. C107D4×SD16semidirect
147. D4×Q8D107semidirect
157. C22×Q8C2×D107semidirect
168. C4⋊Q8C2×D107semidirect
171. C22⋊Q8C2×D107semidirect
175. C4×Q8C2×D107semidirect
238. Q8D4×D107semidirect
247. D4#13133805direct (N × H)
272. C22C2×#13133805semidirect
280. C4C2×#13133805semidirect

Maximal subgroups

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

subgroupindex
C22×#131338052
D4×D4282
#202930012
#202928582
#202928292
#202928272
#202927772
#202927662
C4×#131338052
D4×C107⋊C82
D4×Q8×C1072
D4×SD16107