Finite Group Databasebeta

← #489×D107 #28346787

Subgroups — #489×D107 #28346787

Order 13696.

436 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C4⋊Q8×D107C211yesmaximal, characteristic, complemented
2#124×D107C211yesmaximal, characteristic
3#20292989C211yesmaximal, characteristic, complemented
4#20292988C212yesmaximal, complemented
5#20292987C211yesmaximal, characteristic, complemented
6#20292986C211yesmaximal, characteristic, complemented
7#20292985C212yesmaximal, complemented
8#103×D107C212yesmaximal, complemented
9#102×D107C212yesmaximal
10#97×D107C211yesmaximal, characteristic, complemented
11#489×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107—21no—
13C2×Q8×D107C2211yescharacteristic
14C2×D4×D107C2211yescharacteristic
15C2×C8×D107C2212yes—
16#16812109C2211yescharacteristic, complemented
17#16812108—21no—
18#16812108C2212yescomplemented
19#16812107—21no—
20#16812098C2211yescharacteristic
21#16812097C2211yescharacteristic
22#16812096C2211yescharacteristic
23#16812095—21no—
24#16812095—21no—
25C4⋊C4×D107—21no—
26C4⋊C4×D107C2212yes—
27C22⋊C4×D107—21no—
28C22⋊C4×D107—21no—
29C42×D107C2211yescharacteristic
30#16812034C2212yescomplemented
31#16812033C2212yescomplemented
32#16812032C2212yescomplemented
33#16812031C2212yes—
34#16812030C2212yes—
35#16812029C2212yes—
36#16812020C2211yescharacteristic, complemented
37#16812019C2212yescomplemented
38C4⋊Q8×C107C2211yescharacteristic, complemented
39#124×C107C2211yescharacteristic
40#103×C107C2212yescomplemented
41#102×C107C2212yes—
42#97×C107C2211yescharacteristic, complemented
43C23×D107—21no—
44C2×Dic214—21no—
45C2×Dic214—21no—
46C2×Dic214—22no—
47C2×Dic214C2311yescharacteristic
48Q8×D107—24no—
49Q8×D107—44no—
50C2×D428C2311yescharacteristic
51C2×C22⋊D107—21no—
52D4×D107—24no—
53C2×C4×D107—21no—
54C2×C4×D107—22no—
55C2×C4×D107D411yescharacteristic
56C2×C4×D107D411yescharacteristic
57C2×C4×D107C2311yescharacteristic
58C2×C107⋊C8C2312yes—
59C8×D107—24no—
60C4⋊Dic107—21no—
61C4⋊Dic107C2312yes—
62Dic107⋊C4—22no—
63Dic107⋊C4—22no—
64C22⋊Dic107—21no—
65C22⋊Dic107—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—21no—
69(C2×D107)⋊C4—21no—
70C4×Dic107D411yescharacteristic
71C4×Dic107D411yescharacteristic
72C4×Dic107C2311yescharacteristic
73C2×Q8×C107—21no—
74C2×Q8×C107C2311yescharacteristic
75C2×D4×C107C2311yescharacteristic
76C2×C8×C107C2312yes—
77C4⋊C4×C107—21no—
78C4⋊C4×C107C2312yes—
79C22⋊C4×C107—21no—
80C22⋊C4×C107—21no—
81C42×C107C2311yescharacteristic
82C23×C107—21no—
83C22×D107—21no—
84C22×D107—42no—
85C22×D107—42no—
86C22×D107—42no—
87C22×D107C2×D411yescharacteristic
88Q8×C107—22no—
89Q8×C107—42no—
90D4×C107—22no—
91C2×C4×C107—21no—
92C2×C4×C107—22no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—42no—
97C2×Dic107—21no—
98C2×Dic107—22no—
99C2×Dic107C2×D411yescharacteristic
100C2×Dic107C2×D411yescharacteristic
101C2×Dic107C2×D411yescharacteristic
102D428—22no—
103C4×D107—22no—
104C4×D107—22no—
105C4×D107—42no—
106C4×D107—44no—
107C4×D107#4114yes—
108Dic214—22no—
109Dic214—44no—
110Dic214—42no—
111Dic214—42no—
112C8×C107—22no—
113C107⋊C8—22no—
114C22×C107—41no—
115C22×C107—41no—
116C22×C107—41no—
117C22×C107C22×D411yescharacteristic
118C2×D107—41no—
119C2×D107—41no—
120C2×D107—41no—
121C2×D107—84no—
122C2×D107#12412yes—
123C2×D107AGL(1,ℤ/8)12yes—
124C2×D107#13712yes—
125C4×C107—21no—
126C4×C107—21no—
127C4×C107—41no—
128C4×C107—42no—
129C4×C107C2×#4112yes—
130Dic107—21no—
131Dic107—21no—
132Dic107—41no—
133Dic107—42no—
134Dic107#12412yes—
135C2×C107—81no—
136C2×C107C2×#12411yescharacteristic
137C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C107C2×#13711yescharacteristic
139D107—81no—
140D107#48914yesdirect product factor
141C2×#489—1071nomaximal, Sylow
142C107C2×#48911yescharacteristic, complemented, Sylow
143C2×C4⋊Q8—1071no—
144C2×#124—1071no—
145#489—1071no—
146#489—1071no—
147#489—1072no—
148#489—1072no—
149#489—1071no—
150#489D10711yescharacteristic, direct product factor
151C2×#103—1072no—
152C2×#102—1072no—
153C2×#97—1071no—
154C22×Q8—1071no—
155C22×Q8—2141no—
156C22×D4—1071no—
157C22×C8—1072no—
158C4⋊Q8—1072no—
159C4⋊Q8—1071no—
160C4⋊Q8—2141no—
161C4⋊Q8—2141no—
162C4⋊Q8C2×D10711yescharacteristic, complemented
163#124—1071no—
164#124—1071no—
165#124—1071no—
166#124—2141no—
167#124—2141no—
168#124C2×D10711yescharacteristic
169C2×C4⋊C4—1072no—
170C2×C4⋊C4—2141no—
171C2×C22⋊C4—2141no—
172C2×C22⋊C4—2141no—
173C2×C42—1071no—
174#103—1072no—
175#103—1072no—
176#103—1072no—
177#103C2×D10712yescomplemented
178#102—1072no—
179#102—1072no—
180#102—1072no—
181#102C2×D10712yes—
182#97—1072no—
183#97—1071no—
184#97C2×D10711yescharacteristic, complemented
185C24—2141no—
186C2×Q8—21no—
187C2×Q8—1071no—
188C2×Q8—4284no—
189C2×Q8—2141no—
190C2×Q8—2142no—
191C2×Q8—2144no—
192C2×Q8—2141no—
193C2×Q8C22×D10711yescharacteristic
194C2×D4—1071no—
195C2×D4—2144no—
196C2×D4—2141no—
197C2×D4C22×D10711yescharacteristic
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—1071no—
201C22×C4—2141no—
202C22×C4—2142no—
203C2×C8—1072no—
204C2×C8—2144no—
205C2×C8C22×D10712yes—
206C4⋊C4—21no—
207C4⋊C4—1072no—
208C4⋊C4—2142no—
209C4⋊C4—2142no—
210C4⋊C4—2141no—
211C4⋊C4C22×D10712yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—2141no—
215C22⋊C4—2141no—
216C22⋊C4—2141no—
217C22⋊C4—2141no—
218C22⋊C4—2141no—
219C22⋊C4—2141no—
220C42—1071no—
221C42—1071no—
222C42—1071no—
223C42C22×D10711yescharacteristic
224C23—21no—
225C23—1071no—
226C23—4282no—
227C23—4282no—
228C23—4282no—
229C23—2141no—
230Q8—22no—
231Q8—42no—
232Q8—4282no—
233Q8—4282no—
234Q8—4284no—
235Q8—2142no—
236D4—22no—
237D4—4282no—
238D4—2142no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—1071no—
242C2×C4—1074no—
243C2×C4—1071no—
244C2×C4—1071no—
245C2×C4—4284no—
246C2×C4—4282no—
247C2×C4—2142no—
248C2×C4—2142no—
249C2×C4—2141no—
250C2×C4—2142no—
251C2×C4D4×D10711yescharacteristic
252C2×C4D4×D10711yescharacteristic
253C2×C4C23×D10711yescharacteristic
254C8—22no—
255C8—2142no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—8564no—
260C22—1072no—
261C22—1072no—
262C22—1072no—
263C22—4281no—
264C22—4281no—
265C22—4281no—
266C22C2×D4×D10711yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—1072no—
272C4—4281no—
273C4—4282no—
274C4—2141no—
275C4—2141no—
276C4#41×D10712yes—
277C2—81no—
278C2—8561no—
279C2—1074no—
280C2#124×D10711yescharacteristic
281C2AGL(1,ℤ/8)×D10711yescharacteristic
282C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#489×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×D107C22
hypercenter#489D107
hypocenterC107C2×#489
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D107C2semidirect
3. #20292989C2semidirect
4. #20292988C2semidirect
5. #20292987C2semidirect
6. #20292986C2semidirect
7. #20292985C2semidirect
8. #103×D107C2semidirect
10. #97×D107C2semidirect
11. #489×C107C2semidirect
16. #16812109C22semidirect
18. #16812108C22semidirect
30. #16812034C22semidirect
31. #16812033C22semidirect
32. #16812032C22semidirect
36. #16812020C22semidirect
37. #16812019C22semidirect
38. C4⋊Q8×C107C22semidirect
40. #103×C107C22semidirect
42. #97×C107C22semidirect
140. D107#489direct (N × H)
142. C107C2×#489semidirect
150. #489D107direct (N × H)
162. C4⋊Q8C2×D107semidirect
177. #103C2×D107semidirect
184. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1072
#124×D1072
#202929892
#202929882
#202929872
#202929862
#202929852
#103×D1072
#102×D1072
#97×D1072
#489×C1072
C2×#489107