Finite Group Databasebeta

← #486×D107 #28346764

Subgroups — #486×D107 #28346764

Order 13696.

528 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 145 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#127×D107C211yesmaximal, characteristic
3#20292975C211yesmaximal, characteristic, complemented
4#20292974C211yesmaximal, characteristic, complemented
5#20292973C211yesmaximal, characteristic, complemented
6#20292972C214yesmaximal, complemented
7#102×D107C214yesmaximal, complemented
8C4×C8×D107C211yesmaximal, characteristic, complemented
9#486×C107C211yesmaximal, characteristic, complemented
10C2×Q8×D107—21no—
11C2×D4×D107—21no—
12C2×D4×D107—21no—
13C2×D4×D107C2212yes—
14C2×C8×D107C2212yes—
15#16812109C2211yescharacteristic, complemented
16#16812108—21no—
17#16812108C2212yescomplemented
18#16812107—21no—
19#16812106C2211yescharacteristic
20#16812105—21no—
21#16812105—21no—
22#16812105C2212yes—
23C4⋊C4×D107—21no—
24C4⋊C4×D107C2212yes—
25C42×D107C2211yescharacteristic
26#16812031C2214yescomplemented
27#16812030C2214yescomplemented
28#16812029C2214yescomplemented
29C4×C107⋊C8C2211yescharacteristic, complemented
30C8×Dic107C2212yescomplemented
31C4⋊Q8×C107C2211yescharacteristic, complemented
32#127×C107C2211yescharacteristic
33#102×C107C2214yescomplemented
34C4×C8×C107C2211yescharacteristic, complemented
35C23×D107—22no—
36C2×Dic214—21no—
37C2×Dic214—22no—
38Q8×D107—44no—
39C2×D428—21no—
40C2×D428—21no—
41C2×D428C2312yes—
42C2×C22⋊D107—22no—
43C2×C22⋊D107—22no—
44C2×C22⋊D107—22no—
45D4×D107—28no—
46D4×D107—44no—
47D4×D107—44no—
48C2×C4×D107—22no—
49C2×C4×D107D411yescharacteristic
50C2×C4×D107D411yescharacteristic
51C2×C4×D107C2311yescharacteristic
52C2×C107⋊C8C2312yes—
53C8×D107—24no—
54C4⋊Dic107—21no—
55C4⋊Dic107C2312yes—
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58C4×Dic107D411yescharacteristic
59C4×Dic107D411yescharacteristic
60C4×Dic107C2311yescharacteristic
61C2×Q8×C107—21no—
62C2×D4×C107—21no—
63C2×D4×C107—21no—
64C2×D4×C107C2312yes—
65C2×C8×C107C2312yes—
66C4⋊C4×C107—21no—
67C4⋊C4×C107C2312yes—
68C42×C107C2311yescharacteristic
69C23×C107—22no—
70C22×D107—22no—
71C22×D107—44no—
72C22×D107—44no—
73C22×D107—44no—
74C22×D107C2×D411yescharacteristic
75Q8×C107—42no—
76D4×C107—24no—
77D4×C107—42no—
78D4×C107—42no—
79C2×C4×C107—22no—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2411yescharacteristic
83C22⋊D107—44no—
84C22⋊D107—44no—
85C22⋊D107—44no—
86C2×Dic107—22no—
87C2×Dic107C2×D411yescharacteristic
88C2×Dic107C2×D411yescharacteristic
89C2×Dic107C2×D411yescharacteristic
90D428—24no—
91D428—42no—
92D428—42no—
93C4×D107—44no—
94C4×D107D814yes—
95C4×D107SD1614yes—
96C4×D107#4114yes—
97Dic214—42no—
98Dic214—44no—
99C8×C107—22no—
100C107⋊C8—22no—
101C22×C107—42no—
102C22×C107—42no—
103C22×C107—42no—
104C22×C107C22×D411yescharacteristic
105C2×D107—42no—
106C2×D107—42no—
107C2×D107—42no—
108C2×D107—88no—
109C2×D107#12412yes—
110C2×D107C2×D812yes—
111C2×D107C2×SD1612yes—
112C4×C107—42no—
113C4×C107C2×D812yes—
114C4×C107C2×SD1612yes—
115C4×C107C2×#4112yes—
116Dic107—42no—
117Dic107#12412yes—
118Dic107C2×D812yes—
119Dic107C2×SD1612yes—
120C2×C107—82no—
121C2×C107C2×#12411yescharacteristic
122C2×C107C22×D811yescharacteristic
123C2×C107C22×SD1611yescharacteristic
124D107—82no—
125D107#48614yesdirect product factor
126C2×#486—1071nomaximal, Sylow
127C107C2×#48611yescharacteristic, complemented, Sylow
128C2×C4⋊Q8—1071no—
129C2×#127—1071no—
130#486—1071no—
131#486—1071no—
132#486—1074no—
133#486—1071no—
134#486D10711yescharacteristic, direct product factor
135C2×#102—1074no—
136C2×C4×C8—1071no—
137C22×Q8—2141no—
138C22×D4—1072no—
139C22×D4—2141no—
140C22×D4—2141no—
141C22×C8—1072no—
142C4⋊Q8—1072no—
143C4⋊Q8—1071no—
144C4⋊Q8—2141no—
145C4⋊Q8—2141no—
146C4⋊Q8C2×D10711yescharacteristic, complemented
147#127—1072no—
148#127—1071no—
149#127—2141no—
150#127—2141no—
151#127C2×D10711yescharacteristic
152C2×C4⋊C4—1072no—
153C2×C4⋊C4—2141no—
154C2×C42—1071no—
155#102—1074no—
156#102—1074no—
157#102—1074no—
158#102C2×D10714yescomplemented
159C4×C8—1072no—
160C4×C8—1071no—
161C4×C8C2×D10711yescharacteristic, complemented
162C24—2142no—
163C2×Q8—21no—
164C2×Q8—4284no—
165C2×Q8—2142no—
166C2×Q8—2141no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—1072no—
170C2×D4—4284no—
171C2×D4—4284no—
172C2×D4—2148no—
173C2×D4—2142no—
174C2×D4—2142no—
175C2×D4—2142no—
176C2×D4—2141no—
177C2×D4—2141no—
178C2×D4C22×D10712yes—
179C22×C4—1071no—
180C22×C4—1071no—
181C22×C4—1071no—
182C22×C4—2142no—
183C2×C8—1072no—
184C2×C8—2144no—
185C2×C8C22×D10712yes—
186C4⋊C4—21no—
187C4⋊C4—1072no—
188C4⋊C4—2142no—
189C4⋊C4—2142no—
190C4⋊C4—2141no—
191C4⋊C4C22×D10712yes—
192C42—1071no—
193C42—1071no—
194C42—1071no—
195C42C22×D10711yescharacteristic
196C23—22no—
197C23—1071no—
198C23—4284no—
199C23—4284no—
200C23—4284no—
201C23—2142no—
202Q8—42no—
203Q8—4284no—
204Q8—4282no—
205D4—24no—
206D4—42no—
207D4—42no—
208D4—4284no—
209D4—4284no—
210D4—4284no—
211D4—4282no—
212D4—4282no—
213D4—2144no—
214C2×C4—22no—
215C2×C4—1071no—
216C2×C4—1071no—
217C2×C4—1071no—
218C2×C4—1074no—
219C2×C4—1074no—
220C2×C4—1074no—
221C2×C4—4284no—
222C2×C4—2142no—
223C2×C4D4×D10711yescharacteristic
224C2×C4D4×D10711yescharacteristic
225C2×C4C23×D10711yescharacteristic
226C8—22no—
227C8—2142no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—8568no—
232C22—1072no—
233C22—1072no—
234C22—1072no—
235C22—4282no—
236C22—4282no—
237C22—4282no—
238C22C2×D4×D10711yescharacteristic
239C4—42no—
240C4—1072no—
241C4—1072no—
242C4—1072no—
243C4—4282no—
244C4D8×D10712yes—
245C4SD16×D10712yes—
246C4#41×D10712yes—
247C2—82no—
248C2—8562no—
249C2—1074no—
250C2#124×D10711yescharacteristic
251C2C2×D8×D10711yescharacteristic
252C2C2×SD16×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#486×C107C2
socleC22×C107C22×D4
prime radical#127×D107C2
hypercenter#486D107
hypocenterC107C2×#486
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D107C2semidirect
3. #20292975C2semidirect
4. #20292974C2semidirect
5. #20292973C2semidirect
6. #20292972C2semidirect
7. #102×D107C2semidirect
8. C4×C8×D107C2semidirect
9. #486×C107C2semidirect
15. #16812109C22semidirect
17. #16812108C22semidirect
26. #16812031C22semidirect
27. #16812030C22semidirect
28. #16812029C22semidirect
29. C4×C107⋊C8C22semidirect
30. C8×Dic107C22semidirect
31. C4⋊Q8×C107C22semidirect
33. #102×C107C22semidirect
34. C4×C8×C107C22semidirect
125. D107#486direct (N × H)
127. C107C2×#486semidirect
134. #486D107direct (N × H)
146. C4⋊Q8C2×D107semidirect
158. #102C2×D107semidirect
161. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1072
#127×D1072
#202929752
#202929742
#202929732
#202929722
#102×D1072
C4×C8×D1072
#486×C1072
C2×#486107