Finite Group Databasebeta

← #28345048

Subgroups — #28345048

Order 13696.

492 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×D107C211yesmaximal, characteristic
2C2×#16812067C211yesmaximal, characteristic
3#117×D107C211yesmaximal, characteristic
4#20292670C211yesmaximal, characteristic, complemented
5#20292669C211yesmaximal, characteristic, complemented
6#20292668C212yesmaximal, complemented
7#20292667C211yesmaximal, characteristic, complemented
8#20292666C212yesmaximal, complemented
9#20292664C212yesmaximal, complemented
10#20292662C212yesmaximal, complemented
11#428×C107C211yesmaximal, characteristic, complemented
12C22×C4×D107C2211yescharacteristic
13C2×#13133802C2211yescharacteristic
14C2×#13133801C2212yes—
15M16×D107C418yescomplemented
16C2×C8×D107C2212yes—
17#16812068C2211yescharacteristic
18#16812067—21no—
19#16812067—21no—
20#16812067C2211yescharacteristic
21#16812067C2212yes—
22#16812067C2211yescharacteristic
23#16812067C2212yes—
24#16812066—21no—
25#16812065—21no—
26C2×C4⋊Dic107C2211yescharacteristic
27C4⋊C4×D107C2211yescharacteristic
28C4⋊C4×D107C2211yescharacteristic
29C2×(C2×D107)⋊C4—21no—
30C22⋊C4×D107—21no—
31C2×C4×Dic107—21no—
32C42×D107—21no—
33#16812043C2212yescomplemented
34#16812042C2212yescomplemented
35#16812042C2212yescomplemented
36#16812041C2212yescomplemented
37#16812040C2212yescomplemented
38#16812040C2212yescomplemented
39C2×M16×C107C2211yescharacteristic
40#117×C107C2211yescharacteristic
41C2×C4⋊C4×C107C2211yescharacteristic
42#107×C107C2212yescomplemented
43#106×C107C2212yescomplemented
44C23×D107D411yescharacteristic
45C22×Dic107—21no—
46C22×Dic107D411yescharacteristic
47C2×C4×D107—21no—
48C2×C4×D107—21no—
49C2×C4×D107C2×C414yes—
50C2×C4×D107D411yescharacteristic
51C2×C4×D107D411yescharacteristic
52C2×C4×D107Q814yes—
53C2×C4×D107C2311yescharacteristic
54C2×C4×D107C2311yescharacteristic
55#13133802C2×C414yescomplemented
56#13133801C2×C418yescomplemented
57C2×C107⋊C8C2312yes—
58C8×D107C2×C418yescomplemented
59C4⋊Dic107—21no—
60C4⋊Dic107C2311yescharacteristic
61C4⋊Dic107C2312yes—
62C4⋊Dic107C2311yescharacteristic
63Dic107⋊C4—21no—
64Dic107⋊C4—21no—
65C22⋊Dic107—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—21no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70C4×Dic107—21no—
71C4×Dic107—21no—
72C4×Dic107—21no—
73C4×Dic107—22no—
74C4×Dic107—22no—
75C22×C4×C107—21no—
76C22×C4×C107C2311yescharacteristic
77M16×C107C2×C414yescomplemented
78C2×C8×C107C2312yes—
79C4⋊C4×C107—21no—
80C4⋊C4×C107C2311yescharacteristic
81C4⋊C4×C107C2312yes—
82C4⋊C4×C107C2311yescharacteristic
83C22⋊C4×C107—21no—
84C42×C107—21no—
85C23×C107C2×D411yescharacteristic
86C22×D107—41no—
87C22×D107—41no—
88C22×D107C4⋊C414yes—
89C22×D107C2×D411yescharacteristic
90C22×D107C2×D411yescharacteristic
91C2×C4×C107—21no—
92C2×C4×C107—21no—
93C2×C4×C107—22no—
94C2×C4×C107—42no—
95C2×C4×C107C22×C412yes—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×Q812yes—
98C2×C4×C107C2411yescharacteristic
99C2×Dic107—22no—
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—42no—
103C2×Dic107C4⋊C414yes—
104C2×Dic107C2×D411yescharacteristic
105C2×Dic107C2×D411yescharacteristic
106C4×D107—42no—
107C4×D107—42no—
108C4×D107C4⋊C414yes—
109C4×D107C4⋊C414yes—
110C4×D107C22×C412yes—
111C4×D107C22×C412yes—
112C4×D107C2×Q812yes—
113C4×D107C2×Q812yes—
114C8×C107C22×C414yes—
115C107⋊C8C22×C414yes—
116C22×C107—21no—
117C22×C107—21no—
118C22×C107C2×C4⋊C412yes—
119C22×C107C22×D411yescharacteristic
120C2×D107—21no—
121C2×D107—21no—
122C2×D107—21no—
123C2×D107—21no—
124C2×D107—44no—
125C2×D107C2×C4⋊C412yes—
126C2×D107C2×C4⋊C412yes—
127C4×C107—41no—
128C4×C107—41no—
129C4×C107—42no—
130C4×C107C2×C4⋊C412yes—
131C4×C107C23×C411yescharacteristic
132C4×C107C22×Q811yescharacteristic
133Dic107—41no—
134Dic107—41no—
135Dic107—42no—
136Dic107C2×C4⋊C412yes—
137Dic107C2×C4⋊C412yes—
138C2×C107—22no—
139C2×C107C22×C4⋊C411yescharacteristic
140C2×C107#57511yescharacteristic
141C2×C107#57511yescharacteristic
142D107—22no—
143D107—22no—
144#2911—1071nomaximal, Sylow
145C107#291111yescharacteristic, complemented, Sylow
146C22×M16—1071no—
147C2×#117—1071no—
148C2×#117—1071no—
149#428—1071no—
150#428—1071no—
151#428—1071no—
152#428—1072no—
153#428—1072no—
154#428D10711yescharacteristic, complemented
155#427—1072no—
156#427—1072no—
157C23×C4—1071no—
158C2×M16—1071no—
159C2×M16—1078no—
160C2×M16—1072no—
161C2×M16C2×D10711yescharacteristic
162C22×C8—1072no—
163#117—1071no—
164#117—1071no—
165#117—1072no—
166#117—1071no—
167#117—1072no—
168#117—2141no—
169#117—2141no—
170#117—2141no—
171#117—2141no—
172#117C2×D10711yescharacteristic
173C2×C4⋊C4—1071no—
174C2×C4⋊C4—1071no—
175C2×C4⋊C4—1071no—
176C2×C4⋊C4C2×D10711yescharacteristic
177C2×C22⋊C4—2141no—
178C2×C22⋊C4—2141no—
179C2×C42—2141no—
180C2×C42—2141no—
181#107—1072no—
182#107—1072no—
183#107—1072no—
184#107C2×D10712yescomplemented
185#106—1072no—
186#106—1072no—
187#106—1072no—
188#106C2×D10712yescomplemented
189C24—1071no—
190C22×C4—21no—
191C22×C4—1071no—
192C22×C4—1074no—
193C22×C4—1071no—
194C22×C4—1071no—
195C22×C4—1074no—
196C22×C4—1071no—
197C22×C4—1071no—
198C22×C4—2141no—
199C22×C4—2141no—
200C22×C4—2141no—
201C22×C4C22×D10711yescharacteristic
202M16—1074no—
203M16—1078no—
204M16C4×D10714yescomplemented
205C2×C8—1072no—
206C2×C8—1078no—
207C2×C8C22×D10712yes—
208C4⋊C4—21no—
209C4⋊C4—1071no—
210C4⋊C4—1072no—
211C4⋊C4—1071no—
212C4⋊C4—2141no—
213C4⋊C4—2141no—
214C4⋊C4—2141no—
215C4⋊C4C22×D10711yescharacteristic
216C4⋊C4C22×D10712yes—
217C4⋊C4C22×D10711yescharacteristic
218C22⋊C4—21no—
219C22⋊C4—2141no—
220C22⋊C4—2141no—
221C22⋊C4—2142no—
222C22⋊C4—2142no—
223C22⋊C4—2141no—
224C42—21no—
225C42—2141no—
226C42—2141no—
227C42—2142no—
228C42—2142no—
229C42—2141no—
230C23—1074no—
231C23—1071no—
232C23—1071no—
233C23—4281no—
234C23—4281no—
235C23D4×D10711yescharacteristic
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—1074no—
241C2×C4—1071no—
242C2×C4—1071no—
243C2×C4—1074no—
244C2×C4—1074no—
245C2×C4—1072no—
246C2×C4—1072no—
247C2×C4—1072no—
248C2×C4—1072no—
249C2×C4—4282no—
250C2×C4—4282no—
251C2×C4—4282no—
252C2×C4—2141no—
253C2×C4—2142no—
254C2×C4—2141no—
255C2×C4C2×C4×D10712yes—
256C2×C4D4×D10711yescharacteristic
257C2×C4Q8×D10712yes—
258C2×C4C23×D10711yescharacteristic
259C8—1074no—
260C8C2×C4×D10714yes—
261C22—21no—
262C22—21no—
263C22—1072no—
264C22—1072no—
265C22—4284no—
266C22—2141no—
267C22—2141no—
268C22—2141no—
269C22—2141no—
270C22C4⋊C4×D10712yes—
271C22C2×D4×D10711yescharacteristic
272C4—42no—
273C4—41no—
274C4—41no—
275C4—1072no—
276C4—1072no—
277C4—4281no—
278C4—4281no—
279C4—4282no—
280C4C4⋊C4×D10712yes—
281C4C22×C4×D10711yescharacteristic
282C4C2×Q8×D10711yescharacteristic
283C2—22no—
284C2—2142no—
285C2—2142no—
286C2C2×C4⋊C4×D10711yescharacteristic
287C2#2029342311yescharacteristic
288C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radicalC23×D107D4
cosocleC2×C4×C107C24
hypercenter#428D107
hypocenterC107#2911
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
4. #20292670C2semidirect
5. #20292669C2semidirect
6. #20292668C2semidirect
7. #20292667C2semidirect
8. #20292666C2semidirect
9. #20292664C2semidirect
10. #20292662C2semidirect
11. #428×C107C2semidirect
15. M16×D107C4semidirect
33. #16812043C22semidirect
34. #16812042C22semidirect
35. #16812042C22semidirect
36. #16812041C22semidirect
37. #16812040C22semidirect
38. #16812040C22semidirect
42. #107×C107C22semidirect
43. #106×C107C22semidirect
55. #13133802C2×C4semidirect
56. #13133801C2×C4semidirect
58. C8×D107C2×C4semidirect
77. M16×C107C2×C4semidirect
145. C107#2911semidirect
154. #428D107semidirect
184. #107C2×D107semidirect
188. #106C2×D107semidirect
204. M16C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1072
C2×#168120672
#117×D1072
#202926702
#202926692
#202926682
#202926672
#202926662
#202926642
#202926622
#428×C1072
#2911107