Finite Group Databasebeta

← #28345700

Subgroups — #28345700

Order 13696.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 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×#13133817C211yesmaximal, characteristic, complemented
2C2×#16812139C211yesmaximal, characteristic, complemented
3#20293134C211yesmaximal, characteristic, complemented
4#20292786C214yesmaximal, complemented
5#20292769C214yesmaximal, complemented
6#20292637C211yesmaximal, characteristic, complemented
7#20292632C211yesmaximal, characteristic, complemented
8#20292590C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
10C2×#13133817—24no—
11C2×#13133817—22no—
12C2×#13133817—24no—
13C2×#13133817C2214yes—
14C22×Dic214C2211yescharacteristic
15C22×C22⋊D107—21no—
16C23×Dic107—21no—
17C22×C4×D107C2211yescharacteristic
18#16812139—24no—
19C2×#13133808C2212yescomplemented
20C2×#13133806C2212yescomplemented
21C2×#13133802C2211yescharacteristic
22#16812090—21no—
23#16812085C2212yescomplemented
24#16812080C2212yescomplemented
25#16812077—21no—
26C4×C22⋊D107—22no—
27#16812068C2211yescharacteristic
28#16812034C2214yescomplemented
29#16812030C2214yescomplemented
30(C2×D107)⋊C8C2214yescomplemented
31C2×#41×C107C2211yescharacteristic
32C22×D4×C107C2211yescharacteristic
33AGL(1,ℤ/8)×C107—24no—
34C2×SD16×C107C2212yescomplemented
35C2×D8×C107C2212yescomplemented
36C2×M16×C107C2211yescharacteristic
37C23×D107D411yescharacteristic
38#13133817—216no—
39#13133817—28no—
40#13133817—28no—
41C2×Dic214—21no—
42C2×Dic214—22no—
43C2×Dic214D414yescomplemented
44C2×Dic214C2312yes—
45C2×C22⋊D107—28no—
46C2×C22⋊D107—24no—
47C2×C22⋊D107—22no—
48C22×Dic107—28no—
49C22×Dic107—21no—
50C22×Dic107—24no—
51C22×Dic107D411yescharacteristic
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107D412yes—
55C2×C4×D107C2312yes—
56#13133808—24no—
57#13133806—24no—
58#13133802—22no—
59C2×C107⋊C8C2312yes—
60C4⋊Dic107C2312yes—
61Dic107⋊C4—22no—
62C22⋊Dic107—21no—
63C22⋊Dic107—22no—
64(C2×D107)⋊C4—22no—
65C4×Dic107—21no—
66C24×C107—21no—
67#41×C107—22no—
68#41×C107—22no—
69C2×Q8×C107C2311yescharacteristic
70C2×D4×C107—21no—
71C2×D4×C107—21no—
72C2×D4×C107—22no—
73C2×D4×C107D414yescomplemented
74C2×D4×C107C2311yescharacteristic
75C2×D4×C107C2312yes—
76C22×C4×C107—21no—
77C22×C4×C107C2311yescharacteristic
78SD16×C107—24no—
79D8×C107—24no—
80M16×C107—22no—
81C2×C8×C107C2312yes—
82C23×C107—24no—
83C23×C107—22no—
84C23×C107—21no—
85C23×C107—44no—
86C23×C107C2×D411yescharacteristic
87C22×D107—22no—
88C22×D107—42no—
89C22×D107C2×D412yes—
90Q8×C107—21no—
91Q8×C107—21no—
92D4×C107—24no—
93D4×C107—22no—
94D4×C107—21no—
95D4×C107—21no—
96D4×C107—42no—
97D4×C107C2×D414yes—
98C2×C4×C107—21no—
99C2×C4×C107—42no—
100C2×C4×C107C2×D411yescharacteristic
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2×D412yes—
103C2×C4×C107C2411yescharacteristic
104C22⋊D107—216no—
105C22⋊D107—44no—
106C2×Dic107—216no—
107C2×Dic107—22no—
108C2×Dic107—24no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107C2×D412yes—
112C4×D107—24no—
113C4×D107—22no—
114C4×D107—22no—
115Dic214—24no—
116Dic214—22no—
117Dic214C2×D414yes—
118C8×C107—22no—
119C107⋊C8—22no—
120C22×C107—22no—
121C22×C107—24no—
122C22×C107—48no—
123C22×C107—44no—
124C22×C107—41no—
125C22×C107—42no—
126C22×C107#12012yes—
127C22×C107C22×D411yescharacteristic
128C2×D107—22no—
129C2×D107—22no—
130C2×D107—22no—
131C2×D107—44no—
132C4×C107—41no—
133C4×C107#12012yes—
134C4×C107C22×D411yescharacteristic
135C4×C107C22×D411yescharacteristic
136Dic107—24no—
137Dic107—22no—
138Dic107—42no—
139C2×C107—22no—
140C2×C107—44no—
141C2×C107—81no—
142C2×C107C2×#12011yescharacteristic
143C2×C107#57512yes—
144D107—42no—
145#3005—1071nomaximal, Sylow
146C107#300511yescharacteristic, complemented, Sylow
147C22×#41—1071no—
148C2×#137—1071no—
149C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
150#525—1071no—
151#452—1074no—
152#449—1074no—
153#419—1071no—
154#418—1071no—
155#407—1071no—
156C2×#41—1074no—
157C2×#41—2142no—
158C2×#41—2144no—
159C2×#41—2144no—
160C2×#41C2×D10711yescharacteristic
161C22×Q8—1071no—
162C22×D4—2141no—
163C22×D4C2×D10711yescharacteristic
164C23×C4—1071no—
165C23×C4—2141no—
166#137—2144no—
167AGL(1,ℤ/8)—24no—
168C2×Q16—1072no—
169C2×SD16—1072no—
170C2×SD16C2×D10712yescomplemented
171C2×D8C2×D10712yescomplemented
172C2×M16—1071no—
173C2×M16C2×D10711yescharacteristic
174#123—2141no—
175C22⋊Q8—1072no—
176#121—1072no—
177#120—2141no—
178C4×D4—2142no—
179#117—1071no—
180#103—1074no—
181#102—1074no—
182C22⋊C8—1074no—
183C24—21no—
184C24—1071no—
185#41—22no—
186#41—22no—
187#41—2148no—
188#41—21416no—
189#41—2148no—
190C2×Q8—1074no—
191C2×Q8—1072no—
192C2×Q8—2141no—
193C2×Q8—2142no—
194C2×Q8C22×D10711yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—2142no—
199C2×D4—2144no—
200C2×D4—2148no—
201C2×D4C22⋊D10714yescomplemented
202C2×D4C22×D10711yescharacteristic
203C2×D4C22×D10712yes—
204C22×C4—21no—
205C22×C4—1071no—
206C22×C4—1072no—
207C22×C4—1072no—
208C22×C4—2144no—
209C22×C4—2148no—
210C22×C4—2141no—
211C22×C4—2142no—
212C22×C4—2142no—
213C22×C4C22×D10711yescharacteristic
214Q16—2144no—
215SD16—24no—
216SD16—2144no—
217D8—24no—
218M16—22no—
219M16—2142no—
220C2×C8—1072no—
221C2×C8C22×D10712yes—
222C4⋊C4—1072no—
223C4⋊C4—2142no—
224C22⋊C4—2142no—
225C22⋊C4—2142no—
226C22⋊C4—2141no—
227C42—2141no—
228C23—24no—
229C23—22no—
230C23—21no—
231C23—44no—
232C23—1072no—
233C23—4282no—
234C23—2142no—
235C23D4×D10711yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—1074no—
239Q8—2144no—
240Q8—2142no—
241D4—24no—
242D4—22no—
243D4—21no—
244D4—21no—
245D4—42no—
246D4—4284no—
247D4—21416no—
248D4C2×C22⋊D10714yes—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—1072no—
252C2×C4—2142no—
253C2×C4—21416no—
254C2×C4—2142no—
255C2×C4—2144no—
256C2×C4—2142no—
257C2×C4—2144no—
258C2×C4—2142no—
259C2×C4—2142no—
260C2×C4D4×D10711yescharacteristic
261C2×C4D4×D10712yes—
262C2×C4C2×C22⋊D10712yes—
263C2×C4C23×D10711yescharacteristic
264C8—22no—
265C8—2142no—
266C22—22no—
267C22—24no—
268C22—48no—
269C22—44no—
270C22—42no—
271C22—41no—
272C22—4284no—
273C22—2142no—
274C22—2142no—
275C22—2142no—
276C22#1681207712yes—
277C22C2×D4×D10711yescharacteristic
278C4—41no—
279C4—4282no—
280C4—2142no—
281C4—2144no—
282C4#1681207712yes—
283C4C2×D4×D10711yescharacteristic
284C4C22×C22⋊D10711yescharacteristic
285C2—22no—
286C2—44no—
287C2—81no—
288C2—4282no—
289C2C2×#1681207711yescharacteristic
290C2#2029342312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3005
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×#13133817C2semidirect
2. C2×#16812139C2semidirect
3. #20293134C2semidirect
4. #20292786C2semidirect
5. #20292769C2semidirect
6. #20292637C2semidirect
7. #20292632C2semidirect
8. #20292590C2semidirect
9. C2×AGL(1,ℤ/8)×C107C2semidirect
19. C2×#13133808C22semidirect
20. C2×#13133806C22semidirect
23. #16812085C22semidirect
24. #16812080C22semidirect
28. #16812034C22semidirect
29. #16812030C22semidirect
30. (C2×D107)⋊C8C22semidirect
34. C2×SD16×C107C22semidirect
35. C2×D8×C107C22semidirect
43. C2×Dic214D4semidirect
73. C2×D4×C107D4semidirect
146. C107#3005semidirect
149. C2×AGL(1,ℤ/8)D107semidirect
170. C2×SD16C2×D107semidirect
171. C2×D8C2×D107semidirect
201. C2×D4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338172
C2×#168121392
#202931342
#202927862
#202927692
#202926372
#202926322
#202925902
C2×AGL(1,ℤ/8)×C1072
#3005107