Finite Group Databasebeta

← #28348077

Subgroups — #28348077

Order 13696.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16812135C212yesmaximal, complemented
2#20293298C211yesmaximal, characteristic, complemented
3#20293003C211yesmaximal, characteristic, complemented
4#20292878C212yesmaximal, complemented
5#20292824C211yesmaximal, characteristic, complemented
6#20292822C211yesmaximal, characteristic, complemented
7#20292770C212yesmaximal, complemented
8#20292761C212yesmaximal, complemented
9C4×#13133806C211yesmaximal, characteristic, complemented
10#20292705C211yesmaximal, characteristic, complemented
11D42×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yescomplemented
13#16812135—28no—
14C2×#13133806C2211yescharacteristic
15C2×#13133806C2212yescomplemented
16C2×#13133803C2212yes—
17C2×#13133803C2212yescomplemented
18C2×#13133803C2212yescomplemented
19C2×#13133802C2212yescomplemented
20#16812106C2211yescharacteristic, complemented
21#16812105—21no—
22#16812083C2212yescomplemented
23#16812082—22no—
24C4×Dic214C2211yescharacteristic, complemented
25#16812071—21no—
26#16812040C2211yescharacteristic, complemented
27C4⋊(C107⋊C8)C2211yescharacteristic, complemented
28#16812032C2211yescharacteristic, complemented
29#16812030C2211yescharacteristic
30#16812030C2212yescomplemented
31#16812029C2212yescomplemented
32C22⋊(C107⋊C8)C2212yescomplemented
33C4×C107⋊C8C2211yescharacteristic, complemented
34C22×D4×C107—21no—
35C22×D4×C107C2212yes—
36#127×C107C2211yescharacteristic
37#121×C107—21no—
38#121×C107C2212yes—
39#120×C107—22no—
40C4×D4×C107C2211yescharacteristic
41C4×D4×C107C2211yescharacteristic
42#13133818—24no—
43#13133818—42no—
44C2×Dic214C2311yescharacteristic
45C2×D428—21no—
46C2×D428—21no—
47C2×D428C2312yes—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C4×D107—22no—
51#13133806—24no—
52#13133806D414yescomplemented
53#13133803—24no—
54#13133803—24no—
55#13133803—42no—
56#13133802—24no—
57C2×C107⋊C8C2311yescharacteristic
58C2×C107⋊C8C2312yes—
59C2×C107⋊C8C2311yescharacteristic
60C4⋊Dic107C2311yescharacteristic
61Dic107⋊C4—21no—
62(C2×D107)⋊C4—22no—
63C4×Dic107—21no—
64C24×C107—22no—
65C2×D4×C107—24no—
66C2×D4×C107—22no—
67C2×D4×C107—22no—
68C2×D4×C107—21no—
69C2×D4×C107—21no—
70C2×D4×C107—42no—
71C2×D4×C107—44no—
72C2×D4×C107D411yescharacteristic
73C2×D4×C107C2312yes—
74C2×D4×C107C2311yescharacteristic
75C22×C4×C107—21no—
76C22×C4×C107C2312yes—
77C4⋊C4×C107D411yescharacteristic
78C4⋊C4×C107C2311yescharacteristic
79C22⋊C4×C107—21no—
80C22⋊C4×C107—22no—
81C22⋊C4×C107D412yes—
82C42×C107C2311yescharacteristic
83C23×C107—21no—
84C23×C107—22no—
85C23×C107—44no—
86C23×C107—44no—
87C23×C107—82no—
88C23×C107C2×D412yes—
89C22×D107—22no—
90D4×C107—22no—
91D4×C107—22no—
92D4×C107—21no—
93D4×C107—22no—
94D4×C107—44no—
95D4×C107—44no—
96D4×C107—42no—
97D4×C107—81no—
98D4×C107C2×D412yes—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107—21no—
102C2×C4×C107—42no—
103C2×C4×C107C2×D411yescharacteristic
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2×D411yescharacteristic
106C2×C4×C107C2411yescharacteristic
107C22⋊D107—44no—
108C22⋊D107—82no—
109C2×Dic107—21no—
110C2×Dic107—21no—
111D428—22no—
112D428—22no—
113D428—42no—
114D428—81no—
115C4×D107—44no—
116Dic214—21no—
117Dic214C2×D412yescomplemented
118C107⋊C8—22no—
119C107⋊C8—21no—
120C107⋊C8C2×D412yescomplemented
121C22×C107—22no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107—22no—
125C22×C107—41no—
126C22×C107—41no—
127C22×C107—42no—
128C22×C107—42no—
129C22×C107—42no—
130C22×C107—88no—
131C22×C107C22×D411yescharacteristic
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—42no—
135C4×C107—22no—
136C4×C107—21no—
137C4×C107—41no—
138C4×C107AGL(1,ℤ/8)12yescomplemented
139C4×C107C22×D411yescharacteristic
140C4×C107H(5,𝔽2)11yescharacteristic
141Dic107—22no—
142Dic107—41no—
143C2×C107—42no—
144C2×C107—42no—
145C2×C107—82no—
146C2×C107D4211yescharacteristic
147C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C107#57611yescharacteristic
149D107—82no—
150#3274—1071nomaximal, Sylow
151C107#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1072no—
153#550—1071no—
154D42D10711yescharacteristic, complemented
155#493—1071no—
156#469—1072no—
157#460—1071no—
158#459—1071no—
159#449—1072no—
160#447—1072no—
161C4×SD16—1071no—
162#436—1071no—
163C2×#41—1072no—
164C22×D4—21no—
165C22×D4C2×D10712yes—
166AGL(1,ℤ/8)—2148no—
167C2×SD16—1071no—
168C2×SD16—1072no—
169C2×D8—1072no—
170C2×D8—1072no—
171C2×D8—1072no—
172C2×M16—1072no—
173#127—1071no—
174#127—2141no—
175#127C2×D10711yescharacteristic
176#121—21no—
177#121—1072no—
178#121—2142no—
179#121C2×D10712yes—
180#120—22no—
181C4×Q8—1071no—
182C4×D4—2141no—
183C4×D4C2×D10711yescharacteristic
184C4×D4C2×D10711yescharacteristic
185#106—1071no—
186C4⋊C8—1071no—
187#103—1071no—
188#102—1071no—
189#102—1072no—
190#102—1072no—
191C22⋊C8—1072no—
192C4×C8—1071no—
193C24—22no—
194#41—4282no—
195#41—2144no—
196C2×Q8—1071no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—42no—
203C2×D4—44no—
204C2×D4—1072no—
205C2×D4—2142no—
206C2×D4—2142no—
207C2×D4—2141no—
208C2×D4—2141no—
209C2×D4C22⋊D10711yescharacteristic
210C2×D4C22×D10712yes—
211C2×D4C22×D10711yescharacteristic
212C22×C4—21no—
213C22×C4—2142no—
214C22×C4C22×D10712yes—
215SD16—1074no—
216SD16—2144no—
217D8—4282no—
218D8—2144no—
219D8—2144no—
220M16—2144no—
221C2×C8—1071no—
222C2×C8—1071no—
223C2×C8—1072no—
224C4⋊C4—1071no—
225C4⋊C4—2141no—
226C4⋊C4C22⋊D10711yescharacteristic
227C4⋊C4C22×D10711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—2142no—
231C22⋊C4C22⋊D10712yes—
232C42—2141no—
233C42C22×D10711yescharacteristic
234C23—21no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—82no—
239C23—2142no—
240C23C2×C22⋊D10712yes—
241Q8—1072no—
242Q8—2141no—
243D4—22no—
244D4—22no—
245D4—21no—
246D4—22no—
247D4—44no—
248D4—44no—
249D4—42no—
250D4—81no—
251D4—8561no—
252D4—8562no—
253D4—4284no—
254D4—4282no—
255D4—2142no—
256D4—2142no—
257D4D4×D10712yes—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—4284no—
263C2×C4—2141no—
264C2×C4—2141no—
265C2×C4C2×C22⋊D10711yescharacteristic
266C2×C4C2×C22⋊D10712yes—
267C2×C4C2×C22⋊D10711yescharacteristic
268C2×C4C23×D10711yescharacteristic
269C8—1072no—
270C8—2142no—
271C8—2141no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—88no—
282C22—4282no—
283C22—4282no—
284C22—4282no—
285C22C22×C22⋊D10711yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—4281no—
290C4—2142no—
291C4#1681213512yescomplemented
292C4C2×D4×D10711yescharacteristic
293C4#1681215411yescharacteristic
294C2—42no—
295C2—42no—
296C2—82no—
297C2—8562no—
298C2D4×C22⋊D10711yescharacteristic
299C2C2×#1681213511yescharacteristic
300C2#2029343011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingD42×C107C2
socleC22×C107C22×D4
hypercenterD42D107
hypocenterC107#3274
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. #20293298C2semidirect
3. #20293003C2semidirect
4. #20292878C2semidirect
5. #20292824C2semidirect
6. #20292822C2semidirect
7. #20292770C2semidirect
8. #20292761C2semidirect
9. C4×#13133806C2semidirect
10. #20292705C2semidirect
11. D42×C107C2semidirect
12. C2×#13133818C22semidirect
15. C2×#13133806C22semidirect
17. C2×#13133803C22semidirect
18. C2×#13133803C22semidirect
19. C2×#13133802C22semidirect
20. #16812106C22semidirect
22. #16812083C22semidirect
24. C4×Dic214C22semidirect
26. #16812040C22semidirect
27. C4⋊(C107⋊C8)C22semidirect
28. #16812032C22semidirect
30. #16812030C22semidirect
31. #16812029C22semidirect
32. C22⋊(C107⋊C8)C22semidirect
33. C4×C107⋊C8C22semidirect
52. #13133806D4semidirect
117. Dic214C2×D4semidirect
120. C107⋊C8C2×D4semidirect
138. C4×C107AGL(1,ℤ/8)semidirect
151. C107#3274semidirect
154. D42D107semidirect
291. C4#16812135semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
#202932982
#202930032
#202928782
#202928242
#202928222
#202927702
#202927612
C4×#131338062
#202927052
D42×C1072
#3274107