Finite Group Databasebeta

← #28348081

Subgroups — #28348081

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×#16812134C212yesmaximal, complemented
2D4×D428C211yesmaximal, characteristic, complemented
3#20293003C211yesmaximal, characteristic, complemented
4#20292878C212yesmaximal, complemented
5#20292827C211yesmaximal, characteristic, complemented
6#20292819C211yesmaximal, characteristic, complemented
7#20292772C212yesmaximal, complemented
8#20292760C212yesmaximal, complemented
9C4×#13133805C211yesmaximal, characteristic, complemented
10#20292705C211yesmaximal, characteristic, complemented
11#550×C107C211yesmaximal, characteristic, complemented
12C22×D428C2212yes—
13C2×D4×D107—21no—
14#16812134—28no—
15C2×#13133805C2211yescharacteristic
16C2×#13133805C2212yescomplemented
17C2×#13133803C2212yes—
18C2×#13133803C2212yescomplemented
19C2×#13133803C2212yescomplemented
20C2×#13133802C2212yescomplemented
21#16812106C2211yescharacteristic
22#16812084—21no—
23#16812083C2212yes—
24#16812079—22no—
25C4×D428C2211yescharacteristic
26#16812040C2211yescharacteristic, complemented
27C4⋊(C107⋊C8)C2211yescharacteristic, complemented
28#16812033C2211yescharacteristic, complemented
29#16812030C2212yescomplemented
30#16812029C2211yescharacteristic
31#16812029C2212yescomplemented
32C22⋊(C107⋊C8)C2212yescomplemented
33C4×C107⋊C8C2211yescharacteristic, complemented
34C2×#41×C107C2212yescomplemented
35#127×C107—21no—
36#127×C107C2211yescharacteristic, complemented
37#121×C107—22no—
38#121×C107C2212yescomplemented
39C4×Q8×C107C2211yescharacteristic, complemented
40C4×D4×C107—21no—
41C4×D4×C107C2211yescharacteristic
42C23×D107—22no—
43C2×D428—24no—
44C2×D428—22no—
45C2×D428—21no—
46C2×D428—21no—
47C2×D428—42no—
48C2×D428C2312yes—
49C2×D428C2311yescharacteristic
50C2×C22⋊D107—22no—
51D4×D107—44no—
52C2×C4×D107—21no—
53#13133805—24no—
54#13133805D414yescomplemented
55#13133803—24no—
56#13133803—24no—
57#13133803—42no—
58#13133802—24no—
59C2×C107⋊C8C2311yescharacteristic
60C2×C107⋊C8C2312yes—
61C2×C107⋊C8C2311yescharacteristic
62C4⋊Dic107C2311yescharacteristic
63(C2×D107)⋊C4—21no—
64(C2×D107)⋊C4—22no—
65#41×C107—24no—
66#41×C107—42no—
67C2×Q8×C107C2311yescharacteristic
68C2×D4×C107—22no—
69C2×D4×C107—22no—
70C2×D4×C107—21no—
71C2×D4×C107—21no—
72C2×D4×C107D411yescharacteristic
73C2×D4×C107C2312yes—
74C22×C4×C107—22no—
75C22×C4×C107C2312yes—
76C4⋊C4×C107—21no—
77C4⋊C4×C107D411yescharacteristic
78C4⋊C4×C107C2311yescharacteristic
79C22⋊C4×C107—22no—
80C22⋊C4×C107D412yes—
81C42×C107—21no—
82C42×C107C2311yescharacteristic
83C23×C107—22no—
84C23×C107C2×D412yes—
85C22×D107—21no—
86C22×D107—22no—
87C22×D107—44no—
88C22×D107—44no—
89C22×D107—82no—
90Q8×C107—21no—
91Q8×C107C2×D412yescomplemented
92D4×C107—22no—
93D4×C107—22no—
94D4×C107—22no—
95D4×C107—44no—
96D4×C107—42no—
97D4×C107—81no—
98D4×C107—82no—
99C2×C4×C107—21no—
100C2×C4×C107—22no—
101C2×C4×C107—22no—
102C2×C4×C107—21no—
103C2×C4×C107—44no—
104C2×C4×C107C2×D411yescharacteristic
105C2×C4×C107C2×D412yes—
106C2×C4×C107C2×D411yescharacteristic
107C2×C4×C107C2411yescharacteristic
108C22⋊D107—44no—
109C2×Dic107—21no—
110D428—22no—
111D428—21no—
112D428—22no—
113D428—44no—
114D428—42no—
115D428—81no—
116D428C2×D412yes—
117C4×D107—42no—
118C107⋊C8—22no—
119C107⋊C8—21no—
120C107⋊C8C2×D412yescomplemented
121C22×C107—22no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107—42no—
125C22×C107—42no—
126C22×C107—42no—
127C22×C107C22×D411yescharacteristic
128C2×D107—22no—
129C2×D107—41no—
130C2×D107—41no—
131C2×D107—42no—
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—88no—
135C4×C107—22no—
136C4×C107—22no—
137C4×C107—21no—
138C4×C107—41no—
139C4×C107AGL(1,ℤ/8)12yescomplemented
140C4×C107C22×D411yescharacteristic
141C4×C107H(5,𝔽2)11yescharacteristic
142Dic107—41no—
143C2×C107—42no—
144C2×C107—82no—
145C2×C107D4211yescharacteristic
146C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C107#57611yescharacteristic
148D107—42no—
149D107—82no—
150#3274—1071nomaximal, Sylow
151C107#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1072no—
153#550D10711yescharacteristic, complemented
154D42—1071no—
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×#41C2×D10712yescomplemented
164C22×D4—1072no—
165C22×D4—2141no—
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—21no—
174#127—1071no—
175#127C2×D10711yescharacteristic, complemented
176#121—22no—
177#121—1072no—
178#121—2141no—
179#121C2×D10712yescomplemented
180#120—2142no—
181C4×Q8C2×D10711yescharacteristic, complemented
182C4×D4—21no—
183C4×D4—1071no—
184C4×D4C2×D10711yescharacteristic
185#106—1071no—
186C4⋊C8—1071no—
187#103—1071no—
188#102—1072no—
189#102—1071no—
190#102—1072no—
191C22⋊C8—1072no—
192C4×C8—1071no—
193C24—2142no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D10711yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—1071no—
202C2×D4—1072no—
203C2×D4—4282no—
204C2×D4—4284no—
205C2×D4—2144no—
206C2×D4—2142no—
207C2×D4—2141no—
208C2×D4—2142no—
209C2×D4—2141no—
210C2×D4C22⋊D10711yescharacteristic
211C2×D4C22×D10712yes—
212C22×C4—22no—
213C22×C4—2141no—
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—21no—
225C4⋊C4—1071no—
226C4⋊C4C22⋊D10711yescharacteristic
227C4⋊C4C22×D10711yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—2141no—
230C22⋊C4—2142no—
231C22⋊C4C22⋊D10712yes—
232C42—21no—
233C42C22×D10711yescharacteristic
234C23—22no—
235C23—8562no—
236C23—4284no—
237C23—4284no—
238C23—2142no—
239C23—2141no—
240C23C2×C22⋊D10712yes—
241Q8—21no—
242Q8D4×D10712yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—8561no—
251D4—1072no—
252D4—4284no—
253D4—4284no—
254D4—4282no—
255D4—2142no—
256D4—2142no—
257D4—2141no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—4282no—
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—42no—
276C22—42no—
277C22—42no—
278C22—8568no—
279C22—4282no—
280C22—4282no—
281C22—4282no—
282C22—4281no—
283C22—4281no—
284C22—2142no—
285C22C22×C22⋊D10711yescharacteristic
286C4—21no—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—4281no—
291C4#1681213412yescomplemented
292C4C2×D4×D10711yescharacteristic
293C4#1681215411yescharacteristic
294C2—42no—
295C2—82no—
296C2—8562no—
297C2—4282no—
298C2D4×C22⋊D10711yescharacteristic
299C2C2×#1681213411yescharacteristic
300C2#2029343011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#550×C107C2
socleC22×C107C22×D4
hypercenter#550D107
hypocenterC107#3274
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. D4×D428C2semidirect
3. #20293003C2semidirect
4. #20292878C2semidirect
5. #20292827C2semidirect
6. #20292819C2semidirect
7. #20292772C2semidirect
8. #20292760C2semidirect
9. C4×#13133805C2semidirect
10. #20292705C2semidirect
11. #550×C107C2semidirect
16. C2×#13133805C22semidirect
18. C2×#13133803C22semidirect
19. C2×#13133803C22semidirect
20. C2×#13133802C22semidirect
26. #16812040C22semidirect
27. C4⋊(C107⋊C8)C22semidirect
28. #16812033C22semidirect
29. #16812030C22semidirect
31. #16812029C22semidirect
32. C22⋊(C107⋊C8)C22semidirect
33. C4×C107⋊C8C22semidirect
34. C2×#41×C107C22semidirect
36. #127×C107C22semidirect
38. #121×C107C22semidirect
39. C4×Q8×C107C22semidirect
54. #13133805D4semidirect
91. Q8×C107C2×D4semidirect
120. C107⋊C8C2×D4semidirect
139. C4×C107AGL(1,ℤ/8)semidirect
151. C107#3274semidirect
153. #550D107semidirect
163. C2×#41C2×D107semidirect
175. #127C2×D107semidirect
179. #121C2×D107semidirect
181. C4×Q8C2×D107semidirect
242. Q8D4×D107semidirect
291. C4#16812134semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
D4×D4282
#202930032
#202928782
#202928272
#202928192
#202927722
#202927602
C4×#131338052
#202927052
#550×C1072
#3274107