Finite Group Databasebeta

← #30677757

Subgroups — #30677757

Order 16768.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#22554116C212yesmaximal, complemented
2#22554115C212yesmaximal, complemented
3#22553688C211yesmaximal, characteristic, complemented
4#22553581C212yesmaximal, complemented
5#22553580C212yesmaximal, complemented
6#22553481C212yesmaximal, complemented
7#22553479C211yesmaximal, characteristic, complemented
8#22553426C212yesmaximal, complemented
9#453×C131C211yesmaximal, characteristic, complemented
10#16862292C2212yescomplemented
11#16862291C2211yescharacteristic, complemented
12#16862274C2212yescomplemented
13#16862273—22no—
14#16862272—22no—
15#16862272C2212yescomplemented
16#16862271—22no—
17#16862271C2212yescomplemented
18#16862270—22no—
19#16862267—22no—
20#16862266—22no—
21#16862266C2212yes—
22C2×#14523498—22no—
23C2×#14523498C2212yescomplemented
24C2×#14523498C2212yescomplemented
25D8×D131—22no—
26#16862254C2212yescomplemented
27#16862253C2212yescomplemented
28#16862243C2211yescharacteristic, complemented
29#16862181C2212yescomplemented
30#16862174C2212yescomplemented
31#16862163C2211yescharacteristic, complemented
32#16862162C2212yescomplemented
33C4×C131⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C131C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C131C2212yescomplemented
36#127×C131C2211yescharacteristic, complemented
37#104×C131C2212yescomplemented
38#100×C131C2211yescharacteristic, complemented
39#14523514—22no—
40#14523513—22no—
41#14523513C2312yes—
42#14523512—22no—
43C2×D524—22no—
44C2×D524—41no—
45C2×D524C2312yes—
46C2×C22⋊D131—41no—
47D4×D131—22no—
48D4×D131—22no—
49D4×D131—42no—
50#14523503D412yescomplemented
51#14523502—22no—
52#14523501—22no—
53#14523501D412yescomplemented
54#14523500—22no—
55#14523500D412yescomplemented
56D1048—22no—
57#14523498—22no—
58#14523498—22no—
59#14523498—22no—
60#14523498—22no—
61#14523498—22no—
62#14523498—22no—
63#14523498D412yescomplemented
64#14523497—22no—
65#14523497C2312yes—
66#14523496—22no—
67C2×C131⋊C8—22no—
68C2×C131⋊C8C2312yes—
69C8×D131—22no—
70#41×C131—22no—
71#41×C131C2312yes—
72C2×D4×C131—22no—
73C2×D4×C131—41no—
74C2×D4×C131—41no—
75C2×D4×C131C2311yescharacteristic
76C2×D4×C131C2311yescharacteristic
77SD16×C131—22no—
78D8×C131—22no—
79M16×C131C2312yes—
80C42×C131C2311yescharacteristic
81C22×D131—22no—
82C22×D131—41no—
83C22×D131—42no—
84Dic262C2×D412yes—
85D524—22no—
86D524—22no—
87D524—42no—
88D524—82no—
89D524C2×D412yes—
90C22⋊D131—22no—
91C22⋊D131—42no—
92C22⋊D131—42no—
93C2×Dic131—41no—
94C4×D131—22no—
95C4×D131—42no—
96C131⋊C8—22no—
97C131⋊C8C2×D412yes—
98C131⋊C8C2×D412yes—
99C23×C131—21no—
100C23×C131—21no—
101C23×C131—41no—
102Q8×C131C2×D412yes—
103D4×C131—22no—
104D4×C131—22no—
105D4×C131—22no—
106D4×C131—42no—
107D4×C131—82no—
108D4×C131C2×D412yes—
109C2×C4×C131—21no—
110C2×C4×C131—22no—
111C2×C4×C131—41no—
112C2×C4×C131C2411yescharacteristic
113C8×C131—22no—
114C22×C131—22no—
115C22×C131—41no—
116C22×C131—41no—
117C22×C131—81no—
118C22×C131—81no—
119C22×C131—161no—
120C22×C131H(5,𝔽2)11yescharacteristic
121C2×D131—22no—
122C2×D131—42no—
123C2×D131—82no—
124C2×D131—161no—
125C2×D131—162no—
126C4×C131—22no—
127C4×C131—22no—
128C4×C131C22×D412yes—
129Dic131—22no—
130C2×C131—21no—
131C2×C131—42no—
132C2×C131—81no—
133C2×C131—81no—
134C2×C131D4211yescharacteristic
135D131—42no—
136D131—82no—
137C131#329211yescharacteristic, complemented, Sylow
138#3292—1311nomaximal, Sylow
139#576—1312no—
140#576—1312no—
141#493—1311no—
142#472—1312no—
143#472—1312no—
144#453—1311no—
145#453—1312no—
146#453D13111yescharacteristic, complemented
147#443—1312no—
148H(5,𝔽2)—1312no—
149H(5,𝔽2)—1311no—
150H(5,𝔽2)C2×D13111yescharacteristic, complemented
151AGL(1,ℤ/8)—2622no—
152AGL(1,ℤ/8)—2622no—
153AGL(1,ℤ/8)—2622no—
154AGL(1,ℤ/8)—2622no—
155AGL(1,ℤ/8)—1312no—
156AGL(1,ℤ/8)—1312no—
157AGL(1,ℤ/8)—1312no—
158AGL(1,ℤ/8)C2×D13112yescomplemented
159#135—2622no—
160#135—2622no—
161#135—1312no—
162C2×D8—2622no—
163C2×D8—2622no—
164C2×D8—1312no—
165C2×D8—1312no—
166#131—1312no—
167#131—1312no—
168#127—1311no—
169#127C2×D13111yescharacteristic, complemented
170#108—1312no—
171#104—1312no—
172#104C2×D13112yescomplemented
173#100—1311no—
174#100—1312no—
175#100C2×D13111yescharacteristic, complemented
176C4×C8—1311no—
177#41—22no—
178#41—2622no—
179#41—2622no—
180#41—2622no—
181#41—1312no—
182#41C22×D13112yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2622no—
187C2×D4—2622no—
188C2×D4—2622no—
189C2×D4—5241no—
190C2×D4—5242no—
191C2×D4—5241no—
192C2×D4—1312no—
193C2×D4C22×D13111yescharacteristic
194C2×D4C22×D13111yescharacteristic
195Q16—1312no—
196SD16—22no—
197SD16—2622no—
198SD16—2622no—
199SD16—2622no—
200SD16—1312no—
201SD16—1312no—
202D8—22no—
203D8—2622no—
204D8—2622no—
205D8—2622no—
206D8—2622no—
207D8—2622no—
208D8—2622no—
209D8—2622no—
210D8—1312no—
211M16—2622no—
212M16—2622no—
213M16—1312no—
214M16C22×D13112yes—
215C2×C8—2622no—
216C2×C8—2622no—
217C2×C8—1312no—
218C42C22×D13111yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—2622no—
223C23—5242no—
224C23—5241no—
225Q8—1312no—
226Q8D4×D13112yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—2622no—
232D4—2622no—
233D4—2622no—
234D4—82no—
235D4—5242no—
236D4—5242no—
237D4—5242no—
238D4—10482no—
239D4—1312no—
240D4D4×D13112yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—2622no—
245C2×C4—5242no—
246C2×C4—5241no—
247C2×C4C23×D13111yescharacteristic
248C8—22no—
249C8—2622no—
250C8—1312no—
251C8—1312no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—2622no—
256C22—81no—
257C22—81no—
258C22—5242no—
259C22—161no—
260C22—10482no—
261C22—20962no—
262C22—20961no—
263C22#1686229111yescharacteristic
264C4—22no—
265C4—22no—
266C4—2622no—
267C4C2×D4×D13112yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—5242no—
273C2—10482no—
274C2#2255394411yescharacteristic
subgroupG/H
centerC2#22553944
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#453×C131C2
socleC2×C131D42
hypercenter#453D131
hypocenterC131#3292
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22554116C2semidirect
2. #22554115C2semidirect
3. #22553688C2semidirect
4. #22553581C2semidirect
5. #22553580C2semidirect
6. #22553481C2semidirect
7. #22553479C2semidirect
8. #22553426C2semidirect
9. #453×C131C2semidirect
10. #16862292C22semidirect
11. #16862291C22semidirect
12. #16862274C22semidirect
15. #16862272C22semidirect
17. #16862271C22semidirect
23. C2×#14523498C22semidirect
24. C2×#14523498C22semidirect
26. #16862254C22semidirect
27. #16862253C22semidirect
28. #16862243C22semidirect
29. #16862181C22semidirect
30. #16862174C22semidirect
31. #16862163C22semidirect
32. #16862162C22semidirect
33. C4×C131⋊C8C22semidirect
34. H(5,𝔽2)×C131C22semidirect
35. AGL(1,ℤ/8)×C131C22semidirect
36. #127×C131C22semidirect
37. #104×C131C22semidirect
38. #100×C131C22semidirect
50. #14523503D4semidirect
53. #14523501D4semidirect
55. #14523500D4semidirect
63. #14523498D4semidirect
137. C131#3292semidirect
146. #453D131semidirect
150. H(5,𝔽2)C2×D131semidirect
158. AGL(1,ℤ/8)C2×D131semidirect
169. #127C2×D131semidirect
172. #104C2×D131semidirect
175. #100C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541162
#225541152
#225536882
#225535812
#225535802
#225534812
#225534792
#225534262
#453×C1312
#3292131