Finite Group Databasebeta

← #30677759

Subgroups — #30677759

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#22554117C211yesmaximal, characteristic, complemented
2#22554116C212yesmaximal, complemented
3#22554113C211yesmaximal, characteristic, complemented
4#22553687C211yesmaximal, characteristic, complemented
5#22553583C211yesmaximal, characteristic, complemented
6#22553581C212yesmaximal, complemented
7#22553482C212yesmaximal, complemented
8#22553478C212yesmaximal, complemented
9#22553428C211yesmaximal, characteristic, complemented
10#22553427C211yesmaximal, characteristic, complemented
11#472×C131C211yesmaximal, characteristic, complemented
12#16862292C2212yescomplemented
13#16862291C2212yescomplemented
14#16862274—22no—
15#16862274C2212yescomplemented
16#16862274C2212yescomplemented
17#16862273—22no—
18#16862272C2212yescomplemented
19#16862270—22no—
20#16862270—22no—
21#16862268—21no—
22#16862268C2211yescharacteristic
23#16862267—22no—
24#16862266C2211yescharacteristic
25#16862265—21no—
26C2×D1048—21no—
27C2×D1048C2211yescharacteristic, complemented
28C2×#14523498C2212yescomplemented
29D8×D131—21no—
30D8×D131—22no—
31#16862254C2212yescomplemented
32#16862252C2211yescharacteristic, complemented
33#16862242C2212yescomplemented
34#16862181C2211yescharacteristic, complemented
35#16862173C2211yescharacteristic, complemented
36#16862172C2211yescharacteristic, complemented
37#16862172C2212yescomplemented
38#16862162C2212yescomplemented
39C8×Dic131C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C131C2212yescomplemented
41C2×D8×C131C2211yescharacteristic, complemented
42#131×C131C2211yescharacteristic, complemented
43#108×C131C2211yescharacteristic, complemented
44#100×C131C2212yescomplemented
45#14523514—22no—
46#14523513—22no—
47#14523513C2311yescharacteristic
48#14523513C2312yes—
49#14523512—22no—
50#14523512—22no—
51C2×D524—22no—
52C2×D524C2312yes—
53C2×C22⋊D131—42no—
54C2×C22⋊D131—42no—
55D4×D131—22no—
56D4×D131—22no—
57D4×D131—22no—
58D4×D131—42no—
59Dic524D411yescharacteristic, complemented
60#14523503D411yescharacteristic, complemented
61#14523502—22no—
62#14523502—22no—
63#14523502D412yescomplemented
64#14523501—22no—
65#14523501D411yescharacteristic, complemented
66#14523500D411yescharacteristic, complemented
67D1048—22no—
68D1048—22no—
69D1048—21no—
70D1048—21no—
71D1048D411yescharacteristic, complemented
72#14523498—22no—
73#14523498—22no—
74#14523498—22no—
75#14523498D411yescharacteristic, complemented
76#14523497C2311yescharacteristic
77#14523496—22no—
78#14523496—21no—
79C2×C131⋊C8C2311yescharacteristic
80C8×D131—22no—
81C8×D131—21no—
82C4×Dic131C2311yescharacteristic
83#41×C131C2311yescharacteristic
84C2×D4×C131C2312yes—
85SD16×C131—22no—
86D8×C131—22no—
87D8×C131—21no—
88D8×C131—21no—
89M16×C131—21no—
90M16×C131C2312yes—
91M16×C131C2311yescharacteristic
92C2×C8×C131—21no—
93C2×C8×C131C2311yescharacteristic
94C22×D131—22no—
95C22×D131—42no—
96C22×D131—42no—
97Dic262C2×D412yes—
98Dic262C2×D411yescharacteristic
99D524—22no—
100D524—22no—
101D524—42no—
102D524C2×D412yes—
103D524C2×D411yescharacteristic
104C22⋊D131—22no—
105C22⋊D131—21no—
106C22⋊D131—42no—
107C22⋊D131—42no—
108C22⋊D131—42no—
109C22⋊D131—84no—
110C2×Dic131—21no—
111C2×Dic131—42no—
112C4×D131—21no—
113C4×D131—22no—
114C4×D131—42no—
115C131⋊C8—21no—
116C131⋊C8C2×D411yescharacteristic
117C131⋊C8C2×D411yescharacteristic
118C23×C131—22no—
119Q8×C131C2×D411yescharacteristic
120D4×C131—21no—
121D4×C131—22no—
122D4×C131—22no—
123D4×C131C2×D411yescharacteristic
124C2×C4×C131—21no—
125C2×C4×C131C2411yescharacteristic
126C8×C131—21no—
127C8×C131—22no—
128C8×C131C2×D411yescharacteristic
129C8×C131C2×D411yescharacteristic
130C22×C131—21no—
131C22×C131—42no—
132C22×C131—82no—
133C22×C131H(5,𝔽2)11yescharacteristic
134C2×D131—21no—
135C2×D131—22no—
136C2×D131—42no—
137C2×D131—82no—
138C2×D131—162no—
139C2×D131—162no—
140C4×C131—21no—
141C4×C131C22×D411yescharacteristic
142C4×C131C22×D411yescharacteristic
143Dic131—22no—
144Dic131—22no—
145Dic131—21no—
146C2×C131—21no—
147C2×C131—41no—
148C2×C131—82no—
149C2×C131D4211yescharacteristic
150D131—41no—
151D131—42no—
152D131—82no—
153C131#329211yescharacteristic, complemented, Sylow
154#3292—1311nomaximal, Sylow
155#576—1312no—
156#576—1311no—
157#576—1311no—
158#493—1311no—
159#472—1311no—
160#472—1312no—
161#472D13111yescharacteristic, complemented
162#453—1312no—
163#453—1312no—
164#443—1311no—
165#443—1311no—
166H(5,𝔽2)—1312no—
167H(5,𝔽2)—1312no—
168AGL(1,ℤ/8)—2622no—
169AGL(1,ℤ/8)—2622no—
170AGL(1,ℤ/8)—2622no—
171AGL(1,ℤ/8)—2622no—
172AGL(1,ℤ/8)—1312no—
173AGL(1,ℤ/8)—1312no—
174AGL(1,ℤ/8)—1312no—
175AGL(1,ℤ/8)C2×D13112yescomplemented
176#135—2621no—
177#135—2621no—
178#135—2622no—
179#135—1311no—
180#135—1311no—
181C2×D8—2621no—
182C2×D8—2622no—
183C2×D8—2621no—
184C2×D8—1312no—
185C2×D8—1311no—
186C2×D8C2×D13111yescharacteristic, complemented
187#131—1312no—
188#131—1311no—
189#131C2×D13111yescharacteristic, complemented
190#127—1312no—
191#108—1311no—
192#108C2×D13111yescharacteristic, complemented
193#104—1311no—
194#104—1312no—
195#104—1311no—
196#100—1312no—
197#100C2×D13112yescomplemented
198C4×C8—1311no—
199#41—2622no—
200#41—2622no—
201#41—2622no—
202#41—2622no—
203#41—1311no—
204#41—1312no—
205#41C22×D13111yescharacteristic
206C2×D4—2622no—
207C2×D4—2622no—
208C2×D4—2622no—
209C2×D4—2622no—
210C2×D4—5242no—
211C2×D4—5242no—
212C2×D4—5242no—
213C2×D4—1312no—
214C2×D4C22×D13112yes—
215Q16—1311no—
216Q16—1311no—
217SD16—22no—
218SD16—2622no—
219SD16—2622no—
220SD16—2622no—
221SD16—1311no—
222SD16—1311no—
223SD16—1312no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2622no—
228D8—2622no—
229D8—2622no—
230D8—2622no—
231D8—2622no—
232D8—2621no—
233D8—2621no—
234D8—1311no—
235D8—1311no—
236M16—21no—
237M16—2622no—
238M16—2621no—
239M16—1311no—
240M16C22×D13112yes—
241M16C22×D13111yescharacteristic
242C2×C8—21no—
243C2×C8—2622no—
244C2×C8—2621no—
245C2×C8—1311no—
246C2×C8C22×D13111yescharacteristic
247C42—1311no—
248C23—22no—
249C23—2622no—
250C23—5242no—
251C23—5242no—
252Q8—1311no—
253Q8—1312no—
254Q8D4×D13111yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—2622no—
259D4—2621no—
260D4—2622no—
261D4—2622no—
262D4—5242no—
263D4—5242no—
264D4—5242no—
265D4—5242no—
266D4—10484no—
267D4—1311no—
268D4—1312no—
269D4D4×D13111yescharacteristic
270C2×C4—21no—
271C2×C4—2621no—
272C2×C4—2622no—
273C2×C4—2621no—
274C2×C4—5242no—
275C2×C4—5242no—
276C2×C4C23×D13111yescharacteristic
277C8—21no—
278C8—22no—
279C8—2621no—
280C8—1311no—
281C8—1311no—
282C8D4×D13111yescharacteristic
283C8D4×D13111yescharacteristic
284C22—21no—
285C22—42no—
286C22—2622no—
287C22—2621no—
288C22—82no—
289C22—5242no—
290C22—10482no—
291C22—20962no—
292C22—20962no—
293C22#1686229211yescharacteristic
294C4—21no—
295C4—2622no—
296C4—2622no—
297C4—2621no—
298C4C2×D4×D13111yescharacteristic
299C4C2×D4×D13111yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—5242no—
304C2—5241no—
305C2—10482no—
306C2#2255394611yescharacteristic
subgroupG/H
centerC2#22553946
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#472×C131C2
socleC2×C131D42
hypercenter#472D131
hypocenterC131#3292
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #22554117C2semidirect
2. #22554116C2semidirect
3. #22554113C2semidirect
4. #22553687C2semidirect
5. #22553583C2semidirect
6. #22553581C2semidirect
7. #22553482C2semidirect
8. #22553478C2semidirect
9. #22553428C2semidirect
10. #22553427C2semidirect
11. #472×C131C2semidirect
12. #16862292C22semidirect
13. #16862291C22semidirect
15. #16862274C22semidirect
16. #16862274C22semidirect
18. #16862272C22semidirect
27. C2×D1048C22semidirect
28. C2×#14523498C22semidirect
31. #16862254C22semidirect
32. #16862252C22semidirect
33. #16862242C22semidirect
34. #16862181C22semidirect
35. #16862173C22semidirect
36. #16862172C22semidirect
37. #16862172C22semidirect
38. #16862162C22semidirect
39. C8×Dic131C22semidirect
40. AGL(1,ℤ/8)×C131C22semidirect
41. C2×D8×C131C22semidirect
42. #131×C131C22semidirect
43. #108×C131C22semidirect
44. #100×C131C22semidirect
59. Dic524D4semidirect
60. #14523503D4semidirect
63. #14523502D4semidirect
65. #14523501D4semidirect
66. #14523500D4semidirect
71. D1048D4semidirect
75. #14523498D4semidirect
153. C131#3292semidirect
161. #472D131semidirect
175. AGL(1,ℤ/8)C2×D131semidirect
186. C2×D8C2×D131semidirect
189. #131C2×D131semidirect
192. #108C2×D131semidirect
197. #100C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541172
#225541162
#225541132
#225536872
#225535832
#225535812
#225534822
#225534782
#225534282
#225534272
#472×C1312
#3292131