Finite Group Databasebeta

← #30677756

Subgroups — #30677756

Order 16768.

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

normal subgroups only

This group has 460 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#22554123C212yesmaximal, complemented
2#22554122C212yesmaximal, complemented
3#22553684C211yesmaximal, characteristic, complemented
4#22553574C212yesmaximal, complemented
5#22553573C212yesmaximal, complemented
6#22553496C212yesmaximal, complemented
7#22553479C211yesmaximal, characteristic, complemented
8#22553426C212yesmaximal, complemented
9#453×C131C211yesmaximal, characteristic, complemented
10#16862293C2212yes—
11#16862291C2211yescharacteristic, complemented
12#16862279C2212yescomplemented
13#16862276—22no—
14#16862276C2212yescomplemented
15#16862275—22no—
16#16862272—22no—
17#16862272C2212yes—
18#16862270—22no—
19#16862266—22no—
20#16862266C2212yes—
21#16862265—22no—
22C2×#14523501—22no—
23C2×#14523501C2212yescomplemented
24C2×#14523501C2212yescomplemented
25SD16×D131—22no—
26#16862254C2212yescomplemented
27#16862253C2212yescomplemented
28#16862246C2211yescharacteristic, complemented
29#16862181C2212yescomplemented
30#16862174C2212yescomplemented
31#16862164C2212yescomplemented
32#16862163C2211yescharacteristic, complemented
33C4×C131⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C131C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C131C2212yes—
36#127×C131C2211yescharacteristic
37#104×C131C2212yescomplemented
38#100×C131C2211yescharacteristic, complemented
39#14523513—22no—
40#14523513C2312yes—
41#14523512—22no—
42#14523512—22no—
43#14523512—42no—
44C2×Dic262—22no—
45C2×Dic262C2312yes—
46Q8×D131—22no—
47C2×C22⋊D131—41no—
48D4×D131—22no—
49Dic524—22no—
50#14523503—22no—
51#14523503D412yescomplemented
52#14523502—22no—
53#14523501—22no—
54#14523501—22no—
55#14523501—22no—
56#14523501—22no—
57#14523501—22no—
58#14523501—22no—
59#14523501D412yescomplemented
60#14523500D412yescomplemented
61#14523498—22no—
62#14523498D412yescomplemented
63#14523497—22no—
64#14523497C2312yes—
65#14523496—22no—
66C2×C131⋊C8—22no—
67C2×C131⋊C8C2312yes—
68C8×D131—22no—
69C4⋊Dic131—41no—
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—41no—
82Dic262—22no—
83Dic262—22no—
84Dic262—42no—
85Dic262C2×D412yes—
86D524C2×D412yes—
87C22⋊D131—22no—
88C22⋊D131—42no—
89C22⋊D131—42no—
90C2×Dic131—22no—
91C2×Dic131—41no—
92C2×Dic131—42no—
93C4×D131—22no—
94C4×D131—42no—
95C131⋊C8—22no—
96C131⋊C8C2×D412yes—
97C131⋊C8C2×D412yes—
98C23×C131—21no—
99C23×C131—21no—
100C23×C131—41no—
101Q8×C131C2×D412yes—
102D4×C131—22no—
103D4×C131—22no—
104D4×C131—22no—
105D4×C131—42no—
106D4×C131—82no—
107D4×C131C2×D412yes—
108C2×C4×C131—21no—
109C2×C4×C131—22no—
110C2×C4×C131—41no—
111C2×C4×C131C2411yescharacteristic
112C8×C131—22no—
113C22×C131—22no—
114C22×C131—41no—
115C22×C131—41no—
116C22×C131—81no—
117C22×C131—81no—
118C22×C131—161no—
119C22×C131H(5,𝔽2)11yescharacteristic
120C2×D131—22no—
121C2×D131—161no—
122C4×C131—22no—
123C4×C131—22no—
124C4×C131C22×D412yes—
125Dic131—22no—
126Dic131—42no—
127C2×C131—21no—
128C2×C131—42no—
129C2×C131—81no—
130C2×C131—81no—
131C2×C131D4211yescharacteristic
132D131—42no—
133C131#329111yescharacteristic, complemented, Sylow
134#3291—1311nomaximal, Sylow
135#577—1312no—
136#577—1312no—
137#492—1311no—
138#471—1312no—
139#471—1312no—
140#456—1312no—
141#453—1311no—
142#453D13111yescharacteristic, complemented
143#443—1312no—
1442−1+4—1312no—
145H(5,𝔽2)—1311no—
146H(5,𝔽2)C2×D13111yescharacteristic, complemented
147#137—2622no—
148#137—2622no—
149#137—1312no—
150#137—1312no—
151AGL(1,ℤ/8)—2622no—
152AGL(1,ℤ/8)—2622no—
153AGL(1,ℤ/8)—1312no—
154AGL(1,ℤ/8)C2×D13112yes—
155#135—2622no—
156#135—2622no—
157#135—1312no—
158C2×SD16—2622no—
159C2×SD16—2622no—
160C2×SD16—1312no—
161C2×SD16—1312no—
162#131—1312no—
163#131—1312no—
164C4⋊Q8—1311no—
165#127C2×D13111yescharacteristic
166#108—1312no—
167#104—1312no—
168#104C2×D13112yescomplemented
169#101—1312no—
170#100—1311no—
171#100C2×D13111yescharacteristic, complemented
172C4×C8—1311no—
173#41—22no—
174#41—2622no—
175#41—2622no—
176#41—2622no—
177#41—5242no—
178#41—1312no—
179#41C22×D13112yes—
180C2×Q8—2622no—
181C2×Q8—2622no—
182C2×Q8—1312no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2622no—
187C2×D4—5241no—
188C2×D4C22×D13111yescharacteristic
189C2×D4C22×D13111yescharacteristic
190Q16—2622no—
191Q16—2622no—
192Q16—1312no—
193SD16—22no—
194SD16—2622no—
195SD16—2622no—
196SD16—2622no—
197SD16—2622no—
198SD16—2622no—
199SD16—2622no—
200SD16—2622no—
201SD16—1312no—
202SD16—1312no—
203D8—22no—
204D8—2622no—
205D8—1312no—
206M16—2622no—
207M16—2622no—
208M16—1312no—
209M16C22×D13112yes—
210C2×C8—2622no—
211C2×C8—2622no—
212C2×C8—1312no—
213C4⋊C4—5241no—
214C42C22×D13111yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—5241no—
219Q8—2622no—
220Q8—2622no—
221Q8—5242no—
222Q8—1312no—
223Q8D4×D13112yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—2622no—
229D4—82no—
230D4—5242no—
231D4—5242no—
232D4—1312no—
233D4D4×D13112yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—2622no—
238C2×C4—2622no—
239C2×C4—5242no—
240C2×C4—5241no—
241C2×C4—5242no—
242C2×C4C23×D13111yescharacteristic
243C8—22no—
244C8—2622no—
245C8—1312no—
246C8—1312no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—2622no—
251C22—81no—
252C22—81no—
253C22—161no—
254C22—20961no—
255C22#1686229111yescharacteristic
256C4—22no—
257C4—22no—
258C4—2622no—
259C4—5242no—
260C4C2×D4×D13112yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—5242no—
266C2#2255394411yescharacteristic
subgroupG/H
centerC2#22553944
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#453×C131C2
socleC2×C131D42
hypercenter#453D131
hypocenterC131#3291
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. #22554123C2semidirect
2. #22554122C2semidirect
3. #22553684C2semidirect
4. #22553574C2semidirect
5. #22553573C2semidirect
6. #22553496C2semidirect
7. #22553479C2semidirect
8. #22553426C2semidirect
9. #453×C131C2semidirect
11. #16862291C22semidirect
12. #16862279C22semidirect
14. #16862276C22semidirect
23. C2×#14523501C22semidirect
24. C2×#14523501C22semidirect
26. #16862254C22semidirect
27. #16862253C22semidirect
28. #16862246C22semidirect
29. #16862181C22semidirect
30. #16862174C22semidirect
31. #16862164C22semidirect
32. #16862163C22semidirect
33. C4×C131⋊C8C22semidirect
34. H(5,𝔽2)×C131C22semidirect
37. #104×C131C22semidirect
38. #100×C131C22semidirect
51. #14523503D4semidirect
59. #14523501D4semidirect
60. #14523500D4semidirect
62. #14523498D4semidirect
133. C131#3291semidirect
142. #453D131semidirect
146. H(5,𝔽2)C2×D131semidirect
168. #104C2×D131semidirect
171. #100C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541232
#225541222
#225536842
#225535742
#225535732
#225534962
#225534792
#225534262
#453×C1312
#3291131