Finite Group Databasebeta

← #30676444

Subgroups — #30676444

Order 16768.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16862272C211yesmaximal, characteristic, complemented
2C2×#16862271C211yesmaximal, characteristic, complemented
3C2×#16862266C211yesmaximal, characteristic, complemented
4C22×#14523498C211yesmaximal, characteristic, complemented
5#22553881C211yesmaximal, characteristic, complemented
6#22553702C212yesmaximal, complemented
7#22553699C212yesmaximal, complemented
8#22553694C212yesmaximal, complemented
9#22553688C212yesmaximal, complemented
10#22553268C211yesmaximal, characteristic, complemented
11#535×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C22×D524C2211yescharacteristic
14#16862272—24no—
15#16862271—24no—
16#16862266—24no—
17C2×#14523503C2211yescharacteristic, complemented
18C2×#14523501C2212yescomplemented
19C2×#14523501C2211yescharacteristic, complemented
20C2×#14523500C2212yescomplemented
21C2×#14523500C2211yescharacteristic, complemented
22C2×#14523498—24no—
23C2×#14523498C2212yes—
24C2×#14523498C2212yes—
25C2×#14523498C2211yescharacteristic, complemented
26C2×#14523498C2212yescomplemented
27C2×#14523498C2212yescomplemented
28C2×#14523497C2211yescharacteristic
29C22×C131⋊C8C2211yescharacteristic
30#16862243C2212yescomplemented
31#16862234C2212yescomplemented
32#16862221—22no—
33#16862216—22no—
34#16862157C2212yescomplemented
35C4×C131⋊C8C2212yescomplemented
36C2×#41×C131C2211yescharacteristic
37C22×D4×C131C2211yescharacteristic
38#127×C131C2212yescomplemented
39#124×C131C2212yescomplemented
40#121×C131—22no—
41#120×C131—22no—
42#117×C131C2211yescharacteristic
43C23×D131—21no—
44#14523513—22no—
45#14523513—22no—
46C2×Dic262C2311yescharacteristic
47C2×D524—22no—
48C2×D524—22no—
49C2×D524—21no—
50C2×D524—24no—
51C2×D524C2312yes—
52C2×D524C2311yescharacteristic
53C2×C22⋊D131—21no—
54C2×C4×D131—21no—
55#14523503—22no—
56#14523501—22no—
57#14523501—24no—
58#14523500—22no—
59#14523500—24no—
60#14523498—24no—
61#14523498—24no—
62#14523498—22no—
63#14523498—24no—
64#14523498—24no—
65#14523497D414yescomplemented
66C2×C131⋊C8D414yescomplemented
67C2×C131⋊C8C2311yescharacteristic
68C2×C131⋊C8C2311yescharacteristic
69C2×C131⋊C8C2312yes—
70(C2×D131)⋊C4—24no—
71C24×C131—21no—
72#41×C131—22no—
73#41×C131—22no—
74C2×Q8×C131C2311yescharacteristic
75C2×D4×C131—21no—
76C2×D4×C131—22no—
77C2×D4×C131—22no—
78C2×D4×C131—21no—
79C2×D4×C131—24no—
80C2×D4×C131C2312yes—
81C2×D4×C131C2311yescharacteristic
82C22×C4×C131—21no—
83C22×C4×C131C2311yescharacteristic
84C4⋊C4×C131D412yes—
85C22⋊C4×C131—24no—
86C22⋊C4×C131D412yes—
87C42×C131C2312yes—
88C22×D131—22no—
89C22×D131—21no—
90C22×D131—42no—
91C22×D131—82no—
92Dic262—21no—
93Dic262—21no—
94D524—22no—
95D524—21no—
96D524—22no—
97D524—21no—
98D524—42no—
99D524—84no—
100C22⋊D131—42no—
101C2×Dic131—21no—
102C4×D131—42no—
103C131⋊C8C2×D412yes—
104C131⋊C8C2×D412yes—
105C131⋊C8C2×D414yes—
106C23×C131—22no—
107C23×C131—21no—
108C23×C131—42no—
109C23×C131—82no—
110C23×C131C2×D411yescharacteristic
111Q8×C131—21no—
112Q8×C131—21no—
113D4×C131—22no—
114D4×C131—22no—
115D4×C131—21no—
116D4×C131—21no—
117D4×C131—42no—
118D4×C131—42no—
119D4×C131—84no—
120C2×C4×C131—21no—
121C2×C4×C131—42no—
122C2×C4×C131C2×D412yes—
123C2×C4×C131C2×D412yes—
124C2×C4×C131C2×D414yes—
125C2×C4×C131C2×D411yescharacteristic
126C2×C4×C131C2411yescharacteristic
127C22×C131—22no—
128C22×C131—41no—
129C22×C131—42no—
130C22×C131—42no—
131C22×C131—44no—
132C22×C131—84no—
133C22×C131#12712yes—
134C22×C131C22×D411yescharacteristic
135C2×D131—41no—
136C2×D131—42no—
137C2×D131—42no—
138C2×D131—44no—
139C2×D131—84no—
140C4×C131—24no—
141C4×C131—41no—
142C4×C131#12712yes—
143C4×C131C22×D411yescharacteristic
144C4×C131C22×D411yescharacteristic
145Dic131—41no—
146C2×C131—22no—
147C2×C131—81no—
148C2×C131—82no—
149C2×C131C2×#12711yescharacteristic
150C2×C131#57612yes—
151D131—81no—
152D131—82no—
153C131#315511yescharacteristic, complemented, Sylow
154#3155—1311nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1311no—
156C2×AGL(1,ℤ/8)—1311no—
157C2×#135—1311no—
158C22×D8—1311no—
159#535—1311no—
160#535D13111yescharacteristic, complemented
161#496—1312no—
162#496—1312no—
163#495—1312no—
164#493—1312no—
165#405—1311no—
166C2×#41—1311no—
167C2×#41C2×D13111yescharacteristic
168C22×D4—1311no—
169C22×D4C2×D13111yescharacteristic
170AGL(1,ℤ/8)—2624no—
171AGL(1,ℤ/8)—2624no—
172#135—2624no—
173C2×Q16—1311no—
174C2×SD16—1311no—
175C2×SD16—1312no—
176C2×SD16—1311no—
177C2×SD16—1312no—
178C2×D8—2624no—
179C2×D8—1312no—
180C2×D8—1312no—
181C2×D8—1311no—
182C2×D8—1312no—
183C2×D8—1312no—
184C2×M16—1311no—
185C22×C8—1311no—
186#127—1312no—
187#127C2×D13112yescomplemented
188#124—1312no—
189#124C2×D13112yescomplemented
190#121—22no—
191#121—2622no—
192#120—22no—
193#120—2622no—
194#117C2×D13111yescharacteristic
195#97—1312no—
196C4×C8—1312no—
197C24—21no—
198C24—2621no—
199#41—22no—
200#41—22no—
201#41—2622no—
202#41—2622no—
203C2×Q8—1311no—
204C2×Q8C22×D13111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—24no—
210C2×D4—2621no—
211C2×D4—2622no—
212C2×D4—2622no—
213C2×D4—2621no—
214C2×D4—2624no—
215C2×D4—1312no—
216C2×D4—1311no—
217C2×D4C22×D13112yes—
218C2×D4C22×D13111yescharacteristic
219C22×C4—21no—
220C22×C4—2621no—
221C22×C4C22×D13111yescharacteristic
222Q16—2622no—
223SD16—2622no—
224SD16—2624no—
225SD16—2622no—
226SD16—2624no—
227D8—2624no—
228D8—2624no—
229D8—2622no—
230D8—2624no—
231D8—2624no—
232M16—1314no—
233C2×C8—1314no—
234C2×C8—1311no—
235C2×C8—1312no—
236C2×C8—1311no—
237C4⋊C4C22⋊D13112yes—
238C22⋊C4—24no—
239C22⋊C4—2624no—
240C22⋊C4C22⋊D13112yes—
241C42C22×D13112yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—2622no—
246C23—2621no—
247C23—82no—
248C23—5242no—
249C23—10482no—
250C23C2×C22⋊D13111yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2621no—
254Q8—2621no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—21no—
259D4—42no—
260D4—42no—
261D4—2622no—
262D4—2621no—
263D4—2622no—
264D4—2621no—
265D4—84no—
266D4—5242no—
267D4—5242no—
268D4—10484no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—2621no—
272C2×C4—5242no—
273C2×C4D4×D13112yes—
274C2×C4D4×D13112yes—
275C2×C4C2×C22⋊D13114yes—
276C2×C4C2×C22⋊D13111yescharacteristic
277C2×C4C23×D13111yescharacteristic
278C8—1312no—
279C8—1312no—
280C8—1314no—
281C22—22no—
282C22—42no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—84no—
287C22—5242no—
288C22—5244no—
289C22—5241no—
290C22—5242no—
291C22—10484no—
292C22#1686224212yes—
293C22C22×C22⋊D13111yescharacteristic
294C4—24no—
295C4—41no—
296C4—5241no—
297C4#1686224212yes—
298C4C2×D4×D13111yescharacteristic
299C4C2×D4×D13111yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—10482no—
304C2—10481no—
305C2C2×#1686224211yescharacteristic
306C2#2255411512yes—
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#535×C131C2
socleC22×C131C22×D4
hypercenter#535D131
hypocenterC131#3155
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16862272C2semidirect
2. C2×#16862271C2semidirect
3. C2×#16862266C2semidirect
4. C22×#14523498C2semidirect
5. #22553881C2semidirect
6. #22553702C2semidirect
7. #22553699C2semidirect
8. #22553694C2semidirect
9. #22553688C2semidirect
10. #22553268C2semidirect
11. #535×C131C2semidirect
17. C2×#14523503C22semidirect
18. C2×#14523501C22semidirect
19. C2×#14523501C22semidirect
20. C2×#14523500C22semidirect
21. C2×#14523500C22semidirect
25. C2×#14523498C22semidirect
26. C2×#14523498C22semidirect
27. C2×#14523498C22semidirect
30. #16862243C22semidirect
31. #16862234C22semidirect
34. #16862157C22semidirect
35. C4×C131⋊C8C22semidirect
38. #127×C131C22semidirect
39. #124×C131C22semidirect
65. #14523497D4semidirect
66. C2×C131⋊C8D4semidirect
153. C131#3155semidirect
160. #535D131semidirect
187. #127C2×D131semidirect
189. #124C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622722
C2×#168622712
C2×#168622662
C22×#145234982
#225538812
#225537022
#225536992
#225536942
#225536882
#225532682
#535×C1312
#3155131