Finite Group Databasebeta

← #31792586

Subgroups — #31792586

Order 17536.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#22578206C212yesmaximal, complemented
2#22578205C211yesmaximal, characteristic, complemented
3#22577778C211yesmaximal, characteristic, complemented
4#22577673C211yesmaximal, characteristic, complemented
5#22577671C212yesmaximal, complemented
6#22577670C211yesmaximal, characteristic, complemented
7#22577571C212yesmaximal, complemented
8#22577569C212yesmaximal, complemented
9#22577520C211yesmaximal, characteristic, complemented
10#22577518C211yesmaximal, characteristic, complemented
11#576×C137C211yesmaximal, characteristic, complemented
12#17984134C2212yescomplemented
13#17984115—22no—
14#17984114—22no—
15#17984114C2212yescomplemented
16#17984113—22no—
17#17984113C2212yescomplemented
18#17984113C2212yescomplemented
19#17984111—21no—
20#17984108—22no—
21#17984108C2211yescharacteristic
22C2×D1096C2211yescharacteristic, complemented
23C2×#14555495—22no—
24C2×#14555495C2212yescomplemented
25D8×D137—21no—
26#17984095C2212yescomplemented
27#17984094C2211yescharacteristic, complemented
28#17984084C2212yescomplemented
29#17984024C2211yescharacteristic, complemented
30#17984023C2211yescharacteristic, complemented
31#17984015C2211yescharacteristic, complemented
32#17984015C2212yescomplemented
33#17984014C2211yescharacteristic, complemented
34#17984005C2212yescomplemented
35#17984004C2212yescomplemented
36C8×Dic137C2211yescharacteristic, complemented
37H(5,𝔽2)×C137C2212yescomplemented
38AGL(1,ℤ/8)×C137—22no—
39AGL(1,ℤ/8)×C137C2212yescomplemented
40#135×C137—21no—
41#135×C137C2211yescharacteristic
42C2×D8×C137—21no—
43C2×D8×C137C2211yescharacteristic, complemented
44#131×C137C2211yescharacteristic, complemented
45#14555511—22no—
46#14555510—22no—
47#14555510C2311yescharacteristic
48C2×D548—22no—
49C2×D548C2312yes—
50C2×C22⋊D137—42no—
51D4×D137—22no—
52D4×D137—42no—
53#14555500D411yescharacteristic, complemented
54#14555498—22no—
55#14555498D411yescharacteristic, complemented
56#14555497—22no—
57#14555497—22no—
58#14555497D411yescharacteristic, complemented
59D1096—21no—
60D1096—21no—
61#14555495—22no—
62#14555495—22no—
63#14555495—22no—
64#14555495—22no—
65#14555495—22no—
66#14555495D411yescharacteristic, complemented
67#14555494—22no—
68#14555494C2312yes—
69#14555494C2311yescharacteristic
70#14555493—21no—
71C2×#11764040—22no—
72C2×#11764040C2311yescharacteristic
73C8×D137—21no—
74C4×Dic137C2311yescharacteristic
75#41×C137—22no—
76#41×C137—22no—
77#41×C137C2312yes—
78#41×C137C2311yescharacteristic
79C2×D4×C137—22no—
80C2×D4×C137—22no—
81C2×D4×C137—42no—
82C2×D4×C137C2312yes—
83Q16×C137D411yescharacteristic, complemented
84SD16×C137—22no—
85SD16×C137D412yescomplemented
86D8×C137—22no—
87D8×C137—21no—
88D8×C137—21no—
89D8×C137D411yescharacteristic, complemented
90M16×C137—21no—
91M16×C137C2311yescharacteristic
92C2×C8×C137—21no—
93C2×C8×C137C2311yescharacteristic
94C22×D137—22no—
95C22×D137—42no—
96Dic274C2×D411yescharacteristic
97D548—22no—
98D548—22no—
99D548—42no—
100D548C2×D411yescharacteristic
101C22⋊D137—21no—
102C22⋊D137—42no—
103C22⋊D137—84no—
104C2×Dic137—21no—
105C4×D137—21no—
106C4×D137—42no—
107#11764040—21no—
108#11764040—22no—
109#11764040C2×D411yescharacteristic
110#11764040C2×D411yescharacteristic
111C23×C137—22no—
112C23×C137—42no—
113Q8×C137C2×D411yescharacteristic
114Q8×C137C2×D412yes—
115D4×C137—21no—
116D4×C137—22no—
117D4×C137—22no—
118D4×C137—22no—
119D4×C137—42no—
120D4×C137—42no—
121D4×C137C2×D411yescharacteristic
122D4×C137C2×D412yes—
123C2×C4×C137—21no—
124C2×C4×C137—22no—
125C2×C4×C137—42no—
126C2×C4×C137C2411yescharacteristic
127C8×C137—21no—
128C8×C137C2×D411yescharacteristic
129C8×C137C2×D411yescharacteristic
130C22×C137—21no—
131C22×C137—22no—
132C22×C137—42no—
133C22×C137—82no—
134C22×C137—162no—
135C22×C137H(5,𝔽2)11yescharacteristic
136C2×D137—21no—
137C2×D137—42no—
138C2×D137—82no—
139C2×D137—162no—
140C4×C137—21no—
141C4×C137—22no—
142C4×C137C22×D411yescharacteristic
143C4×C137C22×D411yescharacteristic
144Dic137—22no—
145Dic137—21no—
146C2×C137—21no—
147C2×C137—42no—
148C2×C137—41no—
149C2×C137—82no—
150C2×C137D4211yescharacteristic
151D137—41no—
152D137—82no—
153C137#329211yescharacteristic, complemented, Sylow
154#3292—1371nomaximal, Sylow
155#576—1372no—
156#576—1371no—
157#576D13711yescharacteristic, complemented
158#493—1371no—
159#472—1372no—
160#472—1371no—
161#472—1371no—
162#453—1372no—
163#453—1372no—
164#443—1371no—
165#443—1371no—
166H(5,𝔽2)—1372no—
167H(5,𝔽2)C2×D13712yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—2742no—
170AGL(1,ℤ/8)—2742no—
171AGL(1,ℤ/8)—2742no—
172AGL(1,ℤ/8)—1372no—
173AGL(1,ℤ/8)—1372no—
174AGL(1,ℤ/8)—1372no—
175AGL(1,ℤ/8)C2×D13712yescomplemented
176#135—21no—
177#135—2742no—
178#135—2741no—
179#135—1371no—
180#135C2×D13711yescharacteristic
181C2×D8—21no—
182C2×D8—2742no—
183C2×D8—2741no—
184C2×D8—1372no—
185C2×D8—1371no—
186C2×D8C2×D13711yescharacteristic, complemented
187#131—1372no—
188#131—1371no—
189#131C2×D13711yescharacteristic, complemented
190#127—1372no—
191#108—1371no—
192#108—1371no—
193#104—1371no—
194#104—1372no—
195#104—1371no—
196#100—1372no—
197#100—1372no—
198C4×C8—1371no—
199#41—22no—
200#41—22no—
201#41—2742no—
202#41—2742no—
203#41—1371no—
204#41C22×D13711yescharacteristic
205#41C22×D13712yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—2742no—
210C2×D4—2742no—
211C2×D4—5482no—
212C2×D4—5482no—
213C2×D4—1372no—
214C2×D4C22×D13712yes—
215Q16—1371no—
216Q16C22⋊D13711yescharacteristic, complemented
217SD16—22no—
218SD16—2742no—
219SD16—2742no—
220SD16—2742no—
221SD16—1371no—
222SD16—1371no—
223SD16C22⋊D13712yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2742no—
228D8—2742no—
229D8—2742no—
230D8—2742no—
231D8—2742no—
232D8—2741no—
233D8—2741no—
234D8—1371no—
235D8C22⋊D13711yescharacteristic, complemented
236M16—21no—
237M16—2742no—
238M16—2741no—
239M16—1372no—
240M16—1371no—
241M16C22×D13711yescharacteristic
242C2×C8—21no—
243C2×C8—2742no—
244C2×C8—2741no—
245C2×C8—1371no—
246C2×C8C22×D13711yescharacteristic
247C42—1371no—
248C23—22no—
249C23—42no—
250C23—2742no—
251C23—5482no—
252Q8—1371no—
253Q8D4×D13711yescharacteristic
254Q8C2×C22⋊D13712yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2741no—
262D4—2742no—
263D4—2742no—
264D4—5482no—
265D4—5482no—
266D4—10964no—
267D4—1371no—
268D4D4×D13711yescharacteristic
269D4C2×C22⋊D13712yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—2741no—
274C2×C4—2741no—
275C2×C4—5482no—
276C2×C4C23×D13711yescharacteristic
277C8—21no—
278C8—2742no—
279C8—2741no—
280C8—1371no—
281C8—1371no—
282C8C2×C22⋊D13711yescharacteristic
283C8C2×C22⋊D13711yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—2741no—
290C22—5482no—
291C22—10962no—
292C22—21922no—
293C22#1798413311yescharacteristic
294C4—21no—
295C4—22no—
296C4—2742no—
297C4—2741no—
298C4C2×D4×D13711yescharacteristic
299C4C22×C22⋊D13711yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—5481no—
305C2—10962no—
306C2D4×C22⋊D13711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#576×C137C2
socleC2×C137D42
hypercenter#576D137
hypocenterC137#3292
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #22578206C2semidirect
2. #22578205C2semidirect
3. #22577778C2semidirect
4. #22577673C2semidirect
5. #22577671C2semidirect
6. #22577670C2semidirect
7. #22577571C2semidirect
8. #22577569C2semidirect
9. #22577520C2semidirect
10. #22577518C2semidirect
11. #576×C137C2semidirect
12. #17984134C22semidirect
15. #17984114C22semidirect
17. #17984113C22semidirect
18. #17984113C22semidirect
22. C2×D1096C22semidirect
24. C2×#14555495C22semidirect
26. #17984095C22semidirect
27. #17984094C22semidirect
28. #17984084C22semidirect
29. #17984024C22semidirect
30. #17984023C22semidirect
31. #17984015C22semidirect
32. #17984015C22semidirect
33. #17984014C22semidirect
34. #17984005C22semidirect
35. #17984004C22semidirect
36. C8×Dic137C22semidirect
37. H(5,𝔽2)×C137C22semidirect
39. AGL(1,ℤ/8)×C137C22semidirect
43. C2×D8×C137C22semidirect
44. #131×C137C22semidirect
53. #14555500D4semidirect
55. #14555498D4semidirect
58. #14555497D4semidirect
66. #14555495D4semidirect
83. Q16×C137D4semidirect
85. SD16×C137D4semidirect
89. D8×C137D4semidirect
153. C137#3292semidirect
157. #576D137semidirect
167. H(5,𝔽2)C2×D137semidirect
175. AGL(1,ℤ/8)C2×D137semidirect
186. C2×D8C2×D137semidirect
189. #131C2×D137semidirect
216. Q16C22⋊D137semidirect
223. SD16C22⋊D137semidirect
235. D8C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
#225782062
#225782052
#225777782
#225776732
#225776712
#225776702
#225775712
#225775692
#225775202
#225775182
#576×C1372
#3292137