Finite Group Databasebeta

← #31792578

Subgroups — #31792578

Order 17536.

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

normal subgroups only

This group has 454 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#22578216C211yesmaximal, characteristic, complemented
2#22578212C212yesmaximal, complemented
3#22578208C211yesmaximal, characteristic, complemented
4#22577788C211yesmaximal, characteristic, complemented
5#22577674C211yesmaximal, characteristic, complemented
6#22577669C212yesmaximal, complemented
7#22577579C212yesmaximal, complemented
8#22577578C212yesmaximal, complemented
9#22577519C211yesmaximal, characteristic, complemented
10#22577518C211yesmaximal, characteristic, complemented
11#473×C137C211yesmaximal, characteristic, complemented
12#17984136C2212yescomplemented
13#17984134C2212yes—
14#17984120—22no—
15#17984120—22no—
16#17984119C2212yescomplemented
17#17984116—22no—
18#17984116C2212yes—
19#17984116C2212yescomplemented
20#17984115—22no—
21#17984111—22no—
22#17984111—21no—
23#17984110—21no—
24#17984110C2211yescharacteristic
25#17984108C2211yescharacteristic
26Q16×D137—21no—
27C2×#14555497C2212yescomplemented
28SD16×D137—22no—
29C2×D1096—21no—
30C2×D1096C2211yescharacteristic
31#17984096C2212yescomplemented
32#17984094C2211yescharacteristic, complemented
33#17984077C2212yescomplemented
34#17984023C2211yescharacteristic, complemented
35#17984015C2211yescharacteristic
36#17984014C2211yescharacteristic, complemented
37#17984014C2212yescomplemented
38#17984004C2212yescomplemented
39C8×Dic137C2211yescharacteristic, complemented
40#137×C137C2212yes—
41C2×Q16×C137C2211yescharacteristic, complemented
42#131×C137C2211yescharacteristic
43#108×C137C2211yescharacteristic, complemented
44#101×C137C2212yescomplemented
45#14555511—22no—
46#14555511—22no—
47#14555511—22no—
48#14555510—22no—
49#14555510—42no—
50#14555510C2311yescharacteristic
51#14555510C2312yes—
52Q8×D137—22no—
53Q8×D137—22no—
54C2×D548—22no—
55C2×D548C2312yes—
56D4×D137—22no—
57D4×D137—42no—
58Dic548D411yescharacteristic
59#14555500—22no—
60#14555500D411yescharacteristic
61#14555499—22no—
62#14555499—22no—
63#14555499D412yes—
64#14555498D411yescharacteristic
65#14555497—22no—
66#14555497—22no—
67#14555497—22no—
68#14555497D411yescharacteristic
69D1096—22no—
70D1096—22no—
71D1096—21no—
72D1096—21no—
73D1096D411yescharacteristic
74#14555495D411yescharacteristic
75#14555494C2311yescharacteristic
76#14555493—22no—
77#14555493—21no—
78C2×#11764040C2311yescharacteristic
79C8×D137—22no—
80C8×D137—21no—
81#14555487—42no—
82C4×Dic137C2311yescharacteristic
83#41×C137C2311yescharacteristic
84C2×Q8×C137C2312yes—
85Q16×C137—22no—
86Q16×C137—21no—
87Q16×C137—21no—
88SD16×C137—22no—
89M16×C137—21no—
90M16×C137C2312yes—
91M16×C137C2311yescharacteristic
92C2×C8×C137—21no—
93C2×C8×C137C2311yescharacteristic
94C22×D137—22no—
95C22×D137—42no—
96Dic274—42no—
97Dic274C2×D411yescharacteristic
98Dic274C2×D412yes—
99D548—22no—
100D548—22no—
101D548—42no—
102D548—42no—
103D548C2×D411yescharacteristic
104D548C2×D412yes—
105C22⋊D137—22no—
106C22⋊D137—21no—
107C22⋊D137—42no—
108C2×Dic137—21no—
109C4×D137—22no—
110C4×D137—21no—
111C4×D137—42no—
112C4×D137—42no—
113C4×D137—42no—
114#11764040—21no—
115#11764040C2×D411yescharacteristic
116#11764040C2×D411yescharacteristic
117Q8×C137—22no—
118Q8×C137—22no—
119Q8×C137C2×D411yescharacteristic
120D4×C137—21no—
121D4×C137C2×D411yescharacteristic
122C2×C4×C137—21no—
123C2×C4×C137—22no—
124C2×C4×C137C2411yescharacteristic
125C8×C137—21no—
126C8×C137—22no—
127C8×C137C2×D411yescharacteristic
128C8×C137C2×D411yescharacteristic
129C22×C137—21no—
130C22×C137H(5,𝔽2)11yescharacteristic
131C2×D137—21no—
132C2×D137—22no—
133C2×D137—42no—
134C2×D137—82no—
135C2×D137—162no—
136C4×C137—21no—
137C4×C137—42no—
138C4×C137C22×D411yescharacteristic
139C4×C137C22×D411yescharacteristic
140Dic137—22no—
141Dic137—21no—
142Dic137—41no—
143C2×C137—21no—
144C2×C137—41no—
145C2×C137D4211yescharacteristic
146D137—41no—
147D137—42no—
148D137—82no—
149C137#329011yescharacteristic, complemented, Sylow
150#3290—1371nomaximal, Sylow
151#578—1371no—
152#577—1372no—
153#576—1371no—
154#495—1371no—
155#473D13711yescharacteristic, complemented
156#472—1371no—
157#471—1372no—
158#455—1372no—
159#454—1372no—
160#443—1371no—
161#443—1371no—
1622−1+4—1372no—
163H(5,𝔽2)—1372no—
164#137—2742no—
165#137—2742no—
166#137—1372no—
167#137C2×D13712yes—
168AGL(1,ℤ/8)—2742no—
169AGL(1,ℤ/8)—2742no—
170AGL(1,ℤ/8)—1372no—
171AGL(1,ℤ/8)—1372no—
172#135—2741no—
173#135—2742no—
174#135—2741no—
175#135—1371no—
176#135—1371no—
177C2×Q16—2741no—
178C2×Q16C2×D13711yescharacteristic, complemented
179C2×SD16—2742no—
180C2×SD16—1372no—
181C2×D8—2741no—
182C2×D8—1371no—
183#131—1372no—
184#131—1371no—
185#131C2×D13711yescharacteristic
186#124—1372no—
187#108—1371no—
188#108C2×D13711yescharacteristic, complemented
189#104—1371no—
190#104—1371no—
191#104—1372no—
192#101C2×D13712yescomplemented
193#100—1372no—
194C4×C8—1371no—
195#41—2742no—
196#41—2742no—
197#41—2742no—
198#41—2742no—
199#41—5482no—
200#41—1371no—
201#41—1372no—
202#41C22×D13711yescharacteristic
203C2×Q8—2742no—
204C2×Q8—2742no—
205C2×Q8C22×D13712yes—
206C2×D4—2742no—
207C2×D4—2742no—
208C2×D4—5482no—
209C2×D4—1372no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—2742no—
214Q16—1371no—
215Q16—1371no—
216SD16—22no—
217SD16—2742no—
218SD16—2742no—
219SD16—2742no—
220SD16—2742no—
221SD16—2742no—
222SD16—1371no—
223SD16—1371no—
224SD16—1372no—
225D8—2742no—
226D8—2742no—
227D8—2741no—
228D8—2741no—
229D8—1371no—
230D8—1371no—
231M16—21no—
232M16—2742no—
233M16—2741no—
234M16—1371no—
235M16C22×D13712yes—
236M16C22×D13711yescharacteristic
237C2×C8—21no—
238C2×C8—2742no—
239C2×C8—2741no—
240C2×C8—1371no—
241C2×C8C22×D13711yescharacteristic
242C22⋊C4—5482no—
243C42—1371no—
244C23—2742no—
245C23—5482no—
246Q8—22no—
247Q8—22no—
248Q8—5482no—
249Q8—1371no—
250Q8—1372no—
251Q8D4×D13711yescharacteristic
252D4—21no—
253D4—2742no—
254D4—2741no—
255D4—2742no—
256D4—2742no—
257D4—5482no—
258D4—5482no—
259D4—5482no—
260D4—1371no—
261D4—1372no—
262D4D4×D13711yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—2742no—
266C2×C4—2741no—
267C2×C4—2741no—
268C2×C4—5482no—
269C2×C4—5482no—
270C2×C4—5482no—
271C2×C4C23×D13711yescharacteristic
272C8—21no—
273C8—22no—
274C8—2741no—
275C8—1371no—
276C8—1371no—
277C8D4×D13711yescharacteristic
278C8D4×D13711yescharacteristic
279C22—21no—
280C22—2741no—
281C22—2742no—
282C22—5482no—
283C22—10962no—
284C22—21922no—
285C22#1798413411yescharacteristic
286C4—21no—
287C4—42no—
288C4—2742no—
289C4—2741no—
290C4—5481no—
291C4C2×D4×D13711yescharacteristic
292C4C2×D4×D13711yescharacteristic
293C2—21no—
294C2—41no—
295C2—5482no—
296C2—5481no—
297C2—10962no—
298C2#2257803711yescharacteristic
subgroupG/H
centerC2#22578037
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#473×C137C2
socleC2×C137D42
hypercenter#473D137
hypocenterC137#3290
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #22578216C2semidirect
2. #22578212C2semidirect
3. #22578208C2semidirect
4. #22577788C2semidirect
5. #22577674C2semidirect
6. #22577669C2semidirect
7. #22577579C2semidirect
8. #22577578C2semidirect
9. #22577519C2semidirect
10. #22577518C2semidirect
11. #473×C137C2semidirect
12. #17984136C22semidirect
16. #17984119C22semidirect
19. #17984116C22semidirect
27. C2×#14555497C22semidirect
31. #17984096C22semidirect
32. #17984094C22semidirect
33. #17984077C22semidirect
34. #17984023C22semidirect
36. #17984014C22semidirect
37. #17984014C22semidirect
38. #17984004C22semidirect
39. C8×Dic137C22semidirect
41. C2×Q16×C137C22semidirect
43. #108×C137C22semidirect
44. #101×C137C22semidirect
149. C137#3290semidirect
155. #473D137semidirect
178. C2×Q16C2×D137semidirect
188. #108C2×D137semidirect
192. #101C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782162
#225782122
#225782082
#225777882
#225776742
#225776692
#225775792
#225775782
#225775192
#225775182
#473×C1372
#3290137