Finite Group Databasebeta

← #31792577

Subgroups — #31792577

Order 17536.

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

normal subgroups only

This group has 454 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#22578217C211yesmaximal, characteristic, complemented
2#22578215C212yesmaximal, complemented
3#22578205C211yesmaximal, characteristic, complemented
4#22577787C211yesmaximal, characteristic, complemented
5#22577679C211yesmaximal, characteristic, complemented
6#22577674C211yesmaximal, characteristic, complemented
7#22577666C212yesmaximal, complemented
8#22577582C212yesmaximal, complemented
9#22577577C212yesmaximal, complemented
10#22577521C211yesmaximal, characteristic, complemented
11#443×C137C211yesmaximal, characteristic, complemented
12#17984135C2212yescomplemented
13#17984134C2212yes—
14#17984121—22no—
15#17984121C2212yes—
16#17984118C2212yescomplemented
17#17984117—22no—
18#17984116—22no—
19#17984116C2212yescomplemented
20#17984115—22no—
21#17984113C2212yes—
22#17984111—21no—
23#17984110—22no—
24#17984110C2211yescharacteristic
25#17984107—21no—
26C2×Dic548C2211yescharacteristic, complemented
27Q16×D137—21no—
28C2×#14555499—22no—
29C2×#14555499C2212yescomplemented
30C2×D1096C2211yescharacteristic
31D8×D137—21no—
32#17984094C2211yescharacteristic
33#17984094C2211yescharacteristic, complemented
34#17984076C2212yescomplemented
35#17984024C2211yescharacteristic, complemented
36#17984016C2211yescharacteristic
37#17984016C2211yescharacteristic, complemented
38#17984006C2212yescomplemented
39#17984004C2212yescomplemented
40#135×C137C2211yescharacteristic
41#131×C137C2212yescomplemented
42#108×C137C2211yescharacteristic, complemented
43#104×C137C2212yescomplemented
44C4×C8×C137C2211yescharacteristic, complemented
45#14555511—22no—
46#14555510—22no—
47#14555510—22no—
48#14555510C2311yescharacteristic
49#14555510C2311yescharacteristic
50#14555509—22no—
51#14555509—42no—
52C2×Dic274—22no—
53C2×Dic274C2312yes—
54Q8×D137—22no—
55C2×D548—22no—
56C2×D548C2312yes—
57D4×D137—22no—
58D4×D137—42no—
59Dic548—22no—
60Dic548—21no—
61Dic548—21no—
62Dic548D411yescharacteristic
63#14555500—22no—
64#14555499—22no—
65#14555499—22no—
66#14555499—22no—
67#14555499—22no—
68#14555499D411yescharacteristic
69#14555499D411yescharacteristic
70#14555498—22no—
71#14555497—22no—
72D1096—22no—
73D1096—21no—
74D1096—21no—
75D1096D411yescharacteristic
76#14555495—22no—
77#14555494C2311yescharacteristic
78#14555494C2311yescharacteristic
79#14555493—21no—
80#14555493—21no—
81C8×D137—21no—
82C8×D137—21no—
83#14555487—42no—
84#41×C137C2312yes—
85Q16×C137D411yescharacteristic
86SD16×C137D412yes—
87D8×C137D411yescharacteristic
88M16×C137—22no—
89M16×C137C2312yes—
90C2×C8×C137—22no—
91C2×C8×C137C2311yescharacteristic
92C2×C8×C137C2311yescharacteristic
93C42×C137C2311yescharacteristic
94C22×D137—22no—
95C22×D137—42no—
96Dic274—22no—
97Dic274—22no—
98Dic274—42no—
99Dic274C2×D411yescharacteristic
100Dic274C2×D411yescharacteristic
101D548—22no—
102D548—22no—
103D548—42no—
104D548C2×D411yescharacteristic
105D548C2×D411yescharacteristic
106C22⋊D137—21no—
107C22⋊D137—21no—
108C22⋊D137—42no—
109C22⋊D137—42no—
110C2×Dic137—22no—
111C2×Dic137—42no—
112C4×D137—21no—
113C4×D137—21no—
114C4×D137—42no—
115C4×D137—42no—
116#11764040—21no—
117#11764040—21no—
118Q8×C137C2×D412yes—
119D4×C137—22no—
120D4×C137C2×D412yes—
121C2×C4×C137—22no—
122C2×C4×C137—21no—
123C2×C4×C137C2411yescharacteristic
124C8×C137—22no—
125C8×C137C2×D411yescharacteristic
126C8×C137C2×D411yescharacteristic
127C8×C137C2×D411yescharacteristic
128C8×C137C2×D411yescharacteristic
129C22×C137—22no—
130C22×C137H(5,𝔽2)11yescharacteristic
131C2×D137—21no—
132C2×D137—21no—
133C2×D137—42no—
134C2×D137—82no—
135C2×D137—162no—
136C4×C137—22no—
137C4×C137—41no—
138C4×C137C22×D411yescharacteristic
139C4×C137C22×D411yescharacteristic
140Dic137—21no—
141Dic137—21no—
142Dic137—42no—
143C2×C137—21no—
144C2×C137—42no—
145C2×C137D4211yescharacteristic
146D137—41no—
147D137—41no—
148D137—82no—
149C137#329011yescharacteristic, complemented, Sylow
150#3290—1371nomaximal, Sylow
151#578—1371no—
152#577—1372no—
153#576—1371no—
154#495—1371no—
155#473—1371no—
156#472—1371no—
157#471—1372no—
158#455—1372no—
159#454—1372no—
160#443—1371no—
161#443D13711yescharacteristic, complemented
1622−1+4—1372no—
163H(5,𝔽2)—1372no—
164#137—2742no—
165#137—2742no—
166#137—1372no—
167#137—1372no—
168AGL(1,ℤ/8)—2742no—
169AGL(1,ℤ/8)—2742no—
170AGL(1,ℤ/8)—1372no—
171AGL(1,ℤ/8)—1372no—
172#135—2742no—
173#135—2741no—
174#135—2741no—
175#135—1371no—
176#135C2×D13711yescharacteristic
177C2×Q16—2741no—
178C2×Q16—1371no—
179C2×SD16—2742no—
180C2×SD16—1372no—
181C2×D8—2741no—
182C2×D8—1371no—
183#131—1371no—
184#131—1371no—
185#131C2×D13712yescomplemented
186#124—1372no—
187#108—1371no—
188#108C2×D13711yescharacteristic, complemented
189#104—1371no—
190#104—1371no—
191#104C2×D13712yescomplemented
192#101—1372no—
193#100—1372no—
194C4×C8C2×D13711yescharacteristic, complemented
195#41—2742no—
196#41—2742no—
197#41—2742no—
198#41—2742no—
199#41—5482no—
200#41—1371no—
201#41—1371no—
202#41C22×D13712yes—
203C2×Q8—2742no—
204C2×Q8—2742no—
205C2×Q8—1372no—
206C2×D4—2742no—
207C2×D4—2742no—
208C2×D4—5482no—
209C2×D4—1372no—
210Q16—2742no—
211Q16—2742no—
212Q16—2741no—
213Q16—2741no—
214Q16—1371no—
215Q16D54811yescharacteristic
216SD16—2742no—
217SD16—2742no—
218SD16—2742no—
219SD16—2742no—
220SD16—2742no—
221SD16—2742no—
222SD16—1371no—
223SD16—1371no—
224SD16D54812yes—
225D8—2742no—
226D8—2742no—
227D8—2741no—
228D8—2741no—
229D8—1371no—
230D8D54811yescharacteristic
231M16—22no—
232M16—2741no—
233M16—2741no—
234M16—1371no—
235M16—1371no—
236M16C22×D13712yes—
237C2×C8—22no—
238C2×C8—2741no—
239C2×C8—2741no—
240C2×C8C22×D13711yescharacteristic
241C2×C8C22×D13711yescharacteristic
242C22⋊C4—5482no—
243C42C22×D13711yescharacteristic
244C23—2742no—
245C23—5482no—
246Q8—2742no—
247Q8—2742no—
248Q8—5482no—
249Q8—1371no—
250Q8—1371no—
251Q8C2×D54812yes—
252D4—22no—
253D4—2741no—
254D4—2741no—
255D4—2742no—
256D4—2742no—
257D4—5482no—
258D4—5482no—
259D4—5482no—
260D4—1371no—
261D4—1371no—
262D4C2×D54812yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—2741no—
266C2×C4—2741no—
267C2×C4—2742no—
268C2×C4—5482no—
269C2×C4—5482no—
270C2×C4—5482no—
271C2×C4C23×D13711yescharacteristic
272C8—22no—
273C8—2741no—
274C8—2741no—
275C8D4×D13711yescharacteristic
276C8D4×D13711yescharacteristic
277C8C2×D54811yescharacteristic
278C8C2×D54811yescharacteristic
279C22—22no—
280C22—2741no—
281C22—2741no—
282C22—5482no—
283C22—10962no—
284C22—21922no—
285C22#1798413411yescharacteristic
286C4—22no—
287C4—41no—
288C4—2741no—
289C4—2741no—
290C4—5482no—
291C4C2×D4×D13711yescharacteristic
292C4C22×D54811yescharacteristic
293C2—21no—
294C2—42no—
295C2—5481no—
296C2—5481no—
297C2—10962no—
298C2D4×D54811yescharacteristic
subgroupG/H
centerC2D4×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#443×C137C2
socleC2×C137D42
hypercenter#443D137
hypocenterC137#3290
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #22578217C2semidirect
2. #22578215C2semidirect
3. #22578205C2semidirect
4. #22577787C2semidirect
5. #22577679C2semidirect
6. #22577674C2semidirect
7. #22577666C2semidirect
8. #22577582C2semidirect
9. #22577577C2semidirect
10. #22577521C2semidirect
11. #443×C137C2semidirect
12. #17984135C22semidirect
16. #17984118C22semidirect
19. #17984116C22semidirect
26. C2×Dic548C22semidirect
29. C2×#14555499C22semidirect
33. #17984094C22semidirect
34. #17984076C22semidirect
35. #17984024C22semidirect
37. #17984016C22semidirect
38. #17984006C22semidirect
39. #17984004C22semidirect
41. #131×C137C22semidirect
42. #108×C137C22semidirect
43. #104×C137C22semidirect
44. C4×C8×C137C22semidirect
149. C137#3290semidirect
161. #443D137semidirect
185. #131C2×D137semidirect
188. #108C2×D137semidirect
191. #104C2×D137semidirect
194. C4×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782172
#225782152
#225782052
#225777872
#225776792
#225776742
#225776662
#225775822
#225775772
#225775212
#443×C1372
#3290137