Finite Group Databasebeta

← #31791237

Subgroups — #31791237

Order 17536.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#17984116C211yesmaximal, characteristic, complemented
2C2×#17984114C211yesmaximal, characteristic, complemented
3C2×#17984113C211yesmaximal, characteristic, complemented
4#22577949C211yesmaximal, characteristic, complemented
5C2×#17984084C211yesmaximal, characteristic, complemented
6#22577791C212yesmaximal, complemented
7#22577789C212yesmaximal, complemented
8#22577778C212yesmaximal, complemented
9#22577774C212yesmaximal, complemented
10M16×Dic137C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C22×D548C2211yescharacteristic
14C22×C22⋊D137—22no—
15#17984116—24no—
16#17984114—24no—
17#17984113—24no—
18C2×#14555499C2212yescomplemented
19C2×#14555498C2212yescomplemented
20C2×#14555497C2212yescomplemented
21C2×D1096C2212yescomplemented
22C2×#14555495C2212yescomplemented
23C2×#14555495C2212yescomplemented
24C2×#14555494C2211yescharacteristic
25#17984086C2211yescharacteristic, complemented
26#17984084—22no—
27#17984084C2212yes—
28#17984084C2212yes—
29#17984084C2211yescharacteristic, complemented
30#17984077C2211yescharacteristic, complemented
31#17984075C2211yescharacteristic, complemented
32#17984060—22no—
33C4×C22⋊D137—22no—
34C2×C4×Dic137C2211yescharacteristic
35#17983998C2212yescomplemented
36C8×Dic137C2212yescomplemented
37C2×#41×C137C2211yescharacteristic
38C22×D4×C137C2211yescharacteristic
39AGL(1,ℤ/8)×C137—24no—
40C2×SD16×C137C2212yescomplemented
41C2×D8×C137C2212yescomplemented
42C2×M16×C137C2211yescharacteristic
43C23×D137—21no—
44#14555510—22no—
45#14555510—22no—
46C2×Dic274C2311yescharacteristic
47C2×D548—22no—
48C2×D548—22no—
49C2×D548—21no—
50C2×D548C2311yescharacteristic
51C2×D548C2312yes—
52C2×C22⋊D137—22no—
53C2×C22⋊D137—24no—
54C2×C22⋊D137—24no—
55C2×C22⋊D137—21no—
56C2×C22⋊D137—48no—
57C22×Dic137D412yes—
58C2×C4×D137—21no—
59#14555499—24no—
60#14555498—24no—
61#14555497—24no—
62D1096—24no—
63#14555495—24no—
64#14555495—24no—
65#14555494D414yescomplemented
66C2×#11764040C2312yes—
67#14555489—22no—
68C22⋊Dic137—22no—
69#14555487—22no—
70C4×Dic137D412yes—
71C4×Dic137C2311yescharacteristic
72C4×Dic137C2311yescharacteristic
73C24×C137—21no—
74#41×C137—22no—
75#41×C137—22no—
76C2×Q8×C137C2311yescharacteristic
77C2×D4×C137—21no—
78C2×D4×C137—22no—
79C2×D4×C137—22no—
80C2×D4×C137—21no—
81C2×D4×C137C2311yescharacteristic
82C2×D4×C137C2312yes—
83C22×C4×C137—21no—
84C22×C4×C137C2311yescharacteristic
85SD16×C137—24no—
86D8×C137—24no—
87M16×C137D414yescomplemented
88C2×C8×C137C2312yes—
89C22×D137—21no—
90C22×D137—22no—
91C22×D137—44no—
92C22×D137—42no—
93Dic274—21no—
94Dic274—21no—
95D548—21no—
96D548—22no—
97D548—21no—
98D548—22no—
99D548—42no—
100C22⋊D137—48no—
101C22⋊D137—42no—
102C22⋊D137—44no—
103C22⋊D137—48no—
104C2×Dic137—24no—
105C2×Dic137—21no—
106C2×Dic137C2×D412yes—
107C2×Dic137C2×D412yes—
108C4×D137—42no—
109#11764040C2×D414yescomplemented
110C23×C137—21no—
111C23×C137—22no—
112C23×C137—44no—
113C23×C137—42no—
114C23×C137C2×D411yescharacteristic
115Q8×C137—21no—
116Q8×C137—21no—
117D4×C137—21no—
118D4×C137—22no—
119D4×C137—22no—
120D4×C137—21no—
121D4×C137—42no—
122D4×C137—42no—
123C2×C4×C137—21no—
124C2×C4×C137—42no—
125C2×C4×C137C2×D411yescharacteristic
126C2×C4×C137C2×D412yes—
127C2×C4×C137C2×D412yes—
128C2×C4×C137C2411yescharacteristic
129C8×C137C2×D414yescomplemented
130C22×C137—22no—
131C22×C137—41no—
132C22×C137—42no—
133C22×C137—44no—
134C22×C137—42no—
135C22×C137—84no—
136C22×C137#12712yes—
137C22×C137C22×D411yescharacteristic
138C2×D137—41no—
139C2×D137—42no—
140C2×D137—44no—
141C2×D137—42no—
142C2×D137—84no—
143C4×C137—41no—
144C4×C137#12712yes—
145C4×C137C22×D411yescharacteristic
146C4×C137C22×D411yescharacteristic
147Dic137—22no—
148Dic137—41no—
149Dic137AGL(1,ℤ/8)14yescomplemented
150C2×C137—22no—
151C2×C137—81no—
152C2×C137—82no—
153C2×C137C2×#12711yescharacteristic
154C2×C137C2×AGL(1,ℤ/8)12yes—
155D137—81no—
156D137—82no—
157C137#315111yescharacteristic, complemented, Sylow
158#3151—1371nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1371no—
160C2×AGL(1,ℤ/8)—1371no—
161C2×AGL(1,ℤ/8)—1371no—
162C2×AGL(1,ℤ/8)D13711yescharacteristic, complemented
163#532—1371no—
164C2×#127—1371no—
165#496—1372no—
166#496—1372no—
167#493—1372no—
168#492—1372no—
169C4×M16—1371no—
170C2×#41—1371no—
171C2×#41C2×D13711yescharacteristic
172C22×D4—2742no—
173C22×D4—1371no—
174C22×D4C2×D13711yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2744no—
177AGL(1,ℤ/8)—2744no—
178AGL(1,ℤ/8)—2744no—
179C2×SD16—1372no—
180C2×SD16—1372no—
181C2×SD16—1372no—
182C2×SD16C2×D13712yescomplemented
183C2×D8—1372no—
184C2×D8—1372no—
185C2×D8—1372no—
186C2×D8C2×D13712yescomplemented
187C2×M16—1371no—
188C2×M16C2×D13711yescharacteristic
189C4⋊Q8—1371no—
190#127—2742no—
191#127—1372no—
192#127—1372no—
193#127—1371no—
194#124—1371no—
195#124—1371no—
196#121—2742no—
197C4×D4—2742no—
198C2×C42—1371no—
199#97—1372no—
200C4×C8—1372no—
201C24—21no—
202C24—2741no—
203#41—22no—
204#41—22no—
205#41—2742no—
206#41—2742no—
207C2×Q8—1371no—
208C2×Q8C22×D13711yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—2741no—
214C2×D4—2742no—
215C2×D4—2744no—
216C2×D4—2744no—
217C2×D4—2742no—
218C2×D4—2741no—
219C2×D4—2742no—
220C2×D4—5488no—
221C2×D4—1371no—
222C2×D4—1372no—
223C2×D4C22×D13711yescharacteristic
224C2×D4C22×D13712yes—
225C22×C4—21no—
226C22×C4—2741no—
227C22×C4—1372no—
228C22×C4C22×D13711yescharacteristic
229SD16—24no—
230SD16—2744no—
231SD16—2744no—
232SD16—2744no—
233D8—24no—
234D8—2744no—
235D8—2744no—
236D8—2744no—
237M16—1374no—
238M16C22⋊D13714yescomplemented
239C2×C8—1372no—
240C2×C8C22×D13712yes—
241C4⋊C4—2742no—
242C22⋊C4—2742no—
243C22⋊C4—2742no—
244C42—1372no—
245C42—1371no—
246C42—1371no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—2741no—
252C23—2742no—
253C23—5484no—
254C23—5482no—
255C23D4×D13711yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—2741no—
259Q8—2741no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—2741no—
267D4—2742no—
268D4—2741no—
269D4—2742no—
270D4—5488no—
271D4—5482no—
272D4—5488no—
273D4—5484no—
274D4—5482no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—2744no—
278C2×C4—2741no—
279C2×C4—5482no—
280C2×C4—1372no—
281C2×C4—1372no—
282C2×C4D4×D13711yescharacteristic
283C2×C4D4×D13712yes—
284C2×C4C2×C22⋊D13712yes—
285C2×C4C23×D13711yescharacteristic
286C8—1374no—
287C8C2×C22⋊D13714yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—5484no—
295C22—5482no—
296C22—5481no—
297C22—5482no—
298C22—10964no—
299C22#1798408412yes—
300C22C2×D4×D13711yescharacteristic
301C4—41no—
302C4—2742no—
303C4—5481no—
304C4—1374no—
305C4#1798408412yes—
306C4C2×D4×D13711yescharacteristic
307C4C22×C22⋊D13711yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—10962no—
312C2—10961no—
313C2C2×#1798408411yescharacteristic
314C2AGL(1,ℤ/8)×D13712yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×AGL(1,ℤ/8)×C137C2
socleC22×C137C22×D4
hypercenterC2×AGL(1,ℤ/8)D137
hypocenterC137#3151
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#17984116C2semidirect
2. C2×#17984114C2semidirect
3. C2×#17984113C2semidirect
4. #22577949C2semidirect
5. C2×#17984084C2semidirect
6. #22577791C2semidirect
7. #22577789C2semidirect
8. #22577778C2semidirect
9. #22577774C2semidirect
10. M16×Dic137C2semidirect
11. C2×AGL(1,ℤ/8)×C137C2semidirect
18. C2×#14555499C22semidirect
19. C2×#14555498C22semidirect
20. C2×#14555497C22semidirect
21. C2×D1096C22semidirect
22. C2×#14555495C22semidirect
23. C2×#14555495C22semidirect
25. #17984086C22semidirect
29. #17984084C22semidirect
30. #17984077C22semidirect
31. #17984075C22semidirect
35. #17983998C22semidirect
36. C8×Dic137C22semidirect
40. C2×SD16×C137C22semidirect
41. C2×D8×C137C22semidirect
65. #14555494D4semidirect
87. M16×C137D4semidirect
109. #11764040C2×D4semidirect
129. C8×C137C2×D4semidirect
149. Dic137AGL(1,ℤ/8)semidirect
157. C137#3151semidirect
162. C2×AGL(1,ℤ/8)D137semidirect
182. C2×SD16C2×D137semidirect
186. C2×D8C2×D137semidirect
238. M16C22⋊D137semidirect
287. C8C2×C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841162
C2×#179841142
C2×#179841132
#225779492
C2×#179840842
#225777912
#225777892
#225777782
#225777742
M16×Dic1372
C2×AGL(1,ℤ/8)×C1372
#3151137