Finite Group Databasebeta

← #31790402

Subgroups — #31790402

Order 17536.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D137C211yesmaximal, characteristic, complemented
2#22577910C211yesmaximal, characteristic, complemented
3C2×#17984059C211yesmaximal, characteristic, complemented
4#22577653C212yesmaximal, complemented
5#22577646C212yesmaximal, complemented
6#22577636C212yesmaximal, complemented
7#22577629C212yesmaximal, complemented
8#22577445C211yesmaximal, characteristic, complemented
9#22577408C211yesmaximal, characteristic, complemented
10#22577404C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C137C211yesmaximal, characteristic, complemented
12C22×C22⋊D137—21no—
13C22×C4×D137C2211yescharacteristic
14C2×#14555494C2211yescharacteristic
15C2×#14555493C2212yescomplemented
16M16×D137—24no—
17C2×C8×D137C2212yescomplemented
18#17984069—21no—
19#17984064C2211yescharacteristic, complemented
20#17984064C2211yescharacteristic, complemented
21#17984059—21no—
22#17984059—21no—
23#17984059C2212yes—
24#17984059C2212yes—
25#17984059C2211yescharacteristic, complemented
26#17984059C2211yescharacteristic, complemented
27#17984056—21no—
28C4×C22⋊D137—21no—
29C4×C22⋊D137—21no—
30#17984047C2211yescharacteristic
31C2×C4⋊Dic137C2211yescharacteristic
32C2×C22⋊Dic137—21no—
33#17984022C2212yescomplemented
34#17984020C2212yescomplemented
35#17984012C2212yescomplemented
36#17984009C2212yescomplemented
37#17984009C2212yescomplemented
38#17984009C2212yescomplemented
39C2×#41×C137C2211yescharacteristic
40C22×D4×C137C2211yescharacteristic
41AGL(1,ℤ/8)×C137—24no—
42C2×SD16×C137C2212yescomplemented
43C2×D8×C137C2212yescomplemented
44C2×M16×C137C2211yescharacteristic
45C23×D137D411yescharacteristic
46C2×C22⋊D137—21no—
47C2×C22⋊D137—21no—
48C2×C22⋊D137—22no—
49C2×C22⋊D137—22no—
50C2×C22⋊D137—44no—
51C22×Dic137—21no—
52C22×Dic137D411yescharacteristic
53C2×C4×D137—22no—
54C2×C4×D137—22no—
55C2×C4×D137D411yescharacteristic
56C2×C4×D137D411yescharacteristic
57C2×C4×D137C2311yescharacteristic
58C2×C4×D137C2311yescharacteristic
59#14555494—22no—
60#14555493—24no—
61C2×#11764040C2312yes—
62C8×D137—24no—
63C4⋊Dic137—21no—
64C4⋊Dic137C2311yescharacteristic
65C4⋊Dic137C2311yescharacteristic
66C4⋊Dic137C2312yes—
67#14555489—21no—
68#14555489—21no—
69C22⋊Dic137—22no—
70C22⋊Dic137—22no—
71C22⋊Dic137—21no—
72C22⋊Dic137—21no—
73C22⋊Dic137—21no—
74#14555487—21no—
75#14555487—21no—
76C4×Dic137—21no—
77C24×C137—21no—
78#41×C137—22no—
79#41×C137—22no—
80C2×Q8×C137C2311yescharacteristic
81C2×D4×C137—21no—
82C2×D4×C137—22no—
83C2×D4×C137—21no—
84C2×D4×C137—22no—
85C2×D4×C137C2312yes—
86C2×D4×C137C2311yescharacteristic
87C22×C4×C137—21no—
88C22×C4×C137C2311yescharacteristic
89SD16×C137—24no—
90D8×C137—24no—
91M16×C137D414yescomplemented
92C2×C8×C137C2312yes—
93C22×D137—22no—
94C22×D137—22no—
95C22×D137—41no—
96C22×D137C2×D411yescharacteristic
97C22×D137C2×D411yescharacteristic
98C22⋊D137—42no—
99C22⋊D137—44no—
100C22⋊D137—42no—
101C22⋊D137—44no—
102C2×Dic137—22no—
103C2×Dic137—21no—
104C2×Dic137—21no—
105C2×Dic137—22no—
106C2×Dic137—42no—
107C2×Dic137C2×D411yescharacteristic
108C2×Dic137C2×D411yescharacteristic
109C4×D137—22no—
110C4×D137—22no—
111C4×D137—21no—
112C4×D137—21no—
113C4×D137—21no—
114C4×D137—21no—
115#11764040—22no—
116C23×C137—21no—
117C23×C137—22no—
118C23×C137—42no—
119C23×C137—42no—
120C23×C137—42no—
121C23×C137C2×D411yescharacteristic
122Q8×C137—21no—
123Q8×C137—21no—
124D4×C137—21no—
125D4×C137—22no—
126D4×C137—22no—
127D4×C137—21no—
128D4×C137—42no—
129D4×C137—42no—
130C2×C4×C137—21no—
131C2×C4×C137—42no—
132C2×C4×C137C2×D411yescharacteristic
133C2×C4×C137C2×D412yes—
134C2×C4×C137#4112yes—
135C2×C4×C137C2411yescharacteristic
136C8×C137C2×D414yescomplemented
137C22×C137—21no—
138C22×C137—21no—
139C22×C137—41no—
140C22×C137—42no—
141C22×C137—42no—
142C22×C137—41no—
143C22×C137—42no—
144C22×C137—41no—
145C22×C137—84no—
146C22×C137#12112yes—
147C22×C137C22×D411yescharacteristic
148C2×D137—21no—
149C2×D137—21no—
150C2×D137—21no—
151C2×D137—21no—
152C2×D137—21no—
153C2×D137—44no—
154C2×D137AGL(1,ℤ/8)12yescomplemented
155C4×C137—41no—
156C4×C137#12112yes—
157C4×C137C22×D411yescharacteristic
158C4×C137C2×#4111yescharacteristic
159Dic137—21no—
160Dic137—21no—
161Dic137—42no—
162Dic137—41no—
163Dic137—41no—
164C2×C137—22no—
165C2×C137—81no—
166C2×C137—82no—
167C2×C137C2×#12111yescharacteristic
168C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C137#57511yescharacteristic
170D137—22no—
171D137—41no—
172C137#305511yescharacteristic, complemented, Sylow
173#3055—1371nomaximal, Sylow
174C2×AGL(1,ℤ/8)D13711yescharacteristic, complemented
175C22×M16—1371no—
176#525—1371no—
177C2×#121—1371no—
178#469—1372no—
179#468—1372no—
180#466—1372no—
181#465—1372no—
182#428—1371no—
183#418—1371no—
184#417—1371no—
185C2×#41C2×D13711yescharacteristic
186C22×D4—2741no—
187C22×D4C2×D13711yescharacteristic
188C23×C4—1371no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D13712yescomplemented
191C2×D8C2×D13712yescomplemented
192C2×M16—2744no—
193C2×M16—1371no—
194C2×M16—1372no—
195C2×M16C2×D13711yescharacteristic
196C22×C8—1372no—
197#123—2741no—
198C22⋊Q8—1371no—
199C22⋊Q8—1371no—
200#121—2741no—
201#121—2741no—
202#121—1372no—
203#121—1372no—
204#121—1371no—
205#121—1371no—
206#120—2741no—
207C4×D4—2741no—
208C4×D4—2741no—
209#117—1371no—
210C2×C4⋊C4—1371no—
211C2×C22⋊C4—2741no—
212#107—1372no—
213#106—1372no—
214#103—1372no—
215#102—1372no—
216#102—1372no—
217#102—1372no—
218C24—21no—
219C24—1371no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D13711yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—2741no—
228C2×D4—2741no—
229C2×D4—2742no—
230C2×D4—2742no—
231C2×D4—5484no—
232C2×D4C22×D13711yescharacteristic
233C2×D4C22×D13712yes—
234C22×C4—21no—
235C22×C4—2742no—
236C22×C4—2742no—
237C22×C4—2741no—
238C22×C4—1371no—
239C22×C4—1371no—
240C22×C4—1371no—
241C22×C4—1371no—
242C22×C4—1371no—
243C22×C4C22×D13711yescharacteristic
244SD16—24no—
245D8—24no—
246M16—2742no—
247M16—2744no—
248M16C22⋊D13714yescomplemented
249C2×C8—2744no—
250C2×C8—1372no—
251C2×C8C22×D13712yes—
252C4⋊C4—2741no—
253C4⋊C4—2741no—
254C4⋊C4—2741no—
255C4⋊C4—1371no—
256C4⋊C4—1371no—
257C4⋊C4—1372no—
258C22⋊C4—2741no—
259C22⋊C4—2741no—
260C22⋊C4—2741no—
261C22⋊C4—2742no—
262C22⋊C4—2742no—
263C22⋊C4—2741no—
264C22⋊C4—2741no—
265C42—2741no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—2742no—
272C23—2742no—
273C23—5481no—
274C23—1371no—
275C23—1371no—
276C23D4×D13711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—5484no—
286D4—5482no—
287D4—5484no—
288D4—5482no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—2742no—
292C2×C4—2742no—
293C2×C4—2742no—
294C2×C4—2741no—
295C2×C4—2741no—
296C2×C4—2741no—
297C2×C4—2741no—
298C2×C4—2741no—
299C2×C4—2741no—
300C2×C4—2742no—
301C2×C4—5482no—
302C2×C4—1371no—
303C2×C4—1371no—
304C2×C4D4×D13711yescharacteristic
305C2×C4C2×C22⋊D13712yes—
306C2×C4#1455550912yes—
307C2×C4C23×D13711yescharacteristic
308C8—2742no—
309C8C2×C22⋊D13714yescomplemented
310C22—21no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—2741no—
320C22—2741no—
321C22—2741no—
322C22—2741no—
323C22—2741no—
324C22—5484no—
325C22—1372no—
326C22#1798405912yes—
327C22C2×D4×D13711yescharacteristic
328C4—41no—
329C4—2741no—
330C4—2741no—
331C4—5482no—
332C4—5481no—
333C4—5481no—
334C4#1798405912yes—
335C4C22×C22⋊D13711yescharacteristic
336C4C2×#1455550911yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—2742no—
341C2—5481no—
342C2C2×#1798405911yescharacteristic
343C2AGL(1,ℤ/8)×D13711yescharacteristic
344C2#2257819911yescharacteristic
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#3055
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×M16×D137C2semidirect
2. #22577910C2semidirect
3. C2×#17984059C2semidirect
4. #22577653C2semidirect
5. #22577646C2semidirect
6. #22577636C2semidirect
7. #22577629C2semidirect
8. #22577445C2semidirect
9. #22577408C2semidirect
10. #22577404C2semidirect
11. C2×AGL(1,ℤ/8)×C137C2semidirect
15. C2×#14555493C22semidirect
17. C2×C8×D137C22semidirect
19. #17984064C22semidirect
20. #17984064C22semidirect
25. #17984059C22semidirect
26. #17984059C22semidirect
33. #17984022C22semidirect
34. #17984020C22semidirect
35. #17984012C22semidirect
36. #17984009C22semidirect
37. #17984009C22semidirect
38. #17984009C22semidirect
42. C2×SD16×C137C22semidirect
43. C2×D8×C137C22semidirect
91. M16×C137D4semidirect
136. C8×C137C2×D4semidirect
154. C2×D137AGL(1,ℤ/8)semidirect
172. C137#3055semidirect
174. C2×AGL(1,ℤ/8)D137semidirect
190. C2×SD16C2×D137semidirect
191. C2×D8C2×D137semidirect
248. M16C22⋊D137semidirect
309. C8C2×C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1372
#225779102
C2×#179840592
#225776532
#225776462
#225776362
#225776292
#225774452
#225774082
#225774042
C2×AGL(1,ℤ/8)×C1372
#3055137