Finite Group Databasebeta

← #31790092

Subgroups — #31790092

Order 17536.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984133C211yesmaximal, characteristic, complemented
2C2×#17984108C211yesmaximal, characteristic, complemented
3#22577981C211yesmaximal, characteristic, complemented
4#22577562C212yesmaximal, complemented
5#22577552C212yesmaximal, complemented
6#22577545C212yesmaximal, complemented
7#22577535C212yesmaximal, complemented
8#22577405C211yesmaximal, characteristic, complemented
9C2×#17984009C211yesmaximal, characteristic, complemented
10#22577371C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C137C211yesmaximal, characteristic, complemented
12#17984133—24no—
13#17984133—24no—
14C2×#14555510C2211yescharacteristic
15C2×#14555510C2211yescharacteristic
16C2×#14555509—21no—
17C2×#14555509C2212yes—
18C22×C22⋊D137—22no—
19C2×D4×D137—21no—
20C2×D4×D137C2212yes—
21#17984108—24no—
22C2×#14555500C2211yescharacteristic, complemented
23C2×#14555498C2211yescharacteristic, complemented
24C2×#14555497C2211yescharacteristic, complemented
25C2×#14555495C2211yescharacteristic, complemented
26C22×#11764040C2211yescharacteristic, complemented
27#17984067—21no—
28#17984064C2212yescomplemented
29#17984062—21no—
30#17984060—22no—
31#17984059C2212yescomplemented
32C2×C4⋊Dic137C2211yescharacteristic, complemented
33#17984013C2212yescomplemented
34#17984010C2212yescomplemented
35#17984009C2212yescomplemented
36#17984009C2212yescomplemented
37#17984000C2212yescomplemented
38#17984000C2212yescomplemented
39C2×#41×C137C2211yescharacteristic, complemented
40C22×D4×C137C2211yescharacteristic
41AGL(1,ℤ/8)×C137—24no—
42C2×SD16×C137C2212yescomplemented
43C2×D8×C137C2212yescomplemented
44C2×M16×C137C2211yescharacteristic, complemented
45C23×D137—22no—
46#14555510—22no—
47#14555510—22no—
48#14555510—22no—
49#14555510D414yescomplemented
50#14555509—28no—
51#14555509—24no—
52#14555509—24no—
53C2×Dic274D411yescharacteristic
54C2×Dic274C2311yescharacteristic
55C2×D548D411yescharacteristic
56C2×D548C2311yescharacteristic
57C2×C22⋊D137—216no—
58C2×C22⋊D137—22no—
59C2×C22⋊D137—22no—
60C2×C22⋊D137—22no—
61C2×C22⋊D137—21no—
62C2×C22⋊D137D412yes—
63D4×D137—28no—
64D4×D137—24no—
65D4×D137—24no—
66C22×Dic137—22no—
67C22×Dic137—21no—
68C2×C4×D137—21no—
69C2×C4×D137C2312yes—
70#14555500—22no—
71#14555498—22no—
72#14555497—22no—
73#14555495—22no—
74C2×#11764040—22no—
75C2×#11764040C2311yescharacteristic
76C2×#11764040C2311yescharacteristic
77C4⋊Dic137—21no—
78C4⋊Dic137C2312yes—
79#14555489—22no—
80C22⋊Dic137—22no—
81#14555487—22no—
82C24×C137—21no—
83#41×C137—22no—
84#41×C137—22no—
85C2×Q8×C137C2311yescharacteristic
86C2×D4×C137—21no—
87C2×D4×C137—21no—
88C2×D4×C137—22no—
89C2×D4×C137D414yescomplemented
90C2×D4×C137C2311yescharacteristic
91C2×D4×C137C2312yes—
92C22×C4×C137—21no—
93C22×C4×C137C2311yescharacteristic
94SD16×C137—24no—
95D8×C137—24no—
96M16×C137—22no—
97C2×C8×C137C2312yes—
98C22×D137—21no—
99C22×D137—28no—
100C22×D137—82no—
101C22×D137—82no—
102C22×D137C2×D412yes—
103Dic274—22no—
104Dic274—21no—
105Dic274C2×D412yes—
106D548—22no—
107D548—21no—
108D548C2×D412yes—
109C22⋊D137—28no—
110C22⋊D137—28no—
111C22⋊D137—24no—
112C22⋊D137—24no—
113C22⋊D137—82no—
114C2×Dic137—28no—
115C2×Dic137—21no—
116C2×Dic137—22no—
117C2×Dic137—81no—
118C2×Dic137C2×D412yes—
119C4×D137—24no—
120C4×D137—22no—
121C4×D137—22no—
122#11764040—21no—
123#11764040—21no—
124C23×C137—22no—
125C23×C137—24no—
126C23×C137—21no—
127C23×C137—81no—
128C23×C137—81no—
129C23×C137C2×D411yescharacteristic
130Q8×C137—21no—
131Q8×C137—21no—
132D4×C137—24no—
133D4×C137—22no—
134D4×C137—21no—
135D4×C137—21no—
136D4×C137—42no—
137D4×C137C2×D414yes—
138C2×C4×C137—21no—
139C2×C4×C137—42no—
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2×D412yes—
142C2×C4×C137C2×D412yes—
143C2×C4×C137C2411yescharacteristic
144C8×C137—22no—
145C22×C137—21no—
146C22×C137—21no—
147C22×C137—24no—
148C22×C137—42no—
149C22×C137—42no—
150C22×C137—41no—
151C22×C137—41no—
152C22×C137—41no—
153C22×C137—84no—
154C22×C137#12012yes—
155C22×C137C22×D411yescharacteristic
156C2×D137—22no—
157C2×D137—22no—
158C2×D137—22no—
159C2×D137—22no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137—88no—
163C4×C137—41no—
164C4×C137#12012yes—
165C4×C137C22×D411yescharacteristic
166C4×C137C22×D411yescharacteristic
167Dic137—22no—
168Dic137—22no—
169Dic137—42no—
170C2×C137—22no—
171C2×C137—44no—
172C2×C137—81no—
173C2×C137C2×#12011yescharacteristic
174C2×C137#57611yescharacteristic
175C2×C137#57711yescharacteristic
176D137—42no—
177D137—42no—
178C137#301211yescharacteristic, complemented, Sylow
179#3012—1371nomaximal, Sylow
180C2×H(5,𝔽2)—1371no—
181C2×AGL(1,ℤ/8)D13711yescharacteristic, complemented
182C2×#135—1371no—
183#537—1371no—
184#452—1372no—
185#450—1372no—
186#449—1372no—
187#447—1372no—
188#417—1371no—
189C2×#102—1371no—
190#408—1371no—
191H(5,𝔽2)—2744no—
192H(5,𝔽2)—2744no—
193C2×#41—2741no—
194C2×#41—1372no—
195C2×#41—1371no—
196C2×#41—1371no—
197C2×#41C2×D13711yescharacteristic, complemented
198C22×D4—2741no—
199C22×D4—2742no—
200C22×D4—1372no—
201C22×D4C2×D13711yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—2744no—
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16—1371no—
207C2×SD16C2×D13712yescomplemented
208C2×D8—1371no—
209C2×D8C2×D13712yescomplemented
210C2×M16C2×D13711yescharacteristic, complemented
211C22×C8—1371no—
212C22⋊Q8—2741no—
213C22⋊Q8—1372no—
214#121—2741no—
215#121—2742no—
216#121—1372no—
217C2×C4⋊C4—1371no—
218#103—1372no—
219#102—1372no—
220#102—1372no—
221#102—1372no—
222C22⋊C8—1372no—
223C22⋊C8—1372no—
224C24—21no—
225C24—2742no—
226#41—22no—
227#41—22no—
228#41—2748no—
229#41—2744no—
230#41—2744no—
231#41—2742no—
232#41—2742no—
233#41—2742no—
234#41—1374no—
235C2×Q8—1371no—
236C2×Q8—1371no—
237C2×Q8C22×D13711yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—2748no—
242C2×D4—2744no—
243C2×D4—27416no—
244C2×D4—2742no—
245C2×D4—2741no—
246C2×D4—2744no—
247C2×D4—2742no—
248C2×D4—2742no—
249C2×D4—1372no—
250C2×D4—1371no—
251C2×D4—1371no—
252C2×D4C22⋊D13714yescomplemented
253C2×D4C22×D13711yescharacteristic
254C2×D4C22×D13712yes—
255C22×C4—21no—
256C22×C4—2742no—
257C22×C4—2741no—
258C22×C4—2741no—
259C22×C4—1372no—
260C22×C4C22×D13711yescharacteristic
261Q16—2742no—
262SD16—24no—
263SD16—2742no—
264SD16—2742no—
265D8—24no—
266D8—2742no—
267M16—22no—
268C2×C8—2742no—
269C2×C8—1371no—
270C2×C8—1371no—
271C2×C8C22×D13712yes—
272C4⋊C4—2742no—
273C4⋊C4—2741no—
274C4⋊C4—1372no—
275C22⋊C4—2742no—
276C22⋊C4—2742no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—2748no—
283C23—2741no—
284C23—10962no—
285C23—10962no—
286C23—1372no—
287C23D4×D13711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2742no—
291Q8—2741no—
292Q8—1372no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—2748no—
299D4—2748no—
300D4—2744no—
301D4—2744no—
302D4—2742no—
303D4—2741no—
304D4—10962no—
305D4—1372no—
306D4C2×C22⋊D13714yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2748no—
310C2×C4—2741no—
311C2×C4—2744no—
312C2×C4—2742no—
313C2×C4—2742no—
314C2×C4—2742no—
315C2×C4—10961no—
316C2×C4—1372no—
317C2×C4D4×D13711yescharacteristic
318C2×C4D4×D13712yes—
319C2×C4C2×C22⋊D13712yes—
320C2×C4C23×D13711yescharacteristic
321C8—22no—
322C8—2741no—
323C8—2741no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—2742no—
334C22—2742no—
335C22—2742no—
336C22—2742no—
337C22—5481no—
338C22—5481no—
339C22—10968no—
340C22#1798405612yes—
341C22C2×D4×D13711yescharacteristic
342C4—41no—
343C4—2742no—
344C4—2742no—
345C4—5482no—
346C4#1798405612yes—
347C4C2×D4×D13711yescharacteristic
348C4C22×C22⋊D13711yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—5482no—
353C2—5482no—
354C2C2×#1798405611yescharacteristic
355C2#2257820711yescharacteristic
356C2#2257821311yescharacteristic
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#3012
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×#17984133C2semidirect
2. C2×#17984108C2semidirect
3. #22577981C2semidirect
4. #22577562C2semidirect
5. #22577552C2semidirect
6. #22577545C2semidirect
7. #22577535C2semidirect
8. #22577405C2semidirect
9. C2×#17984009C2semidirect
10. #22577371C2semidirect
11. C2×AGL(1,ℤ/8)×C137C2semidirect
22. C2×#14555500C22semidirect
23. C2×#14555498C22semidirect
24. C2×#14555497C22semidirect
25. C2×#14555495C22semidirect
26. C22×#11764040C22semidirect
28. #17984064C22semidirect
31. #17984059C22semidirect
32. C2×C4⋊Dic137C22semidirect
33. #17984013C22semidirect
34. #17984010C22semidirect
35. #17984009C22semidirect
36. #17984009C22semidirect
37. #17984000C22semidirect
38. #17984000C22semidirect
39. C2×#41×C137C22semidirect
42. C2×SD16×C137C22semidirect
43. C2×D8×C137C22semidirect
44. C2×M16×C137C22semidirect
49. #14555510D4semidirect
89. C2×D4×C137D4semidirect
178. C137#3012semidirect
181. C2×AGL(1,ℤ/8)D137semidirect
197. C2×#41C2×D137semidirect
207. C2×SD16C2×D137semidirect
209. C2×D8C2×D137semidirect
210. C2×M16C2×D137semidirect
252. C2×D4C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841332
C2×#179841082
#225779812
#225775622
#225775522
#225775452
#225775352
#225774052
C2×#179840092
#225773712
C2×AGL(1,ℤ/8)×C1372
#3012137