Finite Group Databasebeta

← #31790098

Subgroups — #31790098

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×#17984114C211yesmaximal, characteristic, complemented
3C2×#17984110C211yesmaximal, characteristic, complemented
4#22577984C211yesmaximal, characteristic, complemented
5#22577566C212yesmaximal, complemented
6#22577555C212yesmaximal, complemented
7#22577549C212yesmaximal, complemented
8#22577538C212yesmaximal, complemented
9#22577402C211yesmaximal, characteristic, complemented
10#22577369C211yesmaximal, characteristic, complemented
11C2×#102×C137C211yesmaximal, characteristic, complemented
12#17984133—24no—
13#17984133—24no—
14C2×#14555510C2211yescharacteristic
15C2×#14555510C2211yescharacteristic
16C2×#14555510C2211yescharacteristic, complemented
17C2×#14555509—21no—
18C2×#14555509C2212yes—
19C22×C22⋊D137—22no—
20C2×D4×D137—21no—
21C2×D4×D137C2212yes—
22#17984114—24no—
23#17984110—24no—
24C2×Dic548C2211yescharacteristic, complemented
25C2×#14555499C2211yescharacteristic, complemented
26C2×#14555499C2211yescharacteristic, complemented
27C2×#14555498C2212yescomplemented
28C2×D1096C2211yescharacteristic, complemented
29C2×#14555495C2212yescomplemented
30C2×#14555494C2211yescharacteristic, complemented
31#17984068—21no—
32#17984068C2212yescomplemented
33#17984063—21no—
34#17984063C2212yescomplemented
35#17984062—22no—
36#17984011C2212yescomplemented
37#17984008C2212yescomplemented
38#17984000C2212yescomplemented
39#17984000C2212yescomplemented
40C22×D4×C137C2211yescharacteristic
41C22×C8×C137C2211yescharacteristic, complemented
42C2×C4⋊C4×C137C2211yescharacteristic, complemented
43#102×C137C2212yescomplemented
44#102×C137C2212yescomplemented
45C23×D137—22no—
46#14555510—22no—
47#14555510—22no—
48#14555510—22no—
49#14555510—22no—
50#14555510—22no—
51#14555510D414yescomplemented
52#14555509—28no—
53#14555509—24no—
54#14555509—24no—
55C2×Dic274D411yescharacteristic
56C2×Dic274C2311yescharacteristic
57C2×Dic274C2311yescharacteristic
58C2×D548—22no—
59C2×D548D411yescharacteristic
60C2×D548C2311yescharacteristic
61C2×D548C2311yescharacteristic
62C2×C22⋊D137—216no—
63C2×C22⋊D137—22no—
64C2×C22⋊D137—22no—
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137D412yes—
68D4×D137—28no—
69D4×D137—24no—
70D4×D137—24no—
71C22×Dic137—22no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137C2312yes—
75Dic548—22no—
76#14555499—22no—
77#14555499—22no—
78#14555498—24no—
79D1096—22no—
80#14555495—24no—
81#14555494—22no—
82C2×#11764040C2312yes—
83C4⋊Dic137—22no—
84#14555487—22no—
85#14555487—22no—
86C24×C137—21no—
87C2×D4×C137—21no—
88C2×D4×C137—22no—
89C2×D4×C137D414yescomplemented
90C2×D4×C137C2312yes—
91C22×C4×C137—21no—
92C22×C4×C137C2311yescharacteristic
93C2×C8×C137—22no—
94C2×C8×C137C2311yescharacteristic
95C2×C8×C137C2311yescharacteristic
96C4⋊C4×C137—21no—
97C4⋊C4×C137C2312yes—
98C22×D137—21no—
99C22×D137—28no—
100C22×D137—21no—
101C22×D137—82no—
102C22×D137—82no—
103C22×D137C2×D412yes—
104Dic274—22no—
105Dic274—21no—
106Dic274—21no—
107Dic274—21no—
108Dic274C2×D412yes—
109D548—22no—
110D548—21no—
111D548—21no—
112D548—21no—
113D548—82no—
114D548C2×D412yes—
115C22⋊D137—28no—
116C22⋊D137—28no—
117C22⋊D137—24no—
118C22⋊D137—24no—
119C22⋊D137—42no—
120C2×Dic137—28no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137C2×D412yes—
124C4×D137—24no—
125C4×D137—22no—
126C4×D137—22no—
127C4×D137—42no—
128#11764040—22no—
129C23×C137—22no—
130C23×C137—24no—
131C23×C137—81no—
132C23×C137—81no—
133C23×C137C2×D411yescharacteristic
134D4×C137—24no—
135D4×C137—22no—
136D4×C137C2×D414yes—
137C2×C4×C137—22no—
138C2×C4×C137—81no—
139C2×C4×C137C2×D411yescharacteristic
140C2×C4×C137C2×D412yes—
141C2×C4×C137C2×D412yes—
142C2×C4×C137C2411yescharacteristic
143C8×C137—21no—
144C8×C137—21no—
145C22×C137—21no—
146C22×C137—21no—
147C22×C137—24no—
148C22×C137—42no—
149C22×C137—42no—
150C22×C137—84no—
151C22×C137#12012yes—
152C22×C137C22×D411yescharacteristic
153C2×D137—22no—
154C2×D137—22no—
155C2×D137—22no—
156C2×D137—22no—
157C2×D137—41no—
158C2×D137—41no—
159C2×D137—41no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137—88no—
163C4×C137—42no—
164C4×C137#12012yes—
165C4×C137C22×D411yescharacteristic
166C4×C137C22×D411yescharacteristic
167Dic137—22no—
168Dic137—22no—
169Dic137—41no—
170C2×C137—22no—
171C2×C137—44no—
172C2×C137C2×#12011yescharacteristic
173C2×C137#57611yescharacteristic
174C2×C137#57711yescharacteristic
175D137—42no—
176D137—42no—
177D137—81no—
178C137#301211yescharacteristic, complemented, Sylow
179#3012—1371nomaximal, Sylow
180C2×H(5,𝔽2)—1371no—
181C2×AGL(1,ℤ/8)—1371no—
182C2×#135—1371no—
183#537—1371no—
184#452—1372no—
185#450—1372no—
186#449—1372no—
187#447—1372no—
188#417—1371no—
189C2×#102D13711yescharacteristic, complemented
190#408—1371no—
191H(5,𝔽2)—2744no—
192H(5,𝔽2)—2744no—
193C2×#41—2741no—
194C2×#41—1372no—
195C2×#41—1371no—
196C2×#41—1371no—
197C2×#41—1371no—
198C22×D4—2741no—
199C22×D4—2742no—
200C22×D4—1372no—
201C22×D4C2×D13711yescharacteristic
202AGL(1,ℤ/8)—2744no—
203#135—2744no—
204C2×Q16—1371no—
205C2×SD16—1372no—
206C2×SD16—1371no—
207C2×SD16—1371no—
208C2×D8—1372no—
209C2×D8—1371no—
210C2×M16—1371no—
211C22×C8C2×D13711yescharacteristic, complemented
212C22⋊Q8—2741no—
213C22⋊Q8—1372no—
214#121—2741no—
215#121—2742no—
216#121—1372no—
217C2×C4⋊C4C2×D13711yescharacteristic, complemented
218#103—1372no—
219#102—1372no—
220#102C2×D13712yescomplemented
221#102C2×D13712yescomplemented
222C22⋊C8—1372no—
223C22⋊C8—1372no—
224C24—21no—
225C24—2742no—
226#41—2748no—
227#41—2744no—
228#41—2744no—
229#41—2742no—
230#41—2742no—
231#41—2742no—
232#41—2742no—
233#41—2742no—
234#41—1374no—
235C2×Q8—1371no—
236C2×Q8—1371no—
237C2×Q8—1371no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—2742no—
241C2×D4—2748no—
242C2×D4—2744no—
243C2×D4—27416no—
244C2×D4—2742no—
245C2×D4—2741no—
246C2×D4—2744no—
247C2×D4—2741no—
248C2×D4—2742no—
249C2×D4—1372no—
250C2×D4—1371no—
251C2×D4—1371no—
252C2×D4—1371no—
253C2×D4D54814yescomplemented
254C2×D4C22×D13712yes—
255C22×C4—21no—
256C22×C4—2742no—
257C22×C4—2741no—
258C22×C4—2741no—
259C22×C4—1372no—
260C22×C4C22×D13711yescharacteristic
261Q16—2742no—
262SD16—2742no—
263SD16—2742no—
264SD16—2744no—
265D8—2742no—
266D8—2744no—
267M16—2742no—
268C2×C8—22no—
269C2×C8—1372no—
270C2×C8C22×D13711yescharacteristic
271C2×C8C22×D13711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2742no—
274C4⋊C4C22×D13712yes—
275C22⋊C4—2742no—
276C22⋊C4—2742no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—2748no—
282C23—2741no—
283C23—2741no—
284C23—10962no—
285C23—10962no—
286C23—1372no—
287C23D4×D13711yescharacteristic
288Q8—2742no—
289Q8—2741no—
290Q8—2741no—
291Q8—2741no—
292Q8—1372no—
293D4—24no—
294D4—22no—
295D4—2748no—
296D4—2748no—
297D4—2744no—
298D4—2744no—
299D4—2742no—
300D4—2741no—
301D4—2741no—
302D4—2741no—
303D4—5482no—
304D4—10962no—
305D4—1372no—
306D4C2×D54814yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—2748no—
310C2×C4—2741no—
311C2×C4—2744no—
312C2×C4—2742no—
313C2×C4—2742no—
314C2×C4—2741no—
315C2×C4—5482no—
316C2×C4—1372no—
317C2×C4D4×D13711yescharacteristic
318C2×C4D4×D13712yes—
319C2×C4C2×D54812yes—
320C2×C4C23×D13711yescharacteristic
321C8—21no—
322C8—21no—
323C8—2742no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—2742no—
331C22—2742no—
332C22—2742no—
333C22—2742no—
334C22—5481no—
335C22—5481no—
336C22—5481no—
337C22—5481no—
338C22—5481no—
339C22—10968no—
340C22#1798405812yes—
341C22C2×D4×D13711yescharacteristic
342C4—42no—
343C4—2742no—
344C4—2742no—
345C4—5481no—
346C4#1798405812yes—
347C4C2×D4×D13711yescharacteristic
348C4C22×D54811yescharacteristic
349C2—22no—
350C2—44no—
351C2—5482no—
352C2—5482no—
353C2—10961no—
354C2C2×#1798405811yescharacteristic
355C2#2257820411yescharacteristic
356C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#102×C137C2
socleC22×C137C22×D4
hypercenterC2×#102D137
hypocenterC137#3012
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#17984133C2semidirect
2. C2×#17984114C2semidirect
3. C2×#17984110C2semidirect
4. #22577984C2semidirect
5. #22577566C2semidirect
6. #22577555C2semidirect
7. #22577549C2semidirect
8. #22577538C2semidirect
9. #22577402C2semidirect
10. #22577369C2semidirect
11. C2×#102×C137C2semidirect
16. C2×#14555510C22semidirect
24. C2×Dic548C22semidirect
25. C2×#14555499C22semidirect
26. C2×#14555499C22semidirect
27. C2×#14555498C22semidirect
28. C2×D1096C22semidirect
29. C2×#14555495C22semidirect
30. C2×#14555494C22semidirect
32. #17984068C22semidirect
34. #17984063C22semidirect
36. #17984011C22semidirect
37. #17984008C22semidirect
38. #17984000C22semidirect
39. #17984000C22semidirect
41. C22×C8×C137C22semidirect
42. C2×C4⋊C4×C137C22semidirect
43. #102×C137C22semidirect
44. #102×C137C22semidirect
51. #14555510D4semidirect
89. C2×D4×C137D4semidirect
178. C137#3012semidirect
189. C2×#102D137semidirect
211. C22×C8C2×D137semidirect
217. C2×C4⋊C4C2×D137semidirect
220. #102C2×D137semidirect
221. #102C2×D137semidirect
253. C2×D4D548semidirect

Maximal subgroups

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

subgroupindex
C2×#179841332
C2×#179841142
C2×#179841102
#225779842
#225775662
#225775552
#225775492
#225775382
#225774022
#225773692
C2×#102×C1372
#3012137