Finite Group Databasebeta

← #31790411

Subgroups — #31790411

Order 17536.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#17984113C211yesmaximal, characteristic, complemented
2C2×M16×D137C211yesmaximal, characteristic, complemented
3#22577910C211yesmaximal, characteristic, complemented
4C2×#17984063C211yesmaximal, characteristic, complemented
5#22577655C212yesmaximal, complemented
6#22577652C212yesmaximal, complemented
7#22577638C212yesmaximal, complemented
8#22577635C212yesmaximal, complemented
9#22577443C211yesmaximal, characteristic, complemented
10#22577409C211yesmaximal, characteristic, complemented
11#417×C137C211yesmaximal, characteristic, complemented
12C22×D548—21no—
13C22×D548C2211yescharacteristic
14C22×C4×D137C2211yescharacteristic
15#17984113—24no—
16C2×#14555497C2212yescomplemented
17C2×#14555495C2212yescomplemented
18C2×#14555494C2211yescharacteristic
19C2×#14555493C2212yescomplemented
20M16×D137—24no—
21C2×C8×D137C2212yescomplemented
22#17984069—21no—
23#17984064C2211yescharacteristic, complemented
24#17984064C2211yescharacteristic, complemented
25#17984063—21no—
26#17984063—21no—
27#17984063C2212yes—
28#17984063C2212yes—
29#17984059C2211yescharacteristic, complemented
30#17984059C2211yescharacteristic, complemented
31#17984056—21no—
32C4×C22⋊D137—21no—
33C4×C22⋊D137—21no—
34#17984047C2211yescharacteristic
35C2×#14555487—21no—
36#17984021C2212yescomplemented
37#17984019C2212yescomplemented
38#17984010C2212yescomplemented
39#17984010C2212yescomplemented
40C2×#41×C137C2211yescharacteristic
41C2×M16×C137C2211yescharacteristic
42C2×C4⋊C4×C137C2211yescharacteristic
43#103×C137C2212yescomplemented
44#102×C137C2212yescomplemented
45C23×D137—21no—
46C23×D137D411yescharacteristic
47C2×D548—22no—
48C2×D548—22no—
49C2×D548—22no—
50C2×D548—21no—
51C2×D548—22no—
52C2×D548—44no—
53C2×D548C2312yes—
54C2×C22⋊D137—21no—
55C2×C22⋊D137—21no—
56C22×Dic137D411yescharacteristic
57C2×C4×D137—22no—
58C2×C4×D137—22no—
59C2×C4×D137D411yescharacteristic
60C2×C4×D137D411yescharacteristic
61C2×C4×D137C2311yescharacteristic
62C2×C4×D137C2311yescharacteristic
63#14555497—24no—
64#14555495—24no—
65#14555494D414yescomplemented
66#14555493—24no—
67C2×#11764040C2312yes—
68C8×D137—24no—
69C4⋊Dic137C2311yescharacteristic
70C4⋊Dic137C2311yescharacteristic
71#14555489—21no—
72#14555489—21no—
73C22⋊Dic137—21no—
74C22⋊Dic137—21no—
75C22⋊Dic137—21no—
76#14555487—21no—
77#14555487—21no—
78#14555487—22no—
79#14555487—22no—
80C4×Dic137—21no—
81#41×C137—22no—
82#41×C137—22no—
83C2×Q8×C137C2311yescharacteristic
84C2×D4×C137—21no—
85C2×D4×C137C2311yescharacteristic
86C22×C4×C137—21no—
87C22×C4×C137—21no—
88C22×C4×C137C2311yescharacteristic
89M16×C137—22no—
90C2×C8×C137C2312yes—
91C4⋊C4×C137—21no—
92C4⋊C4×C137C2312yes—
93C22×D137—22no—
94C22×D137—22no—
95C22×D137—22no—
96C22×D137—41no—
97C22×D137—42no—
98C22×D137—42no—
99C22×D137—42no—
100C22×D137C2×D411yescharacteristic
101C22×D137C2×D411yescharacteristic
102D548—22no—
103D548—22no—
104D548—44no—
105D548—44no—
106D548—42no—
107C22⋊D137—42no—
108C22⋊D137—42no—
109C2×Dic137—22no—
110C2×Dic137—21no—
111C2×Dic137—21no—
112C2×Dic137C2×D411yescharacteristic
113C2×Dic137C2×D411yescharacteristic
114C4×D137—22no—
115C4×D137—22no—
116C4×D137—21no—
117C4×D137—21no—
118C4×D137—21no—
119C4×D137—21no—
120#11764040C2×D414yescomplemented
121C23×C137—21no—
122C23×C137C2×D411yescharacteristic
123Q8×C137—21no—
124Q8×C137—21no—
125D4×C137—21no—
126D4×C137—21no—
127D4×C137—42no—
128C2×C4×C137—22no—
129C2×C4×C137—21no—
130C2×C4×C137—42no—
131C2×C4×C137—42no—
132C2×C4×C137C2×D411yescharacteristic
133C2×C4×C137C2×D412yes—
134C2×C4×C137#4112yes—
135C2×C4×C137C2411yescharacteristic
136C8×C137—22no—
137C22×C137—21no—
138C22×C137—21no—
139C22×C137—41no—
140C22×C137—41no—
141C22×C137—41no—
142C22×C137#12112yes—
143C22×C137C22×D411yescharacteristic
144C2×D137—21no—
145C2×D137—21no—
146C2×D137—21no—
147C2×D137—21no—
148C2×D137—21no—
149C2×D137—44no—
150C2×D137—42no—
151C2×D137—42no—
152C2×D137—42no—
153C2×D137—84no—
154C2×D137AGL(1,ℤ/8)12yescomplemented
155C4×C137—41no—
156C4×C137—42no—
157C4×C137#12112yes—
158C4×C137C22×D411yescharacteristic
159C4×C137C2×#4111yescharacteristic
160Dic137—21no—
161Dic137—21no—
162Dic137—41no—
163Dic137—41no—
164C2×C137—22no—
165C2×C137—81no—
166C2×C137C2×#12111yescharacteristic
167C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C137#57511yescharacteristic
169D137—22no—
170D137—41no—
171D137—82no—
172C137#305511yescharacteristic, complemented, Sylow
173#3055—1371nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1371no—
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#417D13711yescharacteristic, complemented
185C2×#41C2×D13711yescharacteristic
186C22×D4—2741no—
187C22×D4—1371no—
188C23×C4—1371no—
189AGL(1,ℤ/8)—2744no—
190C2×SD16—1372no—
191C2×D8—1372no—
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—1371no—
203#121—1371no—
204#121—1372no—
205#121—1372no—
206#120—2741no—
207C4×D4—2741no—
208C4×D4—2741no—
209#117—1371no—
210C2×C4⋊C4C2×D13711yescharacteristic
211C2×C22⋊C4—2741no—
212#107—1372no—
213#106—1372no—
214#103C2×D13712yescomplemented
215#102—1372no—
216#102—1372no—
217#102C2×D13712yescomplemented
218C24—2741no—
219C24—1371no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D13711yescharacteristic
223C2×D4—21no—
224C2×D4—2742no—
225C2×D4—2742no—
226C2×D4—2741no—
227C2×D4—2741no—
228C2×D4—2742no—
229C2×D4—2741no—
230C2×D4—2742no—
231C2×D4—5484no—
232C2×D4—1372no—
233C2×D4C22×D13711yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—2742no—
237C22×C4—2742no—
238C22×C4—1371no—
239C22×C4—1371no—
240C22×C4—1371no—
241C22×C4—1371no—
242C22×C4—1371no—
243C22×C4C22×D13711yescharacteristic
244SD16—2744no—
245D8—2744no—
246M16—22no—
247M16—2744no—
248M16—1374no—
249C2×C8—2744no—
250C2×C8—1372no—
251C2×C8C22×D13712yes—
252C4⋊C4—21no—
253C4⋊C4—2741no—
254C4⋊C4—2741no—
255C4⋊C4—1371no—
256C4⋊C4—1371no—
257C4⋊C4C22×D13712yes—
258C22⋊C4—2741no—
259C22⋊C4—2741no—
260C22⋊C4—2742no—
261C22⋊C4—2742no—
262C22⋊C4—2741no—
263C22⋊C4—2741no—
264C22⋊C4—2741no—
265C42—2741no—
266C23—21no—
267C23—2742no—
268C23—2742no—
269C23—2742no—
270C23—5481no—
271C23—5482no—
272C23—5482no—
273C23—5482no—
274C23—1371no—
275C23—1371no—
276C23D4×D13711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—2742no—
283D4—2742no—
284D4—5484no—
285D4—5484no—
286D4—5482no—
287D4—5482no—
288D4—5482no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—2742no—
294C2×C4—2742no—
295C2×C4—2742no—
296C2×C4—2741no—
297C2×C4—2741no—
298C2×C4—2741no—
299C2×C4—2741no—
300C2×C4—2741no—
301C2×C4—2741no—
302C2×C4—1371no—
303C2×C4—1371no—
304C2×C4D4×D13711yescharacteristic
305C2×C4D4×D13712yes—
306C2×C4#1455551012yes—
307C2×C4C23×D13711yescharacteristic
308C8—22no—
309C8—1374no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—2741no—
316C22—2741no—
317C22—2741no—
318C22—2741no—
319C22—2741no—
320C22—5484no—
321C22—5482no—
322C22—5482no—
323C22—5482no—
324C22—10964no—
325C22—1372no—
326C22#1798406112yes—
327C22C2×D4×D13711yescharacteristic
328C4—42no—
329C4—41no—
330C4—2741no—
331C4—2741no—
332C4—5481no—
333C4—5481no—
334C4#1798406112yes—
335C4C2×D4×D13711yescharacteristic
336C4C2×#1455551011yescharacteristic
337C2—22no—
338C2—81no—
339C2—2742no—
340C2—5481no—
341C2—10962no—
342C2C2×#1798406111yescharacteristic
343C2AGL(1,ℤ/8)×D13711yescharacteristic
344C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#417×C137C2
socleC22×C137C22×D4
hypercenter#417D137
hypocenterC137#3055
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#17984113C2semidirect
2. C2×M16×D137C2semidirect
3. #22577910C2semidirect
4. C2×#17984063C2semidirect
5. #22577655C2semidirect
6. #22577652C2semidirect
7. #22577638C2semidirect
8. #22577635C2semidirect
9. #22577443C2semidirect
10. #22577409C2semidirect
11. #417×C137C2semidirect
16. C2×#14555497C22semidirect
17. C2×#14555495C22semidirect
19. C2×#14555493C22semidirect
21. C2×C8×D137C22semidirect
23. #17984064C22semidirect
24. #17984064C22semidirect
29. #17984059C22semidirect
30. #17984059C22semidirect
36. #17984021C22semidirect
37. #17984019C22semidirect
38. #17984010C22semidirect
39. #17984010C22semidirect
43. #103×C137C22semidirect
44. #102×C137C22semidirect
65. #14555494D4semidirect
120. #11764040C2×D4semidirect
154. C2×D137AGL(1,ℤ/8)semidirect
172. C137#3055semidirect
184. #417D137semidirect
214. #103C2×D137semidirect
217. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841132
C2×M16×D1372
#225779102
C2×#179840632
#225776552
#225776522
#225776382
#225776352
#225774432
#225774092
#417×C1372
#3055137