Finite Group Databasebeta

← #31790788

Subgroups — #31790788

Order 17536.

492 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 127 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
1#22577951C211yesmaximal, characteristic, complemented
2C4×D4×D137C211yesmaximal, characteristic, complemented
3#22577715C212yesmaximal, complemented
4#22577684C211yesmaximal, characteristic, complemented
5#22577683C211yesmaximal, characteristic, complemented
6#22577595C211yesmaximal, characteristic, complemented
7#22577535C212yesmaximal, complemented
8D8×Dic137C212yesmaximal, complemented
9#22577483C212yesmaximal, complemented
10#22577455C211yesmaximal, characteristic, complemented
11#459×C137C211yesmaximal, characteristic, complemented
12C2×#14555509—21no—
13C2×D4×D137C2211yescharacteristic
14C22×C4×D137—21no—
15C2×#14555495C2212yes—
16#17984088C2211yescharacteristic, complemented
17#17984075—21no—
18#17984059—21no—
19#17984059C2212yes—
20C4×D548C2211yescharacteristic
21D4×Dic137—21no—
22D4×Dic137C2211yescharacteristic
23D4×Dic137C2212yescomplemented
24#17984050C2211yescharacteristic
25C4×C22⋊D137—21no—
26#17984048—21no—
27C4⋊C4×D137C2211yescharacteristic
28C22⋊C4×D137—21no—
29C42×D137C2211yescharacteristic
30#17984022C2212yescomplemented
31#17984021C2212yescomplemented
32C4⋊#11764040C2211yescharacteristic, complemented
33#17984010C2212yes—
34#17984009C2212yes—
35#17984009C2212yescomplemented
36#17984000C2212yes—
37C8×Dic137C2212yescomplemented
38C2×D8×C137C2212yes—
39#127×C137C2211yescharacteristic, complemented
40C4×D4×C137C2211yescharacteristic
41C4⋊C8×C137C2211yescharacteristic, complemented
42#102×C137C2212yescomplemented
43C23×D137—21no—
44#14555509—44no—
45C2×Dic274—21no—
46C2×D548C2311yescharacteristic
47C2×C22⋊D137—22no—
48C2×C22⋊D137—21no—
49D4×D137—22no—
50D4×D137—22no—
51C22×Dic137—21no—
52C22×Dic137—22no—
53C2×C4×D137—21no—
54C2×C4×D137—21no—
55C2×C4×D137—21no—
56C2×C4×D137—44no—
57C2×C4×D137D411yescharacteristic
58C2×C4×D137D411yescharacteristic
59C2×C4×D137C2311yescharacteristic
60#14555495—24no—
61C2×#11764040C2312yes—
62C4⋊Dic137—21no—
63C4⋊Dic137C2312yes—
64C4⋊Dic137C2311yescharacteristic
65#14555489—21no—
66C22⋊Dic137—21no—
67C22⋊Dic137—22no—
68C22⋊Dic137—22no—
69#14555487—21no—
70#14555487—21no—
71C4×Dic137D411yescharacteristic
72C4×Dic137D411yescharacteristic
73C4×Dic137C2311yescharacteristic
74C2×D4×C137—21no—
75C2×D4×C137—21no—
76C2×D4×C137C2312yes—
77C2×D4×C137C2311yescharacteristic
78C22×C4×C137—21no—
79D8×C137—24no—
80C2×C8×C137C2312yes—
81C4⋊C4×C137C2311yescharacteristic
82C22⋊C4×C137—21no—
83C42×C137C2311yescharacteristic
84C22×D137—24no—
85C22×D137—21no—
86C22×D137—42no—
87C22×D137—42no—
88C22×D137C2×D411yescharacteristic
89Dic274—42no—
90D548—21no—
91D548#4112yescomplemented
92C22⋊D137—24no—
93C22⋊D137—44no—
94C2×Dic137—21no—
95C2×Dic137—22no—
96C2×Dic137—42no—
97C2×Dic137—44no—
98C2×Dic137C2×D411yescharacteristic
99C2×Dic137C2×D411yescharacteristic
100C2×Dic137C2×D411yescharacteristic
101C4×D137—21no—
102C4×D137—21no—
103C4×D137—41no—
104C4×D137—42no—
105C4×D137—42no—
106C4×D137—42no—
107C4×D137D814yescomplemented
108#11764040—22no—
109C23×C137—21no—
110C23×C137—22no—
111D4×C137—21no—
112D4×C137—22no—
113D4×C137—22no—
114D4×C137—42no—
115D4×C137—42no—
116D4×C137#4112yescomplemented
117C2×C4×C137—21no—
118C2×C4×C137—42no—
119C2×C4×C137C2×D411yescharacteristic
120C2×C4×C137C2×D411yescharacteristic
121C2×C4×C137C2411yescharacteristic
122C8×C137—22no—
123C22×C137—22no—
124C22×C137—41no—
125C22×C137—41no—
126C22×C137—42no—
127C22×C137—42no—
128C22×C137—42no—
129C22×C137C22×D411yescharacteristic
130C2×D137—21no—
131C2×D137—21no—
132C2×D137—22no—
133C2×D137—48no—
134C2×D137—41no—
135C2×D137—41no—
136C2×D137C2×D812yes—
137C4×C137—21no—
138C4×C137—41no—
139C4×C137C2×D812yes—
140C4×C137C2×#4111yescharacteristic
141C4×C137C2×#4111yescharacteristic
142Dic137—21no—
143Dic137—21no—
144Dic137—41no—
145Dic137—42no—
146Dic137C2×D812yes—
147C2×C137—42no—
148C2×C137—82no—
149C2×C137#52511yescharacteristic
150C2×C137C22×D811yescharacteristic
151C2×C137#57511yescharacteristic
152D137—22no—
153D137—42no—
154C137#310011yescharacteristic, complemented, Sylow
155#3100—1371nomaximal, Sylow
156#532—1371no—
157C2×C4×D4—1371no—
158#480—1372no—
159#474—1371no—
160#474—1371no—
161#459—1371no—
162#459D13711yescharacteristic, complemented
163#447—1372no—
164C4×D8—1372no—
165C4×D8—1372no—
166#431—1371no—
167C2×#41—2741no—
168C22×D4—1371no—
169C23×C4—2741no—
170C2×D8—1372no—
171C2×D8C2×D13712yes—
172C4⋊Q8—1371no—
173#127C2×D13711yescharacteristic, complemented
174#124—2741no—
175#121—2741no—
176#121—1372no—
177C4×D4—2741no—
178C4×D4—2741no—
179C4×D4—2741no—
180C4×D4—1371no—
181C4×D4—1372no—
182C4×D4—1371no—
183C4×D4—1371no—
184C4×D4C2×D13711yescharacteristic
185C2×C4⋊C4—1371no—
186C2×C22⋊C4—2741no—
187C2×C42—1371no—
188#107—1372no—
189#107—1372no—
190C4⋊C8—1371no—
191C4⋊C8C2×D13711yescharacteristic, complemented
192#102—1372no—
193#102—1372no—
194#102—1372no—
195#102C2×D13712yescomplemented
196C22⋊C8—1372no—
197C4×C8—1372no—
198C24—2741no—
199#41—5484no—
200C2×Q8—2741no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—2742no—
204C2×D4—2742no—
205C2×D4—2742no—
206C2×D4—2741no—
207C2×D4—1371no—
208C2×D4C22×D13712yes—
209C2×D4C22×D13711yescharacteristic
210C22×C4—21no—
211C22×C4—2741no—
212C22×C4—2741no—
213C22×C4—2741no—
214C22×C4—2742no—
215C22×C4—2741no—
216C22×C4—5484no—
217C22×C4—1371no—
218C22×C4—1371no—
219C22×C4—1371no—
220D8—24no—
221D8—2744no—
222C2×C8—1372no—
223C2×C8C22×D13712yes—
224C4⋊C4—2741no—
225C4⋊C4—2741no—
226C4⋊C4—1371no—
227C4⋊C4—1372no—
228C4⋊C4C22×D13711yescharacteristic
229C22⋊C4—21no—
230C22⋊C4—2741no—
231C22⋊C4—2741no—
232C22⋊C4—2742no—
233C22⋊C4—2742no—
234C22⋊C4—2741no—
235C42—1371no—
236C42—1371no—
237C42—1371no—
238C42C22×D13711yescharacteristic
239C23—21no—
240C23—22no—
241C23—2744no—
242C23—2741no—
243C23—5482no—
244C23—5482no—
245C23—1371no—
246Q8—5482no—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—2744no—
253D4—2741no—
254D4—5484no—
255D4—1372no—
256D4#1455550912yescomplemented
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—2741no—
260C2×C4—2741no—
261C2×C4—2741no—
262C2×C4—2742no—
263C2×C4—5481no—
264C2×C4—5482no—
265C2×C4—5482no—
266C2×C4—5482no—
267C2×C4—5484no—
268C2×C4—5482no—
269C2×C4—1371no—
270C2×C4—1371no—
271C2×C4—1371no—
272C2×C4—1374no—
273C2×C4D4×D13711yescharacteristic
274C2×C4D4×D13711yescharacteristic
275C2×C4C23×D13711yescharacteristic
276C8—22no—
277C8—2742no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—2741no—
285C22—2741no—
286C22—2742no—
287C22—5488no—
288C22—5481no—
289C22—5481no—
290C22—1372no—
291C22C2×D4×D13711yescharacteristic
292C4—21no—
293C4—41no—
294C4—2741no—
295C4—2741no—
296C4—5481no—
297C4—5482no—
298C4—1372no—
299C4D8×D13712yes—
300C4#41×D13711yescharacteristic
301C4C2×#1455550911yescharacteristic
302C2—42no—
303C2—82no—
304C2—2742no—
305C2—5482no—
306C2#2257791311yescharacteristic
307C2C2×D8×D13711yescharacteristic
308C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#459×C137C2
socleC22×C137C22×D4
hypercenter#459D137
hypocenterC137#3100
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22577951C2semidirect
2. C4×D4×D137C2semidirect
3. #22577715C2semidirect
4. #22577684C2semidirect
5. #22577683C2semidirect
6. #22577595C2semidirect
7. #22577535C2semidirect
8. D8×Dic137C2semidirect
9. #22577483C2semidirect
10. #22577455C2semidirect
11. #459×C137C2semidirect
16. #17984088C22semidirect
23. D4×Dic137C22semidirect
30. #17984022C22semidirect
31. #17984021C22semidirect
32. C4⋊#11764040C22semidirect
35. #17984009C22semidirect
37. C8×Dic137C22semidirect
39. #127×C137C22semidirect
41. C4⋊C8×C137C22semidirect
42. #102×C137C22semidirect
91. D548#41semidirect
107. C4×D137D8semidirect
116. D4×C137#41semidirect
154. C137#3100semidirect
162. #459D137semidirect
173. #127C2×D137semidirect
191. C4⋊C8C2×D137semidirect
195. #102C2×D137semidirect
256. D4#14555509semidirect

Maximal subgroups

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

subgroupindex
#225779512
C4×D4×D1372
#225777152
#225776842
#225776832
#225775952
#225775352
D8×Dic1372
#225774832
#225774552
#459×C1372
#3100137