Finite Group Databasebeta

← #31790791

Subgroups — #31790791

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#22577954C211yesmaximal, characteristic, complemented
2C4×D4×D137C211yesmaximal, characteristic, complemented
3#22577718C212yesmaximal, complemented
4#22577680C211yesmaximal, characteristic, complemented
5#22577599C211yesmaximal, characteristic, complemented
6#22577598C211yesmaximal, characteristic, complemented
7#22577538C212yesmaximal, complemented
8#22577485C212yesmaximal, complemented
9#22577483C212yesmaximal, complemented
10#22577455C211yesmaximal, characteristic, complemented
11#474×C137C211yesmaximal, characteristic, complemented
12C2×#14555511—21no—
13C2×D4×D137C2211yescharacteristic
14C22×C4×D137—21no—
15C2×D1096C2212yes—
16C2×#14555495C2212yes—
17#17984085C2211yescharacteristic, complemented
18#17984077—21no—
19#17984063—21no—
20#17984063C2212yes—
21C4×D548C2211yescharacteristic
22D4×Dic137C2211yescharacteristic
23#17984050—21no—
24#17984050C2211yescharacteristic
25#17984050C2212yescomplemented
26C4×C22⋊D137—21no—
27#17984048—21no—
28C4⋊C4×D137C2211yescharacteristic
29C22⋊C4×D137—21no—
30C42×D137C2211yescharacteristic
31#17984021C2212yescomplemented
32C4⋊#11764040C2211yescharacteristic, complemented
33#17984010C2212yescomplemented
34#17984008C2212yes—
35#17984008C2212yescomplemented
36#17984000C2212yes—
37C8×Dic137C2212yescomplemented
38C4⋊Q8×C137C2211yescharacteristic, complemented
39C4×D4×C137C2211yescharacteristic
40#107×C137C2212yescomplemented
41C4⋊C8×C137C2211yescharacteristic, complemented
42#102×C137C2212yes—
43C23×D137—21no—
44#14555511—44no—
45C2×D548—22no—
46C2×D548—21no—
47C2×D548—21no—
48C2×D548C2312yes—
49C2×D548C2311yescharacteristic
50C2×C22⋊D137—21no—
51D4×D137—22no—
52D4×D137—22no—
53C22×Dic137—21no—
54C2×C4×D137—21no—
55C2×C4×D137—21no—
56C2×C4×D137—22no—
57C2×C4×D137—21no—
58C2×C4×D137—44no—
59C2×C4×D137D411yescharacteristic
60C2×C4×D137D411yescharacteristic
61C2×C4×D137C2311yescharacteristic
62D1096—24no—
63#14555495—24no—
64C2×#11764040C2312yes—
65C4⋊Dic137C2311yescharacteristic
66#14555489—21no—
67C22⋊Dic137—21no—
68#14555487—21no—
69#14555487—22no—
70#14555487—22no—
71#14555487—21no—
72C4×Dic137D411yescharacteristic
73C4×Dic137D411yescharacteristic
74C4×Dic137C2311yescharacteristic
75C2×Q8×C137—21no—
76C2×D4×C137C2311yescharacteristic
77C22×C4×C137—21no—
78C2×C8×C137C2312yes—
79C4⋊C4×C137—21no—
80C4⋊C4×C137C2312yes—
81C4⋊C4×C137C2311yescharacteristic
82C22⋊C4×C137—21no—
83C42×C137C2311yescharacteristic
84C22×D137—24no—
85C22×D137—21no—
86C22×D137—22no—
87C22×D137—42no—
88C22×D137—42no—
89C22×D137C2×D411yescharacteristic
90D548—21no—
91D548—22no—
92D548—22no—
93D548—42no—
94D548—44no—
95D548—42no—
96D548#4112yescomplemented
97C22⋊D137—24no—
98C2×Dic137—21no—
99C2×Dic137—42no—
100C2×Dic137C2×D411yescharacteristic
101C2×Dic137C2×D411yescharacteristic
102C2×Dic137C2×D411yescharacteristic
103C4×D137—21no—
104C4×D137—21no—
105C4×D137—41no—
106C4×D137—42no—
107C4×D137—42no—
108C4×D137—42no—
109C4×D137—44no—
110C4×D137D814yescomplemented
111#11764040—22no—
112C23×C137—21no—
113Q8×C137—42no—
114D4×C137—21no—
115D4×C137#4112yescomplemented
116C2×C4×C137—21no—
117C2×C4×C137—22no—
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×C137C22×D411yescharacteristic
127C2×D137—21no—
128C2×D137—21no—
129C2×D137—22no—
130C2×D137—48no—
131C2×D137—41no—
132C2×D137—41no—
133C2×D137—42no—
134C2×D137—42no—
135C2×D137—42no—
136C2×D137C2×D812yes—
137C4×C137—21no—
138C4×C137—41no—
139C4×C137—42no—
140C4×C137C2×D812yes—
141C4×C137C2×#4111yescharacteristic
142C4×C137C2×#4111yescharacteristic
143Dic137—21no—
144Dic137—21no—
145Dic137—41no—
146Dic137C2×D812yes—
147C2×C137—42no—
148C2×C137#52511yescharacteristic
149C2×C137C22×D811yescharacteristic
150C2×C137#57511yescharacteristic
151D137—22no—
152D137—42no—
153D137—82no—
154C137#310011yescharacteristic, complemented, Sylow
155#3100—1371nomaximal, Sylow
156#532—1371no—
157C2×C4×D4—1371no—
158#480—1372no—
159#474—1371no—
160#474D13711yescharacteristic, complemented
161#459—1371no—
162#459—1371no—
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×D8—1372no—
172C4⋊Q8C2×D13711yescharacteristic, complemented
173#127—1371no—
174#124—2741no—
175#121—2741no—
176#121—1372no—
177C4×D4—2741no—
178C4×D4—2741no—
179C4×D4—2741no—
180C4×D4—1371no—
181C4×D4—1371no—
182C4×D4—1372no—
183C4×D4—1371no—
184C4×D4C2×D13711yescharacteristic
185C2×C4⋊C4—1371no—
186C2×C22⋊C4—2741no—
187C2×C42—1371no—
188#107—1372no—
189#107C2×D13712yescomplemented
190C4⋊C8—1371no—
191C4⋊C8C2×D13711yescharacteristic, complemented
192#102—1372no—
193#102—1372no—
194#102—1372no—
195#102C2×D13712yes—
196C22⋊C8—1372no—
197C4×C8—1372no—
198C24—2741no—
199#41—5484no—
200C2×Q8—21no—
201C2×D4—2742no—
202C2×D4—2742no—
203C2×D4—2742no—
204C2×D4—2741no—
205C2×D4—2741no—
206C2×D4—2741no—
207C2×D4—1372no—
208C2×D4—1371no—
209C2×D4C22×D13711yescharacteristic
210C22×C4—21no—
211C22×C4—2741no—
212C22×C4—2741no—
213C22×C4—2742no—
214C22×C4—2741no—
215C22×C4—2741no—
216C22×C4—5484no—
217C22×C4—1371no—
218C22×C4—1371no—
219C22×C4—1371no—
220D8—2744no—
221D8—2744no—
222C2×C8—1372no—
223C2×C8C22×D13712yes—
224C4⋊C4—21no—
225C4⋊C4—2741no—
226C4⋊C4—1371no—
227C4⋊C4C22×D13711yescharacteristic
228C4⋊C4C22×D13712yes—
229C22⋊C4—21no—
230C22⋊C4—2741no—
231C22⋊C4—2742no—
232C22⋊C4—2742no—
233C22⋊C4—2741no—
234C22⋊C4—2741no—
235C42—1371no—
236C42—1371no—
237C42—1371no—
238C42C22×D13711yescharacteristic
239C23—21no—
240C23—2744no—
241C23—2741no—
242C23—2742no—
243C23—5482no—
244C23—5482no—
245C23—1371no—
246Q8—42no—
247D4—21no—
248D4—2744no—
249D4—2742no—
250D4—2741no—
251D4—2742no—
252D4—5482no—
253D4—5484no—
254D4—5482no—
255D4—1372no—
256D4#1455551112yescomplemented
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—2741no—
261C2×C4—2741no—
262C2×C4—2741no—
263C2×C4—5481no—
264C2×C4—5482no—
265C2×C4—5482no—
266C2×C4—5484no—
267C2×C4—5482no—
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—2741no—
282C22—2741no—
283C22—2742no—
284C22—5488no—
285C22—5481no—
286C22—5481no—
287C22—5482no—
288C22—5482no—
289C22—5482no—
290C22—1372no—
291C22C2×D4×D13711yescharacteristic
292C4—21no—
293C4—41no—
294C4—42no—
295C4—2741no—
296C4—2741no—
297C4—5481no—
298C4—1372no—
299C4D8×D13712yes—
300C4#41×D13711yescharacteristic
301C4C2×#1455551111yescharacteristic
302C2—42no—
303C2—2742no—
304C2—5482no—
305C2—10962no—
306C2#2257791611yescharacteristic
307C2C2×D8×D13711yescharacteristic
308C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#474×C137C2
socleC22×C137C22×D4
hypercenter#474D137
hypocenterC137#3100
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #22577954C2semidirect
2. C4×D4×D137C2semidirect
3. #22577718C2semidirect
4. #22577680C2semidirect
5. #22577599C2semidirect
6. #22577598C2semidirect
7. #22577538C2semidirect
8. #22577485C2semidirect
9. #22577483C2semidirect
10. #22577455C2semidirect
11. #474×C137C2semidirect
17. #17984085C22semidirect
25. #17984050C22semidirect
31. #17984021C22semidirect
32. C4⋊#11764040C22semidirect
33. #17984010C22semidirect
35. #17984008C22semidirect
37. C8×Dic137C22semidirect
38. C4⋊Q8×C137C22semidirect
40. #107×C137C22semidirect
41. C4⋊C8×C137C22semidirect
96. D548#41semidirect
110. C4×D137D8semidirect
115. D4×C137#41semidirect
154. C137#3100semidirect
160. #474D137semidirect
172. C4⋊Q8C2×D137semidirect
189. #107C2×D137semidirect
191. C4⋊C8C2×D137semidirect
256. D4#14555511semidirect

Maximal subgroups

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

subgroupindex
#225779542
C4×D4×D1372
#225777182
#225776802
#225775992
#225775982
#225775382
#225774852
#225774832
#225774552
#474×C1372
#3100137