Finite Group Databasebeta

← #31790784

Subgroups — #31790784

Order 17536.

492 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 129 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#22577948C211yesmaximal, characteristic, complemented
2C2×D4×Dic137C211yesmaximal, characteristic, complemented
3#22577717C211yesmaximal, characteristic, complemented
4#22577714C211yesmaximal, characteristic, complemented
5#22577681C212yesmaximal, complemented
6#22577596C212yesmaximal, complemented
7#22577537C211yesmaximal, characteristic, complemented
8D8×Dic137C212yesmaximal, complemented
9#22577483C212yesmaximal, complemented
10#22577453C211yesmaximal, characteristic, complemented
11#447×C137C211yesmaximal, characteristic, complemented
12C2×#14555509—21no—
13C23×Dic137—21no—
14C2×#14555495C2212yes—
15#17984087C2211yescharacteristic, complemented
16#17984084C2211yescharacteristic, complemented
17#17984074—21no—
18#17984062C2211yescharacteristic
19#17984060—21no—
20D4×Dic137—21no—
21D4×Dic137—21no—
22D4×Dic137C2212yes—
23D4×Dic137C2212yes—
24D4×Dic137C2211yescharacteristic, complemented
25#17984050C2211yescharacteristic, complemented
26#17984048—21no—
27C2×C4⋊Dic137C2211yescharacteristic
28C2×C22⋊Dic137—21no—
29C2×C4×Dic137C2211yescharacteristic
30#17984022C2212yescomplemented
31#17984021C2212yescomplemented
32#17984017C2212yescomplemented
33#17984010C2212yescomplemented
34#17984009C2212yes—
35#17984009C2212yescomplemented
36C22⋊#11764040C2211yescharacteristic
37C8×Dic137C2212yescomplemented
38C22×D4×C137C2211yescharacteristic
39C2×D8×C137C2212yes—
40#121×C137C2211yescharacteristic
41#102×C137C2212yes—
42C22⋊C8×C137C2211yescharacteristic
43#14555509—44no—
44C2×Dic274—21no—
45C2×D548C2311yescharacteristic
46C2×C22⋊D137—21no—
47C2×C22⋊D137—21no—
48C2×C22⋊D137—21no—
49C22×Dic137—21no—
50C22×Dic137—22no—
51C22×Dic137—22no—
52C22×Dic137—21no—
53C22×Dic137—44no—
54C22×Dic137D411yescharacteristic
55C22×Dic137D411yescharacteristic
56C2×C4×D137—21no—
57#14555495—24no—
58C2×#11764040C2312yes—
59C4⋊Dic137—21no—
60C4⋊Dic137C2311yescharacteristic
61C4⋊Dic137C2312yes—
62#14555489—21no—
63C22⋊Dic137—22no—
64C22⋊Dic137—22no—
65C22⋊Dic137—21no—
66#14555487—21no—
67#14555487—21no—
68C4×Dic137D411yescharacteristic
69C4×Dic137D411yescharacteristic
70C4×Dic137C2311yescharacteristic
71C4×Dic137C2311yescharacteristic
72C24×C137—21no—
73C2×D4×C137—24no—
74C2×D4×C137—21no—
75C2×D4×C137—21no—
76C2×D4×C137C2311yescharacteristic
77C2×D4×C137C2312yes—
78C22×C4×C137C2311yescharacteristic
79D8×C137—24no—
80C2×C8×C137C2312yes—
81C4⋊C4×C137C2311yescharacteristic
82C22⋊C4×C137—21no—
83C22×D137—21no—
84Dic274—42no—
85D548—22no—
86C22⋊D137—42no—
87C22⋊D137—42no—
88C22⋊D137—42no—
89C2×Dic137—21no—
90C2×Dic137—22no—
91C2×Dic137—41no—
92C2×Dic137—42no—
93C2×Dic137—44no—
94C2×Dic137—44no—
95C2×Dic137—42no—
96C2×Dic137D814yescomplemented
97C2×Dic137C2×D411yescharacteristic
98C2×Dic137C2×D411yescharacteristic
99C2×Dic137C2×D411yescharacteristic
100C2×Dic137C2×D411yescharacteristic
101C4×D137—42no—
102#11764040—22no—
103C23×C137—24no—
104C23×C137—22no—
105C23×C137—21no—
106C23×C137—42no—
107C23×C137—42no—
108C23×C137C2×D411yescharacteristic
109D4×C137—24no—
110D4×C137—22no—
111D4×C137—22no—
112D4×C137—42no—
113D4×C137#4114yescomplemented
114C2×C4×C137—22no—
115C2×C4×C137—21no—
116C2×C4×C137C2×D411yescharacteristic
117C2×C4×C137C2411yescharacteristic
118C8×C137—22no—
119C22×C137—21no—
120C22×C137—21no—
121C22×C137—24no—
122C22×C137—48no—
123C22×C137—42no—
124C22×C137—42no—
125C22×C137—41no—
126C22×C137—41no—
127C22×C137—41no—
128C22×C137C2×D812yes—
129C22×C137C22×D411yescharacteristic
130C2×D137—41no—
131C2×D137—41no—
132C2×D137—41no—
133C4×C137—21no—
134C4×C137—41no—
135C4×C137C2×#4112yes—
136Dic137—21no—
137Dic137—21no—
138Dic137—42no—
139Dic137—41no—
140Dic137C2×D812yes—
141Dic137C2×D812yes—
142C2×C137—22no—
143C2×C137—44no—
144C2×C137—81no—
145C2×C137#52511yescharacteristic
146C2×C137C22×D811yescharacteristic
147C2×C137#57511yescharacteristic
148D137—81no—
149C137#310011yescharacteristic, complemented, Sylow
150#3100—1371nomaximal, Sylow
151#532—1371no—
152C2×C4×D4—1371no—
153#480—1371no—
154#480—1371no—
155#474—1372no—
156#459—1372no—
157#447—1371no—
158#447D13711yescharacteristic, complemented
159C4×D8—1372no—
160C4×D8—1372no—
161#431—1371no—
162C2×#41—2741no—
163C22×D4C2×D13711yescharacteristic
164C23×C4—2741no—
165C2×D8—1372no—
166C2×D8C2×D13712yes—
167C4⋊Q8—1371no—
168#127—1371no—
169#124—2741no—
170#121—2741no—
171#121—1371no—
172#121C2×D13711yescharacteristic
173C4×D4—2741no—
174C4×D4—2741no—
175C4×D4—2741no—
176C4×D4—1372no—
177C4×D4—1372no—
178C4×D4—1371no—
179C4×D4—1371no—
180C2×C4⋊C4—1371no—
181C2×C22⋊C4—2741no—
182C2×C42—1371no—
183#107—1372no—
184#107—1372no—
185C4⋊C8—1372no—
186#102—1372no—
187#102—1372no—
188#102—1372no—
189#102C2×D13712yes—
190C22⋊C8—1371no—
191C22⋊C8C2×D13711yescharacteristic
192C4×C8—1372no—
193C24—21no—
194#41—5484no—
195C2×Q8—2741no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—2741no—
200C2×D4—2741no—
201C2×D4—2741no—
202C2×D4—1371no—
203C2×D4C22×D13711yescharacteristic
204C2×D4C22×D13712yes—
205C22×C4—2741no—
206C22×C4—2741no—
207C22×C4—2742no—
208C22×C4—2742no—
209C22×C4—2741no—
210C22×C4—5484no—
211C22×C4—1371no—
212C22×C4—1371no—
213C22×C4C22×D13711yescharacteristic
214D8—24no—
215D8—2744no—
216C2×C8—1372no—
217C2×C8C22×D13712yes—
218C4⋊C4—2741no—
219C4⋊C4—2741no—
220C4⋊C4—1372no—
221C4⋊C4—1371no—
222C4⋊C4C22×D13711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2741no—
225C22⋊C4—2742no—
226C22⋊C4—2742no—
227C22⋊C4—2741no—
228C22⋊C4—2741no—
229C42—1371no—
230C42—1371no—
231C42—1371no—
232C42—1371no—
233C23—24no—
234C23—22no—
235C23—21no—
236C23—42no—
237C23—42no—
238C23—2741no—
239C23D4×D13711yescharacteristic
240Q8—5482no—
241D4—24no—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—2742no—
246D4—5482no—
247D4—5482no—
248D4—5482no—
249D4#1455550914yescomplemented
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—2742no—
253C2×C4—2741no—
254C2×C4—5482no—
255C2×C4—5481no—
256C2×C4—5482no—
257C2×C4—5484no—
258C2×C4—5484no—
259C2×C4—5482no—
260C2×C4—1371no—
261C2×C4—1371no—
262C2×C4—1374no—
263C2×C4—1371no—
264C2×C4—1371no—
265C2×C4D4×D13711yescharacteristic
266C2×C4C23×D13711yescharacteristic
267C8—22no—
268C8—2742no—
269C22—21no—
270C22—21no—
271C22—24no—
272C22—48no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—5481no—
279C22—5481no—
280C22—5481no—
281C22D8×D13712yes—
282C22C2×D4×D13711yescharacteristic
283C4—21no—
284C4—41no—
285C4—2741no—
286C4—2741no—
287C4—5482no—
288C4—5481no—
289C4—1372no—
290C4—1372no—
291C4C2×#1455550912yes—
292C2—22no—
293C2—44no—
294C2—81no—
295C2—10961no—
296C2#2257790911yescharacteristic
297C2C2×D8×D13711yescharacteristic
298C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#447×C137C2
socleC22×C137C22×D4
hypercenter#447D137
hypocenterC137#3100
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #22577948C2semidirect
2. C2×D4×Dic137C2semidirect
3. #22577717C2semidirect
4. #22577714C2semidirect
5. #22577681C2semidirect
6. #22577596C2semidirect
7. #22577537C2semidirect
8. D8×Dic137C2semidirect
9. #22577483C2semidirect
10. #22577453C2semidirect
11. #447×C137C2semidirect
15. #17984087C22semidirect
16. #17984084C22semidirect
24. D4×Dic137C22semidirect
25. #17984050C22semidirect
30. #17984022C22semidirect
31. #17984021C22semidirect
32. #17984017C22semidirect
33. #17984010C22semidirect
35. #17984009C22semidirect
37. C8×Dic137C22semidirect
96. C2×Dic137D8semidirect
113. D4×C137#41semidirect
149. C137#3100semidirect
158. #447D137semidirect
249. D4#14555509semidirect

Maximal subgroups

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

subgroupindex
#225779482
C2×D4×Dic1372
#225777172
#225777142
#225776812
#225775962
#225775372
D8×Dic1372
#225774832
#225774532
#447×C1372
#3100137