Finite Group Databasebeta

← #31790792

Subgroups — #31790792

Order 17536.

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

normal subgroups only

This group has 448 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#22577955C211yesmaximal, characteristic, complemented
2C2×D4×Dic137C211yesmaximal, characteristic
3#22577734C211yesmaximal, characteristic, complemented
4#22577731C211yesmaximal, characteristic, complemented
5#22577691C212yesmaximal, complemented
6#22577606C212yesmaximal, complemented
7#22577554C211yesmaximal, characteristic
8#22577494C212yesmaximal, complemented
9SD16×Dic137C212yesmaximal, complemented
10#22577453C211yesmaximal, characteristic, complemented
11#450×C137C211yesmaximal, characteristic
12C23×Dic137—21no—
13C2×#14555498C2212yes—
14#17984087C2211yescharacteristic, complemented
15#17984086C2211yescharacteristic, complemented
16#17984078—21no—
17#17984067C2211yescharacteristic
18#17984065—21no—
19Q8×Dic137C2211yescharacteristic, complemented
20#17984053—21no—
21#17984053C2211yescharacteristic, complemented
22D4×Dic137—21no—
23D4×Dic137—21no—
24D4×Dic137C2212yes—
25D4×Dic137C2212yes—
26#17984044—21no—
27C2×C4⋊Dic137C2211yescharacteristic
28C2×C22⋊Dic137—21no—
29C2×C4×Dic137C2211yescharacteristic
30#17984020C2212yescomplemented
31#17984019C2212yescomplemented
32#17984017C2212yescomplemented
33#17984013C2212yescomplemented
34#17984012C2212yescomplemented
35#17984009C2212yes—
36C22⋊#11764040C2211yescharacteristic
37C8×Dic137C2212yescomplemented
38C22×D4×C137C2211yescharacteristic
39C2×SD16×C137C2212yes—
40C22⋊Q8×C137C2211yescharacteristic
41#102×C137C2212yes—
42C22⋊C8×C137C2211yescharacteristic
43C2×Dic274—21no—
44C2×Dic274C2311yescharacteristic
45C22×Dic137—22no—
46C22×Dic137—22no—
47C22×Dic137—21no—
48C22×Dic137—44no—
49C22×Dic137D411yescharacteristic
50C22×Dic137D411yescharacteristic
51#14555498—24no—
52C2×#11764040C2312yes—
53C4⋊Dic137—21no—
54C4⋊Dic137—21no—
55C4⋊Dic137C2311yescharacteristic
56C4⋊Dic137C2312yes—
57#14555489—21no—
58#14555489—21no—
59#14555489—21no—
60#14555489—21no—
61#14555489—21no—
62C22⋊Dic137—22no—
63C22⋊Dic137—22no—
64C22⋊Dic137—21no—
65C4×Dic137—21no—
66C4×Dic137—21no—
67C4×Dic137D411yescharacteristic
68C4×Dic137D411yescharacteristic
69C4×Dic137C2311yescharacteristic
70C4×Dic137C2311yescharacteristic
71C24×C137—21no—
72C2×Q8×C137C2311yescharacteristic
73C2×D4×C137—24no—
74C2×D4×C137—21no—
75C2×D4×C137C2312yes—
76C22×C4×C137C2311yescharacteristic
77SD16×C137—24no—
78C2×C8×C137C2312yes—
79C4⋊C4×C137—21no—
80C4⋊C4×C137C2311yescharacteristic
81C22⋊C4×C137—21no—
82Dic274—22no—
83Dic274—42no—
84C2×Dic137—21no—
85C2×Dic137—21no—
86C2×Dic137—22no—
87C2×Dic137—41no—
88C2×Dic137—42no—
89C2×Dic137—44no—
90C2×Dic137—44no—
91C2×Dic137SD1614yescomplemented
92C2×Dic137C2×D411yescharacteristic
93C2×Dic137C2×D411yescharacteristic
94C2×Dic137C2×D411yescharacteristic
95C2×Dic137C2×D411yescharacteristic
96#11764040—22no—
97C23×C137—24no—
98C23×C137—22no—
99C23×C137—42no—
100C23×C137—42no—
101C23×C137C2×D411yescharacteristic
102Q8×C137—22no—
103D4×C137—24no—
104D4×C137—22no—
105D4×C137#4114yes—
106C2×C4×C137—22no—
107C2×C4×C137—21no—
108C2×C4×C137—21no—
109C2×C4×C137C2×D411yescharacteristic
110C2×C4×C137C2411yescharacteristic
111C8×C137—22no—
112C22×C137—21no—
113C22×C137—21no—
114C22×C137—24no—
115C22×C137—48no—
116C22×C137—42no—
117C22×C137—42no—
118C22×C137C2×SD1612yes—
119C22×C137C22×D411yescharacteristic
120C4×C137—21no—
121C4×C137—41no—
122C4×C137—41no—
123C4×C137C2×#4112yes—
124Dic137—21no—
125Dic137—21no—
126Dic137—41no—
127Dic137—42no—
128Dic137—41no—
129Dic137C2×SD1612yes—
130Dic137C2×SD1612yes—
131C2×C137—22no—
132C2×C137—44no—
133C2×C137#52511yescharacteristic
134C2×C137C22×SD1611yescharacteristic
135C2×C137#57511yescharacteristic
136C137#310111yescharacteristic, complemented, Sylow
137#3101—1371nomaximal, Sylow
138#533—1371no—
139C2×C4×D4—1371no—
140#483—1371no—
141#483—1371no—
142#476—1372no—
143#461—1372no—
144#450—1371no—
145#450D13711yescharacteristic
146C4×SD16—1372no—
147C4×SD16—1372no—
148#431—1371no—
149C22×D4C2×D13711yescharacteristic
150C23×C4—2741no—
151C2×SD16—1372no—
152C2×SD16C2×D13712yes—
153C4⋊Q8—1371no—
154C4⋊Q8—1371no—
155#125—2741no—
156C22⋊Q8—2741no—
157C22⋊Q8—1371no—
158C22⋊Q8C2×D13711yescharacteristic
159C4×Q8—2741no—
160C4×Q8—1371no—
161C4×Q8—1371no—
162C4×D4—2741no—
163C4×D4—2741no—
164C4×D4—1372no—
165C4×D4—1372no—
166#117—2741no—
167C2×C4⋊C4—1371no—
168C2×C22⋊C4—2741no—
169C2×C42—1371no—
170#106—1372no—
171#106—1372no—
172C4⋊C8—1372no—
173#103—1372no—
174#103—1372no—
175#102—1372no—
176#102C2×D13712yes—
177C22⋊C8—1371no—
178C22⋊C8C2×D13711yescharacteristic
179C4×C8—1372no—
180C24—21no—
181C2×Q8—2741no—
182C2×Q8—1371no—
183C2×Q8C22×D13711yescharacteristic
184C2×D4—24no—
185C2×D4—21no—
186C2×D4C22×D13712yes—
187C22×C4—2741no—
188C22×C4—2742no—
189C22×C4—2742no—
190C22×C4—5484no—
191C22×C4—1371no—
192C22×C4—1371no—
193C22×C4C22×D13711yescharacteristic
194SD16—24no—
195SD16—2744no—
196C2×C8—1372no—
197C2×C8C22×D13712yes—
198C4⋊C4—21no—
199C4⋊C4—2741no—
200C4⋊C4—2741no—
201C4⋊C4—2741no—
202C4⋊C4—2741no—
203C4⋊C4—2741no—
204C4⋊C4—2741no—
205C4⋊C4—2741no—
206C4⋊C4—1371no—
207C4⋊C4—1372no—
208C4⋊C4C22×D13711yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—2741no—
211C22⋊C4—2742no—
212C22⋊C4—2742no—
213C42—2741no—
214C42—2741no—
215C42—1371no—
216C42—1371no—
217C42—1371no—
218C42—1371no—
219C23—24no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23D4×D13711yescharacteristic
224Q8—22no—
225Q8—2742no—
226Q8—5482no—
227D4—24no—
228D4—22no—
229D4#1455550914yes—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—2741no—
234C2×C4—2741no—
235C2×C4—2742no—
236C2×C4—5481no—
237C2×C4—5482no—
238C2×C4—5484no—
239C2×C4—5484no—
240C2×C4—1371no—
241C2×C4—1371no—
242C2×C4—1374no—
243C2×C4—1371no—
244C2×C4—1371no—
245C2×C4D4×D13711yescharacteristic
246C2×C4C23×D13711yescharacteristic
247C8—22no—
248C8—2742no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22SD16×D13712yes—
256C22C2×D4×D13711yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—2741no—
261C4—2741no—
262C4—5481no—
263C4—5481no—
264C4—5482no—
265C4—1372no—
266C4—1372no—
267C4C2×#1455550912yes—
268C2—22no—
269C2—44no—
270C2#2257790911yescharacteristic
271C2C2×SD16×D13711yescharacteristic
272C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#450×C137C2
socleC22×C137C22×D4
prime radicalC22×D4×C137C22
hypercenter#450D137
hypocenterC137#3101
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22577955C2semidirect
3. #22577734C2semidirect
4. #22577731C2semidirect
5. #22577691C2semidirect
6. #22577606C2semidirect
8. #22577494C2semidirect
9. SD16×Dic137C2semidirect
10. #22577453C2semidirect
14. #17984087C22semidirect
15. #17984086C22semidirect
19. Q8×Dic137C22semidirect
21. #17984053C22semidirect
30. #17984020C22semidirect
31. #17984019C22semidirect
32. #17984017C22semidirect
33. #17984013C22semidirect
34. #17984012C22semidirect
37. C8×Dic137C22semidirect
91. C2×Dic137SD16semidirect
136. C137#3101semidirect

Maximal subgroups

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

subgroupindex
#225779552
C2×D4×Dic1372
#225777342
#225777312
#225776912
#225776062
#225775542
#225774942
SD16×Dic1372
#225774532
#450×C1372
#3101137