Finite Group Databasebeta

← #31790794

Subgroups — #31790794

Order 17536.

448 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 125 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×#17984050C211yesmaximal, characteristic
3#22577733C211yesmaximal, characteristic, complemented
4#22577692C212yesmaximal, complemented
5#22577607C212yesmaximal, complemented
6#22577556C211yesmaximal, characteristic
7#22577553C211yesmaximal, characteristic
8#22577490C212yesmaximal, complemented
9#22577488C212yesmaximal, complemented
10#22577453C211yesmaximal, characteristic, complemented
11#483×C137C211yesmaximal, characteristic, complemented
12C22×D548C2211yescharacteristic
13C22×C4×D137—21no—
14C2×#14555499C2212yes—
15C2×#14555497C2212yes—
16#17984087C2211yescharacteristic, complemented
17#17984086C2211yescharacteristic, complemented
18#17984078—21no—
19#17984067C2211yescharacteristic
20#17984065—21no—
21Q8×Dic137C2211yescharacteristic, complemented
22#17984053—21no—
23#17984053C2211yescharacteristic, complemented
24#17984050—21no—
25#17984050—21no—
26#17984050C2212yes—
27#17984050C2212yes—
28#17984044—21no—
29C2×#14555487—21no—
30C2×C4×Dic137C2211yescharacteristic
31#17984019C2212yescomplemented
32#17984017C2212yescomplemented
33#17984011C2212yescomplemented
34#17984010C2212yes—
35#17984008C2212yes—
36C22⋊#11764040C2211yescharacteristic
37C8×Dic137C2212yescomplemented
38C22⋊Q8×C137C2211yescharacteristic
39C2×C4⋊C4×C137C2211yescharacteristic
40#106×C137C2212yescomplemented
41#103×C137C2212yescomplemented
42C22⋊C8×C137C2211yescharacteristic
43C23×D137—21no—
44C2×Dic274—21no—
45C2×Dic274C2311yescharacteristic
46C2×D548—24no—
47C2×D548—21no—
48C2×D548C2312yes—
49C22×Dic137D411yescharacteristic
50C22×Dic137D411yescharacteristic
51C2×C4×D137—22no—
52C2×C4×D137—22no—
53C2×C4×D137—44no—
54#14555499—24no—
55#14555497—24no—
56C2×#11764040C2312yes—
57C4⋊Dic137—21no—
58C4⋊Dic137C2311yescharacteristic
59#14555489—21no—
60#14555489—21no—
61#14555489—21no—
62#14555489—21no—
63#14555489—21no—
64C22⋊Dic137—21no—
65#14555487—22no—
66#14555487—22no—
67C4×Dic137—21no—
68C4×Dic137—21no—
69C4×Dic137D411yescharacteristic
70C4×Dic137D411yescharacteristic
71C4×Dic137C2311yescharacteristic
72C4×Dic137C2311yescharacteristic
73C2×Q8×C137C2311yescharacteristic
74C22×C4×C137—21no—
75C22×C4×C137C2311yescharacteristic
76C2×C8×C137C2312yes—
77C4⋊C4×C137—21no—
78C4⋊C4×C137—21no—
79C4⋊C4×C137C2311yescharacteristic
80C4⋊C4×C137C2312yes—
81C22⋊C4×C137—21no—
82C22×D137—24no—
83C22×D137—22no—
84C22×D137—42no—
85C22×D137—42no—
86Dic274—22no—
87Dic274—42no—
88D548—24no—
89D548—22no—
90D548#4114yes—
91C2×Dic137—21no—
92C2×Dic137—21no—
93C2×Dic137—41no—
94C2×Dic137SD1614yescomplemented
95C2×Dic137C2×D411yescharacteristic
96C2×Dic137C2×D411yescharacteristic
97C2×Dic137C2×D411yescharacteristic
98C2×Dic137C2×D411yescharacteristic
99C4×D137—44no—
100C4×D137—44no—
101#11764040—22no—
102C23×C137C2×D411yescharacteristic
103Q8×C137—22no—
104C2×C4×C137—22no—
105C2×C4×C137—21no—
106C2×C4×C137—21no—
107C2×C4×C137—22no—
108C2×C4×C137—42no—
109C2×C4×C137C2×D411yescharacteristic
110C2×C4×C137C2411yescharacteristic
111C8×C137—22no—
112C22×C137—21no—
113C22×C137—21no—
114C22×C137C2×SD1612yes—
115C22×C137C22×D411yescharacteristic
116C2×D137—24no—
117C2×D137—48no—
118C2×D137—42no—
119C2×D137—42no—
120C4×C137—21no—
121C4×C137—41no—
122C4×C137—42no—
123C4×C137—41no—
124C4×C137C2×#4112yes—
125Dic137—21no—
126Dic137—21no—
127Dic137—41no—
128Dic137—41no—
129Dic137C2×SD1612yes—
130Dic137C2×SD1612yes—
131C2×C137—22no—
132C2×C137#52511yescharacteristic
133C2×C137C22×SD1611yescharacteristic
134C2×C137#57511yescharacteristic
135D137—44no—
136C137#310111yescharacteristic, complemented, Sylow
137#3101—1371nomaximal, Sylow
138#533—1371no—
139C2×C4×D4—1371no—
140#483—1371no—
141#483D13711yescharacteristic, complemented
142#476—1372no—
143#461—1372no—
144#450—1371no—
145#450—1371no—
146C4×SD16—1372no—
147C4×SD16—1372no—
148#431—1371no—
149C22×D4—1371no—
150C23×C4—2741no—
151C2×SD16—1372no—
152C2×SD16—1372no—
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⋊C4C2×D13711yescharacteristic
168C2×C22⋊C4—2741no—
169C2×C42—1371no—
170#106—1372no—
171#106C2×D13712yescomplemented
172C4⋊C8—1372no—
173#103—1372no—
174#103C2×D13712yescomplemented
175#102—1372no—
176#102—1372no—
177C22⋊C8—1371no—
178C22⋊C8C2×D13711yescharacteristic
179C4×C8—1372no—
180C24—2741no—
181C2×Q8—2741no—
182C2×Q8—1371no—
183C2×Q8C22×D13711yescharacteristic
184C2×D4—2744no—
185C2×D4—2741no—
186C2×D4—1372no—
187C22×C4—21no—
188C22×C4—2742no—
189C22×C4—2742no—
190C22×C4—5484no—
191C22×C4—1371no—
192C22×C4—1371no—
193C22×C4C22×D13711yescharacteristic
194SD16—2744no—
195SD16—2744no—
196C2×C8—1372no—
197C2×C8C22×D13712yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—2741no—
201C4⋊C4—2741no—
202C4⋊C4—2741no—
203C4⋊C4—2741no—
204C4⋊C4—2741no—
205C4⋊C4—2741no—
206C4⋊C4—1371no—
207C4⋊C4C22×D13711yescharacteristic
208C4⋊C4C22×D13712yes—
209C22⋊C4—21no—
210C22⋊C4—2742no—
211C22⋊C4—2742no—
212C22⋊C4—2741no—
213C42—2741no—
214C42—2741no—
215C42—1371no—
216C42—1371no—
217C42—1371no—
218C42—1371no—
219C23—2744no—
220C23—2742no—
221C23—5482no—
222C23—5482no—
223C23D4×D13711yescharacteristic
224Q8—22no—
225Q8—2742no—
226Q8—5482no—
227D4—2744no—
228D4—2742no—
229D4—1374no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—2741no—
236C2×C4—2741no—
237C2×C4—5484no—
238C2×C4—5484no—
239C2×C4—5481no—
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—2744no—
252C22—5488no—
253C22—5482no—
254C22—5482no—
255C22SD16×D13712yes—
256C22C2×D4×D13711yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—42no—
261C4—2741no—
262C4—2741no—
263C4—5481no—
264C4—5481no—
265C4—1372no—
266C4—1372no—
267C4#41×D13712yes—
268C2—22no—
269C2—5484no—
270C2#2257791111yescharacteristic
271C2C2×SD16×D13711yescharacteristic
272C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#483×C137C2
socleC22×C137C22×D4
prime radicalC22×D548C22
hypercenter#483D137
hypocenterC137#3101
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #22577955C2semidirect
3. #22577733C2semidirect
4. #22577692C2semidirect
5. #22577607C2semidirect
8. #22577490C2semidirect
9. #22577488C2semidirect
10. #22577453C2semidirect
11. #483×C137C2semidirect
16. #17984087C22semidirect
17. #17984086C22semidirect
21. Q8×Dic137C22semidirect
23. #17984053C22semidirect
31. #17984019C22semidirect
32. #17984017C22semidirect
33. #17984011C22semidirect
37. C8×Dic137C22semidirect
40. #106×C137C22semidirect
41. #103×C137C22semidirect
94. C2×Dic137SD16semidirect
136. C137#3101semidirect
141. #483D137semidirect
171. #106C2×D137semidirect
174. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779552
C2×#179840502
#225777332
#225776922
#225776072
#225775562
#225775532
#225774902
#225774882
#225774532
#483×C1372
#3101137