Finite Group Databasebeta

← #31790747

Subgroups — #31790747

Order 17536.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#22577967C211yesmaximal, characteristic
2#22577727C212yesmaximal, complemented
3#22577727C212yesmaximal, complemented
4#22577721C212yesmaximal, complemented
5#22577716C212yesmaximal, complemented
6#22577443C211yesmaximal, characteristic, complemented
7#22577439C211yesmaximal, characteristic, complemented
8#22577407C211yesmaximal, characteristic, complemented
9C2×#17984008C211yesmaximal, characteristic, complemented
10#22577370C211yesmaximal, characteristic, complemented
11#520×C137C211yesmaximal, characteristic, complemented
12C22×D548C2211yescharacteristic
13C2×#14555494C2211yescharacteristic
14C22×#11764040C2211yescharacteristic
15#17984084—21no—
16#17984077—21no—
17#17984062C2212yes—
18#17984062C2212yes—
19#17984056—22no—
20#17984047C2211yescharacteristic
21#17984021C2211yescharacteristic, complemented
22#17984021C2211yescharacteristic, complemented
23#17984021C2212yescomplemented
24#17984019C2211yescharacteristic, complemented
25#17984019C2211yescharacteristic, complemented
26#17984019C2212yescomplemented
27#17984008C2212yes—
28#17984008C2212yes—
29#17984008C2212yes—
30#17984008C2212yes—
31C22⋊#11764040C2212yescomplemented
32C22⋊#11764040C2212yescomplemented
33C2×#41×C137C2211yescharacteristic
34C4×Q8×C137—21no—
35C4×D4×C137—21no—
36C4×D4×C137—22no—
37#117×C137C2211yescharacteristic
38#117×C137C2212yescomplemented
39C2×C4⋊C4×C137C2211yescharacteristic
40C2×C4⋊C4×C137C2212yescomplemented
41C23×D137—21no—
42C2×D548—21no—
43C2×D548—22no—
44C2×D548—22no—
45C2×D548C2312yes—
46C2×C22⋊D137—22no—
47C2×C22⋊D137—22no—
48#14555494—22no—
49C2×#11764040—22no—
50C2×#11764040C2311yescharacteristic
51C2×#11764040C2312yes—
52C2×#11764040C2311yescharacteristic
53C4⋊Dic137C2311yescharacteristic
54C4⋊Dic137C2311yescharacteristic
55C22⋊Dic137—21no—
56#14555487—22no—
57#14555487—22no—
58C4×Dic137—21no—
59#41×C137—22no—
60#41×C137—22no—
61C2×Q8×C137D411yescharacteristic
62C2×D4×C137D412yes—
63C2×D4×C137D411yescharacteristic
64C22×C4×C137—21no—
65C22×C4×C137—22no—
66C22×C4×C137C2312yes—
67C22×C4×C137C2311yescharacteristic
68C4⋊C4×C137—21no—
69C4⋊C4×C137—22no—
70C4⋊C4×C137C2312yes—
71C4⋊C4×C137C2311yescharacteristic
72C4⋊C4×C137C2311yescharacteristic
73C22⋊C4×C137—21no—
74C22⋊C4×C137—22no—
75C42×C137—21no—
76C42×C137—22no—
77C22×D137—22no—
78C22×D137—42no—
79C22×D137—81no—
80C22×D137—81no—
81D548—22no—
82D548—22no—
83D548—42no—
84C22⋊D137—82no—
85C22⋊D137—82no—
86C2×Dic137—21no—
87C2×Dic137—21no—
88#11764040—21no—
89#11764040—21no—
90#11764040—22no—
91C23×C137C2×D412yes—
92C23×C137C2×D411yescharacteristic
93Q8×C137—22no—
94D4×C137—24no—
95D4×C137—22no—
96C2×C4×C137—22no—
97C2×C4×C137—22no—
98C2×C4×C137—21no—
99C2×C4×C137—21no—
100C2×C4×C137—22no—
101C2×C4×C137—81no—
102C2×C4×C137—82no—
103C2×C4×C137C2×D412yes—
104C2×C4×C137C2×D411yescharacteristic
105C2×C4×C137#4112yes—
106C2×C4×C137#4112yes—
107C2×C4×C137C2411yescharacteristic
108C22×C137—21no—
109C22×C137—21no—
110C22×C137—22no—
111C22×C137—22no—
112C22×C137—22no—
113C22×C137#12312yes—
114C22×C137C22×D411yescharacteristic
115C2×D137—42no—
116C2×D137—42no—
117C2×D137—42no—
118C2×D137—84no—
119C4×C137—22no—
120C4×C137—41no—
121C4×C137—42no—
122C4×C137—41no—
123C4×C137#12312yes—
124C4×C137C2×#4111yescharacteristic
125C4×C137C2×#4111yescharacteristic
126Dic137—41no—
127Dic137—41no—
128C2×C137—22no—
129C2×C137—42no—
130C2×C137C2×#12311yescharacteristic
131C2×C137#57611yescharacteristic
132C2×C137#57711yescharacteristic
133D137—82no—
134C137#309611yescharacteristic, complemented, Sylow
135#3096—1371nomaximal, Sylow
136#535—1371no—
137#520D13711yescharacteristic, complemented
138#482—1372no—
139#482—1372no—
140#481—1372no—
141#480—1372no—
142#428—1371no—
143#427—1371no—
144#418—1371no—
145C2×#102—1371no—
146#408—1371no—
147C2×#41C2×D13711yescharacteristic
148C22×D4—1371no—
149C2×M16—1371no—
150C22×C8—1371no—
151#127—2741no—
152#124—2741no—
153#121—1372no—
154#121—1372no—
155#120—2742no—
156C4×Q8—21no—
157C4×D4—21no—
158C4×D4—22no—
159#117—1371no—
160#117C2×D13711yescharacteristic
161#117C2×D13712yescomplemented
162C2×C4⋊C4C2×D13711yescharacteristic
163C2×C4⋊C4C2×D13712yescomplemented
164#107—1371no—
165#107—1371no—
166#107—1372no—
167#106—1371no—
168#106—1371no—
169#106—1372no—
170#102—1372no—
171#102—1372no—
172#102—1372no—
173#102—1372no—
174C22⋊C8—1372no—
175C22⋊C8—1372no—
176C24—2741no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D13711yescharacteristic
180C2×D4—2742no—
181C2×D4—2742no—
182C2×D4—2741no—
183C2×D4—2742no—
184C2×D4—2742no—
185C2×D4—1372no—
186C2×D4C22⋊D13712yes—
187C2×D4C22⋊D13711yescharacteristic
188C22×C4—21no—
189C22×C4—22no—
190C22×C4C22×D13712yes—
191C22×C4C22×D13711yescharacteristic
192M16—2742no—
193C2×C8—2742no—
194C2×C8—1371no—
195C2×C8—1372no—
196C2×C8—1371no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—1371no—
200C4⋊C4—1371no—
201C4⋊C4C22×D13712yes—
202C4⋊C4C22×D13711yescharacteristic
203C4⋊C4C22×D13711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—2741no—
207C22⋊C4—2742no—
208C22⋊C4—2742no—
209C42—21no—
210C42—22no—
211C42—2741no—
212C23—2742no—
213C23—5482no—
214C23—10961no—
215C23—10961no—
216C23C2×C22⋊D13712yes—
217C23C2×C22⋊D13711yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—2742no—
222D4—2742no—
223D4—5482no—
224D4—10962no—
225D4—10962no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—2741no—
234C2×C4—2741no—
235C2×C4C2×C22⋊D13712yes—
236C2×C4C2×C22⋊D13711yescharacteristic
237C2×C4#1455551012yes—
238C2×C4#1455551012yes—
239C2×C4C23×D13711yescharacteristic
240C8—2741no—
241C8—2741no—
242C8—2742no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—5482no—
249C22—5482no—
250C22—5482no—
251C22—10964no—
252C22#1798407112yes—
253C22C22×C22⋊D13711yescharacteristic
254C4—22no—
255C4—41no—
256C4—42no—
257C4—41no—
258C4—5481no—
259C4—5481no—
260C4#1798407112yes—
261C4C2×#1455551011yescharacteristic
262C4C2×#1455551011yescharacteristic
263C2—22no—
264C2—42no—
265C2—10962no—
266C2C2×#1798407111yescharacteristic
267C2#2257820611yescharacteristic
268C2#2257821111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#520×C137C2
socleC22×C137C22×D4
prime radical#22577967C2
hypercenter#520D137
hypocenterC137#3096
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
2. #22577727C2semidirect
3. #22577727C2semidirect
4. #22577721C2semidirect
5. #22577716C2semidirect
6. #22577443C2semidirect
7. #22577439C2semidirect
8. #22577407C2semidirect
9. C2×#17984008C2semidirect
10. #22577370C2semidirect
11. #520×C137C2semidirect
21. #17984021C22semidirect
22. #17984021C22semidirect
23. #17984021C22semidirect
24. #17984019C22semidirect
25. #17984019C22semidirect
26. #17984019C22semidirect
31. C22⋊#11764040C22semidirect
32. C22⋊#11764040C22semidirect
38. #117×C137C22semidirect
40. C2×C4⋊C4×C137C22semidirect
134. C137#3096semidirect
137. #520D137semidirect
161. #117C2×D137semidirect
163. C2×C4⋊C4C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779672
#225777272
#225777212
#225777162
#225774432
#225774392
#225774072
C2×#179840082
#225773702
#520×C1372
#3096137