Finite Group Databasebeta

← #31790367

Subgroups — #31790367

Order 17536.

520 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#17984108C211yesmaximal, characteristic, complemented
2C23×#11764040C211yesmaximal, characteristic, complemented
3#22577914C211yesmaximal, characteristic, complemented
4#22577642C212yesmaximal, complemented
5#22577637C212yesmaximal, complemented
6#22577634C212yesmaximal, complemented
7#22577630C212yesmaximal, complemented
8#22577439C211yesmaximal, characteristic, complemented
9#22577399C211yesmaximal, characteristic, complemented
10#22577397C211yesmaximal, characteristic, complemented
11#525×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13#17984108—24no—
14C2×#14555500C2211yescharacteristic, complemented
15C2×#14555498C2211yescharacteristic, complemented
16C2×#14555497C2211yescharacteristic, complemented
17C2×#14555495C2211yescharacteristic, complemented
18C22×#11764040—24no—
19C22×#11764040C2212yes—
20C22×#11764040C2212yes—
21C22×#11764040C2211yescharacteristic
22C22×#11764040C2211yescharacteristic
23#17984071—21no—
24#17984067C2212yes—
25#17984062C2212yes—
26#17984057—21no—
27C4×C22⋊D137—22no—
28#17984047C2211yescharacteristic
29#17984021C2212yescomplemented
30#17984019C2212yescomplemented
31#17984012C2212yescomplemented
32#17984011C2212yescomplemented
33#17984009C2212yescomplemented
34#17984008C2212yescomplemented
35C2×#41×C137C2211yescharacteristic
36C23×C4×C137C2211yescharacteristic
37#123×C137—21no—
38C22⋊Q8×C137C2212yes—
39#121×C137C2212yes—
40#120×C137—21no—
41C4×D4×C137—22no—
42#117×C137C2211yescharacteristic
43#14555510—22no—
44#14555510—22no—
45C2×Dic274C2311yescharacteristic
46C2×D548C2311yescharacteristic
47C2×C22⋊D137—22no—
48C2×C22⋊D137—21no—
49C2×C4×D137—21no—
50#14555500—22no—
51#14555498—22no—
52#14555497—22no—
53#14555495—22no—
54C2×#11764040—24no—
55C2×#11764040—24no—
56C2×#11764040—22no—
57C2×#11764040D414yescomplemented
58C2×#11764040C2312yes—
59C2×#11764040C2311yescharacteristic
60C2×#11764040C2311yescharacteristic
61C4⋊Dic137C2312yes—
62#14555489—22no—
63C22⋊Dic137—21no—
64C22⋊Dic137—22no—
65#14555487—22no—
66C4×Dic137—21no—
67C24×C137D411yescharacteristic
68#41×C137—22no—
69#41×C137—22no—
70C2×Q8×C137C2311yescharacteristic
71C2×D4×C137—22no—
72C2×D4×C137—21no—
73C2×D4×C137C2311yescharacteristic
74C22×C4×C137—22no—
75C22×C4×C137—22no—
76C22×C4×C137—21no—
77C22×C4×C137D412yes—
78C22×C4×C137D411yescharacteristic
79C22×C4×C137C2312yes—
80C22×C4×C137C2311yescharacteristic
81C4⋊C4×C137—22no—
82C4⋊C4×C137C2312yes—
83C22⋊C4×C137—21no—
84C22⋊C4×C137—22no—
85C22⋊C4×C137—22no—
86C42×C137—21no—
87C22×D137—21no—
88Dic274—21no—
89Dic274—21no—
90D548—21no—
91D548—21no—
92C22⋊D137—42no—
93C22⋊D137—44no—
94C2×Dic137—22no—
95C2×Dic137—21no—
96C4×D137—42no—
97#11764040—22no—
98#11764040C2×D412yes—
99#11764040C2×D412yes—
100C23×C137—22no—
101C23×C137—24no—
102C23×C137—21no—
103C23×C137C2×D411yescharacteristic
104C23×C137C2×D412yes—
105Q8×C137—21no—
106Q8×C137—21no—
107D4×C137—21no—
108D4×C137—21no—
109D4×C137—42no—
110D4×C137—44no—
111C2×C4×C137—22no—
112C2×C4×C137—24no—
113C2×C4×C137—22no—
114C2×C4×C137—22no—
115C2×C4×C137—22no—
116C2×C4×C137—21no—
117C2×C4×C137—42no—
118C2×C4×C137C2×D412yes—
119C2×C4×C137C2×D412yes—
120C2×C4×C137C2×D411yescharacteristic
121C2×C4×C137#4112yes—
122C2×C4×C137C2411yescharacteristic
123C22×C137—28no—
124C22×C137—22no—
125C22×C137—22no—
126C22×C137—22no—
127C22×C137—41no—
128C22×C137—42no—
129C22×C137#12112yes—
130C22×C137#13514yescomplemented
131C22×C137C22×D411yescharacteristic
132C2×D137—41no—
133C2×D137—42no—
134C4×C137—22no—
135C4×C137—41no—
136C4×C137—42no—
137C4×C137#12112yes—
138C4×C137C22×D411yescharacteristic
139C4×C137C2×#4111yescharacteristic
140Dic137—41no—
141Dic137—42no—
142C2×C137—22no—
143C2×C137—24no—
144C2×C137—81no—
145C2×C137C2×#12111yescharacteristic
146C2×C137C2×#13512yes—
147D137—81no—
148C137#305011yescharacteristic, complemented, Sylow
149#3050—1371nomaximal, Sylow
150C2×#135—1371no—
151C23×C8—1371no—
152#525—1371no—
153#525D13711yescharacteristic, complemented
154#467—1372no—
155#466—1372no—
156#465—1372no—
157#465—1372no—
158#427—1371no—
159#416—1371no—
160#416—1371no—
161C2×#41—1371no—
162C2×#41C2×D13711yescharacteristic
163C23×C4C2×D13711yescharacteristic
164#135—2744no—
165C2×Q16—1371no—
166C2×SD16—1371no—
167C2×SD16—1371no—
168C2×D8—1371no—
169C22×C8—2744no—
170C22×C8—1371no—
171C22×C8—1372no—
172C22×C8—1372no—
173C22×C8—1371no—
174#123—21no—
175#123—2741no—
176C22⋊Q8—1372no—
177C22⋊Q8C2×D13712yes—
178#121—1372no—
179#121C2×D13712yes—
180#120—21no—
181#120—2741no—
182C4×D4—22no—
183C4×D4—2742no—
184#117—1371no—
185#117C2×D13711yescharacteristic
186#107—1372no—
187#106—1372no—
188#103—1372no—
189#103—1372no—
190#102—1372no—
191#102—1372no—
192C24C22⋊D13711yescharacteristic
193#41—22no—
194#41—22no—
195#41—2742no—
196#41—2742no—
197C2×Q8—1371no—
198C2×Q8C22×D13711yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—2742no—
202C2×D4—2741no—
203C2×D4—1371no—
204C2×D4C22×D13711yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—2741no—
209C22×C4C22⋊D13711yescharacteristic
210C22×C4C22⋊D13712yes—
211C22×C4C22×D13712yes—
212C22×C4C22×D13711yescharacteristic
213Q16—2742no—
214SD16—2742no—
215SD16—2742no—
216D8—2742no—
217C2×C8—2742no—
218C2×C8—2744no—
219C2×C8—2744no—
220C2×C8—1372no—
221C2×C8—1374no—
222C2×C8—1371no—
223C2×C8—1371no—
224C4⋊C4—22no—
225C4⋊C4—2742no—
226C4⋊C4—1372no—
227C4⋊C4C22×D13712yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—2741no—
232C22⋊C4—2742no—
233C22⋊C4—2742no—
234C42—21no—
235C42—2741no—
236C23—22no—
237C23—24no—
238C23—21no—
239C23—2741no—
240C23C2×C22⋊D13711yescharacteristic
241C23C2×C22⋊D13712yes—
242Q8—21no—
243Q8—21no—
244Q8—2741no—
245Q8—2741no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—44no—
250D4—2741no—
251D4—2741no—
252D4—5482no—
253D4—5484no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—2742no—
262C2×C4—2741no—
263C2×C4—5482no—
264C2×C4D4×D13712yes—
265C2×C4C2×C22⋊D13712yes—
266C2×C4C2×C22⋊D13711yescharacteristic
267C2×C4#1455550912yes—
268C2×C4C23×D13711yescharacteristic
269C8—2742no—
270C8—1372no—
271C8—1372no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—41no—
278C22—5482no—
279C22—5481no—
280C22#1798406012yes—
281C22#1798410814yescomplemented
282C22C22×C22⋊D13711yescharacteristic
283C4—22no—
284C4—42no—
285C4—41no—
286C4—5482no—
287C4—5481no—
288C4#1798406012yes—
289C4C2×D4×D13711yescharacteristic
290C4C2×#1455550911yescharacteristic
291C2—22no—
292C2—24no—
293C2—81no—
294C2—10961no—
295C2C2×#1798406011yescharacteristic
296C2C2×#1798410812yes—
subgroupG/H
centerC2×C4C2×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#525×C137C2
socleC22×C137C22×D4
hypercenter#525D137
hypocenterC137#3050
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#17984108C2semidirect
2. C23×#11764040C2semidirect
3. #22577914C2semidirect
4. #22577642C2semidirect
5. #22577637C2semidirect
6. #22577634C2semidirect
7. #22577630C2semidirect
8. #22577439C2semidirect
9. #22577399C2semidirect
10. #22577397C2semidirect
11. #525×C137C2semidirect
14. C2×#14555500C22semidirect
15. C2×#14555498C22semidirect
16. C2×#14555497C22semidirect
17. C2×#14555495C22semidirect
29. #17984021C22semidirect
30. #17984019C22semidirect
31. #17984012C22semidirect
32. #17984011C22semidirect
33. #17984009C22semidirect
34. #17984008C22semidirect
57. C2×#11764040D4semidirect
130. C22×C137#135semidirect
148. C137#3050semidirect
153. #525D137semidirect
281. C22#17984108semidirect

Maximal subgroups

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

subgroupindex
C2×#179841082
C23×#117640402
#225779142
#225776422
#225776372
#225776342
#225776302
#225774392
#225773992
#225773972
#525×C1372
#3050137