Finite Group Databasebeta

← #31790467

Subgroups — #31790467

Order 17536.

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

normal subgroups only

This group has 504 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
1C2×#17984114C211yesmaximal, characteristic, complemented
2C2×#17984094C211yesmaximal, characteristic, complemented
3#22577984C211yesmaximal, characteristic, complemented
4#22577970C211yesmaximal, characteristic, complemented
5#22577655C212yesmaximal, complemented
6#22577648C212yesmaximal, complemented
7#22577638C212yesmaximal, complemented
8#22577631C212yesmaximal, complemented
9#22577443C211yesmaximal, characteristic, complemented
10#22577400C211yesmaximal, characteristic, complemented
11C2×#102×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C2×#14555510C2211yescharacteristic
14#17984114—24no—
15C2×#14555498C2212yes—
16C2×#14555495C2212yes—
17#17984094—24no—
18C2×#14555494C2211yescharacteristic
19C2×#14555493C2212yescomplemented
20C2×C8×D137C2212yescomplemented
21#17984084—21no—
22#17984075—21no—
23#17984068—21no—
24#17984068C2212yescomplemented
25#17984063—21no—
26#17984063C2212yescomplemented
27#17984062—22no—
28#17984059C2212yes—
29#17984059C2212yes—
30#17984057—22no—
31#17984047C2211yescharacteristic
32#17984021C2212yescomplemented
33#17984019C2212yescomplemented
34#17984013C2211yescharacteristic, complemented
35#17984013C2211yescharacteristic, complemented
36#17984010C2211yescharacteristic, complemented
37#17984010C2211yescharacteristic, complemented
38C22×D4×C137C2211yescharacteristic
39C22×C8×C137C2211yescharacteristic
40C2×C4⋊C4×C137C2211yescharacteristic
41#102×C137C2212yes—
42#102×C137C2212yes—
43#14555510—22no—
44#14555510—22no—
45#14555510—22no—
46#14555510—22no—
47C2×Dic274D411yescharacteristic
48C2×Dic274C2311yescharacteristic
49C2×D548—22no—
50C2×D548D411yescharacteristic
51C2×D548C2311yescharacteristic
52C2×C22⋊D137—22no—
53C2×C22⋊D137—22no—
54C2×C22⋊D137—21no—
55C2×C22⋊D137D412yes—
56C2×C4×D137—21no—
57C2×C4×D137C2312yes—
58#14555498—24no—
59#14555495—24no—
60#14555494D414yescomplemented
61#14555493—24no—
62C2×#11764040C2312yes—
63C8×D137—24no—
64C4⋊Dic137—22no—
65C4⋊Dic137C2311yescharacteristic
66C4⋊Dic137C2311yescharacteristic
67C22⋊Dic137—22no—
68C22⋊Dic137—22no—
69C22⋊Dic137—21no—
70#14555487—22no—
71#14555487—22no—
72C4×Dic137—21no—
73C24×C137—21no—
74C2×D4×C137—22no—
75C2×D4×C137—21no—
76C2×D4×C137—22no—
77C2×D4×C137C2312yes—
78C22×C4×C137—21no—
79C22×C4×C137C2311yescharacteristic
80C2×C8×C137—22no—
81C2×C8×C137C2311yescharacteristic
82C2×C8×C137C2311yescharacteristic
83C4⋊C4×C137—21no—
84C4⋊C4×C137C2312yes—
85C22×D137—21no—
86C22×D137C2×D412yes—
87Dic274—22no—
88Dic274—21no—
89Dic274—21no—
90D548—22no—
91D548—21no—
92D548—21no—
93D548—82no—
94C22⋊D137—24no—
95C22⋊D137—42no—
96C22⋊D137—82no—
97C22⋊D137—82no—
98C2×Dic137—21no—
99C2×Dic137—21no—
100C2×Dic137—21no—
101C2×Dic137C2×D412yes—
102C4×D137—22no—
103C4×D137—22no—
104C4×D137—42no—
105#11764040C2×D414yes—
106C23×C137—22no—
107C23×C137—42no—
108C23×C137—81no—
109C23×C137—81no—
110C23×C137C2×D411yescharacteristic
111D4×C137—22no—
112D4×C137—22no—
113D4×C137—42no—
114C2×C4×C137—22no—
115C2×C4×C137—81no—
116C2×C4×C137C2×D411yescharacteristic
117C2×C4×C137C2×D412yes—
118C2×C4×C137#4112yes—
119C2×C4×C137C2411yescharacteristic
120C8×C137—21no—
121C8×C137—21no—
122C22×C137—21no—
123C22×C137—21no—
124C22×C137—42no—
125C22×C137—42no—
126C22×C137—42no—
127C22×C137—84no—
128C22×C137#12112yes—
129C22×C137C22×D411yescharacteristic
130C2×D137—22no—
131C2×D137—22no—
132C2×D137—22no—
133C2×D137—41no—
134C2×D137—41no—
135C2×D137—41no—
136C4×C137—42no—
137C4×C137#12112yes—
138C4×C137C22×D411yescharacteristic
139C4×C137C2×#4111yescharacteristic
140Dic137—22no—
141Dic137—41no—
142Dic137—41no—
143Dic137—41no—
144C2×C137—22no—
145C2×C137—82no—
146C2×C137C2×#12111yescharacteristic
147C2×C137#57611yescharacteristic
148C2×C137#57711yescharacteristic
149D137—42no—
150D137—81no—
151C137#306211yescharacteristic, complemented, Sylow
152#3062—1371nomaximal, Sylow
153C2×AGL(1,ℤ/8)—1371no—
154C2×#131—1371no—
155#537—1371no—
156#535—1371no—
157#469—1372no—
158#468—1372no—
159#466—1372no—
160#465—1372no—
161#428—1371no—
162#416—1371no—
163C2×#102D13711yescharacteristic, complemented
164C2×#41—1371no—
165C2×#41—1371no—
166C22×D4C2×D13711yescharacteristic
167AGL(1,ℤ/8)—2744no—
168C2×SD16—1372no—
169C2×D8—1372no—
170#131—2744no—
171C2×M16—1371no—
172C2×M16—1372no—
173C22×C8—1372no—
174C22×C8C2×D13711yescharacteristic
175#127—2741no—
176#124—2741no—
177C22⋊Q8—2741no—
178C22⋊Q8—1372no—
179#121—2741no—
180#121—2742no—
181#121—1372no—
182#121—1372no—
183#121—1372no—
184#120—2742no—
185#117—1371no—
186C2×C4⋊C4C2×D13711yescharacteristic
187#107—1372no—
188#106—1372no—
189#103—1371no—
190#103—1371no—
191#102—1371no—
192#102—1371no—
193#102C2×D13712yes—
194#102C2×D13712yes—
195C24—21no—
196#41—2742no—
197#41—2742no—
198#41—2742no—
199#41—2742no—
200C2×Q8—1371no—
201C2×Q8—1371no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—2742no—
206C2×D4—2741no—
207C2×D4—2742no—
208C2×D4—2742no—
209C2×D4—1372no—
210C2×D4—1371no—
211C2×D4—1371no—
212C2×D4C22×D13712yes—
213C22×C4—21no—
214C22×C4—2741no—
215C22×C4—1372no—
216C22×C4C22×D13711yescharacteristic
217SD16—2744no—
218D8—2744no—
219M16—2744no—
220M16—1374no—
221C2×C8—22no—
222C2×C8—2744no—
223C2×C8—1372no—
224C2×C8C22×D13711yescharacteristic
225C2×C8C22×D13711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—2742no—
228C4⋊C4—1371no—
229C4⋊C4—1371no—
230C4⋊C4C22×D13712yes—
231C22⋊C4—2742no—
232C22⋊C4—2742no—
233C22⋊C4—2741no—
234C22⋊C4—2742no—
235C22⋊C4—2742no—
236C42—2741no—
237C23—22no—
238C23—42no—
239C23—81no—
240C23—81no—
241C23—2741no—
242C23—1372no—
243C23D4×D13711yescharacteristic
244Q8—2742no—
245Q8—2741no—
246Q8—2741no—
247D4—22no—
248D4—22no—
249D4—42no—
250D4—2744no—
251D4—2742no—
252D4—2741no—
253D4—2741no—
254D4—5482no—
255D4—10962no—
256D4—10962no—
257D4—10962no—
258C2×C4—22no—
259C2×C4—81no—
260C2×C4—2742no—
261C2×C4—2742no—
262C2×C4—2741no—
263C2×C4—2741no—
264C2×C4—2741no—
265C2×C4—5482no—
266C2×C4—1372no—
267C2×C4D4×D13711yescharacteristic
268C2×C4D4×D13712yes—
269C2×C4#1455551012yes—
270C2×C4C23×D13711yescharacteristic
271C8—21no—
272C8—21no—
273C8—1374no—
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—84no—
280C22—2742no—
281C22—2742no—
282C22—2742no—
283C22—5481no—
284C22—5481no—
285C22—5481no—
286C22#1798406112yes—
287C22C2×D4×D13711yescharacteristic
288C4—42no—
289C4—2742no—
290C4—5481no—
291C4—5481no—
292C4—5481no—
293C4#1798406112yes—
294C4C2×D4×D13711yescharacteristic
295C4C2×#1455551011yescharacteristic
296C2—22no—
297C2—82no—
298C2—5482no—
299C2—10961no—
300C2C2×#1798406111yescharacteristic
301C2#2257820411yescharacteristic
302C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#102×C137C2
socleC22×C137C22×D4
hypercenterC2×#102D137
hypocenterC137#3062
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#17984114C2semidirect
2. C2×#17984094C2semidirect
3. #22577984C2semidirect
4. #22577970C2semidirect
5. #22577655C2semidirect
6. #22577648C2semidirect
7. #22577638C2semidirect
8. #22577631C2semidirect
9. #22577443C2semidirect
10. #22577400C2semidirect
11. C2×#102×C137C2semidirect
19. C2×#14555493C22semidirect
20. C2×C8×D137C22semidirect
24. #17984068C22semidirect
26. #17984063C22semidirect
32. #17984021C22semidirect
33. #17984019C22semidirect
34. #17984013C22semidirect
35. #17984013C22semidirect
36. #17984010C22semidirect
37. #17984010C22semidirect
60. #14555494D4semidirect
151. C137#3062semidirect
163. C2×#102D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841142
C2×#179840942
#225779842
#225779702
#225776552
#225776482
#225776382
#225776312
#225774432
#225774002
C2×#102×C1372
#3062137