Finite Group Databasebeta

← #31790327

Subgroups — #31790327

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×#17984116C211yesmaximal, characteristic, complemented
2C2×#17984108C211yesmaximal, characteristic, complemented
3#22577967C211yesmaximal, characteristic, complemented
4#22577884C211yesmaximal, characteristic, complemented
5#22577628C212yesmaximal, complemented
6#22577621C212yesmaximal, complemented
7#22577602C212yesmaximal, complemented
8#22577597C212yesmaximal, complemented
9#22577431C211yesmaximal, characteristic, complemented
10C2×#17984008C211yesmaximal, characteristic, complemented
11#417×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C22×D548C2211yescharacteristic
14#17984116—24no—
15#17984108—24no—
16C2×#14555500C2211yescharacteristic, complemented
17C2×#14555499C2212yes—
18C2×#14555498C2211yescharacteristic, complemented
19C2×#14555497C2211yescharacteristic, complemented
20C2×D1096C2212yes—
21C2×#14555495C2211yescharacteristic, complemented
22C22×#11764040C2211yescharacteristic
23#17984084C2212yes—
24#17984077C2212yes—
25#17984062—21no—
26#17984062—21no—
27#17984056—22no—
28#17984053C2212yescomplemented
29#17984050C2212yescomplemented
30C4×C22⋊D137—22no—
31#17984047C2211yescharacteristic
32#17984046—21no—
33C4⋊C4×D137—21no—
34#17984017C2212yescomplemented
35#17984017C2212yescomplemented
36#17984008C2212yes—
37#17984008C2212yes—
38C2×#41×C137C2211yescharacteristic
39C2×M16×C137C2211yescharacteristic
40C2×C4⋊C4×C137C2211yescharacteristic
41#103×C137C2212yescomplemented
42#102×C137C2212yescomplemented
43C23×D137—21no—
44#14555510—22no—
45#14555510D414yescomplemented
46C2×Dic274C2311yescharacteristic
47C2×D548—22no—
48C2×D548—21no—
49C2×D548—22no—
50C2×D548C2312yes—
51C2×D548C2311yescharacteristic
52C2×C22⋊D137—22no—
53C2×C22⋊D137—22no—
54C2×C22⋊D137—21no—
55C2×C4×D137—22no—
56C2×C4×D137—21no—
57#14555500—22no—
58#14555499—24no—
59#14555498—22no—
60#14555497—22no—
61D1096—24no—
62#14555495—22no—
63C2×#11764040—22no—
64C2×#11764040C2311yescharacteristic
65C2×#11764040C2311yescharacteristic
66C4⋊Dic137D411yescharacteristic
67C4⋊Dic137D411yescharacteristic
68#14555489—22no—
69C22⋊Dic137D412yes—
70#14555487—22no—
71#14555487—22no—
72#14555487—22no—
73C4×Dic137—22no—
74C4×Dic137C2312yes—
75#41×C137—22no—
76#41×C137—22no—
77C2×Q8×C137C2311yescharacteristic
78C2×D4×C137—21no—
79C2×D4×C137C2311yescharacteristic
80C22×C4×C137—21no—
81C22×C4×C137—21no—
82C22×C4×C137C2311yescharacteristic
83M16×C137—22no—
84C2×C8×C137C2312yes—
85C4⋊C4×C137—21no—
86C4⋊C4×C137C2312yes—
87C22×D137—21no—
88C22×D137—22no—
89C22×D137—42no—
90C22×D137—81no—
91C22×D137—81no—
92Dic274—21no—
93Dic274C2×D412yes—
94D548—22no—
95D548—21no—
96D548—22no—
97D548—42no—
98D548C2×D412yes—
99C22⋊D137—24no—
100C22⋊D137—82no—
101C22⋊D137—82no—
102C2×Dic137—21no—
103C2×Dic137C2×D412yes—
104C2×Dic137C2×D412yes—
105C4×D137—24no—
106C4×D137—82no—
107#11764040—21no—
108#11764040—21no—
109C23×C137—21no—
110C23×C137C2×D411yescharacteristic
111Q8×C137—21no—
112Q8×C137—21no—
113D4×C137—21no—
114D4×C137—21no—
115D4×C137—42no—
116C2×C4×C137—22no—
117C2×C4×C137—21no—
118C2×C4×C137—42no—
119C2×C4×C137—81no—
120C2×C4×C137C2×D411yescharacteristic
121C2×C4×C137C2×D412yes—
122C2×C4×C137#4112yes—
123C2×C4×C137C2411yescharacteristic
124C8×C137—22no—
125C22×C137—21no—
126C22×C137—21no—
127C22×C137—41no—
128C22×C137—41no—
129C22×C137—41no—
130C22×C137#12112yes—
131C22×C137C22×D411yescharacteristic
132C2×D137—22no—
133C2×D137—41no—
134C2×D137—41no—
135C2×D137—42no—
136C2×D137—42no—
137C2×D137—42no—
138C2×D137—84no—
139C4×C137—41no—
140C4×C137—42no—
141C4×C137#12112yes—
142C4×C137C22×D411yescharacteristic
143C4×C137C2×#4111yescharacteristic
144Dic137—22no—
145Dic137—22no—
146Dic137—22no—
147C2×C137—22no—
148C2×C137—81no—
149C2×C137C2×#12111yescharacteristic
150C2×C137#57611yescharacteristic
151C2×C137#57711yescharacteristic
152D137—42no—
153D137—82no—
154C137#304511yescharacteristic, complemented, Sylow
155#3045—1371nomaximal, Sylow
156C2×AGL(1,ℤ/8)—1371no—
157C2×#135—1371no—
158#535—1371no—
159#520—1371no—
160#464—1372no—
161#463—1372no—
162#460—1372no—
163#459—1372no—
164#424—1371no—
165#417D13711yescharacteristic, complemented
166C2×#102—1371no—
167C2×#41—1371no—
168C2×#41C2×D13711yescharacteristic
169C22×D4—1371no—
170AGL(1,ℤ/8)—2744no—
171#135—2744no—
172C2×Q16—1371no—
173C2×SD16—1371no—
174C2×SD16—1371no—
175C2×SD16—1372no—
176C2×D8—1371no—
177C2×D8—1372no—
178C2×M16C2×D13711yescharacteristic
179C22×C8—1371no—
180#127—1372no—
181#124—1372no—
182#121—2741no—
183#121—2741no—
184#120—2742no—
185C4×Q8—1372no—
186C4×D4—2742no—
187C4×D4—1372no—
188#117—2741no—
189#117—1371no—
190C2×C4⋊C4—2741no—
191C2×C4⋊C4C2×D13711yescharacteristic
192C4⋊C8—1372no—
193C4⋊C8—1372no—
194#103C2×D13712yescomplemented
195#102—1372no—
196#102—1372no—
197#102C2×D13712yescomplemented
198C24—2741no—
199#41—22no—
200#41—22no—
201#41—2742no—
202#41—1374no—
203C2×Q8—1371no—
204C2×Q8C22×D13711yescharacteristic
205C2×D4—21no—
206C2×D4—2741no—
207C2×D4—2742no—
208C2×D4—2742no—
209C2×D4—2742no—
210C2×D4—2741no—
211C2×D4—2742no—
212C2×D4—1372no—
213C2×D4—1371no—
214C2×D4C22×D13711yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—2742no—
218C22×C4—2741no—
219C22×C4C22×D13711yescharacteristic
220Q16—2742no—
221SD16—2744no—
222SD16—2742no—
223SD16—2742no—
224D8—2744no—
225D8—2742no—
226M16—22no—
227C2×C8—2742no—
228C2×C8—1371no—
229C2×C8—1371no—
230C2×C8C22×D13712yes—
231C4⋊C4—21no—
232C4⋊C4—2742no—
233C4⋊C4—1371no—
234C4⋊C4—1371no—
235C4⋊C4C22×D13712yes—
236C22⋊C4—2742no—
237C22⋊C4—2742no—
238C22⋊C4—2742no—
239C22⋊C4—1372no—
240C42—2742no—
241C42—1372no—
242C23—21no—
243C23—2741no—
244C23—2742no—
245C23—5482no—
246C23—10961no—
247C23—10961no—
248C23D4×D13711yescharacteristic
249Q8—21no—
250Q8—21no—
251Q8—2741no—
252Q8—1372no—
253D4—21no—
254D4—21no—
255D4—42no—
256D4—2744no—
257D4—2742no—
258D4—2741no—
259D4—2742no—
260D4—5482no—
261D4—10962no—
262D4—10962no—
263D4—1372no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—2741no—
269C2×C4—2744no—
270C2×C4—10962no—
271C2×C4—1372no—
272C2×C4—1372no—
273C2×C4D4×D13711yescharacteristic
274C2×C4D4×D13712yes—
275C2×C4#1455551012yes—
276C2×C4C23×D13711yescharacteristic
277C8—22no—
278C8—2741no—
279C8—2741no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—2742no—
286C22—5481no—
287C22—5481no—
288C22—5482no—
289C22—5482no—
290C22—5482no—
291C22—10964no—
292C22#1798406112yes—
293C22C2×D4×D13711yescharacteristic
294C4—42no—
295C4—41no—
296C4—2742no—
297C4—2742no—
298C4—2742no—
299C4#1798406112yes—
300C4C2×D4×D13711yescharacteristic
301C4C2×#1455551011yescharacteristic
302C2—22no—
303C2—81no—
304C2—5482no—
305C2—10962no—
306C2C2×#1798406111yescharacteristic
307C2#2257820711yescharacteristic
308C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#417×C137C2
socleC22×C137C22×D4
hypercenter#417D137
hypocenterC137#3045
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#17984116C2semidirect
2. C2×#17984108C2semidirect
3. #22577967C2semidirect
4. #22577884C2semidirect
5. #22577628C2semidirect
6. #22577621C2semidirect
7. #22577602C2semidirect
8. #22577597C2semidirect
9. #22577431C2semidirect
10. C2×#17984008C2semidirect
11. #417×C137C2semidirect
16. C2×#14555500C22semidirect
18. C2×#14555498C22semidirect
19. C2×#14555497C22semidirect
21. C2×#14555495C22semidirect
28. #17984053C22semidirect
29. #17984050C22semidirect
34. #17984017C22semidirect
35. #17984017C22semidirect
41. #103×C137C22semidirect
42. #102×C137C22semidirect
45. #14555510D4semidirect
154. C137#3045semidirect
165. #417D137semidirect
194. #103C2×D137semidirect
197. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841162
C2×#179841082
#225779672
#225778842
#225776282
#225776212
#225776022
#225775972
#225774312
C2×#179840082
#417×C1372
#3045137