Finite Group Databasebeta

← #31790333

Subgroups — #31790333

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×#17984110C211yesmaximal, characteristic, complemented
3#22577970C211yesmaximal, characteristic, complemented
4#22577884C211yesmaximal, characteristic, complemented
5#22577624C212yesmaximal, complemented
6#22577617C212yesmaximal, complemented
7#22577602C212yesmaximal, complemented
8#22577597C212yesmaximal, complemented
9#22577429C211yesmaximal, characteristic, complemented
10#22577402C211yesmaximal, characteristic, complemented
11C2×#102×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C2×#14555510C2211yescharacteristic
14#17984114—24no—
15#17984110—24no—
16C2×Dic548C2211yescharacteristic, complemented
17C2×#14555499C2211yescharacteristic, complemented
18C2×#14555499C2211yescharacteristic, complemented
19C2×#14555498C2212yes—
20C2×D1096C2211yescharacteristic, complemented
21C2×#14555495C2212yes—
22C2×#14555494C2211yescharacteristic
23#17984084C2212yes—
24#17984075C2212yes—
25#17984059—21no—
26#17984059—21no—
27#17984057—22no—
28#17984053C2212yescomplemented
29#17984050C2212yescomplemented
30C4×C22⋊D137—22no—
31#17984047C2211yescharacteristic
32#17984046—21no—
33C4⋊C4×D137—21no—
34#17984017C2212yescomplemented
35#17984017C2212yescomplemented
36#17984011C2212yescomplemented
37#17984008C2212yescomplemented
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#14555510D414yescomplemented
47C2×Dic274C2311yescharacteristic
48C2×Dic274C2311yescharacteristic
49C2×D548C2311yescharacteristic
50C2×D548C2311yescharacteristic
51C2×C22⋊D137—22no—
52C2×C22⋊D137—22no—
53C2×C22⋊D137—21no—
54C2×C22⋊D137—21no—
55C2×C4×D137—22no—
56C2×C4×D137—21no—
57C2×C4×D137—21no—
58Dic548—22no—
59#14555499—22no—
60#14555499—22no—
61#14555498—24no—
62D1096—22no—
63#14555495—24no—
64#14555494—22no—
65C2×#11764040C2312yes—
66C4⋊Dic137D411yescharacteristic
67C4⋊Dic137D411yescharacteristic
68#14555489—22no—
69C22⋊Dic137—22no—
70C22⋊Dic137—22no—
71C22⋊Dic137D412yes—
72#14555487—22no—
73C4×Dic137—22no—
74C4×Dic137C2312yes—
75C24×C137—21no—
76C2×D4×C137—22no—
77C2×D4×C137—21no—
78C2×D4×C137—22no—
79C2×D4×C137C2312yes—
80C22×C4×C137—21no—
81C22×C4×C137C2311yescharacteristic
82C2×C8×C137—22no—
83C2×C8×C137C2311yescharacteristic
84C2×C8×C137C2311yescharacteristic
85C4⋊C4×C137—21no—
86C4⋊C4×C137C2312yes—
87C22×D137—21no—
88C22×D137—21no—
89Dic274—21no—
90Dic274—21no—
91Dic274—21no—
92Dic274C2×D412yes—
93D548—21no—
94D548—21no—
95D548—21no—
96D548C2×D412yes—
97C22⋊D137—24no—
98C22⋊D137—42no—
99C22⋊D137—82no—
100C22⋊D137—82no—
101C2×Dic137—21no—
102C2×Dic137—21no—
103C2×Dic137C2×D412yes—
104C2×Dic137C2×D412yes—
105C4×D137—24no—
106C4×D137—42no—
107C4×D137—82no—
108#11764040—22no—
109C23×C137—22no—
110C23×C137—42no—
111C23×C137—81no—
112C23×C137—81no—
113C23×C137C2×D411yescharacteristic
114D4×C137—22no—
115D4×C137—22no—
116D4×C137—42no—
117C2×C4×C137—22no—
118C2×C4×C137—81no—
119C2×C4×C137C2×D411yescharacteristic
120C2×C4×C137C2×D412yes—
121C2×C4×C137#4112yes—
122C2×C4×C137C2411yescharacteristic
123C8×C137—21no—
124C8×C137—21no—
125C22×C137—21no—
126C22×C137—21no—
127C22×C137—42no—
128C22×C137—42no—
129C22×C137—42no—
130C22×C137—84no—
131C22×C137#12112yes—
132C22×C137C22×D411yescharacteristic
133C2×D137—22no—
134C2×D137—41no—
135C2×D137—41no—
136C2×D137—41no—
137C2×D137—41no—
138C2×D137—41no—
139C4×C137—42no—
140C4×C137#12112yes—
141C4×C137C22×D411yescharacteristic
142C4×C137C2×#4111yescharacteristic
143Dic137—22no—
144Dic137—22no—
145Dic137—22no—
146Dic137—41no—
147C2×C137—22no—
148C2×C137—82no—
149C2×C137C2×#12111yescharacteristic
150C2×C137#57611yescharacteristic
151C2×C137#57711yescharacteristic
152D137—42no—
153D137—81no—
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#417—1371no—
166C2×#102D13711yescharacteristic, complemented
167C2×#41—1371no—
168C2×#41—1371no—
169C22×D4C2×D13711yescharacteristic
170AGL(1,ℤ/8)—2744no—
171#135—2744no—
172C2×Q16—1371no—
173C2×SD16—1372no—
174C2×SD16—1371no—
175C2×SD16—1371no—
176C2×D8—1372no—
177C2×D8—1371no—
178C2×M16—1371no—
179C22×C8C2×D13711yescharacteristic
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#103—1372no—
195#102—1372no—
196#102C2×D13712yes—
197#102C2×D13712yes—
198C24—21no—
199#41—2742no—
200#41—2742no—
201#41—2742no—
202#41—1374no—
203C2×Q8—1371no—
204C2×Q8—1371no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—2741no—
209C2×D4—2741no—
210C2×D4—2742no—
211C2×D4—2742no—
212C2×D4—1371no—
213C2×D4—1371no—
214C2×D4C22×D13712yes—
215C22×C4—21no—
216C22×C4—2742no—
217C22×C4—2741no—
218C22×C4—2741no—
219C22×C4C22×D13711yescharacteristic
220Q16—2742no—
221SD16—2742no—
222SD16—2742no—
223SD16—2744no—
224D8—2742no—
225D8—2744no—
226M16—2742no—
227C2×C8—22no—
228C2×C8—1372no—
229C2×C8C22×D13711yescharacteristic
230C2×C8C22×D13711yescharacteristic
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—22no—
243C23—42no—
244C23—81no—
245C23—81no—
246C23—2741no—
247C23—2741no—
248C23D4×D13711yescharacteristic
249Q8—2741no—
250Q8—2741no—
251Q8—2741no—
252Q8—1372no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—2744no—
257D4—2741no—
258D4—2741no—
259D4—2741no—
260D4—5482no—
261D4—10962no—
262D4—10962no—
263D4—1372no—
264C2×C4—22no—
265C2×C4—81no—
266C2×C4—2741no—
267C2×C4—2744no—
268C2×C4—2741no—
269C2×C4—5482no—
270C2×C4—10962no—
271C2×C4—1372no—
272C2×C4—1372no—
273C2×C4D4×D13711yescharacteristic
274C2×C4D4×D13712yes—
275C2×C4#1455551012yes—
276C2×C4C23×D13711yescharacteristic
277C8—21no—
278C8—21no—
279C8—2742no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—2742no—
287C22—5481no—
288C22—5481no—
289C22—5481no—
290C22—5481no—
291C22—5481no—
292C22#1798406112yes—
293C22C2×D4×D13711yescharacteristic
294C4—42no—
295C4—2742no—
296C4—2742no—
297C4—2742no—
298C4—5481no—
299C4#1798406112yes—
300C4C2×D4×D13711yescharacteristic
301C4C2×#1455551011yescharacteristic
302C2—22no—
303C2—82no—
304C2—5482no—
305C2—10961no—
306C2C2×#1798406111yescharacteristic
307C2#2257820411yescharacteristic
308C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#102×C137C2
socleC22×C137C22×D4
hypercenterC2×#102D137
hypocenterC137#3045
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×#17984110C2semidirect
3. #22577970C2semidirect
4. #22577884C2semidirect
5. #22577624C2semidirect
6. #22577617C2semidirect
7. #22577602C2semidirect
8. #22577597C2semidirect
9. #22577429C2semidirect
10. #22577402C2semidirect
11. C2×#102×C137C2semidirect
16. C2×Dic548C22semidirect
17. C2×#14555499C22semidirect
18. C2×#14555499C22semidirect
20. C2×D1096C22semidirect
28. #17984053C22semidirect
29. #17984050C22semidirect
34. #17984017C22semidirect
35. #17984017C22semidirect
36. #17984011C22semidirect
37. #17984008C22semidirect
46. #14555510D4semidirect
154. C137#3045semidirect
166. 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×#179841102
#225779702
#225778842
#225776242
#225776172
#225776022
#225775972
#225774292
#225774022
C2×#102×C1372
#3045137