Finite Group Databasebeta

← #31790727

Subgroups — #31790727

Order 17536.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#17984059C211yesmaximal, characteristic
2#117×D137C211yesmaximal, characteristic, complemented
3#22577726C212yesmaximal, complemented
4#22577725C212yesmaximal, complemented
5#22577720C212yesmaximal, complemented
6#22577715C212yesmaximal, complemented
7#22577446C211yesmaximal, characteristic, complemented
8#22577445C211yesmaximal, characteristic, complemented
9#22577408C211yesmaximal, characteristic, complemented
10#22577366C211yesmaximal, characteristic, complemented
11#418×C137C211yesmaximal, characteristic, complemented
12C22×C22⋊D137—21no—
13C22×C4×D137C2211yescharacteristic
14C2×#14555494C2211yescharacteristic
15#17984059—21no—
16#17984059—21no—
17#17984059C2212yes—
18#17984059C2212yes—
19#17984047C2211yescharacteristic
20#17984046C2211yescharacteristic, complemented
21#17984046C2211yescharacteristic, complemented
22#17984045—21no—
23#17984044—21no—
24C2×C4⋊Dic137C2211yescharacteristic
25C4⋊C4×D137C2211yescharacteristic, complemented
26C4⋊C4×D137C2211yescharacteristic, complemented
27C2×C22⋊Dic137—21no—
28C22⋊C4×D137—21no—
29C42×D137—21no—
30#17984022C2212yescomplemented
31#17984021C2212yescomplemented
32#17984020C2212yescomplemented
33#17984019C2212yescomplemented
34#17984009C2212yescomplemented
35#17984009C2212yescomplemented
36#17984000C2212yescomplemented
37#17984000C2212yescomplemented
38C22×D4×C137C2211yescharacteristic
39C2×M16×C137C2211yescharacteristic
40#117×C137C2211yescharacteristic
41#102×C137C2212yescomplemented
42#102×C137C2212yescomplemented
43C23×D137D411yescharacteristic
44C2×C22⋊D137—22no—
45C2×C22⋊D137—22no—
46C2×C22⋊D137—44no—
47C22×Dic137—21no—
48C22×Dic137D411yescharacteristic
49C2×C4×D137—21no—
50C2×C4×D137—21no—
51C2×C4×D137—22no—
52C2×C4×D137—22no—
53C2×C4×D137D411yescharacteristic
54C2×C4×D137D411yescharacteristic
55C2×C4×D137C2311yescharacteristic
56C2×C4×D137C2311yescharacteristic
57#14555494—22no—
58C2×#11764040C2312yes—
59C4⋊Dic137—21no—
60C4⋊Dic137C2311yescharacteristic
61C4⋊Dic137C2311yescharacteristic
62C4⋊Dic137C2312yes—
63#14555489—21no—
64#14555489—21no—
65C22⋊Dic137—22no—
66C22⋊Dic137—22no—
67C22⋊Dic137—21no—
68#14555487—21no—
69#14555487—21no—
70C4×Dic137—21no—
71C4×Dic137—21no—
72C4×Dic137—21no—
73C24×C137—21no—
74C2×D4×C137—21no—
75C2×D4×C137—22no—
76C2×D4×C137—22no—
77C2×D4×C137C2312yes—
78C22×C4×C137C2311yescharacteristic
79M16×C137—22no—
80C2×C8×C137C2312yes—
81C4⋊C4×C137C2311yescharacteristic
82C4⋊C4×C137C2311yescharacteristic
83C22⋊C4×C137—21no—
84C42×C137—21no—
85C22×D137—22no—
86C22×D137—22no—
87C22×D137—41no—
88C22×D137C2×D411yescharacteristic
89C22×D137C2×D411yescharacteristic
90C22⋊D137—44no—
91C22⋊D137—44no—
92C2×Dic137—22no—
93C2×Dic137—21no—
94C2×Dic137—21no—
95C2×Dic137—22no—
96C2×Dic137—42no—
97C2×Dic137C2×D411yescharacteristic
98C2×Dic137C2×D411yescharacteristic
99C4×D137—22no—
100C4×D137—22no—
101C4×D137—21no—
102C4×D137—21no—
103C4×D137—21no—
104C4×D137—21no—
105C4×D137—42no—
106C4×D137—42no—
107#11764040—22no—
108C23×C137—22no—
109C23×C137—42no—
110C23×C137—42no—
111C23×C137—42no—
112C23×C137C2×D411yescharacteristic
113D4×C137—22no—
114D4×C137—22no—
115D4×C137—42no—
116C2×C4×C137—21no—
117C2×C4×C137—21no—
118C2×C4×C137C2×D411yescharacteristic
119C2×C4×C137#4112yes—
120C2×C4×C137#4112yes—
121C2×C4×C137C2411yescharacteristic
122C8×C137—22no—
123C22×C137—21no—
124C22×C137—21no—
125C22×C137—42no—
126C22×C137—42no—
127C22×C137—42no—
128C22×C137—84no—
129C22×C137#12312yes—
130C22×C137C22×D411yescharacteristic
131C2×D137—21no—
132C2×D137—21no—
133C2×D137—21no—
134C2×D137—21no—
135C2×D137—21no—
136C2×D137—44no—
137C2×D137AGL(1,ℤ/8)12yes—
138C4×C137—41no—
139C4×C137—41no—
140C4×C137#12312yes—
141C4×C137C2×#4111yescharacteristic
142C4×C137C2×#4111yescharacteristic
143Dic137—21no—
144Dic137—21no—
145Dic137—41no—
146Dic137—42no—
147Dic137—41no—
148C2×C137—22no—
149C2×C137—82no—
150C2×C137C2×#12311yescharacteristic
151C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C137#57511yescharacteristic
153D137—22no—
154D137—41no—
155C137#309411yescharacteristic, complemented, Sylow
156#3094—1371nomaximal, Sylow
157C2×#121—1371no—
158C2×#117—1371no—
159#482—1372no—
160#482—1372no—
161#481—1372no—
162#480—1372no—
163#428—1371no—
164#428—1371no—
165#418—1371no—
166#418D13711yescharacteristic, complemented
167#407—1371no—
168C22×D4—2741no—
169C22×D4C2×D13711yescharacteristic
170C23×C4—1371no—
171C2×M16—1371no—
172C2×M16C2×D13711yescharacteristic
173#121—2741no—
174#121—2741no—
175#121—1372no—
176#121—1372no—
177#117—2741no—
178#117—2741no—
179#117—1371no—
180#117—1371no—
181#117—1371no—
182#117C2×D13711yescharacteristic
183C2×C4⋊C4—1371no—
184C2×C4⋊C4—1371no—
185C2×C4⋊C4—1371no—
186C2×C22⋊C4—2741no—
187C2×C22⋊C4—2741no—
188C2×C42—2741no—
189#107—1372no—
190#107—1372no—
191#106—1372no—
192#106—1372no—
193#102—1372no—
194#102—1372no—
195#102C2×D13712yescomplemented
196#102C2×D13712yescomplemented
197C22⋊C8—1372no—
198C22⋊C8—1372no—
199C24—21no—
200C24—1371no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—2742no—
205C2×D4—2742no—
206C2×D4—5484no—
207C2×D4C22×D13712yes—
208C22×C4—2741no—
209C22×C4—2741no—
210C22×C4—2742no—
211C22×C4—2742no—
212C22×C4—2741no—
213C22×C4—1371no—
214C22×C4—1371no—
215C22×C4—1371no—
216C22×C4—1371no—
217C22×C4—1371no—
218C22×C4C22×D13711yescharacteristic
219M16—22no—
220M16—2742no—
221C2×C8—1372no—
222C2×C8C22×D13712yes—
223C4⋊C4—2741no—
224C4⋊C4—2741no—
225C4⋊C4—2741no—
226C4⋊C4—1371no—
227C4⋊C4—1371no—
228C4⋊C4—1372no—
229C4⋊C4C22×D13711yescharacteristic
230C4⋊C4C22×D13711yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—2741no—
233C22⋊C4—2741no—
234C22⋊C4—2741no—
235C22⋊C4—2742no—
236C22⋊C4—2742no—
237C42—21no—
238C42—2741no—
239C42—2741no—
240C42—2741no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—2742no—
246C23—2742no—
247C23—5481no—
248C23—1371no—
249C23—1371no—
250C23D4×D13711yescharacteristic
251D4—22no—
252D4—22no—
253D4—42no—
254D4—5484no—
255D4—5484no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—2742no—
259C2×C4—2742no—
260C2×C4—2742no—
261C2×C4—2741no—
262C2×C4—2741no—
263C2×C4—2741no—
264C2×C4—2741no—
265C2×C4—2741no—
266C2×C4—2741no—
267C2×C4—2742no—
268C2×C4—5482no—
269C2×C4—5482no—
270C2×C4—5482no—
271C2×C4—1371no—
272C2×C4—1371no—
273C2×C4D4×D13711yescharacteristic
274C2×C4#1455550912yes—
275C2×C4#1455551012yes—
276C2×C4C23×D13711yescharacteristic
277C8—22no—
278C8—2742no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—2741no—
286C22—2741no—
287C22—2741no—
288C22—2741no—
289C22—2741no—
290C22—5484no—
291C22—1372no—
292C22#1798407012yes—
293C22C2×D4×D13711yescharacteristic
294C4—41no—
295C4—41no—
296C4—2741no—
297C4—2741no—
298C4—5482no—
299C4—5481no—
300C4—5481no—
301C4#1798407012yes—
302C4C2×#1455550911yescharacteristic
303C4C2×#1455551011yescharacteristic
304C2—22no—
305C2—82no—
306C2—2742no—
307C2—5481no—
308C2C2×#1798407011yescharacteristic
309C2AGL(1,ℤ/8)×D13711yescharacteristic
310C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#418×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984059C2
hypercenter#418D137
hypocenterC137#3094
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #117×D137C2semidirect
3. #22577726C2semidirect
4. #22577725C2semidirect
5. #22577720C2semidirect
6. #22577715C2semidirect
7. #22577446C2semidirect
8. #22577445C2semidirect
9. #22577408C2semidirect
10. #22577366C2semidirect
11. #418×C137C2semidirect
20. #17984046C22semidirect
21. #17984046C22semidirect
25. C4⋊C4×D137C22semidirect
26. C4⋊C4×D137C22semidirect
30. #17984022C22semidirect
31. #17984021C22semidirect
32. #17984020C22semidirect
33. #17984019C22semidirect
34. #17984009C22semidirect
35. #17984009C22semidirect
36. #17984000C22semidirect
37. #17984000C22semidirect
41. #102×C137C22semidirect
42. #102×C137C22semidirect
155. C137#3094semidirect
166. #418D137semidirect
195. #102C2×D137semidirect
196. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840592
#117×D1372
#225777262
#225777252
#225777202
#225777152
#225774462
#225774452
#225774082
#225773662
#418×C1372
#3094137