Finite Group Databasebeta

← #31790735

Subgroups — #31790735

Order 17536.

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

normal subgroups only

This group has 410 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×#17984064C211yesmaximal, characteristic
2#117×D137C211yesmaximal, characteristic
3#22577743C212yesmaximal, complemented
4#22577742C212yesmaximal, complemented
5#22577737C212yesmaximal, complemented
6#22577732C212yesmaximal, complemented
7#22577446C211yesmaximal, characteristic, complemented
8#22577445C211yesmaximal, characteristic, complemented
9#22577412C211yesmaximal, characteristic, complemented
10#22577366C211yesmaximal, characteristic
11#419×C137C211yesmaximal, characteristic, complemented
12C22×C4×D137C2211yescharacteristic
13C2×#14555494C2211yescharacteristic
14#17984064—21no—
15#17984064—21no—
16#17984064C2212yes—
17#17984064C2212yes—
18#17984047C2211yescharacteristic
19#17984046C2211yescharacteristic
20#17984046C2211yescharacteristic
21#17984045—21no—
22#17984044—21no—
23C2×C4⋊Dic137C2211yescharacteristic
24C2×#14555489—21no—
25C4⋊C4×D137C2211yescharacteristic
26C4⋊C4×D137C2211yescharacteristic
27C2×#14555487—21no—
28C22⋊C4×D137—21no—
29C42×D137—21no—
30#17984022C2212yescomplemented
31#17984021C2212yescomplemented
32#17984020C2212yescomplemented
33#17984019C2212yescomplemented
34#17984012C2212yescomplemented
35#17984012C2212yescomplemented
36#17984000C2212yes—
37#17984000C2212yes—
38C22×Q8×C137C2211yescharacteristic
39C2×M16×C137C2211yescharacteristic
40#117×C137C2211yescharacteristic
41#103×C137C2212yescomplemented
42#103×C137C2212yescomplemented
43C23×D137D411yescharacteristic
44C22×Dic137—21no—
45C22×Dic137D411yescharacteristic
46C2×C4×D137—21no—
47C2×C4×D137—21no—
48C2×C4×D137—22no—
49C2×C4×D137—22no—
50C2×C4×D137D411yescharacteristic
51C2×C4×D137D411yescharacteristic
52C2×C4×D137C2311yescharacteristic
53C2×C4×D137C2311yescharacteristic
54#14555494—22no—
55C2×#11764040C2312yes—
56C4⋊Dic137—21no—
57C4⋊Dic137C2311yescharacteristic
58C4⋊Dic137C2311yescharacteristic
59C4⋊Dic137C2312yes—
60#14555489—21no—
61#14555489—21no—
62#14555489—22no—
63#14555489—22no—
64C22⋊Dic137—21no—
65#14555487—21no—
66#14555487—21no—
67#14555487—22no—
68#14555487—22no—
69C4×Dic137—21no—
70C4×Dic137—21no—
71C4×Dic137—21no—
72C2×Q8×C137—21no—
73C2×Q8×C137—22no—
74C2×Q8×C137—22no—
75C2×Q8×C137C2312yes—
76C22×C4×C137—21no—
77C22×C4×C137C2311yescharacteristic
78M16×C137—22no—
79C2×C8×C137C2312yes—
80C4⋊C4×C137C2311yescharacteristic
81C4⋊C4×C137C2311yescharacteristic
82C22⋊C4×C137—21no—
83C42×C137—21no—
84C22×D137—22no—
85C22×D137—22no—
86C22×D137—41no—
87C22×D137C2×D411yescharacteristic
88C22×D137C2×D411yescharacteristic
89C2×Dic137—22no—
90C2×Dic137—21no—
91C2×Dic137—21no—
92C2×Dic137—22no—
93C2×Dic137—42no—
94C2×Dic137C2×D411yescharacteristic
95C2×Dic137C2×D411yescharacteristic
96C4×D137—22no—
97C4×D137—22no—
98C4×D137—21no—
99C4×D137—21no—
100C4×D137—21no—
101C4×D137—21no—
102C4×D137—42no—
103C4×D137—42no—
104#11764040—22no—
105C23×C137C2×D411yescharacteristic
106Q8×C137—22no—
107Q8×C137—22no—
108Q8×C137—42no—
109C2×C4×C137—21no—
110C2×C4×C137—21no—
111C2×C4×C137—22no—
112C2×C4×C137—42no—
113C2×C4×C137C2×D411yescharacteristic
114C2×C4×C137#4112yes—
115C2×C4×C137#4112yes—
116C2×C4×C137C2411yescharacteristic
117C8×C137—22no—
118C22×C137—21no—
119C22×C137—21no—
120C22×C137#12312yes—
121C22×C137C22×D411yescharacteristic
122C2×D137—21no—
123C2×D137—21no—
124C2×D137—21no—
125C2×D137—21no—
126C2×D137—21no—
127C2×D137—44no—
128C2×D137#13712yes—
129C4×C137—42no—
130C4×C137—41no—
131C4×C137—41no—
132C4×C137#12312yes—
133C4×C137C2×#4111yescharacteristic
134C4×C137C2×#4111yescharacteristic
135Dic137—21no—
136Dic137—21no—
137Dic137—41no—
138Dic137—41no—
139Dic137—42no—
140C2×C137—22no—
141C2×C137C2×#12311yescharacteristic
142C2×C137C2×#13711yescharacteristic
143C2×C137#57511yescharacteristic
144D137—22no—
145D137—41no—
146C137#309511yescharacteristic, complemented, Sylow
147#3095—1371nomaximal, Sylow
148C2×C22⋊Q8—1371no—
149C2×#117—1371no—
150#485—1372no—
151#485—1372no—
152#484—1372no—
153#483—1372no—
154#428—1371no—
155#428—1371no—
156#419—1371no—
157#419D13711yescharacteristic, complemented
158#407—1371no—
159C22×Q8C2×D13711yescharacteristic
160C23×C4—1371no—
161C2×M16—1371no—
162C2×M16C2×D13711yescharacteristic
163C22⋊Q8—2741no—
164C22⋊Q8—2741no—
165C22⋊Q8—1372no—
166C22⋊Q8—1372no—
167#117—2741no—
168#117—2741no—
169#117—1371no—
170#117—1371no—
171#117—1371no—
172#117C2×D13711yescharacteristic
173C2×C4⋊C4—2741no—
174C2×C4⋊C4—1371no—
175C2×C4⋊C4—1371no—
176C2×C4⋊C4—1371no—
177C2×C22⋊C4—2741no—
178C2×C22⋊C4—2741no—
179C2×C42—2741no—
180#107—1372no—
181#107—1372no—
182#106—1372no—
183#106—1372no—
184#103—1372no—
185#103—1372no—
186#103C2×D13712yescomplemented
187#103C2×D13712yescomplemented
188C22⋊C8—1372no—
189C22⋊C8—1372no—
190C24—1371no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8C22×D13712yes—
195C22×C4—21no—
196C22×C4—2741no—
197C22×C4—2741no—
198C22×C4—2742no—
199C22×C4—2742no—
200C22×C4—2741no—
201C22×C4—1371no—
202C22×C4—1371no—
203C22×C4—1371no—
204C22×C4—1371no—
205C22×C4—1371no—
206C22×C4C22×D13711yescharacteristic
207M16—22no—
208M16—2742no—
209C2×C8—1372no—
210C2×C8C22×D13712yes—
211C4⋊C4—2742no—
212C4⋊C4—2742no—
213C4⋊C4—2741no—
214C4⋊C4—2741no—
215C4⋊C4—2741no—
216C4⋊C4—1371no—
217C4⋊C4—1371no—
218C4⋊C4—1372no—
219C4⋊C4C22×D13711yescharacteristic
220C4⋊C4C22×D13711yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—2741no—
223C22⋊C4—2741no—
224C22⋊C4—2742no—
225C22⋊C4—2742no—
226C22⋊C4—2741no—
227C42—21no—
228C42—2741no—
229C42—2741no—
230C42—2741no—
231C23—2742no—
232C23—2742no—
233C23—5481no—
234C23—1371no—
235C23—1371no—
236C23D4×D13711yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—2742no—
245C2×C4—2742no—
246C2×C4—2742no—
247C2×C4—2741no—
248C2×C4—2741no—
249C2×C4—2741no—
250C2×C4—2741no—
251C2×C4—2741no—
252C2×C4—2741no—
253C2×C4—2742no—
254C2×C4—5482no—
255C2×C4—5482no—
256C2×C4—5482no—
257C2×C4—1371no—
258C2×C4—1371no—
259C2×C4D4×D13711yescharacteristic
260C2×C4#1455550912yes—
261C2×C4#1455551012yes—
262C2×C4C23×D13711yescharacteristic
263C8—22no—
264C8—2742no—
265C22—21no—
266C22—21no—
267C22—2741no—
268C22—2741no—
269C22—2741no—
270C22—2741no—
271C22—2741no—
272C22—5484no—
273C22—1372no—
274C22#1798407012yes—
275C22C2×D4×D13711yescharacteristic
276C4—41no—
277C4—41no—
278C4—42no—
279C4—2741no—
280C4—2741no—
281C4—5481no—
282C4—5481no—
283C4—5482no—
284C4#1798407012yes—
285C4C2×#1455550911yescharacteristic
286C4C2×#1455551011yescharacteristic
287C2—22no—
288C2—2742no—
289C2—5481no—
290C2C2×#1798407011yescharacteristic
291C2#137×D13711yescharacteristic
292C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#419×C137C2
socleC22×C137C22×D4
prime radicalC23×D137D4
hypercenter#419D137
hypocenterC137#3095
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #22577743C2semidirect
4. #22577742C2semidirect
5. #22577737C2semidirect
6. #22577732C2semidirect
7. #22577446C2semidirect
8. #22577445C2semidirect
9. #22577412C2semidirect
11. #419×C137C2semidirect
30. #17984022C22semidirect
31. #17984021C22semidirect
32. #17984020C22semidirect
33. #17984019C22semidirect
34. #17984012C22semidirect
35. #17984012C22semidirect
41. #103×C137C22semidirect
42. #103×C137C22semidirect
146. C137#3095semidirect
157. #419D137semidirect
186. #103C2×D137semidirect
187. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840642
#117×D1372
#225777432
#225777422
#225777372
#225777322
#225774462
#225774452
#225774122
#225773662
#419×C1372
#3095137