Finite Group Databasebeta

← #31790796

Subgroups — #31790796

Order 17536.

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

normal subgroups only

This group has 448 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
1#22577958C211yesmaximal, characteristic, complemented
2C4×D4×D137C211yesmaximal, characteristic
3#22577732C212yesmaximal, complemented
4#22577694C211yesmaximal, characteristic, complemented
5#22577693C211yesmaximal, characteristic, complemented
6#22577605C211yesmaximal, characteristic, complemented
7#22577552C212yesmaximal
8SD16×Dic137C212yesmaximal, complemented
9#22577488C212yesmaximal, complemented
10#22577455C211yesmaximal, characteristic, complemented
11#461×C137C211yesmaximal, characteristic, complemented
12C2×D4×D137C2211yescharacteristic
13C22×C4×D137—21no—
14C2×#14555497C2212yes—
15#17984088C2211yescharacteristic, complemented
16#17984079—21no—
17#17984068—21no—
18#17984064C2212yes—
19Q8×Dic137C2212yescomplemented
20#17984053—21no—
21C4×D548C2211yescharacteristic
22D4×Dic137C2211yescharacteristic
23#17984050C2211yescharacteristic
24C4×C22⋊D137—21no—
25#17984048—21no—
26#17984046—21no—
27C4⋊C4×D137C2211yescharacteristic
28C22⋊C4×D137—21no—
29C42×D137C2211yescharacteristic
30#17984020C2212yescomplemented
31#17984019C2212yescomplemented
32C4⋊#11764040C2211yescharacteristic, complemented
33#17984012C2212yescomplemented
34#17984010C2212yes—
35#17984009C2212yes—
36#17984000C2212yes—
37C8×Dic137C2212yescomplemented
38C2×SD16×C137C2212yes—
39C4⋊Q8×C137C2211yescharacteristic, complemented
40C4×D4×C137C2211yescharacteristic
41C4⋊C8×C137C2211yescharacteristic, complemented
42#103×C137C2212yescomplemented
43C23×D137—21no—
44C2×Dic274—21no—
45C2×D548C2311yescharacteristic
46C2×C22⋊D137—21no—
47D4×D137—22no—
48D4×D137—22no—
49C22×Dic137—21no—
50C2×C4×D137—21no—
51C2×C4×D137—21no—
52C2×C4×D137—21no—
53C2×C4×D137—44no—
54C2×C4×D137D411yescharacteristic
55C2×C4×D137D411yescharacteristic
56C2×C4×D137C2311yescharacteristic
57#14555497—24no—
58C2×#11764040C2312yes—
59C4⋊Dic137—22no—
60C4⋊Dic137—21no—
61C4⋊Dic137C2312yes—
62C4⋊Dic137C2311yescharacteristic
63#14555489—21no—
64#14555489—22no—
65C22⋊Dic137—21no—
66#14555487—22no—
67#14555487—21no—
68#14555487—21no—
69C4×Dic137—22no—
70C4×Dic137D411yescharacteristic
71C4×Dic137D411yescharacteristic
72C4×Dic137C2311yescharacteristic
73C2×Q8×C137C2312yes—
74C2×D4×C137C2311yescharacteristic
75C22×C4×C137—21no—
76SD16×C137—24no—
77C2×C8×C137C2312yes—
78C4⋊C4×C137—21no—
79C4⋊C4×C137—21no—
80C4⋊C4×C137C2311yescharacteristic
81C22⋊C4×C137—21no—
82C42×C137C2311yescharacteristic
83C22×D137—24no—
84C22×D137—21no—
85C22×D137—42no—
86C22×D137—42no—
87C22×D137C2×D411yescharacteristic
88Dic274—42no—
89D548—21no—
90D548#4112yes—
91C22⋊D137—24no—
92C2×Dic137—22no—
93C2×Dic137—21no—
94C2×Dic137—42no—
95C2×Dic137C2×D411yescharacteristic
96C2×Dic137C2×D411yescharacteristic
97C2×Dic137C2×D411yescharacteristic
98C4×D137—21no—
99C4×D137—21no—
100C4×D137—41no—
101C4×D137—42no—
102C4×D137—42no—
103C4×D137—42no—
104C4×D137SD1614yescomplemented
105#11764040—22no—
106C23×C137—21no—
107Q8×C137—22no—
108Q8×C137—22no—
109D4×C137—21no—
110D4×C137#4112yes—
111C2×C4×C137—21no—
112C2×C4×C137—22no—
113C2×C4×C137—42no—
114C2×C4×C137C2×D411yescharacteristic
115C2×C4×C137C2×D411yescharacteristic
116C2×C4×C137C2411yescharacteristic
117C8×C137—22no—
118C22×C137—22no—
119C22×C137—41no—
120C22×C137—41no—
121C22×C137C22×D411yescharacteristic
122C2×D137—21no—
123C2×D137—21no—
124C2×D137—22no—
125C2×D137—48no—
126C2×D137—41no—
127C2×D137—41no—
128C2×D137C2×SD1612yes—
129C4×C137—21no—
130C4×C137—42no—
131C4×C137—41no—
132C4×C137C2×SD1612yes—
133C4×C137C2×#4111yescharacteristic
134C4×C137C2×#4111yescharacteristic
135Dic137—21no—
136Dic137—21no—
137Dic137—41no—
138Dic137—42no—
139Dic137C2×SD1612yes—
140C2×C137—42no—
141C2×C137#52511yescharacteristic
142C2×C137C22×SD1611yescharacteristic
143C2×C137#57511yescharacteristic
144D137—22no—
145D137—42no—
146C137#310111yescharacteristic, complemented, Sylow
147#3101—1371nomaximal, Sylow
148#533—1371no—
149C2×C4×D4—1371no—
150#483—1372no—
151#476—1371no—
152#476—1371no—
153#461—1371no—
154#461D13711yescharacteristic, complemented
155#450—1372no—
156C4×SD16—1372no—
157C4×SD16—1372no—
158#431—1371no—
159C22×D4—1371no—
160C23×C4—2741no—
161C2×SD16—1372no—
162C2×SD16C2×D13712yes—
163C4⋊Q8—1371no—
164C4⋊Q8C2×D13711yescharacteristic, complemented
165#125—2741no—
166C22⋊Q8—2741no—
167C22⋊Q8—1372no—
168C4×Q8—2741no—
169C4×Q8—1372no—
170C4×D4—2741no—
171C4×D4—2741no—
172C4×D4—1371no—
173C4×D4—1371no—
174C4×D4—1371no—
175C4×D4C2×D13711yescharacteristic
176#117—2741no—
177C2×C4⋊C4—1371no—
178C2×C22⋊C4—2741no—
179C2×C42—1371no—
180#106—1372no—
181#106—1372no—
182C4⋊C8—1371no—
183C4⋊C8C2×D13711yescharacteristic, complemented
184#103—1372no—
185#103C2×D13712yescomplemented
186#102—1372no—
187#102—1372no—
188C22⋊C8—1372no—
189C4×C8—1372no—
190C24—2741no—
191C2×Q8—2741no—
192C2×Q8C22×D13712yes—
193C2×D4—2742no—
194C2×D4—2742no—
195C2×D4—2741no—
196C2×D4—1371no—
197C2×D4C22×D13711yescharacteristic
198C22×C4—21no—
199C22×C4—2741no—
200C22×C4—2741no—
201C22×C4—2741no—
202C22×C4—2741no—
203C22×C4—5484no—
204C22×C4—1371no—
205C22×C4—1371no—
206C22×C4—1371no—
207SD16—24no—
208SD16—2744no—
209C2×C8—1372no—
210C2×C8C22×D13712yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—2741no—
214C4⋊C4—2742no—
215C4⋊C4—2742no—
216C4⋊C4—2741no—
217C4⋊C4—1372no—
218C4⋊C4—1371no—
219C4⋊C4C22×D13711yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—2742no—
222C22⋊C4—2741no—
223C22⋊C4—2741no—
224C22⋊C4—2741no—
225C42—2742no—
226C42—1371no—
227C42—1371no—
228C42—1371no—
229C42C22×D13711yescharacteristic
230C23—21no—
231C23—2744no—
232C23—2741no—
233C23—5482no—
234C23—5482no—
235C23—1371no—
236Q8—22no—
237Q8—22no—
238Q8—5482no—
239D4—21no—
240D4—2744no—
241D4—2741no—
242D4—1372no—
243D4#1455550912yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—2741no—
248C2×C4—2741no—
249C2×C4—2742no—
250C2×C4—2741no—
251C2×C4—5481no—
252C2×C4—5482no—
253C2×C4—5482no—
254C2×C4—5482no—
255C2×C4—5482no—
256C2×C4—1371no—
257C2×C4—1371no—
258C2×C4—1371no—
259C2×C4—1374no—
260C2×C4D4×D13711yescharacteristic
261C2×C4D4×D13711yescharacteristic
262C2×C4C23×D13711yescharacteristic
263C8—22no—
264C8—2742no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—2741no—
269C22—2741no—
270C22—2742no—
271C22—5488no—
272C22—5481no—
273C22—5481no—
274C22—1372no—
275C22C2×D4×D13711yescharacteristic
276C4—21no—
277C4—42no—
278C4—41no—
279C4—2741no—
280C4—2741no—
281C4—5482no—
282C4—5481no—
283C4—1372no—
284C4SD16×D13712yes—
285C4#41×D13711yescharacteristic
286C4C2×#1455550911yescharacteristic
287C2—42no—
288C2—2742no—
289C2—5482no—
290C2#2257791311yescharacteristic
291C2C2×SD16×D13711yescharacteristic
292C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#461×C137C2
socleC22×C137C22×D4
prime radicalC2×D4×D137C22
hypercenter#461D137
hypocenterC137#3101
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #22577958C2semidirect
3. #22577732C2semidirect
4. #22577694C2semidirect
5. #22577693C2semidirect
6. #22577605C2semidirect
8. SD16×Dic137C2semidirect
9. #22577488C2semidirect
10. #22577455C2semidirect
11. #461×C137C2semidirect
15. #17984088C22semidirect
19. Q8×Dic137C22semidirect
30. #17984020C22semidirect
31. #17984019C22semidirect
32. C4⋊#11764040C22semidirect
33. #17984012C22semidirect
37. C8×Dic137C22semidirect
39. C4⋊Q8×C137C22semidirect
41. C4⋊C8×C137C22semidirect
42. #103×C137C22semidirect
104. C4×D137SD16semidirect
146. C137#3101semidirect
154. #461D137semidirect
164. C4⋊Q8C2×D137semidirect
183. C4⋊C8C2×D137semidirect
185. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779582
C4×D4×D1372
#225777322
#225776942
#225776932
#225776052
#225775522
SD16×Dic1372
#225774882
#225774552
#461×C1372
#3101137