Finite Group Databasebeta

← #31790799

Subgroups — #31790799

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#22577735C212yesmaximal, complemented
4#22577690C211yesmaximal, characteristic, complemented
5#22577609C211yesmaximal, characteristic, complemented
6#22577608C211yesmaximal, characteristic, complemented
7#22577555C212yesmaximal
8#22577494C212yesmaximal, complemented
9#22577490C212yesmaximal, complemented
10#22577455C211yesmaximal, characteristic, complemented
11#476×C137C211yesmaximal, characteristic, complemented
12C2×D4×D137C2211yescharacteristic
13C22×C4×D137—21no—
14C2×#14555499C2212yes—
15C2×#14555498C2212yes—
16#17984088C2211yescharacteristic, complemented
17#17984079—21no—
18#17984068C2212yes—
19#17984064—21no—
20Q8×Dic137—21no—
21#17984053C2212yescomplemented
22C4×D548C2211yescharacteristic
23D4×Dic137C2211yescharacteristic
24#17984050C2211yescharacteristic
25C4×C22⋊D137—21no—
26#17984048—21no—
27#17984046—21no—
28C4⋊C4×D137C2211yescharacteristic
29C22⋊C4×D137—21no—
30C42×D137C2211yescharacteristic
31#17984019C2212yescomplemented
32C4⋊#11764040C2211yescharacteristic, complemented
33#17984013C2212yescomplemented
34#17984011C2212yescomplemented
35#17984008C2212yes—
36#17984000C2212yes—
37C8×Dic137C2212yescomplemented
38C4⋊Q8×C137C2211yescharacteristic, complemented
39C4×D4×C137C2211yescharacteristic
40#106×C137C2212yescomplemented
41C4⋊C8×C137C2211yescharacteristic, complemented
42#102×C137C2212yes—
43C23×D137—21no—
44C2×Dic274C2312yes—
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#14555499—24no—
58#14555498—24no—
59C2×#11764040C2312yes—
60C4⋊Dic137—22no—
61C4⋊Dic137—21no—
62C4⋊Dic137—21no—
63C4⋊Dic137C2311yescharacteristic
64#14555489—21no—
65#14555489—22no—
66C22⋊Dic137—21no—
67#14555487—22no—
68#14555487—21no—
69#14555487—21no—
70C4×Dic137—22no—
71C4×Dic137D411yescharacteristic
72C4×Dic137D411yescharacteristic
73C4×Dic137C2311yescharacteristic
74C2×Q8×C137—21no—
75C2×D4×C137C2311yescharacteristic
76C22×C4×C137—21no—
77C2×C8×C137C2312yes—
78C4⋊C4×C137—21no—
79C4⋊C4×C137C2312yes—
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—22no—
89Dic274—22no—
90D548—21no—
91D548#4112yes—
92C22⋊D137—24no—
93C2×Dic137—22no—
94C2×Dic137—21no—
95C2×Dic137—42no—
96C2×Dic137C2×D411yescharacteristic
97C2×Dic137C2×D411yescharacteristic
98C2×Dic137C2×D411yescharacteristic
99C4×D137—21no—
100C4×D137—21no—
101C4×D137—41no—
102C4×D137—42no—
103C4×D137—42no—
104C4×D137—42no—
105C4×D137SD1614yescomplemented
106#11764040—22no—
107C23×C137—21no—
108Q8×C137—42no—
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—41no—
131C4×C137—42no—
132C4×C137C2×SD1612yes—
133C4×C137C2×#4111yescharacteristic
134C4×C137C2×#4111yescharacteristic
135Dic137—21no—
136Dic137—21no—
137Dic137—42no—
138Dic137—41no—
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#476D13711yescharacteristic, complemented
153#461—1371no—
154#461—1371no—
155#450—1372no—
156C4×SD16—1372no—
157C4×SD16—1372no—
158#431—1371no—
159C22×D4—1371no—
160C23×C4—2741no—
161C2×SD16—1372no—
162C2×SD16—1372no—
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#106C2×D13712yescomplemented
182C4⋊C8—1371no—
183C4⋊C8C2×D13711yescharacteristic, complemented
184#103—1372no—
185#103—1372no—
186#102—1372no—
187#102C2×D13712yes—
188C22⋊C8—1372no—
189C4×C8—1372no—
190C24—2741no—
191C2×Q8—21no—
192C2×Q8—1372no—
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—2744no—
208SD16—2744no—
209C2×C8—1372no—
210C2×C8C22×D13712yes—
211C4⋊C4—21no—
212C4⋊C4—2741no—
213C4⋊C4—2742no—
214C4⋊C4—2742no—
215C4⋊C4—2741no—
216C4⋊C4—2741no—
217C4⋊C4—1371no—
218C4⋊C4C22×D13712yes—
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—42no—
237Q8—2742no—
238Q8—2742no—
239D4—21no—
240D4—2744no—
241D4—2741no—
242D4—1372no—
243D4#1455551112yes—
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×#1455551111yescharacteristic
287C2—42no—
288C2—2742no—
289C2—5482no—
290C2#2257791611yescharacteristic
291C2C2×SD16×D13711yescharacteristic
292C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#476×C137C2
socleC22×C137C22×D4
prime radicalC2×D4×D137C22
hypercenter#476D137
hypocenterC137#3101
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #22577958C2semidirect
3. #22577735C2semidirect
4. #22577690C2semidirect
5. #22577609C2semidirect
6. #22577608C2semidirect
8. #22577494C2semidirect
9. #22577490C2semidirect
10. #22577455C2semidirect
11. #476×C137C2semidirect
16. #17984088C22semidirect
21. #17984053C22semidirect
31. #17984019C22semidirect
32. C4⋊#11764040C22semidirect
33. #17984013C22semidirect
34. #17984011C22semidirect
37. C8×Dic137C22semidirect
38. C4⋊Q8×C137C22semidirect
40. #106×C137C22semidirect
41. C4⋊C8×C137C22semidirect
105. C4×D137SD16semidirect
146. C137#3101semidirect
152. #476D137semidirect
164. C4⋊Q8C2×D137semidirect
181. #106C2×D137semidirect
183. C4⋊C8C2×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
#225777352
#225776902
#225776092
#225776082
#225775552
#225774942
#225774902
#225774552
#476×C1372
#3101137