Finite Group Databasebeta

← #448×D157 #34109321

Subgroups — #448×D157 #34109321

Order 20096.

792 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×Q8×D157C211yesmaximal, characteristic, complemented
2C2×SD16×D157C212yesmaximal, complemented
3#121×D157C211yesmaximal, characteristic
4#23757414C211yesmaximal, characteristic, complemented
5#23757413C212yesmaximal, complemented
6#23757412C211yesmaximal, characteristic, complemented
7#23757411C211yesmaximal, characteristic, complemented
8#23757410C212yesmaximal, complemented
9#103×D157C212yesmaximal, complemented
10C22⋊C8×D157C211yesmaximal, characteristic, complemented
11#448×C157C211yesmaximal, characteristic, complemented
12C22×Dic314—21no—
13C22×Dic314C2211yescharacteristic, complemented
14C2×Q8×D157—28no—
15C2×Q8×D157—21no—
16C2×Q8×D157—21no—
17C2×Q8×D157C2212yes—
18C2×Q8×D157C2212yescomplemented
19C2×D4×D157—21no—
20C2×D4×D157C2211yescharacteristic
21C22×C4×D157—21no—
22C22×C4×D157C2211yescharacteristic
23C2×#14578943C2212yescomplemented
24C2×#14578942C2212yescomplemented
25C2×#14578941C2212yescomplemented
26SD16×D157—28no—
27C2×C8×D157C2212yes—
28#19097746C2211yescharacteristic
29#19097745C2211yescharacteristic
30#19097744—21no—
31#19097743—21no—
32#19097742C2211yescharacteristic
33C4⋊C4×D157C2211yescharacteristic
34C22⋊C4×D157—21no—
35#19097696C2212yescomplemented
36#19097695C2212yescomplemented
37#19097694C2212yescomplemented
38C22⋊#11929434C2211yescharacteristic, complemented
39#19097683C2212yescomplemented
40C22×Q8×C157C2211yescharacteristic, complemented
41C2×SD16×C157C2212yescomplemented
42#121×C157C2211yescharacteristic
43#103×C157C2212yescomplemented
44C22⋊C8×C157C2211yescharacteristic, complemented
45C23×D157—21no—
46C23×D157D411yescharacteristic
47C2×Dic314—28no—
48C2×Dic314—24no—
49C2×Dic314—22no—
50C2×Dic314—22no—
51C2×Dic314—21no—
52C2×Dic314C2312yes—
53Q8×D157—28no—
54Q8×D157—28no—
55Q8×D157—24no—
56Q8×D157—24no—
57Q8×D157—24no—
58Q8×D157D418yescomplemented
59C2×D628—21no—
60C2×D628C2311yescharacteristic
61C2×C22⋊D157—21no—
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64D4×D157—24no—
65D4×D157—44no—
66C22×Dic157—21no—
67C22×Dic157D411yescharacteristic
68C2×C4×D157—28no—
69C2×C4×D157—22no—
70C2×C4×D157—22no—
71C2×C4×D157—24no—
72C2×C4×D157—21no—
73C2×C4×D157D411yescharacteristic
74C2×C4×D157D411yescharacteristic
75C2×C4×D157C2311yescharacteristic
76C2×C4×D157C2311yescharacteristic
77#14578943—24no—
78#14578942—24no—
79#14578941—24no—
80C2×#11929434C2312yes—
81C8×D157—24no—
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
84C22⋊Dic157—21no—
85#14578931—21no—
86#14578931—21no—
87C2×Q8×C157—24no—
88C2×Q8×C157—21no—
89C2×Q8×C157C2312yes—
90C2×D4×C157—21no—
91C2×D4×C157C2311yescharacteristic
92C22×C4×C157—21no—
93C22×C4×C157C2311yescharacteristic
94SD16×C157—24no—
95C2×C8×C157C2312yes—
96C4⋊C4×C157C2311yescharacteristic
97C22⋊C4×C157—21no—
98C22×D157—22no—
99C22×D157—22no—
100C22×D157—21no—
101C22×D157—42no—
102C22×D157—42no—
103C22×D157—42no—
104C22×D157SD1614yescomplemented
105C22×D157C2×D411yescharacteristic
106C22×D157C2×D411yescharacteristic
107Dic314—24no—
108Dic314—28no—
109Dic314—22no—
110Dic314—28no—
111Dic314C2×D414yescomplemented
112D628—22no—
113D628—42no—
114C22⋊D157—42no—
115C22⋊D157—42no—
116C22⋊D157—42no—
117C2×Dic157—24no—
118C2×Dic157—22no—
119C2×Dic157—22no—
120C2×Dic157—21no—
121C2×Dic157C2×D411yescharacteristic
122C2×Dic157C2×D411yescharacteristic
123C4×D157—28no—
124C4×D157—28no—
125C4×D157—22no—
126C4×D157—22no—
127C4×D157—22no—
128C4×D157—42no—
129C4×D157C2×D414yes—
130#11929434—22no—
131C23×C157—21no—
132C23×C157C2×D411yescharacteristic
133Q8×C157—24no—
134Q8×C157—22no—
135Q8×C157C2×D414yescomplemented
136D4×C157—22no—
137D4×C157—42no—
138C2×C4×C157—24no—
139C2×C4×C157—22no—
140C2×C4×C157—22no—
141C2×C4×C157—21no—
142C2×C4×C157C2×D411yescharacteristic
143C2×C4×C157C2411yescharacteristic
144C8×C157—22no—
145C22×C157—21no—
146C22×C157—21no—
147C22×C157—41no—
148C22×C157—41no—
149C22×C157—41no—
150C22×C157C2×SD1612yescomplemented
151C22×C157C22×D411yescharacteristic
152C2×D157—21no—
153C2×D157—21no—
154C2×D157—28no—
155C2×D157—41no—
156C2×D157—41no—
157C2×D157—41no—
158C2×D157—84no—
159C2×D157#12012yes—
160C2×D157C2×SD1612yes—
161C2×D157C2×SD1612yescomplemented
162C2×D157#13712yes—
163C4×C157—24no—
164C4×C157—21no—
165C4×C157—41no—
166C4×C157C22×D412yes—
167Dic157—24no—
168Dic157—21no—
169Dic157—41no—
170Dic157#12012yes—
171C2×C157—22no—
172C2×C157—81no—
173C2×C157C2×#12011yescharacteristic
174C2×C157C22×SD1611yescharacteristic
175C2×C157C2×#13711yescharacteristic
176D157—22no—
177D157—81no—
178D157#44814yesdirect product factor
179C157C2×#44811yescharacteristic, complemented, Sylow
180C2×#448—1571nomaximal, Sylow
181C23×Q8—1571no—
182C22×SD16—1572no—
183C2×#121—1571no—
184#448—1571no—
185#448—1572no—
186#448—1571no—
187#448—1572no—
188#448—1571no—
189#448D15711yescharacteristic, direct product factor
190C2×#103—1572no—
191C2×C22⋊C8—1571no—
192C22×Q8—3141no—
193C22×Q8—3148no—
194C22×Q8—3141no—
195C22×Q8—3141no—
196C22×Q8—1572no—
197C22×Q8—1572no—
198C22×Q8—1571no—
199C22×Q8C2×D15711yescharacteristic, complemented
200C22×D4—3141no—
201C22×D4—1571no—
202C23×C4—3141no—
203C23×C4—1571no—
204C2×SD16—3148no—
205C2×SD16—1572no—
206C2×SD16—1572no—
207C2×SD16—1572no—
208C2×SD16C2×D15712yescomplemented
209C22×C8—1572no—
210#121—3141no—
211#121—3141no—
212#121—1571no—
213#121—1571no—
214#121—1571no—
215#121C2×D15711yescharacteristic
216C2×C4⋊C4—1571no—
217C2×C22⋊C4—3141no—
218#103—1572no—
219#103—1572no—
220#103—1572no—
221#103C2×D15712yescomplemented
222C22⋊C8—1571no—
223C22⋊C8—1572no—
224C22⋊C8C2×D15711yescharacteristic, complemented
225C24—3141no—
226C24—1571no—
227C2×Q8—24no—
228C2×Q8—21no—
229C2×Q8—3148no—
230C2×Q8—3144no—
231C2×Q8—3142no—
232C2×Q8—3142no—
233C2×Q8—3148no—
234C2×Q8—3148no—
235C2×Q8—3144no—
236C2×Q8—3144no—
237C2×Q8—3144no—
238C2×Q8—3141no—
239C2×Q8—1578no—
240C2×Q8—1572no—
241C2×Q8C22×D15712yes—
242C2×D4—21no—
243C2×D4—3141no—
244C2×D4—3141no—
245C2×D4—3141no—
246C2×D4—3144no—
247C2×D4—3141no—
248C2×D4—6284no—
249C2×D4—1571no—
250C2×D4C22×D15711yescharacteristic
251C22×C4—21no—
252C22×C4—3148no—
253C22×C4—3142no—
254C22×C4—3142no—
255C22×C4—3144no—
256C22×C4—3141no—
257C22×C4—3141no—
258C22×C4—1571no—
259C22×C4—1571no—
260C22×C4—1571no—
261C22×C4—1571no—
262C22×C4—1571no—
263C22×C4C22×D15711yescharacteristic
264SD16—24no—
265SD16—3144no—
266SD16—3144no—
267SD16—3144no—
268C2×C8—3144no—
269C2×C8—1572no—
270C2×C8C22×D15712yes—
271C4⋊C4—3141no—
272C4⋊C4—1571no—
273C4⋊C4C22×D15711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—3141no—
276C22⋊C4—3141no—
277C22⋊C4—3141no—
278C23—21no—
279C23—3142no—
280C23—3142no—
281C23—3141no—
282C23—6282no—
283C23—6282no—
284C23—6282no—
285C23—1574no—
286C23—1571no—
287C23—1571no—
288C23D4×D15711yescharacteristic
289Q8—24no—
290Q8—22no—
291Q8—3148no—
292Q8—3144no—
293Q8—3148no—
294Q8—3142no—
295Q8—1574no—
296Q8D4×D15714yescomplemented
297D4—22no—
298D4—42no—
299D4—3142no—
300D4—6282no—
301D4—6282no—
302D4—6282no—
303D4—6282no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—3148no—
309C2×C4—3148no—
310C2×C4—3142no—
311C2×C4—3142no—
312C2×C4—3144no—
313C2×C4—3142no—
314C2×C4—3142no—
315C2×C4—3142no—
316C2×C4—3141no—
317C2×C4—6282no—
318C2×C4—1571no—
319C2×C4—1571no—
320C2×C4—1574no—
321C2×C4D4×D15711yescharacteristic
322C2×C4C23×D15711yescharacteristic
323C8—22no—
324C8—3142no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—3148no—
331C22—3141no—
332C22—3141no—
333C22—6281no—
334C22—6281no—
335C22—6281no—
336C22—1572no—
337C22—1572no—
338C22—1572no—
339C22—1572no—
340C22—12564no—
341C22SD16×D15712yescomplemented
342C22C2×D4×D15711yescharacteristic
343C4—24no—
344C4—21no—
345C4—41no—
346C4—3144no—
347C4—3141no—
348C4—6281no—
349C4—1572no—
350C4C2×D4×D15712yes—
351C2—22no—
352C2—81no—
353C2—3142no—
354C2—1574no—
355C2—12561no—
356C2#120×D15711yescharacteristic
357C2C2×SD16×D15711yescharacteristic
358C2#137×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#448×C157C2
socleC22×C157C22×D4
prime radical#121×D157C2
hypercenter#448D157
hypocenterC157C2×#448
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D157C2semidirect
2. C2×SD16×D157C2semidirect
4. #23757414C2semidirect
5. #23757413C2semidirect
6. #23757412C2semidirect
7. #23757411C2semidirect
8. #23757410C2semidirect
9. #103×D157C2semidirect
10. C22⋊C8×D157C2semidirect
11. #448×C157C2semidirect
13. C22×Dic314C22semidirect
18. C2×Q8×D157C22semidirect
23. C2×#14578943C22semidirect
24. C2×#14578942C22semidirect
25. C2×#14578941C22semidirect
35. #19097696C22semidirect
36. #19097695C22semidirect
37. #19097694C22semidirect
38. C22⋊#11929434C22semidirect
39. #19097683C22semidirect
40. C22×Q8×C157C22semidirect
41. C2×SD16×C157C22semidirect
43. #103×C157C22semidirect
44. C22⋊C8×C157C22semidirect
58. Q8×D157D4semidirect
104. C22×D157SD16semidirect
111. Dic314C2×D4semidirect
135. Q8×C157C2×D4semidirect
150. C22×C157C2×SD16semidirect
161. C2×D157C2×SD16semidirect
178. D157#448direct (N × H)
179. C157C2×#448semidirect
189. #448D157direct (N × H)
199. C22×Q8C2×D157semidirect
208. C2×SD16C2×D157semidirect
221. #103C2×D157semidirect
224. C22⋊C8C2×D157semidirect
296. Q8D4×D157semidirect
341. C22SD16×D157semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1572
C2×SD16×D1572
#121×D1572
#237574142
#237574132
#237574122
#237574112
#237574102
#103×D1572
C22⋊C8×D1572
#448×C1572
C2×#448157