Finite Group Databasebeta

← #448×D197 #38751897

Subgroups — #448×D197 #38751897

Order 25216.

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×D197C211yesmaximal, characteristic, complemented
2C2×SD16×D197C212yesmaximal, complemented
3#121×D197C211yesmaximal, characteristic
4#27161220C211yesmaximal, characteristic, complemented
5#27161219C212yesmaximal, complemented
6#27161218C211yesmaximal, characteristic, complemented
7#27161217C211yesmaximal, characteristic, complemented
8#27161216C212yesmaximal, complemented
9#103×D197C212yesmaximal, complemented
10C22⋊C8×D197C211yesmaximal, characteristic, complemented
11#448×C197C211yesmaximal, characteristic, complemented
12C22×Dic394—21no—
13C22×Dic394C2211yescharacteristic, complemented
14C2×Q8×D197—28no—
15C2×Q8×D197—21no—
16C2×Q8×D197—21no—
17C2×Q8×D197C2212yes—
18C2×Q8×D197C2212yescomplemented
19C2×D4×D197—21no—
20C2×D4×D197C2211yescharacteristic
21C22×C4×D197—21no—
22C22×C4×D197C2211yescharacteristic
23C2×#15691347C2212yescomplemented
24C2×#15691346C2212yescomplemented
25C2×#15691345C2212yescomplemented
26SD16×D197—28no—
27C2×C8×D197C2212yes—
28#20263301C2211yescharacteristic
29#20263300C2211yescharacteristic
30#20263299—21no—
31#20263298—21no—
32#20263297C2211yescharacteristic
33C4⋊C4×D197C2211yescharacteristic
34C22⋊C4×D197—21no—
35#20263251C2212yescomplemented
36#20263250C2212yescomplemented
37#20263249C2212yescomplemented
38C22⋊#13096802C2211yescharacteristic, complemented
39#20263238C2212yescomplemented
40C22×Q8×C197C2211yescharacteristic, complemented
41C2×SD16×C197C2212yescomplemented
42#121×C197C2211yescharacteristic
43#103×C197C2212yescomplemented
44C22⋊C8×C197C2211yescharacteristic, complemented
45C23×D197—21no—
46C23×D197D411yescharacteristic
47C2×Dic394—28no—
48C2×Dic394—24no—
49C2×Dic394—22no—
50C2×Dic394—22no—
51C2×Dic394—21no—
52C2×Dic394C2312yes—
53Q8×D197—28no—
54Q8×D197—28no—
55Q8×D197—24no—
56Q8×D197—24no—
57Q8×D197—24no—
58Q8×D197D418yescomplemented
59C2×D788—21no—
60C2×D788C2311yescharacteristic
61C2×C22⋊D197—21no—
62C2×C22⋊D197—21no—
63C2×C22⋊D197—21no—
64D4×D197—24no—
65D4×D197—44no—
66C22×Dic197—21no—
67C22×Dic197D411yescharacteristic
68C2×C4×D197—28no—
69C2×C4×D197—22no—
70C2×C4×D197—22no—
71C2×C4×D197—24no—
72C2×C4×D197—21no—
73C2×C4×D197D411yescharacteristic
74C2×C4×D197D411yescharacteristic
75C2×C4×D197C2311yescharacteristic
76C2×C4×D197C2311yescharacteristic
77#15691347—24no—
78#15691346—24no—
79#15691345—24no—
80C2×#13096802C2312yes—
81C8×D197—24no—
82C4⋊Dic197C2311yescharacteristic
83#15691337—21no—
84C22⋊Dic197—21no—
85#15691335—21no—
86#15691335—21no—
87C2×Q8×C197—24no—
88C2×Q8×C197—21no—
89C2×Q8×C197C2312yes—
90C2×D4×C197—21no—
91C2×D4×C197C2311yescharacteristic
92C22×C4×C197—21no—
93C22×C4×C197C2311yescharacteristic
94SD16×C197—24no—
95C2×C8×C197C2312yes—
96C4⋊C4×C197C2311yescharacteristic
97C22⋊C4×C197—21no—
98C22×D197—22no—
99C22×D197—22no—
100C22×D197—21no—
101C22×D197—42no—
102C22×D197—42no—
103C22×D197—42no—
104C22×D197SD1614yescomplemented
105C22×D197C2×D411yescharacteristic
106C22×D197C2×D411yescharacteristic
107Dic394—24no—
108Dic394—28no—
109Dic394—22no—
110Dic394—28no—
111Dic394C2×D414yescomplemented
112D788—22no—
113D788—42no—
114C22⋊D197—42no—
115C22⋊D197—42no—
116C22⋊D197—42no—
117C2×Dic197—24no—
118C2×Dic197—22no—
119C2×Dic197—22no—
120C2×Dic197—21no—
121C2×Dic197C2×D411yescharacteristic
122C2×Dic197C2×D411yescharacteristic
123C4×D197—28no—
124C4×D197—28no—
125C4×D197—22no—
126C4×D197—22no—
127C4×D197—22no—
128C4×D197—42no—
129C4×D197C2×D414yes—
130#13096802—22no—
131C23×C197—21no—
132C23×C197C2×D411yescharacteristic
133Q8×C197—24no—
134Q8×C197—22no—
135Q8×C197C2×D414yescomplemented
136D4×C197—22no—
137D4×C197—42no—
138C2×C4×C197—24no—
139C2×C4×C197—22no—
140C2×C4×C197—22no—
141C2×C4×C197—21no—
142C2×C4×C197C2×D411yescharacteristic
143C2×C4×C197C2411yescharacteristic
144C8×C197—22no—
145C22×C197—21no—
146C22×C197—21no—
147C22×C197—41no—
148C22×C197—41no—
149C22×C197—41no—
150C22×C197C2×SD1612yescomplemented
151C22×C197C22×D411yescharacteristic
152C2×D197—21no—
153C2×D197—21no—
154C2×D197—28no—
155C2×D197—41no—
156C2×D197—41no—
157C2×D197—41no—
158C2×D197—84no—
159C2×D197#12012yes—
160C2×D197C2×SD1612yes—
161C2×D197C2×SD1612yescomplemented
162C2×D197#13712yes—
163C4×C197—24no—
164C4×C197—21no—
165C4×C197—41no—
166C4×C197C22×D412yes—
167Dic197—24no—
168Dic197—21no—
169Dic197—41no—
170Dic197#12012yes—
171C2×C197—22no—
172C2×C197—81no—
173C2×C197C2×#12011yescharacteristic
174C2×C197C22×SD1611yescharacteristic
175C2×C197C2×#13711yescharacteristic
176D197—22no—
177D197—81no—
178D197#44814yesdirect product factor
179C197C2×#44811yescharacteristic, complemented, Sylow
180C2×#448—1971nomaximal, Sylow
181C23×Q8—1971no—
182C22×SD16—1972no—
183C2×#121—1971no—
184#448—1971no—
185#448—1972no—
186#448—1971no—
187#448—1972no—
188#448—1971no—
189#448D19711yescharacteristic, direct product factor
190C2×#103—1972no—
191C2×C22⋊C8—1971no—
192C22×Q8—3941no—
193C22×Q8—3948no—
194C22×Q8—3941no—
195C22×Q8—3941no—
196C22×Q8—1972no—
197C22×Q8—1972no—
198C22×Q8—1971no—
199C22×Q8C2×D19711yescharacteristic, complemented
200C22×D4—3941no—
201C22×D4—1971no—
202C23×C4—3941no—
203C23×C4—1971no—
204C2×SD16—3948no—
205C2×SD16—1972no—
206C2×SD16—1972no—
207C2×SD16—1972no—
208C2×SD16C2×D19712yescomplemented
209C22×C8—1972no—
210#121—3941no—
211#121—3941no—
212#121—1971no—
213#121—1971no—
214#121—1971no—
215#121C2×D19711yescharacteristic
216C2×C4⋊C4—1971no—
217C2×C22⋊C4—3941no—
218#103—1972no—
219#103—1972no—
220#103—1972no—
221#103C2×D19712yescomplemented
222C22⋊C8—1971no—
223C22⋊C8—1972no—
224C22⋊C8C2×D19711yescharacteristic, complemented
225C24—3941no—
226C24—1971no—
227C2×Q8—24no—
228C2×Q8—21no—
229C2×Q8—3948no—
230C2×Q8—3944no—
231C2×Q8—3942no—
232C2×Q8—3942no—
233C2×Q8—3948no—
234C2×Q8—3948no—
235C2×Q8—3944no—
236C2×Q8—3944no—
237C2×Q8—3944no—
238C2×Q8—3941no—
239C2×Q8—1978no—
240C2×Q8—1972no—
241C2×Q8C22×D19712yes—
242C2×D4—21no—
243C2×D4—7884no—
244C2×D4—3941no—
245C2×D4—3941no—
246C2×D4—3941no—
247C2×D4—3944no—
248C2×D4—3941no—
249C2×D4—1971no—
250C2×D4C22×D19711yescharacteristic
251C22×C4—21no—
252C22×C4—3941no—
253C22×C4—3948no—
254C22×C4—3942no—
255C22×C4—3942no—
256C22×C4—3944no—
257C22×C4—3941no—
258C22×C4—1971no—
259C22×C4—1971no—
260C22×C4—1971no—
261C22×C4—1971no—
262C22×C4—1971no—
263C22×C4C22×D19711yescharacteristic
264SD16—24no—
265SD16—3944no—
266SD16—3944no—
267SD16—3944no—
268C2×C8—3944no—
269C2×C8—1972no—
270C2×C8C22×D19712yes—
271C4⋊C4—3941no—
272C4⋊C4—1971no—
273C4⋊C4C22×D19711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—3941no—
276C22⋊C4—3941no—
277C22⋊C4—3941no—
278C23—21no—
279C23—7882no—
280C23—7882no—
281C23—7882no—
282C23—3942no—
283C23—3942no—
284C23—3941no—
285C23—1974no—
286C23—1971no—
287C23—1971no—
288C23D4×D19711yescharacteristic
289Q8—24no—
290Q8—22no—
291Q8—3948no—
292Q8—3944no—
293Q8—3948no—
294Q8—3942no—
295Q8—1974no—
296Q8D4×D19714yescomplemented
297D4—22no—
298D4—42no—
299D4—7882no—
300D4—7882no—
301D4—7882no—
302D4—7882no—
303D4—3942no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—7882no—
309C2×C4—3948no—
310C2×C4—3948no—
311C2×C4—3942no—
312C2×C4—3942no—
313C2×C4—3941no—
314C2×C4—3944no—
315C2×C4—3942no—
316C2×C4—3942no—
317C2×C4—3942no—
318C2×C4—1971no—
319C2×C4—1971no—
320C2×C4—1974no—
321C2×C4D4×D19711yescharacteristic
322C2×C4C23×D19711yescharacteristic
323C8—22no—
324C8—3942no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—7881no—
331C22—7881no—
332C22—7881no—
333C22—15764no—
334C22—3948no—
335C22—3941no—
336C22—3941no—
337C22—1972no—
338C22—1972no—
339C22—1972no—
340C22—1972no—
341C22SD16×D19712yescomplemented
342C22C2×D4×D19711yescharacteristic
343C4—24no—
344C4—21no—
345C4—41no—
346C4—7881no—
347C4—3944no—
348C4—3941no—
349C4—1972no—
350C4C2×D4×D19712yes—
351C2—22no—
352C2—81no—
353C2—15761no—
354C2—3942no—
355C2—1974no—
356C2#120×D19711yescharacteristic
357C2C2×SD16×D19711yescharacteristic
358C2#137×D19711yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#448×C197C2
socleC22×C197C22×D4
prime radical#121×D197C2
hypercenter#448D197
hypocenterC197C2×#448
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D197C2semidirect
2. C2×SD16×D197C2semidirect
4. #27161220C2semidirect
5. #27161219C2semidirect
6. #27161218C2semidirect
7. #27161217C2semidirect
8. #27161216C2semidirect
9. #103×D197C2semidirect
10. C22⋊C8×D197C2semidirect
11. #448×C197C2semidirect
13. C22×Dic394C22semidirect
18. C2×Q8×D197C22semidirect
23. C2×#15691347C22semidirect
24. C2×#15691346C22semidirect
25. C2×#15691345C22semidirect
35. #20263251C22semidirect
36. #20263250C22semidirect
37. #20263249C22semidirect
38. C22⋊#13096802C22semidirect
39. #20263238C22semidirect
40. C22×Q8×C197C22semidirect
41. C2×SD16×C197C22semidirect
43. #103×C197C22semidirect
44. C22⋊C8×C197C22semidirect
58. Q8×D197D4semidirect
104. C22×D197SD16semidirect
111. Dic394C2×D4semidirect
135. Q8×C197C2×D4semidirect
150. C22×C197C2×SD16semidirect
161. C2×D197C2×SD16semidirect
178. D197#448direct (N × H)
179. C197C2×#448semidirect
189. #448D197direct (N × H)
199. C22×Q8C2×D197semidirect
208. C2×SD16C2×D197semidirect
221. #103C2×D197semidirect
224. C22⋊C8C2×D197semidirect
296. Q8D4×D197semidirect
341. C22SD16×D197semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1972
C2×SD16×D1972
#121×D1972
#271612202
#271612192
#271612182
#271612172
#271612162
#103×D1972
C22⋊C8×D1972
#448×C1972
C2×#448197