Finite Group Databasebeta

← #38752187

Subgroups — #38752187

Order 25216.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20263350C212yesmaximal, complemented
2#127×D197C211yesmaximal, characteristic, complemented
3C4×#15691357C211yesmaximal, characteristic, complemented
4#27161280C211yesmaximal, characteristic, complemented
5#27161278C212yesmaximal, complemented
6#27161276C211yesmaximal, characteristic, complemented
7#27161274C211yesmaximal, characteristic, complemented
8#27161272C212yesmaximal, complemented
9#27161101C211yesmaximal, characteristic, complemented
10#27161082C212yesmaximal, complemented
11#459×C197C211yesmaximal, characteristic, complemented
12C2×#15691357C2211yescharacteristic
13C2×D4×D197—21no—
14C2×D4×D197—21no—
15C2×D4×D197C2212yes—
16#20263350—28no—
17C2×#15691347C2212yescomplemented
18C2×#15691346C2212yescomplemented
19C2×#15691343C2212yescomplemented
20C2×#15691341C2212yes—
21#20263323C2211yescharacteristic
22#20263322—21no—
23#20263322—21no—
24#20263322C2212yes—
25C4×Dic394C2211yescharacteristic, complemented
26#20263291C2211yescharacteristic, complemented
27D4×Dic197C2211yescharacteristic, complemented
28C4×C22⋊D197—21no—
29#20263286—21no—
30#20263284C2211yescharacteristic
31#20263282—21no—
32C2×C4×Dic197—21no—
33C42×D197C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic, complemented
35#20263255C2212yescomplemented
36#20263248C2212yescomplemented
37#20263247C2212yescomplemented
38#20263246C2212yescomplemented
39C2×D8×C197C2212yescomplemented
40#127×C197C2211yescharacteristic
41C4×D4×C197C2211yescharacteristic, complemented
42C4⋊C8×C197C2211yescharacteristic, complemented
43#102×C197C2212yescomplemented
44C23×D197—22no—
45#15691357—22no—
46#15691357D414yescomplemented
47C2×Dic394C2311yescharacteristic
48C2×D788—21no—
49C2×D788—21no—
50C2×D788C2312yes—
51C2×C22⋊D197—21no—
52C2×C22⋊D197—22no—
53C2×C22⋊D197—22no—
54C2×C22⋊D197—22no—
55D4×D197—24no—
56D4×D197—24no—
57D4×D197—44no—
58D4×D197—44no—
59C22×Dic197—21no—
60C22×Dic197—21no—
61C2×C4×D197D411yescharacteristic
62C2×C4×D197D411yescharacteristic
63C2×C4×D197C2311yescharacteristic
64#15691347—24no—
65#15691346—24no—
66#15691343—24no—
67#15691341—24no—
68C2×#13096802C2312yes—
69C4⋊Dic197C2311yescharacteristic
70#15691337—21no—
71#15691337—21no—
72C22⋊Dic197—21no—
73#15691335—21no—
74C4×Dic197—21no—
75C4×Dic197—21no—
76C4×Dic197—21no—
77C4×Dic197D411yescharacteristic
78C4×Dic197D411yescharacteristic
79C4×Dic197C2311yescharacteristic
80C2×D4×C197—21no—
81C2×D4×C197—21no—
82C2×D4×C197C2312yes—
83C2×D4×C197C2311yescharacteristic
84C22×C4×C197—21no—
85D8×C197—24no—
86C2×C8×C197C2312yes—
87C4⋊C4×C197C2311yescharacteristic
88C22⋊C4×C197—21no—
89C42×C197C2311yescharacteristic
90C22×D197—22no—
91C22×D197—44no—
92C22×D197—44no—
93C22×D197—44no—
94C22×D197C2×D411yescharacteristic
95Dic394—21no—
96Dic394C2×D412yescomplemented
97D788—22no—
98D788—22no—
99D788—42no—
100D788—42no—
101C22⋊D197—24no—
102C22⋊D197—44no—
103C22⋊D197—44no—
104C22⋊D197—44no—
105C2×Dic197—21no—
106C2×Dic197—24no—
107C2×Dic197—21no—
108C2×Dic197—42no—
109C2×Dic197C2×D411yescharacteristic
110C2×Dic197C2×D411yescharacteristic
111C2×Dic197C2×D411yescharacteristic
112C4×D197—22no—
113C4×D197—22no—
114C4×D197C2×D412yes—
115C4×D197#4112yes—
116#13096802—22no—
117C23×C197—21no—
118C23×C197—22no—
119D4×C197—22no—
120D4×C197—21no—
121D4×C197—22no—
122D4×C197—42no—
123D4×C197—42no—
124D4×C197C2×D412yescomplemented
125C2×C4×C197—21no—
126C2×C4×C197—42no—
127C2×C4×C197C2×D411yescharacteristic
128C2×C4×C197C2×D411yescharacteristic
129C2×C4×C197C2411yescharacteristic
130C8×C197—22no—
131C22×C197—22no—
132C22×C197—41no—
133C22×C197—41no—
134C22×C197—42no—
135C22×C197—42no—
136C22×C197—42no—
137C22×C197C22×D411yescharacteristic
138C2×D197—21no—
139C2×D197—21no—
140C2×D197—42no—
141C2×D197—42no—
142C2×D197—42no—
143C2×D197—88no—
144C2×D197#12112yes—
145C4×C197—21no—
146C4×C197—41no—
147C4×C197AGL(1,ℤ/8)12yescomplemented
148C4×C197C22×D411yescharacteristic
149C4×C197C2×#4111yescharacteristic
150Dic197—22no—
151Dic197—21no—
152Dic197—41no—
153Dic197#12112yes—
154Dic197AGL(1,ℤ/8)12yescomplemented
155C2×C197—42no—
156C2×C197—82no—
157C2×C197C2×#12111yescharacteristic
158C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
160D197—22no—
161D197—82no—
162C197#303811yescharacteristic, complemented, Sylow
163#3038—1971nomaximal, Sylow
164C2×AGL(1,ℤ/8)—1972no—
165C2×#127—1971no—
166C4×#41—1971no—
167#460—1971no—
168#460—1971no—
169#460—1972no—
170#459—1972no—
171#459—1971no—
172#459D19711yescharacteristic, complemented
173#423—1971no—
174#418—1972no—
175C2×#41—1971no—
176C22×D4—3941no—
177C22×D4—3941no—
178C22×D4—1972no—
179AGL(1,ℤ/8)—3948no—
180C2×SD16—1972no—
181C2×SD16—1972no—
182C2×D8—1972no—
183C2×D8C2×D19712yescomplemented
184C2×M16—1972no—
185#127—3941no—
186#127—3941no—
187#127—1972no—
188#127—1971no—
189#127C2×D19711yescharacteristic
190C4×Q8—1971no—
191C4×Q8—1971no—
192C4×D4—3941no—
193C4×D4—3941no—
194C4×D4—1971no—
195C4×D4C2×D19711yescharacteristic, complemented
196#117—3941no—
197#117—1971no—
198C2×C42—3941no—
199C2×C42—1971no—
200C4⋊C8—1972no—
201C4⋊C8—1971no—
202C4⋊C8C2×D19711yescharacteristic, complemented
203#102—1972no—
204#102—1972no—
205#102—1972no—
206#102C2×D19712yescomplemented
207C24—3942no—
208#41—3942no—
209#41—1974no—
210C2×Q8—1971no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—7884no—
214C2×D4—7884no—
215C2×D4—3941no—
216C2×D4—3944no—
217C2×D4—3944no—
218C2×D4—3942no—
219C2×D4—3942no—
220C2×D4—3942no—
221C2×D4—3941no—
222C2×D4—3941no—
223C2×D4—1972no—
224C2×D4C22×D19712yes—
225C2×D4C22×D19711yescharacteristic
226C22×C4—21no—
227C22×C4—3941no—
228C22×C4—3941no—
229C22×C4—1971no—
230C22×C4—1971no—
231C22×C4—1971no—
232SD16—3944no—
233SD16—3944no—
234D8—24no—
235D8—3944no—
236M16—3944no—
237C2×C8—1972no—
238C2×C8C22×D19712yes—
239C4⋊C4—3941no—
240C4⋊C4—3941no—
241C4⋊C4—1971no—
242C4⋊C4C22×D19711yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—3941no—
245C22⋊C4—3941no—
246C42—3941no—
247C42—3941no—
248C42—3941no—
249C42—1971no—
250C42—1971no—
251C42—1971no—
252C42C22×D19711yescharacteristic
253C23—21no—
254C23—22no—
255C23—7884no—
256C23—7884no—
257C23—7884no—
258C23—3942no—
259C23—1971no—
260Q8—3941no—
261Q8—1972no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—7882no—
268D4—7884no—
269D4—7884no—
270D4—7884no—
271D4—7882no—
272D4—3944no—
273D4—3942no—
274D4—3942no—
275D4D4×D19712yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—7882no—
279C2×C4—3942no—
280C2×C4—3941no—
281C2×C4—3941no—
282C2×C4—3944no—
283C2×C4—3942no—
284C2×C4—1971no—
285C2×C4—1971no—
286C2×C4—1971no—
287C2×C4—1972no—
288C2×C4—1972no—
289C2×C4D4×D19711yescharacteristic
290C2×C4D4×D19711yescharacteristic
291C2×C4C23×D19711yescharacteristic
292C8—22no—
293C8—3942no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—7882no—
301C22—7882no—
302C22—7882no—
303C22—15768no—
304C22—3941no—
305C22—3941no—
306C22—1972no—
307C22C2×D4×D19711yescharacteristic
308C4—21no—
309C4—41no—
310C4—7881no—
311C4—3941no—
312C4—3942no—
313C4—1972no—
314C4—1972no—
315C4#2026335012yescomplemented
316C4C2×D4×D19711yescharacteristic
317C4#41×D19711yescharacteristic
318C2—42no—
319C2—82no—
320C2—15762no—
321C2—3942no—
322C2#121×D19711yescharacteristic
323C2AGL(1,ℤ/8)×D19711yescharacteristic
324C2C2×#2026335011yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#459×C197C2
socleC22×C197C22×D4
hypercenter#459D197
hypocenterC197#3038
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. #127×D197C2semidirect
3. C4×#15691357C2semidirect
4. #27161280C2semidirect
5. #27161278C2semidirect
6. #27161276C2semidirect
7. #27161274C2semidirect
8. #27161272C2semidirect
9. #27161101C2semidirect
10. #27161082C2semidirect
11. #459×C197C2semidirect
17. C2×#15691347C22semidirect
18. C2×#15691346C22semidirect
19. C2×#15691343C22semidirect
25. C4×Dic394C22semidirect
26. #20263291C22semidirect
27. D4×Dic197C22semidirect
34. C4⋊#13096802C22semidirect
35. #20263255C22semidirect
36. #20263248C22semidirect
37. #20263247C22semidirect
38. #20263246C22semidirect
39. C2×D8×C197C22semidirect
41. C4×D4×C197C22semidirect
42. C4⋊C8×C197C22semidirect
43. #102×C197C22semidirect
46. #15691357D4semidirect
96. Dic394C2×D4semidirect
124. D4×C197C2×D4semidirect
147. C4×C197AGL(1,ℤ/8)semidirect
154. Dic197AGL(1,ℤ/8)semidirect
162. C197#3038semidirect
172. #459D197semidirect
183. C2×D8C2×D197semidirect
195. C4×D4C2×D197semidirect
202. C4⋊C8C2×D197semidirect
206. #102C2×D197semidirect
275. D4D4×D197semidirect
315. C4#20263350semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
#127×D1972
C4×#156913572
#271612802
#271612782
#271612762
#271612742
#271612722
#271611012
#271610822
#459×C1972
#3038197