Finite Group Databasebeta

← #38751941

Subgroups — #38751941

Order 25216.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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×D4×D197C211yesmaximal, characteristic, complemented
2C2×#20263354C211yesmaximal, characteristic, complemented
3C2×#20263352C211yesmaximal, characteristic, complemented
4#27161591C211yesmaximal, characteristic, complemented
5#27161231C214yesmaximal, complemented
6#27161214C214yesmaximal, complemented
7#27161083C211yesmaximal, characteristic, complemented
8#27161042C211yesmaximal, characteristic, complemented
9#418×C197C211yesmaximal, characteristic, complemented
10C24×D197—21no—
11C2×#15691358C2211yescharacteristic, complemented
12C22×D788C2211yescharacteristic
13C22×C22⋊D197—21no—
14C2×D4×D197—24no—
15C2×D4×D197—22no—
16C2×D4×D197—24no—
17C2×D4×D197C2214yes—
18C22×C4×D197C2211yescharacteristic
19#20263354—24no—
20#20263352—24no—
21C2×#15691347C2212yescomplemented
22C2×#15691346C2212yescomplemented
23C2×D1576C2212yescomplemented
24C2×#15691343C2212yescomplemented
25C2×#15691342C2211yescharacteristic, complemented
26#20263310—21no—
27#20263306C2212yescomplemented
28#20263301C2212yescomplemented
29#20263296—21no—
30C4×D788—22no—
31#20263246C2214yescomplemented
32#20263238C2214yescomplemented
33C22×D4×C197C2211yescharacteristic
34C2×M16×C197C2211yescharacteristic, complemented
35#117×C197C2211yescharacteristic, complemented
36#102×C197C2214yescomplemented
37C23×D197—28no—
38C23×D197—24no—
39C23×D197—21no—
40C23×D197—48no—
41C23×D197D411yescharacteristic
42#15691358—22no—
43#15691358—22no—
44C2×Dic394C2311yescharacteristic, complemented
45C2×D788—22no—
46C2×D788—21no—
47C2×D788—22no—
48C2×D788D414yescomplemented
49C2×D788C2312yes—
50C2×D788C2311yescharacteristic, complemented
51C2×C22⋊D197—28no—
52C2×C22⋊D197—24no—
53C2×C22⋊D197—21no—
54D4×D197—216no—
55D4×D197—28no—
56D4×D197—28no—
57C22×Dic197D411yescharacteristic
58C2×C4×D197—22no—
59C2×C4×D197—22no—
60C2×C4×D197—21no—
61C2×C4×D197D412yes—
62C2×C4×D197C2312yes—
63#15691347—24no—
64#15691346—24no—
65D1576—24no—
66#15691343—24no—
67#15691342—22no—
68C2×#13096802C2312yescomplemented
69C4⋊Dic197—22no—
70#15691335—22no—
71#15691335—22no—
72C24×C197—21no—
73C2×D4×C197—21no—
74C2×D4×C197—22no—
75C2×D4×C197D414yescomplemented
76C2×D4×C197C2312yes—
77C22×C4×C197C2311yescharacteristic
78M16×C197—22no—
79C2×C8×C197C2312yescomplemented
80C4⋊C4×C197C2312yescomplemented
81C22⋊C4×C197—21no—
82C42×C197—21no—
83C22×D197—22no—
84C22×D197—24no—
85C22×D197—216no—
86C22×D197—24no—
87C22×D197—22no—
88C22×D197—21no—
89C22×D197—432no—
90C22×D197—48no—
91C22×D197—44no—
92C22×D197—48no—
93C22×D197C2×D412yes—
94Dic394—21no—
95Dic394—21no—
96D788—24no—
97D788—22no—
98D788—21no—
99D788—21no—
100D788—44no—
101D788C2×D414yes—
102C22⋊D197—216no—
103C22⋊D197—42no—
104C2×Dic197—22no—
105C2×Dic197—21no—
106C2×Dic197C2×D412yes—
107C4×D197—24no—
108C4×D197—22no—
109C4×D197—22no—
110C4×D197—42no—
111#13096802—22no—
112C23×C197—24no—
113C23×C197—22no—
114C23×C197—44no—
115C23×C197C2×D411yescharacteristic
116D4×C197—24no—
117D4×C197—22no—
118D4×C197C2×D414yes—
119C2×C4×C197—22no—
120C2×C4×C197C2×D411yescharacteristic
121C2×C4×C197C2×D412yes—
122C2×C4×C197C2×D412yes—
123C2×C4×C197C2411yescharacteristic
124C8×C197—22no—
125C22×C197—22no—
126C22×C197—24no—
127C22×C197—48no—
128C22×C197—44no—
129C22×C197#12012yes—
130C22×C197C22×D411yescharacteristic
131C2×D197—28no—
132C2×D197—22no—
133C2×D197—22no—
134C2×D197—24no—
135C2×D197—432no—
136C2×D197—48no—
137C2×D197—44no—
138C2×D197—41no—
139C2×D197—42no—
140C2×D197AGL(1,ℤ/8)14yescomplemented
141C4×C197—42no—
142C4×C197#12012yes—
143C4×C197C22×D411yescharacteristic
144C4×C197C22×D411yescharacteristic
145Dic197—22no—
146Dic197—41no—
147C2×C197—22no—
148C2×C197—44no—
149C2×C197C2×#12011yescharacteristic
150C2×C197C2×AGL(1,ℤ/8)12yes—
151D197—24no—
152D197—44no—
153D197—81no—
154C197#300311yescharacteristic, complemented, Sylow
155#3003—1971nomaximal, Sylow
156C23×D4—1971no—
157C2×AGL(1,ℤ/8)—1971no—
158C2×AGL(1,ℤ/8)—1971no—
159#525—1971no—
160#450—1974no—
161#447—1974no—
162#418—1971no—
163#418D19711yescharacteristic, complemented
164#407—1971no—
165C25—3941no—
166C2×#41—1971no—
167C22×D4—3944no—
168C22×D4—3944no—
169C22×D4—3942no—
170C22×D4—3941no—
171C22×D4—1974no—
172C22×D4—1971no—
173C22×D4C2×D19711yescharacteristic
174C23×C4—1971no—
175AGL(1,ℤ/8)—3944no—
176AGL(1,ℤ/8)—3944no—
177C2×SD16—1972no—
178C2×SD16—1972no—
179C2×D8—1972no—
180C2×D8—1972no—
181C2×M16—1971no—
182C2×M16C2×D19711yescharacteristic, complemented
183#123—3941no—
184C22⋊Q8—1972no—
185#121—1972no—
186#120—3941no—
187C4×D4—3942no—
188#117C2×D19711yescharacteristic, complemented
189#102—1974no—
190#102C2×D19714yescomplemented
191C22⋊C8—1974no—
192C24—21no—
193C24—7888no—
194C24—3948no—
195C24—3944no—
196C24—3941no—
197C24—1971no—
198#41—3942no—
199#41—3942no—
200C2×Q8—1971no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—3942no—
204C2×D4—39416no—
205C2×D4—3948no—
206C2×D4—3941no—
207C2×D4—3948no—
208C2×D4—3948no—
209C2×D4—3944no—
210C2×D4—3941no—
211C2×D4—3942no—
212C2×D4—1974no—
213C2×D4—1971no—
214C2×D4—1972no—
215C2×D4D78814yescomplemented
216C2×D4C22×D19712yes—
217C22×C4—3941no—
218C22×C4—3942no—
219C22×C4—3942no—
220C22×C4—1971no—
221C22×C4—1972no—
222C22×C4—1972no—
223C22×C4C22×D19711yescharacteristic
224SD16—3944no—
225SD16—3944no—
226D8—3944no—
227D8—3944no—
228M16—22no—
229M16—3942no—
230C2×C8—1972no—
231C2×C8C22×D19712yescomplemented
232C4⋊C4—3942no—
233C4⋊C4C22×D19712yescomplemented
234C22⋊C4—21no—
235C22⋊C4—3942no—
236C22⋊C4—3942no—
237C42—21no—
238C23—24no—
239C23—22no—
240C23—44no—
241C23—7884no—
242C23—78832no—
243C23—7888no—
244C23—7888no—
245C23—39416no—
246C23—3944no—
247C23—3941no—
248C23—3944no—
249C23—3942no—
250C23—3942no—
251C23—1972no—
252C23D4×D19711yescharacteristic
253Q8—3941no—
254Q8—3941no—
255D4—24no—
256D4—22no—
257D4—7884no—
258D4—7882no—
259D4—39416no—
260D4—3944no—
261D4—3941no—
262D4—3941no—
263D4—3942no—
264D4—1974no—
265D4C2×D78814yes—
266C2×C4—22no—
267C2×C4—7882no—
268C2×C4—3941no—
269C2×C4—3942no—
270C2×C4—3944no—
271C2×C4—3942no—
272C2×C4—3942no—
273C2×C4—1972no—
274C2×C4D4×D19711yescharacteristic
275C2×C4D4×D19712yes—
276C2×C4C2×D78812yes—
277C2×C4C23×D19711yescharacteristic
278C8—22no—
279C8—3942no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—7888no—
285C22—78832no—
286C22—7884no—
287C22—7882no—
288C22—7881no—
289C22—3948no—
290C22—3944no—
291C22—3942no—
292C22—3942no—
293C22—1974no—
294C22#2026329612yes—
295C22C2×D4×D19711yescharacteristic
296C4—42no—
297C4—7881no—
298C4—3942no—
299C4#2026329612yes—
300C4C2×D4×D19711yescharacteristic
301C4C22×D78811yescharacteristic
302C2—22no—
303C2—44no—
304C2—7884no—
305C2—15761no—
306C2—3944no—
307C2C2×#2026329611yescharacteristic
308C2AGL(1,ℤ/8)×D19712yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#418×C197C2
socleC22×C197C22×D4
hypercenter#418D197
hypocenterC197#3003
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D197C2semidirect
2. C2×#20263354C2semidirect
3. C2×#20263352C2semidirect
4. #27161591C2semidirect
5. #27161231C2semidirect
6. #27161214C2semidirect
7. #27161083C2semidirect
8. #27161042C2semidirect
9. #418×C197C2semidirect
11. C2×#15691358C22semidirect
21. C2×#15691347C22semidirect
22. C2×#15691346C22semidirect
23. C2×D1576C22semidirect
24. C2×#15691343C22semidirect
25. C2×#15691342C22semidirect
27. #20263306C22semidirect
28. #20263301C22semidirect
31. #20263246C22semidirect
32. #20263238C22semidirect
34. C2×M16×C197C22semidirect
35. #117×C197C22semidirect
36. #102×C197C22semidirect
44. C2×Dic394C23semidirect
48. C2×D788D4semidirect
50. C2×D788C23semidirect
68. C2×#13096802C23semidirect
75. C2×D4×C197D4semidirect
79. C2×C8×C197C23semidirect
80. C4⋊C4×C197C23semidirect
140. C2×D197AGL(1,ℤ/8)semidirect
154. C197#3003semidirect
163. #418D197semidirect
182. C2×M16C2×D197semidirect
188. #117C2×D197semidirect
190. #102C2×D197semidirect
215. C2×D4D788semidirect
231. C2×C8C22×D197semidirect
233. C4⋊C4C22×D197semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1972
C2×#202633542
C2×#202633522
#271615912
#271612312
#271612142
#271610832
#271610422
#418×C1972
#3003197