Finite Group Databasebeta

← #38754507

Subgroups — #38754507

Order 25216.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#27161890C212yesmaximal, complemented
2#27161889C212yesmaximal, complemented
3#27161451C211yesmaximal, characteristic, complemented
4#27161341C212yesmaximal, complemented
5#27161340C212yesmaximal, complemented
6#27161263C212yesmaximal, complemented
7#27161246C211yesmaximal, characteristic, complemented
8#27161193C212yesmaximal, complemented
9#453×C197C211yesmaximal, characteristic, complemented
10#20263373C2212yes—
11#20263371C2211yescharacteristic, complemented
12#20263359C2212yescomplemented
13#20263356—22no—
14#20263356C2212yescomplemented
15#20263355—22no—
16#20263352—22no—
17#20263352C2212yes—
18#20263350—22no—
19#20263346—22no—
20#20263346C2212yes—
21#20263345—22no—
22C2×#15691346—22no—
23C2×#15691346C2212yescomplemented
24C2×#15691346C2212yescomplemented
25SD16×D197—22no—
26#20263334C2212yescomplemented
27#20263333C2212yescomplemented
28#20263326C2211yescharacteristic, complemented
29#20263261C2212yescomplemented
30#20263254C2212yescomplemented
31#20263244C2212yescomplemented
32#20263243C2211yescharacteristic, complemented
33C4×#13096802C2211yescharacteristic, complemented
34H(5,𝔽2)×C197C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C197C2212yes—
36#127×C197C2211yescharacteristic
37#104×C197C2212yescomplemented
38#100×C197C2211yescharacteristic, complemented
39#15691358—22no—
40#15691358C2312yes—
41#15691357—22no—
42#15691357—22no—
43#15691357—42no—
44C2×Dic394—22no—
45C2×Dic394C2312yes—
46Q8×D197—22no—
47C2×C22⋊D197—41no—
48D4×D197—22no—
49Dic788—22no—
50#15691348—22no—
51#15691348D412yescomplemented
52#15691347—22no—
53#15691346—22no—
54#15691346—22no—
55#15691346—22no—
56#15691346—22no—
57#15691346—22no—
58#15691346—22no—
59#15691346D412yescomplemented
60#15691345D412yescomplemented
61#15691343—22no—
62#15691343D412yescomplemented
63#15691342—22no—
64#15691342C2312yes—
65#15691341—22no—
66C2×#13096802—22no—
67C2×#13096802C2312yes—
68C8×D197—22no—
69C4⋊Dic197—41no—
70#41×C197—22no—
71#41×C197C2312yes—
72C2×D4×C197—22no—
73C2×D4×C197—41no—
74C2×D4×C197—41no—
75C2×D4×C197C2311yescharacteristic
76C2×D4×C197C2311yescharacteristic
77SD16×C197—22no—
78D8×C197—22no—
79M16×C197C2312yes—
80C42×C197C2311yescharacteristic
81C22×D197—41no—
82Dic394—22no—
83Dic394—22no—
84Dic394—42no—
85Dic394C2×D412yes—
86D788C2×D412yes—
87C22⋊D197—22no—
88C22⋊D197—42no—
89C22⋊D197—42no—
90C2×Dic197—22no—
91C2×Dic197—41no—
92C2×Dic197—42no—
93C4×D197—22no—
94C4×D197—42no—
95#13096802—22no—
96#13096802C2×D412yes—
97#13096802C2×D412yes—
98C23×C197—21no—
99C23×C197—21no—
100C23×C197—41no—
101Q8×C197C2×D412yes—
102D4×C197—22no—
103D4×C197—22no—
104D4×C197—22no—
105D4×C197—42no—
106D4×C197—82no—
107D4×C197C2×D412yes—
108C2×C4×C197—21no—
109C2×C4×C197—22no—
110C2×C4×C197—41no—
111C2×C4×C197C2411yescharacteristic
112C8×C197—22no—
113C22×C197—22no—
114C22×C197—41no—
115C22×C197—41no—
116C22×C197—81no—
117C22×C197—81no—
118C22×C197—161no—
119C22×C197H(5,𝔽2)11yescharacteristic
120C2×D197—22no—
121C2×D197—161no—
122C4×C197—22no—
123C4×C197—22no—
124C4×C197C22×D412yes—
125Dic197—22no—
126Dic197—42no—
127C2×C197—21no—
128C2×C197—42no—
129C2×C197—81no—
130C2×C197—81no—
131C2×C197D4211yescharacteristic
132D197—42no—
133C197#329111yescharacteristic, complemented, Sylow
134#3291—1971nomaximal, Sylow
135#577—1972no—
136#577—1972no—
137#492—1971no—
138#471—1972no—
139#471—1972no—
140#456—1972no—
141#453—1971no—
142#453D19711yescharacteristic, complemented
143#443—1972no—
1442−1+4—1972no—
145H(5,𝔽2)—1971no—
146H(5,𝔽2)C2×D19711yescharacteristic, complemented
147#137—3942no—
148#137—3942no—
149#137—1972no—
150#137—1972no—
151AGL(1,ℤ/8)—3942no—
152AGL(1,ℤ/8)—3942no—
153AGL(1,ℤ/8)—1972no—
154AGL(1,ℤ/8)C2×D19712yes—
155#135—3942no—
156#135—3942no—
157#135—1972no—
158C2×SD16—3942no—
159C2×SD16—3942no—
160C2×SD16—1972no—
161C2×SD16—1972no—
162#131—1972no—
163#131—1972no—
164C4⋊Q8—1971no—
165#127C2×D19711yescharacteristic
166#108—1972no—
167#104—1972no—
168#104C2×D19712yescomplemented
169#101—1972no—
170#100—1971no—
171#100C2×D19711yescharacteristic, complemented
172C4×C8—1971no—
173#41—22no—
174#41—7882no—
175#41—3942no—
176#41—3942no—
177#41—3942no—
178#41—1972no—
179#41C22×D19712yes—
180C2×Q8—3942no—
181C2×Q8—3942no—
182C2×Q8—1972no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—7881no—
187C2×D4—3942no—
188C2×D4C22×D19711yescharacteristic
189C2×D4C22×D19711yescharacteristic
190Q16—3942no—
191Q16—3942no—
192Q16—1972no—
193SD16—22no—
194SD16—3942no—
195SD16—3942no—
196SD16—3942no—
197SD16—3942no—
198SD16—3942no—
199SD16—3942no—
200SD16—3942no—
201SD16—1972no—
202SD16—1972no—
203D8—22no—
204D8—3942no—
205D8—1972no—
206M16—3942no—
207M16—3942no—
208M16—1972no—
209M16C22×D19712yes—
210C2×C8—3942no—
211C2×C8—3942no—
212C2×C8—1972no—
213C4⋊C4—7881no—
214C42C22×D19711yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—7881no—
219Q8—7882no—
220Q8—3942no—
221Q8—3942no—
222Q8—1972no—
223Q8D4×D19712yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—7882no—
230D4—7882no—
231D4—3942no—
232D4—1972no—
233D4D4×D19712yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—7882no—
238C2×C4—7882no—
239C2×C4—7881no—
240C2×C4—3942no—
241C2×C4—3942no—
242C2×C4C23×D19711yescharacteristic
243C8—22no—
244C8—3942no—
245C8—1972no—
246C8—1972no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—31521no—
254C22—3942no—
255C22#2026337111yescharacteristic
256C4—22no—
257C4—22no—
258C4—7882no—
259C4—3942no—
260C4C2×D4×D19712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—7882no—
266C2#2716171111yescharacteristic
subgroupG/H
centerC2#27161711
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#453×C197C2
socleC2×C197D42
hypercenter#453D197
hypocenterC197#3291
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #27161890C2semidirect
2. #27161889C2semidirect
3. #27161451C2semidirect
4. #27161341C2semidirect
5. #27161340C2semidirect
6. #27161263C2semidirect
7. #27161246C2semidirect
8. #27161193C2semidirect
9. #453×C197C2semidirect
11. #20263371C22semidirect
12. #20263359C22semidirect
14. #20263356C22semidirect
23. C2×#15691346C22semidirect
24. C2×#15691346C22semidirect
26. #20263334C22semidirect
27. #20263333C22semidirect
28. #20263326C22semidirect
29. #20263261C22semidirect
30. #20263254C22semidirect
31. #20263244C22semidirect
32. #20263243C22semidirect
33. C4×#13096802C22semidirect
34. H(5,𝔽2)×C197C22semidirect
37. #104×C197C22semidirect
38. #100×C197C22semidirect
51. #15691348D4semidirect
59. #15691346D4semidirect
60. #15691345D4semidirect
62. #15691343D4semidirect
133. C197#3291semidirect
142. #453D197semidirect
146. H(5,𝔽2)C2×D197semidirect
168. #104C2×D197semidirect
171. #100C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618902
#271618892
#271614512
#271613412
#271613402
#271612632
#271612462
#271611932
#453×C1972
#3291197