Finite Group Databasebeta

← #38754506

Subgroups — #38754506

Order 25216.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#27161891C212yesmaximal, complemented
2#27161886C212yesmaximal, complemented
3#27161452C211yesmaximal, characteristic, complemented
4#27161343C212yesmaximal, complemented
5#27161342C212yesmaximal, complemented
6#27161263C212yesmaximal, complemented
7#27161248C212yesmaximal, complemented
8#27161197C211yesmaximal, characteristic, complemented
9#443×C197C211yesmaximal, characteristic, complemented
10#20263373C2212yes—
11#20263372C2212yescomplemented
12#20263359—22no—
13#20263359C2212yescomplemented
14#20263358—22no—
15#20263356C2212yescomplemented
16#20263354—22no—
17#20263354C2212yes—
18#20263351C2212yes—
19#20263350—22no—
20#20263348—22no—
21#20263348C2211yescharacteristic
22#20263347—22no—
23C2×#15691347—22no—
24C2×#15691347C2212yescomplemented
25C2×#15691347C2212yescomplemented
26SD16×D197—22no—
27#20263332C2212yescomplemented
28#20263326C2211yescharacteristic, complemented
29#20263323C2211yescharacteristic
30#20263262C2211yescharacteristic, complemented
31#20263254C2212yescomplemented
32#20263244C2212yescomplemented
33#20263242C2212yescomplemented
34#135×C197C2211yescharacteristic
35#131×C197C2212yescomplemented
36#108×C197C2211yescharacteristic, complemented
37#104×C197C2212yescomplemented
38C4×C8×C197C2211yescharacteristic, complemented
39#15691359—22no—
40#15691358—22no—
41#15691358—22no—
42#15691358C2312yes—
43#15691357—22no—
44#15691357—42no—
45C2×Dic394—22no—
46C2×Dic394C2312yes—
47Q8×D197—22no—
48C2×D788—22no—
49C2×D788—41no—
50C2×D788C2312yes—
51D4×D197—22no—
52D4×D197—42no—
53Dic788—22no—
54Dic788D411yescharacteristic, complemented
55#15691348—22no—
56#15691347—22no—
57#15691347—22no—
58#15691347—22no—
59#15691347—22no—
60#15691347—22no—
61#15691347—22no—
62#15691347D412yescomplemented
63#15691346—22no—
64#15691345—22no—
65D1576—22no—
66D1576D411yescharacteristic, complemented
67#15691343—22no—
68#15691342C2312yes—
69#15691341—22no—
70C8×D197—22no—
71C4⋊Dic197—41no—
72#41×C197C2312yes—
73Q16×C197D411yescharacteristic, complemented
74SD16×C197D412yescomplemented
75D8×C197D411yescharacteristic, complemented
76M16×C197—22no—
77M16×C197C2312yes—
78C2×C8×C197—22no—
79C2×C8×C197C2311yescharacteristic
80C2×C8×C197C2311yescharacteristic
81C42×C197C2311yescharacteristic
82C22×D197—22no—
83C22×D197—42no—
84Dic394—22no—
85Dic394—22no—
86Dic394—42no—
87Dic394C2×D412yes—
88D788—22no—
89D788—22no—
90D788—42no—
91D788—82no—
92D788C2×D412yes—
93C22⋊D197—22no—
94C22⋊D197—42no—
95C22⋊D197—42no—
96C2×Dic197—22no—
97C2×Dic197—42no—
98C4×D197—22no—
99C4×D197—42no—
100C4×D197—42no—
101#13096802—22no—
102Q8×C197C2×D412yes—
103D4×C197—22no—
104D4×C197C2×D412yes—
105C2×C4×C197—21no—
106C2×C4×C197—22no—
107C2×C4×C197C2411yescharacteristic
108C8×C197—22no—
109C8×C197C2×D411yescharacteristic
110C8×C197C2×D411yescharacteristic
111C8×C197C2×D411yescharacteristic
112C8×C197C2×D411yescharacteristic
113C22×C197—22no—
114C22×C197H(5,𝔽2)11yescharacteristic
115C2×D197—22no—
116C2×D197—42no—
117C2×D197—82no—
118C2×D197—162no—
119C4×C197—22no—
120C4×C197—22no—
121C4×C197C22×D411yescharacteristic
122C4×C197C22×D411yescharacteristic
123Dic197—22no—
124Dic197—42no—
125C2×C197—21no—
126C2×C197—42no—
127C2×C197D4211yescharacteristic
128D197—42no—
129D197—82no—
130C197#329111yescharacteristic, complemented, Sylow
131#3291—1971nomaximal, Sylow
132#577—1972no—
133#577—1972no—
134#492—1971no—
135#471—1972no—
136#471—1972no—
137#456—1972no—
138#453—1972no—
139#443—1971no—
140#443D19711yescharacteristic, complemented
1412−1+4—1972no—
142H(5,𝔽2)—1972no—
143#137—3942no—
144#137—3942no—
145#137—1972no—
146#137—1972no—
147AGL(1,ℤ/8)—3942no—
148AGL(1,ℤ/8)—3942no—
149AGL(1,ℤ/8)—1972no—
150AGL(1,ℤ/8)—1972no—
151#135—3942no—
152#135—3942no—
153#135—1971no—
154#135C2×D19711yescharacteristic
155C2×SD16—3942no—
156C2×SD16—3942no—
157C2×SD16—1972no—
158C2×SD16—1972no—
159#131—1972no—
160#131C2×D19712yescomplemented
161C4⋊Q8—1971no—
162#127—1971no—
163#108—1971no—
164#108C2×D19711yescharacteristic, complemented
165#104—1972no—
166#104C2×D19712yescomplemented
167#101—1972no—
168#100—1972no—
169C4×C8C2×D19711yescharacteristic, complemented
170#41—7882no—
171#41—3942no—
172#41—3942no—
173#41—3942no—
174#41—3942no—
175#41—1972no—
176#41C22×D19712yes—
177C2×Q8—3942no—
178C2×Q8—3942no—
179C2×Q8—1972no—
180C2×D4—7881no—
181C2×D4—7882no—
182C2×D4—3942no—
183C2×D4—3942no—
184C2×D4—1972no—
185Q16—3942no—
186Q16—3942no—
187Q16—1971no—
188Q16D78811yescharacteristic, complemented
189SD16—3942no—
190SD16—3942no—
191SD16—3942no—
192SD16—3942no—
193SD16—3942no—
194SD16—3942no—
195SD16—3942no—
196SD16—3942no—
197SD16—1972no—
198SD16D78812yescomplemented
199D8—3942no—
200D8—3942no—
201D8—1971no—
202D8D78811yescharacteristic, complemented
203M16—22no—
204M16—3942no—
205M16—1972no—
206M16C22×D19712yes—
207C2×C8—22no—
208C2×C8—3942no—
209C2×C8C22×D19711yescharacteristic
210C2×C8C22×D19711yescharacteristic
211C4⋊C4—7881no—
212C42C22×D19711yescharacteristic
213C23—7882no—
214C23—3942no—
215Q8—7882no—
216Q8—3942no—
217Q8—3942no—
218Q8—1972no—
219Q8C2×D78812yes—
220D4—22no—
221D4—7882no—
222D4—7882no—
223D4—7882no—
224D4—15762no—
225D4—3942no—
226D4—3942no—
227D4—3942no—
228D4—1972no—
229D4C2×D78812yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—7882no—
233C2×C4—7882no—
234C2×C4—7882no—
235C2×C4—3942no—
236C2×C4—3942no—
237C2×C4C23×D19711yescharacteristic
238C8—22no—
239C8—3942no—
240C8D4×D19711yescharacteristic
241C8D4×D19711yescharacteristic
242C8C2×D78811yescharacteristic
243C8C2×D78811yescharacteristic
244C22—22no—
245C22—7882no—
246C22—15762no—
247C22—31522no—
248C22—3942no—
249C22#2026337211yescharacteristic
250C4—22no—
251C4—22no—
252C4—7882no—
253C4—3942no—
254C4C2×D4×D19711yescharacteristic
255C4C22×D78811yescharacteristic
256C2—21no—
257C2—42no—
258C2—7882no—
259C2—15762no—
260C2D4×D78811yescharacteristic
subgroupG/H
centerC2D4×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#443×C197C2
socleC2×C197D42
hypercenter#443D197
hypocenterC197#3291
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #27161891C2semidirect
2. #27161886C2semidirect
3. #27161452C2semidirect
4. #27161343C2semidirect
5. #27161342C2semidirect
6. #27161263C2semidirect
7. #27161248C2semidirect
8. #27161197C2semidirect
9. #443×C197C2semidirect
11. #20263372C22semidirect
13. #20263359C22semidirect
15. #20263356C22semidirect
24. C2×#15691347C22semidirect
25. C2×#15691347C22semidirect
27. #20263332C22semidirect
28. #20263326C22semidirect
30. #20263262C22semidirect
31. #20263254C22semidirect
32. #20263244C22semidirect
33. #20263242C22semidirect
35. #131×C197C22semidirect
36. #108×C197C22semidirect
37. #104×C197C22semidirect
38. C4×C8×C197C22semidirect
54. Dic788D4semidirect
62. #15691347D4semidirect
66. D1576D4semidirect
73. Q16×C197D4semidirect
74. SD16×C197D4semidirect
75. D8×C197D4semidirect
130. C197#3291semidirect
140. #443D197semidirect
160. #131C2×D197semidirect
164. #108C2×D197semidirect
166. #104C2×D197semidirect
169. C4×C8C2×D197semidirect
188. Q16D788semidirect
198. SD16D788semidirect
202. D8D788semidirect

Maximal subgroups

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

subgroupindex
#271618912
#271618862
#271614522
#271613432
#271613422
#271612632
#271612482
#271611972
#443×C1972
#3291197