Finite Group Databasebeta

← #38754502

Subgroups — #38754502

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#27161888C212yesmaximal, complemented
2#27161887C212yesmaximal, complemented
3#27161453C211yesmaximal, characteristic, complemented
4#27161345C212yesmaximal, complemented
5#27161344C212yesmaximal, complemented
6#27161261C211yesmaximal, characteristic, complemented
7#27161248C212yesmaximal, complemented
8#27161193C212yesmaximal, complemented
9#456×C197C211yesmaximal, characteristic, complemented
10#20263374C2211yescharacteristic
11#20263372C2212yescomplemented
12#20263358—22no—
13#20263357—22no—
14#20263357C2212yescomplemented
15#20263354C2212yes—
16#20263353—22no—
17#20263351—22no—
18#20263351C2212yes—
19#20263349—22no—
20#20263346—22no—
21#20263346C2212yes—
22C2×#15691345—22no—
23C2×#15691345C2212yescomplemented
24C2×#15691345C2212yescomplemented
25SD16×D197—22no—
26#20263334C2212yescomplemented
27#20263333C2212yescomplemented
28#20263323C2211yescharacteristic
29#20263261C2212yescomplemented
30#20263254C2212yescomplemented
31#20263245C2211yescharacteristic, complemented
32#20263242C2212yescomplemented
33C4×#13096802C2211yescharacteristic, complemented
342−1+4×C197C2211yescharacteristic
35#137×C197C2212yescomplemented
36C4⋊Q8×C197C2211yescharacteristic, complemented
37#104×C197C2212yescomplemented
38#101×C197C2211yescharacteristic, complemented
39#15691359—22no—
40#15691359—22no—
41#15691358—22no—
42#15691358—41no—
43#15691358C2312yes—
44Q8×D197—22no—
45C2×D788—22no—
46C2×D788—41no—
47C2×D788C2312yes—
48D4×D197—22no—
49D4×D197—42no—
50#15691348—22no—
51#15691348D412yescomplemented
52#15691347—22no—
53#15691346D412yescomplemented
54#15691345—22no—
55#15691345—22no—
56#15691345—22no—
57#15691345—22no—
58#15691345—22no—
59#15691345—22no—
60#15691345D412yescomplemented
61D1576—22no—
62#15691343—22no—
63#15691343D412yescomplemented
64#15691342—22no—
65#15691342C2312yes—
66#15691341—22no—
67C2×#13096802—22no—
68C2×#13096802C2312yes—
69C8×D197—22no—
70#41×C197—22no—
71#41×C197—41no—
72#41×C197C2312yes—
73C2×Q8×C197—22no—
74C2×Q8×C197C2311yescharacteristic
75C2×Q8×C197C2311yescharacteristic
76Q16×C197—22no—
77SD16×C197—22no—
78M16×C197C2312yes—
79C4⋊C4×C197—41no—
80C42×C197C2311yescharacteristic
81C22×D197—22no—
82C22×D197—42no—
83Dic394—41no—
84Dic394C2×D412yes—
85D788—22no—
86D788—22no—
87D788—42no—
88D788—41no—
89D788—82no—
90D788C2×D412yes—
91C22⋊D197—22no—
92C22⋊D197—42no—
93C4×D197—22no—
94C4×D197—42no—
95C4×D197—42no—
96#13096802—22no—
97#13096802C2×D412yes—
98#13096802C2×D412yes—
99Q8×C197—22no—
100Q8×C197—22no—
101Q8×C197—41no—
102Q8×C197C2×D412yes—
103D4×C197—22no—
104D4×C197—41no—
105D4×C197C2×D412yes—
106C2×C4×C197—21no—
107C2×C4×C197—22no—
108C2×C4×C197—21no—
109C2×C4×C197—21no—
110C2×C4×C197—42no—
111C2×C4×C197C2411yescharacteristic
112C8×C197—22no—
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×C197—41no—
122C4×C197—41no—
123C4×C197C22×D412yes—
124Dic197—22no—
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—1971no—
138#456D19711yescharacteristic, complemented
139#453—1972no—
140#443—1972no—
1412−1+4—1971no—
1422−1+4C2×D19711yescharacteristic
143H(5,𝔽2)—1972no—
144#137—3942no—
145#137—3942no—
146#137—1972no—
147#137C2×D19712yescomplemented
148AGL(1,ℤ/8)—3942no—
149AGL(1,ℤ/8)—3942no—
150AGL(1,ℤ/8)—1972no—
151AGL(1,ℤ/8)—1972no—
152#135—3942no—
153#135—3942no—
154#135—1972no—
155C2×SD16—3942no—
156C2×SD16—3942no—
157C2×SD16—1972no—
158C2×SD16—1972no—
159#131—1972no—
160#131—1972no—
161C4⋊Q8C2×D19711yescharacteristic, complemented
162#127—1971no—
163#108—1972no—
164#104—1972no—
165#104C2×D19712yescomplemented
166#101—1971no—
167#101C2×D19711yescharacteristic, complemented
168#100—1972no—
169C4×C8—1971no—
170#41—22no—
171#41—41no—
172#41—7881no—
173#41—3942no—
174#41—3942no—
175#41—3942no—
176#41—1972no—
177#41C22×D19712yes—
178C2×Q8—22no—
179C2×Q8—3942no—
180C2×Q8C22×D19711yescharacteristic
181C2×Q8C22×D19711yescharacteristic
182C2×D4—7881no—
183C2×D4—7882no—
184C2×D4—3942no—
185C2×D4—3942no—
186C2×D4—1972no—
187Q16—22no—
188Q16—3942no—
189Q16—1972no—
190SD16—22no—
191SD16—3942no—
192SD16—3942no—
193SD16—3942no—
194SD16—3942no—
195SD16—3942no—
196SD16—3942no—
197SD16—3942no—
198SD16—1972no—
199SD16—1972no—
200D8—3942no—
201D8—3942no—
202D8—1972no—
203M16—3942no—
204M16—3942no—
205M16—1972no—
206M16C22×D19712yes—
207C2×C8—3942no—
208C2×C8—3942no—
209C2×C8—1972no—
210C4⋊C4—41no—
211C42C22×D19711yescharacteristic
212C23—7882no—
213C23—3942no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—7881no—
218Q8—1972no—
219Q8D4×D19712yes—
220D4—22no—
221D4—41no—
222D4—7882no—
223D4—7881no—
224D4—7882no—
225D4—15762no—
226D4—3942no—
227D4—3942no—
228D4—3942no—
229D4—1972no—
230D4D4×D19712yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—7882no—
237C2×C4—7882no—
238C2×C4—3942no—
239C2×C4C23×D19711yescharacteristic
240C8—22no—
241C8—3942no—
242C8—1972no—
243C8—1972no—
244C22—22no—
245C22—7882no—
246C22—15762no—
247C22—31522no—
248C22—3942no—
249C22#2026337111yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—3942no—
255C4C2×D4×D19712yes—
256C2—21no—
257C2—42no—
258C2—7882no—
259C2—15762no—
260C2#2716171111yescharacteristic
subgroupG/H
centerC2#27161711
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#456×C197C2
socleC2×C197D42
hypercenter#456D197
hypocenterC197#3291
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #27161888C2semidirect
2. #27161887C2semidirect
3. #27161453C2semidirect
4. #27161345C2semidirect
5. #27161344C2semidirect
6. #27161261C2semidirect
7. #27161248C2semidirect
8. #27161193C2semidirect
9. #456×C197C2semidirect
11. #20263372C22semidirect
14. #20263357C22semidirect
23. C2×#15691345C22semidirect
24. C2×#15691345C22semidirect
26. #20263334C22semidirect
27. #20263333C22semidirect
29. #20263261C22semidirect
30. #20263254C22semidirect
31. #20263245C22semidirect
32. #20263242C22semidirect
33. C4×#13096802C22semidirect
35. #137×C197C22semidirect
36. C4⋊Q8×C197C22semidirect
37. #104×C197C22semidirect
38. #101×C197C22semidirect
51. #15691348D4semidirect
53. #15691346D4semidirect
60. #15691345D4semidirect
63. #15691343D4semidirect
130. C197#3291semidirect
138. #456D197semidirect
147. #137C2×D197semidirect
161. C4⋊Q8C2×D197semidirect
165. #104C2×D197semidirect
167. #101C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618882
#271618872
#271614532
#271613452
#271613442
#271612612
#271612482
#271611932
#456×C1972
#3291197