Finite Group Databasebeta

← #38754480

Subgroups — #38754480

Order 25216.

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

normal subgroups only

This group has 464 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#27161887C212yesmaximal, complemented
2#27161883C212yesmaximal, complemented
3#27161469C211yesmaximal, characteristic, complemented
4#27161347C212yesmaximal, complemented
5#27161345C212yesmaximal, complemented
6#27161256C211yesmaximal, characteristic, complemented
7#27161248C212yesmaximal, complemented
8#27161198C212yesmaximal, complemented
9#454×C197C211yesmaximal, characteristic, complemented
10#20263372C2212yescomplemented
11#20263371C2211yescharacteristic
12#20263357—22no—
13#20263357C2212yescomplemented
14#20263354C2212yes—
15#20263353—22no—
16#20263351—22no—
17#20263351C2212yescomplemented
18#20263350—22no—
19#20263347—22no—
20#20263346—22no—
21#20263346C2212yes—
22C2×#15691345—22no—
23C2×#15691345C2212yescomplemented
24C2×#15691343C2212yescomplemented
25D8×D197—22no—
26#20263334C2212yescomplemented
27#20263333C2212yescomplemented
28#20263323C2211yescharacteristic, complemented
29#20263261C2212yescomplemented
30#20263254C2212yescomplemented
31#20263243C2211yescharacteristic, complemented
32#20263242C2212yescomplemented
33#20263237C2211yescharacteristic, complemented
342−1+4×C197C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C197C2212yescomplemented
36#124×C197C2211yescharacteristic, complemented
37#104×C197C2212yescomplemented
38#101×C197C2211yescharacteristic, complemented
39#15691359—22no—
40#15691358—22no—
41#15691358C2312yes—
42#15691357—22no—
43C2×D788—22no—
44C2×D788—41no—
45C2×D788C2312yes—
46C2×C22⋊D197—41no—
47D4×D197—22no—
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#15691345D412yescomplemented
59D1576—22no—
60#15691343—22no—
61#15691343—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×D4×C197C2311yescharacteristic
76SD16×C197—22no—
77D8×C197—22no—
78M16×C197C2312yes—
79C22⋊C4×C197—41no—
80C42×C197C2311yescharacteristic
81C22×D197—22no—
82C22×D197—41no—
83C22×D197—42no—
84Dic394C2×D412yes—
85D788—22no—
86D788—22no—
87D788—42no—
88D788—82no—
89D788C2×D412yes—
90C22⋊D197—22no—
91C22⋊D197—42no—
92C22⋊D197—42no—
93C2×Dic197—41no—
94C4×D197—22no—
95C4×D197—42no—
96#13096802—22no—
97#13096802C2×D412yes—
98#13096802C2×D412yes—
99C23×C197—21no—
100Q8×C197—22no—
101Q8×C197—41no—
102Q8×C197C2×D412yes—
103D4×C197—22no—
104D4×C197—22no—
105D4×C197—41no—
106D4×C197C2×D412yes—
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×C197—41no—
115C22×C197—81no—
116C22×C197H(5,𝔽2)11yescharacteristic
117C2×D197—22no—
118C2×D197—42no—
119C2×D197—82no—
120C2×D197—161no—
121C2×D197—162no—
122C4×C197—22no—
123C4×C197—41no—
124C4×C197—41no—
125C4×C197C22×D412yes—
126Dic197—22no—
127C2×C197—21no—
128C2×C197—42no—
129C2×C197—81no—
130C2×C197D4211yescharacteristic
131D197—42no—
132D197—82no—
133C197#328811yescharacteristic, complemented, Sylow
134#3288—1971nomaximal, Sylow
135#577—1972no—
136#576—1972no—
137#496—1971no—
138#472—1972no—
139#471—1972no—
140#455—1971no—
141#454D19711yescharacteristic, complemented
142#453—1972no—
143#444—1972no—
1442−1+4C2×D19711yescharacteristic, complemented
145H(5,𝔽2)—1972no—
146H(5,𝔽2)—1971no—
147#137—3942no—
148#137—1972no—
149AGL(1,ℤ/8)—3942no—
150AGL(1,ℤ/8)—3942no—
151AGL(1,ℤ/8)—3942no—
152AGL(1,ℤ/8)—1972no—
153AGL(1,ℤ/8)—1972no—
154AGL(1,ℤ/8)C2×D19712yescomplemented
155#135—3942no—
156#135—3942no—
157#135—1972no—
158C2×SD16—3942no—
159C2×SD16—1972no—
160C2×D8—3942no—
161C2×D8—1972no—
162#131—1972no—
163#131—1972no—
164#127—1971no—
165#124C2×D19711yescharacteristic, complemented
166#108—1972no—
167#104—1972no—
168#104C2×D19712yescomplemented
169#101C2×D19711yescharacteristic, complemented
170#100—1971no—
171#100—1972no—
172#97—1971no—
173#41—22no—
174#41—41no—
175#41—3942no—
176#41—3942no—
177#41—3942no—
178#41—1972no—
179#41C22×D19712yes—
180C2×Q8—22no—
181C2×Q8C22×D19711yescharacteristic
182C2×D4—7881no—
183C2×D4—7882no—
184C2×D4—7881no—
185C2×D4—3942no—
186C2×D4—3942no—
187C2×D4—3942no—
188C2×D4—1972no—
189C2×D4C22×D19711yescharacteristic
190Q16—3942no—
191Q16—1972no—
192SD16—22no—
193SD16—3942no—
194SD16—3942no—
195SD16—3942no—
196SD16—3942no—
197SD16—3942no—
198SD16—1972no—
199SD16—1972no—
200D8—22no—
201D8—3942no—
202D8—3942no—
203D8—3942no—
204D8—3942no—
205D8—1972no—
206M16—3942no—
207M16—3942no—
208M16—1972no—
209M16C22×D19712yes—
210C2×C8—3942no—
211C2×C8—3942no—
212C2×C8—1972no—
213C22⋊C4—41no—
214C42C22×D19711yescharacteristic
215C23—21no—
216C23—7882no—
217C23—7881no—
218C23—3942no—
219Q8—22no—
220Q8—41no—
221Q8—1972no—
222Q8D4×D19712yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—7882no—
227D4—7882no—
228D4—7882no—
229D4—15762no—
230D4—3942no—
231D4—3942no—
232D4—3942no—
233D4—1972no—
234D4D4×D19712yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—7882no—
240C2×C4—7881no—
241C2×C4—3942no—
242C2×C4C23×D19711yescharacteristic
243C8—22no—
244C8—3942no—
245C8—1972no—
246C8—1972no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—7882no—
251C22—15762no—
252C22—31522no—
253C22—31521no—
254C22—3942no—
255C22#2026337111yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—3942no—
260C4C2×D4×D19712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—7882no—
265C2—15762no—
266C2#2716171111yescharacteristic
subgroupG/H
centerC2#27161711
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#454×C197C2
socleC2×C197D42
hypercenter#454D197
hypocenterC197#3288
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #27161887C2semidirect
2. #27161883C2semidirect
3. #27161469C2semidirect
4. #27161347C2semidirect
5. #27161345C2semidirect
6. #27161256C2semidirect
7. #27161248C2semidirect
8. #27161198C2semidirect
9. #454×C197C2semidirect
10. #20263372C22semidirect
13. #20263357C22semidirect
17. #20263351C22semidirect
23. C2×#15691345C22semidirect
24. C2×#15691343C22semidirect
26. #20263334C22semidirect
27. #20263333C22semidirect
28. #20263323C22semidirect
29. #20263261C22semidirect
30. #20263254C22semidirect
31. #20263243C22semidirect
32. #20263242C22semidirect
33. #20263237C22semidirect
34. 2−1+4×C197C22semidirect
35. AGL(1,ℤ/8)×C197C22semidirect
36. #124×C197C22semidirect
37. #104×C197C22semidirect
38. #101×C197C22semidirect
51. #15691348D4semidirect
53. #15691346D4semidirect
58. #15691345D4semidirect
63. #15691343D4semidirect
133. C197#3288semidirect
141. #454D197semidirect
144. 2−1+4C2×D197semidirect
154. AGL(1,ℤ/8)C2×D197semidirect
165. #124C2×D197semidirect
168. #104C2×D197semidirect
169. #101C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618872
#271618832
#271614692
#271613472
#271613452
#271612562
#271612482
#271611982
#454×C1972
#3288197